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284 Thermoelectrics for Power Generation - A Look at Trends in the Technology

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#### **Simulation of Morphological Effects on Thermoelectric Power, Thermal and Electrical Conductivity in Multi‐ Phase Thermoelectric Materials Simulation of Morphological Effects on Thermoelectric Power, Thermal and Electrical Conductivity in Multi**‐ **Phase Thermoelectric Materials**

Yaniv Gelbstein Yaniv Gelbstein

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/65099

#### **Abstract**

Multi‐phase thermoelectric materials are mainly investigated these days due to their potential of lattice thermal conductivity reduction by scattering of phonons at interfaces of the involved phases, leading to the enhancement of expected thermoelectric efficiency. On the other hand, electronic effects of the involved phases on thermoelectric performance are not always being considered, while developing new multi‐phase thermoelectric materials. In this chapter, electronic effects resulting from controlling the phase distribution and morphology alignment in multi‐phase composite materials is carefully described using the general effective media (GEM) method and analytic approaches. It is shown that taking into account the specific thermoelectric properties of the involved phases might be utilized for estimating expected effective thermoelectric properties of such composite materials for any distribution and relative amount of the phases. An implementation of GEM method for the IV–VI (including SnTe and GeTe), bismuth telluride (Bi2Te3), higher manganese silicides (HMS) and half‐Heusler classes of thermoelectric materials is described in details.

**Keywords:** thermoelectric, GEM, multi‐phase

## **1. Thermoelectrics**

Climate changes, due to fossil fuels combustion and greenhouse gases emission, cause deep concern about environmental conservation. Another pressing issue is sustainable energy

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

production that is coupled with depletion of conventional energy resources. This concern might be tackled by converting the waste heat generated in internal‐combustion vehicles, factories, computers, etc. into electrical energy. Converting this waste heat into electricity will reduce fossil fuel consumption and emission of pollutants. This can be achieved by direct thermoelectric (TE) converters, as was successfully demonstrated by development of various highly efficient TE material classes, including Bi2Te3 [1–3] for temperatures, *T*, of up to ∼300°C, SnTe [4, 5], PbTe [6, 7] and GeTe [8–11], for temperatures range 300 ≤ *T* ≤ 500°C, and higher manganese silicides (HMS) [12–14], half‐Heuslers [15–20], which are capable to operate at higher temperatures. Such materials require unique combination of electronic (i.e. Seebeck coefficient, *α*, electrical resistivity, *ρ*, and electronic thermal conductivity, *κ*e) and lattice (i.e. lattice thermal conductivity, *κ*<sup>l</sup> ) properties, enabling the highest possible TE figure of merit, *ZT* = *α*<sup>2</sup> *T*/[*ρ*(*κ*e + *κ*<sup>l</sup> )], values, for achieving significant heat to electricity conversion efficiencies. Due to the fact, that electronic TE properties are strongly correlated, and follow opposite trends upon modifying charge carriers' concentration, many of recently developed TE materials, were focused on nano‐structuring methods, capable of *κ*<sup>l</sup> reduction due to lattice modifications and correspondingly increasing *ZT*. Such methods included alloying (for PbTe, as an example, alloying with SrTe [21, 22], MgTe [23] and CdTe [24], resulted in strained endotaxial nano‐ structures), applying layered structures with increased interfaces population (e.g. SnSe [25]), and thermodynamically driven phase separation reactions, generating nano‐scale modula‐ tions (e.g. Ge*x*Pb1−*x*Te [26–28] and Ge*x*(Sn*y*Pb1−*y*)1−*x*Te [29, 30]). All of these approaches resulted in significant increase of *ZT* up to ∼2.5 [25] due to effective scattering of phonons by associated generated nano‐features. Nevertheless, although significant enhancement of TE properties was reported due to phonon scattering by nano‐structured phases in such multi‐phase TE materials, most of these researches did not investigate individual electronic contributions of each of the involved phases on effective TE transport properties.

#### **2. Multi‐phase thermoelectric materials**

In the last few decades, major trend is to move from pristine single‐crystal TE compositions towards polycrystalline multi‐phase materials. One of the reasons for that is improved shear mechanical strength of polycrystalline materials compared to single crystals, exhibiting high compression, but very low transverse strengths, required to withstand high thermal and mechanical gradients applied in practical applications. Another reason is the possibility of phonon scattering by the involved interfaces as mentioned above. Most of the TE materials investigated these days are being synthesized by powder metallurgy approach under high uniaxial mechanical pressures, deforming involved grains and phases into anisotropic geometrical morphologies, which affect the electronic transport properties. Besides, a certain amount of porosity (as a second phase) is in many cases unavoidable, adversely affecting TE transport properties. Furthermore, many of currently employed TE materials (e.g. Bi2Te3 and HMS) are crystallographic anisotropic with optimal TE transport properties along preferred orientations. Some researches of such materials for TE applications do not consider crystallo‐ graphic anisotropy, while assuming, that randomly oriented grains of different crystallo‐ graphic planes cancel each other in polycrystalline samples. Yet, some anisotropy can exist also in such materials in case of highly anisotropic specific properties (e.g. mechanical properties), leading to textured polycrystal. For example, texture development of non‐cubic polycrystalline alloys was attributed to multiple deformation modes applied in each grain, twinning resulting in grain reorientation and strong directional grain interactions [12]. Specifically, in Bi2Te3, for example, exhibiting highly anisotropic layered crystal structure consists of 15 parallel layers stacked along crystallographic *c* axis, the presence of van der Waals gap in the crystal lattice, divides crystal into blocks of five mono‐atomic sheets [1]. In this case, retaining the crystallo‐ graphic anisotropy is highly desired. This is due to the fact, that in transverse to crystallo‐ graphic *c* axis, TE power factor (numerator in *ZT* expression) is considerably higher, than in parallel to this direction, mainly due to higher electrical conductivity values. For powder metallurgy synthesized Bi2Te3‐based materials, it was shown that moderate powder grinding pressures, might retain some of the crystallographic anisotropy, due to the weaker van der Waals bonding of atoms located in adjacent layers along *c*‐axis, compared to ionic/covalent bonding between atoms located in each of the layers [31]. In this example, higher *ZT* values in transverse to powder pressing direction are expected as in single crystals. This example highlights the significance of controlling phases' morphology for optimizing TE transport properties.

production that is coupled with depletion of conventional energy resources. This concern might be tackled by converting the waste heat generated in internal‐combustion vehicles, factories, computers, etc. into electrical energy. Converting this waste heat into electricity will reduce fossil fuel consumption and emission of pollutants. This can be achieved by direct thermoelectric (TE) converters, as was successfully demonstrated by development of various highly efficient TE material classes, including Bi2Te3 [1–3] for temperatures, *T*, of up to ∼300°C, SnTe [4, 5], PbTe [6, 7] and GeTe [8–11], for temperatures range 300 ≤ *T* ≤ 500°C, and higher manganese silicides (HMS) [12–14], half‐Heuslers [15–20], which are capable to operate at higher temperatures. Such materials require unique combination of electronic (i.e. Seebeck coefficient, *α*, electrical resistivity, *ρ*, and electronic thermal conductivity, *κ*e) and lattice (i.e.

Due to the fact, that electronic TE properties are strongly correlated, and follow opposite trends upon modifying charge carriers' concentration, many of recently developed TE materials, were

correspondingly increasing *ZT*. Such methods included alloying (for PbTe, as an example, alloying with SrTe [21, 22], MgTe [23] and CdTe [24], resulted in strained endotaxial nano‐ structures), applying layered structures with increased interfaces population (e.g. SnSe [25]), and thermodynamically driven phase separation reactions, generating nano‐scale modula‐ tions (e.g. Ge*x*Pb1−*x*Te [26–28] and Ge*x*(Sn*y*Pb1−*y*)1−*x*Te [29, 30]). All of these approaches resulted in significant increase of *ZT* up to ∼2.5 [25] due to effective scattering of phonons by associated generated nano‐features. Nevertheless, although significant enhancement of TE properties was reported due to phonon scattering by nano‐structured phases in such multi‐phase TE materials, most of these researches did not investigate individual electronic contributions of

In the last few decades, major trend is to move from pristine single‐crystal TE compositions towards polycrystalline multi‐phase materials. One of the reasons for that is improved shear mechanical strength of polycrystalline materials compared to single crystals, exhibiting high compression, but very low transverse strengths, required to withstand high thermal and mechanical gradients applied in practical applications. Another reason is the possibility of phonon scattering by the involved interfaces as mentioned above. Most of the TE materials investigated these days are being synthesized by powder metallurgy approach under high uniaxial mechanical pressures, deforming involved grains and phases into anisotropic geometrical morphologies, which affect the electronic transport properties. Besides, a certain amount of porosity (as a second phase) is in many cases unavoidable, adversely affecting TE transport properties. Furthermore, many of currently employed TE materials (e.g. Bi2Te3 and HMS) are crystallographic anisotropic with optimal TE transport properties along preferred orientations. Some researches of such materials for TE applications do not consider crystallo‐ graphic anisotropy, while assuming, that randomly oriented grains of different crystallo‐

) properties, enabling the highest possible TE figure of merit,

reduction due to lattice modifications and

)], values, for achieving significant heat to electricity conversion efficiencies.

lattice thermal conductivity, *κ*<sup>l</sup>

focused on nano‐structuring methods, capable of *κ*<sup>l</sup>

288 Thermoelectrics for Power Generation - A Look at Trends in the Technology

**2. Multi‐phase thermoelectric materials**

each of the involved phases on effective TE transport properties.

*T*/[*ρ*(*κ*e + *κ*<sup>l</sup>

*ZT* = *α*<sup>2</sup>

Besides of metallurgical phases, individual transport properties of two species (e.g. light and heavy holes in *p*‐type PbTe [32]), in materials with complicated electronic band structures might contribute dramatically to effective TE transport properties.

In this chapter, effective TE properties (Seebeck coefficient, *α* electrical resistivity, *ρ* or conductivity, *σ* = *ρ*−1 and thermal conductivity, *κ*) of general complex structure, consisting of at least two independent phases with any respective relative amount and geometrical alignment are derived by using the GEM method [4] and individual TE properties of each of the involved phases. This approach can be utilized for maximizing TE figure of merit of multi‐ phase composite materials, for example, by intentional alignment of the involved phases along the optimal TE direction.

We consider in this chapter a simple formulation for modelling of multi‐phase TE materials, originated from materials science aspects, such as inter‐diffusion, alloying, dissolution, phase transitions, phase separation, phase segregation, precipitation, recrystallization and other phenomena, that can take place in operation conditions of TE modules, especially TE power generation modules exposed to high thermo‐mechanical stresses.

## **3. TE GEM effective equations for two‐phase materials**

Effective TE properties of two‐phase composites can be accurately predicted by GEM method, Eqs. (1)–(3) [4, 33–35]:

$$\begin{array}{c} \begin{array}{c} \kappa\_{\text{eff}} \\ \stackrel{\text{\cdot}}{\sigma\_{\text{eff}}} \text{\cdot} \text{\cdot} \\ \frac{\alpha\_{\text{eff}}-\alpha\_{\text{2}}}{\alpha\_{1}-\alpha\_{2}} = \frac{\beta\_{\text{2}}}{\kappa\_{1}} \end{array} \\ \begin{array}{c} \alpha\_{\text{eff}}-\alpha\_{2} \\ \stackrel{\cdot}{\sigma\_{\text{2}}} \text{\cdot} \\ \stackrel{\cdot}{\sigma\_{\text{2}}} \text{\cdot} \text{\cdot} \end{array} \end{array} \tag{1}$$

$$\max\_{\mathbf{x}\_{1}} \frac{(\sigma\_{1})\_{\mathbf{y}\_{l}}^{\mathsf{V}} - (\sigma\_{\text{eff}})\_{\mathbf{y}\_{l}}^{\mathsf{V}}}{(\sigma\_{1})\_{\mathbf{y}\_{l}}^{\mathsf{V}} + A(\sigma\_{\text{eff}})\_{\mathbf{y}\_{l}}^{\mathsf{V}}} = (1 - \mathbf{x}\_{1}) \frac{(\sigma\_{\text{eff}})\_{\mathbf{y}\_{l}}^{\mathsf{V}} - (\sigma\_{\text{2}})\_{\mathbf{y}\_{l}}^{\mathsf{V}}}{(\sigma\_{2})\_{\mathbf{y}\_{l}}^{\mathsf{V}} + A(\sigma\_{\text{eff}})\_{\mathbf{y}\_{l}}^{\mathsf{V}}},\tag{2}$$

$$\max\_{\mathbf{k}} \frac{(\mathbf{k}\_{\mathrm{i}})\_{\mathbf{k}}^{\mathbb{K}} - (\mathbf{k}\_{\mathrm{eff}})\_{\mathbf{k}}^{\mathbb{K}}}{(\mathbf{k}\_{\mathrm{i}})\_{\mathbf{k}}^{\mathbb{K}} + A(\mathbf{k}\_{\mathrm{eff}})\_{\mathbf{k}}^{\mathbb{K}}} = (1 - \mathbf{x}\_{\mathrm{i}}) \frac{(\mathbf{x}\_{\mathrm{eif}})\_{\mathbf{k}}^{\mathbb{K}} - (\mathbf{x}\_{\mathrm{i}})\_{\mathbf{k}}^{\mathbb{K}}}{(\mathbf{x}\_{\mathrm{z}})\_{\mathbf{k}}^{\mathbb{K}} + A(\mathbf{x}\_{\mathrm{eif}})\_{\mathbf{k}}^{\mathbb{K}}}.\tag{3}$$

These three GEM equations, Eqs. (1)–(3), are usually employed for calculating effective Seebeck coefficient (*α*eff) and effective electrical and thermal conductivities (*σ*eff and *κ*eff, respectively) for two‐phase materials using individual electrical (*σ*1 and *σ*2) and thermal (*κ*1 and *κ*2) conduc‐ tivity, as well as, individual Seebeck coefficient (*α*1 and *α*2) values of involved phases. Mor‐ phological parameters *A*, *t* can be derived by modelling of experimental results or from percolation equation [33, 34]. Parameter *x*1 is volume fraction of one of the phases. Values of *A* and *t* are strongly affected by phase distribution and morphology. It was shown, that for homogeneously distributed second phase in continuous matrix, *t* value is equal to 1 [4] and the entire morphological alignment possibilities of the second phase related to the matrix phase are bounded by the so‐called 'parallel' and 'series' alignment of the phases (relative to electrical potential or temperature gradients). Parameter *A* varies from 8 for parallel to 0 for series alignments. It can be seen, that for substituting *t* = 1 and *A* = *8* in Eqs. (2) and (3), as in the case of phases distribution in parallel to electrical current direction, reduces equations to Eq. (4), while substituting of Eq. (4) in Eq. (1) leads to Eq. (5):

$$\mathbf{x}\left(\sigma\_{\text{eff}},\kappa\_{\text{eff}}\right) = \left(\sigma\_1,\kappa\_1\right)\mathbf{x}\_1 + \left(\sigma\_2,\kappa\_2\right)\left(1-\mathbf{x}\_1\right),\tag{4}$$

$$\alpha\_{\rm eff} = \frac{\alpha\_1 \sigma\_1 \mathbf{x}\_1 + \alpha\_2 \sigma\_2 \left(1 - \mathbf{x}\_1\right)}{\sigma\_1 \mathbf{x}\_1 + \sigma\_2 \left(1 - \mathbf{x}\_1\right)}.\tag{5}$$

Similarly, substituting *t* = 1 and *A* = 0 in Eqs. (2) and (3), as in the case of series alignment as explained above, reduces them into Eq. (6):

Simulation of Morphological Effects on Thermoelectric Power, Thermal and Electrical Conductivity... http://dx.doi.org/10.5772/65099 291

$$\mathbf{r}\left(\sigma\_{\text{eff}},\kappa\_{\text{eff}}\right) = \frac{\left(\sigma\_{\text{1}},\kappa\_{\text{1}}\right)\left(\sigma\_{\text{2}},\kappa\_{\text{2}}\right)}{\left(\sigma\_{\text{1}},\kappa\_{\text{1}}\right)\left(\mathbf{l}-\mathbf{x}\_{\text{1}}\right) + \left(\sigma\_{\text{2}},\kappa\_{\text{2}}\right)\mathbf{x}\_{\text{1}}}.\tag{6}$$

Please note that although for the case of parallel alignment, effective electrical and thermal conductivity, Eq. (4), follow a simple rule of mixture, a more complicated dependency is apparent for series alignment, Eq. (6). Yet, as shown in Eq. (7), for this latter case, effective electrical resistivity, *ρ*eff = *σ*eff−1, follows the rule of mixture:

$$
\rho\_{\text{eff}} = \rho\_1 \mathbf{x}\_1 + \rho\_2 \left(1 - \mathbf{x}\_1\right). \tag{7}
$$

Substituting of Eq. (6) in Eq. (1), leads in this case to Eq. (8):

eff 2 eff eff 2 2 1 1 2

k

s

k

s

k

s

1 1 1 1 1 eff eff 2 1 1 11 11 1 eff 2 eff

*t t t t tt tt*

( ) ( ) ( ) ( ) ( )

+ +

( ) ( ) ( ) ( ) ( )

+ +

( ) ( ) ( )( ) eff eff 1 1 1 2 2 1

 sk

1 11 2 2 1

+ - <sup>=</sup> + *x x x x*

 as

 s

11 2 1

Similarly, substituting *t* = 1 and *A* = 0 in Eqs. (2) and (3), as in the case of series alignment as

( ) ( )

<sup>1</sup> . <sup>1</sup>

, , ,1 , =+ - *x x* (4)

(5)

 sk

as

s

eff

a

*A A*

*A A*

1 1 1 1 1 eff eff 2 1 1 11 11 1 eff 2 eff

*t t t t tt tt*

1 . - - = -

These three GEM equations, Eqs. (1)–(3), are usually employed for calculating effective Seebeck coefficient (*α*eff) and effective electrical and thermal conductivities (*σ*eff and *κ*eff, respectively) for two‐phase materials using individual electrical (*σ*1 and *σ*2) and thermal (*κ*1 and *κ*2) conduc‐ tivity, as well as, individual Seebeck coefficient (*α*1 and *α*2) values of involved phases. Mor‐ phological parameters *A*, *t* can be derived by modelling of experimental results or from percolation equation [33, 34]. Parameter *x*1 is volume fraction of one of the phases. Values of *A* and *t* are strongly affected by phase distribution and morphology. It was shown, that for homogeneously distributed second phase in continuous matrix, *t* value is equal to 1 [4] and the entire morphological alignment possibilities of the second phase related to the matrix phase are bounded by the so‐called 'parallel' and 'series' alignment of the phases (relative to electrical potential or temperature gradients). Parameter *A* varies from 8 for parallel to 0 for series alignments. It can be seen, that for substituting *t* = 1 and *A* = *8* in Eqs. (2) and (3), as in the case of phases distribution in parallel to electrical current direction, reduces equations to Eq. (4),

 k


a a

a a

( ) ( )

( ) ( )

kk

kk

while substituting of Eq. (4) in Eq. (1) leads to Eq. (5):

s k

explained above, reduces them into Eq. (6):

ss

290 Thermoelectrics for Power Generation - A Look at Trends in the Technology

ss

*x x*

*x x*

2 1 2

k

s

1 , - - = -

 s

1


,

( ) ( )

( ) ( )

kk

ss

 s

 k (1)

(2)

(3)

1


$$\alpha\_{\rm eff} = \frac{\alpha\_1 \kappa\_2 \mathbf{x}\_1 + \alpha\_2 \kappa\_1 (1 - \mathbf{x}\_1)}{\kappa\_1 (1 - \mathbf{x}\_1) + \kappa\_2 \mathbf{x}\_1}. \tag{8}$$

While investigating Eqs. (5) and (8) , for the cases of parallel and series alignment, respectively, it can be easily seen, that for both cases, effective Seebeck coefficient depends not only on individual Seebeck coefficients of the two phases, but also on other electronic transport properties, electrical conductivity of the involved phases for the case of parallel alignment, Eq. (5), and thermal conductivity of the involved phases for the case of series alignment, Eq. (8). An explanation for this observation is given in the next section.

#### **4. Analytical effective equations for multi‐phase materials**

In order to extend GEM, Eqs. (1)–(3) listed above for two‐phase composite materials, into higher‐ordered composites with three or more coexisting phases, a simple analytical model for calculating effective TE properties of several conductors, subjected to external electrical and thermal gradients, can be applied. For this purpose, two boundary conditions explained above, can be examined; one for conductors connected in parallel to both thermal and electrical applied gradients and the other for conductors connected in series.

#### **4.1. Thermoelectric phases in parallel**

In the case of three distributed conductors oriented in parallel to external temperature, Δ*T* = *T*h–*T*c, and electrical potentials, *V*, gradients, shown schematically as 1, 2 and 3 in **Figure 1(a)**, each of them might be considered as a single phase with sample's length and perspective cross‐ section area according to its relative amount (**Figure 1b**). For this case, electrical analogue, shown in **Figure 1(c)**, includes three parallel branches, with power source reflecting the individual open circuit voltage developed according to Seebeck effect (*V*1,2,3 = *α*1,2,3Δ*T*, where *α*1,2,3 – Seebeck coefficients of the involved phases) under applied temperature difference, connected serially to resistor *R*1,2,3, reflecting internal total electrical resistance, of each of the phases. In this case, electrical currents *I*1,2,3, flowing through connectors are given by Eq. (9):

$$I\_{1,2,3} = \frac{V - \int\_{T\_\iota}^{T\_b} \alpha\_{1,2,3} dT}{R\_{1,2,3}}.\tag{9}$$

Total electrical current *I* in three‐phase system is given by Eq. (10):

$$I = I\_1 + I\_2 + I\_3 = V \left(\frac{1}{R\_1} + \frac{1}{R\_2} + \frac{1}{R\_3}\right) - \int\_{T\_s}^{T\_b} \left(\frac{\alpha\_1}{R\_1} + \frac{\alpha\_2}{R\_2} + \frac{\alpha\_3}{R\_3}\right) dT. \tag{10}$$

**Figure 1.** Schematical description of three phases, I–III, oriented in parallel to external temperature and electrical gradi‐ ents, as distributed in the sample (a) and as combined entities with sample's length and perspective cross‐section area according to their relative amount (b). The electrical analogue of this three‐phase material is given in (c).

Considering definition of Seebeck coefficient as derivative of applied voltage with respect to temperature for non‐current flowing condition, Eq. (11), a simple manipulation of Eq. (10) gives Eq. (12), which describes effective Seebeck coefficient, *α*eff, of parallel connected three‐phase structure:

$$\left. \alpha\_{eff} \stackrel{\text{def}}{=} \frac{d\mathcal{V}}{dT} \right|\_{I=0},\tag{11}$$

$$\alpha\_{\rm eff} = \frac{\frac{\alpha\_1}{R\_1} + \frac{\alpha\_2}{R\_2} + \frac{\alpha\_3}{R\_3}}{\frac{1}{R\_1} + \frac{1}{R\_2} + \frac{1}{R\_3}} = \frac{\alpha\_1 R\_2 R\_3 + \alpha\_2 R\_1 R\_3 + \alpha\_3 R\_1 R\_2}{R\_2 R\_3 + R\_1 R\_3 + R\_1 R\_2}.\tag{12}$$

Using specific parameters (resistivity *ρ*1,2,3 and conductivity *σ*1,2,3 = (*ρ*1,2,3) −1) instead of resistan‐ ces *R*1,2,3, as described in Eq. (13), expression for *α*eff for parallel connected three‐phase struc‐ tures can be derived, Eq. (14):

$$R\_{1,2,3} = \frac{\rho\_{1,2,3} l\_{\text{sample}}}{\tilde{A}\_{1,2,3}},\tag{13}$$

$$\begin{aligned} \alpha\_{eff} &= \frac{\left(\frac{a\_1\rho\_2\rho\_3 + a\_2\rho\_1\rho\_3}{\tilde{\lambda}\_2\tilde{\lambda}\_3 + \tilde{\lambda}\_1\tilde{\lambda}\_3 + \tilde{\lambda}\_1\tilde{\lambda}\_2}\right)}{\left(\frac{\rho\_2\rho\_3 + \rho\_1\rho\_3 + \rho\_1\rho\_2}{\tilde{\lambda}\_2\tilde{\lambda}\_3 + \tilde{\lambda}\_1\tilde{\lambda}\_3 + \tilde{\lambda}\_1\tilde{\lambda}\_2}\right)} = \frac{a\_1\tilde{\lambda}\_1\rho\_2\rho\_3 + a\_2\tilde{\lambda}\_2\rho\_1\rho\_3 + a\_3\tilde{\lambda}\_3\rho\_1\rho\_2}{\tilde{\lambda}\_1\rho\_2\rho\_3 + \tilde{\lambda}\_2\rho\_1\rho\_3 + \tilde{\lambda}\_3\rho\_1\rho\_2} = \\\\ \frac{a\_1\sigma\_1\tilde{\lambda}\_1 + a\_2\sigma\_2\tilde{\lambda}\_2 + a\_3\sigma\_3\tilde{\lambda}\_3}{\sigma\_1\tilde{\lambda}\_1 + \sigma\_2\tilde{\lambda}\_2 + \sigma\_3\tilde{\lambda}\_3}, \end{aligned} \tag{14}$$

where, *l*samp = *l*1 = *l*2 = l3 is the sample's length, 1, 2, 3 is the cross‐section area transverse to electrical current flow.

While considering, volume fractions, *x*1,2,3 (= 1, 2, 3. *l*samp/*V*samp, where *V*samp is sample's volume) of the respective phase, Eq. (15) can be easily derived:

$$\left(\left(\alpha\_{\text{eff}}\right)\_{\text{pamile}}\right) = \frac{\alpha\_{\text{1}}\sigma\_{\text{1}}\mathbf{x}\_{\text{1}} + \alpha\_{\text{2}}\sigma\_{\text{2}}\mathbf{x}\_{\text{2}} + \alpha\_{\text{3}}\sigma\_{\text{3}}\mathbf{x}\_{\text{3}}}{\sigma\_{\text{1}}\mathbf{x}\_{\text{1}} + \sigma\_{\text{2}}\mathbf{x}\_{\text{2}} + \sigma\_{\text{3}}\mathbf{x}\_{\text{3}}} = \frac{\sum \alpha\_{i}\sigma\_{i}\mathbf{x}\_{i}}{\sum \sigma\_{i}\mathbf{x}\_{i}}.\tag{15}$$

From electrical analogue shown in **Figure 1(c)**, effective electrical and thermal conductivities can also be easily derived, as expressed in Eqs. (16) and (17), respectively:

$$
\sigma\_1 \left( \sigma\_{\text{eff}} \right)\_{\text{parallel}} = \sigma\_1 \mathbf{x}\_1 + \sigma\_2 \mathbf{x}\_2 + \sigma\_3 \mathbf{x}\_3 = \sum \sigma\_i \mathbf{x}\_i,\tag{16}
$$

$$\left(\mathbf{x}\_{\text{eff}}\right)\_{\text{parallel}} = \mathbf{x}\_1 \mathbf{x}\_1 + \mathbf{x}\_2 \mathbf{x}\_2 + \mathbf{x}\_3 \mathbf{x}\_3 = \sum \mathbf{x}\_i \mathbf{x}\_i. \tag{17}$$

It is noteworthy that applying the same approach for higher *i*‐ordered multi‐phase materials will follow the general‐ordered right‐hand side expressions of Eqs. (15)–(17). Furthermore, it can be easily seen that Eqs. (15)–(17) for the case of two‐phase materials are reduced to Eqs. (5) and (4), respectively, derived from the GEM method.

#### **4.2. Thermoelectric phases in series**

connected serially to resistor *R*1,2,3, reflecting internal total electrical resistance, of each of the phases. In this case, electrical currents *I*1,2,3, flowing through connectors are given by Eq. (9):

1,2,3

.

123 123 <sup>111</sup> . æ öæ ö

*T*

*T*

*c*

è øè ø ò *h*

*RRR RRR*

**Figure 1.** Schematical description of three phases, I–III, oriented in parallel to external temperature and electrical gradi‐ ents, as distributed in the sample (a) and as combined entities with sample's length and perspective cross‐section area

Considering definition of Seebeck coefficient as derivative of applied voltage with respect to temperature for non‐current flowing condition, Eq. (11), a simple manipulation of Eq. (10) gives Eq. (12), which describes effective Seebeck coefficient, *α*eff, of parallel connected three‐phase

1 2 3 123 213 312

. <sup>111</sup>

+ + = = + + + + *R R R RR RR RR*

23 13 12

(12)

*RR RR RR*

aaa

according to their relative amount (b). The electrical analogue of this three‐phase material is given in (c).

123

+ +

aaa

123

*RRR*

eff

a

123

aaa

(9)

(10)

(11)

1,2,3

*h c T <sup>T</sup> V dT*

*R* a


=++= + + - + + ç ÷ç ÷

*II I I V dT*

1,2,3

*I*

Total electrical current *I* in three‐phase system is given by Eq. (10):

123

292 Thermoelectrics for Power Generation - A Look at Trends in the Technology

structure:

Equivalent description for the case of three distributed conductors oriented in series to external temperature and electrical potentials gradients is shown in **Figure 2(a)**.

**Figure 2.** Schematical description of three phases, I–III, oriented in series to external temperature and electrical gradi‐ ents, as distributed in the sample (a) and as combined entities with sample's diameter and perspective lengths accord‐ ing to their relative amount (b).

For this case, a similar analysis is presented, taking into account individual thermal gradients applied on each of the phases. Taking into account that the first, second and third phases are subjected to temperature differences of (*Th* − *T*1), (*T*1 − *T*2) and (*T*2 − *Tc*), respectively, as shown in **Figure 2(b)**, where *T*1,2 are intermediate temperatures (*Th* > *T*1 > *T*2 > *Tc*), effective Seebeck coefficient of such serially aligned three‐phase samples can be described in terms of Eq. (18):

$$\alpha\_{\rm eff} = \frac{\alpha\_1 \left(T\_h - T\_1\right) + \alpha\_2 \left(T\_1 - T\_2\right) + \alpha\_3 \left(T\_2 - T\_c\right)}{T\_h - T\_c}.\tag{18}$$

Under adiabatic heat conduction conditions, where no lateral heat losses are apparent, the heat flow, *Q*, through the entire sample and the individual phases can be described in terms of unidirectional Fourier heat conduction equation, Eq. (19):

$$\underline{Q} = \frac{\kappa\_1 \tilde{A}}{l\_1} \left( T\_h - T\_1 \right) = \frac{\kappa\_2 \tilde{A}}{l\_2} \left( T\_1 - T\_2 \right) = \frac{\kappa\_3 \tilde{A}}{l\_3} \left( T\_2 - T\_c \right) = \frac{\kappa\_{\text{eff}} \tilde{A}}{l\_{\text{amp}}} \left( T\_h - T\_c \right), \tag{19}$$

where *κ*eff is effective thermal conductivity of the three‐phase material, is cross‐section area transverse to heat flow and *κ*1,2,3 and *l*1,2,3 are thermal conductivity and effective length of each of the involved phases, respectively.

Using expression (19), the numerator terms of Eq. (18) can be easily described in terms of expressions (20):

Simulation of Morphological Effects on Thermoelectric Power, Thermal and Electrical Conductivity... http://dx.doi.org/10.5772/65099 295

$$
\alpha\_1 \left( T\_h - T\_1 \right) = \frac{Q l\_1 a\_1}{\kappa\_1 \tilde{A}}, \\
\alpha\_2 \left( T\_1 - T\_2 \right) = \frac{Q l\_2 a\_2}{\kappa\_2 \tilde{A}}, \\
\alpha\_3 \left( T\_2 - T\_c \right) = \frac{Q l\_3 a\_3}{\kappa\_3 \tilde{A}}.\tag{20}
$$

In the rightmost equation of expression (19), *κ*eff/*l*samp represents overall thermal conductance, *K*eff of the three‐phase sample, which is described in Eq. (21), in terms of serially connected thermal resistances, *R*th,1,2,3, specified in Eq. (22):

$$K\_{\rm eff} = \frac{\kappa\_{\rm eff} \tilde{A}}{I\_{\rm amp}} = \frac{1}{\left(R\_{\rm th}\right)\_1 + \left(R\_{\rm th}\right)\_2 + \left(R\_{\rm th}\right)\_3},\tag{21}$$

$$\left(\left(R\_{\text{th}}\right)\_{1,2,3}\right) = \frac{1}{\left(\frac{\boldsymbol{\kappa}\_{1,2,3}\tilde{\boldsymbol{A}}}{I\_{1,2,3}}\right)}.\tag{22}$$

Combining Eqs. (21) and (22) leads to Eq. (23):

**Figure 2.** Schematical description of three phases, I–III, oriented in series to external temperature and electrical gradi‐ ents, as distributed in the sample (a) and as combined entities with sample's diameter and perspective lengths accord‐

For this case, a similar analysis is presented, taking into account individual thermal gradients applied on each of the phases. Taking into account that the first, second and third phases are subjected to temperature differences of (*Th* − *T*1), (*T*1 − *T*2) and (*T*2 − *Tc*), respectively, as shown in **Figure 2(b)**, where *T*1,2 are intermediate temperatures (*Th* > *T*1 > *T*2 > *Tc*), effective Seebeck coefficient of such serially aligned three‐phase samples can be described in terms of Eq. (18):

> 1 1 21 2 32 ( )( )( ) eff . -+ -+ - <sup>=</sup> *h c h c T T TT TT T T*

Under adiabatic heat conduction conditions, where no lateral heat losses are apparent, the heat flow, *Q*, through the entire sample and the individual phases can be described in terms of

( ) ( ) ( ) ( ) <sup>1</sup> <sup>2</sup> <sup>3</sup> eff

where *κ*eff is effective thermal conductivity of the three‐phase material, is cross‐section area transverse to heat flow and *κ*1,2,3 and *l*1,2,3 are thermal conductivity and effective length of each

Using expression (19), the numerator terms of Eq. (18) can be easily described in terms of

= -= -= -= - , %%% % *h c h c*

*AAA A Q T T TT TT T T ll ll*

1 12 2 1 2 3 samp

(18)

 k

(19)

aaa

ing to their relative amount (b).

a

294 Thermoelectrics for Power Generation - A Look at Trends in the Technology

of the involved phases, respectively.

expressions (20):

unidirectional Fourier heat conduction equation, Eq. (19):

kkk

$$K\_{\text{eff}} = \frac{1}{\frac{I\_1}{\kappa\_1 \tilde{A}} + \frac{I\_2}{\kappa\_2 \tilde{A}} + \frac{I\_3}{\kappa\_3 \tilde{A}}}. \tag{23}$$

Substitution of the expression of *K*eff, Eq. (23) in the rightmost term of expression (19) results in the expression of *Th* − *Tc*, presented in Eq. (24):

$$Q = \frac{1}{\frac{I\_1}{\kappa\_1 \tilde{A}} + \frac{I\_2}{\kappa\_2 \tilde{A}} + \frac{I\_3}{\kappa\_3 \tilde{A}}} (T\_h - T\_c) \text{or} \left(T\_h - T\_c\right) = Q \left(\frac{I\_1}{\kappa\_1 \tilde{A}} + \frac{I\_2}{\kappa\_2 \tilde{A}} + \frac{I\_3}{\kappa\_3 \tilde{A}}\right). \tag{24}$$

Substitution of temperature differences derived in Eqs. (20) and (24) into Eq. (18) results in the expression of *α*eff for serially connected three‐phase structures, Eq. (25):

$$\left(\alpha\_{eff}\right)\_{series} = \frac{\left(\frac{\mathcal{Q}l\_1a\_1}{\kappa\_1\overline{A}} + \frac{\mathcal{Q}l\_2a\_2}{\kappa\_2\overline{A}} + \frac{\mathcal{Q}l\_3a\_3}{\kappa\_3\overline{A}}\right)}{\mathcal{Q}\left(\frac{l\_1}{\kappa\_1\overline{A}} + \frac{l\_2}{\kappa\_2\overline{A}} + \frac{l\_3}{\kappa\_3\overline{A}}\right)} = \frac{\left(\frac{a\_1\overline{x\_1}}{\kappa\_1} + \frac{a\_2\overline{x\_2}}{\kappa\_2} + \frac{a\_3\overline{x\_3}}{\kappa\_3}\right)}{\left(\frac{\overline{x\_1}}{\kappa\_1} + \frac{\overline{x\_2}}{\kappa\_2} + \frac{\overline{x\_3}}{\kappa\_3}\right)} = \frac{\Sigma\frac{a\_1\overline{x\_1}}{\kappa\_1}}{\Sigma\frac{\overline{x\_1}}{\kappa\_1}}\tag{25}$$

Applying the same considerations described above, effective electrical and thermal conduc‐ tivities can also be derived, as expressed in Eqs. (26) and (27), respectively:

$$\left(\left(\sigma\_{\text{eff}}\right)\_{\text{series}} = \frac{1}{\frac{\mathbf{x}\_1}{\sigma\_1} + \frac{\mathbf{x}\_2}{\sigma\_2} + \frac{\mathbf{x}\_3}{\sigma\_3}} = \frac{1}{\left(\sum \frac{\mathbf{x}\_i}{\sigma\_i}\right)},\tag{26}$$

$$\left(\kappa\_{\text{eff}}\right)\_{\text{semic}} = \frac{1}{\frac{\underline{\boldsymbol{X}\_1}}{\underline{\boldsymbol{\kappa}\_1}} + \frac{\underline{\boldsymbol{X}\_2}}{\underline{\boldsymbol{\kappa}\_2}} + \frac{\underline{\boldsymbol{X}\_3}}{\underline{\boldsymbol{\kappa}\_3}}} = \frac{1}{\left(\sum \frac{\underline{\boldsymbol{X}\_i}}{\underline{\boldsymbol{\kappa}\_i}}\right)^{\cdot}}.\tag{27}$$

Similarly to the previous case of parallel‐connected phases, *i*‐ordered multi‐phase materials will follow general‐ordered right‐hand side expressions of Eqs. (25)–(27). Furthermore, it can be easily seen, that Eq. (25) and Eqs. (26) and (27) for the case of two‐phase materials are reduced to Eqs. (8) and (6), respectively, derived from the GEM method, highlighting validity of the analytic approach described here.

#### **5. Practical examples and applications**

Prior to describing the full potential of the GEM concept on optimizing performance of multi‐ phase TE materials, two general examples highlighting the potential of the method for monitoring the microstructure and phase morphology are described.

While analysing measured electrical and thermal conductivities of Cu following different spark plasma sintering (SPS) conditions, resulting in porosity levels in the range of 0–30%, a good agreement to GEM equations, Eqs. (2) and (3), was observed while assuming homoge‐ neous dispersion (*t* = 1) and nearly spherical morphology (*A* = 2), as were observed by electronic microscopy, as well as *σ*1, *κ*1 values of pure Cu (the matrix phase), and *σ*2, *κ*2 equal to zero (the pores phase) [36]. This approach not just validated experimentally the GEM equations described above, but also paved a route for monitoring porosity amount during SPS consoli‐ dation process, which is widely applied in the synthesis of TE materials, as pointed out above, just by measuring electrical resistivity of the samples. For the SnTe system in the two‐phase compositional range between pure Sn and SnTe compound, a parallel morphological align‐ ment of the phases was identified both by electronic microscopy and by measuring Seebeck coefficient values of the samples [4]. The latter was validated by comparing measured *α*eff to values, calculated by GEM equation, Eq. (1), with various *A* values. The best agreement was obtained for *A* = *8*, indicating a parallel alignment of the phases. This approach validated the possibility to identify geometrical alignment of the phases just by measuring Seebeck coefficient values without any requirement of advanced electron microscopy.

Specifically, for TE materials, it was recently shown that upon introduction of MoSe2 phase into layered *n*‐type Bi2Te2.4Se0.6 alloy for optimizing its TE performance, the best performance was obtained for oriented samples with *A* = 0.3, in Eqs. (1)–(3), as shown, for example, for *ρ*eff, in **Figure 3(a)** [3]. In this figure, the agreement of red experimental points with *A* = 0.3 curve can be clearly seen.

Simulation of Morphological Effects on Thermoelectric Power, Thermal and Electrical Conductivity... http://dx.doi.org/10.5772/65099 297

( ) eff series <sup>123</sup>

( ) eff series <sup>123</sup>

monitoring the microstructure and phase morphology are described.

coefficient values without any requirement of advanced electron microscopy.

Specifically, for TE materials, it was recently shown that upon introduction of MoSe2 phase into layered *n*‐type Bi2Te2.4Se0.6 alloy for optimizing its TE performance, the best performance was obtained for oriented samples with *A* = 0.3, in Eqs. (1)–(3), as shown, for example, for *ρ*eff, in **Figure 3(a)** [3]. In this figure, the agreement of red experimental points with *A* = 0.3 curve

s

296 Thermoelectrics for Power Generation - A Look at Trends in the Technology

k

of the analytic approach described here.

can be clearly seen.

**5. Practical examples and applications**

123

123

kkk

sss

1 1 = = , æ ö + + ç ÷

*xx x x*

1 1 = = . æ ö + + ç ÷

*xxx x*

Similarly to the previous case of parallel‐connected phases, *i*‐ordered multi‐phase materials will follow general‐ordered right‐hand side expressions of Eqs. (25)–(27). Furthermore, it can be easily seen, that Eq. (25) and Eqs. (26) and (27) for the case of two‐phase materials are reduced to Eqs. (8) and (6), respectively, derived from the GEM method, highlighting validity

Prior to describing the full potential of the GEM concept on optimizing performance of multi‐ phase TE materials, two general examples highlighting the potential of the method for

While analysing measured electrical and thermal conductivities of Cu following different spark plasma sintering (SPS) conditions, resulting in porosity levels in the range of 0–30%, a good agreement to GEM equations, Eqs. (2) and (3), was observed while assuming homoge‐ neous dispersion (*t* = 1) and nearly spherical morphology (*A* = 2), as were observed by electronic microscopy, as well as *σ*1, *κ*1 values of pure Cu (the matrix phase), and *σ*2, *κ*2 equal to zero (the pores phase) [36]. This approach not just validated experimentally the GEM equations described above, but also paved a route for monitoring porosity amount during SPS consoli‐ dation process, which is widely applied in the synthesis of TE materials, as pointed out above, just by measuring electrical resistivity of the samples. For the SnTe system in the two‐phase compositional range between pure Sn and SnTe compound, a parallel morphological align‐ ment of the phases was identified both by electronic microscopy and by measuring Seebeck coefficient values of the samples [4]. The latter was validated by comparing measured *α*eff to values, calculated by GEM equation, Eq. (1), with various *A* values. The best agreement was obtained for *A* = *8*, indicating a parallel alignment of the phases. This approach validated the possibility to identify geometrical alignment of the phases just by measuring Seebeck

è ø å *<sup>i</sup> i*

è ø å *<sup>i</sup> i*

k

s

(26)

(27)

**Figure 3.** (a) Variations of effective electrical resistivity values upon introduction of MoSe2 in Bi2Te2.4Se0.6‐MoSe2 two‐ phase system [3]. (b) Room temperature GEM analysis of effective Seebeck coefficient upon homogeneous mixing (*t* =  1) of *c*‐axis and *a*‐axis oriented grains of HMS for different geometrical alignment (0,*series*<*A*<∞*,parallel*) conditions [12]. (c) Interaction of *ZT* surfaces and volumes between three phases, solution treated (ST) matrix (B), Pb‐rich (A) and Ge‐ rich (C) phases of Pb0.25Sn0.25Ge0.5Te. The entire interaction volumes are bounded by ABC points, where each volume is bounded by two surfaces of series (S1‐S2‐S3) and parallel (P1‐P2‐P3) alignments [37].

A similar approach was recently applied for investigation of the morphological effects on TE properties of Ti0.3Zr0.35Hf0.35Ni1+δSn alloys following phase separation into half‐Heulser Ti0.3Zr0.35Hf0.35NiSn and Heusler Ti0.3Zr0.35Hf0.35Ni2Sn phases [15]. In this research, it was found that although phases' orientation was aligned in intermediate level (*A* = 0.8) between parallel (*A* = 8) and spherical (*A* = 2) alignments, enhanced TE performance is expected in a series alignment while substituting *A*=0 in Eqs. (1)–(3).

Another very interesting implementation of GEM approach was recently applied to estimate effective room temperature Seebeck coefficient and electrical resistivity values of a randomly morphological oriented homogeneous mixture of (001) and (hk0) grains in anisotropic polycrystalline HMS TE samples [12]. Applying GEM analysis to homogeneous distribution of (001) and (hk0) oriented grains (*t* = 1), for different alignment (*A*) conditions, resulted in the blue curves shown in **Figure 3(b)**. In this figure, the upper and lower blue curves represent series and parallel alignments of two configurations, respectively, points 2 and 3 represent *c*‐ and *a*‐axis‐oriented crystals, respectively, and intermediate dashed blue curve indicates a spherical distribution of two directions. Point 1 indicates 50% mixture of the directions for a spherical alignment, representing mixture of two directions, as in the case of non‐textured polycrystalline HMS powder. The black and red curves of **Figure 3(b)** indicate interaction between *c*‐ and *a*‐axis‐oriented grains with randomly distributed polycrystalline powder (point 1 in **Figure 3b**), as was calculated by GEM approach. In that case, a partial *c*‐axis preferred orientated powder, embedded in a homogeneous surrounding of macroscopic non‐ preferred‐orientated powder is expected to exhibit *α*eff values that are bounded in between the series2 and parallel2 black curves of the figures. Similarly, *α*eff values for partial *a*‐axis preferred orientated powder are expected to be bounded between series3 and parallel3 red curves of the figure. The experimentally measured *α*‖, *α*∟ values while considering 10% preferred orienta‐ tion, as was identified by XRD, are also shown in the figure. It can be seen that experimental points lie in the interaction zone between *c*‐ and *a*‐axis‐orientated powder and a randomly distributed powder, bounded by the black and red curves, respectively. This indicates the validity of proposed calculation route to estimate electronic transport properties of textured polycrystalline materials. It can be also seen that for HMS, ∼10% preferred orientation of both of investigated directions is almost independent of the orientation of the grains, and, therefore, controlling the alignment of the grains morphology is not expected to affect the effective Seebeck coefficient.

Implementation of GEM concept in three‐phase TE materials, based on Eqs. (15)–(17) and (25)– (27), was recently shown for quasi‐ternary GeTe‐PbTe‐SnTe system [37, 38]. Specifically, it was shown that phase separation of solution‐treated (ST) Pb0.25Sn0.25Ge0.5Te composition (phase B in **Figure 3c**) into Pb‐rich, Pb0.33Sn0.3Ge0.37Te (phase A), and Ge‐rich, Pb0.1Sn0.17Ge0.73Te (phase C) phases is apparent in the system. In this system, prolonged thermal treatments at each temperature resulted, at the first stages, in three phases, parent B phase and two decomposed A and C phases. This stage is terminated by full decomposition into A and C, where only these phases are apparent. Furthermore, a lamellar alignment of the phases was observed at the first 24 h of thermal treatment, while prolonged treatments were resulted in spheroidization, due to reduced surface area free energy at this configuration. It was also observed that *ZT* values were increased during the first 24 h while reduced at more prolonged durations. For explaining these experimental evidences, GEM approach was applied, as shown in **Figure 3(c)**. In this figure, triangle BDE indicates the specific interaction surface for separation of the phase B into the phases A and C, where BD side of triangle represents series ('lamellar') alignment mor‐ phology and BE represents parallel alignment of the phases. The dashed BO line represents spherical alignment. It can be easily shown that measured *ZT* values, indicated by the blue line, indeed follow the series alignment (BD line) at the first decomposition stages, but from this point on approach the dashed BO line until a full spheroidization is occurred (at point o). From this analysis, it was concluded that any theoretical possibility for retaining the lamellar morphology in this system would result in even higher *ZT* values of up to ∼1.8 after a complete decomposition of the matrix into the two involved separation phases.

## **6. Concluding remarks**

In this chapter, the potential of GEM approach to optimize electronic properties of multi‐phase thermoelectric materials in terms of compositional or morphological considerations is shown in details. This approach already proved itself in monitoring of the densification rate of powder metallurgy processed materials, as well as in the determination of compositional modifications in binary systems just by measuring one of the transport properties. It is just beginning to approach the true potential to optimize thermoelectric transport properties of multi‐phase materials, such as those containing embedded nano‐features for reduction of the lattice thermal conductivity, where electronic contribution of the involved phase is usually neglected. It was shown that method does not just explain unexpected electronic trends in such materials, but might be employed for prediction of synthesis routes for optimizing thermoelectric figure of merit based on different compositions or alignment morphologies.

Based on the pointed above examples, it is obvious that for TE power generators operating at low (<300°C), intermediate (300–500°C) and high (>500°C) temperature ranges, Bi2Te3, PbTe/ GeTe and HMS/half‐Heusler‐based compositions might be employed. In such systems, identifying compositions enabling phase separation or precipitation into multi‐phases, according to specific phase diagram, has a potential to reduce lattice thermal conductivity. Yet, for maximizing TE potential, optimal geometrical alignment of the phases should be identified. Using the proposed approach, based on individual TE transport properties of the involved phases, optimal geometrical alignment direction might be identified, leading to enhanced TE performance, enabling a real contribution to the society by reducing our dependence on fossil fuels and by minimizing emission of greenhouse gases.

## **Acknowledgements**

validity of proposed calculation route to estimate electronic transport properties of textured polycrystalline materials. It can be also seen that for HMS, ∼10% preferred orientation of both of investigated directions is almost independent of the orientation of the grains, and, therefore, controlling the alignment of the grains morphology is not expected to affect the effective

298 Thermoelectrics for Power Generation - A Look at Trends in the Technology

Implementation of GEM concept in three‐phase TE materials, based on Eqs. (15)–(17) and (25)– (27), was recently shown for quasi‐ternary GeTe‐PbTe‐SnTe system [37, 38]. Specifically, it was shown that phase separation of solution‐treated (ST) Pb0.25Sn0.25Ge0.5Te composition (phase B in **Figure 3c**) into Pb‐rich, Pb0.33Sn0.3Ge0.37Te (phase A), and Ge‐rich, Pb0.1Sn0.17Ge0.73Te (phase C) phases is apparent in the system. In this system, prolonged thermal treatments at each temperature resulted, at the first stages, in three phases, parent B phase and two decomposed A and C phases. This stage is terminated by full decomposition into A and C, where only these phases are apparent. Furthermore, a lamellar alignment of the phases was observed at the first 24 h of thermal treatment, while prolonged treatments were resulted in spheroidization, due to reduced surface area free energy at this configuration. It was also observed that *ZT* values were increased during the first 24 h while reduced at more prolonged durations. For explaining these experimental evidences, GEM approach was applied, as shown in **Figure 3(c)**. In this figure, triangle BDE indicates the specific interaction surface for separation of the phase B into the phases A and C, where BD side of triangle represents series ('lamellar') alignment mor‐ phology and BE represents parallel alignment of the phases. The dashed BO line represents spherical alignment. It can be easily shown that measured *ZT* values, indicated by the blue line, indeed follow the series alignment (BD line) at the first decomposition stages, but from this point on approach the dashed BO line until a full spheroidization is occurred (at point o). From this analysis, it was concluded that any theoretical possibility for retaining the lamellar morphology in this system would result in even higher *ZT* values of up to ∼1.8 after a complete

decomposition of the matrix into the two involved separation phases.

merit based on different compositions or alignment morphologies.

In this chapter, the potential of GEM approach to optimize electronic properties of multi‐phase thermoelectric materials in terms of compositional or morphological considerations is shown in details. This approach already proved itself in monitoring of the densification rate of powder metallurgy processed materials, as well as in the determination of compositional modifications in binary systems just by measuring one of the transport properties. It is just beginning to approach the true potential to optimize thermoelectric transport properties of multi‐phase materials, such as those containing embedded nano‐features for reduction of the lattice thermal conductivity, where electronic contribution of the involved phase is usually neglected. It was shown that method does not just explain unexpected electronic trends in such materials, but might be employed for prediction of synthesis routes for optimizing thermoelectric figure of

Based on the pointed above examples, it is obvious that for TE power generators operating at low (<300°C), intermediate (300–500°C) and high (>500°C) temperature ranges, Bi2Te3, PbTe/

Seebeck coefficient.

**6. Concluding remarks**

The work was supported by the Ministry of National Infrastructures, Energy and Water Resources grant (3/15), No. 215‐11‐050.

## **Author details**

Yaniv Gelbstein

Address all correspondence to: yanivge@bgu.ac.il

Department of Materials Engineering, Ben‐Gurion University of the Negev, Beer‐Sheva, Israel

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## **Thermal Conductivity and Non-Newtonian Behavior of Complex Plasma Liquids Thermal Conductivity and Non-Newtonian Behavior of Complex Plasma Liquids**

Aamir Shahzad and Maogang He Aamir Shahzad and Maogang He

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/65563

#### **Abstract**

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38.

101:113707.

Understanding of thermophysical properties of complex liquids under various conditions is of practical interest in the field of science and technology. Thermal conductivity of nonideal complex (dusty) plasmas (NICDPs) is investigated by using homogeneous nonequilibrium molecular dynamics (HNEMD) simulation method. New investigations have shown, for the first time, that Yukawa dusty plasma liquids (YDPLs) exhibit a non-Newtonian behavior expressed with the increase of plasma conductivity with increasing external force field strength *F*ext. The observations for lattice correlation functions Ψ (*t*) show, that our YDPL system remains in strongly coupled regime for a complete range of plasma states of (Γ, *κ*), where (Γ) Coulomb coupling and (κ) Debye screening length. It is demonstrated, that the present NICDP system follows a simple scaling law of thermal conductivity. It has been shown, that our new simulations extend the range of *F*ext used in the earlier studies in order to find out the size of the linear ranges. It has been shown that obtained results at near equilibrium (*F*ext = 0.005) are in satisfactory agreement with the earlier simulation results and with the presented reference set of data showed deviations within less than ±15% for most of the present data points and generally overpredicted thermal conductivity by 3–22%, depending on (Γ, *κ*).

**Keywords:** thermophysical properties, thermal conductivity, nonlinear effects, lattice correlation functions, nonideal complex (dusty) plasmas, nonequilibrium molecular dynamics

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **1. Introduction**

The exact numerical investigation of transport properties of complex liquids is a fundamental research task in the field of thermophysics, as various transport data are closely related with setup and the confirmation of equations of state. A reliable knowledge of transport data is also important for optimization of technological processes and apparatus design in various engineering and science fields (incl. thermoelectric devices) and, in particular, when provision of precise data for parameters of heat, mass, and momentum transport is required [1–3]. In thermophysical properties of fluids, chemical properties remain unaffected, but physical properties of material are changed by variable temperature, composition, and pressure. These properties of simple and complex liquids explain the phase transition [4]. These fluids can be examined experimentally, theoretically, and by simulation techniques. Thermophysical properties (thermodynamics and transport coefficients) include thermal conductivity, thermal expansion, thermal radiative properties, thermal diffusivity, enthalpy internal energy, Joule-Thomson coefficients, and heat capacity, as well as, thermal diffusion coefficients, mass coefficients, viscosity, speed of sound, and interfacial and surface tension. Thermophysical properties of gases and liquids, such as hydrogen, H2; oxygen, O2; nitrogen, N2; and water, H2O are different from ideal gas at high pressure and low temperature. Specific models are required for the calculation of these properties in the widest range of pressure and temperature. Different fluids, such as gases and liquids are used as a power generation source in different power plants. For example, heavy water, steam, air, and different gases are used for power generation in nuclear power plants, gas turbine plants, and internal combustion plants. Also, for cooling in refrigerators and fast nuclear reactors, ammonia and sodium in liquid phase are used as a cooling agent.

#### **1.2. Dusty plasma**

Nowadays dusty plasma refers to as complex plasma in analogy to the condensed matter field of "complex liquids" in soft matter (colloidal suspensions, polymers, surfactants, etc.). The dust particles combine physics of nonideal plasmas and condensed matter, and this field has played an important role in both newly system designs and advance development micro- and nanotechnology. This complex plasma system has four components, i.e., ions, electrons, neutral atoms, and dust particles with high charges, as compared to other species, which are responsible for the extraordinary plasma properties. The study of complex (dusty) plasmas reveals rich variety of interesting phenomena and extends knowledge on fundamental aspects of plasma physics at the microscopic level. Among these, the freezing (gaseous-liquid-solid) phase transition is of particular interest. Complex plasma is called strongly coupled plasma, in which thermal energy (kinetic energy) of nearest neighbors is much smaller than their Coulomb interaction potential energy, whereas plasma is called weakly coupled when Coulomb interparticle potential energy of nearest particles is much smaller than their kinetic energy [5–7].

Plasma is the fourth state of matter, and usually, it is said, that there are three states of matter, but another state was also found to exist, named as plasma. Irving Langmuir (American physicist) defined plasma as "it is a quasi-neutral gas of charged and neutral particles, which exhibits collective behavior," and he got the Nobel Prize in 1927 because firstly he was using the term plasma [8]. In this definition, quasi-neutral means that plasma is electrically neutral and has approximately equal ion and electron density (*ni* ≈ *ne*). The term collective behavior shows, that due to Coulomb potential or electric field, plasma's particles collide with each other. Simply, plasma is an ionized gas. In a gas, sufficient energy is given to eject free electrons from atoms or molecules, and, as a result, ions and electrons, both species, coexist. There are several ways to convert a gas into plasma, but all include pumping the gas with energy. For instance, plasma can be created due to a spark in a gas. Usually, neutral plasmas are relatively hot, such as the solar corona (1000000 K), a candle flame (1000 K), or the ionosphere around our planet (300 K). In this universe, the main source of plasma is sun in which the large number of electrons of hydrogen and helium molecules is removed. So, the sun is a great big ball of plasma like other stars.

#### *1.2.1. Types of plasma*

**1. Introduction**

304 Thermoelectrics for Power Generation - A Look at Trends in the Technology

phase are used as a cooling agent.

**1.2. Dusty plasma**

energy [5–7].

The exact numerical investigation of transport properties of complex liquids is a fundamental research task in the field of thermophysics, as various transport data are closely related with setup and the confirmation of equations of state. A reliable knowledge of transport data is also important for optimization of technological processes and apparatus design in various engineering and science fields (incl. thermoelectric devices) and, in particular, when provision of precise data for parameters of heat, mass, and momentum transport is required [1–3]. In thermophysical properties of fluids, chemical properties remain unaffected, but physical properties of material are changed by variable temperature, composition, and pressure. These properties of simple and complex liquids explain the phase transition [4]. These fluids can be examined experimentally, theoretically, and by simulation techniques. Thermophysical properties (thermodynamics and transport coefficients) include thermal conductivity, thermal expansion, thermal radiative properties, thermal diffusivity, enthalpy internal energy, Joule-Thomson coefficients, and heat capacity, as well as, thermal diffusion coefficients, mass coefficients, viscosity, speed of sound, and interfacial and surface tension. Thermophysical properties of gases and liquids, such as hydrogen, H2; oxygen, O2; nitrogen, N2; and water, H2O are different from ideal gas at high pressure and low temperature. Specific models are required for the calculation of these properties in the widest range of pressure and temperature. Different fluids, such as gases and liquids are used as a power generation source in different power plants. For example, heavy water, steam, air, and different gases are used for power generation in nuclear power plants, gas turbine plants, and internal combustion plants. Also, for cooling in refrigerators and fast nuclear reactors, ammonia and sodium in liquid

Nowadays dusty plasma refers to as complex plasma in analogy to the condensed matter field of "complex liquids" in soft matter (colloidal suspensions, polymers, surfactants, etc.). The dust particles combine physics of nonideal plasmas and condensed matter, and this field has played an important role in both newly system designs and advance development micro- and nanotechnology. This complex plasma system has four components, i.e., ions, electrons, neutral atoms, and dust particles with high charges, as compared to other species, which are responsible for the extraordinary plasma properties. The study of complex (dusty) plasmas reveals rich variety of interesting phenomena and extends knowledge on fundamental aspects of plasma physics at the microscopic level. Among these, the freezing (gaseous-liquid-solid) phase transition is of particular interest. Complex plasma is called strongly coupled plasma, in which thermal energy (kinetic energy) of nearest neighbors is much smaller than their Coulomb interaction potential energy, whereas plasma is called weakly coupled when Coulomb interparticle potential energy of nearest particles is much smaller than their kinetic

Plasma is the fourth state of matter, and usually, it is said, that there are three states of matter, but another state was also found to exist, named as plasma. Irving Langmuir (American

There are different types of plasma that are described by many characteristics, such as temperature, degree of ionization, and density.

#### *1.2.1.1. Cold plasma*

In laboratory, in the positive column of a glow discharge tube, there exists plasma in which the same number of ions and electrons is present. When gas pressure is low, collision between electrons and gas molecules is not frequent. So, nonthermal equilibrium between energy of electrons and gas molecules does not exist. So, energy of electrons is very high as compared to gas molecules and the motion of gas molecules can be ignored. We have *T*e≫*T*i≫*T*g, where *T*e, *Ti* , and *T*g represent temperature of electron, ion, and gas molecules and such type of plasma is known as cold plasma. In cold plasma, the magnetic force can be ignored and only the electric force is considered to act on the particles. In cold plasma technique, cold gases are used to disinfect surfaces of packaging or food products. Vegetative microorganisms and spores on packaging materials can be inactivate at temperature below 40°C. This process can have a clear advantage compared to heat treatment for temperature-sensitive products. Also it can reduce the amount of water used for disinfection of packaging materials. Because cold plasma is in the form of gas, so, the irregularly shaped packages, such as bottles can be treated easily as compared to UV or pulsed light where shadowing occurs.

#### *1.2.1.2. Hot plasma*

When gas pressure is high in the discharge tube, then electrons collide with gas molecules very frequently and thermal equilibrium exists between electrons and gas molecules. We have *T*e ≅ *T*i . Such type of plasma is known as hot plasma and it is also known as thermal plasma. Hot plasma is one which approaches to a local thermodynamic equilibrium (LTE). Atmospheric arcs, sparks, and flames are used for the production of such type of plasmas.

#### *1.2.1.3. Ultracold plasma*

If plasma occurs at temperature as low as 1 K, then such type of plasma is known as ultracold plasma, and it can be formed by photoionizing laser-cooled atoms and pulsed lasers. In ultracold plasmas, the particles are strongly interacting because their thermal energy is less than Coulomb energy between neighboring particles [9].

#### *1.2.1.4. Ideal plasma*

There are mainly two types of plasma according to plasmas' ideality and properties study, nonideal plasmas (weakly coupled and strongly coupled plasmas) and ideal plasmas (very weakly coupled plasmas). Whenever the kinetic energy of plasma is much larger than the potential energy and plasma has a low temperature and high density, then such type of plasma is known as ideal plasma. Ideal plasma is one in which Coulomb collisions are negligible. If the average distance among the interacting particles is large, then the interaction potential can be ignored due to this large-distance ideal plasma that does not have any arrangement of particles [2].

#### *1.2.1.5. Nonideal (complex) plasma*

Nonideal plasmas are often found in nature, as well as, in technological services. They can be shown as electron plasma in solid and liquid metals and electrolytes, the superdense plasma of the matter of white dwarfs, the sun and the interiors (deep layers) of the giant planets of the solar system, and astrophysical objects, whose structure and evolution are defined by plasma characteristics [2]. Further examples of nonideal plasmas are brown dwarfs, laser-generated plasmas, capillary discharges, plasma-opening switches, high-power electrical fuses, exploding wires, etc. On the bases of Coulomb coupling, nonideal plasma can be divided into two families: weakly coupled plasma (WCP, Г < 1) and strongly coupled plasma (SCP, Г ≥ 1).

Nonideal complex plasmas are found in daily life and can be found in processing industries to manufacture many products that we deal in our everyday life directly or indirectly at moderate temperature, such as plastic bags, automobile bumpers, airplane turbine blades, artificial joints, and, most importantly, in semiconductor circuits. Moreover, nonideal plasmas (terrestrial plasmas) are not hard to find. They occur in gas-discharge lighting, such as neon lighting used for commercial purposes and fluorescent lamps, for instance, compact fluorescence light sources, which have a higher performance than the traditional incandescent light sources, a variety of laboratory experiments, and a growing array of industrial processes. Modern display methods contain plasma screens, in which small plasma discharges are used to stimulate a phosphor layer, which then emits light [2, 7].

#### *1.2.2. Complex (dusty) plasma*

Dust is present everywhere in the universe and mostly it is present in solid form. It is also present in gaseous form, which is often ionized, and thus the dust coexists with plasma and forms "dusty plasma." In dusty plasma, dust particles are immersed in plasma, in which ions, electrons, and neutrals are present. These dust particles are charged and then affected by electric or magnetic fields and can cause different changes in the properties of plasma. The presence of dust component gives rise to new plasma phenomena and allows study of fundamental aspects of plasma physics at the microscopic level. Dust particles are charged due to the interaction between dust particles and the surrounding plasmas. Due to this interaction, grains are charged very rapidly. The charge on grains depends on the flow of ions and electrons. These charged grains enhance plasma environments, for example, setting up space charges. Also, to determine the charge on dust grain, it is assumed, that a spherically symmetric isolated dust grain is injected in plasma and only the effect of ion and electron is considered. Moreover, there are many other charging processes, such as secondary emission, electron emission, thermionic emission, field emission, radioactivity, and impact ionization. Complex plasma is condensed plasma characterized by strong interaction between existing molecules and atoms; it is also called strongly coupled complex plasma. Dusty plasma is complex plasma which includes many components: ions, electrons, neutral particle, and dust particles. Last 20–25 years, strongly coupled plasmas were mainly studied theoretically, due to lack of suitable laboratory tools and equipment. However, experimental strongly coupled plasma studies became more common with the discovery of ways to find dusty plasma [10], laser-cooled ion plasmas in a penning trap [11], and ultracold neutral plasmas [12]. Plasma systems can be treated theoretically in a straightforward way in the extreme limits of both weakly coupled and strongly coupled plasmas [2].

The main goals of this chapter are to study thermal conductivity (*λ*) along with lattice correlation (long-range crystalline order) at the corresponding plasma states and to extend the set of plasma states (Γ, *κ*) by using the same method as introduced by Shahzad and He [3] with different system sizes (*N*). The effect of external force field strengths on Ψ(*t*) and corresponding *λ* values of complex Yukawa liquids of dust particles is another interesting task that is calculated under near-equilibrium condition.

## **2. HNEMD model and simulation approach**

*1.2.1.3. Ultracold plasma*

*1.2.1.4. Ideal plasma*

particles [2].

*1.2.1.5. Nonideal (complex) plasma*

*1.2.2. Complex (dusty) plasma*

than Coulomb energy between neighboring particles [9].

306 Thermoelectrics for Power Generation - A Look at Trends in the Technology

to stimulate a phosphor layer, which then emits light [2, 7].

If plasma occurs at temperature as low as 1 K, then such type of plasma is known as ultracold plasma, and it can be formed by photoionizing laser-cooled atoms and pulsed lasers. In ultracold plasmas, the particles are strongly interacting because their thermal energy is less

There are mainly two types of plasma according to plasmas' ideality and properties study, nonideal plasmas (weakly coupled and strongly coupled plasmas) and ideal plasmas (very weakly coupled plasmas). Whenever the kinetic energy of plasma is much larger than the potential energy and plasma has a low temperature and high density, then such type of plasma is known as ideal plasma. Ideal plasma is one in which Coulomb collisions are negligible. If the average distance among the interacting particles is large, then the interaction potential can be ignored due to this large-distance ideal plasma that does not have any arrangement of

Nonideal plasmas are often found in nature, as well as, in technological services. They can be shown as electron plasma in solid and liquid metals and electrolytes, the superdense plasma of the matter of white dwarfs, the sun and the interiors (deep layers) of the giant planets of the solar system, and astrophysical objects, whose structure and evolution are defined by plasma characteristics [2]. Further examples of nonideal plasmas are brown dwarfs, laser-generated plasmas, capillary discharges, plasma-opening switches, high-power electrical fuses, exploding wires, etc. On the bases of Coulomb coupling, nonideal plasma can be divided into two families: weakly coupled plasma (WCP, Г < 1) and strongly coupled plasma (SCP, Г ≥ 1).

Nonideal complex plasmas are found in daily life and can be found in processing industries to manufacture many products that we deal in our everyday life directly or indirectly at moderate temperature, such as plastic bags, automobile bumpers, airplane turbine blades, artificial joints, and, most importantly, in semiconductor circuits. Moreover, nonideal plasmas (terrestrial plasmas) are not hard to find. They occur in gas-discharge lighting, such as neon lighting used for commercial purposes and fluorescent lamps, for instance, compact fluorescence light sources, which have a higher performance than the traditional incandescent light sources, a variety of laboratory experiments, and a growing array of industrial processes. Modern display methods contain plasma screens, in which small plasma discharges are used

Dust is present everywhere in the universe and mostly it is present in solid form. It is also present in gaseous form, which is often ionized, and thus the dust coexists with plasma and forms "dusty plasma." In dusty plasma, dust particles are immersed in plasma, in which ions, electrons, and neutrals are present. These dust particles are charged and then affected by In this section, we will introduce theoretical background needed in this work. We start by introducing the model system, which is used in our HNEMD simulations. We consider a cubic box of edge length *L* and have *N* number of particles or millions of atoms. In MD technique, the range of particle number is chosen as *N* = 500–1000 [13]. Periodic boundary conditions (PBCs) are used for selection of the size or dimension of the box. Practically, PBCs avoid the surface size effects. The particles present in the box interact with each other with known interaction potential. This potential may be Yukawa, Coulomb, and Lennard-Jones potential depends on the type of the system considered.

The particles interact through screened Coulomb potential, which depends on the physical parameters and the background plasmas. Average interparticle interaction is frequently considered to be isotropic and basically repulsive and approximated by Yukawa interaction potential [1–3]. Yukawa model has been employed in many physical and chemical systems (for instance, biological and pharmaceutical sciences, colloidal and ionic systems, space and environment sciences, physics and chemistry of polymers and materials, etc.) [1–8]. In the present case, the interaction of potential energy of particles is in Yukawa form:

$$\phi\_{\mathbf{v}}(|\mathbf{r}|) = \frac{Q\_d^2}{4\pi\varepsilon\_0} \frac{e^{-|\mathbf{v}|\cdot\lambda\_0}}{|\mathbf{r}|},\tag{1}$$

where charge on dust particle is *Q*d, magnitude of interparticle division is *r*, and Debye length *λ*D accounts for the screening of interaction by other plasma species. Due to the long-range interaction between the particles, Yukawa potential energy cannot be solved directly. Ewald sum method is used with periodic boundary conditions to calculate Yukawa potential energy, force, and heat energy current. Improved nonequilibrium molecular dynamics (NEMD) method proposed by Evans has been employed to estimate thermal conductivity of strongly coupled complex (dusty) plasmas. During the simulation of Yukawa systems, sufficient number of particles *N* is to be selected to study the size effect of system. It comes to know, that there is no effect of system size on thermal conductivity or on any other properties under limited statistical uncertainties. Negative divergence of Yukawa potential *F* = (−∇*φ*) and Newton's equation of motion integrated by predictor-corrector algorithm are used for calculation of force exerted by the particles on each other or in minimum image convection [14].

In HNEMD technique, in order to measure thermal conductivity and nonlinearity of NICDPs, the system will be perturbed by applying the external field along *z*-axis. In three-dimensional systems, we use standard Green-Kubo relations (GKR) for calculation of thermal conductivity coefficient of uncharged particles [15]:

$$\mathcal{X} = \frac{1}{3k\_B VT^2} \Big| \Big\langle \mathbf{J}\_{\varrho}(t)\mathbf{J}\_{\varrho}(0) \Big\rangle dt. \tag{2}$$

Here, **J***Q* is vector of heat flux, *V* is the volume, *T* is the temperature of system, and *k*B is Boltzmann constant. At microscopic level, heat flux vector has value:

$$\mathbf{J}\_{\varrho}V = \sum\_{i=1}^{N} E\_i \frac{\mathbf{p}\_i}{m} - \frac{1}{2} \sum\_{i \neq j} \mathbf{r}\_y (\frac{\mathbf{p}\_i}{m}.\mathbf{F}\_y). \tag{3}$$

In this equation, r*ij* = r*<sup>i</sup>* − r*<sup>j</sup>* is the position vector, F*ij* is the force of interaction on particle *i* due to *j*, and p*<sup>i</sup>* represents the momentum vector of the *i*th particle. The energy *Ei* of *i*th particle is given by:

$$E\_i = \frac{\mathbf{p}\_i^2}{2m} + \frac{1}{2} \sum\_{i \neq j} \phi\_{ij},\tag{4}$$

where ∅*ij* is Yukawa pair potential between particles *i* and *j*. According to non-Hamiltonian dynamics, generalization of linear response theory, proposed by Evans, for system moving with equations of motion and recently detailed understanding of Ewald-Yukawa sums [1–3] allows to present thermal conductivity as:

$$\mathcal{A} = \lim\_{F\_z \to 0} \lim\_{t \to \nu} \frac{-\left< \mathbf{J}\_{\mathcal{Q}\_z}(t) \right>}{T F\_z},\tag{5}$$

where J*Qz* is *z*-component of heat energy flux vector for strongly coupled complex (dusty) plasma liquids. Thermal conductivity coefficient for charged particle of plasma according to GKR is given by Eq. (5), and further detail is provided in Ref. [3]. Plasma states of Yukawa systems can be illustrated fully by three reduced parameters: plasma coupling parameter Γ = (*Q*<sup>2</sup> /4πε0)(*a*ws*k*B*T*), screening parameter *κ* ≡ *a*ws/*λD*, and reduced external force *Fext* for HNEMD model, where *a*ws is Wigner-Seitz radius, *Q* is charge on dust particle, and *ε*0 is permittivity of free space [1–3]. Gaussian thermostat is used to control temperature of systems [14]. Simulation time step is d*t* = 0.001/*ω*p, where *ω*p = (*nQ*<sup>2</sup> /ε0*m*) 1/2 is dust plasma frequency with *m* is dust particle's mass and *n* is number density. Reported simulations are performed between 3.0 × 105 /*ω*p and 1.5 × 105 /*ω*p time units in the series of data recording of thermal conductivity (*λ*) [16, 17].

## **3. Computer simulation outcomes**

#### **3.1. Particle lattice correlation**

environment sciences, physics and chemistry of polymers and materials, etc.) [1–8]. In the

2 / <sup>D</sup> d


**r**

l

**<sup>r</sup>** (1)

0 () , <sup>4</sup>

where charge on dust particle is *Q*d, magnitude of interparticle division is *r*, and Debye length *λ*D accounts for the screening of interaction by other plasma species. Due to the long-range interaction between the particles, Yukawa potential energy cannot be solved directly. Ewald sum method is used with periodic boundary conditions to calculate Yukawa potential energy, force, and heat energy current. Improved nonequilibrium molecular dynamics (NEMD) method proposed by Evans has been employed to estimate thermal conductivity of strongly coupled complex (dusty) plasmas. During the simulation of Yukawa systems, sufficient number of particles *N* is to be selected to study the size effect of system. It comes to know, that there is no effect of system size on thermal conductivity or on any other properties under limited statistical uncertainties. Negative divergence of Yukawa potential *F* = (−∇*φ*) and Newton's equation of motion integrated by predictor-corrector algorithm are used for calculation of force exerted by the particles on each other or in minimum image convection [14].

In HNEMD technique, in order to measure thermal conductivity and nonlinearity of NICDPs, the system will be perturbed by applying the external field along *z*-axis. In three-dimensional systems, we use standard Green-Kubo relations (GKR) for calculation of thermal conductivity

> 2 B 0

¥

*k VT*

Boltzmann constant. At microscopic level, heat flux vector has value:

1

*<sup>E</sup> <sup>m</sup>*

*N*

*V E*

− r*<sup>j</sup>*

l

<sup>1</sup> ( ). (0) . <sup>3</sup> *Q Q t dt*

Here, **J***Q* is vector of heat flux, *V* is the volume, *T* is the temperature of system, and *k*B is

*i i Q i ij ij i i j*

<sup>=</sup> *m m* <sup>¹</sup>

represents the momentum vector of the *i*th particle. The energy *Ei*

<sup>2</sup> <sup>1</sup> , 2 2 *i i ij i j*

¹

f

<sup>1</sup> ( . ). <sup>2</sup>

<sup>=</sup> ò **J J** (2)

= - å å **p p <sup>J</sup> r F** (3)

is the position vector, F*ij* is the force of interaction on particle *i* due

= + å **<sup>p</sup>** (4)

of *i*th particle is

pe

present case, the interaction of potential energy of particles is in Yukawa form:

Y

**r**

f

308 Thermoelectrics for Power Generation - A Look at Trends in the Technology

coefficient of uncharged particles [15]:

In this equation, r*ij* = r*<sup>i</sup>*

to *j*, and p*<sup>i</sup>*

given by:

The structural information of Yukawa system is given by lattice correlation. For the calculation of lattice correlation, density of given material at point *r* can be calculated as:

$$\rho\left(r\right) = \sum\_{j=1}^{N} \mathcal{S}\left(r - r\_j\right),\tag{6}$$

where *ρ* is density of system, *N* is number of particles, *δ* represents distribution of particles, and *rj* is the position of particle corresponding to particle at position *r*. Eq. (6) gives information about the system being in ordered state and then it may be in solid or crystal form depending on *ρ*. The lattice correlation equation according to Fourier transform is:

$$\Psi = \frac{1}{N} \sum\_{i=1}^{N} \exp\left(-i\mathbf{k}.\mathbf{r}\_{\cdot}\right). \tag{7}$$

System arrangement (ordered or disordered) is calculated by simulation based on Eq. (7). When value of lattice correlation approaches to |Ψ| ≈ 1, then the system will be in ordered state, and if value becomes |Ψ| ≈ 0, then the system will be in liquid or gas (nonideal gas) state. In Eq. (7), *k* is lattice correlation vector for ordered state, and its value is different for different lattice structures. Its value for face-centered cubic (FCC) is *k* = 2π/(1,−1,1)*l*, for body-centered cubic (BCC) is *k* = 2π/(1,0,1)*l*, and for simple cubic (SC) is *k* = 2π/(1,0,0)*l*; here, *l* is edge length [14].

Lattice correlation was examined in 3D NICDPs in the limit of appropriate constant near equilibrium external force field strength *F*ext = 0.005. **Figure 1** illustrates lattice correlation in NICDPs versus simulation time at normalized *F*ext = 0.005. In this case, additional parameter includes the heat energy flux J*Q* and the external force field strength *F*ext(*t*) = (0,0,*F*z) is selected along *z*-axis, in the limit of *t* → ∞, and its normalized value *F*ext = (*Fz*)(*a*ws/J*Q*) [17].

**Figure 1.** Dependences of lattice correlation |Ψ (*t*)| on simulation time (Δ*t*) at external force field strength *F*ext = 0.005 imposed to Yukawa systems, for three values of coupling states Γ = 10, Γ = 50, and Γ = 100 and system size *N* = 500, (a) at *κ* = 1.4 and (b) at *κ* = 2.

#### **3.2. Normalized thermal conductivity**

We now turn attention to the key results obtained through HNEMD simulations. Obtained computer-simulated data confirm, that thermal conductivity of Yukawa system can be calculated with satisfactory statistics by an extended HNEMD approach. **Figures 2**–**5** display the main results calculated from HNEMD method for various plasma states for Yukawa liquids at *κ* = 1.4 and *κ* = 2 and *κ* = 4, respectively. HNEMD simulation is used to compute the thermal conductivity normalized by plasma frequency (*ωp*) as *λ*0 = *λ*/*nkBω*p*aws*, or by Einstein frequency (*ω*E) as *λ*\* = *λ*/√3*nk*B*ω*E*aws* of YDPLs, at the normalized external field strength. These normalizations of transport properties, including *λ*0, were widely used in earlier studies of one-component complex plasma (OCCP) [18] and NICDPs [1–3, 19–21]. HNEMD method is employed to investigate *λ*0 of 3D NICDPs at reduced external force field *F*ext = 0.005 over suitable domain of plasma parameters of coupling (1 ≤ Γ ≤ 300) and screening (1 ≤ *κ* ≤ 4).

state, and if value becomes |Ψ| ≈ 0, then the system will be in liquid or gas (nonideal gas) state. In Eq. (7), *k* is lattice correlation vector for ordered state, and its value is different for different lattice structures. Its value for face-centered cubic (FCC) is *k* = 2π/(1,−1,1)*l*, for body-centered cubic (BCC) is *k* = 2π/(1,0,1)*l*, and for simple cubic (SC) is *k* = 2π/(1,0,0)*l*; here, *l* is edge length [14]. Lattice correlation was examined in 3D NICDPs in the limit of appropriate constant near equilibrium external force field strength *F*ext = 0.005. **Figure 1** illustrates lattice correlation in NICDPs versus simulation time at normalized *F*ext = 0.005. In this case, additional parameter includes the heat energy flux J*Q* and the external force field strength *F*ext(*t*) = (0,0,*F*z) is selected

**Figure 1.** Dependences of lattice correlation |Ψ (*t*)| on simulation time (Δ*t*) at external force field strength *F*ext = 0.005 imposed to Yukawa systems, for three values of coupling states Γ = 10, Γ = 50, and Γ = 100 and system size *N* = 500, (a)

We now turn attention to the key results obtained through HNEMD simulations. Obtained computer-simulated data confirm, that thermal conductivity of Yukawa system can be calculated with satisfactory statistics by an extended HNEMD approach. **Figures 2**–**5** display

at *κ* = 1.4 and (b) at *κ* = 2.

**3.2. Normalized thermal conductivity**

along *z*-axis, in the limit of *t* → ∞, and its normalized value *F*ext = (*Fz*)(*a*ws/J*Q*) [17].

310 Thermoelectrics for Power Generation - A Look at Trends in the Technology

**Figure 2.** (a) Data for thermal conductivity *λ*0 normalized by plasma frequency, calculated by different MD methods for Coulomb coupling parameter (1 ≤ Γ ≤ 300) at *N* = 500 and *κ* = 1.4. Data obtained by Shahzad and He for homogeneous perturbed MD (HPMD)-SH (HPMD) [1], Salin and Caillol for equilibrium MD (EMD)-SC (EMD) [19], Donko and Hartmann for inhomogeneous NEMD-DH NEMD [20], and Faussurier and Murillo for variance procedure (VP)-FM VP [21]. (b) Dependences of *λ*0 normalized by *λ*REF at *F*ext = 0.005 on Coulomb coupling parameter (1 ≤ Γ ≤ 100) at *N* = 500 and *κ* = 1.4. Our present and earlier normalized results calculated at different *F*ext. Results obtained by Shahzad and He [3] with *N* = 13500 at *F*ext = 0.005 are taken as reference set of data *λ*REF. Dashed lines represent spreading ±15 ranges around the reference data.

**Figure 3.** (a) Data for thermal conductivity *λ*0 normalized by plasma frequency, calculated by different MD methods for Coulomb coupling parameter (1 ≤ Γ ≤ 300) at *N* = 864 and *κ* = 1.4. Data obtained by Shahzad and He for homogeneous perturbed MD (HPMD)-SH (HPMD) [1], Salin and Caillol for equilibrium MD (EMD)-SC (EMD) [19], Donko and Hartmann for inhomogeneous NEMD-DH NEMD [20], and Faussurier and Murillo for variance procedure (VP)-FM VP [21]. (b) Dependences of *λ*0 reduced by *λ*REF at *F*ext = 0.005 on Coulomb coupling parameter (1 ≤ Γ ≤ 100) at *N* = 864 and *κ* = 1.4. Our present and earlier reduced results calculated at different *F*ext. Results obtained by Shahzad and He [3] with *N* = 13500 at *F*ext = 0.005 are taken as reference set of data *λ*REF. Dashed lines represent spreading ±15 ranges around the reference data.

Different sequences of *λ*0 corresponding to decreasing sequence of *F*ext are calculated to determine the linear regime of YDPLs under the action of reduced force field strength. The earlier data of thermal conductivities of YDPLs has been limited to the small values of plasma parameters at different *F*ext. The present HNEMD simulation enables study over the whole domain (Γ, *κ*) of plasma-state parameters with variation of *F*ext. In this case, possible low value of the force field strength *F*ext = 0.005 for determination of the near-equilibrium values of Yukawa thermal conductivity is to be chosen, for small reasonable system size. This possible reasonable external force field gives the near-equilibrium thermal conductivity measurements, which are acceptable for the whole domain of plasma parameters (Γ, *κ*).

Thermal Conductivity and Non-Newtonian Behavior of Complex Plasma Liquids http://dx.doi.org/10.5772/65563 313

**Figure 4.** (a) Data for thermal conductivity *λ*0 normalized by plasma frequency, calculated by different MD methods for Coulomb coupling parameter (1 ≤ Γ ≤ 300) at *N* = 500 and *κ* = 2. Data obtained by Shahzad and He for homogeneous perturbed MD (HPMD)-SH (HPMD) [1], Salin and Caillol for equilibrium MD (EMD)-SC (EMD) [19], Donko and Hartmann for inhomogeneous NEMD-DH NEMD [20], and Faussurier and Murillo for variance procedure (VP)-FM VP [21]. (b) Dependences of *λ*0 reduced by *λ*REF at *F*ext = 0.005 on Coulomb coupling parameter (1 ≤ Γ ≤ 100) at *N* = 500 and *κ* = 2. Our present and earlier reduced results calculated at different *F*ext. Results obtained by Shahzad and He [3] with *N* = 13500 at *F*ext = 0.005 are taken as reference set of data *λ*REF. Dashed lines represent spreading ±15 ranges around the reference data.

**Figure 3.** (a) Data for thermal conductivity *λ*0 normalized by plasma frequency, calculated by different MD methods for Coulomb coupling parameter (1 ≤ Γ ≤ 300) at *N* = 864 and *κ* = 1.4. Data obtained by Shahzad and He for homogeneous perturbed MD (HPMD)-SH (HPMD) [1], Salin and Caillol for equilibrium MD (EMD)-SC (EMD) [19], Donko and Hartmann for inhomogeneous NEMD-DH NEMD [20], and Faussurier and Murillo for variance procedure (VP)-FM VP [21]. (b) Dependences of *λ*0 reduced by *λ*REF at *F*ext = 0.005 on Coulomb coupling parameter (1 ≤ Γ ≤ 100) at *N* = 864 and *κ* = 1.4. Our present and earlier reduced results calculated at different *F*ext. Results obtained by Shahzad and He [3] with *N* = 13500 at *F*ext = 0.005 are taken as reference set of data *λ*REF. Dashed lines represent spreading ±15 ranges

Different sequences of *λ*0 corresponding to decreasing sequence of *F*ext are calculated to determine the linear regime of YDPLs under the action of reduced force field strength. The earlier data of thermal conductivities of YDPLs has been limited to the small values of plasma parameters at different *F*ext. The present HNEMD simulation enables study over the whole domain (Γ, *κ*) of plasma-state parameters with variation of *F*ext. In this case, possible low value of the force field strength *F*ext = 0.005 for determination of the near-equilibrium values of Yukawa thermal conductivity is to be chosen, for small reasonable system size. This possible reasonable external force field gives the near-equilibrium thermal conductivity measurements,

which are acceptable for the whole domain of plasma parameters (Γ, *κ*).

312 Thermoelectrics for Power Generation - A Look at Trends in the Technology

around the reference data.

**Figures 2**–**5** show, that measured thermal conductivity is in satisfactory agreement with earlier HPMD simulations by Shahzad and He [1], inhomogeneous NEMD computations by Donko and Hartmann [20], and EMD measurements by Salin and Caillol [19]. The present results are also higher than Salin and Caillol [19] results at lower Γ = 1 and 2, for *κ* = 1.4 (*N* = 500 and 864). The minimum value of *λ*0 is *λ*min ≈ 0.4533 at Γ = 20 and *κ* = 1.4. Deviation of the present data from the earlier calculated results based on different techniques of EMD, HPMD, and NEMD is also calculated. It is observed, that the results of *λ*0 are within the range of ~7–50% for EMD, ~10–16% for NEMD, and ~10–40% for HPMD. Moreover, **Figure 3(a)** shows, that obtained thermal conductivity at *F*ext = 0.005 for *N* = 864 is in good agreement with HPMD simulation of Shahzad and He [1], but it is noted, that our results are slightly greater than EMD of Salin and Caillol [19], NEMD of Donko and Hartmann [20], and VP of Faussurier and Murillo [21] at lower values of Γ. Deviation of the present data from EMD, HPMD, and inhomogeneous NEMD is within the range of ~4–14%, ~3–16%, and ~4–30%. The minimum value of *λ*0 = 0.4331 at Γ = 20 and *κ* = 1.4.

**Figure 5.** (a) Data for thermal conductivity (*λ*0) normalized by plasma frequency, calculated by different MD methods for Coulomb coupling parameter (1 ≤ Γ ≤ 300) at *N* = 500 and *κ* = 4. Data obtained by Shahzad and He for homogeneous perturbed MD (HPMD)-SH (HPMD) [1], Salin and Caillol for equilibrium MD (EMD)-SC (EMD) [19], Donko and Hartmann for inhomogeneous NEMD-DH NEMD [20], and Faussurier and Murillo for variance procedure (VP)-FM VP [21]. (b) Dependences of *λ*0 reduced by *λ*REF at *F*ext = 0.005 on Coulomb coupling parameter (1 ≤ Γ ≤ 100) at *N* = 500 and *κ* = 4. Our present and earlier reduced results calculated at different *F*ext. Results obtained by Shahzad and He [3] with *N* = 13500 at *F*ext = 0.005 are taken as reference set of data *λ*REF. Dashed lines represent spreading ±15 ranges around the reference data.

This comparison shows, that our data remain within the limited statistical uncertainty range. Panels (b) of **Figures 2**–**5** compare the present simulation results of thermal conductivity, normalized by reference data, calculated here from HNEMD approach for different sets of external force field strengths with reference set of data and earlier known simulation data of HPMD, EMD, inhomogeneous NEMD, and VP techniques [1, 19–21]. A series of different sequences of HNEMD simulations are performed with *N* = 500 and 864 at various normalized *F*ext values for varying screening parameters. In our case, results measured by Shahzad and He [3] with *N* = 13500 at *F*ext = 0.005 are taken as reference set of data *λ*REF and are shown panels (b) of the respective figures. It is important, that computationally noted linear regime is traced between 0.001 ≤ *F*ext and ≤ 0.009, which depends on the plasma state points (Γ, *κ*). It observed, that differences between the simulation data calculated by different authors are large at some state points and differences with their own present data sets are much smaller. Plasma's thermal conductivity is generally overpredicted within ~3–17% (~8–15%), ~5–30% (~5–38%), and ~5–40% (~3–17%) relative to the data of EMD by Salin and Caillol [19], inhomogeneous NEMD by Donko and Hartman [20], and HPMD by Shahzad and He [1], respectively, for *N* = 500 (864). It is concluded, that most of data points of presented results fall under ±15 range around the reference data.

For the cases of *κ* = 2 and *κ* = 4, the minimum value of *λ*0 is *λ*min ≈ 0.3413 (for *κ* = 2) at Γ = 50 and *λ*min ≈ 0.2218 (for *κ* = 4) at Γ = 200. The comparison of the present data with the earlier investigated results obtained from different techniques of EMD, HPMD, and NEMD is also shown in respective figure panels (a) for *κ* = 2 and *κ* = 4. It is observed, that the results of *λ*0 are within the range ~1–15% (~2–13%, for *κ* = 4) for EMD, ~2–20% (~3–13%, for *κ* = 4) for HPMD, ~15–22% for inhomogeneous NEMD, and ~1–28% for HPMD. It is observed, that for *κ* = 4, thermal conductivity agrees relatively well with the simulation data in general within ~5–35% for EMD and ~2–23% for HPMD.

It is concluded from figures, that obtained results agree well with earlier results at intermediate and high Γ values; however, some data points deviate at the lower Γ values. **Figures 2**–**5** depict, that extended HNEMD approach can accurately predict thermal conductivity of Yukawa system (dusty plasma). We have used the present developed homogeneous NEMD method, which has an excellent performance; its accuracy is comparable to that of EMD and inhomogeneous NEMD techniques. The first conclusion from above **Figure 5** is that thermal conductivity depends on plasma parameters Γ and *κ*, confirming the earlier numerical results. Furthermore, it is examined, that the position of minimum value of *λ*min shifts toward higher Γ with increase in *κ*, as expected. The minimum value of *λ*0 decreases with increasing *κ*, as *λ*<sup>0</sup> = 0.4533 at *κ* = 1 to *λ*0 = 0.2218 at *κ* = 4 and *λ*0 = 0.4331 at *κ* = 1 to *λ*0 = 0.2099 at *κ* = 4 for *N* = 500 and 864, respectively. Our numerical data are considerably more comprehensive covering the full range of coupling strengths from nearly nonideal gaseous state to strongly coupled liquid (SCL) as shown in **Figure 1**. The present approach has shown excellent results of *λ*0 at steadystate value of *F*ext at lower, intermediate, and higher coupling values, and it also gives more comparable performance of normalized thermal conductivity at lower *N*. This approach yields the practical accuracy, compared to EMD, for relatively small total simulation times due to the act of finite nonzero external force field; the signal-to-noise ratio of thermal response is high. It is significant from our simulation data, that the external force field strength of *F*ext increases with increase in *κ*.

#### **4. Summary**

NEMD is within the range of ~4–14%, ~3–16%, and ~4–30%. The minimum value of *λ*0 = 0.4331

**Figure 5.** (a) Data for thermal conductivity (*λ*0) normalized by plasma frequency, calculated by different MD methods for Coulomb coupling parameter (1 ≤ Γ ≤ 300) at *N* = 500 and *κ* = 4. Data obtained by Shahzad and He for homogeneous perturbed MD (HPMD)-SH (HPMD) [1], Salin and Caillol for equilibrium MD (EMD)-SC (EMD) [19], Donko and Hartmann for inhomogeneous NEMD-DH NEMD [20], and Faussurier and Murillo for variance procedure (VP)-FM VP [21]. (b) Dependences of *λ*0 reduced by *λ*REF at *F*ext = 0.005 on Coulomb coupling parameter (1 ≤ Γ ≤ 100) at *N* = 500 and *κ* = 4. Our present and earlier reduced results calculated at different *F*ext. Results obtained by Shahzad and He [3] with *N* = 13500 at *F*ext = 0.005 are taken as reference set of data *λ*REF. Dashed lines represent spreading ±15 ranges around the

This comparison shows, that our data remain within the limited statistical uncertainty range. Panels (b) of **Figures 2**–**5** compare the present simulation results of thermal conductivity, normalized by reference data, calculated here from HNEMD approach for different sets of external force field strengths with reference set of data and earlier known simulation data of HPMD, EMD, inhomogeneous NEMD, and VP techniques [1, 19–21]. A series of different sequences of HNEMD simulations are performed with *N* = 500 and 864 at various normalized *F*ext values for varying screening parameters. In our case, results measured by Shahzad and

at Γ = 20 and *κ* = 1.4.

314 Thermoelectrics for Power Generation - A Look at Trends in the Technology

reference data.

Thermal conductivity of NICDP system was investigated for wide range of Coulomb coupling parameter (1 ≤ Γ ≤ 300) and screening parameter (1 ≤ *κ* ≤ 4), applying external force field, by

using HNEMD method. This HNEMD simulation method reveals, that our present results are in good agreement with the earlier results obtained by equilibrium MD and homogeneous and inhomogeneous NEMD simulations for NICDPs. It is confirmed, that lattice correlation is not affected by system size, while lattice correlation decreases with increment of *κ* and at high temperature (1/Γ). It is confirmed from presented HNEMD simulation results, in which normalized thermal conductivities follow simple universal (temperature) scaling law. Observations show, that the minimum value of thermal conductivity shifts toward higher Coulomb couplings with increasing screening strength, confirming earlier numerical results. It has been shown, that thermal conductivity depends on both temperature (plasma coupling) and density (screening) in 3D Yukawa systems, which shows previous data for NICDPs. Presented HNEMD approach was particularly a powerful numerical technique, which involves fast calculation of thermal conductivity, on small and intermediate system sizes, in contexts, applicable to the study of matter at microscopic level for 3D NICDPs. The second major contribution of this chapter is that it provides, for the first time, understanding and determination of non-Newtonian behavior of Yukawa liquid. It is important to note, that thermal conductivity is in increasing behavior at higher values of external force field. These indications show, that increasing behavior of field dependence of thermal conductivity decreases with increasing *κ* and decreasing Γ. Described simulation technique can be helpful for estimation of thermal conductivity regularities in novel liquid, organic (polymer), and multiphase solidstate thermoelectric materials.

## **Acknowledgements**

This work was sponsored by the National Natural Science Fund for Distinguished Young Scholars of China (NSFC no. 51525604) and partially sponsored by the Higher Education Commission (HEC) of Pakistan (no. IPFP/HRD/HEC/2014/916). The authors thank Z. Donkó (Hungarian Academy of Sciences) for providing his thermal conductivity data of Yukawa liquids for the comparisons of our simulation results and useful discussions. We are grateful to the National High Performance Computing Center of Xi'an Jiaotong University and National Advanced Computing Center of the National Centre for Physics (NCP), Pakistan, for allocating computer time to test and run our MD code.

#### **Abbreviations**



## **Author details**

using HNEMD method. This HNEMD simulation method reveals, that our present results are in good agreement with the earlier results obtained by equilibrium MD and homogeneous and inhomogeneous NEMD simulations for NICDPs. It is confirmed, that lattice correlation is not affected by system size, while lattice correlation decreases with increment of *κ* and at high temperature (1/Γ). It is confirmed from presented HNEMD simulation results, in which normalized thermal conductivities follow simple universal (temperature) scaling law. Observations show, that the minimum value of thermal conductivity shifts toward higher Coulomb couplings with increasing screening strength, confirming earlier numerical results. It has been shown, that thermal conductivity depends on both temperature (plasma coupling) and density (screening) in 3D Yukawa systems, which shows previous data for NICDPs. Presented HNEMD approach was particularly a powerful numerical technique, which involves fast calculation of thermal conductivity, on small and intermediate system sizes, in contexts, applicable to the study of matter at microscopic level for 3D NICDPs. The second major contribution of this chapter is that it provides, for the first time, understanding and determination of non-Newtonian behavior of Yukawa liquid. It is important to note, that thermal conductivity is in increasing behavior at higher values of external force field. These indications show, that increasing behavior of field dependence of thermal conductivity decreases with increasing *κ* and decreasing Γ. Described simulation technique can be helpful for estimation of thermal conductivity regularities in novel liquid, organic (polymer), and multiphase solid-

316 Thermoelectrics for Power Generation - A Look at Trends in the Technology

This work was sponsored by the National Natural Science Fund for Distinguished Young Scholars of China (NSFC no. 51525604) and partially sponsored by the Higher Education Commission (HEC) of Pakistan (no. IPFP/HRD/HEC/2014/916). The authors thank Z. Donkó (Hungarian Academy of Sciences) for providing his thermal conductivity data of Yukawa liquids for the comparisons of our simulation results and useful discussions. We are grateful to the National High Performance Computing Center of Xi'an Jiaotong University and National Advanced Computing Center of the National Centre for Physics (NCP), Pakistan, for allocating

state thermoelectric materials.

computer time to test and run our MD code.

NICDP nonideal complex (dusty) plasma

Γ Coulomb coupling *κ* Debye screening length *F*ext external force field strength

HNEMD homogeneous nonequilibrium molecular dynamics

**Acknowledgements**

**Abbreviations**

Aamir Shahzad1,2\* and Maogang He2

\*Address all correspondence to: aamirshahzad\_8@hotmail.com; aamir.awan@gcuf.edu.pk

1 Department of Physics, Government College University Faisalabad (GCUF), Faisalabad, Pakistan

2 Key Laboratory of Thermo-Fluid Science and Engineering, Ministry of Education (MOE), Xi'an Jiaotong University, Xi'an, PR China

## **References**


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[3] Shahzad A, He M-G. Thermal conductivity of three-dimensional Yukawa liquids (dusty plasmas). Contrib. Plasma Phys. 2012;52(8):667. DOI: 10.1002/ctpp.201200002

[4] Fortov VE, Vaulina OS, Lisin EA, Gavrikov AV, Petrov OF. Analysis of pair interparticle interaction in nonideal dissipative systems. J. Exp. Theor. Phys. 2010;110:662–674.DOI:

[5] Shahzad A, He M-G. Thermodynamic characteristics of dusty plasma studied by using molecular dynamics simulation. Plasma Sci.Technol. 2012;14(9):771–777. DOI:

[6] Shahzad A, He M-G. Interaction contributions in thermal conductivity of threedimensional complex liquids. In: Liejin GUO, editor. AIP Conference Proceedings; 26–

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[8] Chen FF. Introduction to Plasma Physics and Controlled Fusion. 2nd ed. New York:

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[10] Kalman GJ, Rommel JM, Blagoev K. Strongly Coupled Coulomb Systems. New York:

[11] Jensen MJ, Hasegawa T, Bollinger JJ, Dubin DHE, et al. Rapid heating of a strongly coupled plasma near the solid-liquid phase transition. Phys. Rev. Lett. 2005;94:025001.

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[14] Rapaport DC. The Art of Molecular Dynamics Simulation. New York: Cambridge

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[16] Shahzad A, He M-G. Homogeneous nonequilibrium molecular dynamics evaluations of thermal conductivity 2D Yukawa liquids. Int. J. Thermophys. 2015;36(10–11):2565.

[17] Shahzad A, He M-G. Calculations of thermal conductivity of complex (dusty) plasmas using homogeneous nonequilibrium molecular simulations. Radiat. Eff. Defect. S.

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DOI: 10.1007/s10765-014-1671-8


**Constructional Nanomaterials**

## **Nitrogen-Doped Carbon Nanotube/Polymer Nanocomposites Towards Thermoelectric Applications Nitrogen-Doped Carbon Nanotube/Polymer Nanocomposites Towards Thermoelectric Applications**

Mohammad Arjmand and Soheil Sadeghi Mohammad Arjmand and Soheil Sadeghi

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/65675

#### **Abstract**

This study investigates the impact of nitrogen doping on the performance of carbon nanotube (CNT)/polymer nanocomposites for thermoelectric applications; this was performed through measurement of conductivity of the generated nanocomposites. Three different catalysts (Co, Fe, and Ni) were used to synthesize nitrogen-doped CNTs (N-CNTs) by chemical vapor deposition technique. Synthesized N-CNTs were meltmixed with a polyvinylidene fluoride (PVDF) matrix with a small-scale mixer at a broad range of loadings from 0.3 to 3.5 wt.% and then compression molded. Measurement of electrical conductivity of the generated nanocomposites showed superior properties in the following order of the synthesis catalyst: Co > Fe > Ni. We employed various characterization techniques to figure out the reasons behind dissimilar electrical conductivity of the generated nanocomposites, i.e., transmission electron microscopy, X-ray photoelectron spectroscopy, Raman spectroscopy, thermogravimetric analysis, light microscopy, and rheometry. It was found out, that the superior electrical conductivity of (N-CNT)Co nanocomposites was due to a combination of high synthesis yield, high aspect ratio, low nitrogen content, and high crystallinity of N-CNTs coupled with a good state of N-CNT dispersion. Moreover, it was revealed, that nitrogen doping had an adverse impact on electrical conductivity and, thus, on thermoelectric performance of CNT/polymer nanocomposites.

**Keywords:** carbon nanotube, nitrogen doping, polymer nanocomposites, electrical conductivity, thermoelectric

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **1. Introduction**

## **1.1. Conductive filler/polymer nanocomposites**

Conductive filler/polymer nanocomposites (CPNs) have recently drawn great interest to be employed in various applications due to their unique properties, such as tunable electrical conductivity, light weight, low cost, corrosion resistance, and processability [1, 2]. CPNs are generated by incorporating conductive filler into a polymer matrix. Conventional polymers, such as polycarbonate and polystyrene are insulative; however, incorporating conductive fillers to these polymer matrices can provide them with a broad range of conductivities through the formation of a two- or three-dimensional conductive network (**Figure 1**). Tunable electrical conductivity of CPNs entitles them to be used in a broad spectrum of applications, such as thermoelectric, charge storage, antistatic dissipation, electrostatic discharge (ESD) protection, and electromagnetic interference (EMI) shielding [3–8]. In fact, the level of electrical conductivity defines the applications in which CPNs can be employed. Charge storage and ESD protection are the major applications of CPNs necessitating low and medium electrical conductivity, respectively, whereas thermoelectric and EMI shielding require high electrical conductivity.

**Figure 1.** The approximate range of electrical conductivity covered by CPNs [1].

#### **1.2. Conductive filler/polymer nanocomposites for thermoelectric applications**

Thermoelectric devices provide an all solid-state means of heat to electricity conversion. These devices feature many advantages in heat pumps and electrical power generators, such as possessing no moving parts, generating zero noise, being easy to maintain, extended lifetime, and being highly reliable [9, 10]. However, their limited efficiency has restricted their usage to specialized applications, where cost and efficiency are of great importance. Thermoelectric efficiency is expressed in terms of dimensionless figure of merit ( , defined as 2 , in which α is Seebeck coefficient, *σ* is electrical conductivity, *κ* is thermal con-

ductivity, and *T* is absolute temperature. The upper limit for thermoelectric energy conver-

sion efficiency is Carnot limit. Several recently developed thermoelectric materials exhibit a *ZT* near unity, resulting in efficiency equal to 10 % of the Carnot efficiency limit. By reducing the physical dimensionality of the thermoelectric materials (quantum confinement) [11], it is possible to significantly increase Seebeck coefficient and decrease the thermal conductivity. This can lead to *ZT*s as high as 3 at 550 K in n-type PbSe0.98Te0.02/PbTe quantum-dot superlattices [12]. However, application of these heavy metal thermoelectric materials is associated with high cost of material and production processes, poor processability, and huge adverse environmental impacts [13–15]. Accordingly, polymeric materials have drawn great interest to be used in thermoelectric applications. Improved processability, low cost, low thermal conductivity, and low density of polymeric materials are among key potential benefits stimulating the development of polymer-based thermoelectric materials [16, 17]. It is worth mentioning, that the figure of merit in polymer-based thermoelectric materials is *ZT* ~ *0*(10− 3), which necessitates further investigations in this area [18].

**1. Introduction**

as 2

**1.1. Conductive filler/polymer nanocomposites**

324 Thermoelectrics for Power Generation - A Look at Trends in the Technology

**Figure 1.** The approximate range of electrical conductivity covered by CPNs [1].

**1.2. Conductive filler/polymer nanocomposites for thermoelectric applications**

Thermoelectric devices provide an all solid-state means of heat to electricity conversion. These devices feature many advantages in heat pumps and electrical power generators, such as possessing no moving parts, generating zero noise, being easy to maintain, extended lifetime, and being highly reliable [9, 10]. However, their limited efficiency has restricted their usage to specialized applications, where cost and efficiency are of great importance. Thermoelectric efficiency is expressed in terms of dimensionless figure of merit ( , defined

ductivity, and *T* is absolute temperature. The upper limit for thermoelectric energy conver-

, in which α is Seebeck coefficient, *σ* is electrical conductivity, *κ* is thermal con-

Conductive filler/polymer nanocomposites (CPNs) have recently drawn great interest to be employed in various applications due to their unique properties, such as tunable electrical conductivity, light weight, low cost, corrosion resistance, and processability [1, 2]. CPNs are generated by incorporating conductive filler into a polymer matrix. Conventional polymers, such as polycarbonate and polystyrene are insulative; however, incorporating conductive fillers to these polymer matrices can provide them with a broad range of conductivities through the formation of a two- or three-dimensional conductive network (**Figure 1**). Tunable electrical conductivity of CPNs entitles them to be used in a broad spectrum of applications, such as thermoelectric, charge storage, antistatic dissipation, electrostatic discharge (ESD) protection, and electromagnetic interference (EMI) shielding [3–8]. In fact, the level of electrical conductivity defines the applications in which CPNs can be employed. Charge storage and ESD protection are the major applications of CPNs necessitating low and medium electrical conductivity, respectively, whereas thermoelectric and EMI shielding require high electrical conductivity.

Organic polymers exhibit poor electrical conductivity, which necessitates addition of conductive fillers in order to provide high electrical conductivity and reasonable thermoelectric performance. Recent studies suggested, that carbon-based nanofiller/polymer nanocomposites hold a significant promise in development of lightweight, low-cost thermoelectric materials [19–23]. As an instance, Yu et al. [17] demonstrated, that by forming a segregated network of carbon nanotubes (CNTs), electrical conductivities as high as 48 S cm−1 were achievable, while Seebeck coefficient and thermal conductivity were marginally impacted by CNT presence. This resulted in a figure of merit of 0.006 at room temperature. These findings render CNT-based polymer nanocomposite as a basis for development of thermoelectric functional materials for future green energy applications.

#### **1.3. Nitrogen-doped carbon nanotube/polymer nanocomposites for thermoelectric applications**

Different types of conductive nanofillers have been employed to develop CPNs, viz., carbonaceous nanofillers and metallic nanowires [4, 5, 24], among which, CNT has appealed remarkable attention due to its large surface area and outstanding electrical, thermal, and mechanical properties [25, 26]. The luminous era of CNTs initiated in 1991 by their discovery from soot using an arc-discharge apparatus [27]. Like fullerene and graphene, CNTs consist of a *sp*<sup>2</sup> network of carbon atoms. Among these, three carbonaceous poly-types, CNT is the only one produced in large industrial scale. There are two general types of CNTs: multi-walled CNT (MWCNT) and single-walled CNT (SWCNT). MWCNT consists of multiple rolled layers of graphite coaxially arranged around a central hollow core with van der Waals forces between contiguous layers, while SWCNT is made of a single rolled graphene [28]. The global market for CNT primary grades was \$158.6 million in 2014 and is anticipated to reach \$670.6 million in 2019. CNT/polymer nanocomposites represent, by far, the largest segment in the overall market of CNTs [29].

Manipulating the electronic energy gap of CNTs could lead to their superior performance. Since CNTs are *sp*<sup>2</sup> carbon systems, theoretical [30] and experimental [31] studies showed, that substituting carbon atoms with heteroatoms can result in adjustment of electronic and structural patterns of carbon nanotubes. Nitrogen is the best choice for heteroatom substitution owing to its size proximity to carbon [32]. Bearing in mind, that nitrogen comprises one additional electron as compared to carbon, doping CNTs with nitrogen has emerged as an attractive research topic to improve the electronic properties of CNTs.

Essentially, there are three common nitrogen bonding configurations in nitrogen-doped CNTs (N-CNTs), viz., quaternary, pyridinic, and pyrrolic. As depicted in **Figure 2**, quaternary nitrogen is directly replaced for C atom in the hexagonal network, is *sp*<sup>2</sup> hybridized, and creates electron-donor state. The pyridinic nitrogen is a part of sixfold ring structure and is *sp*<sup>2</sup> hybridized, and two of its five electrons are localized sole pair. Pyrrolic nitrogen is a portion of a five-membered ring structure, is *sp*<sup>3</sup> hybridized, and gives its remaining two electrons to a π orbital, integrating the aromatic ring [33]. Whereas the quaternary and pyridinic nitrogen lead to side-wall defects, the pyrrolic nitrogen is believed to form internal cappings, generating bamboo-like sections [34]. Besides these three types, there is also possibility for N2 molecules to get trapped inside the tube axis or intercalated into the graphitic layers of N-CNTs.

**Figure 2.** Major types of nitrogen bonding in N-CNTs [35].

Basically, as yet, most of the research studies have investigated the influence of nitrogen doping on electronic properties of CNTs via factors, such as density of states (DOS) and Fermi level; nevertheless, inspecting the electrical properties of polymer nanocomposites containing N-CNTs is still at its infancy [35–38]. Hence, the current study aims to research the impact of nitrogen doping on the performance of N-CNT/polymer nanocomposites for thermoelectric applications by studying its influence on electrical conductivity of N-CNT/polymer nanocomposites. N-CNTs were synthesized with different types of catalyst (Co, Fe, and Ni) to obtain diverse nitrogen contents. Afterward, synthesized N-CNTs were melt mixed into polyvinylidene fluoride (PVDF) and compression molded. We evaluated electrical conductivity of the nanocomposites at different loadings and scrutinized the underlying causes behind dissimilar conductivities of the generated nanocomposites. As high electrical conductivity of CPNs is an important factor on their performance as thermoelectric materials, this study goes significantly beyond the state of the art and gives new insight on the role of nitrogen doping on conductivity and therefore performance of CNT nanocomposites for thermoelectric applications.

## **2. Experimental**

structural patterns of carbon nanotubes. Nitrogen is the best choice for heteroatom substitution owing to its size proximity to carbon [32]. Bearing in mind, that nitrogen comprises one additional electron as compared to carbon, doping CNTs with nitrogen has emerged as an

Essentially, there are three common nitrogen bonding configurations in nitrogen-doped CNTs (N-CNTs), viz., quaternary, pyridinic, and pyrrolic. As depicted in **Figure 2**, quaternary

electron-donor state. The pyridinic nitrogen is a part of sixfold ring structure and is *sp*<sup>2</sup> hybridized, and two of its five electrons are localized sole pair. Pyrrolic nitrogen is a portion

a π orbital, integrating the aromatic ring [33]. Whereas the quaternary and pyridinic nitrogen lead to side-wall defects, the pyrrolic nitrogen is believed to form internal cappings, generating bamboo-like sections [34]. Besides these three types, there is also possibility for N2 molecules to get trapped inside the tube axis or intercalated into the graphitic layers of N-CNTs.

Basically, as yet, most of the research studies have investigated the influence of nitrogen doping on electronic properties of CNTs via factors, such as density of states (DOS) and Fermi level; nevertheless, inspecting the electrical properties of polymer nanocomposites containing N-CNTs is still at its infancy [35–38]. Hence, the current study aims to research the impact of nitrogen doping on the performance of N-CNT/polymer nanocomposites for thermoelectric applications by studying its influence on electrical conductivity of N-CNT/polymer nanocomposites. N-CNTs were synthesized with different types of catalyst (Co, Fe, and Ni) to obtain diverse nitrogen contents. Afterward, synthesized N-CNTs were melt mixed into polyvinylidene fluoride (PVDF) and compression molded. We evaluated electrical conductivity of the nanocomposites at different loadings and scrutinized the underlying causes behind dissimilar conductivities of the generated nanocomposites. As high electrical conductivity of CPNs is an important factor on their performance as thermoelectric materials, this study goes significantly

hybridized, and creates

hybridized, and gives its remaining two electrons to

attractive research topic to improve the electronic properties of CNTs.

326 Thermoelectrics for Power Generation - A Look at Trends in the Technology

nitrogen is directly replaced for C atom in the hexagonal network, is *sp*<sup>2</sup>

of a five-membered ring structure, is *sp*<sup>3</sup>

**Figure 2.** Major types of nitrogen bonding in N-CNTs [35].

#### **2.1. Materials synthesis**

We employed the incipient wetness impregnation technique to produce catalyst precursors. The catalyst precursors were dissolved in water and then impregnated onto aluminum oxide support (Sasol Catalox Sba-200). Thereafter, the developed materials were dried, calcinated, and reduced. Having high solubility and diffusion rate in carbon, Co, Fe, and Ni were chosen as the catalysts [39, 40]. Accordingly, we employed cobalt nitrate hexahydrate, iron (III) nitrate nonahydrate, and nickel (II) sulfate hexahydrate as the catalyst precursors. We set the metal loading at 20 wt.%. The catalyst calcination, reduction, and N-CNT synthesis were performed in CVD setup, detailed in a former study [41]. CVD setup comprised a quartz tubular reactor with an inner diameter of 4.5 cm encapsulated within a furnace. The following steps were implemented for the preparation of the catalysts: first, the catalysts were calcinated under air atmosphere with a flow rate of 100 sccm at 350 °C for 4 h. In this stage, metallic salts were translated into metal oxides. Thereafter, we used a mortar and pestle to achieve a fine powder. Hydrogen gas at a flow rate of 100 sccm at 400 °C for 1 h was utilized to obtain alumina-supported metal catalysts. Afterward, we conveyed a combination of ethane (50 sccm), ammonia (50 sccm), and argon (50 sccm) over the synthesized catalysts. Ethane played the role of carbon source, whereas ammonia and argon were nitrogen source and inert gas carrier, respectively. The synthesis temperature, synthesis time, and catalyst mass were set at 750 °C, 2 h, and 0.6 g, respectively. Catalyst preparation process is elucidated further elsewhere [39].

The polymer matrix utilized for the nanocomposite preparation was semicrystalline PVDF 11008/0001, purchased from 3M Canada, with an average density of 1.78 g/cm3 and melting point of 160 °C. PVDF was opted as the polymer matrix owing to its ferroelectricity, high dielectric strength (~13 kV mm−1), corrosion resistance, good mechanical properties, thermal stability, good chemical resistance (excellent with acid and alkali), and robust interaction of electrophilic fluorine groups with CNTs [42–44]. The mixing of synthesized N-CNTs with PVDF matrix was carried out with Alberta Polymer Asymmetric Minimixer (APAM) at 240 °C and 235 rpm. PVDF matrix was first masticated within the mixing cup for 3 min, and then N-CNTs were inserted and mixed for an additional 14 min. For each catalyst, the nanocomposites with different N-CNT concentrations, i.e., 0.3, 0.5, 1.0, 2.0, 2.7, and 3.5 wt.%, were prepared. The nanocomposites were molded into circular cavities with 0.5 mm thickness using Carver compression molder (Carver Inc.) at 220 °C under 38 MPa pressure for 10 min. The molded samples were used for electrical, morphological, and rheological characterizations.

## **2.2. Materials characterization**

#### *2.2.1. N-CNT characterization*

#### *2.2.1.1. Transmission electron microscopy of N-CNTs*

High-resolution transmission electron microscopy (HRTEM) was used to inspect the morphology of synthesized N-CNTs. HRTEM was conducted on Tecnai TF20 G2 FEG-TEM (FEI) at 200 kV acceleration voltage with a standard single-tilt holder. The images were taken with Gatan UltraScan 4000 CCD camera at 2048 × 2048 pixels. For HRTEM, around 1.0 mg of N-CNT powder was dispersed in 10 mL ethanol and bath sonicated for 15 min. A drop of the dispersion was mounted on the carbon side of a standard TEM grid covered with a ~40 nm holey carbon film (EMS). Measurement of the geometrical dimensions of N-CNTs was conducted for over 100 individual ones utilizing MeasureIT software (Olympus Soft Imaging Solutions GmbH).

#### *2.2.1.2. X-ray photoelectron spectroscopy, Raman spectroscopy and thermogravimetric analysis*

PHI VersaProbe 5000-XPS was used to obtain X-ray photoelectron spectra. The spectra were achieved employing monochromatic Al source at 1486.6 eV and 49.3 W with a beam diameter of 200.0 μm. The structural defects of N-CNTs were inspected using Raman spectroscopy. Renishaw inVia Raman microscope was used to obtain Raman spectra. Excitation was provided by the radiation of an argon-ion laser beam with 514 nm wavelength. A 5× objective was used to get Raman spectra. The yield of the synthesis process was inspected with Thermogravimetric Analyzer (TA instruments, Model: Q500). The samples were heated under air atmosphere (Praxair AI INDK) from ambient temperature to 950 °C at a rate of 10 °C/min. The samples were kept at 950 °C for 10 min before cooling.

#### *2.2.2. Nanocomposite characterization*

#### *2.2.2.1. Light microscopy*

The microdispersion state of the nanofillers within PVDF matrix was enumerated using light transmission microscopy (LM) on thin cuts (5 μm thickness) of the compression-molded samples, prepared with Leica Microtome RM 2265 (Leica Microsystems GmbH). Olympus BH2 optical microscope (Olympus Deutschland GmbH) equipped with CCD camera DP71 was used to capture images with dimensions of 600 μm × 800 μm from different cut sections. The software Stream Motion (Olympus) was used to analyze the images. The agglomerate area ratio (in %) was defined by dividing the spotted area of non-dispersed nanofillers (with equivalent circle diameter > 5 μm, area > 19.6 μm2 ) over the whole sample area (15 cuts, ca. 7.2 mm2 ). Mean value and standard deviation, demonstrating the differences between the cuts and thus heterogeneity, were reckoned. The relative transparency of the cuts provided added information about the amount of dispersed nanofillers in the samples. The relative transparency was quantified by dividing the transparency of the cut over the transparency of the glass slide/cover glass assembly. Ten various areas per sample were used to obtain mean values and standard deviations. Further information on employing LM to evaluate microdispersion state of nanofillers within nanocomposites is presented elsewhere [45, 46].

#### *2.2.2.2. TEM*

**2.2. Materials characterization**

*2.2.1.1. Transmission electron microscopy of N-CNTs*

328 Thermoelectrics for Power Generation - A Look at Trends in the Technology

samples were kept at 950 °C for 10 min before cooling.

equivalent circle diameter > 5 μm, area > 19.6 μm2

*2.2.2. Nanocomposite characterization*

*2.2.2.1. Light microscopy*

mm2

High-resolution transmission electron microscopy (HRTEM) was used to inspect the morphology of synthesized N-CNTs. HRTEM was conducted on Tecnai TF20 G2 FEG-TEM (FEI) at 200 kV acceleration voltage with a standard single-tilt holder. The images were taken with Gatan UltraScan 4000 CCD camera at 2048 × 2048 pixels. For HRTEM, around 1.0 mg of N-CNT powder was dispersed in 10 mL ethanol and bath sonicated for 15 min. A drop of the dispersion was mounted on the carbon side of a standard TEM grid covered with a ~40 nm holey carbon film (EMS). Measurement of the geometrical dimensions of N-CNTs was conducted for over 100 individual ones utilizing MeasureIT software (Olympus Soft Imaging

*2.2.1.2. X-ray photoelectron spectroscopy, Raman spectroscopy and thermogravimetric analysis*

PHI VersaProbe 5000-XPS was used to obtain X-ray photoelectron spectra. The spectra were achieved employing monochromatic Al source at 1486.6 eV and 49.3 W with a beam diameter of 200.0 μm. The structural defects of N-CNTs were inspected using Raman spectroscopy. Renishaw inVia Raman microscope was used to obtain Raman spectra. Excitation was provided by the radiation of an argon-ion laser beam with 514 nm wavelength. A 5× objective was used to get Raman spectra. The yield of the synthesis process was inspected with Thermogravimetric Analyzer (TA instruments, Model: Q500). The samples were heated under air atmosphere (Praxair AI INDK) from ambient temperature to 950 °C at a rate of 10 °C/min. The

The microdispersion state of the nanofillers within PVDF matrix was enumerated using light transmission microscopy (LM) on thin cuts (5 μm thickness) of the compression-molded samples, prepared with Leica Microtome RM 2265 (Leica Microsystems GmbH). Olympus BH2 optical microscope (Olympus Deutschland GmbH) equipped with CCD camera DP71 was used to capture images with dimensions of 600 μm × 800 μm from different cut sections. The software Stream Motion (Olympus) was used to analyze the images. The agglomerate area ratio (in %) was defined by dividing the spotted area of non-dispersed nanofillers (with

). Mean value and standard deviation, demonstrating the differences between the cuts and thus heterogeneity, were reckoned. The relative transparency of the cuts provided added information about the amount of dispersed nanofillers in the samples. The relative transparency was quantified by dividing the transparency of the cut over the transparency of the glass slide/cover glass assembly. Ten various areas per sample were used to obtain mean values and

) over the whole sample area (15 cuts, ca. 7.2

*2.2.1. N-CNT characterization*

Solutions GmbH).

Ultrathin sections of the samples were cut using ultramicrotome EM UC6/FC6 (Leica) setup with an ultrasonic diamond knife at ambient temperature. The sections were floated off water and thereafter transferred on carbon-filmed TEM copper grids. TEM characterizations were carried out employing TEM LIBRA 120 (Carl Zeiss SMT) with an acceleration voltage of 120 kV.

#### *2.2.2.3. Rheology*

Rheological measurements were performed using Anton-Paar MCR 302 rheometer at 240 °C using 25 mm cone-plate geometry with a cone angle of 1° and truncation of 47 μm. The thermal stability of the prepared samples was validated by conducting small-amplitude oscillatory shear measurements prior to and following the long-time exposure of the samples to elevated temperatures. Various rheological properties were measured at 240 °C to characterize the linear and nonlinear response for the neat and nanocomposite samples.

#### *2.2.2.4. Electrical conductivity*

Two conductivity meters with 90 V as the applied voltage were employed to measure the electrical conductivity of the generated materials. For nanocomposites with an electrical conductivity higher than 10−2 S m−1, the measurements were conducted according to ASTM 257-75 standards employing Loresta GP resistivity meter (MCP-T610 model, Mitsubishi Chemical Co.). An ESP probe was used to avert the effect of contact resistance. For electrical conductivities less than 10−2 S m−1, the measurements were carried out with Keithley 6517A electrometer connected to Keithley 8009 test fixture (Keithley Instruments).

## **3. Results and discussion**

## **3.1. General background**

#### *3.1.1. Mechanisms of electrical conductivity*

Electrical conductivity derives from ordered movement of charge carriers (electric current). In the absence of an electric field, the conduction electrons are scattered freely in a solid owing to their thermal energy. If an electric field, *E*, is applied, the force on an electron, *e*, is –*eE*, and the electron is accelerated in the opposite direction to the electric field because of its negative charge. Accordingly, there is a net velocity and the current density is presented by [47]:

$$J = N\_e \times \ e \times \mu \times E,\tag{1}$$

where *J* is the current density, *Ne* is concentration of electrons, *e* is charge of electron, *μ* is the electron mobility, and *E* is the applied electric field. The applied electric field equals to the applied voltage over the thickness of a sample. Hence, the electrical conductivity can be determined as:

$$
\sigma = \frac{J}{E},
\tag{2}
$$

where *σ* is electrical conductivity and its SI unit is Siemens per meter (S m−1). Electrical conductivity of materials is an intrinsic property, which spans a very wide range. The conductivity of insulators is typically less than 10−10 S m−1, that of semiconductive materials covers the range 10−10 to around 10−2 S m−1, and for semimetals and metals is more than 10−2 S m−1.

Electrical conductivity of materials can be elucidated employing the band theory [48]. In the band theory, the energy level of each electron is reflected as a horizontal line. As any solid possesses a large number of electrons with various energy levels, the sets of energy levels form two continuous energy bands, named valence band and conduction band. The energy gap between the two bands signifies the forbidden region for electrons. Electrons restrained to individual atoms or interatomic bonds are, in the band theory, said to be in valence band. Those electrons, that can move freely in substance upon applying electric field lie in conduction band. **Figure 3** depicts a schematic of the bands in a solid identifying three main types of materials: insulators, semiconductors, and metals. Valence and conduction bands in metals overlap each other; therefore, metals indicate very high conductivity. In intrinsic semiconductors, the valence-conduction band gap is adequately small, so that, electrons in valence band can be excited to conduction band by thermal energy. Among the three types of materials illustrated in **Figure 3**, insulators show the largest valence-conduction band gap, and, therefore, fewer electrons can be excited to their conduction band by thermal energy. This results in a very low conductivity in insulators.

**Figure 3.** Simplified diagram of the electronic band structure in the band theory, reproduced from [34].

#### *3.1.2. Electrical conductivity in CPNs*

where *J* is the current density, *Ne* is concentration of electrons, *e* is charge of electron, *μ* is the electron mobility, and *E* is the applied electric field. The applied electric field equals to the applied voltage over the thickness of a sample. Hence, the electrical conductivity can be

330 Thermoelectrics for Power Generation - A Look at Trends in the Technology

, *J E* s

where *σ* is electrical conductivity and its SI unit is Siemens per meter (S m−1). Electrical conductivity of materials is an intrinsic property, which spans a very wide range. The conductivity of insulators is typically less than 10−10 S m−1, that of semiconductive materials covers the range 10−10 to around 10−2 S m−1, and for semimetals and metals is more than 10−2 S m−1.

Electrical conductivity of materials can be elucidated employing the band theory [48]. In the band theory, the energy level of each electron is reflected as a horizontal line. As any solid possesses a large number of electrons with various energy levels, the sets of energy levels form two continuous energy bands, named valence band and conduction band. The energy gap between the two bands signifies the forbidden region for electrons. Electrons restrained to individual atoms or interatomic bonds are, in the band theory, said to be in valence band. Those electrons, that can move freely in substance upon applying electric field lie in conduction band. **Figure 3** depicts a schematic of the bands in a solid identifying three main types of materials: insulators, semiconductors, and metals. Valence and conduction bands in metals overlap each other; therefore, metals indicate very high conductivity. In intrinsic semiconductors, the valence-conduction band gap is adequately small, so that, electrons in valence band can be excited to conduction band by thermal energy. Among the three types of materials illustrated in **Figure 3**, insulators show the largest valence-conduction band gap, and, therefore, fewer electrons can be excited to their conduction band by thermal energy. This results in a very low

**Figure 3.** Simplified diagram of the electronic band structure in the band theory, reproduced from [34].

= (2)

determined as:

conductivity in insulators.

High electrical conductivity, i.e., conductive network formation, at very low filler contents has made CPNs distinctive materials for industrial applications [25, 26]. Conductive network formation in CPNs is better understood with the concept of percolation threshold [49, 50]. Percolation means, that at least one conductive pathway forms to allow electrical current to pass across CPNs, thereby transforming CPNs from insulative to conductive. Percolation happens at a narrow filler concentration range, where the electrical conductivity of CPNs drastically increases by several orders of magnitude. Low electrical percolation threshold in CPNs leads to the production of cost-effective composites.

Many statistical, geometric, thermodynamic, and structure-based models have been introduced to anticipate the percolation threshold and electrical conductivity of CPNs [49, 51]. Although the percolation theory is just valid at conductive filler concentrations above the percolation threshold, it is the most acceptable one. Statistical percolation theory estimates the percolation threshold of CPNs as:

$$
\sigma = \sigma\_0 \cdot \left( V - V\_c \right)', \tag{3}
$$

where *σ* is electrical conductivity of CPN, *σ*0 is electrical conductivity of conductive filler, *V* is dimensionless volume content of conductive filler, and *Vc* and *t* are percolation threshold and critical exponent, respectively [49]. The equation is valid for filler concentration above the percolation threshold, i.e., *V > Vc*. Higher *t* value and lower percolation threshold correspond to well-dispersed, high-aspect-ratio fillers [52–54].

**Figure 4** illustrates a typical percolation curve of CPNs [55]. In general, percolation curve of CPNs can be divided into three regions: (1) region far below the percolation threshold (insulative region), (2) region where percolation occurs (percolation region), and (3) region far above the percolation threshold (conductive region). In the insulative region, the conductive filler loading is very low with the fillers far from each other; thus, polymer matrix dominates the charge transfer. As a matter of fact, at low filler concentrations, the insulating gaps are very large and the chance, that nomadic charge carriers are transferred between conductive fillers is very low.

By enhancing filler loading, the gaps between conductive fillers decrease, and a drastic increase in electrical conductivity is observed over a narrow concentration range (percolation region). In this region, hopping and direct-contact mechanisms become significant. When the mean particle-particle distance reaches below 1.8 nm, the dominant electron transfer mechanism become hopping mechanism [56–58]. It is reported, that the presence of large conductive agglomerates in CPN results in a very high secondary internal electric field between the conductive islands [57, 59]. This high field strength assists free electrons in conductive filler having adequate energy to hop over the insulative gaps. Nevertheless, hopping takes place when an electron receives sufficient energy to pass over distance to nearest free site with lower energy to alter its lattice site. In the percolation region, due to proximity or direct contact of conductive fillers, the nomadic charge carriers in conductive fillers play the dominant role in conduction mechanism. Since these free charge carriers belong to the conduction band, the conductivity of the nanocomposite rises by several orders of magnitude in the percolation region. Next, by adding more filler loading, a well-developed, 3D conductive network initiates to form, but the electrical conductivity increases only marginally. This is due to substantial current dissipation at the contact spots between conductive fillers, i.e., the constriction resistance, leading to a plateau in the percolation curve [26].

**Figure 4.** Percolation curve of compression-molded CNT/polystyrene nanocomposite (a typical percolation curve of CPNs) [55].

#### **3.2. Electrical conductivity of N-CNT/PVDF nanocomposites**

The percolation curves of N-CNT/PVDF nanocomposites are depicted in **Figure 5**. It was observed, that (N-CNT)Co/PVDF nanocomposites presented the lowest percolation threshold (1.5 wt.%) and highest electrical conductivity (3 S m−1 at 3.5 wt.%). However, it was revealed, that Ni-based nanocomposites were insulative up to 2.7 wt.% and experienced a slight increase in electrical conductivity at 3.5 wt.%. The Fe-based nanocomposites presented an increase in electrical conductivity from 1.0 wt.% to 3.5 wt.% with a mild slope.

**Figure 5.** Electrical conductivity of N-CNT/PVDF nanocomposites as a function of N-CNT content. N-CNTs were synthesized over Co, Fe, and Ni catalysts.

There are many factors impacting the electrical conductivity of CPNs, such as loading, intrinsic conductivity, size, and aspect ratio of conductive filler, inherent properties of polymer medium, interfacial properties of CPN constituents, dispersion and distribution of filler, blending method, and crystalline structure of the matrix. The impacts of the aforementioned parameters on electrical conductivity of CNT/polymer nanocomposites have been well reviewed in the literature [1, 60–62]. Accordingly, in succeeding section, we scrutinize structural and morphological features of N-CNTs and their nanocomposites to figure out the reasons behind different electrical behaviors of the generated nanocomposites.

#### **3.3. Morphological and structural characterization of N-CNTs**

conduction mechanism. Since these free charge carriers belong to the conduction band, the conductivity of the nanocomposite rises by several orders of magnitude in the percolation region. Next, by adding more filler loading, a well-developed, 3D conductive network initiates to form, but the electrical conductivity increases only marginally. This is due to substantial current dissipation at the contact spots between conductive fillers, i.e., the constriction

**Figure 4.** Percolation curve of compression-molded CNT/polystyrene nanocomposite (a typical percolation curve of

The percolation curves of N-CNT/PVDF nanocomposites are depicted in **Figure 5**. It was observed, that (N-CNT)Co/PVDF nanocomposites presented the lowest percolation threshold (1.5 wt.%) and highest electrical conductivity (3 S m−1 at 3.5 wt.%). However, it was revealed, that Ni-based nanocomposites were insulative up to 2.7 wt.% and experienced a slight increase in electrical conductivity at 3.5 wt.%. The Fe-based nanocomposites presented an increase in

**Figure 5.** Electrical conductivity of N-CNT/PVDF nanocomposites as a function of N-CNT content. N-CNTs were syn-

resistance, leading to a plateau in the percolation curve [26].

332 Thermoelectrics for Power Generation - A Look at Trends in the Technology

**3.2. Electrical conductivity of N-CNT/PVDF nanocomposites**

electrical conductivity from 1.0 wt.% to 3.5 wt.% with a mild slope.

CPNs) [55].

thesized over Co, Fe, and Ni catalysts.

The morphology and graphitic structure of N-CNTs were analyzed using TEM images. **Figure 6** indicates, that the type of synthesis catalyst played a leading role in creating the final morphology of N-CNTs. As depicted in **Figure 6**, we observed an open-channel morphology for (N-CNT)Co and a bamboo-like morphology for (N-CNT)Fe and (N-CNT)Ni. Surface roughness is observed in bamboo-like N-CNTs, deriving from defected bonding of bamboo-like sections. The flawed parts in the wall of N-CNTs are attributed to replacement of nitrogen atoms [63, 64]. Since an open-channel structure was formed for (N-CNT)Co, we can say, that other factors, than nitrogen bonding, are involved in creation of bamboo-like morphology, such as type of catalyst.

**Figure 6.** TEM micrographs of N-CNTs synthesized over Co, Fe, and Ni catalysts.

**Table 1** tabulates average length and diameter, nitrogen content, Raman feature, and synthesis yield of synthesized N-CNTs. We perceived, that N-CNTs synthesized over Fe catalyst had the largest diameter, over double that of (N-CNT)Ni. Statistical analysis of particle size of the catalysts revealed a good correlation between diameter of N-CNTs and size of catalyst particles. Discrepancies in original size of the catalyst particles and also dissimilar tendencies of the catalyst particles to sinter at synthesis temperatures are among significant parameters affecting the variation in diameter of N-CNTs. It should be noted, that metallic nanoparticles with sizes below 10 nm experience a drastic drop in melting point [65]. High synthesis temperature range (600–1000 °C) coupled with exothermic thermal decomposition of the precursor molecules


results in higher temperature than nominal reaction temperature, contributing to metal liquefaction and coalescence of catalyst particles [66, 67].

**Table 1.** Physical and structural features of N-CNTs synthesized over Co, Fe, and Ni catalysts.

**Table 1** shows, that (N-CNT)Co and (N-CNT)Fe had average length about 2.6 μm, while (N-CNT)Ni exhibited considerably lower average length about 1.2 μm. The shorter length of (N-CNT)Ni can be attributed to either inferior activity of Ni catalyst, as will be shown by TGA, or the presence of larger amount of nitrogen in their structure, as will be exhibited by X-ray photoelectron spectroscopy (XPS) analysis. The presence of nitrogen can be envisaged as an important factor to bend, close, and cap N-CNTs. It is worth noting, that average length and diameter of N-CNTs are of high significance for electrical applications, since CNTs with high aspect ratio provide CPNs with superior electrical performance [68, 69].

The amount of nitrogen content can have a weighty effect on morphological, physical, and electronic properties of N-CNTs. The achieved data revealed, that atomic content of nitrogen incorporated into (N-CNT)Ni was 3.3 at.%, whereas (N-CNT)Co and (N-CNT)Fe had considerably lower nitrogen content, i.e., 2.2 at.%. As nitrogen could have the effect of closing the tube structures and thereby developing more disordered, bent, and capped structures, the larger nitrogen content of N-CNTNi could be a contributing factor to its lower length.

In Raman spectra of CNTs, tangential mode (*G* band) and defect-active mode (*D* band) offer valuable information about physical and electronic structure of CNTs [70, 71]. Hence, Raman spectroscopy was used to inspect the influence of nitrogen doping on physical and morphological features of N-CNTs. *G* band (~1600 cm−1) derives from the stretching of C─C bond in graphitic materials and is mutual to all *sp*<sup>2</sup> carbon forms. *D* band (~1400 cm−1) is double-Raman scattering process, which requires lattice distortion to break the basic symmetry of the graphitic structure [72]. Therefore, the presence of structural defects stimulates *D* band feature. Accordingly, the ratio of *D* and *G* band intensities is often used as indicative tool to validate the structural perfection of CNTs [73]. We observed, that (N-CNT)Ni had the uppermost *ID*/*IG* ratio, signifying the poorest crystallinity. These results are in line with TEM images of (N-CNT)Ni, indicating poorer crystalline morphology than the other forms of N-CNTs. Moreover, Villalpando-Paez et al. [74] and Ibrahim et al. [75] reported good correlation between nitrogen concentration and *ID*/*IG* ratio. This is in agreement with our study and shows the opposing influence of nitrogen doping on the crystalline structure of N-CNTs. We also observed, that (N-CNT)Ni went through more breakage during the melt mixing process, ascribed to its poorer crystallinity.

TGA analysis helped investigate the synthesis yield. We obtained residues of 11.5 %, 14.9 %, and 36.1 %, relative to original mass, for (N-CNT)Co, (N-CNT)Fe, and (N-CNT)Ni, respectively. The residue consists of metallic oxide particles and alumina substrate [41, 76]. The higher the yield of the synthesis process, the lower is the amount of the remaining residue. Thus, we can claim, that Ni catalyst had an inferior performance compared to Co and Fe catalysts. The catalyst particles contained 80 wt.% alumina and 20 wt.% metallic particles. Alumina is insulative and metallic particles have much less surface area than synthesized N-CNTs, and their surface area even further reduced due to sintering phenomenon. This justifies the significance of synthesis yield on electrical properties.

#### **3.4. Morphological characterization of N-CNT/PVDF nanocomposites**

results in higher temperature than nominal reaction temperature, contributing to metal

**Table 1** shows, that (N-CNT)Co and (N-CNT)Fe had average length about 2.6 μm, while (N-CNT)Ni exhibited considerably lower average length about 1.2 μm. The shorter length of (N-CNT)Ni can be attributed to either inferior activity of Ni catalyst, as will be shown by TGA, or the presence of larger amount of nitrogen in their structure, as will be exhibited by X-ray photoelectron spectroscopy (XPS) analysis. The presence of nitrogen can be envisaged as an important factor to bend, close, and cap N-CNTs. It is worth noting, that average length and diameter of N-CNTs are of high significance for electrical applications, since CNTs with

The amount of nitrogen content can have a weighty effect on morphological, physical, and electronic properties of N-CNTs. The achieved data revealed, that atomic content of nitrogen incorporated into (N-CNT)Ni was 3.3 at.%, whereas (N-CNT)Co and (N-CNT)Fe had considerably lower nitrogen content, i.e., 2.2 at.%. As nitrogen could have the effect of closing the tube structures and thereby developing more disordered, bent, and capped structures, the larger

In Raman spectra of CNTs, tangential mode (*G* band) and defect-active mode (*D* band) offer valuable information about physical and electronic structure of CNTs [70, 71]. Hence, Raman spectroscopy was used to inspect the influence of nitrogen doping on physical and morphological features of N-CNTs. *G* band (~1600 cm−1) derives from the stretching of C─C bond in

scattering process, which requires lattice distortion to break the basic symmetry of the graphitic structure [72]. Therefore, the presence of structural defects stimulates *D* band feature. Accordingly, the ratio of *D* and *G* band intensities is often used as indicative tool to validate the structural perfection of CNTs [73]. We observed, that (N-CNT)Ni had the uppermost *ID*/*IG* ratio, signifying the poorest crystallinity. These results are in line with TEM images of (N-CNT)Ni, indicating poorer crystalline morphology than the other forms of N-CNTs. Moreover, Villalpando-Paez et al. [74] and Ibrahim et al. [75] reported good correlation between nitrogen concentration and *ID*/*IG* ratio. This is in agreement with our study and shows the opposing influence of nitrogen doping on the crystalline structure of N-CNTs. We also observed, that

carbon forms. *D* band (~1400 cm−1) is double-Raman

Length (μm) 2.6 2.6 1.2 Diameter (nm) 25 46 20 Nitrogen content (at.%) 2.2 2.2 3.3 *ID*/*IG* 0.79 0.73 0.81 Synthesis yield % 89.5 85.1 63.9

**Table 1.** Physical and structural features of N-CNTs synthesized over Co, Fe, and Ni catalysts.

high aspect ratio provide CPNs with superior electrical performance [68, 69].

nitrogen content of N-CNTNi could be a contributing factor to its lower length.

graphitic materials and is mutual to all *sp*<sup>2</sup>

**Co Fe Ni**

liquefaction and coalescence of catalyst particles [66, 67].

334 Thermoelectrics for Power Generation - A Look at Trends in the Technology

The dispersion state of conductive filler within polymer matrix is intensely influential on electrical properties. Hence, we inspected the dispersion state at three various scales. Microdispersion state of N-CNTs within the polymer medium was investigated via LM. LM talks about the portion of fillers, that appears as big agglomerates and is not disentangled well, and was enumerated as the agglomerate area ratio in our study. Moreover, gray appearance of LM samples helps us quantify the agglomerates with sizes equal to or slightly larger than the wavelength of visible light, ca. 400–700 nm, but smaller than visually identifiable agglomerates. Darker background denotes more nanotubes dispersed in this range. We also employed TEM to obtain information about nanodispersion state of carbon nanotubes, i.e., how well carbon nanotubes disentangle individually.

**Figure 7** portrays examples for LM images of three different nanocomposites, corresponding to different synthesis catalysts, with 2.0 wt.% N-CNT content. Quantification of the agglomerate area ratio, as shown in **Table 2**, illustrates the lowest agglomerate area ratio for samples containing Fe-based N-CNTs, followed by Co and Ni. The corresponding relative transparency values indicate the lowest value for Co-based N-CNTs, followed by Fe-based and Ni-based.

**Figure 7.** LM images of microtomed sections of 2.0 wt.% N-CNT/PVDF nanocomposites. N-CNTs were synthesized over Co, Fe, and Ni catalysts. The red squares represent areas employed for relative transparency quantifications.


**Table 2.** LM microdispersion parameters of microtomed N-CNT/PVDF nanocomposites with N-CNTs synthesized over different catalysts.

TEM images look into nanodispersion state of N-CNTs in PVDF medium (**Figure 8**). The images clearly show, that (N-CNT)Ni had the worst dispersion state. TEM image of (N-CNT)Ni/ PVDF nanocomposite shows a few individual nanotubes beside fairly large agglomerates. (N-CNT)Co presented the best state of nanodispersion, while (N-CNT)Fe/PVDF nanocomposites held small agglomerates with sizes around 500 nm. In conclusion, microscopy images showed, that (N-CNT)Co and (N-CNT)Fe had better both microdispersion and nanodispersion than their Ni-based counterpart. Co-based and Fe-based N-CNTs indicated only marginal discrepancies in their dispersion state.

**Figure 8.** TEM images of 2.0 wt.% N-CNT/PVDF nanocomposites with N-CNTs synthesized with different catalysts, illustrating nanodispersion state of N-CNTs.

#### **3.5. Linear and nonlinear melt-state rheological response of N-CNT/PVDF nanocomposites**

**Figure 9** depicts storage modulus (*G*′) and loss modulus (*G*″) of N-CNT/PVDF nanocomposites as a function of frequency under small-amplitude oscillatory shear (*γ* = 1 %) for a frequency range from 0.1 rad/s to 625 rad/s at 240 °C.

As shown in **Figure 9**, *G*′ in low frequency region (ω ~ 0.1 rad/s) is significantly larger than *G*″ (a damping factor smaller than unity) for (N-CNT)Co/PVDF and (N-CNT)Ni/PVDF nanocomposites at concentrations as low as 0.5 wt.%. (N-CNT)Fe/PVDF nanocomposite samples, however, exhibited an elastic dominant response (*G*′ > *G*″) only at very high nanofiller concentrations (~2.0 wt.%). It is noticeable, that all N-CNT/PVDF nanocomposites, regardless of synthesis catalyst, showed a signature for the existence of an ultraslow relaxation process (a near-zero slope for *G*′ in low frequency region) at concentrations as low as 0.5 wt.%. This indicates, that linear rheological response was affected by the presence of N-CNTs at concentrations, that no significant enhancement in electrical conductivity was observable in N-CNT nanocomposites.

The linear melt-state rheological response is mainly controlled by several factors, such as intertube van der Waals interactions, micro- and nanodispersion states, individual CNT stiffness, and CNT network stiffness [77–79]. Individual CNT stiffness is mainly controlled by intra-wall C─C bond strength and graphitic interlayer load transfer [80]. This suggests, that structural imperfections and defects in CNT graphitic walls can deteriorate their elastic properties. Moreover, stiffness of the network structure formed by CNT bundles is mainly determined by the load transfer across CNT/polymer and CNT-CNT interface [79]. In this context, it could be mentioned, that a scenario entirely based on individual CNT stiffness may not be able to fully describe the observations for elastic response of N-CNT nanocomposites in low frequency region. As can be seen in **Figure 9**, Co-, Fe-, and Ni-based N-CNT nanocomposites reached a storage modulus of 9640 Pa, 2290 Pa, and 3860 Pa at 0.1 rad, respectively. The presence of higher amount of structural imperfections may not be responsible for lower elasticity observed for Fe-based N-CNT nanocomposite as (N-CNT)Fe showed the lowest *ID*/*IG*. Therefore, the main contributing factors to linear melt-state rheological observations could be considered as the dispersion state and load transfer across the interfacial region.

**Co Fe Ni**

Agglomerate area ratio % 2.3 1.8 2.8 Relative transparency % 37 53 86

336 Thermoelectrics for Power Generation - A Look at Trends in the Technology

**Table 2.** LM microdispersion parameters of microtomed N-CNT/PVDF nanocomposites with N-CNTs synthesized

TEM images look into nanodispersion state of N-CNTs in PVDF medium (**Figure 8**). The images clearly show, that (N-CNT)Ni had the worst dispersion state. TEM image of (N-CNT)Ni/ PVDF nanocomposite shows a few individual nanotubes beside fairly large agglomerates. (N-CNT)Co presented the best state of nanodispersion, while (N-CNT)Fe/PVDF nanocomposites held small agglomerates with sizes around 500 nm. In conclusion, microscopy images showed, that (N-CNT)Co and (N-CNT)Fe had better both microdispersion and nanodispersion than their Ni-based counterpart. Co-based and Fe-based N-CNTs indicated only marginal discrepancies

**Figure 8.** TEM images of 2.0 wt.% N-CNT/PVDF nanocomposites with N-CNTs synthesized with different catalysts,

**3.5. Linear and nonlinear melt-state rheological response of N-CNT/PVDF nanocomposites**

**Figure 9** depicts storage modulus (*G*′) and loss modulus (*G*″) of N-CNT/PVDF nanocomposites as a function of frequency under small-amplitude oscillatory shear (*γ* = 1 %) for a frequency

As shown in **Figure 9**, *G*′ in low frequency region (ω ~ 0.1 rad/s) is significantly larger than *G*″ (a damping factor smaller than unity) for (N-CNT)Co/PVDF and (N-CNT)Ni/PVDF nanocomposites at concentrations as low as 0.5 wt.%. (N-CNT)Fe/PVDF nanocomposite samples, however, exhibited an elastic dominant response (*G*′ > *G*″) only at very high nanofiller concentrations (~2.0 wt.%). It is noticeable, that all N-CNT/PVDF nanocomposites, regardless of synthesis catalyst, showed a signature for the existence of an ultraslow relaxation process (a near-zero slope for *G*′ in low frequency region) at concentrations as low as 0.5 wt.%. This indicates, that linear rheological response was affected by the presence of N-CNTs at concentrations, that no significant enhancement in electrical conductivity was observable in

over different catalysts.

in their dispersion state.

illustrating nanodispersion state of N-CNTs.

N-CNT nanocomposites.

range from 0.1 rad/s to 625 rad/s at 240 °C.

**Figure 9.** Small amplitude oscillatory shear response at *γ* = 1 % and *T* = 240 °C for neat PVDF and N-CNT/PVDF nanocomposites with N-CNTs synthesized over different catalysts.

**Figure 10** depicts oscillatory amplitude sweep response of neat PVDF and N-CNT/PVDF nanocomposites containing 3.5 wt.% N-CNTs synthesized over different catalysts over a range of applied strain amplitudes from 0.1 to 1000.0 % at an angular frequency of 0.1 rad/s. The responses observed for neat PVDF and N-CNT/PVDF nanocomposite samples demonstrated a transition from a linear regime to a nonlinear regime and also a drop in *G*′ as strain amplitude increases. It is noticeable, that in low-strain region, all nanocomposite samples exhibited elastic dominant response (*G*′ > *G*″). These results also feature a crossover strain amplitude *γx* (*G*′ = *G*″), which is a measure of N-CNT network sensitivity to deformationinduced microstructural changes. As shown in **Figure 10**, Co-, Fe-, and Ni-based N-CNT nanocomposites exhibited crossover strain amplitudes of 32.0 %, 5.8 %, and 5.6 %, respectively. This implies, that Co-based N-CNTs featured a very resilient behavior toward the applied deformation field and the stress-bearing backbone of the fractal clusters survived up to strain amplitudes one order of magnitude larger than N-CNTs synthesized over Fe and Ni. It is noticeable, that Ni-based N-CNT nanocomposite showed a multistep transition into a nonlinear regime as *G*′ dropped to an intermediate plateau and then significantly decreased. Moreover, the first step decrease in *G*′ in (N-CNT)Ni nanocomposite was accompanied by a dissipation process signified by a weak local peak in *G*″.

**Figure 10.** Oscillatory amplitude sweep response of neat PVDF and N-CNT/PVDF nanocomposites containing 3.5 wt.% N-CNTs synthesized over different catalysts for strain amplitudes of *γ*0 = 0.1–1000 % at an angular frequency of *ω* = 0.1 rad/s. The insets show the non-dimensionalized elastic Lissajous loops for strain amplitudes indicated by solid lines.

Insets in **Figure 10** depict non-dimensionalized elastic Lissajous loops [81–83], in which normalized torque *Mnorm.* is plotted as a function of normalized deflection angle *φnorm*. . At small strain amplitudes (*γ* ~ 1.0 %), for neat PVDF and nanocomposites, elastic Lissajous loops were elliptical, corresponding to a linear viscoelastic response. The area enclosed by elastic Lissajous loops is significantly smaller in N-CNT nanocomposites than neat PVDF, indicating an elastic dominant response in this region. As strain amplitude increased, Lissajous loops in N-CNT nanocomposite samples became distorted, indicative of thixotropy and a yielding process in nanocomposite samples. Furthermore, observed patterns for nanocomposites suggest that N-CNT at-rest microstructure partially survived in both weakly (*γ* ~ *γx*) and strongly (*γ* > *γx*) nonlinear regimes as the area enclosed by Lissajous loops is relatively smaller, than that observed for neat PVDF. The area enclosed by elastic Lissajous loops in weakly and strongly nonlinear regimes for N-CNT nanocomposites showed the following order: Ni > Fe ~ Co.

As demonstrated by LM observations, Ni-based N-CNT nanocomposite presented the highest agglomerate area and relative transparency, indicating a poor dispersion quality of N-CNTs within PVDF matrix. This could be responsible for observing a multistep transition into a nonlinear regime and strongly nonlinear response at intermediate strain amplitudes in Nibased N-CNT samples. As explained in the preceding section, N-CNTs synthesized over different catalysts demonstrated fundamentally different dispersion states at different scales. The presence of densely aggregated N-CNT structures in Ni-based N-CNT nanocomposite led to poor load transfer across polymer-aggregate interface, resulting in deformation-induced microstructural changes initiated from aggregate-aggregate boundaries at intermediate strain amplitudes (*γ* ~ *γx*). This was followed by widespread disintegration of (N-CNT)Ni aggregates, marked by multistep transition into a nonlinear regime. However, in Co-based and Fe-based N-CNT nanocomposites, the transition into nonlinear regime occurred by *stochastic erosion* [84, 85] of network structures formed by individually dispersed N-CNTs bound polymer chains and polymer matrix entanglement network [86, 87].

In this context, it could be added, that Fe-based N-CNT nanocomposites demonstrated a dual nature in a sense, that it showed an almost one-step transition into a nonlinear regime; however, the crossover point occurred at fairly small strain amplitude (*γx* = 5.8 %). This dual behavior can be explained in conjunction with poorer load transfer across interfacial region than Cobased N-CNT nanocomposite as a result of denser N-CNT clusters present in Fe-based N-CNT nanocomposite. Moreover, one-step transition to a nonlinear regime compared to the multistep transition observed for Ni-based N-CNT nanocomposite can be attributed to better nanodispersion state achieved in Fe-based N-CNT nanocomposite (see TEM images in **Figure 8** and relative transparency values in **Table 2**). Overall, it can be expressed, that no direct link between individual N-CNT structural features and rheological response was detectable, and thus N-CNT dispersion state played the main role in determining the melt-state rheological response.

## **4. Conclusions**

range of applied strain amplitudes from 0.1 to 1000.0 % at an angular frequency of 0.1 rad/s. The responses observed for neat PVDF and N-CNT/PVDF nanocomposite samples demonstrated a transition from a linear regime to a nonlinear regime and also a drop in *G*′ as strain amplitude increases. It is noticeable, that in low-strain region, all nanocomposite samples exhibited elastic dominant response (*G*′ > *G*″). These results also feature a crossover strain amplitude *γx* (*G*′ = *G*″), which is a measure of N-CNT network sensitivity to deformationinduced microstructural changes. As shown in **Figure 10**, Co-, Fe-, and Ni-based N-CNT nanocomposites exhibited crossover strain amplitudes of 32.0 %, 5.8 %, and 5.6 %, respectively. This implies, that Co-based N-CNTs featured a very resilient behavior toward the applied deformation field and the stress-bearing backbone of the fractal clusters survived up to strain amplitudes one order of magnitude larger than N-CNTs synthesized over Fe and Ni. It is noticeable, that Ni-based N-CNT nanocomposite showed a multistep transition into a nonlinear regime as *G*′ dropped to an intermediate plateau and then significantly decreased. Moreover, the first step decrease in *G*′ in (N-CNT)Ni nanocomposite was accompa-

**Figure 10.** Oscillatory amplitude sweep response of neat PVDF and N-CNT/PVDF nanocomposites containing 3.5 wt.% N-CNTs synthesized over different catalysts for strain amplitudes of *γ*0 = 0.1–1000 % at an angular frequency of *ω* = 0.1 rad/s. The insets show the non-dimensionalized elastic Lissajous loops for strain amplitudes indicated by solid lines.

Insets in **Figure 10** depict non-dimensionalized elastic Lissajous loops [81–83], in which

strain amplitudes (*γ* ~ 1.0 %), for neat PVDF and nanocomposites, elastic Lissajous loops were

. At small

normalized torque *Mnorm.* is plotted as a function of normalized deflection angle *φnorm*.

nied by a dissipation process signified by a weak local peak in *G*″.

338 Thermoelectrics for Power Generation - A Look at Trends in the Technology

In brief, this study revealed, that electrical conductivity of N-CNT/PVDF nanocomposites is highly dependent on N-CNT synthesis catalyst. Measuring electrical conductivity of the generated nanocomposites showed superior electrical conductivity and, thus, thermoelectric performance in the following order of the synthesis catalyst: Co > Fe > Ni. It was observed, that a combination of high synthesis yield, high aspect ratio, low structural defects, enhanced network formation, and good state of N-CNT dispersion can provide N-CNT/PVDF nanocomposites with superior electrical conductivity. Moreover, it was revealed, that nitrogen doping had an adverse impact on electrical conductivity of CNT/polymer nanocomposites and, therefore, their performance as thermoelectric materials.

## **Acknowledgements**

Financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) is highly appreciated. We would like to thank Prof. Uttandaraman Sundararaj for his supervision to perform this project. We are grateful to Dr. Lars Laurentius for his assistance with Raman spectroscopy. In addition, we thank Dr. Petra Pötschke and Ms. Uta Reuter from IPF Dresden for LM and TEM investigations. Dr. Mohammad Arjmand thanks IPF Dresden for granting a research stay.

## **Author details**

Mohammad Arjmand\* and Soheil Sadeghi

\*Address all correspondence to: arjmand64@yahoo.com

University of Calgary, Calgary, Canada

## **References**


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a combination of high synthesis yield, high aspect ratio, low structural defects, enhanced network formation, and good state of N-CNT dispersion can provide N-CNT/PVDF nanocomposites with superior electrical conductivity. Moreover, it was revealed, that nitrogen doping had an adverse impact on electrical conductivity of CNT/polymer nanocomposites

Financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) is highly appreciated. We would like to thank Prof. Uttandaraman Sundararaj for his supervision to perform this project. We are grateful to Dr. Lars Laurentius for his assistance with Raman spectroscopy. In addition, we thank Dr. Petra Pötschke and Ms. Uta Reuter from IPF Dresden for LM and TEM investigations. Dr. Mohammad Arjmand thanks IPF Dresden

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\*Address all correspondence to: arjmand64@yahoo.com

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348 Thermoelectrics for Power Generation - A Look at Trends in the Technology

i2006-10156-y.

10.1122/1.3571554.

#### **Methods and Apparatus for Measuring Thermopower and Electrical Conductivity of Thermoelectric Materials at High Temperatures** Methods and Apparatus for Measuring Thermopower and Electrical Conductivity of Thermoelectric Materials at High Temperatures

Alexander T. Burkov, Andrey I. Fedotov and Sergey V. Novikov Alexander T. Burkov, Andrey I. Fedotov and Sergey V. Novikov

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/66290

#### Abstract

The principles and methods of thermopower and electrical conductivity measurements at high temperatures (100–1000 K) are reviewed. These two properties define the socalled power factor of thermoelectric materials. Moreover, in combination with thermal conductivity, they determine efficiency of thermoelectric conversion. In spite of the principal simplicity of measurement methods of these properties, their practical realization is rather complicated, especially at high temperatures. This leads to large uncertainties in determination of the properties, complicates comparison of the results, obtained by different groups, and hinders realistic estimate of potential thermoelectric efficiency of new materials. The lack of commonly accepted reference material for thermopower measurements exaggerates the problem. Therefore, it is very important to have a clear understanding of capabilities and limitations of the measuring methods and set-ups. The chapter deals with definitions of thermoelectric parameters and principles of their experimental determination. Metrological characteristics of state-of-the-art experimental set-ups for high temperature measurements are analyzed.

Keywords: thermopower, electrical conductivity, high temperature, thermoelectric material, measurement

## 1. Introduction

Thermoelectric energy conversion is based on two effects discovered in the nineteenth century: Seebeck effect and Peltier effect [1]. Historically, Seebeck effect was the first discovered thermoelectric effect, which consists in appearance of electrical current in the circuit of two different conductors at the presence of temperature difference. In year 1821, Thomas Johann Seebeck discovered, that magnetic field is generated in closed circuit consisting of bismuth (or antimony) and copper in

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons

distribution, and eproduction in any medium, provided the original work is properly cited.

the presence of temperature difference between two contacts. He first announced this discover in year 1825 in the writings of Berlin Academy of Sciences. Seebeck called this phenomenon thermomagnetism. The term "thermoelectricity" was proposed by Hans Christian Oersted approximately at the same time. There are indications, that this effect was observed and correctly interpreted in years 1794–1795 by Alessandro Volta [2]. Peltier effect was discovered in 1834 by Jean-Charles Peltier. When electrical current is forced to flow through circuit of two conductors, one contact gives out heat, while another absorbs heat. These two physical effects have become the basis for thermoelectric converters. For a long time, their practical application was limited by the use of simple thermoelectric sources for research and metal thermocouples for temperature measurement. The situation changed, when Abram F. Ioffe suggested to use semiconductors instead of metals. Based on PbS and ZnSb compounds, generator for vacuum tube radios was developed.

In the early 1950s, projects on creation of thermoelectric coolers started, and new effective materials based on compounds (Bi,Sb)2Te3 were discovered. Alloys, based on these compounds, are still the basic materials for thermoelectric refrigeration units. In the 1950–1960s, the complete elementary theory of thermoelectric conversion was created [3–5]. It was shown, that efficiency is determined by parameter ZT <sup>¼</sup> <sup>T</sup> <sup>α</sup>2<sup>σ</sup> <sup>κ</sup> , where T, α, σ and κ are absolute temperature, Seebeck coefficient (or thermopower), electrical and thermal conductivity, respectively. Almost all thermoelectric materials currently used in the industry were discovered, technology for their production was developed: synthesis, crystal growth, metal-ceramic technology (Figure 1) [6].

Design and production technology of multi-element assembly of thermocouples, which are called thermoelectric (TE) cells or modules, have been developed. These modules may consist of one or more stages (cascade module); they are used to create different variants of thermoelectric coolers (TEC) and thermoelectric generators (TEG).

After relatively rapid development in years 1950–1960s, further improving of thermoelectric parameters and TE devices progressed more slowly.

Figure 1. ZT dependence on temperature for the main thermoelectric semiconductor materials according to data of the 1960s: left side is p-type materials; right side is n-type materials.

For a long time, the maximum value of dimensionless parameter ZT does not exceed the value of 1. Areas of application of thermoelectric energy conversion techniques have been largely limited to special applications, such as power sources for spacecraft and military applications, where cost is not a major limiting factor.

In the 1980s, mass application of TE cooling for variety of purposes had started, and the market for thermoelectric cooling continues to expand today. In general, we can say, that method of thermoelectric conversion definitively established itself as one of the high-end technologies, especially for cooling purposes. This is due to its technical advantages. However, its wider application is constrained by insufficiently high efficiency of thermoelectric conversion of modern thermoelectric materials, that make the method economically ineffective. Therefore, the ultimate goal of basic research in physics and chemistry of thermoelectric materials is the development of more efficient thermoelectric materials for TEC and TEG. In view of this problem, precise and reliable measurement of thermoelectric properties (thermoelectric power, electrical and thermal conductivity) of new TE materials plays important role. These measurements must satisfy a number of requirements. Naturally, measurement results must be reliable and sufficiently accurate, measurements must be performed over a wide range of temperatures comparable with a typical range of applications. Despite the relative simplicity of fundamental measuring methods of thermoelectric properties, their practical implementation, accounting of above requirements, is a difficult task. For example, requirements for measurement accuracy are determined by the minimum of practically meaningful change of ZT parameter, which is about 10%. In order to reliably detect such small change of this parameter, measurement accuracy of thermoelectric coefficients α, σ, and κ must be not worse than 3%.

## 2. Thermoelectric coefficients and principles of experimental determination

#### 2.1. Electrical conductivity

the presence of temperature difference between two contacts. He first announced this discover in year 1825 in the writings of Berlin Academy of Sciences. Seebeck called this phenomenon thermomagnetism. The term "thermoelectricity" was proposed by Hans Christian Oersted approximately at the same time. There are indications, that this effect was observed and correctly interpreted in years 1794–1795 by Alessandro Volta [2]. Peltier effect was discovered in 1834 by Jean-Charles Peltier. When electrical current is forced to flow through circuit of two conductors, one contact gives out heat, while another absorbs heat. These two physical effects have become the basis for thermoelectric converters. For a long time, their practical application was limited by the use of simple thermoelectric sources for research and metal thermocouples for temperature measurement. The situation changed, when Abram F. Ioffe suggested to use semiconductors instead of metals. Based on PbS and ZnSb compounds, generator for vacuum tube radios was developed.

In the early 1950s, projects on creation of thermoelectric coolers started, and new effective materials based on compounds (Bi,Sb)2Te3 were discovered. Alloys, based on these compounds, are still the basic materials for thermoelectric refrigeration units. In the 1950–1960s, the complete elementary theory of thermoelectric conversion was created [3–5]. It was shown,

temperature, Seebeck coefficient (or thermopower), electrical and thermal conductivity, respectively. Almost all thermoelectric materials currently used in the industry were discovered, technology for their production was developed: synthesis, crystal growth, metal-ceramic tech-

Design and production technology of multi-element assembly of thermocouples, which are called thermoelectric (TE) cells or modules, have been developed. These modules may consist of one or more stages (cascade module); they are used to create different variants of thermo-

After relatively rapid development in years 1950–1960s, further improving of thermoelectric

Figure 1. ZT dependence on temperature for the main thermoelectric semiconductor materials according to data of the

<sup>κ</sup> , where T, α, σ and κ are absolute

that efficiency is determined by parameter ZT <sup>¼</sup> <sup>T</sup> <sup>α</sup>2<sup>σ</sup>

352 Thermoelectrics for Power Generation - A Look at Trends in the Technology

electric coolers (TEC) and thermoelectric generators (TEG).

parameters and TE devices progressed more slowly.

1960s: left side is p-type materials; right side is n-type materials.

nology (Figure 1) [6].

Electrical resistivity <sup>ρ</sup>, or conductivity <sup>σ</sup> <sup>¼</sup> <sup>1</sup> <sup>ρ</sup>, which is inverse value, determines electrical current density j in conductor, when external electric field E is applied: j ¼ σE (Ohm's law). Coefficient σ does not depend on current value. In general, σ is a second rank tensor, the number of independent components of this tensor depends on the sample material crystallographic symmetry. For crystals with cubic symmetry, tensor σ has only diagonal components, and they are all identical. Thus, it degenerates into scalar in this case. Detailed information on tensor structure for crystal lattice of different symmetry can be found in [7].

Figure 2 shows electrical scheme for measuring electrical conductivity. When electrical current I is forced through the uniform conductor under isothermal conditions, an electric field arises. The sample electrical resistance R can be found from potential difference ΔV between two points on the sample surface and electrical current magnitude: <sup>R</sup> <sup>¼</sup> <sup>Δ</sup><sup>V</sup> <sup>I</sup> . Resistance R depends on parameters of the sample material and its geometrical dimensions: <sup>R</sup> <sup>¼</sup> <sup>1</sup> <sup>σ</sup> · <sup>l</sup> a · b , where l, a, b is distance between potential probes, the sample's width and thickness, respectively. Thus, electrical conductivity σ of the sample material can be determined from measured values: its resistance R and geometrical parameters l, a, and b:

$$
\sigma = \frac{1}{R} \times \frac{l}{a \times b}. \tag{1}
$$

It should be noted, that geometric parameters do not necessarily coincide with dimensions of the sample. Electrical conductivity value is always positive, in linear approximation it does not depend on electric field (or magnitude of electric current), but depends on temperature. Depending on type of material, electrical conductivity value changes over very wide range. For metals at room temperature, σ is in the range of 106 –104 S cm−<sup>1</sup> , and for good insulators, it falls to 10−<sup>20</sup> S cm−<sup>1</sup> . Electrical conductivity of typical conductor at room temperature or above is inversely proportional to temperature and has finite value as temperature approaches absolute zero 0 K (Figure 3a). Electrical conductivity of insulators increases exponentially with increasing temperature and vanishes at low temperatures (Figure 3b).

Figure 2. Scheme of electric circuit for measuring electrical conductivity value.

Figure 3. Temperature dependence of electrical conductivity σ of metal (a) and insulator (b).

#### 2.2. Thermoelectric effects

electrical conductivity σ of the sample material can be determined from measured values: its

It should be noted, that geometric parameters do not necessarily coincide with dimensions of the sample. Electrical conductivity value is always positive, in linear approximation it does not depend on electric field (or magnitude of electric current), but depends on temperature. Depending on type of material, electrical conductivity value changes over very wide range.

is inversely proportional to temperature and has finite value as temperature approaches absolute zero 0 K (Figure 3a). Electrical conductivity of insulators increases exponentially with

: (1)

, and for good insulators, it

–104 S cm−<sup>1</sup>

. Electrical conductivity of typical conductor at room temperature or above

<sup>σ</sup> <sup>¼</sup> <sup>1</sup> R · l a · b

resistance R and geometrical parameters l, a, and b:

354 Thermoelectrics for Power Generation - A Look at Trends in the Technology

For metals at room temperature, σ is in the range of 106

increasing temperature and vanishes at low temperatures (Figure 3b).

Figure 2. Scheme of electric circuit for measuring electrical conductivity value.

Figure 3. Temperature dependence of electrical conductivity σ of metal (a) and insulator (b).

falls to 10−<sup>20</sup> S cm−<sup>1</sup>

Seebeck effect is occurrence of electromotive force in conductor, which has temperature gradient inside. It can be observed in a simple circuit consisting of two different conductors (x and l), when contacts of these conductors have different temperatures (Figure 4). Under these conditions, there will be a potential difference in a circuit: ΔV∝αxlðT2−T1Þ, where T2−T<sup>1</sup> is a temperature difference between contacts and coefficient α is known as Seebeck coefficient or thermoelectric power. Seebeck coefficient is formally defined as follows: E ¼ α∇T, here E – electric field induced in conductor in the presence of temperature gradient ∇T. Seebeck coefficient α is a second rank tensor. In contrast to electrical conductivity, Seebeck coefficient can be either positive or negative. Potential difference measured by voltmeter in the circuit shown in Figure 4, ΔV ¼ ϕ2−ϕ1, where ϕ<sup>2</sup> and ϕ<sup>1</sup> are input voltmeter potentials "1" and "2" at the same

Figure 4. Thermoelectric circuit consisting of two conductors connected in series. Contacts of conductors are maintained at temperatures T<sup>1</sup> and T2.

The circuit shown in Figure 4 consists of two different conductors, x ("sample") and l (wires connecting the sample with voltmeter). We assume, that both conductors are uniform. Seebeck coefficient of the sample and wires are denoted as α<sup>x</sup> and αl, respectively. For homogeneous and isotropic conductors, coefficient α is independent on position along the wire and direction of temperature gradient, but usually it depends on temperature. Therefore:

$$
\Delta V = -\int\_{1}^{T\_1} \alpha\_l \nabla T dl - \int\_{T\_1}^{T\_2} \alpha\_x \nabla T dl - \int\_{T\_2}^{2} \alpha\_l \nabla T dl \tag{2}
$$

$$
\Delta V = -\int\_{T\_0}^{T\_1} \alpha\_l dT + \int\_{T\_0}^{T\_1} \alpha\_l dT - \int\_{T\_1}^{T\_2} \alpha\_l dT + \int\_{T\_1}^{T\_2} \alpha\_l dT = -\int\_{T\_1}^{T\_2} (\alpha\_x - \alpha\_l) dT. \tag{3}
$$

When temperature difference T2−T<sup>1</sup> is small compared to average temperature ðT<sup>2</sup> þ T1Þ=2, then:

$$
\Delta V = -(T\_2 - T\_1) \times (\alpha\_\text{x} - \alpha\_l). \tag{4}
$$

Hence, experimentally measured potential difference is proportional to temperature difference between the sample and probe contacts and Seebeck coefficient difference of the sample material and probes. It means, that in this kind of experiment only difference αx−α<sup>l</sup> can be measured, and it is called relative thermoelectric power of "x" and "l" conductors αxl. In order to determine absolute thermopower of the sample αx, it is necessary to know thermopower of probe α<sup>l</sup> (usually called as reference electrode probe). The magnitude of thermopower of metals ranges from ±10−<sup>6</sup> to ±5×10−<sup>5</sup> V/K (at room temperature), while thermopower of thermoelectric semiconductors can reach ±10−<sup>3</sup> V/K.

Peltier effect can be observed in a similar circuit by replacing voltmeter to current source. When electrical current flows through the circuit, then at one contact, heat is emitted and at another heat is absorbed. Quantity of heat (Q) emitted or absorbed per unit time at contact of two materials is given by formula: Q ¼ Πlx · I, here Πlx – Peltier coefficient of materials l and x, I – current flowing through the contacts. Similar to thermopower, Peltier coefficient of each material can be determined: Πlx ¼ Πl−Πx. Thermopower and Peltier coefficient are interrelated by Thompson relation [8, 9]:

$$
\Pi = T \times \alpha.\tag{5}
$$

Another important thermoelectric effect is Thompson effect. When electrical current passes through homogeneous conductor in the presence of temperature gradient, then some heat energy is released or absorbed depending on mutual orientation of current and temperature gradient. In contrast to Joule heat, in this effect, heat can be emitted, leading to additional heating of conductor or absorbed, leading to cooling. When electrical current with density j flows through conductor, then quantity of heat (q), emitted in unit volume of conductor per unit time, equals to [8, 9]: q ¼ −τ<sup>T</sup> · j · ∇T. In contrast to Seebeck and Peltier coefficients, Thompson coefficient τ<sup>T</sup> can be measured for individual conductor. Thompson coefficient is interrelated to two other thermoelectric coefficients by second Thompson relation [8, 9]:

$$
\pi\_T = T \frac{d\alpha}{dT}.\tag{6}
$$

This important relation allows to determine thermopower:

$$\alpha(T) = \int\_0^T \frac{\tau\_T}{T} dT\tag{7}$$

and to build the absolute thermoelectric scale, which we will discuss further.

## 3. Measurement principles

#### 3.1. Electrical conductivity

ΔV ¼ −

moelectric semiconductors can reach ±10−<sup>3</sup> V/K.

by Thompson relation [8, 9]:

T ð1

356 Thermoelectrics for Power Generation - A Look at Trends in the Technology

T0

αldT þ

T ð1

T0

αldT−

T ð2

T1

αxdT þ

When temperature difference T2−T<sup>1</sup> is small compared to average temperature ðT<sup>2</sup> þ T1Þ=2, then:

Hence, experimentally measured potential difference is proportional to temperature difference between the sample and probe contacts and Seebeck coefficient difference of the sample material and probes. It means, that in this kind of experiment only difference αx−α<sup>l</sup> can be measured, and it is called relative thermoelectric power of "x" and "l" conductors αxl. In order to determine absolute thermopower of the sample αx, it is necessary to know thermopower of probe α<sup>l</sup> (usually called as reference electrode probe). The magnitude of thermopower of metals ranges from ±10−<sup>6</sup> to ±5×10−<sup>5</sup> V/K (at room temperature), while thermopower of ther-

Peltier effect can be observed in a similar circuit by replacing voltmeter to current source. When electrical current flows through the circuit, then at one contact, heat is emitted and at another heat is absorbed. Quantity of heat (Q) emitted or absorbed per unit time at contact of two materials is given by formula: Q ¼ Πlx · I, here Πlx – Peltier coefficient of materials l and x, I – current flowing through the contacts. Similar to thermopower, Peltier coefficient of each material can be determined: Πlx ¼ Πl−Πx. Thermopower and Peltier coefficient are interrelated

Another important thermoelectric effect is Thompson effect. When electrical current passes through homogeneous conductor in the presence of temperature gradient, then some heat energy is released or absorbed depending on mutual orientation of current and temperature gradient. In contrast to Joule heat, in this effect, heat can be emitted, leading to additional heating of conductor or absorbed, leading to cooling. When electrical current with density j flows through conductor, then quantity of heat (q), emitted in unit volume of conductor per unit time, equals to [8, 9]: q ¼ −τ<sup>T</sup> · j · ∇T. In contrast to Seebeck and Peltier coefficients, Thompson coefficient τ<sup>T</sup> can be measured for individual conductor. Thompson coefficient is interrelated to two other thermoelectric coefficients by second Thompson relation [8, 9]:

<sup>τ</sup><sup>T</sup> <sup>¼</sup> <sup>T</sup> <sup>d</sup><sup>α</sup>

ð T

τT

0

αðTÞ ¼

and to build the absolute thermoelectric scale, which we will discuss further.

This important relation allows to determine thermopower:

T ð2

T1

αldT ¼ −

T ð2

T1

ΔV ¼ −ðT2−T1Þ · ðαx−αlÞ: (4)

Π ¼ T · α: (5)

dT : (6)

<sup>T</sup> dT (7)

ðαx−αlÞdT: (3)

If the sample is homogeneous (electrical conductivity is the same everywhere inside the sample), then under uniform electrical current distribution inside the sample, electrical conductivity of the material can be determined by formula (1) on the base of experimentally determined values R, l, a and b. In conductivity measurements, some heat energy is always generated in the sample volume due to Joule heating. Amount of heat generated in unit volume of the sample is determined by Joule-Lenz's law: qj ¼ j · E ¼ j <sup>2</sup> · ρ. This heat energy can affect the accuracy of conductivity measurement, changing sample temperature, and inducing thermoelectric contribution to measured potential difference (ΔV). In order to reduce influence of Joule heat on conductivity measurement, one has to use lower current density and provide good thermal contact of the sample with environment.

Eq. (1) is applicable, if the sample is in isothermal conditions. In actual practice, this condition is almost never fulfilled. Moreover, electrical conductivity is often measured simultaneously with thermoelectric coefficient, which requires temperature gradient. Under these conditions, potential difference, measured in the circuit shown in Figure 2, will include two components: <sup>Δ</sup><sup>V</sup> <sup>¼</sup> <sup>R</sup> · <sup>I</sup> <sup>þ</sup> <sup>α</sup>lxΔ<sup>T</sup> <sup>¼</sup> <sup>1</sup> <sup>σ</sup> · <sup>l</sup> <sup>a</sup> · <sup>b</sup> I þ αlxΔT, where αlx and ΔT – thermopower of the sample and temperature difference between potential probes, respectively. For thermoelectric materials, both contributions can be of the same order of magnitude. To eliminate the influence of thermal gradient in the sample on electrical conductivity, two methods are used:


In the following sections, different methods of measurement σ are described. To exclude the contribution of thermoelectric effects in all of them, AC or DC measurements may be used.

#### 3.1.1. Classic measurement scheme

Classic measurement scheme of electrical conductivity is presented in Figure 5. In this method, the sample should be prepared in the form of long thin and uniform wire with diameter d. Potential difference ΔV12, is measured between points "1" and "2" separated by distance "l", when current I passes through the wire. Electrical conductivity is determined by formula: <sup>σ</sup> <sup>¼</sup> <sup>I</sup> <sup>Δ</sup>V<sup>12</sup> · <sup>4</sup><sup>l</sup> πd2.

The wire must be placed into electrically non-conductive medium having sufficiently high thermal conductivity, which absorbs heat generated in the sample, and minimizes temperature gradient in it. When the sample is prepared in the form of long thin wire, then low measuring current density j can be used. In this case, along with reducing quantity of Joule heat, it is possible to maintain large enough potential difference ΔV<sup>12</sup> by increasing distance l between potential probes, which improves measurements accuracy. However, this method is rarely used in practice.

First, long wire samples are inconvenient, when measurements are performed in chambers with limited volume, such as cryostat for measurements at low temperatures or vacuum chamber at high temperature measurements. Second, majority of materials is difficult or impossible to prepare in the form of thin homogeneous wire. Therefore, usually, short samples are used in the form of cylinder or parallelepiped, thin plate or film. The accuracy of resistance measurement of such samples is lower than in classic configuration.

Figure 5. Classic method of electrical resistance measurement. The sample is prepared as a homogeneous wire, contacts with current leads are maintained at temperature Tx.

#### 3.1.2. Samples of regular geometric shape

Figure 6 shows the scheme for measuring electrical conductivity of short samples with a regular geometric shape. The sample for such measurements must have simple geometric shape allowing accurate determination of electrical current density and potential gradient in the sample, which have to be uniform. Current contacts should provide uniform current distribution in the sample.

Figure 6. Resistance measuring scheme of short samples.

3.1.1. Classic measurement scheme

358 Thermoelectrics for Power Generation - A Look at Trends in the Technology

<sup>σ</sup> <sup>¼</sup> <sup>I</sup>

<sup>Δ</sup>V<sup>12</sup> · <sup>4</sup><sup>l</sup> πd2.

used in practice.

3.1.2. Samples of regular geometric shape

with current leads are maintained at temperature Tx.

distribution in the sample.

Classic measurement scheme of electrical conductivity is presented in Figure 5. In this method, the sample should be prepared in the form of long thin and uniform wire with diameter d. Potential difference ΔV12, is measured between points "1" and "2" separated by distance "l", when current I passes through the wire. Electrical conductivity is determined by formula:

The wire must be placed into electrically non-conductive medium having sufficiently high thermal conductivity, which absorbs heat generated in the sample, and minimizes temperature gradient in it. When the sample is prepared in the form of long thin wire, then low measuring current density j can be used. In this case, along with reducing quantity of Joule heat, it is possible to maintain large enough potential difference ΔV<sup>12</sup> by increasing distance l between potential probes, which improves measurements accuracy. However, this method is rarely

First, long wire samples are inconvenient, when measurements are performed in chambers with limited volume, such as cryostat for measurements at low temperatures or vacuum chamber at high temperature measurements. Second, majority of materials is difficult or impossible to prepare in the form of thin homogeneous wire. Therefore, usually, short samples are used in the form of cylinder or parallelepiped, thin plate or film. The accuracy of resistance

Figure 6 shows the scheme for measuring electrical conductivity of short samples with a regular geometric shape. The sample for such measurements must have simple geometric shape allowing accurate determination of electrical current density and potential gradient in the sample, which have to be uniform. Current contacts should provide uniform current

Figure 5. Classic method of electrical resistance measurement. The sample is prepared as a homogeneous wire, contacts

measurement of such samples is lower than in classic configuration.

Since electrical conductivity depends on temperature, then, during measuring it, temperature of the sample must be set and determine precisely. Electrical conductivity is determined by formula: <sup>σ</sup> <sup>¼</sup> <sup>I</sup> <sup>Δ</sup><sup>V</sup> · <sup>l</sup> <sup>A</sup>, here, A is cross-section area of the sample in plane, perpendicular to electric current direction.

#### 3.1.3. Four-probe method of electrical resistivity measurement

In both schemes of measuring electrical conductivity σ, described above, geometric parameters coincide with cross-section of the sample A = a×b and distance between potential probes (l). There are modifications of these schemes, in which geometric parameters do not match sizes of the sample. They include four-probe method [10–12] and van der Pauw method [12, 13]. Note that all methods of measuring electrical conductivity described here are essentially four-probe methods in the sense, that potential probes are separated from electrical current leads. However, this term is also used as the name of specific embodiment of methods for measuring electrical conductivity. In the most common variant of this method, all four electrodes are arranged along straight line on flat surface of the sample (Figure 7). If electrodes are arranged symmetrically, and thickness (d) and minimum distance from electrodes to the edge of the sample is much greater than distance between electrodes (l) (semi-infinite space approximation), then electrical conductivity is determined by simple expression [10–12]: <sup>σ</sup> <sup>¼</sup> <sup>2</sup><sup>I</sup> <sup>π</sup> ·Δ<sup>V</sup> · <sup>1</sup> <sup>S</sup>−<sup>l</sup> − <sup>1</sup> Sþl h i, here <sup>I</sup> – electrical current flowing through the sample, <sup>S</sup> – distance between outermost electrodes (current contacts), l – distance between potential probes. Practical criterion of applicability of this approximation is S/d < 5. If electrodes are arranged at the same distance from each other, that is, S = <sup>3</sup>l, we get <sup>σ</sup> <sup>¼</sup> <sup>I</sup> <sup>2</sup><sup>π</sup> · <sup>l</sup> ·Δ<sup>V</sup>. In another limit case d << l, expression for determining electrical conductivity takes the form [10–12]: <sup>σ</sup> <sup>¼</sup> <sup>I</sup> <sup>π</sup>�d�Δ<sup>V</sup> ln <sup>S</sup>þ<sup>l</sup> S−l � �.

If S = <sup>3</sup>l, we obtain: <sup>σ</sup> <sup>¼</sup> <sup>I</sup> <sup>π</sup> · <sup>d</sup> ·Δ<sup>V</sup> ln2. This formula is applicable for S/d > 5.

For arbitrary thickness of the sample, expression for σ is as follows [10–12]:

$$\sigma = \frac{2I}{\pi \times \Delta V} \left\{ \frac{1}{S-l} - \frac{1}{S+l} + 2 \times \sum\_{n=1}^{N} \left[ \frac{1}{\sqrt{(S-l)^2 + (4nd)^2}} - \frac{1}{\sqrt{(S+l)^2 + (4nd)^2}} \right] \right\}. \tag{8}$$

$$\left[ \underbrace{\Delta V}\_{\text{max}} \right] = \frac{1}{\Delta V} \left[ \underbrace{\Delta V}\_{\text{max}} \right]$$

$$\Delta V = \frac{1}{S} \left[ \underbrace{\Delta V}\_{\text{max}} \right]$$

Figure 7. Four-probe method to measure electrical conductivity.

Four-probe method is a convenient way to determine quickly and accurately electrical conductivity and does not require preparation of samples with regular geometric shape. It requires one flat surface only. However, the sample surface area needs to be large enough to satisfy condition Lmin > 10S for any distance (L) from measuring probes to the edge of the sample. Otherwise, measured potential difference ΔV will depend on type and shape of the sample boundaries.

#### 3.1.4. Van der Pauw method

Van der Pauw method is applied for measuring electrical conductivity of the samples with irregular shape [12–15]. To measure electrical conductivity by van der Pauw method, it is necessary to form four contacts at arbitrary points A, B, C and D on the edge of flat sample (Figure 8).

Figure 8. Schematic view of arbitrary shape flat plate (sample) with four contacts A, B, C, D for measuring electrical conductivity by van der Pauw method.

By passing electrical current IAB between contacts A and B, one can determine resistance RAB,CD as follows: RAB,CD <sup>¼</sup> <sup>Δ</sup>VCD IAB , where ΔVCD is potential difference between contacts C and D. Similarly: RBC,DA <sup>¼</sup> <sup>Δ</sup>VDA IBC .

If the following conditions are fulfilled:

<sup>σ</sup> <sup>¼</sup> <sup>2</sup><sup>I</sup> π · ΔV

boundaries.

(Figure 8).

3.1.4. Van der Pauw method

conductivity by van der Pauw method.

1 S−l <sup>−</sup> <sup>1</sup> S þ l

360 Thermoelectrics for Power Generation - A Look at Trends in the Technology

Figure 7. Four-probe method to measure electrical conductivity.

8 ><

>:

þ 2 · ∑ ∞ n¼1

2 6 4

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>2</sup> þ ð4nd<sup>Þ</sup>

2 <sup>q</sup> <sup>−</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

q

ðS þ lÞ

<sup>2</sup> þ ð4nd<sup>Þ</sup>

2

3 7 5

9 >=

>;

: (8)

ðS−lÞ

Four-probe method is a convenient way to determine quickly and accurately electrical conductivity and does not require preparation of samples with regular geometric shape. It requires one flat surface only. However, the sample surface area needs to be large enough to satisfy condition Lmin > 10S for any distance (L) from measuring probes to the edge of the sample. Otherwise, measured potential difference ΔV will depend on type and shape of the sample

Van der Pauw method is applied for measuring electrical conductivity of the samples with irregular shape [12–15]. To measure electrical conductivity by van der Pauw method, it is necessary to form four contacts at arbitrary points A, B, C and D on the edge of flat sample

Figure 8. Schematic view of arbitrary shape flat plate (sample) with four contacts A, B, C, D for measuring electrical


then RAB,CD and RBC,DA satisfy to equation [13, 14]:

$$\exp\left(-\pi d \sigma R\_{AB,CD}\right) + \exp\left(-\pi d \sigma R\_{BC,DA}\right) = 1,\tag{9}$$

where d is thickness of the sample plate. Since resistances RAB,CD, RBC,DA and d are known, σ is the only one unknown quantity in Eq. (9), and it can be found by solving this equation.

Solution of Eq. (9) can be written in the form [13]:

$$
\sigma = \frac{2\ln 2}{\pi d (R\_{AB, CD} + R\_{BC, DA})} \frac{1}{f \left(\frac{R\_{AB, CD}}{R\_{BC, DA}}\right)},\tag{10}
$$

where f is function depending only on the ratio RAB,CD RBC,DA. Graph of this function is shown in Figure 9.

When RAB,CD RBC,DA<sup>≈</sup> 1, then <sup>f</sup> can be approximated by expression:

$$f \approx 1 - \left(\frac{R\_{AB,CD} - R\_{BC,DA}}{R\_{AB,CD} + R\_{BC,DA}}\right)^2 \frac{\ln 2}{2} - \left(\frac{R\_{AB,CD} - R\_{BC,DA}}{R\_{AB,CD} + R\_{BC,DA}}\right)^4 \left[\frac{(\ln 2)^2}{4} - \frac{(\ln 2)^3}{12}\right].\tag{11}$$

Situation is considerably simplified, if the sample has symmetry axis [14]. Assume that contacts A and C are located on the symmetry axis and contacts B and D are placed symmetrically relative to this axis (Figure 10). Then RAB,CD ¼ RAD,CB. According to the theorem of reciprocity for passive four poles [16], we have RAD,CB ¼ RCB,AD ¼ RBC,DA, and it follows from Eq. (9): <sup>σ</sup> <sup>¼</sup> ln2 <sup>π</sup> d RAB,CD.

#### 3.2. Thermopower

Figure 11 shows principle of measuring thermoelectric power. There are two direct methods of thermopower measurement: the integral method is historically the first and is conceptually simpler (Eq. (3)); and the differential method, which is practically the most used (Eq. (4)).

Figure 9. Dependence of function f on ratio RAB,CD RBC,DA [14].

Figure 10. Contacts configuration for the sample having axis of symmetry.

Figure 11. Scheme of integral method for thermopower measurements: voltage of thermocouple, consisting of sample and reference wires, is measured as function of thermocouple junction temperature T by voltmeter ΔV. Temperature is measured with help of another thermocouple by voltmeter VT. Junctions of the sample, reference and thermocouple wires with cupper wires, connected to voltmeters are kept at fixed temperature T0. Ta is ambient temperature.

#### 3.2.1. Integral method

Figure 11 shows electrical circuit for measuring thermopower by integral method. Voltage of thermocouple, consisting of the sample and reference electrode wires, is measured as function

$$\text{of-temperature:} \,\Delta V = -\prod\_{T\_0}^{T} (\alpha\_x - \alpha\_l) dT. \text{ Hence, } \alpha\_x(T) - a\_l(T) = -\frac{d\Delta V}{dT}.$$

In this method, besides ΔV, it is necessary to measure temperature T of contacts of material under study and reference electrode. This can be done by using additional electrode with known thermopower, which forms thermocouple with reference electrode. Note, that serious disadvantage of this method is that samples must be prepared in the form of homogeneous wires. But, many materials, which are considered as prospective thermoelectrics, are very difficult or impossible to prepare in form of wire. Therefore, integral method of thermopower measurement is used very rarely now.

#### 3.2.2. Differential method

Figure 9. Dependence of function f on ratio RAB,CD

362 Thermoelectrics for Power Generation - A Look at Trends in the Technology

RBC,DA [14].

Figure 11. Scheme of integral method for thermopower measurements: voltage of thermocouple, consisting of sample and reference wires, is measured as function of thermocouple junction temperature T by voltmeter ΔV. Temperature is measured with help of another thermocouple by voltmeter VT. Junctions of the sample, reference and thermocouple wires

with cupper wires, connected to voltmeters are kept at fixed temperature T0. Ta is ambient temperature.

Figure 10. Contacts configuration for the sample having axis of symmetry.

In contrast to integral method, differential one is designed for measuring thermopower of short samples of any shape, including thin films. Therefore, the vast majority of thermopower measurements have been performed by this method. Figure 12 shows a scheme of differential method. Temperature difference between two points on the sample is measured with two thermocouples (or other temperature sensors), and thermopower signal ΔV can be measured by the same branches of thermocouples. Using Eq. (4), expression for determining absolute thermopower of the sample can be written as follows:

$$
\alpha\_x = -\frac{\Delta V}{\Delta T} + \alpha\_l. \tag{12}
$$

#### 3.3. Absolute thermoelectric scale

In order to determine absolute thermopower of the material αx, it is necessary to know absolute thermopower of reference electrode αl. This is a key point in thermopower measurements. There is no direct method for measuring absolute thermoelectric power. Determination of absolute thermopower is based on two physical phenomena:

Thomson's relationship between Seebeck (α) and Thomson coefficients (τT) [9].

Property of superconductors: electric field E = 0 inside superconductor. Hence, it follows, that thermopower of superconductor is zero.

Based on these two phenomena, absolute thermopower of some materials was determined. Currently, lead, copper, and platinum are the main materials of reference electrodes. Dataset of absolute thermoelectric power of these metals establish absolute thermoelectric scale. This scale is based on experimental data of Thomson coefficient τT. Absolute thermoelectric power can be calculated according to second Thompson relation (Eqs. (6) and (7)).

Figure 12. Scheme of differential method for thermopower measurements. Heat flow generated by gradient heater passes through the sample and creates temperature gradient in it. Temperature difference between two points on surface of the sample is measured using thermocouples. The same thermocouple branches are used to measure potential difference between points on the sample.

However, in practice, we cannot determine by Eq. (7) absolute thermopower of studied material, since it requires information about Thomson coefficient in temperature range from absolute zero to T, which is fundamentally impossible. This problem can be solved with superconducting materials. In superconducting state, that is, at T < Tc, thermopower α = 0. Hence, for such materials, it is sufficient to know Thomson coefficient value at temperature T>Tc only.

Nyström [17] created first absolute thermoelectric scale, which was based on his measurements of Thomson coefficient of copper in temperature range from 723 to 1023 K and Borelius's low-temperature data [18, 19]. Using data of absolute thermoelectric power of copper, Nyström determined absolute thermoelectric power of platinum. Later, Rudnitskii [20] has extrapolated Nyström's data for platinum up to 1473 K. Cusack and Kendall [21] have processed Thompson coefficient data and calculated absolute thermoelectric power of number of metals in wide temperature range, including platinum up to 2000 K, and molybdenum and tungsten up to 2400 K (using Thomson coefficient data obtained by Lander [22]). The most accurate thermoelectric scale was created by Roberts, who carried out measurements of Thomson coefficient of lead, copper, and platinum [23–25]. Thomson coefficient was measured for lead in temperature range from 7 K (i.e., from superconducting transition temperature) to 600 K (up to nearly melting temperature). Thomson coefficient of copper was measured up to 873 K, and for platinum and tungsten up to 1600 K. On the basis of these data, thermoelectric scale overlapping temperature range between 0 and 1600 K was created. According to Roberts estimations, his thermoelectric scale has error not more than ±0.01 μV/K at room temperature, ±0.02 μV/K at 600 K, ±0.05 μV/K at 900 K, and ±0.2 μV/K at 1600 K. At higher temperatures, absolute thermopower data is much less precise. Accuracy of the data at 2000 K is about ±2 μV/K. The results of these studies are summarized in Table 1.


Table 1. Thermopower of lead (αPb), copper (αCu), and platinum (αPt).

However, in practice, we cannot determine by Eq. (7) absolute thermopower of studied material, since it requires information about Thomson coefficient in temperature range from absolute zero to T, which is fundamentally impossible. This problem can be solved with superconducting materials. In superconducting state, that is, at T < Tc, thermopower α = 0. Hence, for such materials, it is sufficient to know Thomson coefficient value at temperature

Figure 12. Scheme of differential method for thermopower measurements. Heat flow generated by gradient heater passes through the sample and creates temperature gradient in it. Temperature difference between two points on surface of the sample is measured using thermocouples. The same thermocouple branches are used to measure potential difference

Nyström [17] created first absolute thermoelectric scale, which was based on his measurements of Thomson coefficient of copper in temperature range from 723 to 1023 K and Borelius's low-temperature data [18, 19]. Using data of absolute thermoelectric power of copper, Nyström determined absolute thermoelectric power of platinum. Later, Rudnitskii [20] has extrapolated Nyström's data for platinum up to 1473 K. Cusack and Kendall [21] have processed Thompson coefficient data and calculated absolute thermoelectric power of number of metals in wide temperature range, including platinum up to 2000 K, and molybdenum and tungsten up to 2400 K (using Thomson coefficient data obtained by Lander [22]). The most accurate thermoelectric scale was created by Roberts, who carried out

T>Tc only.

between points on the sample.

364 Thermoelectrics for Power Generation - A Look at Trends in the Technology

In practice, for measuring thermopower at high temperature (above 100 K) are used thermocouples copper-constantan and platinum-platinum/rhodium and reference electrode of platinum or copper, respectively. For both, platinum and copper, the absolute thermopower was accurately determined by Roberts only above room temperature. Therefore, it was necessary to expand temperature range of accurate determination of absolute thermopower of these metals to lower temperature region. Absolute thermoelectric power of platinum in temperature range from 25 to 1600 K was determined in [26] using Roberts's data and Moore's and Grave's low temperature data [27], which were adjusted using Roberts's data for lead [23], so that, corrected data are consistent with Roberts' high temperature data. These data and experimental results for platinum are shown in Figure 13 and summarized in Table 1 [26].

By using combined experimental data obtained in temperature range 70–1500 K, thermopower of platinum can be described by empirical interpolation formula αPtðTÞ:

$$
\Delta a\_{Pl}(T) = 0.186T \left[ \exp\left(-\frac{T}{88}\right) \text{-0.0786} + \frac{0.43}{1 + \left(\frac{T}{84.3}\right)^4} \right] - 2.57.\tag{13}
$$

This function and its deviation from experimental points are shown in Figure 13.

Figure 13. The top panel shows absolute thermoelectric power of platinum: • – Moore's data [27]; + −Roberts's data [24, 25]; ○ – combined data (not all data points are depicted). Solid line shows interpolation function. The bottom panel presents deviation of interpolation function from experimental data Δα ¼ αexp er−αPtðTÞ.

Absolute thermoelectric power of copper was determined in [26] using Roberts' data for temperature range 273–900 K [24], and at temperatures below 273 K using Cusack's and Kendall's results [21]. Small correction was introduced in the data, so that, this low-temperature dependence smoothly joints with Roberts' high-temperature data. Adjusted and original experimental data and empirical interpolation formula αCu for temperature range 70–1000 K are shown in Figure 14 and Table 1 [26]. Interpolation function αCu is given by:

$$\alpha\_{\rm Cu}(T) = 0.041 \ T \left[ \exp\left(-\frac{T}{93}\right) - 0.123 + \frac{0.442}{1 + \left(\frac{T}{172.4}\right)^3} \right] + 0.804. \tag{14}$$

The error of this practical thermoelectric scale (considering the interpolation error) is estimated as follows [26]:

In temperature range 70–900 K: ±0.1 μV/K and 1000–1500 K: ±0.5 μV/K.

In practice, for measuring thermopower at high temperature (above 100 K) are used thermocouples copper-constantan and platinum-platinum/rhodium and reference electrode of platinum or copper, respectively. For both, platinum and copper, the absolute thermopower was accurately determined by Roberts only above room temperature. Therefore, it was necessary to expand temperature range of accurate determination of absolute thermopower of these metals to lower temperature region. Absolute thermoelectric power of platinum in temperature range from 25 to 1600 K was determined in [26] using Roberts's data and Moore's and Grave's low temperature data [27], which were adjusted using Roberts's data for lead [23], so that, corrected data are consistent with Roberts' high temperature data. These data and experimental results for platinum are shown in Fig-

By using combined experimental data obtained in temperature range 70–1500 K, thermopower

−0:0786 þ

" #

0:43 <sup>1</sup> <sup>þ</sup> <sup>T</sup> 84:3 � �<sup>4</sup>

−2:57: (13)

88 � �

Figure 13. The top panel shows absolute thermoelectric power of platinum: • – Moore's data [27]; + −Roberts's data [24, 25]; ○ – combined data (not all data points are depicted). Solid line shows interpolation function. The bottom panel

presents deviation of interpolation function from experimental data Δα ¼ αexp er−αPtðTÞ.

This function and its deviation from experimental points are shown in Figure 13.

of platinum can be described by empirical interpolation formula αPtðTÞ:

<sup>α</sup>PtðTÞ ¼ <sup>0</sup>:186<sup>T</sup> exp <sup>−</sup> <sup>T</sup>

ure 13 and summarized in Table 1 [26].

366 Thermoelectrics for Power Generation - A Look at Trends in the Technology

In formulas (13) and (14), thermopower is expressed in μV/K, and temperature is expressed in Kelvin degree.

Figure 14. The top panel shows absolute thermoelectric power of copper: ● – Cusack's data [21]; + <sup>−</sup> Roberts's data [24]; ○ – adjusted data. Solid line is interpolation function. The bottom panel shows deviation of interpolation function from experimental data Δα ¼ αexp er−αCuðTÞ.

## 4. Error analysis

#### 4.1. Electrical conductivity

Errors in measurements of electrical conductivity can be divided into three categories. First, it is electrical signal measurement errors, that is, potential difference and current magnitude. Second, it is errors associated with shape of the sample and of measuring electrodes. And third, there are errors associated with change in temperature of the sample during measurement process.

The first kind of errors is common to all measurements of electrical signals and are not specific for measuring electron transport properties. When modern measuring equipment is used and proper organization of measuring system and procedure are applied, then these errors generally are not a factor limiting the accuracy of measurements. Possible exceptions are measurements of electrical conductivity of high pure metals at very low temperatures. However, these cases are not typical for high-temperature measurements of thermoelectric materials, and not analyzed here.

#### 4.1.1. Errors associated with shape

Errors associated with sample's and electrode's shape are, perhaps, the main problem in most cases. When measuring conductivity, actual measured value is a total resistance of the sample between potential probes <sup>R</sup> <sup>¼</sup> <sup>Δ</sup><sup>V</sup> <sup>I</sup> . In order to obtain electrical conductivity of the sample, it is necessary to know cross-section of the sample (A) and distance between potential probes (l) <sup>σ</sup> <sup>¼</sup> <sup>1</sup> <sup>R</sup> · <sup>l</sup> A. There are four sources of errors associated with geometric factor. The easiest is inaccuracy in determining size and shape of the sample. Assume, that the sample has parallelepiped shape with typical dimensions 2 × 2 × 10 mm3 . In ordinary methods of sample machining and measurement of lengths, typical error of size determination is of the order 0.01 mm. This error includes distance measurement inaccuracy, and shape and surface imperfections of the sample as well. This error causes error of determining the section ΔA/A equals to 1%. The error of determining distance l, which includes both error in measurement of length and finite size of potential contact, is of the order 0.1 mm. Thus, total error in determining geometric factor is equal to <sup>Δ</sup><sup>A</sup> <sup>A</sup> <sup>þ</sup> <sup>Δ</sup><sup>l</sup> <sup>l</sup> ¼ 0:02, that is, 2%. It is accuracy limit of measuring resistance by four-probe method using bulk samples. Of course, accuracy can be improved by using a special highprecision technology for manufacturing of the sample and measuring its dimensions. However, these methods are not applicable for mass measurements.

A second important factor, determining accuracy of resistivity measurements, is electrical current distribution in the sample. Ideally, electrical current distribution in the sample must be uniform (Figure 15a). In this case, electrical current lines are parallel to axis of the sample and potential distribution on sample surface, where it can be measured, is the same as in the bulk. However, in most cases, point current contacts are used for measuring resistance and, in such case, current distribution is not uniform in the sample (Figure 15b).

As a result, potential distribution on surface of the sample may differ significantly from distribution in volume. To minimize this error, distance between the nearest current and potential probes must be (for highly conductive samples) more, than the maximum transverse dimension of the sample. With increasing resistance of the sample material, this distance must be also increased. Potential probes must be arranged along electrical current lines. If potential probes are arranged along line directed at angle ψ with respect to current lines, then effective length is l � ¼ l cosψ. For small angles ψ, error can be expressed as follows: Δl ¼ jl−l � j ¼ <sup>l</sup>ð1<sup>−</sup> cosψÞ≈<sup>l</sup> ·ψ<sup>2</sup> and <sup>Δ</sup><sup>l</sup> <sup>l</sup> <sup>¼</sup> <sup>ψ</sup><sup>2</sup> . The probe position error of 6° results in resistance error of 1%.

4. Error analysis

ment process.

analyzed here.

<sup>σ</sup> <sup>¼</sup> <sup>1</sup> <sup>R</sup> · <sup>l</sup>

equal to <sup>Δ</sup><sup>A</sup>

<sup>A</sup> <sup>þ</sup> <sup>Δ</sup><sup>l</sup>

4.1.1. Errors associated with shape

between potential probes <sup>R</sup> <sup>¼</sup> <sup>Δ</sup><sup>V</sup>

ped shape with typical dimensions 2 × 2 × 10 mm3

ever, these methods are not applicable for mass measurements.

such case, current distribution is not uniform in the sample (Figure 15b).

4.1. Electrical conductivity

368 Thermoelectrics for Power Generation - A Look at Trends in the Technology

Errors in measurements of electrical conductivity can be divided into three categories. First, it is electrical signal measurement errors, that is, potential difference and current magnitude. Second, it is errors associated with shape of the sample and of measuring electrodes. And third, there are errors associated with change in temperature of the sample during measure-

The first kind of errors is common to all measurements of electrical signals and are not specific for measuring electron transport properties. When modern measuring equipment is used and proper organization of measuring system and procedure are applied, then these errors generally are not a factor limiting the accuracy of measurements. Possible exceptions are measurements of electrical conductivity of high pure metals at very low temperatures. However, these cases are not typical for high-temperature measurements of thermoelectric materials, and not

Errors associated with sample's and electrode's shape are, perhaps, the main problem in most cases. When measuring conductivity, actual measured value is a total resistance of the sample

necessary to know cross-section of the sample (A) and distance between potential probes (l)

curacy in determining size and shape of the sample. Assume, that the sample has parallelepi-

and measurement of lengths, typical error of size determination is of the order 0.01 mm. This error includes distance measurement inaccuracy, and shape and surface imperfections of the sample as well. This error causes error of determining the section ΔA/A equals to 1%. The error of determining distance l, which includes both error in measurement of length and finite size of potential contact, is of the order 0.1 mm. Thus, total error in determining geometric factor is

method using bulk samples. Of course, accuracy can be improved by using a special highprecision technology for manufacturing of the sample and measuring its dimensions. How-

A second important factor, determining accuracy of resistivity measurements, is electrical current distribution in the sample. Ideally, electrical current distribution in the sample must be uniform (Figure 15a). In this case, electrical current lines are parallel to axis of the sample and potential distribution on sample surface, where it can be measured, is the same as in the bulk. However, in most cases, point current contacts are used for measuring resistance and, in

As a result, potential distribution on surface of the sample may differ significantly from distribution in volume. To minimize this error, distance between the nearest current and potential

A. There are four sources of errors associated with geometric factor. The easiest is inac-

<sup>l</sup> ¼ 0:02, that is, 2%. It is accuracy limit of measuring resistance by four-probe

<sup>I</sup> . In order to obtain electrical conductivity of the sample, it is

. In ordinary methods of sample machining

Figure 15. Errors associated with geometry of the sample and current leads. Dashed lines indicate electrical current flow; doted lines represent equipotential surfaces: a. ideal current lead contacts-homogeneous current distribution; b. point current lead contacts-nonhomogeneous current distribution.

Errors related to mechanical imperfection of samples are the most common and bring the greatest trouble. This may be pores, cracks, non-uniformity in composition, and so on. There is no general recipe to minimize such errors. Errors associated with the presence of pores can be reduced in part by corrections proportional to deviation of actual density of the sample from theoretical, calculated on the basis of structural data. It should be noted, that geometrical factor leads also to errors in determining temperature coefficient of electrical conductivity <sup>d</sup><sup>σ</sup> dT.

#### 4.1.2. Errors associated with changes in thermal regime of the sample during measurement

Two types of phenomena leading to such kind of errors can be distinguished: sample temperature changes due to Joule heating and changes of temperature distribution in the sample due to thermoelectric effects.

Since Joule heat released in the sample is equal to I 2 R, then measuring at lower current and improving heat transfer from sample to the environment can effectively solve the problem of temperature changes. More difficult task is to eliminate the influence of thermoelectric effects, namely, Peltier and Seebeck effects. This influence arises, when measurement of electrical conductivity is performed with direct current (DC), whereas in measurements of electrical conductivity with alternative current (AC), thermoelectric effects do not affect measurement accuracy.

Since the sample and connected to it electrical current leads represent nonuniform electrical circuit, Peltier heat will be released at one contact of current lead with the sample, while at another contact it will be absorbed. This will produce temperature difference across the sample. Figure 16 shows time diagram of potential difference across the sample during measurement of resistance with taking into account Peltier effect. We assume, that in the initial state, when electrical current is turned off, there is no temperature gradient in the sample, so the potential difference ΔV<sup>0</sup> ¼ 0. Due to finite sample heat capacity, immediately after electrical current is switch-on, temperature of the contact is not changed and measured voltage is equal to ΔV<sup>þ</sup> ¼ R · I <sup>þ</sup>. However, due to Peltier effect, heat flow from one contact to another creates in the sample temperature gradient. Therefore, additional potential difference arises, so that the total potential difference between probes is ΔV<sup>þ</sup> ¼ R · I <sup>þ</sup> þ α ·ΔTðtÞ, here α is relative thermoelectric power of the pair "sample-potential probe," ΔT is temperature difference between potential probes. This difference increases with time at rate depending on heat capacity of the system and rate of release and absorption of heat on the contacts due to Peltier effect.

Figure 16. Timing diagram of potential difference across the sample, when measuring electrical conductivity with DC current.

ΔT is stabilized at level, which is determined by balance between rate of heat generation at the contacts, thermal conductivity of the sample and conditions of heat exchange between sample and the environment. To estimate the maximum value of the effect, we assume, that there is no heat exchange between sample and the environment. When electrical current with density j flows, then the amount of Peltier heat qP, generated at contact between current lead and the sample, is equal to: qP ¼ Π · j, where Π is Peltier coefficient of the pair "sample-current lead". At stationary conditions and in the absence of heat exchange with the environment, whole heat flow passes through the sample due to thermal conductivity: q ¼ −κ∇T, here κ is a thermal conductivity of the sample. The flow balance qP ¼ −q determines temperature gradient: <sup>∇</sup><sup>T</sup> <sup>¼</sup> <sup>Π</sup> · <sup>I</sup> <sup>κ</sup> . The effect of this temperature gradient on conductivity measurement precision depends on ratio of voltage drop across the sample <sup>Δ</sup>V<sup>ρ</sup> <sup>¼</sup> <sup>I</sup> ·<sup>R</sup> <sup>¼</sup> <sup>j</sup> · <sup>A</sup> · <sup>ρ</sup> · <sup>l</sup> <sup>A</sup> ¼ j · l · ρ, which occurs when electrical current passes, to potential difference related to temperature gradient ΔVthermo ¼ α · l · ∇T:

Methods and Apparatus for Measuring Thermopower and Electrical Conductivity of Thermoelectric... http://dx.doi.org/10.5772/66290 371

$$\frac{\Delta V\_{\text{thermo}}}{\Delta V\_{\rho}} = \frac{\Pi \times j \times \alpha \times l}{\rho \times j \times \kappa \times l} = T \frac{\alpha^2 \sigma}{\kappa} \,. \tag{15}$$

In deriving the latter expression, Thompson relation (5) was used. As we can see, <sup>Δ</sup>Vthermo <sup>Δ</sup>V<sup>ρ</sup> is determined by dimensionless figure of merit ZT <sup>¼</sup> <sup>T</sup> <sup>α</sup>2<sup>σ</sup> <sup>κ</sup> . For good thermoelectric materials, this value can be of the order of unity. It is important, that error related to Peltier effect does not depend on direction or magnitude of electrical current or sample geometry. Therefore, it cannot be eliminated by changing these parameters of experiment. The error can be significantly reduced in two ways:


#### 4.1.3. Measurement errors in four-probe method

Since the sample and connected to it electrical current leads represent nonuniform electrical circuit, Peltier heat will be released at one contact of current lead with the sample, while at another contact it will be absorbed. This will produce temperature difference across the sample. Figure 16 shows time diagram of potential difference across the sample during measurement of resistance with taking into account Peltier effect. We assume, that in the initial state, when electrical current is turned off, there is no temperature gradient in the sample, so the potential difference ΔV<sup>0</sup> ¼ 0. Due to finite sample heat capacity, immediately after electrical current is switch-on, temperature of the contact is not changed and measured voltage is equal

the sample temperature gradient. Therefore, additional potential difference arises, so that the

moelectric power of the pair "sample-potential probe," ΔT is temperature difference between potential probes. This difference increases with time at rate depending on heat capacity of the

ΔT is stabilized at level, which is determined by balance between rate of heat generation at the contacts, thermal conductivity of the sample and conditions of heat exchange between sample and the environment. To estimate the maximum value of the effect, we assume, that there is no heat exchange between sample and the environment. When electrical current with density j flows, then the amount of Peltier heat qP, generated at contact between current lead and the sample, is equal to: qP ¼ Π · j, where Π is Peltier coefficient of the pair "sample-current lead". At stationary conditions and in the absence of heat exchange with the environment, whole heat flow passes through the sample due to thermal conductivity: q ¼ −κ∇T, here κ is a thermal conductivity of the sample. The flow balance qP ¼ −q determines temperature gradient:

Figure 16. Timing diagram of potential difference across the sample, when measuring electrical conductivity with DC

<sup>κ</sup> . The effect of this temperature gradient on conductivity measurement precision

occurs when electrical current passes, to potential difference related to temperature gradient

depends on ratio of voltage drop across the sample <sup>Δ</sup>V<sup>ρ</sup> <sup>¼</sup> <sup>I</sup> ·<sup>R</sup> <sup>¼</sup> <sup>j</sup> · <sup>A</sup> · <sup>ρ</sup> · <sup>l</sup>

system and rate of release and absorption of heat on the contacts due to Peltier effect.

total potential difference between probes is ΔV<sup>þ</sup> ¼ R · I

370 Thermoelectrics for Power Generation - A Look at Trends in the Technology

<sup>þ</sup>. However, due to Peltier effect, heat flow from one contact to another creates in

<sup>þ</sup> þ α ·ΔTðtÞ, here α is relative ther-

<sup>A</sup> ¼ j · l · ρ, which

to ΔV<sup>þ</sup> ¼ R · I

<sup>∇</sup><sup>T</sup> <sup>¼</sup> <sup>Π</sup> · <sup>I</sup>

current.

ΔVthermo ¼ α · l · ∇T:

When conditions of applicability of four-probe method are fulfilled, then errors of electrical conductivity measurements will be caused by inaccuracy of determining the distance between potential contacts and sample thickness. Distance between potential contacts is limited by conditions of method applicability, and it should be much less than linear dimensions of the sample. For typical sample having flat surface area 10×10 mm<sup>2</sup> , the distance between the contacts must be less than 1 mm. Typically, the diameter of contact area of potential probe is of order of 0.01 mm, therefore, the error in determining distance between contacts will be Δl/l ≥ 1%. The error in determining of average thickness of the sample is of the same order. Thus, accuracy of determining electrical conductivity with four-probe method will usually be at least 2%.

#### 4.1.4. Error estimation: van der Pauw method

Measurement errors in van der Pauw method are associated with non-ideal contacts, that is, with their finite size and offset from the edge of the sample. Estimation of errors has been done for three typical cases of non-ideal contacts and is shown in Figure 17 [14]. For simplicity, let us consider circle shape sample with diameter D, electrical contacts to which are arranged at equal distance from each other. Assume, that only one contact is imperfect. In practice, there are no ideal contacts. To the first approximation, the total error is the sum of errors on each contact. Advantage of van der Pauw method is applicability to samples of different (including irregular) forms, because in many cases test material is available in the form of small plates. Such samples do not require further processing and can be used for other purposes after van der Pauw measurement. However, in cases, where high measurement accuracy is required, the

samples of special form should be used [12]. They can be divided into two groups. The first group includes the samples having the shape of cloverleaf. Such form allows to increase the length of the border, so that imperfect contacts make negligible error in measurement results. The second group includes samples, having symmetrical shape and extended contacts, which respective correction functions have already been calculated for.

Figure 17. The relative errors Δσ/σ when measuring electrical conductivity of circle shape sample [14]: a. One of contacts has length l along the edge of the sample; b. One of contacts has length l perpendicular to the edge of the sample; c. One of contacts is point contact located at distance l from the sample edge.

#### 4.2. Measurement errors of thermopower

Measurement errors of thermopower by differential method are mainly related to incorrect determination of temperature difference ΔT. We can distinguish two sources of errors in determination of ΔT:

1. Temperature sensors and calibration are non-ideal. Thermocouples are almost exclusively used as temperature sensors at high-temperature thermopower measurements. To ensure precise determination of ΔT, thermocouple must satisfy very rigid requirements, such as the branches homogeneity and stability of their properties. Typical value of ΔT is about 10 K. At sample temperature of about 1000 K, just 0.1% difference in average thermopower of two thermocouples will lead to errors in determining ΔT of 10%. For example, average thermopower of one of the most commonly used thermocouple, consisting of platinum wire and wire of alloy Pt+10% Rh, is about 10 μV/K in temperature range 300–1000 K. Deviation of average thermopower of one thermocouple on another of the order of 0.01 μV/K will result in error in ΔT of 10%. Therefore, for measurements of thermopower, highquality thermocouple wires should be used only and their homogeneity should be monitored during operation.

samples of special form should be used [12]. They can be divided into two groups. The first group includes the samples having the shape of cloverleaf. Such form allows to increase the length of the border, so that imperfect contacts make negligible error in measurement results. The second group includes samples, having symmetrical shape and extended contacts, which

Measurement errors of thermopower by differential method are mainly related to incorrect determination of temperature difference ΔT. We can distinguish two sources of errors in

Figure 17. The relative errors Δσ/σ when measuring electrical conductivity of circle shape sample [14]: a. One of contacts has length l along the edge of the sample; b. One of contacts has length l perpendicular to the edge of the sample; c. One of

1. Temperature sensors and calibration are non-ideal. Thermocouples are almost exclusively used as temperature sensors at high-temperature thermopower measurements. To ensure precise determination of ΔT, thermocouple must satisfy very rigid requirements, such as the branches homogeneity and stability of their properties. Typical value of ΔT is about 10 K. At sample temperature of about 1000 K, just 0.1% difference in average thermopower of two thermocouples will lead to errors in determining ΔT of 10%. For example, average thermopower of one of the most commonly used thermocouple, consisting of platinum wire and wire of alloy Pt+10% Rh, is about 10 μV/K in temperature range 300–1000 K. Deviation of average thermopower of one thermocouple on another of the order of 0.01 μV/K will result in error in ΔT of 10%. Therefore, for measurements of thermopower, high-

respective correction functions have already been calculated for.

372 Thermoelectrics for Power Generation - A Look at Trends in the Technology

4.2. Measurement errors of thermopower

contacts is point contact located at distance l from the sample edge.

determination of ΔT:

2. The main source of errors in thermopower measurements associated with mismatch between the points, where ΔT and ΔV are measured (see, e.g., [28]). Junction of thermocouple used for measuring temperature at the point of electrical contact of reference electrode (which is usually one of used thermocouple branches) has finite dimensions. In real conditions of high temperature measurements, significant heat flow may occur along thermocouple branches. Combination of these factors leads to the fact, that average temperature of the junction and real temperature of electrical contact of reference electrode with the sample differ, that leads to error in determining of thermopower. For this type of errors, it is difficult to make general numerical estimate, because errors depend on several factors, which are difficult to control: size of thermocouple junction, cross-section and thermal conductivity of thermocouple branches, the value of thermal resistance at contact of thermocouple with the sample, temperature distribution in contact area. Error evaluation can be done by measuring thermopower of well-known materials, which have stable properties. Unfortunately, as we have already noted, so far, there is no standard for thermopower at high temperatures. Some metals can be used as reference samples. Due to the combination of the properties, platinum and nickel are the most suitable for high temperatures. It should be noted, that if platinum is used as a reference electrode for thermopower measurements, platinum sample is not suitable as a reference for evaluation of measurement error. In this case, as follows from Eq. (4) ΔV ¼ 0 (since α<sup>x</sup> ¼ αl). Thermopower <sup>α</sup><sup>x</sup> <sup>¼</sup> <sup>Δ</sup><sup>V</sup> <sup>Δ</sup><sup>T</sup> þ αl, determined in such measurements, will have correct value, regardless of the accuracy of determining ΔT.

Specifications analysis of set-ups for thermopower measurements and experience allow to state, that accuracy of determination of thermopower at high temperatures is limited by about ±5%. This estimate includes also uncertainty of modern absolute thermoelectric scale, which at high temperatures reaches ±0.5 μV/K. However, for thermoelectric materials, in which thermopower value is of the order of 100 μV/K or more, this uncertainty is not significant. Note also, that errors associated with inhomogeneity of thermocouple wires may be partially removed, when using alternating temperature difference [29–32]. At the same time, the second-type errors cannot be eliminated with alternating temperature difference and/or by use of differential thermocouple for measuring temperature difference, as it is sometimes assumed [30].

## 5. Devices for measuring thermopower and electrical conductivity

Devices realizing differential thermopower measurement technique can be divided into two classes: with variable (modulated) and static temperature difference. Measurements with variable temperature difference allow to eliminate or significantly reduce errors associated with inhomogeneity of branches of thermocouples, with slow instrumental drift or constant voltages caused by inhomogeneity of electrical circuits due to thermoelectric effects. This method has an advantage comparing to measurements with static temperature difference at low temperatures, when amplitude of ΔT is very small, because condition, which must be satisfied is ΔT << T. Therefore, there have been numerous variants of its implementation, designed for measuring thermopower at low temperatures [31–36]. At high temperatures, gradient modulation does not bring significant increase in accuracy, and implementation of this method is more difficult. Nevertheless, variable gradient method has been used at high temperatures as well [37–39].

Further, we describe in detail two experimental set-ups for measuring thermopower and electrical conductivity in temperature range from 80 to 2000 K [26, 40] and give brief overview of other devices for measurement of these properties.

## 5.1. Set-up for thermopower and electrical conductivity measurements at 80–1300 K

General view of measuring apparatus shown in photograph (Figure 18). Set-up was built to provide the fast and high quality electrical conductivity and Seebeck coefficient measurements using samples of any shape, including thin films. These objectives were fully achieved [26, 40].

Figure 18. Experimental set-up for thermopower and electrical conductivity measurements at 80–1300 K.

The set-up consists of four main parts:

1. Sample holder.

satisfied is ΔT << T. Therefore, there have been numerous variants of its implementation, designed for measuring thermopower at low temperatures [31–36]. At high temperatures, gradient modulation does not bring significant increase in accuracy, and implementation of this method is more difficult. Nevertheless, variable gradient method has been used at high

Further, we describe in detail two experimental set-ups for measuring thermopower and electrical conductivity in temperature range from 80 to 2000 K [26, 40] and give brief overview

General view of measuring apparatus shown in photograph (Figure 18). Set-up was built to provide the fast and high quality electrical conductivity and Seebeck coefficient measurements using samples of any shape, including thin films. These objectives were fully achieved

5.1. Set-up for thermopower and electrical conductivity measurements at 80–1300 K

Figure 18. Experimental set-up for thermopower and electrical conductivity measurements at 80–1300 K.

temperatures as well [37–39].

[26, 40].

of other devices for measurement of these properties.

374 Thermoelectrics for Power Generation - A Look at Trends in the Technology


Sample holder is located inside vacuum chamber, which can be pumped out using turbo molecular pump to residual pressure down to 10−<sup>4</sup> Pa. Typically, chamber is filled with helium gas to pressure slightly above atmospheric. Measurements can be performed in vacuum, but in this case, accuracy of measurement of thermoelectric power decreases. Moreover, it must be borne in mind, that metallization of isolators may occur at high temperatures due to vaporization of metals.

General view of sample holder is shown in Figure 19 [26]. The basis of the holder is two coaxial tubes made of high-temperature steel, which are mounted on vacuum flange (19). Inner tube (16) is mounted on top of the flange (19). Gradient heater (11), supporting plate (8) and heat sink (4) mounted on other end of inner tube. Outer tube (15) is centered relative to inner tube with steel disks (14), which are mounted on inner tube at distance of 50 mm from each other. This system of two coaxial tubes is rigid and stable, which is especially important at high temperatures. All current and thermocouple wires are arranged in the space between inner and outer tubes and, therefore, they are well protected from mechanical damage and contamination. Outer tube can be easily removed, allowing access to the wires in case of repair. Sample supporting plate (8) is located between gradient heater (11) and heat sink (4) made of molybdenum. Selection of molybdenum as material for heater and heat sink is motivated by its high thermal conductivity and mechanical stability at high temperatures. The sample (5) is pressed against supporting plate (8) by press arm (10), pressure plate (9) and steel spring (13). These parts are made of special high-temperature steel. Cold junctions of thermocouples are made in the form of copper block (17), inside of which is made connection of thermocouple branches with copper wires, connecting thermocouple with the measuring equipment.

In this case, two conditions should be fulfilled:


Connection of sample holder with measuring equipment is carried out by means of connector made of conductors with low thermopower relative to thermopower of copper. Temperature of reference point (17) is measured by thermistor (18).

Selection of thermocouple is mainly determined by temperature measurement interval. For temperature range from 80 to 600 K, the best choice is thermocouple copper-constantan, it has good sensitivity, stable enough, thermopower of copper is well-known and it is rather low. For temperatures from 300 K to ≈1600 K Pt-Pt/Rh thermocouples are the best choice, where the second branch is alloy of platinum and rhodium. Usually, as the second branch of these thermocouples, alloys of platinum with 10 and 13% rhodium are used. Thermopower of platinum, which is normally used here as reference electrode in measurements of thermopower, is also well known.

Figure 19. General view of sample holder (a), and sample supporting plate (b). The dimensions are given in millimeter.

Figures 19b and 20 show details of mounting and pressing mechanism of thermocouples on the supporting plate. The basis for mounting thermocouples (6) and current contacts (7) are two-channel tube (21) made of Al2O3 of 1 mm diameter. Tubes are pressed against the sample (5) using small springs (22) made of iridium wire. The springs are welded to the supporting plate (8). Such system provides reliable contact of thermocouples and current contacts with the sample within the whole operating temperature range. The choice of material for the springs (22) is important for providing reliable and stable contacts. The most important condition is to maintain elasticity of the material up to about 1300 K, as well as, mechanical and chemical stability. Iridium satisfies in full these requirements. Other good materials are tungsten-rhenium alloys; however, they cannot be used in oxidizing atmosphere. For electrical isolation of the sample from supporting (8) and pressing (9) plate, thin mica sheets (20) are used. Gradient heater (12) (Figure 19a) is used to regulate temperature gradient in the sample. Temperature gradient is mainly formed by slightly asymmetric sample's position relative to the center of the heater (Figure 18). Typical value of temperature gradient between measuring thermocouples is in the range from 5 to 20 K (depends on temperature).

good sensitivity, stable enough, thermopower of copper is well-known and it is rather low. For temperatures from 300 K to ≈1600 K Pt-Pt/Rh thermocouples are the best choice, where the second branch is alloy of platinum and rhodium. Usually, as the second branch of these thermocouples, alloys of platinum with 10 and 13% rhodium are used. Thermopower of platinum, which is normally used here as reference electrode in measurements of

Figures 19b and 20 show details of mounting and pressing mechanism of thermocouples on the supporting plate. The basis for mounting thermocouples (6) and current contacts (7) are two-channel tube (21) made of Al2O3 of 1 mm diameter. Tubes are pressed against the sample (5) using small springs (22) made of iridium wire. The springs are welded to the supporting plate (8). Such system provides reliable contact of thermocouples and current contacts with the sample within the whole operating temperature range. The choice of

Figure 19. General view of sample holder (a), and sample supporting plate (b). The dimensions are given in millimeter.

thermopower, is also well known.

376 Thermoelectrics for Power Generation - A Look at Trends in the Technology

Figure 20. Detailed view of mechanical contact mechanism: (a) side view of the sample supporting plate with ceramic tubes (21) and springs (22); distance between tubes is not in scale with their diameter; (b) cross-sectional view of the sample supporting plate.

Five electrodes are used in the sample holder: 3 – for thermocouple contacts and 2 – for current contacts. The distance between Th1 and Th2 equals to 3 mm, between Th1 and Th3 – 10 mm (Figure 19). Such configuration allows measuring properties of the samples of various sizes with optimal accuracy. The sample holder allows measurements with both bulk samples and thin films as well.

#### 5.1.1. Measurement procedure

Standard four-probe DC current method is used for measurements of electrical conductivity. Differential method with constant temperature gradient is utilized for thermopower measurement. Performing reliable measurements of thermopower requires accurate temperature measuring and availability of precise and detailed information about thermoelectric power of reference electrodes depending on temperature. As shown above, thermopower in differential measurement method is given by Eq. (12). To determine ΔT, precise calibration data for thermocouples must be used: T ¼ FðVÞ. For standard thermocouples, calibration dependences are usually presented in the form of tables or dependencies of VðTÞ. If measurement is automated, it is more convenient to have calibration dependence in the form of analytic functions. In this case, it is important to choose the most natural analytic representation. In rough approximation, metal thermopower is linear function of temperature (generally, this is incorrect statement, but for metals and alloys used in thermocouples it is true), and then thermopower of thermocouple can be

$$\begin{array}{cccc} \text{expressed} & \text{as} & \text{follows:} & V(T) = \int\_{T\_0}^{T} a\_{12}(T)dT \propto \int\_{T\_0}^{T} kTdT = \frac{1}{2}k(T^2 - T\_0^2) & \text{and,} & \text{hence,} \\ \end{array}$$

T∝ <sup>2</sup> k � �<sup>1</sup>=<sup>2</sup> · ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>V</sup> <sup>þ</sup> <sup>k</sup> <sup>2</sup> <sup>T</sup><sup>2</sup> 0 q . Therefore, we represent thermocouple calibration dependence in the form of: T ¼ ∑ n i¼0 bi<sup>ð</sup> ffiffiffiffiffiffiffiffiffiffiffiffi <sup>V</sup> <sup>þ</sup> <sup>a</sup> <sup>p</sup> <sup>Þ</sup><sup>i</sup> .

Coefficients b<sup>i</sup> of interpolating polynomials for four standard thermocouples are shown in Table 2.


Table 2. Coefficients of interpolating polynomials for thermocouples [26].

These polynomials can be used in the following temperature ranges:

Pt-Pt+13% Rh from 273 to 1873 K;

with optimal accuracy. The sample holder allows measurements with both bulk samples and

Standard four-probe DC current method is used for measurements of electrical conductivity. Differential method with constant temperature gradient is utilized for thermopower measurement. Performing reliable measurements of thermopower requires accurate temperature measuring and availability of precise and detailed information about thermoelectric power of reference electrodes depending on temperature. As shown above, thermopower in differential measurement method is given by Eq. (12). To determine ΔT, precise calibration data for thermocouples must be used: T ¼ FðVÞ. For standard thermocouples, calibration dependences are usually presented in the form of tables or dependencies of VðTÞ. If measurement is automated, it is more convenient to have calibration dependence in the form of analytic functions. In this case, it is important to choose the most natural analytic representation. In rough approximation, metal thermopower is linear function of temperature (generally, this is incorrect statement, but for metals and alloys used in thermocouples it is true), and then thermopower of thermocouple can be

> ð T

α12ðTÞdT∝

Coefficients b<sup>i</sup> of interpolating polynomials for four standard thermocouples are shown in

ð T

kTdT <sup>¼</sup> <sup>1</sup>

<sup>2</sup> <sup>k</sup>ðT<sup>2</sup> −T<sup>2</sup>

<sup>0</sup>Þ and, hence,

T0

Pt–Pt+13% Rh Pt–Pt+10% Rh Chromel-alumel Copper-constantan

. Therefore, we represent thermocouple calibration dependence in the

T0

a 0.1676 0.2045 6.4 6.1 b<sup>0</sup> 237.54 230.43 28.5 34.4 b<sup>1</sup> 000 0 b<sup>2</sup> 273.7 269.884 135.5 116.9675 b<sup>3</sup> −179.79 −162.308 −90.6 −73 b<sup>4</sup> 85.454 65.836 33.363 29.5 b<sup>5</sup> −22.35955 −9.851 −6.6509 −7.02 b<sup>6</sup> 2.9002 −1.207 0.7315 1.488 b<sup>7</sup> −0.13503 0.524 −0.04125 −0.9712 b<sup>8</sup> −0.0015 −0.04238 −0.00092 0.0347

thin films as well.

T∝ <sup>2</sup> k � �<sup>1</sup>=<sup>2</sup> ·

Table 2.

5.1.1. Measurement procedure

expressed as follows: VðTÞ ¼

bi<sup>ð</sup> ffiffiffiffiffiffiffiffiffiffiffiffi <sup>V</sup> <sup>þ</sup> <sup>a</sup> <sup>p</sup> <sup>Þ</sup><sup>i</sup>

.

378 Thermoelectrics for Power Generation - A Look at Trends in the Technology

Table 2. Coefficients of interpolating polynomials for thermocouples [26].

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>V</sup> <sup>þ</sup> <sup>k</sup> <sup>2</sup> <sup>T</sup><sup>2</sup> 0

n i¼0

q

form of: T ¼ ∑

Pt-Pt+10% Rh from 233 to 1883 K;

chromel-alumel from 43 to 1543 K;

copper-constantan from 53 to 673 K.

Polynomial coefficients were obtained by fitting polynomials to calibration tables recommended by International Electrotechnical Commission for standard thermocouples. Deviation from calibration tables in specified temperature ranges does not exceed 0.1 K for copper-constantan and Pt-Pt+13% Rh thermocouples; 0.15 K for thermocouple Pt-Pt+10% Rh; and 1.5 K for chromel-alumel thermocouple. Additional error in determining temperature difference across the sample due to these deviations is within ±1% for copper-constantan thermocouple and both platinum thermocouples, and ±3% for chromel-alumel thermocouple.

Measurements of both properties are performed simultaneously. When measuring temperature dependence of the parameters, it is not required to establish steady temperature at each point. Measurements are performed with a continuous change in temperature at rate up to 10 K/min.

#### 5.2. Set-up for measuring thermopower and electrical conductivity at 300–2000 K

Thermopower measurements at very high temperatures, particularly above 1500 K, are rather difficult due to several factors:


Described apparatus allows measurements of thermopower and resistance of bulk samples of conductors at temperatures from 300 K to temperature slightly above 2000 K with good accuracy. This system allows to work with samples of various shapes and sizes. Perhaps, this is the most high-temperature experimental device for direct measurement of thermopower described in the literature. The exception is the device used Lander [22] for measurement of Thompson coefficient of some metals up to 2400 K.

The main original part of this set-up is sample holder [41]; scheme of this holder is shown in Figure 21. The basis of holder is molybdenum tube (2), in the lower part of which is fixed molybdenum massive heat sink (1), where replaceable molybdenum bottom sample support (4) is installed. At the top of the tube (working position of the holder is vertical), molybdenum pusher (8) is located, which is isolated from tube by ceramic (Al2O3) rings (7). The lower ring is held by molybdenum stop (6), which also protects ceramic ring from metallization by metal vapors from the hot zone at the bottom. The sample (3) is clamped between the upper (5) and lower (4) supports under weight of gravity transmitted through molybdenum (8) and stainless steel (9) pushers. The holder is mounted in vacuum chamber with heater through ceramic insulating tube (10).

Figure 21. Sample holder for measuring thermopower and resistance at 300–2000 K: (1) – heat sink; (2) – outer molybdenum tube; (3) – sample; (4) – bottom sample support; (5) – upper sample support; (6) – molybdenum support for insulation; (7) – insulating ring made of Al2O3; (8) – molybdenum pusher; (9) – stainless steel pusher; (10) – ceramic insulation.

To measure temperature and thermopower in this set-up, thermocouples of tungsten-rhenium alloys are used: WR10-WR20. These are alloys of W + 10% Re and W + 20% Re, respectively. WR20 alloy is used as the reference electrode in measuring thermopower. For thermocouple WR10-WR20, there is standard calibration, however, the absolute thermopower of the branches is not known. The absolute thermoelectric power of WR20 alloy was determined by measuring thermopower of reference metal samples. As standards were used: platinum in temperature range 300–1700 K, and molybdenum at 1700–2100 K. Thermopower of highpurity molybdenum sample was beforehand accurately measured in temperature range from 80 to 1600 K relative to copper and platinum. Cusack and Kendall data [21] were used at higher temperatures. However, in order to provide a smooth joining of low-temperature data with Cusack's data, it must be entered temperature-independent correction of 2 μV/K in these data. A possible reason for this difference is insufficient purity of metal, which was used by Lander [22] in measurement of Thomson coefficient of molybdenum. Thermopower of molybdenum and WR20 alloy are shown in Figure 22. At temperatures from 100 to 2000 K, thermopower of WR20 can be calculated using interpolation polynomial:

$$\alpha\_{WR20} = 1.6337 \times 10^{-12} \times T^4 - 1.2669 \times 10^{-8} \times T^3 + 2.6192 \times 10^{-5} \times T^2 - 1.6889 \times 10^{-2} \times T + 3.111. \tag{16}$$

Figure 22. Thermopower of molybdenum and WR20 alloy: ● – molybdenum thermopower according Cusack [21]; ■ – adjusted molybdenum thermopower; ▲ – thermopower of WR20 alloy.

#### 5.3. Other techniques

molybdenum massive heat sink (1), where replaceable molybdenum bottom sample support (4) is installed. At the top of the tube (working position of the holder is vertical), molybdenum pusher (8) is located, which is isolated from tube by ceramic (Al2O3) rings (7). The lower ring is held by molybdenum stop (6), which also protects ceramic ring from metallization by metal vapors from the hot zone at the bottom. The sample (3) is clamped between the upper (5) and lower (4) supports under weight of gravity transmitted through molybdenum (8) and stainless steel (9) pushers. The holder is mounted in vacuum chamber with heater through ceramic

380 Thermoelectrics for Power Generation - A Look at Trends in the Technology

To measure temperature and thermopower in this set-up, thermocouples of tungsten-rhenium alloys are used: WR10-WR20. These are alloys of W + 10% Re and W + 20% Re, respectively. WR20 alloy is used as the reference electrode in measuring thermopower. For thermocouple WR10-WR20, there is standard calibration, however, the absolute thermopower of the branches is not known. The absolute thermoelectric power of WR20 alloy was determined by measuring thermopower of reference metal samples. As standards were used: platinum in temperature range 300–1700 K, and molybdenum at 1700–2100 K. Thermopower of highpurity molybdenum sample was beforehand accurately measured in temperature range from 80 to 1600 K relative to copper and platinum. Cusack and Kendall data [21] were used at

Figure 21. Sample holder for measuring thermopower and resistance at 300–2000 K: (1) – heat sink; (2) – outer molybdenum tube; (3) – sample; (4) – bottom sample support; (5) – upper sample support; (6) – molybdenum support for insulation; (7) – insulating ring made of Al2O3; (8) – molybdenum pusher; (9) – stainless steel pusher; (10) – ceramic

insulating tube (10).

insulation.

Petrov [42] built set-up for simultaneous measurement of thermopower, thermal conductivity and electrical conductivity of thermoelectric materials (i.e., materials with very low thermal conductivity), at temperatures from 100 to 1300 K, which operates successfully (in upgraded form) up to nowadays. In this device, method of electrical conductivity measurement with DC current, differential method of thermopower measurement and classic steady-state method of thermal conductivity measurement are used. The measurements at each value of temperature must be carried out in stationary temperature conditions. Since, achievement of thermal equilibrium, especially at low temperatures, is slow, measurements over whole temperature range takes several days. To suppress the heat loss by radiation, active heat shield and special ceramic filling with very low and known thermal conductivity are used. This system allows to determine parameter ZT as a result of simultaneous measurement of α, σ, and κ with accuracy of ±5%.

In contrast to electrical conductivity measurements, thermopower measurement is difficult to automate using analog methods. Therefore, before the advent of personal computers, these measurements were very time-consuming. There are several original analog automated devices for measuring thermoelectric power [30, 33]; however, they were not widely used.

Interesting device for measuring thermopower at high temperatures has been developed by Wood et al. [37]. This device uses a differential method for measuring thermoelectric power with modulation of temperature difference over the sample. Interchangeable heating the ends of the sample by light flash lamps was used for the modulation of temperature difference. Light beam energy was applied to the sample by means of sapphire optical fibers, between which the sample was clamped. The device allowed to measure thermopower up to 1900 K, with amplitude of temperature difference modulation of a few degrees. Author estimates measurement error of thermopower as ±1%, but does not specify experimental evidence of stated accuracy.

In apparatus for measuring thermal conductivity and thermoelectric power at temperatures 300–750 K, described in Ref. [43], stationary method of measuring thermal conductivity and differential method of thermopower measurement are used. Measurement of thermal conductivity is based on the comparison between temperature difference of heat source and heat sink in the presence of the sample and without the sample. At each temperature, after thermal stabilization, measurements of ΔT with the sample in contact with heat source and heat sink are performed. Then, heat source is disconnected from the sample and ΔT is measured again. Assumed, that heat losses in the system are the same in both states, and losses due to radiation from the sample are not considered. This put in question the correctness of the measurement. Thermopower is measured by differential method with constant temperature gradient.

In set-up for measuring electrical conductivity and thermoelectric power at 300–1300 K [44], electrical conductivity is measured with AC current at frequency 16 Hz, and for measuring thermopower, differential method with constant temperature difference is used. Thermocouples, which are used for the measurement of temperature gradient and thermopower, are mounted in holes drilled in the sample by using graphite paste. After installation of the sample, paste must be heat treated to ensure proper contact. This, as well as, current leads design, which cannot provide stable electrical contact, is a serious disadvantage of the system. Extremely small thermopower measurement error 0.3%, stated by authors, has not been experimentally confirmed.

AC electrical conductivity measurement procedure and differential method with temperature gradient modulation for measuring thermopower are utilized in set-up for thermopower and electrical conductivity measurements at 300–1273 K [38]. The publication, however, contained only measurement principles, which are not original. No details of measuring device were presented.

Interesting sample holder design for measuring thermoelectric power at temperatures up to 1200 K was suggested in Ref. [45]. This is further development of Wood's system [37], but with significant changes. Distinctive feature of the design is axial location of thermocouples. Thermocouples, supported by four-channel thin tubes, extend along the central axis of gradient heaters, between which is clamped the sample. Working junctions of thermocouples are pressed against the ends of the sample by springs. Therefore, sample does not require special preparation for measurement. Thermopower is measured by differential method with temperature gradient modulation; amplitude of modulation is up to 20 K. The article provides fairly detailed analysis of measurement errors of thermopower.

A feature of the holder for measuring thermopower and electrical conductivity proposed in Ref. [46] is the material: the main parts of this device are made of ceramics (Al2O3). Therefore, this device can be used for high temperature (1200 K) measurements in oxidizing atmosphere in the case of using platinum thermocouples. Thermoelectric power is measured by differential method with variable temperature gradient.

Relatively detailed overview of methods and devices for measurement of thermopower and electrical conductivity was published by Martin et al. [47].

Apart from temperature, pressure and magnetic field are accessible experimental parameters affecting the material properties. Dependences of electrical conductivity and thermoelectric power on magnetic field and pressure provide important information about electronic structure and conductivity mechanisms. Generally, studies of thermoelectric and conductivity dependencies on pressure and magnetic field are carried out at low temperatures. However, for thermoelectric materials, dependence of their properties on pressure and magnetic field at high temperatures is of considerable interest. Therefore, considerable effort has been directed toward the study of these dependences and development of devices for such measurements [48–52].

## 6. Conclusion

measurements were very time-consuming. There are several original analog automated devices for measuring thermoelectric power [30, 33]; however, they were not widely used.

382 Thermoelectrics for Power Generation - A Look at Trends in the Technology

Interesting device for measuring thermopower at high temperatures has been developed by Wood et al. [37]. This device uses a differential method for measuring thermoelectric power with modulation of temperature difference over the sample. Interchangeable heating the ends of the sample by light flash lamps was used for the modulation of temperature difference. Light beam energy was applied to the sample by means of sapphire optical fibers, between which the sample was clamped. The device allowed to measure thermopower up to 1900 K, with amplitude of temperature difference modulation of a few degrees. Author estimates measurement error of thermopower as ±1%, but does not specify experimental evidence of

In apparatus for measuring thermal conductivity and thermoelectric power at temperatures 300–750 K, described in Ref. [43], stationary method of measuring thermal conductivity and differential method of thermopower measurement are used. Measurement of thermal conductivity is based on the comparison between temperature difference of heat source and heat sink in the presence of the sample and without the sample. At each temperature, after thermal stabilization, measurements of ΔT with the sample in contact with heat source and heat sink are performed. Then, heat source is disconnected from the sample and ΔT is measured again. Assumed, that heat losses in the system are the same in both states, and losses due to radiation from the sample are not considered. This put in question the correctness of the measurement. Thermopower is measured by differential method with constant

In set-up for measuring electrical conductivity and thermoelectric power at 300–1300 K [44], electrical conductivity is measured with AC current at frequency 16 Hz, and for measuring thermopower, differential method with constant temperature difference is used. Thermocouples, which are used for the measurement of temperature gradient and thermopower, are mounted in holes drilled in the sample by using graphite paste. After installation of the sample, paste must be heat treated to ensure proper contact. This, as well as, current leads design, which cannot provide stable electrical contact, is a serious disadvantage of the system. Extremely small thermopower measurement error 0.3%, stated by authors, has not been

AC electrical conductivity measurement procedure and differential method with temperature gradient modulation for measuring thermopower are utilized in set-up for thermopower and electrical conductivity measurements at 300–1273 K [38]. The publication, however, contained only measurement principles, which are not original. No details of measuring device were

Interesting sample holder design for measuring thermoelectric power at temperatures up to 1200 K was suggested in Ref. [45]. This is further development of Wood's system [37], but with significant changes. Distinctive feature of the design is axial location of thermocouples. Thermocouples, supported by four-channel thin tubes, extend along the central axis of gradient heaters, between which is clamped the sample. Working junctions of

stated accuracy.

temperature gradient.

experimentally confirmed.

presented.

Research and successful development of novel effective materials for thermoelectric energy converters is critically dependent on obtaining accurate and reliable information about properties of these materials. The most important characteristics of thermoelectric materials are thermopower and electrical conductivity. They determine potential effectiveness of thermoelectric material and provide important information on its electronic structure. Measurements of these properties must meet a number of requirements. Measurement results must be reliable and sufficiently accurate. Measurements must be performed over a wide range of temperatures comparable with a typical range of applications. In experimental research for new thermoelectric materials, the versatility of measurement set-ups is especially important. They should make affordable measurements of samples of different shapes and dimensions in a wide range of temperatures. Despite relative simplicity of fundamental methods of measuring thermoelectric properties of materials, their practical implementation is a difficult task. Additional difficulty is the lack of commonly accepted reference materials for measuring thermopower at high temperatures, making it difficult to compare the results obtained by independent groups. In such circumstances, it is crucial to understand clearly possibilities and limitations of different methods for measuring thermoelectric properties and unconditional implementation of some basic requirements by researchers. When measuring

thermopower, the most important points are: (1) thermoelectric signal and temperature difference must be measured between the same points of the sample; (2) potential contacts and temperature sensors must be in good thermal and electrical contact with the sample; (3) when using thermocouples, special attention must be given to thermoelectric homogeneity of their branches.

## Author details

Alexander T. Burkov\*, Andrey I. Fedotov and Sergey V. Novikov

\*Address all correspondence to: a.burkov@mail.ioffe.ru

Ioffe Institute, Saint Petersburg, Russian Federation, Russia

#### References


[11] Bowler N.: Four-point potential drop measurements for materials characterization. Measurement Science and Technology. 2011;22:012001-1-11. DOI:10.1088/0957-0233/22/1/ 012001.

thermopower, the most important points are: (1) thermoelectric signal and temperature difference must be measured between the same points of the sample; (2) potential contacts and temperature sensors must be in good thermal and electrical contact with the sample; (3) when using thermocouples, special attention must be given to thermoelectric homogeneity of their

[1] Stilbans LS. Physics of Semiconductors. Moscow: Sovetskoe Radio; 1967. 451 p. (in Rus-

[2] Anatychuk LI. On the discovery of thermoelectricity by Volta. Journal of Thermoelectric-

[3] Ioffe AF, Stilbans LS, Iordanishvili EK, Stavitskaya TS. Thermoelectric Cooling. Moscow,

[5] Ioffe AF. Semiconductor Thermoelements and Thermoelectric Cooling. London:

[6] Manasian YuG. Sudovye termoelektricheskie ustroystva i ustanocki. Leningrad:

[7] Nye JF. Physical Properties of Crystals: Their Representation by Tensors and Matrices.

[8] Landau LD, Lifshitz EM. Course of Theoretical Physics, v. 10: Pitaevskii LP, Lifshitz EM,

[9] Barnard RD. Thermoelectricity in Metals and Alloys. London: Taylor & Francis; 1972. 259 p. [10] Bowler N. Theory of Four-Point Direct-Current Potential Drop Measurements on a Metal Plate. Research in Nondestructive Evaluation. 2006;17:29–48. DOI: 10.1080/

Leningrad: USSR Academy of Sciences Publishing; 1956. 114 p. (in Russian).

[4] Ioffe AF. Physics of Semiconductors. London: Infosearch; 1960. 436 p.

Alexander T. Burkov\*, Andrey I. Fedotov and Sergey V. Novikov

\*Address all correspondence to: a.burkov@mail.ioffe.ru

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Ioffe Institute, Saint Petersburg, Russian Federation, Russia

branches.

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[40] Burkov AT. Measurements of resistivity and thermopower: principles and practical realization. In: M. Rowe, editor. Thermoelectrics Handbook: Macro to Nano. Boca Raton: CRC Press; 2006. p. 22-1-12.

[27] Moore JP, Graves RS. Absolute Seebeck coefficient of platinum from 80 to 340 K and the thermal and electrical conductivities of lead from 80 to 400 K. Journal of Applied Physics.

[28] Horne RA. Errors associated with thermoelectric power measurements using small temperature differences. Review of Scientific Instruments. 1960;31:459–460. DOI: 10.1063/

[29] Testardi LR, McConnell GK.: Measurement of the seebeck coefficient with small temperature differences. Review of Scientific Instruments. 1961;32:1067–1068. DOI: 10.1063/

[30] Berglund CN, Beairsto RC. An automatic technique for accurate measurements of Seebeck coefficient. Review of Scientific Instruments. 1967;38:66–68. DOI: 10.1063/

[31] Aubin M, Ghamlouch H, Fournier P. Measurement of the Seebeck coefficient by an ac technique: application to high-temperature superconductors. Review of Scientific Instru-

[32] Resel R, Gratz E, Burkov AT. et al. Thermopower measurements in magnetic fields up to 17 tesla using the toggled heating method. Review of Scientific Instruments.

[33] Caskey GR, Sellmver DJ, Rubin LG. A technique for the rapid measurement of thermoelectric power. Review of Scientific Instruments. 1969;40:1280–1282. DOI: 10.1063/

[34] Chaikin PM, Kwak JF. Apparatus for thermopower measurements on organic conductors. Review of Scientific Instruments. 1975;46:218–220. DOI: 10.1063/1.1134171.

[35] Putti M, Cimberle MR, Canesi A, Foglia C, Siri AS. Thermopower measurements of hightemperature superconductors: experimental artifacts due to applied thermal gradient and

[36] Chen F, Cooley JC, Hults WL, Smith JL. Low-frequency ac measurement of the Seebeck coefficient. Review of Scientific Instruments. 2001;72:4201–4206. DOI: 10.1063/1.1406930.

[37] Wood C, Zoltan D, Stapfer G.: Measurement of Seebeck coefficient using a light pulse.

[38] D'Angelo J, Downey A, Hogan T. Temperature dependent thermoelectric material power factor measurement system. Review of Scientific Instruments. 2010;81:075107-1-4. DOI:

[39] Ravichandran J, Kardel JT, Scullin ML, Bahk J-H, Heijmerikx H, Bowers JE, Majumdar A. An apparatus for simultaneous measurement of electrical conductivity and thermopower of thin films in the temperature range of 300–750 K. Review of Scientific Instruments.

a technique for avoiding them. Physical Review B. 1998; 58:12344–12349.

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ments. 1993;64:2938–2941. DOI: 10.1063/1.1144387.

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#### **Novel Measurement Methods for Thermoelectric Power Generator Materials and Devices** Novel Measurement Methods for Thermoelectric Power Generator Materials and Devices

Patrick J. Taylor, Adam Wilson, Jay R. Maddux, Theodorian Borca-Tasciuc, Samuel P. Moran, Eduardo Castillo and Diana Borca-Tasciuc Patrick J. Taylor, Adam Wilson, Jay R. Maddux, Theodorian Borca-Tasciuc, Samuel P. Moran, Eduardo Castillo and Diana Borca-Tasciuc

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/65443

#### Abstract

Thermoelectric measurements are notoriously challenging. In this work, we outline new thermoelectric characterization methods that are experimentally more straightforward and provide much higher accuracy, reducing error by at least a factor of 2. Specifically, three novel measurement methodologies for thermal conductivity are detailed: steadystate isothermal measurements, scanning hot probe, and lock-in transient Harman technique. These three new measurement methodologies are validated using experimental measurement results from standards, as well as candidate materials for thermoelectric power generation. We review thermal conductivity measurement results from new half-Heusler (ZrNiSn-based) materials, as well as commercial (Bi,Sb)2(Te,Se)3 and mature PbTe samples. For devices, we show characterization of commercial (Bi,Sb)2(Te,Se)3 modules, precommercial PbTe/TAGS modules, and new high accuracy numerical device simulation of Skutterudite devices. Measurements are validated by comparison to wellestablished standard reference materials, as well as evaluation of device performance, and comparison to theoretical prediction obtained using measurements of individual properties. The new measurement methodologies presented here provide a new, compelling, simple, and more accurate means of material characterization, providing better agreement with theory.

Keywords: thermal conductivity, Seebeck coefficient, electrical resistivity, ZT, device efficiency

## 1. Introduction

The efficiency with which a thermoelectric (TE) power generator can convert heat energy to electricity is determined, in part, by thermal conductivity, κ, of the materials used for fabricating TE devices. Experimental measurement of that property usually results in surprisingly

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons

distribution, and eproduction in any medium, provided the original work is properly cited.

significant error, > 10% [1]. There are many causes of that error, and in this chapter, we provide new solutions which are experimentally faster, yield results which are more consistent with physical devices, and address several sources of experimental error, which may reduce uncertainty by a factor of 2 or more.

In this work, we describe several new, more accurate techniques to measure thermal conductivity, κ, of thermoelectric materials. The first is based on detailed control of the heat flows within a sample under steady-state conditions; the so-called steady-state isothermal technique. The second is nondestructive microscale analysis technique called scanning hot-probe (or scanning thermal microscopy). And the third is lock-in transient Harman method, which is a comprehensive modification of transient Harman technique, employing a lock-in procedure and considers detailed contact effects. A new interesting follow-on is frequency-dependent Nyquist analysis, which presages a different perspective on the material analysis.

The truest test of the accuracy of measurement is comparison with fabricated devices. To support the validation of measurements of individual material properties, we outline a new device metrics, which allows comparison between theoretical and measured device efficiency. We outline a new slope-efficiency method, which can be used to determine informative index ZTmaximum of any device. The second method of device evaluation is a numerical device model called the discretized heat balance model, which considers a piecewise continuous collection of discrete layers within a device, where boundary heat flows have energy and current continuity relationships, and enable incredibly easy determination of device efficiency.

## 2. Novel measurements of thermal conductivity

#### 2.1. Steady-state isothermal technique

This new measurement of κ leverages Peltier heat, QΠ, an electronically controlled internal heat source unique to thermoelectric materials. Peltier heat causes either heating (+QΠ) or cooling (−QΠ) at the junction between a thermoelectric material and a metal by passing the proper polarity of electric current, I.

Q<sup>Π</sup> was first employed to roughly estimate κ by Putley [2]. In Putley's experiment, convective, parasitic, and nonsymmetric heat flows required correction factors larger than 20%. Harman dramatically improved upon Putley's demonstration by performing the measurements in vacuum, and by reducing other parasitic heat flows. Despite these improvements, error is still obtained, because parasitic heat flows are nonzero [3].

In these past studies, principal parasitic heat flows causing error include conduction along lead wires, conduction along thermocouples, Joule heating within lead wires, and radiation. The magnitudes of these parasitic heat flows can be as large as 30% of Peltier heat. Penn quantified the significance of parasitic heat flows in Harman's technique, and showed, that they induced error of more than 10% [4]. Bowley and Goldsmid [5], as well as Buist [6] reported, that parasitic heat flows cause error, usually larger than 20%.

The focus of the present work is description of a new, correctionless method to measure κ by balancing two independently controlled heat sources: QΠ, and a radiatively coupled input heat as per the Stefan-Boltzmann law, QSB. In this new method, Q<sup>Π</sup> and QSB can be independently balanced. A finite temperature difference across the sample imposed by QSB can be cancelled and even inverted by application of QΠ. When exactly cancelled, there is no temperature difference across the sample (i.e., ΔT = 0) and a steady-state isothermal condition is obtained leaving the steady-state temperature of the sample exactly equal to that of the surroundings, Te. Because there is no ΔT, parasitic heat flows, such as those along lead-wires/thermocouples (Qwires) and radiative heat loss, (Qradiation-error), which would otherwise cause significant error [7], converge exactly to zero. Analysis of this technique using a Peltier cooler demonstrates error of less than �1%, an improvement of over an order of magnitude [8]. When considering thermoelectric power generators, other considerations must be taken into account to determine experimental uncertainty (such as view angle for radiative heat flows from the environment, material emissivity, etc.), which are more complicated and beyond the scope of the work presented here. However, as is demonstrated qualitatively here, experimental uncertainty is reduced using the steady-state isothermal technique by significantly more than a factor of 2.

significant error, > 10% [1]. There are many causes of that error, and in this chapter, we provide new solutions which are experimentally faster, yield results which are more consistent with physical devices, and address several sources of experimental error, which may reduce

In this work, we describe several new, more accurate techniques to measure thermal conductivity, κ, of thermoelectric materials. The first is based on detailed control of the heat flows within a sample under steady-state conditions; the so-called steady-state isothermal technique. The second is nondestructive microscale analysis technique called scanning hot-probe (or scanning thermal microscopy). And the third is lock-in transient Harman method, which is a comprehensive modification of transient Harman technique, employing a lock-in procedure and considers detailed contact effects. A new interesting follow-on is frequency-dependent

The truest test of the accuracy of measurement is comparison with fabricated devices. To support the validation of measurements of individual material properties, we outline a new device metrics, which allows comparison between theoretical and measured device efficiency. We outline a new slope-efficiency method, which can be used to determine informative index ZTmaximum of any device. The second method of device evaluation is a numerical device model called the discretized heat balance model, which considers a piecewise continuous collection of discrete layers within a device, where boundary heat flows have energy and current

Nyquist analysis, which presages a different perspective on the material analysis.

continuity relationships, and enable incredibly easy determination of device efficiency.

This new measurement of κ leverages Peltier heat, QΠ, an electronically controlled internal heat source unique to thermoelectric materials. Peltier heat causes either heating (+QΠ) or cooling (−QΠ) at the junction between a thermoelectric material and a metal by passing the

Q<sup>Π</sup> was first employed to roughly estimate κ by Putley [2]. In Putley's experiment, convective, parasitic, and nonsymmetric heat flows required correction factors larger than 20%. Harman dramatically improved upon Putley's demonstration by performing the measurements in vacuum, and by reducing other parasitic heat flows. Despite these improvements, error is still

In these past studies, principal parasitic heat flows causing error include conduction along lead wires, conduction along thermocouples, Joule heating within lead wires, and radiation. The magnitudes of these parasitic heat flows can be as large as 30% of Peltier heat. Penn quantified the significance of parasitic heat flows in Harman's technique, and showed, that they induced error of more than 10% [4]. Bowley and Goldsmid [5], as well as Buist [6] reported, that

2. Novel measurements of thermal conductivity

obtained, because parasitic heat flows are nonzero [3].

parasitic heat flows cause error, usually larger than 20%.

2.1. Steady-state isothermal technique

proper polarity of electric current, I.

uncertainty by a factor of 2 or more.

390 Thermoelectrics for Power Generation - A Look at Trends in the Technology

Assume a sample having the temperature of one end anchored to the temperature of the environment, Te, by a large heat-sink, and the opposite end, Ttop, having small thermal mass and capable of temperature diversion by the application of QSB. Vacuum is used to obviate convective heat flow, and QSB is applied by small heater having approximately the same subtended area as that of the cross-sectional area (A) of the sample, such that QSB is localized to the top and there is no direct line of sight along the length of the sample (ℓ). Thermocouples are attached to each end to determine the temperature difference across the sample, and electrical leads are used for passing �I to control QΠ. Figure 1 depicts this experimental setup.

When QSB is applied, the temperature of the heated contact will increase with respect to T<sup>e</sup> and some magnitude of Q<sup>κ</sup> will be conducted through the sample. At the heated contact, contributing heat flows include QSB, Qκ, Qradiation-error, and Qwires. Radiation-error flow Qradiation-error is governed by the emissivity (ε), Stefan-Boltzmann constant (σ), sidewall temperature (Tsidewall), and sidewall area (Asidewall) of the sample. Qwires follows the usual Fourier's law description where, for simplicity, aspect ratio (Awires/ℓwires) and thermal conductivity (κwires) of all wires and thermocouples are combined into one lumped parasitic term. When electrical current flows through the sample, Q<sup>Π</sup> is absorbed at the heated contact, where the first Kelvin relation gives Q<sup>Π</sup> = (αsumIT) and αsum is the sum of Seebeck coefficients of the sample and the contact metal, and T is the temperature of the heated contact. An equal and opposite value of Q<sup>Π</sup> is liberated at the contact between the sample and large heat-sink, but is too small to cause any measurable temperature change of the heat-sink, and is therefore negligible. Including Q<sup>Π</sup> at the heated contact yields Eq. (1), which, under steady-state conditions sums to zero:

$$
\sum Q = Q\_{\rm SB} - Q\_{\rm II} - Q\_{\kappa} - Q\_{\rm radiation-error} - Q\_{\rm wires} = 0. \tag{1}
$$

To quantify the magnitude of Qκ, a range of electrical currents can be passed, which enables Q<sup>Π</sup> to absorb a corresponding range of QSB at the contact, so we get:

Figure 1. Experimental schematic representation of steady-state isothermal technique.

For progressively larger I, Q<sup>Π</sup> absorbs increasingly more of QSB and temperature of the heated contact begins to converge to Te, such that the overall ΔT across the sample goes to zero. As ΔT becomes smaller with increasing I, the only relevant heat flows are QSB, QΠ, and Q<sup>κ</sup> because Qradiation-error and Qwires are only statistically significant [7] for larger ΔT, say >10 K, and all parasitic heat flows converge to zero. Therefore, under these conditions at any given I, ΔT across the sample is required to satisfy Eq. (3):

$$Q\_{\rm SB} = (\alpha\_{\rm sum} \text{IT}) + \left[ \kappa \left( \frac{A}{\ell} \right) \Delta T \right]. \tag{3}$$

From the requirement imposed by Eq. (3), a new method for measuring κ is obtained. Eq. (3) is solved to show the dependence of ΔTon electrical current. By taking the derivative of Eq. (3), prior knowledge of QSB is not required, because it is a constant, and the analysis yields the following:

$$\frac{\partial \Delta T}{\partial \mathbf{I}} = -\left[\frac{\alpha\_{\text{sum}} T}{\kappa \left(\frac{A}{\ell}\right)}\right].\tag{4}$$

To determine κ, Eq. (4) is solved using the slope at ΔT = 0 of steady-state ΔTas function of I and that slope is combined with αsum, as well as geometrical aspect ratio (A/ℓ) of the sample [8]. Figure 2 shows dependence of the temperature difference between ends of the thermoelement sample on value of passing current (left panel) and real picture measurement configuration (right panel).

Figure 2. (Left) Linear ΔT decrease across sample by application of milliamps of current (I) for Peltier cooling and (right) picture showing a fully connected sample.

#### 2.1.1. Thermal conductivity of n-type half-Heusler

Thermal conductivity of a half-Heusler alloy (Figure 3) was collected (triangles) and is presented with respect to previously published data (squares, circles, and diamonds) measured by the laser-flash thermal diffusivity technique and reported by researchers from GMZ corporation [9]. Because of the speed, ease, and simplicity of the new technique presented, there is opportunity for significantly more collected data. One data point can be collected in seconds. However, as can be seen in Figure 3, it is consistent to within experimental error and falls within the bounds of the published laser-flash data.

#### 2.1.2. Thermal conductivity of PbTe

QSB ¼ ð∝sumITÞ þ κ

across the sample is required to satisfy Eq. (3):

(right panel).

A ℓ � � ΔT � �

392 Thermoelectrics for Power Generation - A Look at Trends in the Technology

<sup>þ</sup> εσAsidewallðT<sup>4</sup>

For progressively larger I, Q<sup>Π</sup> absorbs increasingly more of QSB and temperature of the heated contact begins to converge to Te, such that the overall ΔT across the sample goes to zero. As ΔT becomes smaller with increasing I, the only relevant heat flows are QSB, QΠ, and Q<sup>κ</sup> because Qradiation-error and Qwires are only statistically significant [7] for larger ΔT, say >10 K, and all parasitic heat flows converge to zero. Therefore, under these conditions at any given I, ΔT

From the requirement imposed by Eq. (3), a new method for measuring κ is obtained. Eq. (3) is solved to show the dependence of ΔTon electrical current. By taking the derivative of Eq. (3), prior knowledge of QSB is not required, because it is a constant, and the analysis yields the following:

> <sup>∂</sup><sup>I</sup> <sup>¼</sup> <sup>−</sup> <sup>α</sup>sum<sup>T</sup> κ <sup>A</sup> ℓ � � " #

To determine κ, Eq. (4) is solved using the slope at ΔT = 0 of steady-state ΔTas function of I and that slope is combined with αsum, as well as geometrical aspect ratio (A/ℓ) of the sample [8]. Figure 2 shows dependence of the temperature difference between ends of the thermoelement sample on value of passing current (left panel) and real picture measurement configuration

A ℓ � � ΔT � �

QSB ¼ ðαsumITÞþ κ

Figure 1. Experimental schematic representation of steady-state isothermal technique.

∂ΔT

sidewallTe 4

Þ þ κwires

Awires ℓwires � �

: (3)

: (4)

� �

ΔT

: (2)

One category of high performance thermoelectric materials is near-degenerate semiconductors. Such materials do not directly obey the Wiedemann-Franz relationship between electrical resistivity and thermal conductivity, due to the significant contribution of lattice thermal conduction to the total thermal conductivity. However, utilizing a modified Wiedemann-Franz relationship to find the thermal conductivity due to electron flow allows direct, real-time deconvolution of lattice thermal conductivity (κlattice) from electronic contribution (κelectronic). For charge carriers concentrations near degeneracy, and random scattering of charge carriers, Rosi et al. [10] describe how κelectronic can be determined using electrical resistivity, ρ, by:

$$
\kappa\_{\text{electronic}} = \left(\frac{\pi^2}{3}\right) \left(\frac{k\_B}{q}\right)^2 \left(\frac{T}{\rho}\right). \tag{5}
$$

If thermal conductivity and electrical resistivity are measured, then κlattice can be determined by κlattice = [κ−κelectronic].

Figure 4 shows temperature dependences of PbTe thermal conductivity (including measured data) and deconvolution of electronic and lattice contributions to total thermal conductivity.

Figure 3. Thermal conductivity of n-type half-Heusler measured by steady-state isothermal technique as compared to published data [9].

#### 2.2. Scanning hot probe

The scanning hot probe technique provides measurement of local thermal conductivity and Seebeck coefficient of a sample by measuring average probe temperature when probe tip and sample are in "thermal contact", i.e., when probe tip is in physical contact with the sample or is at known distance near enough to the sample to induce measurable heat exchange between probe and sample. Average probe temperature is also measured far from the sample, to account for the amount of heat lost to the surroundings and through the probe contacts. Difference in average probe temperature between these two cases is due to the heat transferred to the sample, which may be quantified through the following analytical derivation. Figure 5 depicts the thermal exchange between probe, sample, and surroundings, as well as the series thermal resistance network between probe and sample.

For steady-state probe heating using DC current, (or AC current at low frequency, when the heat capacity effects are negligible, and temperature rise amplitude is frequency independent and equivalent to DC temperature rise to good approximation) the governing equation describing amplitude of the temperature profile of the probe shown in Figure 5 is given by [12]:

Novel Measurement Methods for Thermoelectric Power Generator Materials and Devices http://dx.doi.org/10.5772/65443 395

$$\left(\frac{d^2T^\*}{d\mathbf{x}^2} - \left(\frac{2h\_{\rm eff}}{\kappa\_\mathbf{P}r} - \frac{I^2\rho\_0 \mathbf{T}\mathbf{CR}}{\kappa\_\mathbf{P}\pi\mathbf{7}^2r^4}\right)T^\* + \frac{I^2\rho\_0}{\kappa\_\mathbf{P}\pi\mathbf{7}^2r^4} = 0,\tag{6}$$

where <sup>T</sup>� <sup>¼</sup> <sup>T</sup>ðxÞ−T0, <sup>h</sup>eff <sup>¼</sup> <sup>h</sup> <sup>þ</sup> <sup>4</sup>ϵσT<sup>3</sup> <sup>0</sup> (here, h is the convective heat transfer coefficient, ϵ is the probe's emissivity, T<sup>3</sup> <sup>0</sup> is an approximation for the exact (T<sup>4</sup> <sup>−</sup>T<sup>o</sup> 4 ) term and σ is Stefan-Boltzmann constant. ρ<sup>0</sup> and κ<sup>P</sup> are the probe's electrical resistivity and thermal conductivity, respectively, TCR is the probe's temperature coefficient of resistance, I is root-mean-square electrical current passed through the probe, and r is the radius of the probe. Contribution from radiation is negligible for ΔT < 100 K [13].

Figure 4. Measurement of thermal conductivity of PbTe, and deconvolution of electronic and lattice contributions to the total thermal conductivity.

2.2. Scanning hot probe

published data [9].

thermal resistance network between probe and sample.

394 Thermoelectrics for Power Generation - A Look at Trends in the Technology

The scanning hot probe technique provides measurement of local thermal conductivity and Seebeck coefficient of a sample by measuring average probe temperature when probe tip and sample are in "thermal contact", i.e., when probe tip is in physical contact with the sample or is at known distance near enough to the sample to induce measurable heat exchange between probe and sample. Average probe temperature is also measured far from the sample, to account for the amount of heat lost to the surroundings and through the probe contacts. Difference in average probe temperature between these two cases is due to the heat transferred to the sample, which may be quantified through the following analytical derivation. Figure 5 depicts the thermal exchange between probe, sample, and surroundings, as well as the series

Figure 3. Thermal conductivity of n-type half-Heusler measured by steady-state isothermal technique as compared to

For steady-state probe heating using DC current, (or AC current at low frequency, when the heat capacity effects are negligible, and temperature rise amplitude is frequency independent and equivalent to DC temperature rise to good approximation) the governing equation describing

amplitude of the temperature profile of the probe shown in Figure 5 is given by [12]:

Figure 5. Diagram showing thermal phenomenology around probe tip. Reproduced from [11] with permission from The Royal Society of Chemistry.

To obtain an analytical solution to the second order differential equation, two boundary conditions are employed. The first assumption is that the ends of the probe are at ambient temperature (i.e., Tð0Þ ¼ T0). The second assumption is that the tip region of the probe of length 2b is of uniform temperature, and by energy balance at the probe tip region, we get:

$$-\kappa\_{\rm P} A \frac{dT^\*}{d\mathbf{x}}\big|\_{\mathbf{x}=L/2-b} + I^2 \rho\_0 (1 + T\mathcal{R} \mathbf{x} \times T^\*\big|\_{\mathbf{x}=L/2-b}) \frac{b}{A} = \frac{Q\_s}{2},\tag{7}$$

where the left-hand side is heat conduction and Joule heating of the probe, <sup>A</sup> <sup>¼</sup> <sup>π</sup>r<sup>2</sup> is the probe's cross-sectional area, and L is the length of the probe, and where the right-hand side is heat transfer through one leg of the probe (thus half the total heat transfer to the sample, by symmetry). Finally, heat transfer rate between probe and sample, Qs, is:

$$Q\_s = \frac{\Delta T\_{\text{tip}}}{R\_{\text{C}}^{\text{th}} + R\_{\text{S}}^{\text{th}}} = \frac{\Delta T\_{\text{S}}}{R\_{\text{S}}^{\text{th}}},\tag{8}$$

where ΔTtip ¼ Ttip−T<sup>0</sup> and ΔT<sup>S</sup> ¼ TS−T<sup>0</sup> are temperature of the probe and sample, respectively, at the tip region, and Rth <sup>S</sup> is samples thermal resistance. Solving Eq. (6) to obtain temperature profile along the probe for a given value of Q<sup>s</sup> yields the following expression:

$$
\Delta T\_P(\mathbf{x}) = \mathbb{C}\_1 e^{\lambda \mathbf{x}} - \mathbb{C}\_2 e^{-\lambda \mathbf{x}} + \frac{\Gamma}{\lambda^2},
\tag{9}
$$

where <sup>λ</sup> <sup>¼</sup> <sup>I</sup> 2 ρ0 <sup>κ</sup>Pπ2r4, <sup>Γ</sup> <sup>¼</sup> <sup>2</sup>heff <sup>κ</sup>P<sup>r</sup> <sup>−</sup> <sup>I</sup> 2 ρ0TCR <sup>κ</sup>Pπ2r<sup>4</sup> , and constants C1 and C2 are easily obtained by applying boundary condition Tð0Þ ¼ T0.

If the sample is bulk, or has bulk-like thickness, thermal conductivity is found from Rth <sup>S</sup> by employing semiinfinite medium assumption and 2D bulk sample assumption [14]:

$$R\_S^{\text{th}} = \frac{1}{4\kappa\_{\text{sample}}b}.\tag{10}$$

If the sample is a thin film of thickness l on substrate, and is thin enough, that there is negligible heat spreading in the in-plane directions of the sample, then thermal conductivity is found by solving the expression for the series thermal resistance across substrate and film, with 1D heat transfer across the thickness of the film [14]:

$$R\_S^{\text{th}} = \frac{1}{4\kappa\_{\text{substrate}}b} + \frac{l}{\pi\kappa\_{\text{film}}b^2} \,. \tag{11}$$

When heat transfer may be multidimensional and anisotropic, models developed by Son et al. [15] for laser heating may be used to predict thermal resistance of the sample, based on the respective values of thermal conductivity for the film and substrate.

Data collected from scanning hot thermoelectric probe experiment are probe voltage, voltage across a reference resistor, Seebeck voltage, and photodetector voltage for position sensing. The value of current passing through the system is obtained by dividing voltage across the reference resistor by known electrical resistance of that resistor. Probe resistance is then found by dividing probe voltage by that value of current. Often, instead of single reference resistor, Wheatstone bridge is utilized. Figure 6 depicts the difference in circuit between measurement taken (a) with and (b) without Wheatstone bridge.

To obtain an analytical solution to the second order differential equation, two boundary conditions are employed. The first assumption is that the ends of the probe are at ambient temperature (i.e., Tð0Þ ¼ T0). The second assumption is that the tip region of the probe of length 2b is of uniform temperature, and by energy balance at the probe tip region, we get:

ρ0ð1 þ TCR· T�

where the left-hand side is heat conduction and Joule heating of the probe, <sup>A</sup> <sup>¼</sup> <sup>π</sup>r<sup>2</sup> is the probe's cross-sectional area, and L is the length of the probe, and where the right-hand side is heat transfer through one leg of the probe (thus half the total heat transfer to the sample, by

where ΔTtip ¼ Ttip−T<sup>0</sup> and ΔT<sup>S</sup> ¼ TS−T<sup>0</sup> are temperature of the probe and sample, respec-

temperature profile along the probe for a given value of Q<sup>s</sup> yields the following expression:

If the sample is bulk, or has bulk-like thickness, thermal conductivity is found from Rth

If the sample is a thin film of thickness l on substrate, and is thin enough, that there is negligible heat spreading in the in-plane directions of the sample, then thermal conductivity is found by solving the expression for the series thermal resistance across substrate and film,

employing semiinfinite medium assumption and 2D bulk sample assumption [14]:

Rth <sup>S</sup> <sup>¼</sup> <sup>1</sup> 4κsampleb

λx −C2e <sup>−</sup>λ<sup>x</sup> <sup>þ</sup> Γ

j

<sup>¼</sup> <sup>Δ</sup>T<sup>S</sup> Rth S

<sup>x</sup>¼L=2−b<sup>Þ</sup> <sup>b</sup>

<sup>S</sup> is samples thermal resistance. Solving Eq. (6) to obtain

<sup>κ</sup>Pπ2r<sup>4</sup> , and constants C1 and C2 are easily obtained by applying

<sup>A</sup> <sup>¼</sup> Qs

<sup>2</sup> , (7)

, (8)

<sup>λ</sup><sup>2</sup> , (9)

: (10)

πκfilmb<sup>2</sup> : (11)

<sup>S</sup> by

<sup>−</sup>κP<sup>A</sup> dT� dx j

396 Thermoelectrics for Power Generation - A Look at Trends in the Technology

<sup>κ</sup>P<sup>r</sup> <sup>−</sup> <sup>I</sup> 2 ρ0TCR

with 1D heat transfer across the thickness of the film [14]:

Rth

respective values of thermal conductivity for the film and substrate.

<sup>S</sup> <sup>¼</sup> <sup>1</sup>

4κsubstrateb

When heat transfer may be multidimensional and anisotropic, models developed by Son et al. [15] for laser heating may be used to predict thermal resistance of the sample, based on the

Data collected from scanning hot thermoelectric probe experiment are probe voltage, voltage across a reference resistor, Seebeck voltage, and photodetector voltage for position sensing. The value of current passing through the system is obtained by dividing voltage across the

þ

l

tively, at the tip region, and Rth

2 ρ0 <sup>κ</sup>Pπ2r4, <sup>Γ</sup> <sup>¼</sup> <sup>2</sup>heff

boundary condition Tð0Þ ¼ T0.

where <sup>λ</sup> <sup>¼</sup> <sup>I</sup>

<sup>x</sup>¼L=2−<sup>b</sup> <sup>þ</sup> <sup>I</sup>

symmetry). Finally, heat transfer rate between probe and sample, Qs, is:

2

Qs <sup>¼</sup> <sup>Δ</sup>Ttip Rth <sup>C</sup> <sup>þ</sup> <sup>R</sup>th S

ΔTPðxÞ ¼ C1e

Figure 6. Schematic scanning thermal microscopy circuits (left) with Wheatstone bridge and (right) with a reference resistor.

In DC mode, with resistor wired in series with the probe, measured probe voltage, VP, may be expressed in terms of voltage across reference resistor (Vr) as:

$$V\_{\rm P} = R\_{\rm P}I = \frac{R\_{\rm P}V\_{\rm r}}{R\_{\rm r}},\tag{12}$$

where I is DC current passing through the circuit, R<sup>P</sup> is electrical resistance of the probe, and R<sup>r</sup> is known electrical resistance of reference resistor. Probe's electrical resistance is proportional to temperature rise above ambient, when the probe undergoes Joule heating. It may be expressed as:

$$R\_{\rm P} = \Delta T\_{\rm P}(R\_0 \text{TCR}) + R\_0 + R\_{\rm c},\tag{13}$$

where ΔT<sup>P</sup> is average probe temperature rise, R<sup>0</sup> is nominal probe electrical resistance at 19.9° C, (not including electrical contacts to the circuit), when the probe is not being heated, R<sup>c</sup> is electrical resistance arise from contacts and circuit's wiring, and TCR is probe's temperature coefficient of resistance, in terms of 1/°C.

Defining average probe thermal resistance as average probe temperature rise divided by Joule heating power, we can write average probe thermal resistance as:

$$R\_P^{\text{th}} = \frac{\Delta T\_P}{I^2 R\_0 (1 + (T \text{CR}) \Delta T\_P)}. \tag{14}$$

Eq. (14) allows determination of Rth <sup>P</sup> by the slope of probe temperature rise with power applied, reducing the overall experimental uncertainty compared with a single value of temperature at a given power. If the circuit uses Wheatstone bridge, equations differ only by the method of finding electrical resistance of the probe. In this case, R<sup>P</sup> reduces to:

$$R\_{\rm P} = R\_0 + \frac{V\_{\rm B}}{V\_{\rm A}} \frac{(R\_{\rm B} + R\_0)^2}{R\_{\rm B}} + R\_{\rm c},\tag{15}$$

where V<sup>B</sup> and V<sup>A</sup> are voltages across bridge side and probe side, respectively. R<sup>B</sup> is the total resistance of bridge side of the circuit, and R<sup>0</sup> is no heating resistance of the probe side of the bridge. With the probe resistance obtained, remaining equations are left unchanged. When AC current of amplitude I<sup>0</sup> is passed through the circuit, then measured probe resistance can be expressed as:

$$R\_P = \underbrace{R\_0 \left(1 + (T\text{CR})\Delta T\_{P,DC}\right)}\_{\text{DC component}} + \underbrace{R\_0 (T\text{CR})\Delta T\_{P,2\omega}\cos(2\omega t + \phi)}\_{\text{AC component}}\tag{16}$$

The probe tip voltage is expressed as:

$$V = I\_0 R\_P \left( 1 + (T \text{CR}) \Delta T\_{P, \text{DC}} \right) \cos \left( \omega t \right) + \frac{I\_0 R\_P (T \text{CR}) \Delta T\_{P, 2\omega}}{2} [\cos \left( 3\omega t + \phi \right) + \cos(\omega t + \phi) . \tag{17}$$

Thus, temperature amplitude is determined to be:

$$
\Delta T\_{\text{ave}} = \frac{2V\_{3\omega}}{(T\text{CR})V\_{1\omega}}.\tag{18}
$$

To obtain sample thermal conductivity, the probe must be calibrated. Quantities in Eqs. (6)–(11), which are not determined directly from experimental measurement are: heff, TCR, kP, A, L, ρ0, b, and Rth <sup>C</sup> . To be fully calibrated, these quantities must be known. The probe manufacturer specifies values for TCR, k<sup>P</sup> and ρ0, and these values are used in this work. Values A and L may be found by determining probe's geometry (typically from SEM or microscope images, but may also be determined by measuring Rth <sup>P</sup> in a vacuum and in air). heff is determined by measuring Rth <sup>P</sup> far from contact, and matching the value predicted by the analytical model by adjusting heff and integrating Eq. (9) from x = 0 to L and dividing by L, with Q<sup>s</sup> ¼ 0 to obtain the average probe temperature when no heat is transferred to the sample. Finally, probe-to-sample thermal exchange parameters, b and Rth <sup>C</sup> , must be determined. Typically, these values have been assumed to be sample-independent for given probe-to-sample contact force or probe-to-sample distance. As such, calibration strategies utilize measurements on two samples. However, these parameters are now shown to change with sample thermal conductivity. Figure 7 demonstrates change in b and Rth <sup>C</sup> with sample thermal conductivity. Alternatively, if the sample is electrically grounded and probe tip is capable of making good electrical contact with the sample, then sample with known thermal conductivity and Seebeck coefficient may be used to determine both b and Rth C simultaneously. Figure 7 demonstrates this calibration strategy, together with the typical "intersection method" using two or more samples. Care must be taken to calibrate in the correct range of thermal conductivity, as samples with thermal conductivity of higher than 1.1 W/mK yield a different pair of b and Rth <sup>C</sup> values compared with samples with thermal conductivity of 1.1 W/mK and lower. Table 1 presents the results of measurements taken with properly calibrated Wollaston probe tips, showing good agreement with independent measurements.

Novel Measurement Methods for Thermoelectric Power Generator Materials and Devices http://dx.doi.org/10.5772/65443 399

R<sup>P</sup> ¼ R<sup>0</sup> þ

1 þ ðTCRÞΔTP,DC


�

expressed as:

V ¼ I0R<sup>P</sup>

and Rth

Rth

and Rth

�

R<sup>P</sup> ¼ R<sup>0</sup>

The probe tip voltage is expressed as:

also be determined by measuring Rth

exchange parameters, b and Rth

different pair of b and Rth

1 þ ðTCRÞΔTP, DC

Thus, temperature amplitude is determined to be:

�

398 Thermoelectrics for Power Generation - A Look at Trends in the Technology

V<sup>B</sup> V<sup>A</sup>

where V<sup>B</sup> and V<sup>A</sup> are voltages across bridge side and probe side, respectively. R<sup>B</sup> is the total resistance of bridge side of the circuit, and R<sup>0</sup> is no heating resistance of the probe side of the bridge. With the probe resistance obtained, remaining equations are left unchanged. When AC current of amplitude I<sup>0</sup> is passed through the circuit, then measured probe resistance can be

�

cosðωtÞ þ <sup>I</sup>0RPðTCRÞΔTP, <sup>2</sup><sup>ω</sup>

<sup>Δ</sup>Tave <sup>¼</sup> <sup>2</sup>V<sup>3</sup><sup>ω</sup>

To obtain sample thermal conductivity, the probe must be calibrated. Quantities in Eqs. (6)–(11), which are not determined directly from experimental measurement are: heff, TCR, kP, A, L, ρ0, b,

<sup>P</sup> far from contact, and matching the value predicted by the analytical model by adjusting heff and integrating Eq. (9) from x = 0 to L and dividing by L, with Q<sup>s</sup> ¼ 0 to obtain the average probe temperature when no heat is transferred to the sample. Finally, probe-to-sample thermal

to be sample-independent for given probe-to-sample contact force or probe-to-sample distance. As such, calibration strategies utilize measurements on two samples. However, these parameters are now shown to change with sample thermal conductivity. Figure 7 demonstrates change in b

simultaneously. Figure 7 demonstrates this calibration strategy, together with the typical "intersection method" using two or more samples. Care must be taken to calibrate in the correct range of thermal conductivity, as samples with thermal conductivity of higher than 1.1 W/mK yield a

and lower. Table 1 presents the results of measurements taken with properly calibrated Wollas-

ton probe tips, showing good agreement with independent measurements.

<sup>C</sup> with sample thermal conductivity. Alternatively, if the sample is electrically grounded and probe tip is capable of making good electrical contact with the sample, then sample with known thermal conductivity and Seebeck coefficient may be used to determine both b and Rth

ðTCRÞV<sup>1</sup><sup>ω</sup>

<sup>C</sup> . To be fully calibrated, these quantities must be known. The probe manufacturer specifies values for TCR, k<sup>P</sup> and ρ0, and these values are used in this work. Values A and L may be found by determining probe's geometry (typically from SEM or microscope images, but may

ðR<sup>B</sup> þ R0Þ

R<sup>B</sup>

2

þ R0ðTCRÞΔTP, <sup>2</sup>ωcosð2ωt þ φÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} AC Component

þ Rc, (15)

<sup>2</sup> <sup>½</sup> cosð3ω<sup>t</sup> <sup>þ</sup> <sup>φ</sup>Þ þ cosðω<sup>t</sup> <sup>þ</sup> <sup>φ</sup>Þ: (17)

<sup>P</sup> in a vacuum and in air). heff is determined by measuring

<sup>C</sup> , must be determined. Typically, these values have been assumed

<sup>C</sup> values compared with samples with thermal conductivity of 1.1 W/mK

: (18)

: (16)

C

Figure 7. Empirical probe calibration strategies for determining b and Rth <sup>C</sup> using (top) intersections of curves taken from samples with known κ for (left) low thermal conductivity and (right) high thermal conductivity and (bottom) determining b and Rth <sup>C</sup> using a single sample with known κ and α. Adapted from [11] with permission from The Royal Society of Chemistry.

#### 2.3. Transient and lock-in Harman techniques to decouple material ZT and thermoelectric properties

Finally, a new method of measuring material thermal conductivity by simultaneously measuring thermoelectric figure of merit (ZT), electrical resistivity (ρ), and Seebeck coefficient (α) is proposed. ZT is dimensionless measure of the efficiency of material at converting thermal into electrical energy, or vice versa, at a given temperature, T, and may be expressed as:

$$\mathbf{Z}\mathbf{T} = \alpha^2 \mathbf{T}/\rho\mathbf{x}.\tag{19}$$

Thus, thermal conductivity is obtained if the other terms in Eq. (19) are known. This new method also allows for measuring intrinsic ZT with reduced experimental error by accounting for losses through nonideal contacts and geometry.


Table 1. Tabular results for a range of materials [16–20]. Reproduced from [11] with permission from The Royal Society of Chemistry. Conventional application of Harman method uses four probes–two to pass current, and two to measure the voltage response of the sample. Harman demonstrated that, while electrical response of the sample was nearly instantaneous, voltage generated by Seebeck effect, which is thermally driven, is much slower. By taking advantage of this fact, thermal signal could be determined from voltage response of the sample to a sudden change in voltage over time, or response to an AC current passed, locking into thermally driven signal. It was shown, by letting α ¼ Vα=ΔT, κ ¼ −ΔTA=ðαTIÞ, and ρ ¼ VρA=ðLIÞ, that ZT could be reduced to the ratio of resistive voltage to Seebeck voltage (i.e., ZT ¼ VS=Vρ). This assumes ideal contacts (negligible thermal and electrical losses through the contact leads) and that temperature rise is due only to Peltier heating, neglecting effects of Joule heating. However, these effects are often difficult to mitigate, and may be accounted for by appropriate modeling (see Figure 8).

From resistive voltage, one may be able to determine electrical resistivity of the material; however, Seebeck coefficient and thermal conductivity remain coupled in equation for ZT. To decouple them, Seebeck coefficient may be simultaneously determined by adding a pair of thermocouple wires at the top and bottom surfaces of the sample as per Figure 9. If we label electric potential in each corner of the sample E<sup>1</sup> – E4, respectively, using Ivory technique [17], we may find Seebeck coefficient from taking voltage measurements across the sample. If voltage is measured at opposite corners, then voltage values measured are E<sup>13</sup> and E24. Value α is determined from expression below, where m is the slope of E<sup>13</sup> vs E24:

$$
\alpha\_{\text{sample}} = \lim\_{E\_{13} \to E\_{24}} \frac{1}{1 - m} \alpha\_{ba} + \alpha\_b. \tag{20}
$$

This technique for measuring Seebeck coefficient reduces the required number of voltage measurements to determine α from three to two and mitigates mismatch in thermocouples, since DC offsets are removed by using a slope. It also allows for AC measurements of the total voltage, and determination of ZT from Nyquist diagrams.

#### 2.3.1. Transient Harman technique–analytical model

Sample SiGe film on glass

1.8 µm

 2.8

 0.3 µm 14.948

 54 K/W

 44.927

7820 K/W

 76.134

 9494 K/W

 1.22

 0.21 W/K·m // 1.23

 0.12 W/K·m[16]

substrate

Fe-doped PCDTBT

3.0 µm

 2.8

 0.3 µm 15.220

 155 K/W

 44.927

7820 K/W

 87.022

 14.631 K/W

 1.03

 0.15 W/K·m [17]

> (1:1 doping

concentration)

PCDTBT Tellurium Film Au film on silicon

substrate

PEDOT CAL

PANI-5 % GNP CAL

PANI-7 % GNP CAL

p-typeBi2Te3 CAL

Borosilicate

AISI 304 Steel CAL

Goodfellow®99.9

pure Niobium CAL

Table 1. Tabular results for a range of materials [16–20].

 %

Bulk

 428

 24 nm

 12.194

 140 K/W

Reproduced

 from [11] with permission

 from The Royal Society of Chemistry.

 40.191

 1532 K/W

 10.632

 2329 K/W

 Glass CAL

Bulk

Bulk

 428

 24 nm

 13.811

 119 K/W

 40.191

 1532 K/W

 37.511

 3511 K/W

 2.8

 0.3 µm 15.516

 134 K/W

 44.927

 7820 K/W

 82.313

 9787 K/W

Bulk

Bulk

Bulk

Bulk

 2.8

 0.3 µm 15.700

 145 K/W

 44.927

 7820 K/W

 92.113

 11.911 K/W

 2.8

 0.3 µm 16.314

 118 K/W

 44.927

 7820 K/W

 131.760

 25.913 K/W

 0.68

 0.97

 1.08

 15.6

 54.9

 8.9 W/K·m // 53.7 W/K·m

 2.2 W/K·m // 16.2 W/K·m

 0.11 W/K·m // 1.1 W/K·m

 0.11 W/K·m // 1.0 W/K·m

 0.08 W/K·m //0.65 W/ K·m [20]

 2.8

 0.3 µm 17.018

 115 K/W

 44.927

 7820 K/W

 188.595

 27.836 K/W

 0.47

 0.06 W/K·m // 0.49 W/K·m [20]

 2.8

 0.3 µm 17.429

 217 K/W

 44.927

 7820 K/W

 241.732

 37.672 K/W

 0.37

 0.05 W/K·m // 0.36 W/K·m

(non-doped)

 3.0 µm

 2.74 µm 2.8

150 nm

 428

 24 nm

 11.624

 157 K/W

 40.191

1532 K/W

 5505

 253 K/W

 0.3 µm 15.749

 75.5 K/W

 44.927

7820 K/W

 112.476

 6.480 K/W

 0.79

 104.2

67.4W/K·m

 //110

 2 W/ K·m [19]

400 Thermoelectrics for Power Generation - A Look at Trends in the Technology

 0.04 W/K·m //0.78

 0.08 W/K·m [18]

 2.8

 0.3 µm 17.866

 204 K/W

 44.927

7820 K/W

 358.859

 66.204 K/W

 0.25

 0.04 W/K·m //0.20

 0.02 W/K·m [17]

lb

 Rth

P

Rth

Rth

S

κf ilm, this work // κf ilm, expected

> C

> > Several experimental setups used by the research community for thermoelectric characterization of thin films employ clean room microfabrication techniques to pattern a metallic electrode on top of the sample, while others use bonded wires or micromanipulated probes to make electrical contact with the top surface of the film sample [3, 21–24]. Configuration modeled in this work is similar to these situations, as shown in Figure 8. Thermoelectric film (3) with cross-sectional area, A3, is deposited on substrate; metallic electrode (2) covers the top film's surface; and electrically conductive probe wire (1) of diameter, d1, is brought in contact with the top surface of the sample. Substrate electrode (4) situated at the interface between thermoelectric film and substrate is used to close the loop and pass current into the film. The substrate electrode is assumed to have negligible electrical and thermal resistance. Its contribution to thermoelectric transport is therefore neglected, with exception of Peltier effect. Substrate electrode temperature is assumed to be the same as the top surface of the substrate (Tb). Electrical and thermal contact resistances expressed as specific values RC\_i-j and Rth\_i-j are assumed at interfaces between adjacent layers indexed by i and j, with (j=i+1). Classical thermoelectric transport model, which neglects electron-phonon nonequilibrium effects, is

developed by assuming, that the thickness of thermoelectric film is much larger than phononelectron thermalization length [25]. Under these conditions, thermoelectric transport in the probe, electrode, and sample is considered one-dimensional. In each layer i, x is the spatial coordinate; h, κ, and ρ are convection heat transfer coefficient, thermal conductivity and electrical resistivity, respectively; P is perimeter, T is absolute temperature, α is Seebeck coefficient, A<sup>3</sup> is area perpendicular to thermoelectric transport direction, and J is current density.

Figure 8. Schematic representation of lock-in Harman technique, under assumptions of (top) ideal contacts and (bottom) considering nonnegligible thermal and electrical resistances arise from contacts. Reprinted from [21] with the permission of AIP Publishing.

developed by assuming, that the thickness of thermoelectric film is much larger than phononelectron thermalization length [25]. Under these conditions, thermoelectric transport in the probe, electrode, and sample is considered one-dimensional. In each layer i, x is the spatial coordinate; h, κ, and ρ are convection heat transfer coefficient, thermal conductivity and electrical resistivity, respectively; P is perimeter, T is absolute temperature, α is Seebeck coefficient, A<sup>3</sup> is area perpendicular to thermoelectric transport direction, and J is current density.

402 Thermoelectrics for Power Generation - A Look at Trends in the Technology

Figure 8. Schematic representation of lock-in Harman technique, under assumptions of (top) ideal contacts and (bottom) considering nonnegligible thermal and electrical resistances arise from contacts. Reprinted from [21] with the permission

of AIP Publishing.

Figure 9. Schematic representation of sample measurement, including thermocouple contacts (ivory technique). (Left) sectional view (reprinted from [21] with the permission of AIP Publishing) and (right) view of sample from above.

Eq. (21) represents steady state energy balance in layers 1–3; the first term on the left side represents heat conduction, the second represents lateral convection and the third represents Joule heating. Thermal radiation and temperature dependence of thermoelectric properties have been neglected as small temperature differences are assumed to occur during the experiments.

$$\frac{d^2T\_i}{d\mathbf{x}\_i^2} - \frac{h\_i P\_i}{\kappa\_i A\_i}(T\_i - T\_0) + \frac{f\_i^2 \rho\_i}{\kappa\_i} = 0; \quad i = 1, 2, 3\,\tag{21}$$

General solution of Eq. (21) for temperature profile in the layer i is of the form,

$$T\_i(\mathbf{x}\_i) = T\_0 + c\_{1,i} \varepsilon \sqrt{\frac{\kappa\_{i,i}^p x\_i}{\kappa\_i^{\kappa\_i}}} + c\_{2,i} \varepsilon \sqrt{\frac{h\_i \rho\_i}{\kappa\_i h\_i}}; + \frac{f\_i^2 \rho\_i A\_i}{\kappa\_i h\_i}; \quad i = 1-3. \tag{22}$$

Integration constants c1, i and c2, i are determined using six boundary conditions. First, the temperature at free end of the probe of length l<sup>1</sup> is assumed to be ambient temperature as stated by:

$$T\_1 = T\_0; x\_1 = l\_1. \tag{23}$$

Second, the temperature of the end of the probe in contact with the surface is assumed to be constant temperature (see expression in the ensuing discussion, Eq. (31)).

The third boundary condition considers energy conservation at the interface between the film and the substrate:

$$\begin{aligned} & \left. \left. -\kappa\_3 A\_3 \frac{dT\_3}{dx\_3} \right|\_{x\_3=0} + J\_3 \alpha\_3 A\_3 T\_3 \right|\_{x\_3=0} = \frac{T\_b \left| -T\_3 \right|\_{x\_3=0}}{R\_{\text{fl}\_{-3}=4}/A\_3} + \frac{1}{2} J\_4^2 A\_4 R\_{\text{C}\_{-3}=4} = \\ & \frac{T\_0 - T\_b}{\Theta\_{\text{solv}4}} + J\_4^2 A\_4 R\_{\text{C}\_{-3}=4} + J\_4 \alpha\_4 A\_4 T\_b. \end{aligned} \tag{24}$$

In Eq. (24), the left side represents heat transfer rate out of the interface. It includes heat conduction and Peltier terms in layer 3, respectively. The middle section of Eq. (24) represents one way to express heat transfer rate entering the interface. It is written as the sum of heat conduction across the interface thermal contact resistance and contact Joule heating term deposited at the interface in layer 3. It is assumed, that the total contact Joule heating is split equally on both sides of the interface. The right side of Eq. (24) represents the second way to express heat transfer rate entering the interface and includes: (1) substrate heat conduction transfer rate written as the temperature difference across the substrate divided by substrate's thermal conduction resistance; (2) total Joule heating due to electrical contact resistance; and (3) Peltier contribution due to electric current flowing through the substrate electrode. It is assumed, that the bottom surface of the substrate is at ambient temperature. Thermal conduction resistance of the substrate, Θsubst, can be determined by conduction shape factor. For instance, for sample of diameter d<sup>3</sup> on semiinfinite substrate with thermal conductivity κ<sup>3</sup> shape factor is 0.5 κ<sup>3</sup> -1d<sup>3</sup> -1.

Similar to Eq. (23), the fourth boundary condition is energy balance at the interface between probe and electrode:

$$\begin{aligned} &-\kappa\_1 A\_1 \frac{dT\_1}{dx\_1}|\_{x\_1=0} + f\_1 a\_1 A\_1 T\_1|\_{x\_1=0} + h\_2 (A\_2 - A\_1)(T\_2|\_{x\_2=l\_2} - T\_0) = \\ &\frac{T\_2|\_{x\_2=l\_2} - T\_1|\_{x\_3=0}}{R\_{\underline{dl}\_-1-2}/A\_1} + \frac{1}{2} f\_2^2 A\_2 R\_{\underline{C}\_-1-2} = -\kappa\_2 A\_2 \frac{dT\_2}{dx\_2}|\_{x\_2=l\_2} + f\_2 a\_2 A\_2 T\_2|\_{x\_2=l\_2} + f\_2^2 A\_2 R\_{\underline{C}\_-1-2}. \end{aligned} \tag{25}$$

Here, heat transfer rate exiting the interface (the left side of the equation) also includes convection from the top surface of the electrode to the ambient. Similar analysis is performed at the electrode-sample interface, as stated in (26):

$$\begin{split} & -\kappa\_2 A\_2 \frac{dT\_2}{dx\_2}|\_{x\_2=0} + I\_2 \alpha\_2 A\_2 T\_2|\_{x\_2=0} = \frac{T\_3|\_{x\_3=l\_3} - T\_2|\_{x\_2=0}}{R\_{\text{fl}\_- 1-2}/A\_1} + \frac{1}{2} I\_3^2 A\_3 R\_{\text{C\\_2-3}} = \\ & -\kappa\_3 A\_3 \frac{dT\_3}{dx\_3}|\_{x\_3=l\_3} + I\_3 \alpha\_3 A\_3 T\_3|\_{x\_3=l\_3} + I\_3^2 A\_3 R\_{\text{C\\_2-2}-3}. \end{split} \tag{26}$$

Finally, continuity of electrical current in the layers of the sample requires:

$$J\_1A\_1 = J\_2A\_2 = J\_3A\_3 = J\_4A\_4. \tag{27}$$

Modeling approach discussed above can be used to study in detail effects on temperature profile due to thermal and electrical properties of individual layers and contacts.

Thermal conductivity of the thermoelectric film is typically determined from relationship between temperature rise (usually the measured surface temperature) and dissipated power. In addition, difference between the surface and substrate temperature together with Seebeck voltage developed across the film is used to calculate Seebeck coefficient of the film. Practitioners in thermoelectric field need a way to evaluate steady state surface temperature before electrical current is switched off for transient Harman method under nonideal boundary conditions.

The main strategy pursued here is to use the superposition principle to calculate the total temperature rise by solving separately for temperature solutions under Joule heating and Peltier effects. Rather than using full set of Eqs. (21)–(27), several assumptions are made in this section in order to arrive at an easy to use expression for the surface temperature, which still reflects the main thermoelectric transport mechanisms in many practical situations. These assumptions are: (1) electrode's contributions (layer 2) to thermoelectric transport are neglected because metallic electrode layers typically have low Seebeck coefficient similar to the probe and much lower electrical and thermal resistances compared to thermoelectric films; (2) Seebeck coefficient of the current probe is neglected; (3) convection terms on the film surfaces are neglected; (4) substrate thermal resistance and film-substrate electrode thermal contact resistances are neglected when compared to film thermal resistance, since thermoelectric film samples are typically low thermal conductivity films on high thermal conductivity substrates; and (5) Joule heating at film-substrate electrode contact is neglected because under assumption (4) the substrate acts as a heat sink.

Total temperature, T, in thermoelectric film is then divided into two components as: T = T '+T\* , where T' is linear temperature component, LTC, that is independent of Joule heating terms and includes Peltier effects, while T\* is nonlinear component, NLTC, and takes into account Joule heating effects, including electrical contact resistance heating. Then the set of Eqs. (21)–(27) for T' becomes:

$$-\kappa\_3 A\_3 \frac{d^2 T\_3'}{dx\_3^2} = 0,\tag{28}$$

$$T\_3^{'} = T\_0; \mathbf{x}\_3 = \mathbf{0},\tag{29}$$

$$-\kappa\_3 A\_3 \frac{dT\_3'}{d\mathbf{x}\_3}|\_{\mathbf{x} = l\_3} + J\_3 A\_3 \alpha\_3 T\_S' = \frac{T\_S' - T\_w'}{R\_{th\_-1-3}/A\_1} = q\_w'; \mathbf{x}\_3 = l\_3,\tag{30}$$

where Ts' and qw' are temperature of the top surface of the sample and heat transfer rate through the probe, respectively. Heat transfer rate through the probe is calculated using a fin model with ambient temperature at free end and constant temperature Tw' at its base [16]:

$$\boldsymbol{q}\_w^{'} = \boldsymbol{a}(\boldsymbol{T}\_w^{'} - \boldsymbol{T}\_0),\tag{31}$$

where the constant a is defined as:

one way to express heat transfer rate entering the interface. It is written as the sum of heat conduction across the interface thermal contact resistance and contact Joule heating term deposited at the interface in layer 3. It is assumed, that the total contact Joule heating is split equally on both sides of the interface. The right side of Eq. (24) represents the second way to express heat transfer rate entering the interface and includes: (1) substrate heat conduction transfer rate written as the temperature difference across the substrate divided by substrate's thermal conduction resistance; (2) total Joule heating due to electrical contact resistance; and (3) Peltier contribution due to electric current flowing through the substrate electrode. It is assumed, that the bottom surface of the substrate is at ambient temperature. Thermal conduction resistance of the substrate, Θsubst, can be determined by conduction shape factor. For instance, for sample of diameter d<sup>3</sup> on semiinfinite substrate with thermal conductivity κ<sup>3</sup>

Similar to Eq. (23), the fourth boundary condition is energy balance at the interface between

dT<sup>2</sup> dx<sup>2</sup>

Here, heat transfer rate exiting the interface (the left side of the equation) also includes convection from the top surface of the electrode to the ambient. Similar analysis is performed

> x3¼l<sup>3</sup> −T2j x2¼0

2 <sup>3</sup>A3RC\_2−3:

Modeling approach discussed above can be used to study in detail effects on temperature

Thermal conductivity of the thermoelectric film is typically determined from relationship between temperature rise (usually the measured surface temperature) and dissipated power. In addition, difference between the surface and substrate temperature together with Seebeck voltage developed across the film is used to calculate Seebeck coefficient of the film. Practitioners in thermoelectric field need a way to evaluate steady state surface temperature before electrical current is switched off for transient Harman method under nonideal boundary

The main strategy pursued here is to use the superposition principle to calculate the total temperature rise by solving separately for temperature solutions under Joule heating and Peltier effects. Rather than using full set of Eqs. (21)–(27), several assumptions are made in this section in

Rth\_1−2=A<sup>1</sup>

x2¼l<sup>2</sup>

−T0Þ ¼

þ 1 2 J 2

J1A<sup>1</sup> ¼ J2A<sup>2</sup> ¼ J3A<sup>3</sup> ¼ J4A4: (27)

<sup>x</sup>2¼l<sup>2</sup> <sup>þ</sup> <sup>J</sup> 2

<sup>3</sup>A3RC\_2−<sup>3</sup> ¼

<sup>2</sup>A2RC\_1−2: (25)

(26)

<sup>j</sup><sup>x</sup>2¼l<sup>2</sup> <sup>þ</sup> <sup>J</sup>2α2A2T2<sup>j</sup>

<sup>x</sup>1¼<sup>0</sup> <sup>þ</sup> <sup>h</sup>2ðA2−A1ÞðT2<sup>j</sup>

<sup>2</sup>A2RC\_1−<sup>2</sup> ¼ −κ2A<sup>2</sup>

dx<sup>2</sup> <sup>j</sup><sup>x</sup>2¼<sup>0</sup> <sup>þ</sup> <sup>J</sup>2α2A2T2j<sup>x</sup>2¼<sup>0</sup> <sup>¼</sup> <sup>T</sup>3<sup>j</sup>

dx<sup>3</sup> <sup>j</sup><sup>x</sup>3¼l<sup>3</sup> <sup>þ</sup> <sup>J</sup>3α3A3T3j<sup>x</sup>3¼l<sup>3</sup> <sup>þ</sup> <sup>J</sup>

Finally, continuity of electrical current in the layers of the sample requires:

profile due to thermal and electrical properties of individual layers and contacts.

shape factor is 0.5 κ<sup>3</sup>

probe and electrode:

−κ1A<sup>1</sup> dT<sup>1</sup> dx<sup>1</sup> j

Rth\_1−2=A<sup>1</sup>

−κ2A<sup>2</sup> dT<sup>2</sup>

−κ3A<sup>3</sup> dT<sup>3</sup>

T2j x2¼l<sup>2</sup> −T1j x3¼0

conditions.


<sup>x</sup>1¼<sup>0</sup> <sup>þ</sup> <sup>J</sup>1α1A1T1<sup>j</sup>

404 Thermoelectrics for Power Generation - A Look at Trends in the Technology

þ 1 2 J 2

at the electrode-sample interface, as stated in (26):

$$a = \frac{\sqrt{h\_1 P\_1 \kappa\_1 A\_1}}{\tanh\left(\frac{\vec{l}\_1^2 h\_1 P\_1}{\kappa\_1 A\_1}\right)^{0.5}}.\tag{32}$$

Then, the solution for LTC of the top surface of the sample can be calculated as:

$$T\_S' = \frac{(a\_{\rm th} l\_3 / A\_3 + \kappa\_3) T\_0}{a\_{\rm th} l\_3 / A\_3 + \kappa\_3 - l\_3 l\_3 \alpha\_3},\tag{33}$$

where ath is the total heat conductance through the contact and the probe defined as:

$$a\_{th} = \frac{a}{1 + aR\_{th\\_1\text{-}3}/A\_1}.\tag{34}$$

Next, T\* is calculated from the following equations:

$$\frac{d^2T\_3^\*}{dx\_3^2} + \frac{J\_3^2\rho\_3}{\kappa\_3} = 0,\tag{35}$$

$$T\_3^\* = 0; x\_3 = 0,\tag{36}$$

$$\begin{cases} -\kappa\_3 A\_3 \frac{dT\_3^\*}{dx\_3}|\_{x\_3=l\_3} + f\_3 A\_3 \alpha\_3 T\_3^\*|\_{x\_3=l\_3} + f\_3^2 A\_3 R\_{C\_-1-3} = \\ \frac{T\_S^\* - T\_w^\*}{R\_{th\_-1-3}/A\_1} + \frac{1}{2} f\_1^2 A\_1 R\_{C\_-1-2} = q\_w^\*; x\_3 = l\_3. \end{cases} \tag{37}$$

Heat transfer along the probe is calculated by solving the fin model with volumetric Joule heating and a temperature rise equal to zero (relative to the ambient) at free end of the fin (away from the sample). The equation for the probe heat transfer is:

$$
\mathfrak{q}\_w^\* = aT\_w^\* \mathfrak{-b},
\tag{38}
$$

where the constant b is expressed as:

$$b = -a \frac{f\_3^2 A\_3^2 \rho\_1}{h\_1 P\_1 A\_1} \left[ 1 - \frac{1}{\cosh[\left(l\_1^2 h\_1 P\_1 / \kappa\_1 A\_1\right)^{0.5}]} \right]. \tag{39}$$

Then, the solution for NLTC is given by:

$$T\_S^\* = \frac{J\_3^2 l\_3 (R\_{\subset\_-1-3} \frac{A\_3}{A\_1} + \frac{\rho\_3 l\_3}{2}) - \frac{l\_3 b\_{th} b\_{cut}}{A\_3}}{a\_{th} l\_3 / A\_3 + \kappa\_3 - l\_3 l\_3 \alpha\_3},\tag{40}$$

where,

$$b\_{\rm th} = \frac{b}{1 + aR\_{\rm th\\_1-3}/A\_1} \qquad b\_{\rm cont} = 1 + \frac{\frac{1}{2}l\_3^2 \frac{A\_3^2}{A\_1} R\_{\rm C\\_1-3}}{b} \frac{aR\_{\rm th\\_1-3}}{A\_1} \tag{41}$$

Finally, total temperature of the top surface of the sample, Ts, is:

$$T\_S = \frac{1}{a\_{th}l\_3/A\_3 + \kappa\_3 - l\_3l\_3\alpha\_3} \begin{bmatrix} (a\_{th}l\_3/A\_3 + \kappa\_3)T\_0 + \\ l\_3^2 l\_3 (R\_{\subset \subset -3} \frac{A\_3}{A\_1} + \frac{\rho\_3 l\_3}{2}) - b\_{th}b\_{cont} \frac{l\_3}{A\_3} \end{bmatrix},\tag{42}$$

where the first term contains Peltier effect's induced contributions to the surface temperature, the second term includes Joule heating effects from the sample and contact, and the third term includes Joule heating contribution from the probe wire.

Temperature of the probe at junction with the sample surface is then calculated as:

$$T\_w = \frac{T\_S + \frac{R\_{\rm th, 1-3}}{A\_1} \left(\frac{1}{2} f\_3^2 \frac{A\_3^2}{A\_1} R\_{\gets, 1-3} + aT\_0 - b\right)}{1 + aR\_{\scriptstyle h, 1-3}/A\_1}.\tag{43}$$

Understanding how to eliminate or reduce the effects due to heat loss and electrical and thermal contact resistances is critical in designing test structures amenable for accurate thermoelectric transport measurements. Parasitic effects are expected to be different for macroscale versus microscale samples, and this section focuses on microscale samples. To illustrate these effects, the surface temperature predictions as a function of current density are discussed for thermoelectric sample of 10 +10 +10 µm3 in contact with copper probe of 5 µm diameter and 1.3 mm length. Thermoelectric properties of thermoelectric film are similar to n-type Bi2Te2.7Se0.3 and are listed in Table 2.

d2 T� 3 dx<sup>2</sup> 3 þ J 2 <sup>3</sup>ρ<sup>3</sup> κ3

T�

<sup>x</sup>3¼l<sup>3</sup> <sup>þ</sup> <sup>J</sup>3A3α3T�

q� <sup>w</sup> ¼ aT�

þ 1 2 J 2

(away from the sample). The equation for the probe heat transfer is:

b ¼ −a J 2 3A<sup>2</sup> <sup>3</sup>ρ<sup>1</sup> h1P1A<sup>1</sup>

> T� S ¼ J 2 <sup>3</sup>l3ðRC\_1−<sup>3</sup>

Finally, total temperature of the top surface of the sample, Ts, is:

athl3=A<sup>3</sup> þ κ3−J3l3α<sup>3</sup>

includes Joule heating contribution from the probe wire.

Tw ¼

1 þ aRth\_1−3=A<sup>1</sup> bcont ¼ 1 þ

J 2 <sup>3</sup>l3ðRC\_1−<sup>3</sup>

where the first term contains Peltier effect's induced contributions to the surface temperature, the second term includes Joule heating effects from the sample and contact, and the third term

1 þ aRth\_1−3=A<sup>1</sup>

Understanding how to eliminate or reduce the effects due to heat loss and electrical and thermal contact resistances is critical in designing test structures amenable for accurate

2 4

Temperature of the probe at junction with the sample surface is then calculated as:

TS <sup>þ</sup> Rth\_1−<sup>3</sup> A1 1 2 J 2 3 A2 3

3j <sup>x</sup>3¼l<sup>3</sup> <sup>þ</sup> <sup>J</sup> 2

<sup>1</sup>A1RC\_1−<sup>2</sup> ¼ q�

Heat transfer along the probe is calculated by solving the fin model with volumetric Joule heating and a temperature rise equal to zero (relative to the ambient) at free end of the fin

> <sup>1</sup><sup>−</sup> <sup>1</sup> cosh½ðl 2

> > A3 <sup>A</sup><sup>1</sup> <sup>þ</sup> <sup>ρ</sup>3l<sup>3</sup>

athl3=A<sup>3</sup> þ κ3−J3l3α<sup>3</sup>

<sup>1</sup>h1P1=κ1A1Þ

<sup>2</sup> <sup>Þ</sup><sup>−</sup> <sup>l</sup>3bthbcont A3

ðathl3=A<sup>3</sup> þ κ3ÞT0þ

A3 A1

<sup>A</sup><sup>1</sup> RC\_1−<sup>3</sup> þ aT0−b � �

" #

0:5

RC\_1−<sup>3</sup>

<sup>2</sup> <sup>Þ</sup>−bthbcont

aRth\_1−<sup>3</sup> A1

> l3 A3

3

b

<sup>þ</sup> <sup>ρ</sup>3l<sup>3</sup>

−κ3A<sup>3</sup> dT� 3 dx<sup>3</sup> j

406 Thermoelectrics for Power Generation - A Look at Trends in the Technology

T� S−T� w Rth\_1−3=A<sup>1</sup>

where the constant b is expressed as:

Then, the solution for NLTC is given by:

bth <sup>¼</sup> <sup>b</sup>

TS <sup>¼</sup> <sup>1</sup>

where,

¼ 0, (35)

<sup>w</sup>; <sup>x</sup><sup>3</sup> <sup>¼</sup> <sup>l</sup>3: (37)

<sup>w</sup>−b, (38)

: (39)

, (40)

: (41)

5, (42)

: (43)

<sup>3</sup> ¼ 0; x<sup>3</sup> ¼ 0, (36)

<sup>3</sup>A3RC\_1−<sup>3</sup> ¼


a = Diameter, b = Width, c = The convection heat transfer coefficient for 5 μm probe is 3000 (W/m2 K).

Table 2. Sample parameters and thermal/thermoelectric properties. Reprinted from [21] with the permission of AIP Publishing.

Figure 10 shows rise of surface temperature with respect to ambient temperature, calculated from Eq. (42) for a range of specific thermal contact resistances. Electrical resistance of contact was assumed to be equal to theoretical limit predicted for electrical boundary resistance between Bi2Te3 and metal electrode [25]. Direction of electrical current was chosen such, that the sample surface undergoes Peltier cooling. At low current densities, Peltier cooling term dominates over Joule heating terms and temperature of the top surface of the sample decreases linearly as electrical current density increases. After reaching the maximum cooling temperature at optimum current density, Joule heating terms start to dominate over Peltier terms. Parasitic conduction heat transfer effect is apparent even at very low current densities, as shown by inset in Figure 10. It leads to reduction of temperature difference across the sample as compared to predictions of an ideal Harman model. On the other hand, as thermal contact resistance increases, the sample cooling is stronger because the thermal barrier created at the contact reduces heat transfer rate with the probe. Importance of thermal barrier effect is gauged by comparison between thermal resistances of the probe, probe-sample contact, and the sample itself. The modeled probe has thermal resistance of ∼5 +104 K/W, which is similar to 6 +104 K/ W thermal resistance of the sample; therefore, a significant heat transfer occurs through the probe. As the thermal contact resistance increases, the probe heat transfer is reduced, particularly after contact thermal resistance becomes of the same order as thermal resistance of the probe. Alternative way to minimize the probe heat transfer rate could be realized by reducing diameter of the probe. However, besides practical challenges, this may have a negative impact associated with increase in resistance of electrical contact, as shown in Figure 10.

Figure 10. Calculated temperature response for specified Rth and Rc. Reprinted from [21] with the permission of AIP Publishing.

Heat transfer through the substrate could also play a major role in establishing the surface temperature. For large current densities, the strength of heat transfer through the substrate is indicated by large positive temperature difference measured across the sample. This difference is in contrast with the predictions of an ideal Harman model, where Joule heating effects never generate a temperature difference across the sample.

The effect of electrical contact resistance in absence of thermal contact resistance is investigated in Figure 10. The modeled probe has electrical resistance of ∼1 Ohm, which is similar to electrical resistance of the sample, therefore Joule heating effects can occur simultaneously in the sample and the probe. In addition, even for very low specific electrical contact resistance of 1 +10-11 Ohm +m2 , electrical contact resistance for the modeled probe is considerable (0.5 Ohm). Electrical contact resistance increases by two orders of magnitude if the probe diameter is reduced by a factor of 10. This illustrates strong requirements to control the probe and contact electrical resistances in transient Harman experiments performed on film on substrate samples. This is because good thermoelectric samples have low electrical resistances, so Joule heating effect in contacts and probe can easily become dominant. Inspection of the probe and wire Joule heating terms suggests mitigation of the electrical contact resistance problem may be achieved by preparing thick film samples (large l3) for measurements, but with small cross-sectional area (A3) in order to increase the relative importance of electrical resistance of the sample. Since this strategy leads to an increase in thermal resistance of the sample, one must simultaneously address the need to mitigate parasitic probe heat conduction effects by choosing a probe with thermal resistance larger than that of the sample. The probe's thermal resistance may be calculated by knowing the material properties and diameter of the probe.

One proposed strategy for determining the properties under nonideal conditions is the bipolar method, where transient Harman experiments are performed using direct and reversed current directions and where measured Seebeck and resistive voltages across the sample are averaged. This is believed to eliminate Joule heating effects and reveal the intrinsic Peltier effects in the sample [21, 23]. However, as demonstrated in this work, when nonideal boundary conditions are present, parasitic effects cannot be always completely eliminated by this strategy. Nevertheless, the analysis below demonstrates the ability to exploit this behavior to determine both thermal and electrical transport properties of the samples and their contacts.

Bipolar resistive voltage difference measured across the sample ΔVρ� is related to the total electrical resistance through the expression:

$$\frac{V\_{\rho+} - V\_{\rho-}}{I\_3 A\_3} = \frac{\Delta V\_{\rho \pm}}{I\_3 A\_3} = 2 \left( R\_{\mathbb{C}\_- 1 - 3} / A\_1 + R\_{\mathbb{C}\_- 3 - 4} / A\_3 + \frac{\rho\_3 l\_3}{A\_3} \right). \tag{44}$$

To find expression for bipolar Seebeck voltage, first Seebeck coefficient, αs, is expressed as measured V<sup>S</sup> as a function of the surface temperature of the sample as:

$$
\alpha\_S = \frac{-V\_S}{T\_S - T\_0},
\tag{45}
$$

where V<sup>s</sup> is experimental Seebeck voltage measured between the probe and electrode situated at the bottom of the sample. If the probe temperature is measured, then:

Figure 10. Calculated temperature response for specified Rth and Rc. Reprinted from [21] with the permission of AIP

408 Thermoelectrics for Power Generation - A Look at Trends in the Technology

Publishing.

$$\alpha\_S = \frac{-V\_S}{(T\_w - T\_0)\left(1 + \frac{aR\_{\rm th\\_1\cdot 3}}{A\_1}\right) + \frac{R\_{\rm th\\_1\cdot 3}}{A\_1}\left(b - \frac{1}{2}f\_3^2 A\_3^2 \frac{R\_\odot\cdot 1 \cdot 3}{A\_1}\right)}}\tag{46}$$

When bipolar method is used, then α<sup>s</sup> can be extracted from bipolar Seebeck voltage difference ΔVS� and temperature difference ΔTw� using expression:

$$\alpha\_S = \frac{-\Delta V\_{S\pm}}{\Delta T\_{S\pm}} = \frac{-\Delta V\_{S\pm}}{\Delta T\_{w\pm}\left(1 + \frac{a\mathcal{R}\_{\text{th}\perp 1\text{-}3}}{A\_1}\right)}.\tag{47}$$

Next, under small current approximations:

$$\kappa \kappa + a\_{\rm fl} \frac{l\_3}{A\_3} >> J\_3 l\_3 a\_3 \text{ \&\quad T\_0 \left(\kappa \gamma + a\_{\rm fl} \frac{l\_3}{A\_3}\right) >> J\_3^2 l\_3 (R\_{\rm C-1-3} \frac{A\_3}{A\_1} + \frac{\rho\_3 l\_3}{2}) \cdot \frac{l\_3 b\_{\rm fl} b\_{\rm cont}}{A\_3},\tag{48}$$

a simplified expression for bipolar surface temperature difference is obtained as:

$$
\Delta T\_{S\pm} = \frac{2T\_0}{\kappa\_3 + a\_{th}l\_3/A\_3} I\_3 l\_3 \alpha\_3. \tag{49}
$$

Expression for ZT obtained through bipolar technique is found as:

$$ZT\_0 = \frac{-\Delta V\_{S\pm}}{\Delta V\_{\rho\pm}} \left( 1 + \frac{a\_{\theta l} l\_3}{\kappa\_3 A\_3} \right) \times \left( 1 + \frac{R\_{\subset \subset -3} l\_3}{\rho\_3 A\_3 A\_1} + \frac{R\_{\subset \subset -4} l\_3}{\rho\_3 A\_3} \right). \tag{50}$$

Another strategy is to perform experiments over a range of currents and use differential changes in VS and V<sup>ρ</sup> with current I. Under small current conditions, the following expression is then obtained for the figure of merit:

$$ZT\_0 = \frac{-dV\_S/dI}{dV\_\rho/dI} \left(1 + \frac{a\_{th}l\_3}{\kappa\_3 A\_3}\right) \times \left(1 + \frac{R\_{\subset\_-1-3}l\_3}{\rho\_3 A\_3 A\_1} + \frac{R\_{\subset\_-3-4}l\_3}{\rho\_3 A\_3}\right). \tag{51}$$

Neither bipolar nor differential current methods alone are able to account for all parasitic effects. As a result, these effects must be considered in data reduction or otherwise minimized. A variable thickness method [21, 23, 24] is used to account for electrical contact resistance effects, while heat losses and thermal resistance effects are neglected. A different method to determine all thermoelectric properties without the need for extensive sample preparations is outlined below.

The strategy explored here is to use bipolar experiments performed over a wide range of currents rather, than small current regime required by above methods. It is expected, that at large currents, experimental Seebeck voltage and temperature signals become sensitive to electrical transport properties of the sample and contacts and could be used to determine the sample and contact thermoelectric properties. In addition to Seebeck and resistive voltage drops, method requires measurement of the sample surface temperature or the probe temperature (at the contact with the samples surface).

Proposed strategy takes into consideration selective sensitivity of thermal signals to Peltier and combined Peltier and Joule heating effects under low and high current regimes, respectively. Under small current approximations, temperatures of the probe and sample surface are linear with current, and thermal conductivity can be expressed as a function of experimentally measured slope of the probe temperature as:

$$\kappa\_3 = \frac{l\_3}{A\_3} \left[ \frac{\alpha\_3 T\_0}{\left(1 + \frac{a R\_{th} \cdot 1 \cdot 3}{A\_1}\right) d T\_{v/d I}} - a\_{th} \right]. \tag{52}$$

In Eq. (52), value of Seebeck coefficient is substituted from Eq. (47), which is valid at any current. Next, Eqs. (44), (47), and (52) are substituted in Eq. (43). For the sake of discussion it is assumed, that specific electrical contact resistance is similar at the top and bottom contacts (other assumptions are discussed in Section 3). After the above substitutions, predicted probe temperature and Seebeck coefficient become a function of two unknowns, specific electrical and thermal contact resistances, which are then used to fit experimental signals under large current regime for both direct and reverse currents. This strategy allows the unique determination of all thermoelectric properties of the sample and electrical and thermal contact resistances. Details of the fitting procedure are presented in experimental validation section.

#### 2.3.2. Lock-in Harman technique–analytical model

<sup>α</sup><sup>S</sup> <sup>¼</sup> <sup>−</sup>VS <sup>ð</sup>Tw−T0<sup>Þ</sup> <sup>1</sup> <sup>þ</sup> aRth\_1−<sup>3</sup>

> <sup>α</sup><sup>S</sup> <sup>¼</sup> <sup>−</sup>ΔVS� ΔTS�

>> J3l3α<sup>3</sup> & T<sup>0</sup> κ<sup>3</sup> þ ath

Expression for ZT obtained through bipolar technique is found as:

dVρ=dI <sup>1</sup> <sup>þ</sup>

1 þ

ZT<sup>0</sup> <sup>¼</sup> <sup>−</sup>ΔVS� ΔV<sup>ρ</sup>�

ZT<sup>0</sup> <sup>¼</sup> <sup>−</sup>dVS=dI

is then obtained for the figure of merit:

outlined below.

ΔVS� and temperature difference ΔTw� using expression:

410 Thermoelectrics for Power Generation - A Look at Trends in the Technology

Next, under small current approximations:

l3 A3

κ<sup>3</sup> þ ath

A1 

When bipolar method is used, then α<sup>s</sup> can be extracted from bipolar Seebeck voltage difference

l3 A3

κ<sup>3</sup> þ athl3=A<sup>3</sup>

· 1 þ

· 1 þ

Another strategy is to perform experiments over a range of currents and use differential changes in VS and V<sup>ρ</sup> with current I. Under small current conditions, the following expression

Neither bipolar nor differential current methods alone are able to account for all parasitic effects. As a result, these effects must be considered in data reduction or otherwise minimized. A variable thickness method [21, 23, 24] is used to account for electrical contact resistance effects, while heat losses and thermal resistance effects are neglected. A different method to determine all thermoelectric properties without the need for extensive sample preparations is

The strategy explored here is to use bipolar experiments performed over a wide range of currents rather, than small current regime required by above methods. It is expected, that at large currents, experimental Seebeck voltage and temperature signals become sensitive to electrical transport properties of the sample and contacts and could be used to determine the sample and contact thermoelectric properties. In addition to Seebeck and resistive voltage

RC\_1−3l<sup>3</sup> ρ3A3A<sup>1</sup>

RC\_1−3l<sup>3</sup> ρ3A3A<sup>1</sup>

þ

þ

RC\_3−4l<sup>3</sup> ρ3A<sup>3</sup> 2

RC\_3−4l<sup>3</sup> ρ3A<sup>3</sup> 2

a simplified expression for bipolar surface temperature difference is obtained as:

<sup>Δ</sup>TS� <sup>¼</sup> <sup>2</sup>T<sup>0</sup>

athl<sup>3</sup> κ3A<sup>3</sup> 

athl<sup>3</sup> κ3A<sup>3</sup>  <sup>þ</sup> Rth\_1−<sup>3</sup> <sup>A</sup><sup>1</sup> <sup>b</sup><sup>−</sup> <sup>1</sup> 2 J 2 3A<sup>2</sup> 3 RC\_1−<sup>3</sup> A1

<sup>¼</sup> <sup>−</sup>ΔVS� <sup>Δ</sup>Tw� <sup>1</sup> <sup>þ</sup> aRth\_1−<sup>3</sup>

> >> J 2 <sup>3</sup>l3ðRC\_1−<sup>3</sup>

A1

: (46)

: (47)

<sup>þ</sup> <sup>ρ</sup>3l<sup>3</sup> <sup>2</sup> <sup>Þ</sup><sup>−</sup>

J3l3α3: (49)

l3bthbcont A3

: (50)

: (51)

, (48)

A3 A1

> To find frequency-dependent temperature solution in the sample, governing equation and boundary conditions for the problem were first expressed as a function of time and then transformed to frequency domain. Governing thermal transport equation for the sample with attached wires was derived by balancing the energy in infinitesimal length dx of wire or sample domain treated as Joule heated fin and neglecting Thompson effects. Approach is similar to the steady state model [25]. The governing equation is:

$$\frac{\partial^2 T\_i(\mathbf{x}, t)}{\partial \mathbf{x}\_i^2} - m\_i^2 (T\_i - T\_0) + \frac{f^2 \rho\_i}{2\kappa\_i} = \frac{1}{\Delta\_i} \frac{\partial T\_i(\mathbf{x}, t)}{\partial t}. \tag{53}$$

The first term on the left side accounts for conduction through the wire/sample, where T<sup>i</sup> is temperature along length of the wire/sample as a function of position, x, and time, t. The second term represents convective heating or cooling from the environment, where m<sup>i</sup> <sup>2</sup> =h p/(κ A), with h being heat transfer coefficient, p wire/samples circumference, A its cross-sectional area, and T<sup>0</sup> ambient temperature. The third term accounts for Joule heating, and the right side of equation is transient heat storage term. J is electrical current density, and Δ is thermal diffusivity.

Figure 11 represents the same configuration of sample and wires as considered before, but here shows heat transfer domains used in the model. 1D heat transfer was modeled in each of seven domains, one each for the six wires plus another for the sample. Details of the boundary conditions are given below. Temperature solution requires a total of 14 boundary conditions, two for the sample and four for each set of wires. The wire boundary conditions are as follows: two boundary conditions per wire (or six total) were defined by assuming, that the end of each wire was at room temperature, since the wires in the experiment were relatively long compared to their width and measurements were conducted in ambient conditions. The remaining two boundary conditions per wire (summing to twelve in total) are that the ends of each wire in contact with the sample are at a fixed temperature. The two boundary conditions across the sample come from the fact, that heat transfer at the interfaces must be balanced. For the interface between the first set of wires (domains 1–3) and the sample (domain 4), the energy balance yields Eq. (54):

q1−<sup>3</sup> þ J 2 <sup>4</sup>A4R<sup>14</sup> þ J4α34A4T4ð0, tÞ ¼ −κ4A<sup>4</sup> ∂T4ð0,tÞ ∂x<sup>4</sup> (54)

Figure 11. Schematic representation of the sample showing heat transfer domains.

The first term is the rate of heat conduction to and from the wires and is a function of temperature gradient at the wire-sample interface and thermal conductivity of the wires. This is calculated using the pin fin equation below for the experimental results. The second term accounts for Joule heating due to electrical contact resistivity between the current lead and the sample, R14. The third term is Peltier heating at the interface, where α<sup>34</sup> ¼ α3−α4, relative Seebeck coefficient between the wire (3) and the sample (4). The right side of the equation is the heat conducted through the sample. Form of boundary condition for the other wire-sample interface is identical. To transform the problem from a partial differential equation in time to an ordinary differential equation in frequency, ω, we used Fourier transform. Before applying Fourier transformation, it was convenient to represent the sample temperature as a Fourier series. This was possible since temperature is a function of periodic excitation signal and is therefore itself periodic. Temperature as the sum of its DC component (η = 0) and all of the harmonics (all other values of η) of fundamental frequency, ω0, is given by Eq. (55):

Novel Measurement Methods for Thermoelectric Power Generator Materials and Devices http://dx.doi.org/10.5772/65443 413

$$T\_{\mathbf{i}}(\mathbf{x},t) = \sum\_{\cdots}^{\infty} T\_{\mathbf{i}}(\mathbf{x}, n\omega\_{\mathbf{0}}) e^{i\eta\omega\_{\mathbf{0}}t}.\tag{55}$$

Substituting this into governing equation and applying Fourier transform gives transformed governing equation:

$$\frac{\partial^2 T\_i(\mathbf{x}, \omega)}{\partial \mathbf{x}\_i^2} - m\_i^2 T\_i(\mathbf{x}, \omega) + m\_i^2 T\_0 \sqrt{2\pi} \delta(\omega) + \frac{\int^2 \rho\_i}{8\kappa\_i} \sqrt{2\pi} (2\delta(\omega) + \delta(\omega - 2\omega\_0) + \delta(\omega + 2\omega\_0) - \frac{\dot{\mathbf{i}}\omega}{\Delta\_i} T\_i(\mathbf{x}, \omega) \cdot \mathbf{n}). \tag{56}$$

Dirac delta function, δ, is employed because signals of constant frequency in the time domain become delta functions in frequency domain. Conduction, convection, and heat storage terms are present at each harmonic with additional convection term present at ω = 0. Joule heating occurs only at ω = 0 and ω = 2·ω0. The transformation was next applied to the boundary conditions. The transformed boundary condition for the first interface is given by Eq. (57):

$$\begin{split} &q\_{1-3} + \frac{I\_4 A\_4 R\_{14}}{2} \sqrt{2\pi} \Big( 2\delta(\omega) + \delta(\omega - 2\omega\_0) + \delta(\omega + 2\omega\_0) \Big) \\ &+ \frac{I\_4 S\_{34} A\_4}{2} \sum\_{-\mathbf{w}}^{\mathbf{w}} T\_4(0, \mathbf{n} \boldsymbol{\omega}\_0) \sqrt{2\pi} \Big( \delta \Big( \omega \cdot \boldsymbol{\omega}\_0(\mathbf{n} + 1) \Big) - \delta \Big( \omega \cdot \boldsymbol{\omega}\_0(\mathbf{n} - 1) \Big) \Big) = -\kappa\_4 A\_4 \frac{\partial T\_4(0, \boldsymbol{\omega})}{\partial \mathbf{x}}. \end{split} \tag{57}$$

Joule heating term is again present at ω = 0 and 2·ω0, while Peltier heating term occurs only at the fundamental frequency. This demonstrates mathematically how Joule and Peltier components of heat transfer are separated by measuring the harmonics of temperature. All measurements in this work use Peltier component. Joule component is not used in measurements described here. Since this work focuses on the first harmonic measurements, the solution for the first harmonic and its derivative are given by Eq. (58) and Eq. (59), where R is the root of homogenous form of governing equation and is given by Eq. (60):

$$T\_{\mathbf{i}}(\mathbf{x}\_{\mathbf{i}},\omega) = \mathfrak{c}\_{\mathbf{i}1,\mathbf{1}\omega} \mathfrak{e}^{\mathbb{R}\_{\mathbf{l}\omega}\mathbf{x}} + \mathfrak{c}\_{\mathbf{i}2,\mathbf{1}\omega} \mathfrak{e}^{-\mathbb{R}\_{\mathbf{l}\omega}\mathbf{x}},\tag{58}$$

$$\frac{\partial T\_i(\mathbf{x}\_i, \omega)}{\partial \mathbf{x}\_i} = \mathbf{c}\_{i1, 1\omega} R\_{1\omega} e^{R\_{1\omega}\mathbf{x}} \mathbf{-}\_{i2, 1\omega} R\_{1\omega} e^{-R\_{1\omega}\mathbf{x}},\tag{59}$$

$$R = \sqrt{m^2 + \frac{i\omega}{\Delta}}.\tag{60}$$

The undetermined coefficients were solved numerically using transformed boundary conditions.

#### 2.3.3. Experimental results–transient Harman

seven domains, one each for the six wires plus another for the sample. Details of the boundary conditions are given below. Temperature solution requires a total of 14 boundary conditions, two for the sample and four for each set of wires. The wire boundary conditions are as follows: two boundary conditions per wire (or six total) were defined by assuming, that the end of each wire was at room temperature, since the wires in the experiment were relatively long compared to their width and measurements were conducted in ambient conditions. The remaining two boundary conditions per wire (summing to twelve in total) are that the ends of each wire in contact with the sample are at a fixed temperature. The two boundary conditions across the sample come from the fact, that heat transfer at the interfaces must be balanced. For the interface between the first set of wires (domains 1–3) and the sample (domain 4), the energy

<sup>4</sup>A4R<sup>14</sup> þ J4α34A4T4ð0, tÞ ¼ −κ4A<sup>4</sup>

The first term is the rate of heat conduction to and from the wires and is a function of temperature gradient at the wire-sample interface and thermal conductivity of the wires. This is calculated using the pin fin equation below for the experimental results. The second term accounts for Joule heating due to electrical contact resistivity between the current lead and the sample, R14. The third term is Peltier heating at the interface, where α<sup>34</sup> ¼ α3−α4, relative Seebeck coefficient between the wire (3) and the sample (4). The right side of the equation is the heat conducted through the sample. Form of boundary condition for the other wire-sample interface is identical. To transform the problem from a partial differential equation in time to an ordinary differential equation in frequency, ω, we used Fourier transform. Before applying Fourier transformation, it was convenient to represent the sample temperature as a Fourier series. This was possible since temperature is a function of periodic excitation signal and is therefore itself periodic. Temperature as the sum of its DC component (η = 0) and all of the harmonics (all other values of η) of

Figure 11. Schematic representation of the sample showing heat transfer domains.

∂T4ð0,tÞ ∂x<sup>4</sup>

(54)

balance yields Eq. (54):

q1−<sup>3</sup> þ J 2

412 Thermoelectrics for Power Generation - A Look at Trends in the Technology

fundamental frequency, ω0, is given by Eq. (55):

Figure 12 shows experimentally measured total and Seebeck voltages as a function of current.

Resistive voltage drop obtained after subtracting Seebeck voltage from total voltage includes contributions from the sample, probe-sample contact, and sample-substrate electrode contact. Inset in Figure 12 shows an example of measured voltage as a function of time during an experiment at 163 mA. The figure of merit calculated according to classical Harman method yields an average value of 0.11, much smaller than the manufacturer value of 0.85. This discrepancy is due to parasitic effects neglected in classical technique.

Figure 12. The deconvolution of resistive and Seebeck voltage contributions. Reprinted from [21] with the permission of AIP Publishing.

Measured probe temperature is linear with current for the smallest two bipolar currents and yields a slope of −55.1 K/A and was substituted in Eq. (52). Originally, only one probe was modeled in contact with the sample, the energy loss across constantan wire was evaluated and found to be in an order of magnitude smaller than for copper wire, therefore its effect is expected to be negligible.

Transient Harman experiments performed under the highest direct and reverse current conditions were used in the fitting of sample's thermoelectric properties and contact resistances. In the fitting procedure, thermal contact resistance Rth\_1-3 was varied over a wide range, typically between 1.0 +10−<sup>8</sup> m<sup>2</sup> +K/W and 1.0 +10−<sup>5</sup> m<sup>2</sup> +K/W with step of 1.0 +10−<sup>8</sup> m<sup>2</sup> K/W. For each Rth\_1-3 value, sample's Seebeck coefficient was determined from Eq. (47) and sample's thermal conductivity was determined from Eq. (52). In addition, two sets of solutions for electrical contact resistance as function of Rth\_1-3 are generated by fitting experimental Seebeck coefficient and the probe temperatures. The first set of solutions is obtained by minimizing the mean square deviation between Seebeck coefficient values calculated from Eqs. (46)–(47). The second set of solutions is obtained by minimizing the mean square deviation between experimental temperatures of the probe and the predictions of Eq. (43). In Eq. (43), T<sup>S</sup> was substituted from Eq. (42), thermal conductivity was substituted from Eq. (52), Seebeck coefficient from Eq. (47) and resistivity of the sample from Eq. (41). It was found, that the first set of solutions is more sensitive to thermal contact resistance, while the second set was more sensitive to electrical contact resistance. The intersection of the two sets of solutions leads to the unique solution for thermal and electrical contact resistances and thermoelectric properties of the sample.

yields an average value of 0.11, much smaller than the manufacturer value of 0.85. This

Measured probe temperature is linear with current for the smallest two bipolar currents and yields a slope of −55.1 K/A and was substituted in Eq. (52). Originally, only one probe was modeled in contact with the sample, the energy loss across constantan wire was evaluated and found to be in an order of magnitude smaller than for copper wire, therefore its effect is expected to be negligible. Transient Harman experiments performed under the highest direct and reverse current conditions were used in the fitting of sample's thermoelectric properties and contact resistances. In the fitting procedure, thermal contact resistance Rth\_1-3 was varied over a wide range, typically

Figure 12. The deconvolution of resistive and Seebeck voltage contributions. Reprinted from [21] with the permission of

AIP Publishing.

discrepancy is due to parasitic effects neglected in classical technique.

414 Thermoelectrics for Power Generation - A Look at Trends in the Technology

Since the original sample-substrate interface from commercial Peltier device has a negligible electrical contact resistance compared with the sample, RC,3-4 was taken as zero in Eq. (41). It was also assumed, that Rc,1-3 and Rth,1-3 represent respectively lump electrical and lump thermal contact resistance contributions due to the probe-indium interface, indium layer (negligible contribution), and indium-sample interfaces. However, measured resistive voltage drop through the constantan probe does not include electrical resistance contribution due to the probe-indium interface. Therefore, the fitting procedure was repeated several times under four different assumptions regarding contact electrical resistance. The assumptions and the fitting results are presented and discussed below.

The thermoelectric properties arise from each of the following four cases are summarized in Table 3. In case 1, electrical contact resistance Rc\_1-3 was assumed to originate only from the probe-indium interface. In case 2, electrical contact resistance Rc\_1-3 was assumed to originate only from the indium-sample interface. In case 3, electrical contact resistance Rc\_1-3 was assumed to split equally between probe-indium and indium-sample interfaces. Finally, in case 4, manufacturer's value for sample resistivity ρ<sup>3</sup> =1 +10−<sup>5</sup> Ohm +m, was used in the fitting process, which allowed the exact determination of the split of probe-sample electrical contact resistance between two contributions. In this case, Rc,1-2 <sup>=</sup> 4.9 +10-10 Ohm +m<sup>2</sup> may be split as 1.1 +10−<sup>10</sup> Ohm +m2 due to the probe-indium interface and 3.8 +10−<sup>10</sup> Ohm +m<sup>2</sup> due to indium-sample interface. Relatively large values of measured electrical and thermal contact resistances are due to the imperfections of the mechanical contact under the small contact load used in this proof-of-concept experiment. The indium-pellet interface dominates the contact resistance.

The highest deviations between measured sample's Seebeck and thermal conductivity as compared with manufacture's values are respectively 6 μV/K (3%) and 0.12 W/(m +K) (8 %). These deviations are smaller than the experimental uncertainty. The uncertainty in thermal conductivity due to propagation of the uncertainty in temperature and voltage measurements was calculated to be 0.26 W/(m +K). Similarly, for Seebeck coefficient, the uncertainty is equal to 9.9 μV/K. To accurately determine the sample resistivity, resistive voltage drop should be measured through the copper probe, which is used to pass electrical current, at the same time as through the constantan wire, so total electrical contact resistance and sample resistivity can be accurately determined. When correct resistivity of the sample was employed in the fitting, the sample thermal conductivity was within 3 % of manufacture's values.


Table 3. Summary of cases simulated to explore effect of contacts in transient Harman measurements.

Figure 13 shows comparison between measured and calculated temperature of the probe as a function of electrical current passed through it.

The theoretical predictions use the fitted thermoelectric properties and employ Eq. (43) with either all terms or only Peltier terms. The theoretical predictions were performed for all cases 1–4 and, since they superpose along the same line, they are not individually distinguishable in Figure 13. Joule heating effects are important at large currents in tested sample, as demonstrated by the discrepancy between Peltier heating only predictions and combined Peltier and Joule heating model. Predictions based on solving Eq. (20)–(27), that also include the convection on the sample surface and the contributions from indium electrode, show no significant difference with prediction based on Eq. (43). There is an excellent agreement between experimental and modeling data over entire electrical current range. Data sets for intermediate current values (∼85 mA) show also excellent agreement, although they have not been used in the fitting.

#### 2.3.4. Experimental results–lock-in Harman

ZT and individual thermoelectric properties may also be characterized experimentally using a Nyquist plot (plotting imaginary vs real parts of the complex voltage signal or sample temperature rise). Nyquist analysis of voltage measurements across the sample allows for direct calculation of the slope m, which, in turn, yields α. ZT is obtained from finding each of VR and V<sup>S</sup> from the different regimes represented in the plot (see Figure 14). V<sup>R</sup> is obtained from the value of the real part of the voltage response when the imaginary part is equal to zero, and V<sup>S</sup> is the radius of the circular portion of the Nyquist plot.

The samples measured were bulk bismuth telluride alloys with dimensions of 4.5 +3.8 +3.8 mm<sup>3</sup> . Thin layer of gold was deposited on either end to improve adhesion and current spreading between the sample and lead wires. One lead wire and one thermocouple were soldered to either end of the sample. Current was applied through un-insulated 50.8 µm diameter copper wire, and voltage was measured using 50.8 µm E-type thermocouples. Two sets of voltage measurements were made across the sample using each set of thermocouple wires for excitation frequencies between 10 mHz and 10 Hz. Amplitudes of resulting voltages, E<sup>13</sup> and E24, are plotted in Figure 14. When the signal is applied at low frequencies, then measured voltage is sum of total voltage across the sample and Seebeck voltage in thermocouple's wires. As frequency is increased, thermal component in the sample and wires decays and voltage approaches resistive voltage of the sample, and E13 approaches E24.

measured through the copper probe, which is used to pass electrical current, at the same time as through the constantan wire, so total electrical contact resistance and sample resistivity can be accurately determined. When correct resistivity of the sample was employed in the fitting,

+

+

+

+

Ohm) ρ<sup>3</sup> (Ohm m)

+10-5

10−<sup>10</sup> Nonphysical (negative)

+10−<sup>5</sup>

10−<sup>5</sup> (manufacturer specified)

+

10−<sup>10</sup> 5.7

10−<sup>10</sup> 3.3

10−<sup>10</sup> 1

Figure 13 shows comparison between measured and calculated temperature of the probe as a

The theoretical predictions use the fitted thermoelectric properties and employ Eq. (43) with either all terms or only Peltier terms. The theoretical predictions were performed for all cases 1–4 and, since they superpose along the same line, they are not individually distinguishable in Figure 13. Joule heating effects are important at large currents in tested sample, as demonstrated by the discrepancy between Peltier heating only predictions and combined Peltier and Joule heating model. Predictions based on solving Eq. (20)–(27), that also include the convection on the sample surface and the contributions from indium electrode, show no significant difference with prediction based on Eq. (43). There is an excellent agreement between experimental and modeling data over entire electrical current range. Data sets for intermediate current values (∼85 mA) show also excellent agreement, although they have not been used in

ZT and individual thermoelectric properties may also be characterized experimentally using a Nyquist plot (plotting imaginary vs real parts of the complex voltage signal or sample temperature rise). Nyquist analysis of voltage measurements across the sample allows for direct calculation of the slope m, which, in turn, yields α. ZT is obtained from finding each of VR and V<sup>S</sup> from the different regimes represented in the plot (see Figure 14). V<sup>R</sup> is obtained from the value of the real part of the voltage response when the imaginary part is equal to zero, and

. Thin layer of gold was deposited on either end to improve adhesion and current spreading between the sample and lead wires. One lead wire and one thermocouple were soldered to either end of the sample. Current was applied through un-insulated 50.8 µm diameter copper wire, and voltage was measured using 50.8 µm E-type thermocouples. Two sets of voltage measurements were made across the sample using each set of thermocouple

+3.8 +3.8

The samples measured were bulk bismuth telluride alloys with dimensions of 4.5

the sample thermal conductivity was within 3 % of manufacture's values.

10−<sup>6</sup> 1.38 −212 2.9

10−<sup>7</sup> 1.48 −218 6.2

10−<sup>7</sup> 1.43 −215 3.9

10−<sup>7</sup> 1.46 −217 4.9

Table 3. Summary of cases simulated to explore effect of contacts in transient Harman measurements.

Case # Rth,1-3 (K/W) κ<sup>3</sup> (W/mK) α<sup>3</sup> (μV/K) Rc,1-3 (m<sup>2</sup>

416 Thermoelectrics for Power Generation - A Look at Trends in the Technology

function of electrical current passed through it.

2.3.4. Experimental results–lock-in Harman

V<sup>S</sup> is the radius of the circular portion of the Nyquist plot.

the fitting.

1 1.1

2 6.6

3 9.4

4 7.8

+

+

+

+

mm<sup>3</sup>

Figure 13. Measured versus calculated probe temperature. Reprinted from [16] with the permission of AIP Publishing.

These two voltages were used to find Seebeck coefficient by Ivory's technique [22] using Eq. (21). Non-imaginary values of two voltages are plotted against each other in Figure 15 and resulting in Seebeck coefficient αsample = 202.6 1.4 μV/K. Real parts of signals are used, as these are components, that are in phase with excitation signal and as a result are in phase with each other. Amplitudes may be out of phase with each other and imaginary part is much smaller. Advantage of Ivory technique is that the magnitudes of measured voltages are greater than in traditional technique, if αsample is larger than average value of α<sup>a</sup> and αb, which is often the case, when measuring thermoelectric materials, because voltage measured across the sample is equal to ΔT(αa-αsample), whereas that measured in thermocouples is ΔT(αa–αb)/2. This assumes, that the sample is symmetric and that ΔT is total temperature gradient across the sample. Thus, temperature gradient measured by one set of thermocouples is ΔT/2. Since temperatures on the two sides of the sample are 180° out of phase, the total temperature difference is twice the temperature amplitude registered on one side. The larger signal results in better signal-to-noise ratios and less error in the final calculation.

Figure 14. E<sup>13</sup> and E<sup>24</sup> voltages as a function of frequency.

Total voltage in the sample Vsample was calculated using Eq. (61) and then plotted on Nyquist diagram, shown in Figure 16:

$$V\_{\text{sample}} = \frac{\alpha\_{\text{b}} E\_{13} - \alpha\_{\text{a}} E\_{24}}{\alpha\_{\text{b}} - \alpha\_{\text{a}}}.\tag{61}$$

Obtained data is again shown as superposition of resistive V<sup>R</sup> and Seebeck V<sup>S</sup> voltages, and Vsample and V<sup>R</sup> can be found by extrapolating the data to the real axis as described in introduction. High frequency behavior of real devices may not obey the −45° assumption, if contacts have significant heat capacity [25]. As seen in Figure 16, behavior of the sample deviates somewhat from −45°, which can be attributed to heat capacity of the solder between the wires and the sample. From Figure 16, Vsample is 0.17 mV and V<sup>R</sup> is 0.088 mV. Extrinsic ZT of the device is 0.93 for this measurement. To find intrinsic ZT of the material, then nonidealities in measurement system must be accounted for.

Novel Measurement Methods for Thermoelectric Power Generator Materials and Devices http://dx.doi.org/10.5772/65443 419

Figure 15. Real parts of E<sup>24</sup> and E<sup>13</sup> to find the slope, m, and Seebeck coefficient.

smaller. Advantage of Ivory technique is that the magnitudes of measured voltages are greater than in traditional technique, if αsample is larger than average value of α<sup>a</sup> and αb, which is often the case, when measuring thermoelectric materials, because voltage measured across the sample is equal to ΔT(αa-αsample), whereas that measured in thermocouples is ΔT(αa–αb)/2. This assumes, that the sample is symmetric and that ΔT is total temperature gradient across the sample. Thus, temperature gradient measured by one set of thermocouples is ΔT/2. Since temperatures on the two sides of the sample are 180° out of phase, the total temperature difference is twice the temperature amplitude registered on one side. The larger signal results

Total voltage in the sample Vsample was calculated using Eq. (61) and then plotted on Nyquist

<sup>V</sup>sample <sup>¼</sup> <sup>α</sup>bE13−αaE<sup>24</sup>

Obtained data is again shown as superposition of resistive V<sup>R</sup> and Seebeck V<sup>S</sup> voltages, and Vsample and V<sup>R</sup> can be found by extrapolating the data to the real axis as described in introduction. High frequency behavior of real devices may not obey the −45° assumption, if contacts have significant heat capacity [25]. As seen in Figure 16, behavior of the sample deviates somewhat from −45°, which can be attributed to heat capacity of the solder between the wires and the sample. From Figure 16, Vsample is 0.17 mV and V<sup>R</sup> is 0.088 mV. Extrinsic ZT of the device is 0.93 for this measurement. To find intrinsic ZT of the material, then nonidealities in

αb−α<sup>a</sup>

: (61)

diagram, shown in Figure 16:

measurement system must be accounted for.

Figure 14. E<sup>13</sup> and E<sup>24</sup> voltages as a function of frequency.

in better signal-to-noise ratios and less error in the final calculation.

418 Thermoelectrics for Power Generation - A Look at Trends in the Technology

Figure 16. Voltage measurements plotted on Nyquist diagram.

To find equation for intrinsic ZT, the derivation of ZT performed by Harman can be repeated to include terms for heat loss and contact resistance. This adds two correction factors as shown in Eq. (62). The first is the ratio of heat lost from the sample to Peltier heat generation at the wiresample interface. This heat loss may be due to either convection or radiation from the end of the sample or conduction through the contacts. The second correction factor is the ratio of voltage drop across the contacts to that in the sample, which is equal to resistance of the contacts divided by that of the sample:

$$\text{ZT} = \frac{V\_{\text{S}}}{V\_{\text{R}}} \left[ \left( 1 - \frac{q\_{\text{loss}}}{q\_{\text{Peller}}} \right) \left( 1 - \frac{V\_{\text{C}}}{V\_{\text{R}}} \right) \right]^{-1}. \tag{62}$$

For this experiment, radiation and convection from the sample itself were negligible compared to heat loss through the contacts and only the latter was considered. Since ZT was calculated using voltages approximating DC, where heat transfer is in steady state, and high frequency AC, which is not affected by heat losses, steady-state equation can be used to account for heat loss. Conduction through the wires is described by Eq. (63), which is the fin equation, where the base temperature is equal to that of the wire-sample interface, Ts, and T<sup>0</sup> is the ambient temperature:

$$q\_{\rm w} = \sqrt{hp\kappa A}(T\_S - T\_0)\tanh\left(\sqrt{\frac{hP}{\kappa A}}L\right). \tag{63}$$

For long thin wires, the hyperbolic tangent goes to one and may be neglected. We considered losses by convection through the wire and temperature at the end of the wire away from the sample was assumed to have reached ambient (since wires were long and thin). Even though, the sample temperature was not measured directly, the temperature gradient across the sample was found from measured Seebeck coefficient value, αsample, and Seebeck voltage, VS. Once T<sup>s</sup> was found this way and substituted in Eq. (63), q<sup>w</sup> was determined for each of two wires, summed up and the value used in Eq. (62). In the equation, qPeltier ¼ aIT. After all these substitutions, intrinsic ZT of the material was calculated as 1.04.

Samples resistivity was determined as 7.0 +10−<sup>6</sup> Ohm +m based on resistive voltage, VR, and neglecting the contribution of the contacts. This was confirmed in independent measurement probing resistive voltage profile along the length of the sample, which resulted in value of 7.25 +10−<sup>6</sup> Ohm +m. Using intrinsic ZT, measured Seebeck coefficient and the first value of resistivity, thermal conductivity of 1.55 W/(m +K) is obtained, while if the second resistivity value is used κ is determined as 1.6 W/(m +K).

The same material properties were also found by fitting the predictions based on the numerical model described above in Eqs. (58)–(60) to experimental data. The fitting is shown in Figure 17, where Seebeck voltage data was converted to temperature amplitude using measured Seebeck coefficient. Adjusting thermal conductivity in the model and using the least squares fit, thermal conductivity was 1.55 W/(m +K) and thermal diffusivity was 9.5 +10−<sup>3</sup> cm<sup>2</sup> /s.

Figure 17. Sample temperature and fitted prediction plotted on Nyquist diagram.

To find equation for intrinsic ZT, the derivation of ZT performed by Harman can be repeated to include terms for heat loss and contact resistance. This adds two correction factors as shown in Eq. (62). The first is the ratio of heat lost from the sample to Peltier heat generation at the wiresample interface. This heat loss may be due to either convection or radiation from the end of the sample or conduction through the contacts. The second correction factor is the ratio of voltage drop across the contacts to that in the sample, which is equal to resistance of the

> <sup>1</sup><sup>−</sup> <sup>q</sup>loss qPeltier � �

For this experiment, radiation and convection from the sample itself were negligible compared to heat loss through the contacts and only the latter was considered. Since ZT was calculated using voltages approximating DC, where heat transfer is in steady state, and high frequency AC, which is not affected by heat losses, steady-state equation can be used to account for heat loss. Conduction through the wires is described by Eq. (63), which is the fin equation, where the base temperature is equal to that of the wire-sample interface, Ts, and T<sup>0</sup> is the ambient

hpκ<sup>A</sup> <sup>p</sup> <sup>ð</sup>TS−T0Þtan<sup>h</sup>

+

+K). The same material properties were also found by fitting the predictions based on the numerical model described above in Eqs. (58)–(60) to experimental data. The fitting is shown in Figure 17, where Seebeck voltage data was converted to temperature amplitude using measured Seebeck coefficient. Adjusting thermal conductivity in the model and using the

10−<sup>6</sup> Ohm

neglecting the contribution of the contacts. This was confirmed in independent measurement probing resistive voltage profile along the length of the sample, which resulted in value of

+

+

m. Using intrinsic ZT, measured Seebeck coefficient and the first value of

+

For long thin wires, the hyperbolic tangent goes to one and may be neglected. We considered losses by convection through the wire and temperature at the end of the wire away from the sample was assumed to have reached ambient (since wires were long and thin). Even though, the sample temperature was not measured directly, the temperature gradient across the sample was found from measured Seebeck coefficient value, αsample, and Seebeck voltage, VS. Once T<sup>s</sup> was found this way and substituted in Eq. (63), q<sup>w</sup> was determined for each of two wires, summed up and the value used in Eq. (62). In the equation, qPeltier ¼ aIT. After all these

1− V<sup>C</sup> V<sup>R</sup>

> ffiffiffiffiffiffi hP κA r

L !

: (62)

: (63)

m based on resistive voltage, VR, and

K) and thermal diffusivity was

K) is obtained, while if the second resistivity

� � � � <sup>−</sup><sup>1</sup>

contacts divided by that of the sample:

420 Thermoelectrics for Power Generation - A Look at Trends in the Technology

temperature:

7.25 +

9.5 + 10−<sup>6</sup> Ohm

10−<sup>3</sup> cm<sup>2</sup>

/s.

ZT <sup>¼</sup> <sup>V</sup><sup>S</sup> V<sup>R</sup>

<sup>q</sup><sup>w</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffi

substitutions, intrinsic ZT of the material was calculated as 1.04.

Samples resistivity was determined as 7.0

resistivity, thermal conductivity of 1.55 W/(m

least squares fit, thermal conductivity was 1.55 W/(m

value is used κ is determined as 1.6 W/(m

+

Overall, calculated uncertainty in this experiment was small with that in Seebeck coefficient and extrinsic ZT measurements being lower, than that of calculated intrinsic ZT. The latter is due to the uncertainty in calculating the heat loss through the wires, specifically calculating the heat transfer coefficient between the wires and the air. The heat transfer coefficient was calculated assuming a horizontal cylinder in air. While the uncertainty for the heat transfer coefficient was determined to be about 2%, its uncertainty was assumed to be closer to 10%. This was done because there is some additional uncertainty surrounding the assumptions of free convection and due to its dependence on lab conditions. The uncertainty could be improved by testing the sample in an evacuated chamber, eliminating entirely the need to calculate heat transfer coefficient. The uncertainty of extrinsic ZT is due to that of voltage measurements, and was assumed to be 1% of the measured value, and temperature, assumed to be 296 2 K. The uncertainty in voltage measurement was assumed 1% as conservative estimate. The error of the device was lower, but noise in the system and variation between measurements was closer to 1%. Some error in Seebeck coefficient measurement will be present due to the assumption, that temperature gradients across all the wires in each thermocouple are identical. Since the junctions of thermocouples were somewhat embedded in solder, there may be slight temperature gradient between two wire-solder interfaces. However, this difference was assumed to be negligible compared to temperature gradient along lengths of wires, and the uncertainty in Seebeck coefficient measurement was calculated as less than 1%. The uncertainty in Vsample was similarly low, while that in determining VR from the sample voltage was calculated to be about 2–3%. These were calculated using error propagation from the uncertainty of the measurements. For extrinsic value of ZT, the uncertainty was calculated as 2.7%. With addition of the uncertainty in the heat loss calculation, that for intrinsic ZT increased to 5.9%.

#### 3. Verification strategies for measurements

#### 3.1. Slope-efficiency method: rapid measurement of device ZTmaximum.

Maximum electrical power output, Pmax, of any thermoelectric generator (TEG) depends on open-circuit voltage, Voc, and occurs when internal device resistance, Rint, exactly equals to resistance of external load. When Rint = Rload, then total system resistance = 2Rint, and Voc drops exactly by half leading to:

$$P\_{\text{max}} = \frac{V\_{\text{oc}}}{4R\_{\text{int}}}.\tag{64}$$

For TEG consisting of some number "i" of individual "thermocouples" connected in series and each having n-type thermoelement and p-type thermoelement, Seebeck effect relates Voc to the temperature difference, ΔT, induced by the heat source as described:

$$V\_{\infty} = \sum\_{0}^{\mathrm{i}} (\alpha\_{\mathrm{n}} + \alpha\_{\mathrm{p}})\_{\mathrm{i}} \Delta T,\tag{65}$$

where α<sup>n</sup> and α<sup>p</sup> are values of n-type and p-type Seebeck coefficients from each individual thermoelement, respectively. Thus, the sum of Seebeck coefficients from i thermocouples is ensemble-average proportionality between Voc and ΔT. Likewise, Rint is the sum of resistances from i thermocouples, and it is the ensemble-average electrical resistivity of n-type (ρn) and ptype (ρp) thermoelements times their respective area (A)-to-length (ℓ) values:

$$R\_{\rm int} = \sum\_{0}^{\rm i} \left( \rho\_{\rm n} \frac{\ell}{\rm A} + \rho\_{\rm p} \frac{\ell}{\rm A} \right)\_{\rm i} \tag{66}$$

Pmax can be expressed in terms of Seebeck coefficients:

$$P\_{\text{max}} = \frac{\left(\sum\_{0}^{\text{i}} (a\_{\text{n}} + a\_{\text{p}})\_{\text{i}}\right)^{2} \Delta T^{2}}{4R\_{\text{int}}}.\tag{67}$$

This expression highlights first important point: Pmax increases with ΔT<sup>2</sup> . So, for large electrical power output, the largest possible ΔT is desired.

The efficiency, Φ, with which TEG can convert heat flow, Q, to electrical power is also important because the most electrical power possible from a given amount of heat flow is desirable. A new expression for the efficiency of a TEG can be obtained starting with expression for Pmax. Novel Measurement Methods for Thermoelectric Power Generator Materials and Devices http://dx.doi.org/10.5772/65443 423

The ratio of electrical energy generated per given amount of input heat energy is the definition of efficiency:

$$
\Phi = \frac{P\_{\text{max}}}{Q}.\tag{68}
$$

Eq. (68) can be rewritten, assuming for simplicity a unicouple (i = 1), as:

$$
\Phi = \frac{\left(\alpha\_{\rm n} + \alpha\_{\rm p}\right)^2 \Delta T^2}{4\mathcal{R}\_{\rm int}Q}.\tag{69}
$$

The flow of heat is dominated by thermal conductivity of the materials from which TEG is constructed, so Fourier's law can be used to express Q:

$$\Phi = \frac{\left(\alpha\_{\rm n} + \alpha\_{\rm p}\right)^{2} \Delta T^{2}}{4 \mathsf{R}\_{\rm int} \left( \left[ \kappa\_{\rm n} + \kappa\_{\rm p} \right] \frac{A}{T} \Delta T \right)}. \tag{70}$$

Then expressing Rint as described earlier:

These were calculated using error propagation from the uncertainty of the measurements. For extrinsic value of ZT, the uncertainty was calculated as 2.7%. With addition of the uncertainty in

Maximum electrical power output, Pmax, of any thermoelectric generator (TEG) depends on open-circuit voltage, Voc, and occurs when internal device resistance, Rint, exactly equals to resistance of external load. When Rint = Rload, then total system resistance = 2Rint, and Voc drops

> <sup>P</sup>max <sup>¼</sup> <sup>V</sup>oc<sup>2</sup> 4Rint

For TEG consisting of some number "i" of individual "thermocouples" connected in series and each having n-type thermoelement and p-type thermoelement, Seebeck effect relates Voc to the

ðα<sup>n</sup> þ αpÞ<sup>i</sup>

ℓ A

2 ΔT<sup>2</sup>

i

<sup>0</sup>ðα<sup>n</sup> þ αpÞ<sup>i</sup>

The efficiency, Φ, with which TEG can convert heat flow, Q, to electrical power is also important because the most electrical power possible from a given amount of heat flow is desirable. A new expression for the efficiency of a TEG can be obtained starting with expression for Pmax.

4Rint

where α<sup>n</sup> and α<sup>p</sup> are values of n-type and p-type Seebeck coefficients from each individual thermoelement, respectively. Thus, the sum of Seebeck coefficients from i thermocouples is ensemble-average proportionality between Voc and ΔT. Likewise, Rint is the sum of resistances from i thermocouples, and it is the ensemble-average electrical resistivity of n-type (ρn) and p-

: (64)

ΔT, (65)

: (66)

: (67)

. So, for large electrical

the heat loss calculation, that for intrinsic ZT increased to 5.9%.

3.1. Slope-efficiency method: rapid measurement of device ZTmaximum.

temperature difference, ΔT, induced by the heat source as described:

<sup>V</sup>oc <sup>¼</sup> ∑ i

type (ρp) thermoelements times their respective area (A)-to-length (ℓ) values:

<sup>R</sup>int <sup>¼</sup> ∑ i

Pmax ¼

This expression highlights first important point: Pmax increases with ΔT<sup>2</sup>

Pmax can be expressed in terms of Seebeck coefficients:

power output, the largest possible ΔT is desired.

0 ρn ℓ <sup>A</sup> <sup>þ</sup> <sup>ρ</sup><sup>p</sup>

 ∑i

0

3. Verification strategies for measurements

422 Thermoelectrics for Power Generation - A Look at Trends in the Technology

exactly by half leading to:

$$\Phi = \frac{\left(\alpha\_{\rm n} + \alpha\_{\rm p}\right)^{2} \Delta T^{2}}{4\left(\rho\_{\rm n}\frac{\ell}{A} + \rho\_{\rm p}\frac{\ell}{A}\right)\left([\kappa\_{\rm n} + \kappa\_{\rm p}]\frac{A}{\ell}\Delta T\right)}.\tag{71}$$

For planar TEG devices, the values of ℓ of both n-type and p-type thermoelements are equal; however, cross-sectional areas of n-type and p-type may be quite different. Identifying crosssectional area of n-type as A<sup>n</sup> and that of p-type as A<sup>p</sup> allows a simplification, yielding Φ in terms of measurable materials properties and temperature difference:

$$\Phi = \frac{1}{4} \left( \frac{(\alpha\_{\rm n} + \alpha\_{\rm p})^2}{\left(\frac{\rho\_{\rm n}}{A\_{\rm n}} + \frac{\rho\_{\rm p}}{A\_{\rm p}}\right)(\kappa\_{\rm n} A\_{\rm n} + \kappa\_{\rm p} A\_{\rm p})} \right) \Delta T. \tag{72}$$

The proportionality between Φ and ΔT will be termed "Zdevice":

$$Z\_{\rm device} = \left(\frac{\left(\alpha\_{\rm n} + \alpha\_{\rm p}\right)^2}{\left(\frac{\rho\_{\rm n}}{A\_{\rm n}} + \frac{\rho\_{\rm p}}{A\_{\rm p}}\right)\left(\kappa\_{\rm n}A\_{\rm n} + \kappa\_{\rm p}A\_{\rm p}\right)}\right). \tag{73}$$

Note, that when area-to-length ratios are optimized for maximum efficiency, this relationship reduces to the common, well-known expression for device ZT:

$$Z\_{\rm max} = \left(\frac{(\alpha\_{\rm n} + \alpha\_{\rm p})}{\left(\sqrt{\kappa\_{\rm n} \rho\_{\rm p}} + \sqrt{\kappa\_{\rm p} \rho\_{\rm n}}\right)}\right)^2. \tag{74}$$

TEG efficiency can be measured as function of ΔT, and the slope of that data should be equal to:

#### 424 Thermoelectrics for Power Generation - A Look at Trends in the Technology

$$\frac{\partial \Phi}{\partial \Delta T} = \frac{1}{4} \left( \frac{\left(\alpha\_{\text{n}} + \alpha\_{\text{p}}\right)^{2}}{\left(\frac{\rho\_{\text{n}}}{A\_{\text{n}}} + \frac{\rho\_{\text{p}}}{A\_{\text{p}}}\right) \left(\kappa\_{\text{n}} \mathbf{A}\_{\text{n}} + \kappa\_{\text{p}} \mathbf{A}\_{\text{p}}\right)} \right) = \frac{Z\_{\text{device}}}{4}.\tag{75}$$

This expression highlights a second important point, that Φ should linearly increase as a function of ΔT according to the slope indicated by ¼ of the quantity in parentheses. This makes sense, because Φ increases linearly with ΔT, and Pmax increases as ΔT<sup>2</sup> . Taking the ratio yields a simple linear dependence on ΔT. Note, that the material properties are all temperature dependent, so taking the derivative would necessarily yield higher-order terms. However, we make use of the following assumptions: (1) the temperature dependence of the electrical component of thermal conductivity depends on the mobility of charge carriers, and the electrical resistivity depends on the inverse of that mobility, so these dependencies can be assumed to be first-order to cancel completely. (2) Seebeck coefficient does have a relatively small, but finite temperature dependence; however the derivative of Seebeck coefficient should yield a temperature dependence of T-1 which approximately cancels with the temperature dependence of lattice thermal conductivity in the denominator. So, to first order the linearity of the slope would be expected, and is in fact experimentally observed as will be shown.

A new index to determine maximum ZT of TEG device can be obtained by measuring the slope of TEG efficiency. To calculate maximum ZT, four times the slope of TEG efficiency multiplied by maximum temperature, under which TEG displays linear behavior with respect to ΔT. Outside the linear regime of TEGs, the basic properties can no longer be described by these functions. Therefore, measure of maximum ZT of TEG device can be determined by:

$$(\text{Maximum})Z\_{device}T = 4\left(\frac{(\alpha\_n + \alpha\_p)^2}{\left(\frac{\rho\_n}{A\_n} + \frac{\rho\_p}{A\_p}\right)(\kappa\_n A\_n + \kappa\_p A\_p)}\right)T\_{maximum}.\tag{76}$$

The significance of this analysis is that it allows unique means to rapidly obtain ZTmaximum and confirm properties and individual measurements. Measurements can be confirmed by measuring slope of efficiency as function of ΔT and ZT can be obtained and compared to theoretical ZT as calculated by individual measurements.

#### 3.1.1. Analysis of commercial (Bi,Sb)2(Te,Se)3 module

Efficiency of commercial (Bi,Sb)2(Te,Se)3 device is presented in Figure 18. This device is designed for high thermal impedance and has optimum performance window from nearly room-temperature to roughly 425 K. Slope of efficiency was determined and is shown in inset of Figure 18, and is given as 0.0004/K. As expected, the slope is highly linear function of ΔT until deviation from non-linearity begins at 405 K. The maximum ZT can be obtained by the simple relationship, and observed maximum temperature of roughly 405 K:

$$Z\_{\text{device}} T\_{\text{maximum}} = 4 \left( \frac{\partial \mathcal{O}}{\partial \Delta T} \right) T\_{\text{maximum}}.\tag{77}$$

Obtained value of ZTmaximum is equal to 0.7, which is consistent with established values for commercial devices.

Figure 18. Slope of efficiency from (Bi,Sb)2(Te,Se)3 to determine ZTmax.

#### 3.1.2. Analysis of PbTe/TAGS module

∂Φ <sup>∂</sup>Δ<sup>T</sup> <sup>¼</sup> <sup>1</sup> 4

424 Thermoelectrics for Power Generation - A Look at Trends in the Technology

and is in fact experimentally observed as will be shown.

ZT as calculated by individual measurements.

commercial devices.

3.1.1. Analysis of commercial (Bi,Sb)2(Te,Se)3 module

ðα<sup>n</sup> þ αpÞ

This expression highlights a second important point, that Φ should linearly increase as a function of ΔT according to the slope indicated by ¼ of the quantity in parentheses. This makes

simple linear dependence on ΔT. Note, that the material properties are all temperature dependent, so taking the derivative would necessarily yield higher-order terms. However, we make use of the following assumptions: (1) the temperature dependence of the electrical component of thermal conductivity depends on the mobility of charge carriers, and the electrical resistivity depends on the inverse of that mobility, so these dependencies can be assumed to be first-order to cancel completely. (2) Seebeck coefficient does have a relatively small, but finite temperature dependence; however the derivative of Seebeck coefficient should yield a temperature dependence of T-1 which approximately cancels with the temperature dependence of lattice thermal conductivity in the denominator. So, to first order the linearity of the slope would be expected,

A new index to determine maximum ZT of TEG device can be obtained by measuring the slope of TEG efficiency. To calculate maximum ZT, four times the slope of TEG efficiency multiplied by maximum temperature, under which TEG displays linear behavior with respect to ΔT. Outside the linear regime of TEGs, the basic properties can no longer be described by these

2

1

ðκnAn þ κpApÞ

functions. Therefore, measure of maximum ZT of TEG device can be determined by:

0 @

ρn An <sup>þ</sup> <sup>ρ</sup><sup>p</sup> Ap � �

The significance of this analysis is that it allows unique means to rapidly obtain ZTmaximum and confirm properties and individual measurements. Measurements can be confirmed by measuring slope of efficiency as function of ΔT and ZT can be obtained and compared to theoretical

Efficiency of commercial (Bi,Sb)2(Te,Se)3 device is presented in Figure 18. This device is designed for high thermal impedance and has optimum performance window from nearly room-temperature to roughly 425 K. Slope of efficiency was determined and is shown in inset of Figure 18, and is given as 0.0004/K. As expected, the slope is highly linear function of ΔT until deviation from non-linearity begins at 405 K. The maximum ZT can be obtained by the

Obtained value of ZTmaximum is equal to 0.7, which is consistent with established values for

∂ΔT � �

<sup>ð</sup>MaximumÞZdeviceT <sup>¼</sup> <sup>4</sup> <sup>ð</sup>α<sup>n</sup> <sup>þ</sup> <sup>α</sup>p<sup>Þ</sup>

simple relationship, and observed maximum temperature of roughly 405 K:

<sup>Z</sup>deviceTmaximum <sup>¼</sup> <sup>4</sup> <sup>∂</sup><sup>Φ</sup>

ρn An <sup>þ</sup> <sup>ρ</sup><sup>p</sup> A<sup>p</sup> � �

sense, because Φ increases linearly with ΔT, and Pmax increases as ΔT<sup>2</sup>

0 @

2

1

<sup>A</sup> <sup>¼</sup> <sup>Z</sup>device

<sup>4</sup> : (75)

. Taking the ratio yields a

ATmaximum: (76)

Tmaximum: (77)

ðκnAn þ κpApÞ

Efficiency of PbTe/TAGS device is presented below. Figure 19 shows temperature dependence of PbTe/TAGS module efficiency. At low temperature, the slope is somewhat a nonlinear function of ΔT because, it is well known, that properties of these materials are uninteresting at low-temperature, but optimum at elevated temperature. So, slope of efficiency is measured in temperature range >500 K, where properties are linear. The maximum ZT can, therefore, be obtained by Eq. (77), and observed maximum temperature for linear device behavior, which for the device being measured is equal to 873 K. Obtained slope is 0.0002/K resulting in value ZTmaximum = 0.7, which is consistent with established values for well-known PbTe/TAGS modules.

#### 3.2. Discretized heat-balance model and analysis

More detailed device analysis and performance modeling including effects of temperaturedependent material properties may be accomplished through the use of numerical methods. One technique for performing numerical analysis on TEG was reported by Lau and Buist [26] and later confirmed and expanded upon by Hogan and Shih [27]. It involves partitioning the legs of TEG into virtual segments for computational purposes, where each segment is taken to be isothermal. Neighboring segments then vary in temperature such, that governing thermoelectric heat balance equations based on constant parameter theory are satisfied [28, 29]. This process is illustrated in Figure 20.

Figure 19. Slope of efficiency from pre-commercial PbTe/TAGS to determine ZTmax.

Figure 20. Discrete communicating layers having thermal and electrical flux continuity. Expanded view of i th segment explicitly showing the heat flows that must be balanced to maintain continuity through the bulk of the TEG device.

Based on constant-parameter theory [29] and heat balance at the top surface of the ith segment, we have:

$$\mathbf{Q}\_{i} = \mathbf{a}\_{i}\mathbf{I}\mathbf{T}\_{i} - \mathbf{I}^{2}\mathbf{R}\_{i}/2 + \mathbf{K}\_{i}(T\_{i} - T\_{i-1}),\tag{78}$$

where the total heat flux Q<sup>i</sup> into segment is the sum of individual components indicated.

Here α<sup>i</sup> is Seebeck coefficient of the ith segment at temperature T, R<sup>i</sup> is the electrical resistance of the ith segment at temperature T, Ki is thermal conductance of the ith segment at temperature T. Last, I, and T<sup>i</sup> are the electrical current and temperature of the ith segment, respectively. The electrical power, P, is determined by:

$$Q\_{i-1} = Q\_i \text{--} P,\tag{79}$$

where power P is delivered to the external load resistor, RL:

$$P = I^2 \mathcal{R}\_L.\tag{80}$$

The discrete heat balance equations (Eqs. (78) and (79)) derived in this manner can be easily solved for single leg of TEG with an iterative technique [26, 27]. For a given TC, an initial estimate is made for the heat delivered to the cold junction, QC, and temperature and heat flow in each segment is determined sequentially, ending with a numerical solution for heat absorbed at the hot junctionQ<sup>H</sup> and the hot-side temperature TH. If calculated T<sup>H</sup> is not equal to the desired TH boundary condition, then Q<sup>C</sup> is adjusted accordingly and the process is repeated until the desired T<sup>H</sup> is achieved.

The initial hot-side temperature of segment is taken as a uniform temperature for the entire segment and its thermoelectric properties are then determined from a curve fit to measured data. Adjacent segments attain different temperatures as the system is solved according to the energy balance requirements. Thus, temperature-dependent effects are fully incorporated into the model. In fact, Hogan and Shih [27] were able to demonstrate excellent agreement using the discrete approximation as compared with an exact analysis of temperature-dependent TEG performance by Sherman et.al. [30].

Figure 21 shows temperature profiles calculated for n-type Skutterudite material with temperature-dependent properties, operating at the indicated boundary conditions. For simplicity, electrical current is treated here as though there were an external load resistance matched to the internal resistance of the thermoelectric material leg, thus producing maximum output power.

It is also instructive to examine calculated heat flow through the leg as this helps to illustrate thermal-to-electrical conversion process. Figure 22 shows heat flow corresponding to temperature profiles depicted in Figure 21. From the hot side to the cold side of the leg (or right to left on the plot), heat flow is reduced as thermal energy is converted to electrical power and delivered to the load. Examining the specific case of T<sup>H</sup> = 750 K and T<sup>C</sup> = 300 K, there is approximately 5.7 W of thermal power incident on the hot side and 5.0 W rejected at the cold side, leaving 0.7 W which is delivered as electrical power to the load. So the conversion efficiency within just one of the legs of the TEG itself is simply 0.7 W/5.7 W = 12.3%. The same methodology may be applied to the companion p-type leg to complete the analysis of a full TEG at a given set of temperature boundary conditions. Proper temperature profiles of n-type and p-type legs together are shown in Figure 23. Taking T<sup>H</sup> = 800 K and T<sup>C</sup> = 300 K, the combined incident thermal power is equal to 6.42 Wattsthermal (n-type) + 7.85 Wattsthermal (ptype) = 14.27 Wattsthermal (TEG). And that rejected to the cold side is equal to 5.58 Wattsthermal (n-type) + 7.16 Wattsthermal (p-type) = 12.74 Wattsthermal (TEG). This calculation finds the overall efficiency = 1.53 W/14.27 W = 10.7%.

Based on constant-parameter theory [29] and heat balance at the top surface of the ith segment,

Ri=2 þ KiðTi−Ti<sup>−</sup>1Þ, (78)

th segment

2

Figure 20. Discrete communicating layers having thermal and electrical flux continuity. Expanded view of i

explicitly showing the heat flows that must be balanced to maintain continuity through the bulk of the TEG device.

where the total heat flux Q<sup>i</sup> into segment is the sum of individual components indicated.

Qi ¼ aiITi–I

Figure 19. Slope of efficiency from pre-commercial PbTe/TAGS to determine ZTmax.

426 Thermoelectrics for Power Generation - A Look at Trends in the Technology

we have:

Figure 21. Temperature profile in n-type Skutterudite at maximum power. T<sup>C</sup> = 300 K.

Figure 22. Heat flow profile in n-type Skutterudite sample at maximum output power, corresponding to temperature profiles depicted in Figure 21. TC = 300 K.

Figure 23. Temperature profiles in both n- and p-type legs of TEG. TC = 300 K.

Figure 21. Temperature profile in n-type Skutterudite at maximum power. T<sup>C</sup> = 300 K.

428 Thermoelectrics for Power Generation - A Look at Trends in the Technology

profiles depicted in Figure 21. TC = 300 K.

Figure 22. Heat flow profile in n-type Skutterudite sample at maximum output power, corresponding to temperature

This discussion has highlighted a simple, but powerful temperature-dependent phenomenological model for precisely calculating temperature profiles, heat flows, power outputs, and efficiencies in a single leg of TEG. Full TEG device modeling is accomplished by simultaneously solving the discrete heat balances as described for each leg (subject to hot side and cold side boundary conditions) along with the simultaneous energy balance relationship required for electrical power being delivered to the load.

Figure 24 shows calculated efficiency using the discretized heat balance theory of idealized Skutterudite n-type and Skutterudite p-type devices.

Electrical and thermal contact resistivities are defined to be zero, but could easily be included as finite quantities, which would add penalties to the efficiency. Slope of efficiency identified in the best-fit is quantified first using unitless efficiency data. The equation is then re-included on the plot after converting to percent. This is so that ZTmaximum can be calculated using slope/ efficiency method described in Section 3.1, and for overall clarity in the final plot. Slope of efficiency is 0.0002/K, and the upper-limit temperature is 800 K, so, following Eq. (77), value ZTmaximum= 0.64 is obtained for the device.

Therefore, to confirm measurements for device fabricated using materials, from which measurements were collected, it could be assembled and the efficiency is measured. If measurements are accurate and not overestimated, then performance should be consistent with ZTmaximum value equals to 0.64. The slope of the data should be roughly 0.0002/K.

Figure 24. Calculated efficiency of single p-n couple TEG. TC = 300 K and Th(max) = 800 K.

## 4. Conclusions

In conclusion, we have presented several novel approaches to the significant challenge of accurately determining the thermal conductivity of thermoelectric materials. The new solutions can be much faster experimentally, and they successfully address several sources of experimental error. The overall result is significantly reduced error, which may reduce uncertainty by a factor of 2 or more. Further, we introduce new approaches to compare device performance with physical property measurements as a novel means of confirming measurements. Using this approach, the new measurements can be clearly seen to yield physical property measurements which are more consistent with physical device performance.

The first new thermal conductivity measurement method, steady-state isothermal technique, improves accuracy by collecting data under conditions where thermal losses and errors are unimportant. The validity was confirmed by comparing the thermal conductivity extracted from a Peltier cooling device with the lab measured value: the error was ∼2% [8]. The second is nondestructive micro-scale analysis technique called the scanning hot-probe, and the third is lock-in transient Harman method, which is a comprehensive modification of transient Harman technique. The second and third methods reduce error by highly detailed treatment of interfacial contact effects including electrical contact resistance and thermal contact effects. The high accuracy for both of these methods is obtained by comparison with established standard reference materials whose properties are well-known and accepted. A new interesting follow-on is frequency-dependent Nyquist analysis, which presages a different perspective on the materials analysis, and even further simplified measurement.

The truest test of the accuracy of measurement is comparison with fabricated devices. To support the validation of measurements of individual material properties, we have outlined new device metrics, which allows comparison between theoretical and measured device efficiency. We outline a new slope-efficiency method, which can be used to determine informative index ZTmaximum of any device. The second method of device evaluation is a numerical device model called the discretized heat balance model. Using this modeling approach, we showed, that a piecewise continuous collection of discrete layers within a device, where boundary heat flows have energy and current continuity relationships, can yield an incredibly easy theoretical determination of device efficiency, which can be compared with experimental values.

## Author details

Patrick J. Taylor1 \*, Adam Wilson1,2, Jay R. Maddux1,3, Theodorian Borca-Tasciuc2 , Samuel P. Moran<sup>2</sup> , Eduardo Castillo<sup>2</sup> and Diana Borca-Tasciuc2

\*Address all correspondence to: patrick.j.taylor36.civ@mail.mil

1 US Army Research Laboratory, Sensors and Electron Devices Directorate, Adelphi, MD, USA

2 Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY, USA

3 General Technical Services, NJ, USA

## References

4. Conclusions

In conclusion, we have presented several novel approaches to the significant challenge of accurately determining the thermal conductivity of thermoelectric materials. The new solutions can be much faster experimentally, and they successfully address several sources of experimental error. The overall result is significantly reduced error, which may reduce uncertainty by a factor of 2 or more. Further, we introduce new approaches to compare device performance with physical property measurements as a novel means of confirming measurements. Using this approach, the new measurements can be clearly seen to yield physical

Figure 24. Calculated efficiency of single p-n couple TEG. TC = 300 K and Th(max) = 800 K.

430 Thermoelectrics for Power Generation - A Look at Trends in the Technology

property measurements which are more consistent with physical device performance.

spective on the materials analysis, and even further simplified measurement.

The first new thermal conductivity measurement method, steady-state isothermal technique, improves accuracy by collecting data under conditions where thermal losses and errors are unimportant. The validity was confirmed by comparing the thermal conductivity extracted from a Peltier cooling device with the lab measured value: the error was ∼2% [8]. The second is nondestructive micro-scale analysis technique called the scanning hot-probe, and the third is lock-in transient Harman method, which is a comprehensive modification of transient Harman technique. The second and third methods reduce error by highly detailed treatment of interfacial contact effects including electrical contact resistance and thermal contact effects. The high accuracy for both of these methods is obtained by comparison with established standard reference materials whose properties are well-known and accepted. A new interesting follow-on is frequency-dependent Nyquist analysis, which presages a different per-


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**Thermoelectric Generators Simulation, Modeling and Design**

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9718.ch39

#### **Thermoelectric Power Generation Optimization by Thermal Design Means Thermoelectric Power Generation Optimization by Thermal Design Means**

Patricia Aranguren and David Astrain Patricia Aranguren and David Astrain

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/65849

#### **Abstract**

One of the biggest challenges of the twenty‐first century is to satisfy the demand for electrical energy in an environmentally speaking clean way. Thus, it is very important to search for new alternative energy sources along with increasing the efficiency of current processes. Thermoelectric power generation, by means of harvesting waste heat and converting it into electricity, can help to achieve above‐mentioned goal. Nowadays, efficiency of thermoelectric power generators limits them to become key technology in electric power generation, but their performance has potential of being optimized, if thermal design of such generators is optimized. Heat exchangers located on both sides of thermoelectric modules (TEMs), mass flow of refrigerants and occupancy ratio (the area covered by TEMs related to base area), among others, need to be fine‐tuned in order to obtain the maximum net power generation (thermoelectric power generation minus consumption of auxiliary equipment). Finned dissipator, cold plate, heat pipe and thermosiphon are experimentally tested to maximize net thermoelectric generation on real‐working furnace based on computational model. Maximum generation of 137  MWh/year using thermosiphons is achieved with 32% of area covered by TEMs.

**Keywords:** thermoelectric generator, optimization, computational model, heat ex‐ changer, occupancy ratio

## **1. Introduction**

The excessive use of fossil fuels has lead into severe environmental issues. Consequently, global warming, greenhouse gases emissions, climate change, acid rain and ozone depletion are commonly heard on the media. Moreover, combustible resources are limited, and more

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

restrict environmental regulations are arising. Hence, one of the biggest challenges of the twenty‐first century is to satisfy energetic demand in environmentally friendly manner.

In order to fulfill the previous aim, new tendencies are springing, such as smart utilization of energy throughout boosting savings, avoiding waste and developing more efficient, so as less fuel consuming, equipment and through the development of renewable energies. Thermo‐ electric generation contributes to diminish the impact that fossil fuels generate. A better exploitation of fossil fuels is possible due to their potential to harvest waste heat and convert it into electricity, improving efficiency of energy generating systems.

Nowadays, thermoelectrics is an emerging technology, which converts waste heat into electricity. Solid‐state operation of thermoelectric generators (TEGs) eliminates the presence of moving parts and/or chemical reactions, and thus the maintenance is reduced to minimum. It cancels greenhouse gases emissions to environment, and long lives are achieved due to safe operation of thermoelectric generators.

Waste heat is defined as by‐product heat of a process, which is not exploited afterward, but it is emitted to the ambient. Nowadays, great amount of produced energy is lavished as waste heat, and at least 40% of the primary energy utilized in industrialized countries is emitted to the ambient as waste heat [1]. Nevertheless, most of this waste heat presents low temperature levels (low temperature grade heat), as **Figure 1** presents, explaining the most studied use up to the moment, heating of fluids for heating or other purposes [2–4]. It has been estimated, that double the heating needs of the United States, the 16.4 % of the primary energy consumed worldwide, could be supplied with waste heat [1].

Particular temperature grade, that waste heat presents, restricts applicable technologies to harvest it with effective conversion to electricity. However, thermoelectricity is a promising technology to recover low temperature grade waste heat [5]. Several studies have ratified promising future, that TEGs demonstrate ability to produce electric energy from waste heat of different applications. Some of them are introduced here: Bi2Te3‐PbTe TEG obtains 211 kW electrical power from waste heat of Portland Cement Rotary Kilns [6]; study conducted in Japan presents potential of recovering radiant heat from steelmaking processes with 10‐kW‐class grid‐connected TEG system [7]; thermoelectric power density of approximately 193.1 W/m2 is obtained from waste heat of biomass gasifier [8], while power density nearly 100 W/m2 is obtained from combustion chamber [9]; thermoelectric generator integrated within photovol‐ taic/thermal absorber improves total efficiency of generating system [10]; nearly 5 kWh/year‐ m2 can be produced from solar ponds [11]; and the most common and studied TEGs, which recover waste heat from exhaust gas of vehicles in order to improve their efficiency [12–14].

restrict environmental regulations are arising. Hence, one of the biggest challenges of the twenty‐first century is to satisfy energetic demand in environmentally friendly manner.

In order to fulfill the previous aim, new tendencies are springing, such as smart utilization of energy throughout boosting savings, avoiding waste and developing more efficient, so as less fuel consuming, equipment and through the development of renewable energies. Thermo‐ electric generation contributes to diminish the impact that fossil fuels generate. A better exploitation of fossil fuels is possible due to their potential to harvest waste heat and convert

Nowadays, thermoelectrics is an emerging technology, which converts waste heat into electricity. Solid‐state operation of thermoelectric generators (TEGs) eliminates the presence of moving parts and/or chemical reactions, and thus the maintenance is reduced to minimum. It cancels greenhouse gases emissions to environment, and long lives are achieved due to safe

Waste heat is defined as by‐product heat of a process, which is not exploited afterward, but it is emitted to the ambient. Nowadays, great amount of produced energy is lavished as waste heat, and at least 40% of the primary energy utilized in industrialized countries is emitted to the ambient as waste heat [1]. Nevertheless, most of this waste heat presents low temperature levels (low temperature grade heat), as **Figure 1** presents, explaining the most studied use up to the moment, heating of fluids for heating or other purposes [2–4]. It has been estimated, that double the heating needs of the United States, the 16.4 % of the primary energy consumed

Particular temperature grade, that waste heat presents, restricts applicable technologies to harvest it with effective conversion to electricity. However, thermoelectricity is a promising technology to recover low temperature grade waste heat [5]. Several studies have ratified promising future, that TEGs demonstrate ability to produce electric energy from waste heat

it into electricity, improving efficiency of energy generating systems.

438 Thermoelectrics for Power Generation - A Look at Trends in the Technology

operation of thermoelectric generators.

**Figure 1.** Temperature grade of waste heat [1].

worldwide, could be supplied with waste heat [1].

Efficiency, that normally TEGs present between 5 and 10% [15, 16], is deterrent to make these systems attractive enough to pass the thin line between laboratory experimentation and simulation and commercialization and expansion of this technology. Nowadays, the two issues that are the main objectives are to improve efficiency of thermoelectric generation systems: the first objective is development and improvement of thermoelectric materials through modifi‐ cation of conventional materials with new technologies, such as introducing nanostructures into conventional semiconductors [17, 18] or creating novel thermoelectric materials, such as polymers [19], oxides [20], half‐heusler [21] or skutterudites [22]; the second objective is to optimize thermal design of the system. To achieve the latter objective, different approaches can be studied and implemented, for example, heat exchangers located on both sides of thermo‐ electric modules can be optimized through many different approaches that will be detailed afterwards, and also the number of thermoelectric modules (TEMs) has to be properly selected to reach the maximum thermoelectric generation. Occupancy ratio δ, parameter that includes number of used TEMs *MTEM*, that is, ratio between area covered by TEMs *ATEM* and base area *Ab* of heat exchangers (Eq. (1)), is crucial parameter to optimize thermoelectric generation:

$$\mathfrak{G} = \frac{M\_{\rm TEM} A\_{\rm TEM}}{A\_b}.\tag{1}$$

Although it seems, that higher number of thermoelectric modules would mean higher thermoelectric power generation, thermal resistance per thermoelectric module of heat exchangers worsens, if occupancy ratio rises, resulting in reduction in thermoelectric power generation per TEM. Each application presents optimum point, where thermoelectric power generation is maximum [23–25]. Moreover, reduction in the number of modules does not only imply increase in thermoelectric power generation, but also decrease in initial investment.

Optimization of heat exchangers attached to hot and cold sides of TEMs is very important to maximize thermoelectric power generation, and improvement in thermal resistances will result in higher temperature difference between hot and cold TEM sides close to temperature difference between heat exchangers, and, hence, will provide higher thermoelectric power generation [26–29]. Optimization of heat dissipation systems can be done by modifying their geometry, such as increasing number, height or spacing of fins of finned dissipator [30, 31] or by properly selecting channel's diameter, internal distribution and/or internal inserts of cold plates [32–35]. Besides, inclusion of novel heat exchangers, such as heat pipes [23, 36] or thermosiphons [37–39], could procure higher thermoelectric power generation. Nevertheless, increase in power generation does not necessarily mean improvement in net generation (usable energy obtained from any application) due to increase in the consumption of auxiliary equipment, coolant pump or fans, in order to optimize the thermal behavior of the systems [9, 33, 34, 40].

In this chapter, computational optimization of real furnace located in Spain is performed giving experimental data of thermal resistances of different kinds of heat exchangers (finned dissi‐ pator, cold plate, heat pipe and thermosiphon) as function of occupancy ratio, mass flow of refrigerants and heat power to dissipate. Net power generation, that is, thermoelectric power generation minus power consumption of auxiliary equipment Eq. (2), is computed and maximized by means of previously mentioned parameters:

$$
\dot{\mathcal{W}}\_{net} = \dot{\mathcal{W}}\_{\text{TEM}} - \dot{\mathcal{W}}\_{\text{aux}}.\tag{2}
$$

## **2. Computational methodology**

Thermoelectric generators produce electricity when there is temperature gradient between hot and cold sides of TEM. Therefore, harvesting of waste heat to produce electricity by thermo‐ electric generation is becoming very interesting field of studying. Gratuity of waste heat and its great presence in numerous applications overcome low efficiency values, that TEGs present; however, until to date not many applications have been materialized. Initial investment and payback time (due to low efficiency) are deterrents for the development of this technology. This is the reason why computational models are playing very important role in the develop‐ ment of thermoelectric power generation. Due to complicated physical phenomena, that take place in TEGs, knowledge of TEG‐based systems' behavior in different conditions is crucial to evaluate their potential, as well as to improve their performance, basing on both thermoelectric material properties and properties and dimensions of heat exchangers located on both sides of TEMs.

Modeling of each component of TEG is essential to perform accurate simulation of behavior of TEG‐based systems. TEG is formed by TEMs (which present thermoelectric material, ceramic plates, joints…), the heat exchangers that are located on both sides of thermoelectric modules, as well as by any elementary component for correct assembly of the whole system; consequently, everything needs to be included into the model [41, 42]. Moreover, each thermoelectric phenomenon (Seebeck effect, Thomson effect, Peltier effect and Joule effect) needs to be taken into account, especially in thermoelectric generation due to significant temperature difference between hot and cold sides of TEMs, to obtain accurate results [43– 45]; likewise, thermoelectric properties need to be defined as function of temperature not to commit big errors [46]. Furthermore, resolution has to bear in mind transient state of operation [47, 48], especially if trying to model combustion systems with permanent changes in per‐ formance and, thus, permanent changes in temperature and mass flow, as vehicles or com‐ bustion stoves. The latest applications are very precious due to gratuity of waste heat.

plates [32–35]. Besides, inclusion of novel heat exchangers, such as heat pipes [23, 36] or thermosiphons [37–39], could procure higher thermoelectric power generation. Nevertheless, increase in power generation does not necessarily mean improvement in net generation (usable energy obtained from any application) due to increase in the consumption of auxiliary equipment, coolant pump or fans, in order to optimize the thermal behavior of the systems [9,

In this chapter, computational optimization of real furnace located in Spain is performed giving experimental data of thermal resistances of different kinds of heat exchangers (finned dissi‐ pator, cold plate, heat pipe and thermosiphon) as function of occupancy ratio, mass flow of refrigerants and heat power to dissipate. Net power generation, that is, thermoelectric power generation minus power consumption of auxiliary equipment Eq. (2), is computed and

Thermoelectric generators produce electricity when there is temperature gradient between hot and cold sides of TEM. Therefore, harvesting of waste heat to produce electricity by thermo‐ electric generation is becoming very interesting field of studying. Gratuity of waste heat and its great presence in numerous applications overcome low efficiency values, that TEGs present; however, until to date not many applications have been materialized. Initial investment and payback time (due to low efficiency) are deterrents for the development of this technology. This is the reason why computational models are playing very important role in the develop‐ ment of thermoelectric power generation. Due to complicated physical phenomena, that take place in TEGs, knowledge of TEG‐based systems' behavior in different conditions is crucial to evaluate their potential, as well as to improve their performance, basing on both thermoelectric material properties and properties and dimensions of heat exchangers located on both sides

Modeling of each component of TEG is essential to perform accurate simulation of behavior of TEG‐based systems. TEG is formed by TEMs (which present thermoelectric material, ceramic plates, joints…), the heat exchangers that are located on both sides of thermoelectric modules, as well as by any elementary component for correct assembly of the whole system; consequently, everything needs to be included into the model [41, 42]. Moreover, each thermoelectric phenomenon (Seebeck effect, Thomson effect, Peltier effect and Joule effect) needs to be taken into account, especially in thermoelectric generation due to significant temperature difference between hot and cold sides of TEMs, to obtain accurate results [43– 45]; likewise, thermoelectric properties need to be defined as function of temperature not to commit big errors [46]. Furthermore, resolution has to bear in mind transient state of operation [47, 48], especially if trying to model combustion systems with permanent changes in per‐

= - && & *WW W net TEM aux*. (2)

maximized by means of previously mentioned parameters:

440 Thermoelectrics for Power Generation - A Look at Trends in the Technology

**2. Computational methodology**

33, 34, 40].

of TEMs.

Computational model developed to optimize any thermoelectric application, especially TEGs, which harvest waste heat to produce electricity, includes each thermoelectric phenomenon, each component of the system, temperature dependence of thermoelectric properties and transient state of operation. Moreover, it includes novel parameters, such as occupancy ratio, that is, the ratio between area covered by thermoelectric modules and dissipative base area (Eq. (1)), mass flow of refrigerants and temperature decrease in flue gases when flowing along TEG. Previously mentioned parameters are determinant of net thermoelectric power genera‐ tion (Eq. (2)), the main parameter to optimize in any application.

Computational methodology uses finite differences approach to solve behavior of the system. It solves each thermoelectric phenomenon, Seebeck effect Eq. (3), Peltier effect Eq. (4), Thomson effect Eq. (5) and Joule effect Eq. (6), and it includes Fourier law of heat conduction used in one‐dimensional form, when heat power generation Eq. (7) takes place:

$$
\alpha\_{AB} = \frac{dE\_t}{dT} = \alpha\_A - \alpha\_{B'} \tag{3}
$$

$$
\dot{Q}\_{Peltiter} = \pm \pi\_{AB} l = \pm lT(a\_A - a\_B) \tag{4}
$$

$$
\dot{Q}\_{Thomson} = -\sigma \vec{I} (\overline{\Delta T}) \tag{5}
$$

$$
\dot{Q}\_{Joule} = \mathbb{R}\_0 I^2 \, , \tag{6}
$$

$$
\rho c\_p \frac{\delta T}{\delta t} = k \left( \frac{\delta^2 T}{\delta \mathbf{x}\_2} \right) + \overline{q}. \tag{7}
$$

Resolution methodology is based on previously published and validated computational model [26, 49].

Temperature decrease in flue gases is achieved by discretizing pipe, where flue gases circulate. Within each block, thermoelectric phenomenon is solved. To that objective, temperature of flue gases must be known. Temperature of heat source in each block is selected as the mean temperature between entry and exit temperatures of each block, = <sup>=</sup> <sup>1</sup> <sup>2</sup> + . Exit temperature is obtained using heat power extracted from flue gases by TEG in that block, Eq. (8). As blocks are located sequentially, exit temperature of previous block coincides with entry temperature of the following block, + 1 = :

$$T\_s^i = T\_e^i - \frac{\dot{\mathbf{Q}}^i}{\dot{m}\_{gas}c\_p}.\tag{8}$$

**Figure 2** presents block "i" of pipe and discretization of that block in order to apply finite differences method to solve thermoelectric phenomena. There are totally 16 nodes, which represent the whole TEG: node 1 is heat source, while node 16 is heat sink; nodes 2 and 15 are hot side and cold side heat exchangers, respectively; and nodes 3–14 represent TEM, where nodes 3 and 14 are hot and cold sides and from node 4 to node 13 thermoelectric material is represented. Electrical analogy is composed by thermal resistances, thermal capacities and absorbed or generated heat fluxes. and stand for resistances of hot side and cold side heat dissipators, respectively, are contact resistances and and stand for two alternative ways for heat power to reach heat sink. The best scenario would be where the total amount of heat power circulates through thermoelectric modules, but in real application there are parasitic heats that circulate along other elements. In this case, heat power that reaches cold sink directly from hot source is born in mind through , and heat power that flows through assembling screws attaching cold and hot dissipators is represented by .

**Figure 2.** Thermoelectric generator discretization of block "i".

Particularly, this model considers temperature loss of flue gases, while they circulate along TEG, occupancy ratio and mass flow of refrigerants. Methodology used can be seen in **Figure 3**. The first step is to choose the number of blocks, in which the pipe is discretized, . Once this parameter is selected and information of application is introduced into the model, resolution starts from the first block, where the mean temperature of the block is supposed to be entry temperature of the block, in the case of first block's temperature is that of flue gases. The next step is to suppose heat power that needs to be dissipated by heat exchangers, ˙ , parameter that determines thermal resistance of dissipation systems, as it will be seen in the next section. Thermal resistances of dissipation systems are now determined, so the finite differences method can be used to solve thermoelectric phenom‐ ena, obtaining heat power to dissipate and closing the most interior iteration loop. As heat dissipators are function of heat power to dissipate, and at the same time, they define amount of heat that TEG is extracting from flue gases, this issue is solved through an iteration process, which obtains the heat power to dissipate. Once known, the mean tem‐ perature of the block needs to be obtained. The mean temperature is computed as the mean value between entry and exit temperatures of flue gases, and exit temperature is obtained using Eq. (8), so new iteration loop solves this situation. Finally, when everything has converged, thermoelectric generation is saved and resolution follows to the next block. This procedure keeps on until each block has been solved and the total power generation has been computed. **Figure 3** presents schematic of the methodology used to obtain ther‐ moelectric power generation.

**Figure 3.** Computational model for thermoelectric power generation.

= - . & &

*i i s e*

absorbed or generated heat fluxes.

sink directly from hot source is born in mind through

442 Thermoelectrics for Power Generation - A Look at Trends in the Technology

**Figure 2.** Thermoelectric generator discretization of block "i".

heat dissipators, respectively,

*<sup>Q</sup> T T*

and

assembling screws attaching cold and hot dissipators is represented by

*i*

*gas p*

**Figure 2** presents block "i" of pipe and discretization of that block in order to apply finite differences method to solve thermoelectric phenomena. There are totally 16 nodes, which represent the whole TEG: node 1 is heat source, while node 16 is heat sink; nodes 2 and 15 are hot side and cold side heat exchangers, respectively; and nodes 3–14 represent TEM, where nodes 3 and 14 are hot and cold sides and from node 4 to node 13 thermoelectric material is represented. Electrical analogy is composed by thermal resistances, thermal capacities and

are contact resistances and

alternative ways for heat power to reach heat sink. The best scenario would be where the total amount of heat power circulates through thermoelectric modules, but in real application there are parasitic heats that circulate along other elements. In this case, heat power that reaches cold

Particularly, this model considers temperature loss of flue gases, while they circulate along TEG, occupancy ratio and mass flow of refrigerants. Methodology used can be seen in **Figure 3**. The first step is to choose the number of blocks, in which the pipe is discretized, . Once this parameter is selected and information of application is introduced into the model, resolution starts from the first block, where the mean temperature of the block is

*m c* (8)

stand for resistances of hot side and cold side

and

, and heat power that flows through

.

stand for two

Net power generation, Eq. (2), is afterward obtained, giving power consumption of auxiliary equipment, determined by the test conducted to thermally characterize the different types of heat exchangers studied, finned dissipator, cold plate, heat pipe and thermosiphon. Experi‐ mental thermal characterization of these systems is explained in the next section, very important data that are essential to include into the computational model in order to calculate accurate results about thermoelectric power generation from any application.

#### **3. Thermal characterization of heat exchangers**

Thermal characterization of heat dissipation systems is crucial to obtain accurate results using computational model presented in the above section. Four different heat dissipation systems (cold plate, finned dissipator, heat pipe and thermosiphon) have been experimentally tested in order to obtain their thermal resistances as function of influential parameters in thermo‐ electric power generation: occupancy ratio, mass flow of refrigerants and heat power to dissipate.

Thermal resistances are presented as thermal resistances per thermoelectric module and computed through experimentation as Eq. (9) presents:

$$R^{TEM} = \frac{T\_m^{HX} - T\_{amb}}{\frac{\dot{Q}\_{\mathbb{C}}}{M\_{TEM}}}.\tag{9}$$

 stands for average temperature of each heat exchanger, where heat is applied and represents ambient temperature, it is selected constant, 22°C, as heat dissipation systems are placed into climatic chamber ensuring constant temperature during experiments. To test different occupancy ratios, heat plates of the same size of TEMs have been used. Heat power to dissipate corresponds to electric power supplied to heat plates (˙ = ). One side of heat plate is thermally isolated to assure that total supplied electrical power is transformed into heat power directed to heat exchangers. Finally, the number of TEMs (MTEM) contributes to get medium thermal resistance per thermoelectric module of each heat exchanger.

Variable parameters during experiments are occupancy ratio, heat power to dissipate and mass flow of the refrigerants. Each configuration has been replicated three times ( = 3) to reduce the random standard uncertainty of the mean ( ). Expanded uncertainty of measured resistances (Eq. (10)) is composed of previously mentioned uncertainty (Eqs. (11) and (12)), systematic standard uncertainty (Eq. (13)) and level of confidence, in this case chosen to be the 95% [50]:

$$\mathcal{U}\_{\mathcal{R}^{TEM}} = 2 \left( b\_{\mathcal{R}^{TEM}}^2 + s\_{\overline{\mathcal{R}}^{TEM}}^2 \right)^{\frac{1}{2}} \, \, \, \, \tag{10}$$

Thermoelectric Power Generation Optimization by Thermal Design Means http://dx.doi.org/10.5772/65849 445

$$s\_{\overline{R}^{TEM}}^2 = \frac{1}{M\_{sample} \left(M\_{sample} - 1\right)} \sum\_{k=1}^{M\_{sample}} \left(R\_k^{TEM} - \overline{R}^{TEM}\right)^2,\tag{11}$$

$$\overline{R}^{TEM} = \frac{1}{M\_{sample}} \sum\_{k=1}^{M\_{sample}} R\_k^{TEM} \, , \tag{12}$$

$$b\_{R^{TEM}}^2 = \left(\frac{\partial \mathbb{R}^{TEM}}{\partial T\_{\text{m}}^{\text{HK}}}\right)^2 b\_{T\_{\text{m}}}^2 + \left(\frac{\partial \mathbb{R}^{TEM}}{\partial T\_{\text{amb}}}\right)^2 b\_{T\_{\text{amb}}}^2 + \left(\frac{\partial \mathbb{R}^{TEM}}{\partial V\_{\text{HP}}}\right)^2 b\_{V\_{\text{HP}}}^2 + \left(\frac{\partial \mathbb{R}^{TEM}}{\partial I\_{\text{HP}}}\right)^2 b\_{I\_{\text{HP}}}^2. \tag{13}$$

#### **3.1. Cold plate**

important data that are essential to include into the computational model in order to calculate

Thermal characterization of heat dissipation systems is crucial to obtain accurate results using computational model presented in the above section. Four different heat dissipation systems (cold plate, finned dissipator, heat pipe and thermosiphon) have been experimentally tested in order to obtain their thermal resistances as function of influential parameters in thermo‐ electric power generation: occupancy ratio, mass flow of refrigerants and heat power to

Thermal resistances are presented as thermal resistances per thermoelectric module and

*C TEM*

 stands for average temperature of each heat exchanger, where heat is applied and represents ambient temperature, it is selected constant, 22°C, as heat dissipation systems are placed into climatic chamber ensuring constant temperature during experiments. To test different occupancy ratios, heat plates of the same size of TEMs have been used. Heat power

heat plate is thermally isolated to assure that total supplied electrical power is transformed into heat power directed to heat exchangers. Finally, the number of TEMs (MTEM) contributes

Variable parameters during experiments are occupancy ratio, heat power to dissipate and mass flow of the refrigerants. Each configuration has been replicated three times ( = 3) to

measured resistances (Eq. (10)) is composed of previously mentioned uncertainty (Eqs. (11) and (12)), systematic standard uncertainty (Eq. (13)) and level of confidence, in this case chosen

1

2 2 <sup>2</sup> 2 , (10)

to get medium thermal resistance per thermoelectric module of each heat exchanger.

*U bs R RR TEM* = + ( *TEM TEM* )

. (9)

=

). Expanded uncertainty of

). One side of


*T T*

*Q M*

accurate results about thermoelectric power generation from any application.

**3. Thermal characterization of heat exchangers**

444 Thermoelectrics for Power Generation - A Look at Trends in the Technology

computed through experimentation as Eq. (9) presents:

*R*

to dissipate corresponds to electric power supplied to heat plates (˙

reduce the random standard uncertainty of the mean (

dissipate.

to be the 95% [50]:

Use of fluids as heat carrier enhances thermal transfer. In the case of tested system, water has been used in order to characterize heat dissipation system thermally and to analyze results, if net thermoelectric generation increases. Heat dissipation system is formed by cold plate (cold side heat exchanger), fan‐coil composed by core and fans to make air circulate through its fins (the secondary heat exchanger in charge of reducing temperature of heat carrier fluid), pump, necessary elements to direct fluid flow and secure safe performance of dissipation system and sensors to obtain the data, as shown in **Figure 4** [51]. Cold plate has 26 transversal channels with diameter 6.2 mm and two manifolds to distribute water coolant along the channels. Plate exterior dimensions are 190 mm × 230 mm. The fan‐coil presents core formed by two 8 mm diameter pipes with total number of 12 passes. It is provided with wind tunnel, which presents three fans to make air circulate through its fins. Pump used in the system has been specially chosen, and pumping level can be chosen from one to four using switch.

**Figure 4.** Test bench used to obtain experimentally thermal resistance of cold plate system [51].

Power consumption of pump and fans of fan‐coil as function of water and air mass flows, respectively, is shown in **Figure 8**, which represents all relations between mass flows and consumption of auxiliary equipment used for different heat exchangers.

Thermal characterization has been performed using experimental data and validated compu‐ tational model, enabling to obtain thermal resistances on test bench. Description of the model and validation details can be found in publications, Aranguren et al. [40, 51]. Thermal resist‐ ance of cold plate is not function of heat power to dissipate due to small influence of this parameter on temperature of water coolant, the term that could influence on thermal resistance. **Figure 5a** depicts influence of heat power for specific water mass flow. **Figure 5b** presents the influence of occupancy ratio on thermal resistance per thermoelectric module for fixed heat power and water mass flow. As ratio grows, implying that number of TEMs grows, thermal resistance worsens due to the reduction in dissipative area per thermoelectric module. **Figure 5c** presents dependence of thermal resistance air and water mass flows at different occupancy ratio. Occupancy ratio has a great influence on thermal resistance, showing that increasing number of modules harms thermal resistance. Within the same occupancy ratio, water and air mass flow show influence on thermal resistance, most notable for high occupancy ratios, where dissipative area is reduced and any improvement in convective coefficients procures important benefits to thermal resistance.

**Figure 5.** Thermal resistance per thermoelectric module of cold plate. (a) Thermal resistance as function of heat power to dissipate for ˙ = 0.055 kg/s, (b) thermal resistance as function of occupancy ratio for ˙ = 0.044 kg/s, (c) thermal resistance as function of air and water mass flows.

#### **3.2. Finned dissipator**

Finned dissipators up to date have been the most used heat exchangers in thermoelectricity due to their simplicity. Studied finned dissipator has external dimensions of 190 mm × 230 mm, base thickness of 14.5 mm and height, thickness and spacing of fins of 39.5, 1.5 and 3.3 mm, respectively. It is provided with wind tunnel, which includes two fans to make air circulate along its fins. The finned dissipator is shown in **Figure 7a**. Relation between power consump‐ tion of fans and air mass flow is presented in **Figure 8**.

**Figure 6** presents thermal resistance of finned dissipator as function of heat power to dissipate, occupancy ratio and mass flow of air. Heat power to dissipate does not determine thermal resistance, as (shown in **Figure 6a** and **b**). Each panel of **Figure 6** presents thermal resistance of finned dissipator as function of heat power to dissipate, the first one for fixed air mass flow of m˙ ai = 0.024 kg/s and the second one for m˙ ai = 0.060 kg/s. Occupancy ratio influences highly thermal resistance, and higher occupancy ratios procure higher thermal resistances per thermoelectric module, due to the reduction in dissipative area per TEM, as presented in **Figure 6**. **Figure 6d** shows the influence of air mass flow, and for high occupancy ratios, influence is more remarkable than for low ones, due to the higher benefits that improvement in convection coefficients has for small convective areas.

**Figure 6.** Thermal resistance per thermoelectric module of finned dissipator. (a) Thermal resistance as function of heat power to dissipate for ˙ = 0.024 kg/s, (b) thermal resistance as function of heat power to dissipate for ˙ = 0.060 kg/s, (c) thermal resistance as function of occupancy ratio, (d) thermal resistance as function of air mass flow.

The expanded uncertainty of thermal resistance RTEM is equal to ±10.80%.

#### **3.3. Heat pipe**

Power consumption of pump and fans of fan‐coil as function of water and air mass flows, respectively, is shown in **Figure 8**, which represents all relations between mass flows and

Thermal characterization has been performed using experimental data and validated compu‐ tational model, enabling to obtain thermal resistances on test bench. Description of the model and validation details can be found in publications, Aranguren et al. [40, 51]. Thermal resist‐ ance of cold plate is not function of heat power to dissipate due to small influence of this parameter on temperature of water coolant, the term that could influence on thermal resistance. **Figure 5a** depicts influence of heat power for specific water mass flow. **Figure 5b** presents the influence of occupancy ratio on thermal resistance per thermoelectric module for fixed heat power and water mass flow. As ratio grows, implying that number of TEMs grows, thermal resistance worsens due to the reduction in dissipative area per thermoelectric module. **Figure 5c** presents dependence of thermal resistance air and water mass flows at different occupancy ratio. Occupancy ratio has a great influence on thermal resistance, showing that increasing number of modules harms thermal resistance. Within the same occupancy ratio, water and air mass flow show influence on thermal resistance, most notable for high occupancy ratios, where dissipative area is reduced and any improvement in convective coefficients

**Figure 5.** Thermal resistance per thermoelectric module of cold plate. (a) Thermal resistance as function of heat power

Finned dissipators up to date have been the most used heat exchangers in thermoelectricity due to their simplicity. Studied finned dissipator has external dimensions of 190 mm × 230 mm, base thickness of 14.5 mm and height, thickness and spacing of fins of 39.5, 1.5 and 3.3 mm,

= 0.044 kg/s, (c)

= 0.055 kg/s, (b) thermal resistance as function of occupancy ratio for ˙

consumption of auxiliary equipment used for different heat exchangers.

446 Thermoelectrics for Power Generation - A Look at Trends in the Technology

procures important benefits to thermal resistance.

thermal resistance as function of air and water mass flows.

to dissipate for ˙

**3.2. Finned dissipator**

Heat pipes are passive devices able to transfer great amount of heat with small temperature differences. Heat pipes present sealed volumes provided with porous media and divided into three regions: evaporator, where heat is absorbed; condenser, where heat is emitted; and adiabatic region. Working fluid evaporates due to heat gained and flows into the condenser, where it condensates and returns to evaporator due to capillary lift. Tested heat pipe is composed by 10 8 mm diameter pipes with length of 350 mm and spaced 7 mm. Base external dimension of heat pipe is 90 × 192.5 mm2 , and pipes are inserted, being the region, where heat arrives. To help condensation of working fluid water, heat pipe includes wind tunnel provided with fan, as **Figure 7b** presents. Air mass flow as function of power consumption is shown in **Figure 8**.

**Figure 7.** Heat exchanger devices: (a) Finned dissipator; (b) heat pipe.

**Figure 8.** Power consumption of fans of finned dissipator, heat pipe and fan‐coil as function of air mass flow and pow‐ er consumption of the pump of cold plate heat dissipation system as function of water mass flow.

Thermal resistance of heat pipe is function of heat power to dissipate. Condensation and boiling coefficients depend on temperature of fluid and walls, and, therefore, thermal resist‐ ance is function of heat power that has to be dissipated, as shown **Figure 9a**. Occupancy ratio and air mass flow present the same tendency as in previous cases, as shown in **Figure 9b**, **c** and **d**. The expanded uncertainty of thermal resistance RTEM is equal to ±7.88%.

**Figure 9.** Thermal resistance per thermoelectric module of heat pipe. (a) Thermal resistance as function of heat power to dissipate for ˙ = 0.061 kg/s, (b) thermal resistance as function of occupancy ratio for ˙ = 100 W, (c) ther‐ mal resistance as function of air mass flow for ˙ = 100 W, and (d) thermal resistance as function of air mass flow for ˙ = 150 W.

#### **3.4. Thermosiphon**

three regions: evaporator, where heat is absorbed; condenser, where heat is emitted; and adiabatic region. Working fluid evaporates due to heat gained and flows into the condenser, where it condensates and returns to evaporator due to capillary lift. Tested heat pipe is composed by 10 8 mm diameter pipes with length of 350 mm and spaced 7 mm. Base external

arrives. To help condensation of working fluid water, heat pipe includes wind tunnel provided with fan, as **Figure 7b** presents. Air mass flow as function of power consumption is shown in

**Figure 8.** Power consumption of fans of finned dissipator, heat pipe and fan‐coil as function of air mass flow and pow‐

er consumption of the pump of cold plate heat dissipation system as function of water mass flow.

, and pipes are inserted, being the region, where heat

dimension of heat pipe is 90 × 192.5 mm2

448 Thermoelectrics for Power Generation - A Look at Trends in the Technology

**Figure 7.** Heat exchanger devices: (a) Finned dissipator; (b) heat pipe.

**Figure 8**.

Thermosiphons with phase change present the same physical phenomena, than heat pipes, but they do not present porous media. Hence, they need gravitational forces to ensure that condensate heat carrier returns to evaporator. Tested thermosiphon has vessel of 160 × 200  mm2 and 22 mm diameter pipe that connects the circuit. Pipe is divided into six channels with diameter of 10 mm. Condenser area is composed by seven levels extended along 850 mm with width of 240 mm and depth of 500 mm. This area has 8 mm spaced fins in order to help working fluid, R134a, to condensate. Thermosiphon does not present auxiliary equipment as previous heat exchangers presented. **Figure 10** shows heat dissipation system. Thermosiphon test does not present any fans to help working fluid to condensate, so thermal resistance depends only on heat power to dissipate and occupancy ratio, as displayed in **Figure 11**. Due to natural convection to exterior space and boiling and condensation coefficients, thermal resistance depends on calorific power to a higher extent to dissipate. Higher heat power to dissipate procures higher temperatures, which benefit transfer coefficients involved, and procuring lower thermal resistances, especially high occupancy ratios, is more affected due to high occupation, as presented in **Figure 11b**. The expanded uncertainty of thermal resistance is equal to ±8.42%.

**Figure 10.** Tested thermosiphon.

**Figure 11.** Thermal resistance per thermoelectric module of thermosiphon. (a) Thermal resistance as function of heat power to dissipate, (b) thermal resistance as function of occupancy ratio.

## **4. Thermoelectric computational optimization of waste heat energy harvesting from real application**

convection to exterior space and boiling and condensation coefficients, thermal resistance depends on calorific power to a higher extent to dissipate. Higher heat power to dissipate procures higher temperatures, which benefit transfer coefficients involved, and procuring lower thermal resistances, especially high occupancy ratios, is more affected due to high occupation, as presented in **Figure 11b**. The expanded uncertainty of thermal resistance

450 Thermoelectrics for Power Generation - A Look at Trends in the Technology

**Figure 11.** Thermal resistance per thermoelectric module of thermosiphon. (a) Thermal resistance as function of heat

power to dissipate, (b) thermal resistance as function of occupancy ratio.

is equal to ±8.42%.

**Figure 10.** Tested thermosiphon.

Computational model, which enables determining behavior of any TEG and thermal charac‐ terization of four different types of studied heat exchangers used to optimize net thermoelectric power generation of real application, is tested on furnace located in Spain. Computational model computes thermoelectric power generation, including calculation power consumption of auxiliary equipment shown in **Figure 8**, and net power generation can be computed (Eq. 2) as well, which is a real target of optimization in any application.

Selected application is furnace, which works 24 h a day, 350 days a year. Temperature of flue gases is 187°C and mass flow is 5.49 kg/s. Chimney has diameter of 0.8 m, transversal area of 0.5 m2 and height of 12 m. Therefore, available surface to locate TEG is 33.6 m2 . Flue gases emitted to ambient atmosphere are heat source of TEG, while ambient air is heat sink. Temperature of heat sink has been chosen as medium temperature of the year of the location of the furnace, Tamb = 17 °C. TEMs simulated are TG12‐8‐01L from Marlow Industries [52], where hot and cold sides area equals to 40 × 40 mm2 and TEMs can work up to 250°C on hot side.

Optimization is done for cold side of TEG, where four types of heat exchangers are simulated as function of occupancy ratio and mass flow of refrigerants in order to look for the maximum net power generation. On hot side, that is, interior of chimney, finned dissipator with base thickness of 4 mm and height, thickness and spacing of fins of 50, 6 and 1.5 mm, respectively, have been simulated. Thermal resistance of the latter heat exchanger has been computed as function of occupancy ratio and velocity of flue gases using a Computational Fluid Dynamics program, ANSYS Fluent. Eq. (14) presents thermal resistance of hot side of TEG per thermo‐ electric module:

$$\begin{aligned} R^{\text{TEM}} &= 0.046127 - 0.887591\delta - 0.000251v\_{gas} + 0.385376 / \ln\{v\_{gas}\} + \\ 0.304593\delta^2 - 0.281665 / \ln^2\{v\_{gas}\} + 4.35262\delta / \ln\{v\_{gas}\}. \end{aligned} \tag{14}$$

Total consumption of auxiliary equipment is essential to obtain net thermoelectric power generation, the goal of this optimization. Consumption of auxiliary equipment is obtained as function of mass flow presented in **Figure 8** and with accounting for number of TEM units necessary to cover the whole available surface of the chimney, totally 769 units. Cold plate, finned dissipator and heat pipe present auxiliary consumption, while thermosyphon does not, as explained in previous section. As it can be seen in **Figures 5**, **6** and **9**, increment in mass flow of refrigerants, with simultaneous increment in auxiliary equipment consumption, causes improvement in thermal resistances. This fact leads to higher thermoelectric power generation, due to improvement in heat transfer on both sides of the TEMs, obtaining higher difference of temperature between their sides and consequently higher thermoelectric power generation. Nevertheless, consumption of auxiliary equipment grows, so it is not so clear as in the case, when increasing mass flow of refrigerants leads to increase in net power generation. **Figure 12**

shows thermoelectric and net power generation as function of occupancy ratio, when finned dissipators are located on cold side of TEG. It can be seen, that higher air mass flow produces higher thermoelectric power generation; however, net power generation has optimum near the second smallest mass flow simulated, and after this value, net power generation decreases significantly, even obtaining negative values for small occupancy ratios and high mass flow of the air.

**Figure 12.** Thermoelectric and net power generation as function of occupancy ratio, when finned dissipators are simu‐ lated on cold side of the chimney.

Occupancy ratio is also determined for thermoelectric power generation. Higher occupancy ratio leads to higher thermal resistances per thermoelectric module and, therefore, less power generation per module unit; however, number of units to produce electricity is higher. Once more, it is necessary to elaborate optimization to get the maximum net power generation point. **Figure 12** presents the influence of this parameter, when finned dissipators are simulated on cold side. Occupation ratio value that provides maximum power generation is equal to δ ≈ 0.4, that is, optimum is reached, when approximately 40% of available surface is covered by TEMs only. This optimization is crucial to obtain the highest thermoelectric power generation and to optimize initial investment as well, because reduction in number of TEMs, which is necessary to install, reduces the cost of the application.

**Figure 13** presents optimum points for net power generation for each occupancy ratio simulated. These points have been obtained by optimizing mass flow of refrigerants of every heat exchanger at each value of occupancy ratio. **Figure 13** shows that the best cold side heat exchanger for this case is thermosiphon, obtaining up to 16280 W from waste heat of the furnace. The optimum occupancy ratio that provides this value equals to = 0.32. The number of TEMs necessary to cover 32% of chimney surface is 6720, and the smallest TEMs number is required, if compared with other optimum values as function of occupancy ratio. Therefore, thermosiphons are heat exchangers that not only provide the highest net thermoelectric power generation, but also the ones that require the smallest initial investment. Moreover, these systems have no moving parts, so they are completely robust and lack of maintenance.

Thermosiphons produce 30% more net optimal power than the second best option, heat pipes. Besides, occupancy ratio to maximize net power generation for heat pipes is higher, 0.42, so the initial investment has to be approximately 30% higher to obtain the maximum electrical energy. If optimal heat exchangers are compared with cold plates and finned dissipators, electrical energy production is 72 and 86% higher. Cold plates present optimum values of higher than thermosiphons, while finned dissipators present approximately the same occu‐ pancy ratio, so the same initial investment. Consumption of auxiliary equipment is deterrent, and small thermal resistances that cold plate presents produce higher thermoelectric power generation, but large consumption of auxiliary equipment negatively influences on net power generation, as **Figure 13** shows, even when auxiliary equipment consumption has been optimized to obtain the maximum net power generation for each occupancy ratio.

**Figure 13.** Optimal net thermoelectric power generation for four studied heat exchangers as function of occupancy ratio.

Optimized TEG is able to generate 137 MWh/year, taking into account that furnace works 8400 h a year, with power generation of 484.5 W/m2 and average of 2.42 W/TEM. Produced electrical energy could supply 40 Spanish dwellings, just harvesting waste heat that furnace emits to the ambient with TEG formed by finned dissipators on hot side and thermosiphons on cold side.

## **5. Conclusions**

shows thermoelectric and net power generation as function of occupancy ratio, when finned dissipators are located on cold side of TEG. It can be seen, that higher air mass flow produces higher thermoelectric power generation; however, net power generation has optimum near the second smallest mass flow simulated, and after this value, net power generation decreases significantly, even obtaining negative values for small occupancy ratios and high mass flow of

452 Thermoelectrics for Power Generation - A Look at Trends in the Technology

**Figure 12.** Thermoelectric and net power generation as function of occupancy ratio, when finned dissipators are simu‐

Occupancy ratio is also determined for thermoelectric power generation. Higher occupancy ratio leads to higher thermal resistances per thermoelectric module and, therefore, less power generation per module unit; however, number of units to produce electricity is higher. Once more, it is necessary to elaborate optimization to get the maximum net power generation point. **Figure 12** presents the influence of this parameter, when finned dissipators are simulated on cold side. Occupation ratio value that provides maximum power generation is equal to δ ≈ 0.4, that is, optimum is reached, when approximately 40% of available surface is covered by TEMs only. This optimization is crucial to obtain the highest thermoelectric power generation and to optimize initial investment as well, because reduction in number of TEMs, which is necessary

**Figure 13** presents optimum points for net power generation for each occupancy ratio simulated. These points have been obtained by optimizing mass flow of refrigerants of every heat exchanger at each value of occupancy ratio. **Figure 13** shows that the best cold side heat exchanger for this case is thermosiphon, obtaining up to 16280 W from waste heat of the furnace. The optimum occupancy ratio that provides this value equals to = 0.32. The number of TEMs necessary to cover 32% of chimney surface is 6720, and the smallest TEMs number is required, if compared with other optimum values as function of occupancy ratio. Therefore,

the air.

lated on cold side of the chimney.

to install, reduces the cost of the application.

Harvesting of waste heat to produce electrical energy via TEGs is a promising technology to help mitigate environmental issues that nowadays society is facing. TEGs are solid‐state systems, which barely present moving parts, and, therefore, they are very robust, reliable, silent and long‐lasting.

Developed general computational model allows predicting behavior of any TEG. It does not include the most common simplifications that the rest of models from publications have and besides it includes new parameters, such as occupancy ratio, mass flow of refrigerants and temperature reduction in flue gases, while they flow along the system. The latter parameters are determinant for thermoelectric power generation and, therefore, very important to bear in mind for optimization study.

Thermal resistances of different heat exchange systems are function of novel parameter, occupancy ratio, included in computational model. Occupancy ratio has negative influence on thermal resistance per thermoelectric module, if it increases, due to the reduction in available dissipative area per TEM. Calorific power to dissipate influences just heat exchangers, where phase change is involved and mass flow of refrigerants determines thermal resistance, but in greater extent, when occupancy ratio has high values.

Thermosiphon with phase change is a dissipation system that provides the highest net thermoelectric power generation, 137 MWh/year, which is equivalent to supply 40 Spanish dwellings, when 32% of chimney surface is covered by TEMs. This production is 30, 72 and 87% higher than optimal productions of heat pipe, cold plate and finned dissipators, respec‐ tively. Moreover, the number of TEMs required for use in TEG with thermosiphons is lower or similar to that for the rest of heat dissipation systems, so not only power generation is optimum, but initial investment also.

The absence of moving parts for TEG built with thermosiphon procures really robust, reliable and silent power generation system that can produce electrical energy from waste heat of any system, improving their efficiency and, therefore, collaborating to satisfy demand for electrical energy in green manner.

#### **Nomenclature**



systems, which barely present moving parts, and, therefore, they are very robust, reliable,

Developed general computational model allows predicting behavior of any TEG. It does not include the most common simplifications that the rest of models from publications have and besides it includes new parameters, such as occupancy ratio, mass flow of refrigerants and temperature reduction in flue gases, while they flow along the system. The latter parameters are determinant for thermoelectric power generation and, therefore, very important to bear in

Thermal resistances of different heat exchange systems are function of novel parameter, occupancy ratio, included in computational model. Occupancy ratio has negative influence on thermal resistance per thermoelectric module, if it increases, due to the reduction in available dissipative area per TEM. Calorific power to dissipate influences just heat exchangers, where phase change is involved and mass flow of refrigerants determines thermal resistance, but in

Thermosiphon with phase change is a dissipation system that provides the highest net thermoelectric power generation, 137 MWh/year, which is equivalent to supply 40 Spanish dwellings, when 32% of chimney surface is covered by TEMs. This production is 30, 72 and 87% higher than optimal productions of heat pipe, cold plate and finned dissipators, respec‐ tively. Moreover, the number of TEMs required for use in TEG with thermosiphons is lower or similar to that for the rest of heat dissipation systems, so not only power generation is

The absence of moving parts for TEG built with thermosiphon procures really robust, reliable and silent power generation system that can produce electrical energy from waste heat of any system, improving their efficiency and, therefore, collaborating to satisfy demand for electrical

silent and long‐lasting.

mind for optimization study.

optimum, but initial investment also.

energy in green manner.

δ Occupancy ratio ρ Density, kg/m3

*ATEM* Area of a TEM, m2

*I* Current , A

σ Thomson coefficient, V/K α Seebeck coefficient, V/K π Peltier coefficient, V

*K* Thermal conductivity, W/(m × K)

*Ab* Area of the heat exchanger base, m2

*TEM* Systematic standard uncertainty

*c*<sup>p</sup> Specific heat at constant pressure, J/(kg × K)

**Nomenclature**

*bR*

greater extent, when occupancy ratio has high values.

454 Thermoelectrics for Power Generation - A Look at Trends in the Technology


## **Author details**

Patricia Aranguren1,2\* and David Astrain1,2

\*Address all correspondence to: patricia.arangureng@unavarra.es

1 Mechanical, Energy and Materials Engineering Department, Public University of Navarre, Pamplona, Spain

2 Smart Cities Institute, Pamplona, Spain

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#### **Modeling of a Thermoelectric Generator Device** Modeling of a Thermoelectric Generator Device

Eurydice Kanimba and Zhiting Tian Eurydice Kanimba and Zhiting Tian

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/65741

## Abstract

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460 Thermoelectrics for Power Generation - A Look at Trends in the Technology

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Thermoelectric generators (TEGs) are devices that employ Seebeck effect in thermopile to convert temperature gradient induced by waste heat into electrical power. Recently, TEGs have enticed increasing attention as green and flexible source of electricity able to meet wide range of power requirements from thermocouple sensors to power generators in satellites. Thermoelectric generators suffer from low-conversion efficiency; however, they could be promising solutions, when they are used to harvest waste heat coming from industry processes or central-heating systems. This chapter covers the working principles behind TEGs, depicts numerous schematics explaining functionality of TEGs, and investigates performance of TEGs. A detailed derivation process, which provides performance expressions dictating operation of TEGs, is exposed in this chapter. In addition, thermal resistance network is shown to explain thermal connection of thermocouples in TEGs in parallel and electrical connection of thermocouples in series. Performance features shown in this chapter are power output, efficiency, and voltage induced within TEG as functions of numerous parameters.

Keywords: Seebeck effect, Peltier effect, Thomson effect, Joule heating, thermal resistance network, electrical resistance network, structure of TEGs, TEGs performance expressions derivation, analytical model, performance analysis of TEGs

## 1. Introduction

Increase in greenhouse gases emissions in the atmosphere due to burning of fossil fuels for the production of electricity and heat energy has motivated the development of alternative efficient and clean-energy-generation systems including that for the recovery of waste heat into electrical power. Numerous power-generation systems, such as solar panels, wind turbines, and geothermal power plants, which utilize renewable energies, have been designed to reduce dependency on fossil fuels, thus reducing greenhouse gases emissions. However, such powergeneration systems require high maintenance and are often expensive as compared to thermoelectric generator devices (TEGs). Thermoelectric generator device (TEG) is a device that directly converts heat into electricity. Essentially, TEG is thermoelectric module (TEM), which

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons

distribution, and eproduction in any medium, provided the original work is properly cited.

consists of thermopiles, that is, a set of thermocouples built by legs of p- and n-type semiconductors, which are connected electrically in series and thermally in parallel [1, 2]. Thermocouples built by legs of p- and n-type semiconductors are sandwiched between two ceramic plates, which are to be held at two different temperatures to realize generation regime. Temperature gradient induced between top and bottom ceramic plates originates voltage on TEG poles due to Seebeck effect in thermocouples built by legs of p- and n-type semiconductors.

Employing waste heat as heat source for TEGs is cost-effective due to waste heat being free of charge and already available. About 70% of the world energy production is known to be wasted into atmosphere through heat dissipation, which is one of significant contributions in global warming [3]. Therefore, the utilization of waste heat by converting into electricity using TEGs can contribute to energy savings and preservation of the environment as well. Thermoelectric device can also operate in reverse mode as thermoelectric cooler (TEC) and produce reverse temperature gradient between top and bottom ceramic plates due to Peltier effect, if electrical bias is applied. Depending on operation mode, applying bias voltage to thermoelectric module (TEM) and hence initiating flow of electrical current result in the production of temperature difference between top and bottom plates and TEM acts as thermoelectric cooler (TEC) and vice versa; the placement of TEM in temperature gradient results in the occurrence of voltage on TEM poles and TEM acts as heat pump with the function of thermoelectric generator (TEG) [4].

Thermoelectric devices possess various advantages compared to other power-generation systems [5]. TEGs are branded attractive power-generation systems, because they are silent solidstate devices with no moving parts, environmental friendly, scalable from small to giant heat sources, and highly reliable. They also have extended lifetime and ability to utilize low-grade thermal energy to generate electrical energy.

## 2. TEG-working principle

#### 2.1. Seebeck effect

Seebeck effect describes the induction of voltage, when junctions of two different conducting materials are maintained at different temperatures as shown in Figure 1. Seebeck effect increases in magnitude, when Seebeck coefficient of conducting materials and/or temperature difference between their connections increases. Voltage induced through Seebeck effect is defined as below:

$$V = \alpha \Delta T,\tag{1}$$

where α is Seebeck coefficient and ΔT is the temperature difference between hot junction and cold junction.

#### 2.2. Peltier effect

Peltier effect describes heat dissipation or absorption at the connection of two conducting materials, when current flows through the junction as shown in Figure 2. Depending on the direction of current flow, heat is either absorbed or dissipated at connection.

Figure 1. Seebeck effect.

consists of thermopiles, that is, a set of thermocouples built by legs of p- and n-type semiconductors, which are connected electrically in series and thermally in parallel [1, 2]. Thermocouples built by legs of p- and n-type semiconductors are sandwiched between two ceramic plates, which are to be held at two different temperatures to realize generation regime. Temperature gradient induced between top and bottom ceramic plates originates voltage on TEG poles due to Seebeck effect in thermocouples built by legs of p- and n-type semiconductors.

Employing waste heat as heat source for TEGs is cost-effective due to waste heat being free of charge and already available. About 70% of the world energy production is known to be wasted into atmosphere through heat dissipation, which is one of significant contributions in global warming [3]. Therefore, the utilization of waste heat by converting into electricity using TEGs can contribute to energy savings and preservation of the environment as well. Thermoelectric device can also operate in reverse mode as thermoelectric cooler (TEC) and produce reverse temperature gradient between top and bottom ceramic plates due to Peltier effect, if electrical bias is applied. Depending on operation mode, applying bias voltage to thermoelectric module (TEM) and hence initiating flow of electrical current result in the production of temperature difference between top and bottom plates and TEM acts as thermoelectric cooler (TEC) and vice versa; the placement of TEM in temperature gradient results in the occurrence of voltage on TEM poles and TEM acts as heat pump with the function of thermoelectric generator (TEG) [4]. Thermoelectric devices possess various advantages compared to other power-generation systems [5]. TEGs are branded attractive power-generation systems, because they are silent solidstate devices with no moving parts, environmental friendly, scalable from small to giant heat sources, and highly reliable. They also have extended lifetime and ability to utilize low-grade

Seebeck effect describes the induction of voltage, when junctions of two different conducting materials are maintained at different temperatures as shown in Figure 1. Seebeck effect increases in magnitude, when Seebeck coefficient of conducting materials and/or temperature difference between their connections increases. Voltage induced through Seebeck effect is

where α is Seebeck coefficient and ΔT is the temperature difference between hot junction and

Peltier effect describes heat dissipation or absorption at the connection of two conducting materials, when current flows through the junction as shown in Figure 2. Depending on the

direction of current flow, heat is either absorbed or dissipated at connection.

V ¼ αΔT, (1)

thermal energy to generate electrical energy.

462 Thermoelectrics for Power Generation - A Look at Trends in the Technology

2. TEG-working principle

2.1. Seebeck effect

defined as below:

cold junction.

2.2. Peltier effect

Figure 2. Peltier effect.

#### 2.3. Thomson effect

Thomson effect describes the dissipation or absorption of heat, when electric current passes through a circuit composed of a single material, which has temperature variation along its length, as shown in Figure 3. ΔQ represents heat dissipation, when electrical current flows through a homogeneous conductor. Thomson coefficient is given by second Kelvin relationship [6–9]:

$$
\mu = T \frac{d\alpha}{dT},
\tag{2}
$$

where μ and T, respectively, symbolize Thomson coefficient and temperature. If Seebeck coefficient, α, is temperature independent, then Thomson coefficient is equal to zero.

Figure 3. Thomson effect.

#### 2.4. Joule heating

Joule-heating effect defines heat dissipated by material with nonzero electrical resistance in the presence of electrical current, as shown in Figure 4,

Figure 4. Joule heating.

## 3. Structure of TEG

#### 3.1. Three-dimensional representation of comprehensive operation of TEG

TEGs are composed of numerous legs (slabs) made of p- and n-type semiconductors forming thermocouples, all connected electrically in series and thermally in parallel. Semiconductor legs are connected to each other through conductive copper tabs, and they are sandwiched between two ceramic plates, which conduct heat, but behave as insulators to electrical current. Schematic diagram of three-dimensional (3-D) multielement thermoelectric generator is shown in Figure 5.

Waste heat from various sources, such as automobile engines exhaust, industry and infrastructure-heating activities, geothermal, and others, can be supplied to top ceramic plate of TEGs. As shown in Figure 5, heat flows through ceramic plate and copper-conductive tabs before reaching the top surface of p- and n-type legs made of proper semiconductors, which is defined as the hot side of TEG. Heat flows through both semiconductor's legs and then again through copper-conductive tabs and bottom ceramic plate. Through heat sink, the bottom ceramic plate is maintained at significantly lower temperature than top ceramic in order to produce high-temperature gradient, which will lead to high-power output. Allowed temperature applied on top and bottom ceramic plates depends on materials of p- and n-type legs. Also, p- and n-type materials are designed to possess low thermal conductivity in order to restrict, as much as possible, heat flow through semiconductors and maintain temperature difference between hot and cold sides of TEG.

Pictorial distribution of temperature along legs of TEG at conditional difference of temperature ΔT between hot and cold sides is shown in Figure 6.

Figure 5. 3-D schematic of multielement TEG.

2.4. Joule heating

Figure 3. Thomson effect.

3. Structure of TEG

Figure 4. Joule heating.

in Figure 5.

presence of electrical current, as shown in Figure 4,

464 Thermoelectrics for Power Generation - A Look at Trends in the Technology

Joule-heating effect defines heat dissipated by material with nonzero electrical resistance in the

3.1. Three-dimensional representation of comprehensive operation of TEG

TEGs are composed of numerous legs (slabs) made of p- and n-type semiconductors forming thermocouples, all connected electrically in series and thermally in parallel. Semiconductor legs are connected to each other through conductive copper tabs, and they are sandwiched between two ceramic plates, which conduct heat, but behave as insulators to electrical current. Schematic diagram of three-dimensional (3-D) multielement thermoelectric generator is shown

Waste heat from various sources, such as automobile engines exhaust, industry and infrastructure-heating activities, geothermal, and others, can be supplied to top ceramic plate of TEGs.

Figure 6. Temperature gradient within TEG.

After temperature gradient has been induced between hot and cold sides of TEG, voltage occurred on TEG-positive and -negative poles due to Seebeck effect, as depicted in Figure 7.

Voltage generated in TEG due to Seebeck effect induces the movement of charge carriers within p-and n-type semiconductor legs and, hence, electrical current in electrical circuit including load resistor RL connected to TEG poles, current density formed, is displayed in Figure 8.

Figure 7. Voltage distribution within TEG.

Figure 8. Current density within TEG.

#### 3.2. 1-D representation of TEG

Establishing one-dimensional (1-D) representation of TEG is helpful in determining analytical expressions of heat absorbed and heat rejected, as the power output of TEG is defined as the difference between heat absorbed and heat rejected. Figure 9 represents 1-D schematic of TEG with heat source and heat sink, respectively, applied on top and bottom sides of TEG.

TH, QH, and KH are, respectively, heat source temperature, heat supplied from heat source to TEG, and thermal conductance of hot side of TEG. TL, QL, and KL are, respectively, heat sink temperature, heat rejected from TEG to heat sink, and thermal conductance of TEG cold side. Th and Qh define the temperature of hot junction of thermocouples and heat flow through hot junctions of TEG. Tc and Qc describe the temperature at cold junction of thermocouples and heat flow through cold junctions of TEG. Assuming thermoelectric properties to be temperature independent, α, k, ρ can, respectively, be defined as constant Seebeck coefficient, constant thermal conductivity, and constant electrical resistivity.

Figure 9. 1-D schematic of multielement TEG.

#### 3.3. Electrical network resistance

After temperature gradient has been induced between hot and cold sides of TEG, voltage occurred on TEG-positive and -negative poles due to Seebeck effect, as depicted in Figure 7. Voltage generated in TEG due to Seebeck effect induces the movement of charge carriers within p-and n-type semiconductor legs and, hence, electrical current in electrical circuit including load

Establishing one-dimensional (1-D) representation of TEG is helpful in determining analytical expressions of heat absorbed and heat rejected, as the power output of TEG is defined as the difference between heat absorbed and heat rejected. Figure 9 represents 1-D schematic of TEG

TH, QH, and KH are, respectively, heat source temperature, heat supplied from heat source to TEG, and thermal conductance of hot side of TEG. TL, QL, and KL are, respectively, heat sink temperature, heat rejected from TEG to heat sink, and thermal conductance of TEG cold side. Th and Qh define the temperature of hot junction of thermocouples and heat flow through hot junctions of TEG. Tc and Qc describe the temperature at cold junction of thermocouples and

with heat source and heat sink, respectively, applied on top and bottom sides of TEG.

resistor RL connected to TEG poles, current density formed, is displayed in Figure 8.

466 Thermoelectrics for Power Generation - A Look at Trends in the Technology

3.2. 1-D representation of TEG

Figure 8. Current density within TEG.

Figure 7. Voltage distribution within TEG.

Electrical resistance network of TEG is shown in Figure 10. P-type and n-type semiconductor legs are connected to each other electrically in series through copper-conductive tabs.

Rp and Rn are electrical resistance associated, respectively, with p- and n-type semiconductor legs. Rcpeh, Rcpec, and RL are, respectively, electrical resistance of copper-conductive strips on the hot side, electrical resistance of copper-conductive strips on the cold side, and external load resistance.

Figure 10. Electrical network resistance.

#### 3.4. Thermal network resistance

Thermal resistance of TEG is shown in Figure 11 and it assists in determining heat transfer rate through ceramic plates, copper strips, and p- and n-type semiconductor legs. The number of thermocouples is N.

Teceh, Ticeh, and Rceh are, respectively, external temperature of hot ceramic plate, internal temperature of hot ceramic plate, and thermal resistance associated with ceramic plate on the hot side. Th, Rcph, and Rteg are, respectively, the temperature at the hot junction of p- and n-type semiconductor legs, thermal resistance of copper strip on the hot side, and thermal resistance of both pand n-type semiconductor legs. Tc, Rcpc, and Ticec, are, respectively, the temperature at cold junction of p- and n-type semiconductor legs, thermal resistance of ceramic plate on the cold side, and internal temperature of cold ceramic plate. Rcec and Tecec are, respectively, thermal resistance of cold ceramic plate and external temperature of cold ceramic plate.

Figure 11. Thermal resistance network.

#### 4. Theoretical model

#### 4.1. Analysis of thermoelectric material properties and geometry of TEG

Thermoelectric materials of TEG legs, p- and n-type semiconductors, are characterized by parameter called the figure of merit Z, which measures the ability of thermoelectric materials to convert heat into electrical power. The figure of merit is expressed as follows:

$$Z = \frac{\alpha^2}{\rho k},\tag{3}$$

where α, ρ, and k are, respectively, Seebeck coefficient, electrical resistivity, and thermal conductivity of thermoelectric materials. Great thermoelectric materials possess high Seebeck coefficient, low electrical resistivity, and low thermal conductivity [10].

In order to obtain maximum figure of merit, when designing TEG, the geometry of semiconductor legs and properties of thermoelectric materials need to satisfy the following equation [1, 11]:

$$\frac{A\_p^2 L\_n^2}{A\_n^2 L\_p^2} = \frac{k\_n \rho\_p}{k\_p \rho\_n},\tag{4}$$

where Ap, An, Lp, Ln, kp, kn, ρp, and ρ<sup>n</sup> are, respectively, the cross-sectional area, length, thermal conductivity, and electrical resistivity of p- and n-type semiconductor legs.

To reduce manufacturing costs, p- and n-type semiconductor legs are fabricated with the same geometry, that is, Ap ¼ An ¼ A, and Lp ¼ Ln ¼ L. Similarly, p- and n-type semiconductor legs are made of doped alloys to produce the same thermoelectric properties, that is, ρ<sup>p</sup> ¼ ρn, kp ¼ kn, and α<sup>p</sup> ¼ −α<sup>n</sup> [12].

#### 4.2. TEG performance analysis

3.4. Thermal network resistance

468 Thermoelectrics for Power Generation - A Look at Trends in the Technology

thermocouples is N.

4. Theoretical model

Figure 11. Thermal resistance network.

Thermal resistance of TEG is shown in Figure 11 and it assists in determining heat transfer rate through ceramic plates, copper strips, and p- and n-type semiconductor legs. The number of

Teceh, Ticeh, and Rceh are, respectively, external temperature of hot ceramic plate, internal temperature of hot ceramic plate, and thermal resistance associated with ceramic plate on the hot side. Th, Rcph, and Rteg are, respectively, the temperature at the hot junction of p- and n-type semiconductor legs, thermal resistance of copper strip on the hot side, and thermal resistance of both pand n-type semiconductor legs. Tc, Rcpc, and Ticec, are, respectively, the temperature at cold junction of p- and n-type semiconductor legs, thermal resistance of ceramic plate on the cold side, and internal temperature of cold ceramic plate. Rcec and Tecec are, respectively, thermal

resistance of cold ceramic plate and external temperature of cold ceramic plate.

4.1. Analysis of thermoelectric material properties and geometry of TEG

to convert heat into electrical power. The figure of merit is expressed as follows:

Thermoelectric materials of TEG legs, p- and n-type semiconductors, are characterized by parameter called the figure of merit Z, which measures the ability of thermoelectric materials

> <sup>Z</sup> <sup>¼</sup> <sup>α</sup><sup>2</sup> ρk

, (3)

In order to obtain expressions describing TEG performance, thermocouple built by legs of p- and n-type semiconductors is extracted from Figure 9 and represented in Figure 12. Figure 12 represents heat transfer within single thermocouple. The length and cross-sectional area of both p- and n-type semiconductor legs are equal and symbolize as L and A, respectively. The junction of thermocouple is fixed at thermal conducting and electricalinsulating ceramic plate.

Figure 12. Heat transfer within TEG thermocouple.

Qh, Qc, Qkin, Qkout, Qj, Lp, Ln, and δcu are, respectively, heat absorbed at hot junction, heat rejected at cold junction, Fourier heat conduction transferred inside of control volume, Fourier heat conduction transferred out of control volume, Joule heating generated within control volume, the length of p- and n-type legs, and the thickness of copper electrical-conducting strips.

Employing the conservation of energy and assuming one-dimensional steady-state condition, the energy equation of differential control volume inside of p-type semiconductor leg can be expressed as follows:

$$Q\_{\rm kin} \mathbf{-Q}\_{\rm kout} + Q\_{\rm j} = \mathbf{0},\tag{5}$$

$$\mathbf{Q(x)-Q(x+dx)} + Q\_{\rangle} = \mathbf{0}.\tag{6}$$

Using Taylor expansion:

$$Q(\mathbf{x}) - \left(Q(\mathbf{x}) + \frac{\partial Q(\mathbf{x})}{\partial \mathbf{x}} d\mathbf{x}\right) + \frac{I^2 \rho\_p}{A\_p} d\mathbf{x} = \mathbf{0},\tag{7}$$

I represents electrical current induced within TEG device:

$$-\frac{\partial Q(\mathbf{x})}{\partial \mathbf{x}}d\mathbf{x} + \frac{\mathbf{i}^2 \rho\_p}{A\_p}d\mathbf{x} = \mathbf{0}.\tag{8}$$

Fourier's law of conduction for one-dimensional heat conduction states:

$$Q(\mathbf{x}) = -k\_p A\_p \frac{\partial T\_p}{\partial \mathbf{x}}.\tag{9}$$

Substituting Eq. (9) into Eq. (8):

$$-\frac{\partial}{\partial \mathbf{x}} \left( -k\_p A\_p \frac{\partial T\_p}{\partial \mathbf{x}} \right) d\mathbf{x} + \frac{I^2 \rho\_p}{A\_p} d\mathbf{x} = \mathbf{0}.\tag{10}$$

Provided that thermoelectric properties are temperature independent, kp can be taken out of derivative and Eq. (10) can be expressed as follows:

$$k\_p A\_p \frac{d^2 T\_p}{d\mathbf{x}^2} d\mathbf{x} + \frac{I^2 \rho\_p}{A\_p} d\mathbf{x} = 0.\tag{11}$$

Integrating Eq. (11):

$$\int\_0^\chi k\_p A\_p \frac{d^2 T\_p}{d\mathbf{x}^2} d\mathbf{x} + \int\_0^\chi \frac{I^2 \rho\_p}{A\_p} d\mathbf{x} = 0,\tag{12}$$

$$k\_p A\_p \left(\frac{dT\_p}{d\mathbf{x}}|\_{\mathbf{x}} \frac{dT\_p}{d\mathbf{x}}|\_0\right) + \frac{I^2 \rho\_p}{A\_p} \mathbf{x} = \mathbf{0},\tag{13}$$

$$\mathbf{x} = \mathbf{0} \to T\_p(\mathbf{0}) = T\_h,\tag{14}$$

$$k\_p A\_p \frac{dT\_p}{d\mathbf{x}}\Big|\_{0} = Q\_p(0),\tag{15}$$

where Qpð0Þ is Fourier heat conduction transferred inside of top p-type leg:

#### Modeling of a Thermoelectric Generator Device http://dx.doi.org/10.5772/65741 471

$$\int\_{0}^{L\_p} k\_p A\_p \frac{dT\_p}{d\mathbf{x}} d\mathbf{x} + \int\_{0}^{L\_p} \frac{I^2 \rho\_p}{A\_p} \mathbf{x} d\mathbf{x} = -\int\_{0}^{L\_p} Q\_p(\mathbf{0}) d\mathbf{x},\tag{16}$$

$$k\_p A\_p \left( T\_p(L\_p) - T\_p(0) \right) + \frac{I^2 \rho\_p}{A\_p} \frac{L\_p}{2} = -Q\_p(0) L\_p,\tag{17}$$

$$\mathbf{x} = \mathbf{0} \to T\_p(\mathbf{0}) = T\_h,\tag{18}$$

$$\infty = L\_p \to T\_p(L\_p) = T\_c,\tag{19}$$

$$Q\_p(0) = \frac{k\_p A\_p}{L\_p} (T\_h - T\_c) \text{--} 0.5 \frac{I^2 \rho\_p L\_p}{A\_p} \,. \tag{20}$$

Considering Peltier effect happening at the hot junction of p-type leg:

heat conduction transferred out of control volume, Joule heating generated within control volume, the length of p- and n-type legs, and the thickness of copper electrical-conducting strips. Employing the conservation of energy and assuming one-dimensional steady-state condition, the energy equation of differential control volume inside of p-type semiconductor leg can be

<sup>Q</sup>ðxÞ<sup>−</sup> <sup>Q</sup>ðxÞ þ <sup>∂</sup>Qðx<sup>Þ</sup>

− ∂QðxÞ ∂x

Fourier's law of conduction for one-dimensional heat conduction states:

<sup>∂</sup><sup>x</sup> <sup>−</sup>kpAp

kpAp d2 Tp dx<sup>2</sup> dx <sup>þ</sup>

∫ x

0 kpAp d2 Tp dx<sup>2</sup> dx <sup>þ</sup> <sup>∫</sup>

kpAp

dTp dx j x− dTp dx j 0

kpAp dTp dx 0

where Qpð0Þ is Fourier heat conduction transferred inside of top p-type leg:

− ∂

derivative and Eq. (10) can be expressed as follows:

I represents electrical current induced within TEG device:

470 Thermoelectrics for Power Generation - A Look at Trends in the Technology

∂x dx

dx þ I 2 ρp Ap

QðxÞ ¼ −kpAp

∂Tp ∂x 

dx þ I 2 ρp Ap

Provided that thermoelectric properties are temperature independent, kp can be taken out of

I 2 ρp Ap

x

0 I 2 ρp Ap

> þ I 2 ρp Ap

þ I 2 ρp Ap

∂Tp

Qkin−Qkout þ Qj ¼ 0, (5)

dx ¼ 0, (7)

dx ¼ 0: (8)

<sup>∂</sup><sup>x</sup> : (9)

dx ¼ 0: (10)

dx ¼ 0: (11)

dx ¼ 0, (12)

x ¼ 0, (13)

x ¼ 0 ! Tpð0Þ ¼ Th, (14)

¼ Qpð0Þ, (15)

QðxÞ−Qðx þ dxÞ þ Qj ¼ 0: (6)

expressed as follows:

Using Taylor expansion:

Substituting Eq. (9) into Eq. (8):

Integrating Eq. (11):

$$Q\_{\rm ph} = \alpha\_p I T\_h + \frac{k\_p A\_p}{L\_p} (T\_h - T\_c) \text{--} 0.5 \frac{I^2 \rho\_p L\_p}{A\_p},\tag{21}$$

where Qph is the total heat absorbed at the hot junction of p-type leg.

Employing the same procedure with the same boundary conditions to derive heat flow through n-type leg leads to the expression of Qnh as follows:

$$Q\_{\rm nh} = -\alpha\_n I T\_h + \frac{k\_n A\_n}{L\_n} (T\_h - T\_c) - 0.5 \frac{I^2 \rho\_n L\_n}{A\_n},\tag{22}$$

where Qnh is the total heat absorbed at the hot junction of n-type leg. The total heat absorbed at the hot junction of both p- and n-type semiconductor legs is, therefore:

$$Q\_{\rm h} = Q\_{\rm ph} + Q\_{\rm rh},\tag{23}$$

$$Q\_h = (a\_p - \alpha\_n)IT\_h + \left(\frac{k\_p A\_p}{L\_p} + \frac{k\_n A\_n}{L\_n}\right)(T\_h - T\_c) - 0.5\left(\frac{\rho\_p L\_p}{A\_p} + \frac{\rho\_n L\_n}{A\_n}\right)I^2. \tag{24}$$

We use the same method to derive expression for heat rejected at the cold junction of p-type and n-type legs. Consequently, the following expression is obtained:

$$Q\_c = (a\_p - \alpha\_n)IT\_c + \left(\frac{k\_p A\_p}{L\_p} + \frac{k\_n A\_n}{L\_n}\right)(T\_h - T\_c) + 0.5(\frac{\rho\_p L\_p}{A\_p} + \frac{\rho\_n L\_n}{A\_n})I^2. \tag{25}$$

#### 4.3. TEG performance expressions

TEG is characterized by numerous performance expressions, including heat absorbed on the hot side, heat rejected on the cold side, power output, voltage induced, and current flowing in the electrical circuit with load resistor. Defining symbols below from Eqs. (24) and (25):

$$K = \frac{k\_p A\_p}{L\_p} + \frac{k\_n A\_n}{L\_n},\tag{26}$$

$$r = \frac{\rho\_p L\_p}{A\_p} + \frac{\rho\_n L\_n}{A\_n},\tag{27}$$

$$
\alpha = \alpha\_p \text{--} \alpha\_n. \tag{28}
$$

Expressions of heat flow through the hot and cold junctions for N semiconductor thermocouples can therefore be expressed as follows:

$$Q\_h = N(\alpha I T\_h \text{--} 0.5 rI^2 + K(T\_h - T\_c)),\tag{29}$$

$$Q\_c = N(aIT\_c + 0.5rI^2 + K(T\_h - T\_c)).\tag{30}$$

As stated previously, the power generated by TEG is defined as the difference between heat absorbed at the hot junction and heat rejected at the cold junction:

$$P = Q\_h - Q\_c = N(aI(T\_h - T\_c) - rI^2). \tag{31}$$

Optimal current generated in TEG is obtained by first deriving Eq. (31) with respect to current as follows:

$$\frac{dP}{dI} = N(\alpha(T\_h - T\_c) \text{--} 2Ir). \tag{32}$$

Eq. (32) is equated to zero to determine the following expression of optimal current:

$$I\_{\rm opt} = \frac{\alpha (T\_h - T\_c)}{2r}.\tag{33}$$

Generally speaking, voltage, current, and output power induced in TEG consisting of set of thermocouples similar to the one represented in Figure 9 are, respectively, defined as:

$$I = \frac{\alpha (T\_h - T\_c)}{r + R\_L},\tag{34}$$

$$P = I^2 R\_L = \left(\frac{\alpha (T\_h - T\_c)}{r + R\_L}\right)^2 R\_L,\tag{35}$$

$$V = IR\_L = \frac{\alpha (T\_h - T\_c)}{r + R\_L} R\_L,\tag{36}$$

where RL, is the external resistance load. To get optimum electrical current induced, and output power generated in the electrical circuit with TEG consisting of set of thermocouples, external resistance needs to be equal to the total internal electrical resistance of p- and n-type semiconductor legs. The efficiency of TEG is given by:

$$
\eta = \frac{P}{Q\_h}.\tag{37}
$$

In actual TEG, two thermoelectric materials are used, that is, p- and n-type semiconductors. The maximum efficiency provided by TEG is expressed as follows:

$$\eta\_{\text{max}} = \left( 1 - \frac{T\_h}{T\_c} \right) \frac{\sqrt{1 + Z\overline{T}} - 1}{\sqrt{1 + Z\overline{T}} + \frac{T\_h}{T\_c}},\tag{38}$$

where Z and T are, respectively, the figure of merit of p- and n-type semiconductors and averaged temperature between temperatures at the hot and cold sides.

#### 4.4. Performance simulation example of a TEG

Numerical example is adopted in order to optimize and analyze effects of heat transfer governing equations on output power, efficiency, and induced voltage of TEG.

In numerical analysis, the following geometry is adopted (Table 1).

The following thermoelectric properties are adopted (Table 2).


Table 1. Geometry of TEG.

<sup>K</sup> <sup>¼</sup> kpAp Lp þ knAn Ln

<sup>r</sup> <sup>¼</sup> <sup>ρ</sup>pLp Ap

ples can therefore be expressed as follows:

472 Thermoelectrics for Power Generation - A Look at Trends in the Technology

as follows:

<sup>þ</sup> <sup>ρ</sup>nLn An

Expressions of heat flow through the hot and cold junctions for N semiconductor thermocou-

As stated previously, the power generated by TEG is defined as the difference between heat

<sup>P</sup> <sup>¼</sup> Qh−Qc <sup>¼</sup> <sup>N</sup>ðαIðTh−TcÞ−rI<sup>2</sup>

Optimal current generated in TEG is obtained by first deriving Eq. (31) with respect to current

<sup>I</sup>opt <sup>¼</sup> <sup>α</sup>ðTh−Tc<sup>Þ</sup>

Generally speaking, voltage, current, and output power induced in TEG consisting of set of

<sup>I</sup> <sup>¼</sup> <sup>α</sup>ðTh−Tc<sup>Þ</sup> r þ RL

RL <sup>¼</sup> <sup>α</sup>ðTh−Tc<sup>Þ</sup> r þ RL <sup>2</sup>

where RL, is the external resistance load. To get optimum electrical current induced, and output power generated in the electrical circuit with TEG consisting of set of thermocouples, external resistance needs to be equal to the total internal electrical resistance of p- and n-type

r þ RL

<sup>V</sup> <sup>¼</sup> IRL <sup>¼</sup> <sup>α</sup>ðTh−Tc<sup>Þ</sup>

absorbed at the hot junction and heat rejected at the cold junction:

dP

P ¼ I 2

semiconductor legs. The efficiency of TEG is given by:

Eq. (32) is equated to zero to determine the following expression of optimal current:

thermocouples similar to the one represented in Figure 9 are, respectively, defined as:

, (26)

, (27)

Þ: (31)

α ¼ αp−αn: (28)

Qh <sup>¼</sup> <sup>N</sup>ðαITh−0:5rI<sup>2</sup> <sup>þ</sup> <sup>K</sup>ðTh−TcÞÞ, (29)

Qc <sup>¼</sup> <sup>N</sup>ðαITc <sup>þ</sup> <sup>0</sup>:5rI<sup>2</sup> <sup>þ</sup> <sup>K</sup>ðTh−TcÞÞ: (30)

dI <sup>¼</sup> <sup>N</sup>ðαðTh−TcÞ−2IrÞ: (32)

<sup>2</sup><sup>r</sup> : (33)

, (34)

RL, (35)

RL, (36)


Table 2. Thermoelectric properties.

All obtained performance curves are computed at the hot-side temperature up to Th ¼ 673 K and the cold-side temperature of Tc ¼ 373 K.

#### 4.4.1. Power and efficiency as function of electrical current

By fixing the cold side at temperature Tc ¼ 373 K and varying the hot-side temperature from 473 to 673 K with increment of 100 K, the power generated behaves as follow:

One can observe that the power as a function of current behaves as a parabola with optimum power value at specific current. Figure 13 shows the existence of maximal current value, which corresponds to optimum power. Any current higher or lower than the maximum current value generates power output less than optimum power. Also, as temperature at the hot side increases, then power produced increases as well.

Figure 13. TEG output power as a function of electrical current.

Efficiency curves shown in Figure 14 behave as parabola as well, with specific current value maximizing efficiency for each temperature difference. In real devices, TEGs are always operated at an optimal current. One thing to note is that the efficiency of TEG is still low compared to other energy-conversion techniques. A lot of effort has been made to enhance efficiency [13, 14]. Given that heat sources are plenty and free, TEGs could be promising solutions, when they are employed to harvest waste heat from industry activities and central-heating systems.

Figure 14. Efficiency as a function of current.

#### 4.4.2. I-V dependences

Efficiency curves shown in Figure 14 behave as parabola as well, with specific current value maximizing efficiency for each temperature difference. In real devices, TEGs are always operated at an optimal current. One thing to note is that the efficiency of TEG is still low compared to other energy-conversion techniques. A lot of effort has been made to enhance efficiency [13, 14]. Given that heat sources are plenty and free, TEGs could be promising solutions, when they are employed to harvest waste heat from industry activities and cen-

tral-heating systems.

Figure 14. Efficiency as a function of current.

Figure 13. TEG output power as a function of electrical current.

474 Thermoelectrics for Power Generation - A Look at Trends in the Technology

Employing various temperature differences, while maintaining the cold-side temperature at 373 K, voltage induced as a function of current behaves as shown in Figure 15.

One can observe from Figure 15 that voltage induced for each temperature difference is decreasing and the linear function of output electrical current. Slopes of I–V dependences are the same.

Figure 15. Voltage as a function of current (I-V dependences of TEG).

#### 4.4.3. Power and efficiency as a function of hot-side temperature

While still maintaining the cold side at a temperature of 373 K and replacing current in output power equation (Eq. (31)) by optimal current expression (Eq. (33)), power expression becomes a function of temperature at the hot side, and Figure 16 shows the behavior of output power as a function of the hot-side temperature.

Figure 16. Power as a function of hot-side temperature.

Output power as a function of hot-side temperature behaves as nonlinear curve increasing as the hot-side temperature increases.

The efficiency of TEG as a function of hot-side temperature is shown in Figure 17.

Figure 17. Efficiency of TEG as a function of hot-side temperature.

4.4.4. Power as a function of external load resistance

Figure 18 depicts variations of output power as a function of external load resistance. Eq. (35) is used to obtain dependences shown in Figure 18.

Figure 18. Output power as a function of external load resistance.

Optimal output power occurs when load resistance equates to internal electrical resistance of the total number of p- and n-type semiconductor legs.

#### 4.4.5. Efficiency as a function of the figure of merit (ZT)

Output power as a function of hot-side temperature behaves as nonlinear curve increasing as

Figure 18 depicts variations of output power as a function of external load resistance. Eq. (35)

The efficiency of TEG as a function of hot-side temperature is shown in Figure 17.

the hot-side temperature increases.

476 Thermoelectrics for Power Generation - A Look at Trends in the Technology

4.4.4. Power as a function of external load resistance

Figure 17. Efficiency of TEG as a function of hot-side temperature.

is used to obtain dependences shown in Figure 18.

Figure 18. Output power as a function of external load resistance.

ZT value is modified figure of merit, where T represents averaged temperature between the hot-side and cold-side temperatures. For each temperature difference, efficiency increases as ZT value increases. Therefore, employing thermoelectric materials possessing high ZT values leads to great TEG efficiency (Figure 19).

Figure 19. Efficiency as a function of ZT value.

#### 5. Conclusion

In this chapter, the basics of thermoelectric generator devices are covered including phenomena that guide their operation. State-of-the-art modeling efforts are summarized. The presented modeling is crucial for comprehensive understanding of heat to electric energy conversion in TEGs. Simulation results are very useful in predicting the maximum ratings of TEGs during operation under different ambient conditions.

## Acknowledgements

This work was funded by the startup fund from Virginia Polytechnic Institute and State University.

## Author details

Eurydice Kanimba and Zhiting Tian\*

\*Address all correspondence to: zhiting@vt.edu

Virginia Polytechnic Institute and State University, USA

## References


[13] Ebling D, et al. Module geometry and contact resistance of thermoelectric generators analyzed by multiphysics simulation. Journal of Electronic Materials, 2010. 39(9): p. 1376–1380. DOI: 10.1007/s11664-010-1331-0

Author details

References

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Sons, Inc.

00012-7

jpowsour.2005.12.084

Eurydice Kanimba and Zhiting Tian\*

\*Address all correspondence to: zhiting@vt.edu

Energy, 2015. 90: p. 1239–1250.

University Press on Demand, Oxford, 1997.

InTech Open Access Publisher, Croatia, 2011.

DOI: http://dx.doi.org/10.1016/j.energy.2011.03.057

Virginia Polytechnic Institute and State University, USA

478 Thermoelectrics for Power Generation - A Look at Trends in the Technology

[1] Rowe DM. CRC handbook of thermoelectric, CRC press, Boca Raton, 1995.

[7] Sutton GW. Direct energy conversion, McGraw-Hill, New York, 1966.

[2] Rowe DM and Bhandari CM. Modern thermoelectrics, Prentice Hall, Upper Saddle River,

[3] Zevenhoven R and Beyene A. The relative contribution of waste heat from power plants to global warming. Energy, 2011. 36(6): p. 3754–3762. DOI: 10.1016/j.energy.2010.10.010 [4] Moh'd AA-N, Tashtoush BM and Jaradat AA. Modeling and simulation of thermoelectric device working as a heat pump and an electric generator under Mediterranean climate.

[5] Mamur H and Ahiska R. A review: Thermoelectric generators in renewable energy. International Journal of Renewable Energy Research (IJRER), 2014. 4(1): p. 128–136. [6] Ioffe A, Kaye J and Welsh JA. Direct conversion of heat to electricity. 1960: John Wiley and

[8] Decher R. Direct energy conversion: fundamentals of electric power production, Oxford

[9] Riffat SB and Ma X. Thermoelectrics: A review of present and potential applications. Applied Thermal Engineering, 2003. 23(8): p. 913–935. DOI: 10.1016/S1359-4311(03)

[10] Dziurdzia P. Modeling and simulation of thermoelectric energy harvesting processes,

[11] Thomas JP, Qidwai MA and Kellogg JC. Energy scavenging for small-scale unmanned systems. Journal of Power Sources, 2006. 159(2): p. 1494–1509. DOI: 10.1016/j.

[12] Meng F, Chen L, and Sun F. A numerical model and comparative investigation of a thermoelectric generator with multi-irreversibilities. Energy, 2011. 36(5): p. 3513–3522. [14] Priya S and Inman DJ. Energy harvesting technologies. Vol. 21, Springer, New York 2009.

#### **Calculation Methods for Thermoelectric Generator Performance Calculation Methods for Thermoelectric Generator Performance**

Fuqiang Cheng Fuqiang Cheng

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/65596

#### **Abstract**

This chapter aims to build one-dimensional thermoelectric model for device-level thermoelectric generator (TEG) performance calculation and prediction under steady heat transfer. Model concept takes into account Seebeck, Peltier, Thomson effects, and Joule conduction heat. Thermal resistances between heat source, heat sink, and thermocouple are also considered. Then, model is simplified to analyze influences of basic thermal and electrical parameters on TEG performance, when Thomson effect is neglected. At last, an experimental setup is introduced to gauge the output power and validate the model. Meantime, TEG simulation by software ANSYS is introduced briefly.

**Keywords:** thermoelectric generator, thermoelectric model, output power, thermoelement

## **1. Introduction**

Output power *P*out and energy conversion efficiency *η* are the primary parameters to characterize TEG performance. They are intensively influenced by such factors as temperature of heat source and sink, thermoelectric materials physical properties, thermocouple geometries, thermal and electrical contact properties, and load factor. Therefore, it is necessary to build physical model formulating these factors concisely, to conduct realistic TEG design. At present, many significant works have been undertaken for modeling device-level TEG precisely [1–3]. In addition, comprehensive three-dimensional (3D) thermoelectric model has been successfully developed in software ANSYS [4]. In Refs. [5–7], quasi-one-dimensional thermoelectric model is established, where Thomson effect and thermal resistances between thermocouple and heat source, heat sink are neglected. In Ref. [8], improved one-dimensional model

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

including Thomson coefficient and thermal resistances is used to analyze the matched load, the limit of energy conversion efficiency, and the influence of Peltier effect. It shows, that expression of matched load contains not only the inner electrical resistance of TEG, but also the terms resulting from Peltier and Joule effects. In Ref. [9], one-dimensional model to analyze the influence of Thomson heat is built and experimentally validated.

In this chapter, Seebeck, Peltier, Thomson effect, and Joule conduction heat are formulated in thermoelectric generation module model. By model simplification, analytical expressions of output power and energy efficiency are introduced. Essential factors for enhancing the output power are extracted. Then, an experimental setup is built to measure the output power and validate the model. And TEG simulation by software ANSYS is presented.

## **2. Thermoelectric model for device-level TEG**

#### **2.1. TEG cell structure**

TEG cell consisting of thermocouple is shown in **Figure 1**, where basic thermoelectric effects including Peltier and Joule heat and a circuit with load *R*L are included. The p and n thermoelements are cuboids of the same thickness and bridged by an electrode in series. Practical devices usually make use of thermoelectric modules containing a number of TEG cells connected electrically in series and thermally in parallel. Cross-sectional area and thickness of thermocouple are marked as *A* and *l*. Subscripts 'n' and 'p' are used to discriminate conductivity type of thermoelements. Temperature of heat source and heat sink is *T*1 and *T*0, and that of hot and cold side of thermocouple is *T*h and *T*c. ∆*T*g = *T*h − *T*c is temperature difference on thermoelements, and ∆*T* = *T*1 − *T*0 is the one of heat source and heat sink.

**Figure 1.** Structure and circuit sketch of TEG cell.

There are Joule heat flowing out and Peltier heat flowing in at hot end of thermoelements, and at cold end, Peltier heat flows out and Joule heat flows out. In addition, there is thermal resistance *R*th,h and *R*th,c between thermoelements, heat source and heat sink. Heat flow *q*h passes from heat source to hot side of thermocouple and the counterpart *q*<sup>c</sup> outflows from cold side of thermocouple to heat sink.

#### **2.2. Basic model**

including Thomson coefficient and thermal resistances is used to analyze the matched load, the limit of energy conversion efficiency, and the influence of Peltier effect. It shows, that expression of matched load contains not only the inner electrical resistance of TEG, but also the terms resulting from Peltier and Joule effects. In Ref. [9], one-dimensional model to analyze

In this chapter, Seebeck, Peltier, Thomson effect, and Joule conduction heat are formulated in thermoelectric generation module model. By model simplification, analytical expressions of output power and energy efficiency are introduced. Essential factors for enhancing the output power are extracted. Then, an experimental setup is built to measure the output power and

TEG cell consisting of thermocouple is shown in **Figure 1**, where basic thermoelectric effects including Peltier and Joule heat and a circuit with load *R*L are included. The p and n thermoelements are cuboids of the same thickness and bridged by an electrode in series. Practical devices usually make use of thermoelectric modules containing a number of TEG cells connected electrically in series and thermally in parallel. Cross-sectional area and thickness of thermocouple are marked as *A* and *l*. Subscripts 'n' and 'p' are used to discriminate conductivity type of thermoelements. Temperature of heat source and heat sink is *T*1 and *T*0, and that of hot and cold side of thermocouple is *T*h and *T*c. ∆*T*g = *T*h − *T*c is temperature difference on

the influence of Thomson heat is built and experimentally validated.

482 Thermoelectrics for Power Generation - A Look at Trends in the Technology

validate the model. And TEG simulation by software ANSYS is presented.

thermoelements, and ∆*T* = *T*1 − *T*0 is the one of heat source and heat sink.

**2. Thermoelectric model for device-level TEG**

**2.1. TEG cell structure**

**Figure 1.** Structure and circuit sketch of TEG cell.

It is assumed, that thermoelements are physically homogeneous and insulated from the surroundings both electrically and thermally, except at junction-reservoir contacts [8–9]. Variable *x* is defined as location in the thickness direction of thermoelements. According to nonequilibrium thermodynamics under steady heat transfer, energy conservative equations of temperature distributions *T*n(*x*) and *T*p(*x*) are:

$$\begin{cases} K\_{\boldsymbol{n}} l\_{\boldsymbol{n}} \frac{d^2 T\_{\boldsymbol{n}}(\mathbf{x})}{d\mathbf{x}^2} - \tau\_{\boldsymbol{n}} I \frac{dT\_{\boldsymbol{n}}(\mathbf{x})}{d\mathbf{x}} + \frac{R\_{\boldsymbol{n}} I^2}{l\_{\boldsymbol{n}}} = 0\\ K\_{\boldsymbol{p}} l\_{\boldsymbol{p}} \frac{d^2 T\_{\boldsymbol{p}}(\mathbf{x})}{d\mathbf{x}^2} + \tau\_{\boldsymbol{p}} I \frac{dT\_{\boldsymbol{p}}(\mathbf{x})}{d\mathbf{x}} + \frac{R\_{\boldsymbol{p}} I^2}{l\_{\boldsymbol{p}}} = 0 \end{cases} \tag{1}$$

Three terms in the above equations represent thermal conduction, Thomson and Joule heat. *K*, *R,* and *τ* are thermal conductance, electrical resistance and Thomson coefficient (V∙K−1), respectively. Relationship of *K*, *R* and *A*, *l* is = and = , where *λ* and *ρ* are thermal conductivity and electrical resistivity of thermoelectric materials. To solve Eq. (1) analytically, material parameters *K*, *R,* and *τ* are considered to be constant. The boundary conditions of Eq. (1) are:

$$T\_n\left(\mathbf{0}\right) = T\_p\left(\mathbf{0}\right) = T\_\varepsilon,\tag{2}$$

$$T\_n\left(I\_n\right) = T\_p\left(I\_p\right) = T\_h. \tag{3}$$

Electrical current *I* is determined by formula:

$$I = \frac{U\_0}{R\_\text{L} + R\_\text{g}},\tag{4}$$

$$U\_o = \int\_{T\_4}^{T\_b} \alpha(T) \mathrm{d}T,\tag{5}$$

where *U*0 is the voltage of thermocouple, *R*g is the electrical resistance of TEG cell, which contains resistance of thermocouple and contact resistance, and *α*(*T*) = *α*p(*T*) − *α*n(*T*) is Seebeck coefficient (V∙K−1) of thermocouple.

In practice, temperature of heat source *T*1 and heat sink *T*0 can be measured and determined. To acquire *T*h and *T*c, relationship of *T*1, *T*0 and *T*h, *T*c is necessary. That is:

$$\begin{cases} T\_{\text{l}} - T\_{\text{h}} = R\_{\text{th},\text{h}} q\_{\text{h}}\\ T\_{\text{c}} - T\_{\text{o}} = R\_{\text{th},\text{c}} q\_{\text{c}} \end{cases} \tag{6}$$

In Eq. (6), heat flows *q*h and *q*c are:

$$q\_{\mathbf{h}} = K\_{\mathbf{n}} l\_{\mathbf{n}} \frac{\mathbf{d}T\_{\mathbf{n}}(\mathbf{x})}{\mathbf{d}\mathbf{x}}\Big|\_{\mathbf{x}=l\_{\mathbf{n}}} + K\_{\mathbf{p}} l\_{\mathbf{p}} \frac{\mathbf{d}T\_{\mathbf{p}}(\mathbf{x})}{\mathbf{d}\mathbf{x}}\Big|\_{\mathbf{x}=l\_{\mathbf{p}}} + \alpha(T\_{\mathbf{h}})T\_{\mathbf{h}}I - I^{2}R\_{\mathbf{ch}},\tag{7}$$

$$q\_{\varepsilon} = K\_{\text{n}} l\_{\text{n}} \frac{\text{d}T\_{\text{n}}(\text{x})}{\text{d}\mathbf{x}}\Big|\_{\text{x}=0} + K\_{\text{p}} l\_{\text{p}} \frac{\text{d}T\_{\text{p}}(\text{x})}{\text{d}\mathbf{x}}\Big|\_{\text{x}=0} + \alpha(T\_{\text{c}})T\_{\text{c}}I + I^{2}R\_{\text{c}},\tag{8}$$

wherein *R*ch and *R*cc are contact electrical resistances at hot and cold side of the thermocouple. Thermal conduction heat, Peltier heat (the third term), and contact Joule heat are within Eqs. (7) and (8). By solving Eq. (1) with Eqs. (2)–(5), *T*n(*x*) and *T*p(*x*) only relating to *T*h, *T*c, and *R*<sup>L</sup> can be obtained. And flows *q*h and *q*c can be formulated with *T*h, *T*c, and *R*L in Eqs. (7) and (8). Then, *T*h and *T*c can be determined for a given *R*L by solving Eq. (6) numerically, which is presented in detail [9].

Finally, the output power *P*out and energy conversion efficiency *η* are calculated by the basic equations of thermoelectricity:

$$P\_{\rm out} = q\_{\rm h} - q\_{\rm c} = I^2 R\_{\rm L},\tag{9}$$

$$
\eta = \frac{P}{q\_{\text{h}}}.\tag{10}
$$

When neglecting Thomson heat, the problem will be much simplified. By solving Eqs. (1)–(8) with *τ* = 0, an cubic equation about Δ*T*g can be yielded as:

$$a\_1 \Delta T\_g^3 + b\_1 \Delta T\_g^2 + c\_1 \Delta T\_g + \mathbf{l} = \mathbf{0},\tag{11}$$

where:

$$a\_{\mathrm{l}} = \frac{\overline{\alpha}^4 R\_{\mathrm{l}} R\_{\mathrm{th},\mathrm{c}} R\_{\mathrm{th},\mathrm{h}}}{\left(R\_{\mathrm{g}} + R\_{\mathrm{L}}\right)^3 \Delta T},$$

where *U*0 is the voltage of thermocouple, *R*g is the electrical resistance of TEG cell, which contains resistance of thermocouple and contact resistance, and *α*(*T*) = *α*p(*T*) − *α*n(*T*) is Seebeck

In practice, temperature of heat source *T*1 and heat sink *T*0 can be measured and determined.

n p 2 h nn p p h h ch

( ) <sup>p</sup> ( ) <sup>n</sup> <sup>2</sup> c nn 0 pp 0 cc cc

wherein *R*ch and *R*cc are contact electrical resistances at hot and cold side of the thermocouple. Thermal conduction heat, Peltier heat (the third term), and contact Joule heat are within Eqs. (7) and (8). By solving Eq. (1) with Eqs. (2)–(5), *T*n(*x*) and *T*p(*x*) only relating to *T*h, *T*c, and *R*<sup>L</sup> can be obtained. And flows *q*h and *q*c can be formulated with *T*h, *T*c, and *R*L in Eqs. (7) and (8). Then, *T*h and *T*c can be determined for a given *R*L by solving Eq. (6) numerically, which is

Finally, the output power *P*out and energy conversion efficiency *η* are calculated by the basic

h . *P q* h

3 2

with *τ* = 0, an cubic equation about Δ*T*g can be yielded as:

When neglecting Thomson heat, the problem will be much simplified. By solving Eqs. (1)–(8)

2

out h c L *P q q IR* =-= , (9)

<sup>111</sup> 1 0, *ggg aT bT cT* D +D +D += (11)

= (10)

() , d d *x x*

() , d d *<sup>p</sup> x l x l*

a

a

(7)

(8)

(6)

1 h th,h h c 0 th,c c . *TT Rq T T Rq*

ì - = ï í ï - = î

( ) ( ) n

*q Kl K l T TI I R x x* = + +- <sup>=</sup> <sup>=</sup>

*q Kl K l T TI I R x x* = + ++ <sup>=</sup> <sup>=</sup>

*T x T x*

d d

d d

*T x T x*

To acquire *T*h and *T*c, relationship of *T*1, *T*0 and *T*h, *T*c is necessary. That is:

coefficient (V∙K−1) of thermocouple.

484 Thermoelectrics for Power Generation - A Look at Trends in the Technology

In Eq. (6), heat flows *q*h and *q*c are:

presented in detail [9].

where:

equations of thermoelectricity:

$$b\_{\rm 1} = -\frac{\overline{\alpha}^2 R\_{\rm th,c} R\_{\rm th,h}}{\left(R\_{\rm g} + R\_{\rm L}\right)^2 \Delta T} \left[ R\_{\rm g} \left(\frac{\varepsilon}{R\_{\rm th,c}} + \frac{\varepsilon - 1}{R\_{\rm th,h}}\right) + R\_{\rm L} \left(\frac{1}{R\_{\rm th,c}} - \frac{1}{R\_{\rm th,h}}\right) \right],$$

$$\mathcal{L}\_{1} = -\frac{R\_{\text{th},\text{c}}R\_{\text{th},\text{h}}}{\Delta T} \left[ \left( \frac{T\_{\text{l}}}{R\_{\text{th},\text{h}}} + \frac{T\_{\text{o}}}{R\_{\text{th},\text{c}}} \right) \frac{\alpha^{2}}{R\_{\text{g}} + R\_{\text{L}}} + \frac{K\_{\text{g}}}{R\_{\text{th},\text{h}}} + \frac{K\_{\text{g}}}{R\_{\text{th},\text{c}}} + \frac{1}{R\_{\text{th},\text{c}}R\_{\text{th},\text{h}}} \right],$$

$$
\varepsilon = \frac{0.5R\_0 + R\_\infty}{R\_\text{g}},
$$

and Seebeck coefficient *α* becomes a constant. Equation (11) suits thermoelectric module consisting of *m* thermocouples, as well, where *α* and *K*g are *m* times of those of a single thermocouple, but *R*th,h and *R*th,c are exactly on the contrary.

Generally, *c*1 is far larger than *a*1 and *b*1 in absolute value. Mainly, because of practical module, Seebeck coefficient *α* has a very small value of about 10−2 V∙K−1, which is much less than unity. For example, taking module TEG-127-150-9 in Ref. [8], *α* = 0.05 V∙K−1, *R*g = 3.4 Ohm, *R*L = 4 Ohm, *R*th,c = 6 K∙W−1, *R*th,h = 0.1 K∙W−1, *K*g = 2.907 W∙K−1, *ε* ≈ 0.5, *T*0 = 297 K, and *T*1 = 323 K, calculation result is *a*1 ≈ 1.423 × 10−9 K−3, *b*1 ≈ 5.905 × 10−5 K−2, and *c*1 ≈ −0.7461 K−1. So, the terms with Δ*T*<sup>g</sup> order higher than unity can be neglected. At last, here is:

$$
\Delta T\_{\rm g} = \frac{\Delta T}{1 + R\_{\rm th,c} K\_{\rm g} + R\_{\rm th,h} K\_{\rm g} + \alpha^2 \frac{\left(R\_{\rm th,c} T\_{\rm l} + R\_{\rm th,h} T\_0\right)}{R\_{\rm g} + R\_{\rm L}}} \tag{12}
$$

It can be seen, that Δ*T*g is influenced not only by thermal resistances *R*th,c and *R*th,h, but also by Peltier effect, which is presented in the last term of the denominator and functions to decrease Δ*T*g. Because it is tantamount to accelerate heat conduction in thermocouple, Peltier heat flows in and out on two sides of thermocouple. By combing Eqs. (12) and (4), (5), (9), the output power *P*out is:

$$P\_{\rm out} = \frac{\alpha^2 \Delta T\_{\rm g}^2 R\_{\rm L}}{\left(R\_{\rm g} + R\_{\rm L}\right)^2} = \frac{\alpha^2 \Delta T^2 R\_{\rm L}}{\left(1 + R\_{\rm th,c} K\_{\rm g} + R\_{\rm th,h} K\_{\rm g}\right)^2 \left[R\_{\rm g} + R\_{\rm L} + \frac{\alpha^2 \left(R\_{\rm th,c} T\_{\rm l} + R\_{\rm th,h} T\_{\rm o}\right)^2}{1 + R\_{\rm th,c} K\_{\rm g} + R\_{\rm th,h} K\_{\rm g}}\right]^2}.\tag{13}$$

*R*L, *T*1, *T*0, *R*th,c, *R*th,h, *α*, *R*g, and *K*g directly affect *P*out. In those parameters, *α*, *R*g and *K*g are TEG internal factors, and *R*L, *T*1, and *T*0 are the external ones, and *R*th,c and *R*th,h originate from both the internal and external. From the form of Eq. (13), it is obvious, that reducing *T*1, *T*0, *R*th,c, and *R*th,h can increase *P*out, if Δ*T* is constant, owing to influence of Peltier effect on Δ*T*g. On the other hand, *P*out has a maximum along with *R*g and *K*g.

#### **2.3. Matched load, output power and energy efficiency**

First of all, influence of *R*L on *P*out is analyzed. In Eq. (13), *P*out reaches maximum, when *R*L is:

$$R\_L = R\_\text{g} + \frac{\alpha^2 \left(R\_{\text{th},\text{s}}T\_\text{l} + R\_{\text{th,h}}T\_\text{o}\right)}{1 + R\_{\text{th},\text{s}}K\_\text{g} + R\_{\text{th,h}}K\_\text{g}},\tag{14}$$

which is the matched load and marked as *R*L,m. Indeed, *R*L,m is slightly larger than *R*g due to the very small value of *α*<sup>2</sup> . It means, that existence of Peltier effect increases irreversible heat in thermoelectric module. And reducing *T*1 and *T*0 helps to cut down this irreversible heat. When *K*g→+∞, *R*L,m is equal to *R*g, since at this moment heat conduction in thermocouple runs under infinitesimal temperature difference and the irreversibility of heat transfer disappears. However, this irreversibility exists with finite *K*g, leading to heat loss in thermocouple, that is equivalent to increase in internal resistance. Define *R*L/*R*g as the load factor *s*L. So, *s*L is:

$$\mathcal{S}\_{\rm L} = 1 + \frac{ZK\_{\rm g}\left(R\_{\rm th,s}T\_1 + R\_{\rm th,h}T\_0\right)}{1 + R\_{\rm th,s}K\_{\rm g} + R\_{\rm th,h}K\_{\rm g}},\tag{15}$$

when *R*L is equal to matched load and = 2 g g is the figure of merit. For thermoelectric module, the output power is:

$$P\_{\rm out} = \frac{m^2 \alpha^2 \Delta T^2 R\_{\rm L}}{\left(1 + R\_{\rm th,c} K\_{\rm g} + R\_{\rm th,h} K\_{\rm g}\right)^2 \left\{R\_{\rm L} + m \left[R\_{\rm g} + \frac{\alpha^2 \left(R\_{\rm th,c} T\_{\rm l} + R\_{\rm th,h} T\_{\rm o}\right)}{1 + R\_{\rm th,c} K\_{\rm g} + R\_{\rm th,h} K\_{\rm g}}\right]\right\}^2},\tag{16}$$

where *m* is the number of thermocouples. And the corresponding matched load is L,m = g + 2(th,c1 + th,h0) 1 + th,c <sup>g</sup> + th,h g .

As for energy efficiency *η*, by Eq. (7), which can be expressed as function of Δ*T*g and Eqs. (12) and (13), it is:

$$\eta = \frac{P}{q\_{\text{h}}} = \frac{\alpha^2 \Delta T^2 R\_{\text{L}}}{c\_{\text{2}}^2 \left(R\_{\text{L}} + R\_{\text{g}}\right)^2} \left\{ \frac{\alpha^2 \Delta T \left[T\_{\text{h}} \left(R\_{\text{L}} + R\_{\text{g}}\right) - \frac{\varepsilon \Delta T R\_{\text{g}}}{c\_{\text{2}}}\right]}{c\_{\text{2}} \left(R\_{\text{L}} + R\_{\text{g}}\right)^2} + \frac{\Delta T K\_{\text{g}}}{c\_{\text{2}}} \right\},\tag{17}$$

where

*R*L, *T*1, *T*0, *R*th,c, *R*th,h, *α*, *R*g, and *K*g directly affect *P*out. In those parameters, *α*, *R*g and *K*g are TEG internal factors, and *R*L, *T*1, and *T*0 are the external ones, and *R*th,c and *R*th,h originate from both the internal and external. From the form of Eq. (13), it is obvious, that reducing *T*1, *T*0, *R*th,c, and *R*th,h can increase *P*out, if Δ*T* is constant, owing to influence of Peltier effect on Δ*T*g. On the other

First of all, influence of *R*L on *P*out is analyzed. In Eq. (13), *P*out reaches maximum, when *R*L is:

, <sup>1</sup> *<sup>L</sup>*

( ) <sup>2</sup> th,c 1 th,h 0

*RT RT*

*RK RK*

which is the matched load and marked as *R*L,m. Indeed, *R*L,m is slightly larger than *R*g due to

in thermoelectric module. And reducing *T*1 and *T*0 helps to cut down this irreversible heat. When *K*g→+∞, *R*L,m is equal to *R*g, since at this moment heat conduction in thermocouple runs under infinitesimal temperature difference and the irreversibility of heat transfer disappears. However, this irreversibility exists with finite *K*g, leading to heat loss in thermocouple, that is equivalent to increase in internal resistance. Define *R*L/*R*g as the load factor *s*L. So, *s*L is:

g th,c 1 th,h 0 ( )

+

th,c g th,h g 1 , <sup>1</sup> *ZK R T R T <sup>s</sup> RK RK*

> g g

22 2

out <sup>2</sup> <sup>2</sup> <sup>2</sup> th,c 1 th,h 0

a

( ) <sup>1</sup>

+ + ++ í ý ê ú

where *m* is the number of thermocouples. And the corresponding matched load is

As for energy efficiency *η*, by Eq. (7), which can be expressed as function of Δ*T*g and Eqs. (12)

th,c g th,h g L g

*R K R K R mR*

g . L

1

a

ï ï ì ü é ù +

+ + ï ï î þ ë û

th,c g th,h g

+ + (14)

+ + (15)

th,c g th,h g

*RT RT*

*RK RK*

is the figure of merit. For thermoelectric

,

(16)

. It means, that existence of Peltier effect increases irreversible heat

hand, *P*out has a maximum along with *R*g and *K*g.

the very small value of *α*<sup>2</sup>

module, the output power is:

L,m =

and (13), it is:

g +

**2.3. Matched load, output power and energy efficiency**

486 Thermoelectrics for Power Generation - A Look at Trends in the Technology

g

= +

a+

*R R*

L

when *R*L is equal to matched load and = 2

( )

2(th,c1 + th,h0)

<sup>g</sup> + th,h

1 + th,c

*m TR <sup>P</sup>*

<sup>D</sup> <sup>=</sup>

= +

$$\mathcal{L}\_2 = 1 + R\_{\text{th},\text{c}} K\_{\text{g}} + R\_{\text{th},\text{h}} K\_{\text{g}} + \frac{\alpha^2 \left( R\_{\text{th},\text{c}} T\_{\text{l}} + R\_{\text{th},\text{h}} T\_0 \right)}{R\_{\text{L}} + R\_{\text{g}}}.$$

By solving Eq. (17) about the partial derivative of *R*L, it can be obtained, that when load factor *s*L is:

$$\mathbf{s}\_{\rm L} = \sqrt{\mathbf{l} + ZT\_{\rm h} + \frac{Z\varepsilon\Delta T + ZK\_{\rm g}\left(\mathbf{l} + ZT\_{\rm h}\right)\left(R\_{\rm th,c}T\_{\rm i} + R\_{\rm th,h}T\_{\rm o}\right)}{\mathbf{l} + R\_{\rm th,c}K\_{\rm g} + R\_{\rm th,h}K\_{\rm g}}}},\tag{18}$$

then *η* reaches maximum. Equation (18) is downright different from Eq. (15) in the expressions, so achieving maximum of output power and energy efficiency simultaneously is impossible. Actually, the corresponding load factor of the former is smaller than that of the latter. When the ideal state is considered (*R*th,c = *R*th,h = 0), *s*L = 1 is for the former and L <sup>=</sup> 1 + h + , which is larger than 1, is for the latter.

#### **2.4. Influence of** *K***g on TEG performance**

*K*g is important internal factor that influences the output performance in TEG. When matched load is reached, the corresponding output power *P*out,m is:

$$P\_{\rm out,n} = \frac{ZK\_{\rm g}\Delta T^2}{4\left(1 + R\_{\rm th,c}K\_{\rm g} + R\_{\rm th,h}K\_{\rm g}\right)^2 \left[1 + \frac{ZK\_{\rm g}\left(R\_{\rm th,c}T\_1 + R\_{\rm th,h}T\_0\right)}{1 + R\_{\rm th,c}K\_{\rm g} + R\_{\rm th,h}K\_{\rm g}}\right]}.\tag{19}$$

For a common thermoelectric module, thermoelements have the same size, *l*n = *l*p and *A*n = *A*p, so the figure of merit *Z* is not related to their size, but material physical parameters. From Eq. (19), we can see, that increase in *Z* will enhance the output power. By solving Eq. (19) regarding the partial derivative of *K*g, when:

$$K\_{\rm g} = \frac{1}{\sqrt{\left(R\_{\rm th,c} + R\_{\rm th,h}\right)^2 + Z\left(R\_{\rm th,c}T\_1 + R\_{\rm th,h}T\_0\right)\left(R\_{\rm th,c} + R\_{\rm th,h}\right)}} = \left(\mathcal{A}\_{\rm p} + \mathcal{A}\_{\rm n}\right)\frac{A\_{\rm c}}{I\_{\rm c}},\tag{20}$$

then *P*out,m reaches maximum, where *l*e and *A*e are thickness and cross-sectional area of thermoelements. Since *R*th,c and *R*th,h are related to *A*e, but not to *l*e, there is an optimal *l*e to maximize *P*out,m:

$$P\_{\rm out,n} = \frac{Z\Delta T^2}{4c\_3 \left(1 + \frac{R\_{\rm th,c}}{c\_3} + \frac{R\_{\rm th,h}}{c\_3}\right)^2 \left[1 + \frac{Z\left(R\_{\rm th,c}T\_1 + R\_{\rm th,h}T\_0\right)}{c\_3 \left(1 + \frac{R\_{\rm th,c}}{c\_3} + \frac{R\_{\rm th,h}}{c\_3}\right)}\right]},\tag{21}$$

and

$$\mathcal{L}\_3 = \sqrt{\left(R\_{\text{th},\text{c}} + R\_{\text{th},\text{h}}\right)^2 + Z\left(R\_{\text{th},\text{c}}^2 T\_{\text{l}} + R\_{\text{th},\text{h}}^2 T\_0 + R\_{\text{th},\text{c}} R\_{\text{th},\text{h}} T\_{\text{l}} + R\_{\text{th},\text{c}} R\_{\text{th},\text{h}} T\_0\right)}.$$

#### **2.5. Influence of Peltier effect on TEG performance**

When Peltier effect is neglected, the relation of Δ*T*g and Δ*T* is:

$$
\Delta T\_{\rm g} = \frac{R\_{\rm th,g}}{R\_{\rm th,g} + R\_{\rm th,h} + R\_{\rm th,c}} \Delta T,\tag{22}
$$

and the corresponding output power with matched load *R*L = *R*g and constant Seebeck coefficient is:

$$P\_{\rm out,m} = \frac{\left(a\Delta T\right)^2}{4R\_{\rm g}\left(1 + R\_{\rm th,h}K\_{\rm g} + R\_{\rm th,c}K\_{\rm g}\right)}.\tag{23}$$

Meantime, the output power Eq. (13) is for the condition without Peltier effect, and the ratio of Eq. (23) and Eq. (19), ηPelt, reflects influence degree of Peltier effect on the output power:

$$\eta\_{\rm reh} = \left( 1 + \frac{ZT\_0}{2} \frac{R\_{\rm th,c} + R\_{\rm th,h}}{R\_{\rm th,g} + R\_{\rm th,h} + R\_{\rm th,c}} + \frac{Z}{2} \frac{R\_{\rm th,c} \Delta T}{R\_{\rm th,g} + R\_{\rm th,h} + R\_{\rm th,c}} \right)^2. \tag{24}$$

It is known, that when *R*th,g ≪ *R*th,c + *R*th,h, ηPelt is approximately (1 + 0.5*ZT*0 + 0.25*Z*Δ*T*)2 with *R*th,c ≈ *R*th,h, and even with Δ*T* → 0, the output power calculated without Peltier effect is more than the output power considering Peltier effect, by over 120% for a common Bi2Te3-based module with *ZT* ≈ 1. That means, the influence of Peltier effect must be considered. Similar status is obtained, where the difference is more than 50%, when *R*th,g ≈ *R*th,c + *R*th,h. On the contrary, when *R*th,g ≫ *R*th,c + *R*th,h, ηPelt is approximately equal to 1, which means the influence of Peltier effect is negligible. Hence, the smaller the thermal resistance of thermocouple *R*th,g, the stronger is the influence of Peltier effect.

Eventually, basic factors for enhancing TEG output power are summarized as:


## **3. Test validation**

( )( )( ) ( ) <sup>e</sup> <sup>g</sup> p n <sup>2</sup> <sup>e</sup> th,c th,h th,c 1 th,h 0 th,c th,h <sup>1</sup> <sup>=</sup> = + ,

then *P*out,m reaches maximum, where *l*e and *A*e are thickness and cross-sectional area of thermoelements. Since *R*th,c and *R*th,h are related to *A*e, but not to *l*e, there is an optimal *l*e to

2

th,c th,h th,c 1 th,h 0

ê ú æ ö <sup>+</sup> ç ÷ ++ + æ ö è ø ç ÷ + +

3 3 th,c th,h 3

( )( ) <sup>2</sup> 2 2 3 th,c th,h th,c 1 th,h 0 th,c th,h 1 th,c th,h 0 *c R R ZR T R T R R T R R T* = ++ ++ + .

1

2

*R R ZR T R T <sup>c</sup> c c R R <sup>c</sup> c c*

th,

th,g th,h th,c Δ Δ , *<sup>R</sup> <sup>g</sup> T T*

and the corresponding output power with matched load *R*L = *R*g and constant Seebeck

( ) ( )

*R RK RK* a

g th,h g th,c g <sup>Δ</sup> . 4 1 *T*

Meantime, the output power Eq. (13) is for the condition without Peltier effect, and the ratio of Eq. (23) and Eq. (19), ηPelt, reflects influence degree of Peltier effect on the output power:

> th,g th,h th,c th,g th,h th,c 1 .

0 th,c th,h th,c

*ZT R R RT Z RR RR R R*

æ ö + D = + ç ÷ + ++ ++ è ø

2 2

2

<sup>D</sup> <sup>=</sup> é ù

l l

,

( )

ë û è ø

3 3

*RRR* <sup>=</sup> + + (22)

<sup>=</sup> + + (23)

2

(20)

(21)

(24)

++ + + *<sup>A</sup> <sup>K</sup> <sup>l</sup> R R ZR T R T R R*

maximize *P*out,m:

and

coefficient is:

out,m

3

488 Thermoelectrics for Power Generation - A Look at Trends in the Technology

**2.5. Influence of Peltier effect on TEG performance**

When Peltier effect is neglected, the relation of Δ*T*g and Δ*T* is:

out,m

*P*

Pelt

h

g

4 1 1

*Z T <sup>P</sup>*

#### **3.1. Materials property**

P-type and n-type Bi2Te3-based materials are, respectively, Bi0.5Sb1.5Te3 and Bi2Te2.85Se0.15, which are prepared by mechanical alloy + spark plasma sintering method. Seebeck coefficient and resistivity of the materials are tested by HGTE-II thermoelectric material performance test system (Chinese patent no. ZL200510018806.4) with test temperature up to 1073 K, relative error of not more than 6%. Thermal conductivity of the materials is measured by laser perturbation method (Type TC-7000 of ULVAC RIKO®). As shown in **Table 1**, parameters are obtained by polynomial fitting of the experimental data. In temperature range 273 K < *T* < 493 K, Seebeck coefficient value *α* is between 170 × 10−6 V∙K−1 and 220 × 10−6 V∙K−1, decreasing with rising temperature. Electrical resistivity *ρ* is (8.3–20.0) × 10−6 Ohm∙m and thermal conductivity *λ* is 1.4–2.1 W∙m−1∙K−1, which both show obvious increase with temperature rise.


**Table 1.** Physical parameters of Bi2Te3-based materials (273 K < *T* < 493 K).

In practice, it is difficult to measure thermal resistances *R*th,c and *R*th,h, and contact electrical resistances *r*cc and *r*ch. Their values are determined according to empirical formulas. Contact electrical resistivity *ρ*c (Ohm∙m2 ) at leg-strap junctions and thermal conductivity *λ*c (W m−1 K−1) of thermal conductive layer (≈1.2 mm thick) are according to the empirical formulas given by Rowe et al. [5]:

$$\frac{2\,\rho\_{\text{c}}}{\rho} = 0.1 \text{ mm}, \frac{\lambda}{\lambda\_{\text{c}}} = 0.2. \tag{25}$$

Here *ρ* and *λ* are electrical resistivity and thermal conductivity of thermoelements, respectively. In our calculation, the mean values of *ρ* and *λ* over the temperature range are taken as references. As *ρ* and *λ* vary with temperature, values of *ρ*c and *λ* c are also different as the temperature varies. Experiments under four temperature conditions are carried out and the corresponding parameter values are shown in **Table 2**.


**Table 2.** Thermal resistances and contact electrical resistances under different temperatures.

#### **3.2. Test setup**

System for measuring output performance of thermoelectric modules was established, mainly including electric heating plate controlled by PID, adjustable load, circulatory cooling unit, thermal imaging device, temperature and voltage data acquisition units, etc., with its basic structure as shown in **Figure 2**. Electric heating plate is used as heat source, with temperature control precision of ±0.1 K and temperature ranging from room temperature to 773 K. Cooling unit, which consists of heat sink, water tank, flow meter and flow valve, etc., takes cold water as the coolant. Heat sink is made of red-copper and its temperature could be adjusted by controlling flowrate of cooling water. In addition, some thermal conductive filler is pasted on both sides of module to reduce thermal resistance between module, heat source and heat sink. Electrical current in the circuit is obtained via measuring voltage on both ends of sampling resistor (metal film precision resistor: 0.2 Ohm, precision of ±1%).

Voltage and temperature signal are acquired by 9207 and 9214 acquisition card of National Instruments (NI) Company, with precision of ±0.5%. Data to be acquired include as follows: (1) temperature of heat source and heat sink; (2) temperature of the coolant (water) inside heat sink and water tank; (3) voltage on adjustable load and sampling resistor. K-type thermocouples with diameter of 1 mm are inserted in heat source and heat sink to measure temperature values. Actually, even though electric heating plate is controlled by PID, heat source temperature still fluctuates during the change of load resistance. In order to eliminate impacts of such transient effect, data shall not be acquired until the heat source and heat sink temperatures are stable.

**Figure 2.** Configuration of the output performance test system for the thermoelectric modules.

Test of energy efficiency is not undertaken due to its complexity, where the heat flow into the hot side of the module must be measured or evaluated. An effective way is to adopt heat flux sensor and bury it just under the module. But that would impact heat conduction between heat source and the module, leading to higher thermal resistance. And heat flux sensors of high temperature enduring are really costly. Another useful method is by calculating electrically generated heat in heat source, and at the same time, radiation and convective heat loss must be subtracted, as is introduced in Ref. [10].

#### **3.3. Comparison of test results with calculation**

In practice, it is difficult to measure thermal resistances *R*th,c and *R*th,h, and contact electrical resistances *r*cc and *r*ch. Their values are determined according to empirical formulas.

*λ*c (W m−1 K−1) of thermal conductive layer (≈1.2 mm thick) are according to the empirical

c

 l

 l

Here *ρ* and *λ* are electrical resistivity and thermal conductivity of thermoelements, respectively. In our calculation, the mean values of *ρ* and *λ* over the temperature range are taken as references. As *ρ* and *λ* vary with temperature, values of *ρ*c and *λ* c are also different as the temperature varies. Experiments under four temperature conditions are carried out and the

*T***1 = 81**

System for measuring output performance of thermoelectric modules was established, mainly including electric heating plate controlled by PID, adjustable load, circulatory cooling unit, thermal imaging device, temperature and voltage data acquisition units, etc., with its basic structure as shown in **Figure 2**. Electric heating plate is used as heat source, with temperature control precision of ±0.1 K and temperature ranging from room temperature to 773 K. Cooling unit, which consists of heat sink, water tank, flow meter and flow valve, etc., takes cold water as the coolant. Heat sink is made of red-copper and its temperature could be adjusted by controlling flowrate of cooling water. In addition, some thermal conductive filler is pasted on both sides of module to reduce thermal resistance between module, heat source and heat sink. Electrical current in the circuit is obtained via measuring voltage on both ends of sampling

Voltage and temperature signal are acquired by 9207 and 9214 acquisition card of National Instruments (NI) Company, with precision of ±0.5%. Data to be acquired include as follows: (1) temperature of heat source and heat sink; (2) temperature of the coolant (water) inside heat sink and water tank; (3) voltage on adjustable load and sampling resistor. K-type thermocou-

*r*cc/mOhm 0.6 0.7 0.7 0.8 *r*ch/mOhm 0.6 0.7 0.7 0.8 *R*th,c/K∙W−1 31.0 31.5 30.9 31.0 *R*th,h/K∙W−1 23.5 23.3 22.8 23.0

**Table 2.** Thermal resistances and contact electrical resistances under different temperatures.

resistor (metal film precision resistor: 0.2 Ohm, precision of ±1%).

*T***0 = 23** *T***1 = 111** *T***0 = 27** *T***1 = 147** *T***0 = 27** *T***1 = 177**

<sup>2</sup> 0.1 mm, 0.2.

c

r

r

corresponding parameter values are shown in **Table 2**.

**Temperatures, °C Parameters** *T***0 = 23**

) at leg-strap junctions and thermal conductivity

= = (25)

Contact electrical resistivity *ρ*c (Ohm∙m2

490 Thermoelectrics for Power Generation - A Look at Trends in the Technology

formulas given by Rowe et al. [5]:

**3.2. Test setup**

**Figures 3** and **4** show variations of output power *P*out with load *R*L at four temperature conditions, acquired by physical model calculation, ANSYS simulation and experiment. AN-SYS method will be introduced in the next part. **Figure 3** shows data at heat sink temperature *T*0 = 300 K, while **Figure 4**—at *T*0 = 296 K. **Figures 5** and **6** are the corresponding current-voltage (I-V) characteristics. *R*L results are disposed in the same way. From the results, it is found, that the output power has maximum value with the increase of load. And current is linearly related to voltage. Calculation results are well coincident with AN-SYS results, and they are both a little higher than experimental data. Under the four temperature conditions, values of maximum output power are 2.5, 2.6, 2.8 and 1.1% higher than experimental results with *T*1 changing from high to low. They are especially coincident well, when temperature difference Δ*T* is small. From the analysis follows, deviation of calculated results is caused mainly by taking thermal conductivity and electrical resistivity as constant (i.e., using the mean values), when solving Eq. (1). When Δ*T* is small, then material physical parameters vary within a narrow range. So, values of parameters are close to the real values. Otherwise, when Δ*T* is large, material physical parameters change within a large scale, leading to a great deviation of calculations.

**Figure 3.** Dependences of output power on load resistance: calculations, experiments and ANSYS at *T*0 = 300 K.

**Figure 4.** Dependences of output power on load resistance: calculations, experiments and ANSYS at *T*0 = 296 K.

**Figure 5.** I-V characteristics of thermoelectric module: calculations, experiments and ANSYS at *T*0 = 300 K.

**Figure 6.** I-V characteristics of thermoelectric module: calculations, experiments and ANSYS at *T*0 = 296 K.

## **4. Introduction to TEG simulation in ANSYS**

#### **4.1. TEG cell model**

the real values. Otherwise, when Δ*T* is large, material physical parameters change within a

**Figure 3.** Dependences of output power on load resistance: calculations, experiments and ANSYS at *T*0 = 300 K.

**Figure 4.** Dependences of output power on load resistance: calculations, experiments and ANSYS at *T*0 = 296 K.

**Figure 5.** I-V characteristics of thermoelectric module: calculations, experiments and ANSYS at *T*0 = 300 K.

large scale, leading to a great deviation of calculations.

492 Thermoelectrics for Power Generation - A Look at Trends in the Technology

By software simulation, TEG performance can be achieved both in thermal and in electrical aspects. But it is not direct to cognize and understand the influence of thermoelectric effects, when compared with the above physical model. In this part, TEG cell model is set up by ANSYS, and geometry and meshing methods are illustrated in **Figure 7**. Thickness and crosssectional area of thermoelements are 1.6 mm and 1.4 mm × 1.4 mm, respectively. Other geometry parameters are shown in **Figure 7**. Thermoelectric module consists mainly of p-n thermoelements, current-conducting copper straps and ceramic substrates for heat conducting and electric insulation. Thermoelements and copper strap are meshed by element SOLID226 in ANSYS. This type of element contains 20 nodes with voltage and temperature as the degrees of freedom. It can simulate 3D thermal-electrical coupling field. Element SOLID90 is used to mesh ceramic substrate. It has 20 nodes with temperature as the degree of freedom. Load resistance is simulated by element CIRCU124.

**Figure 7.** The geometry of TEG cell in ANSYS and its mesh.

Contact properties of the leg-strap junction are implemented with element pairs CON-TACT174/TARGET170. Detailed finite element formulations in ANSYS are introduced in [4], and the range of contact thermal conductivity and electrical resistivity is explicated in [11].

#### **4.2. APDL codes for TEG simulation**

ANSYS Parametric Design Language (APDL) is widely used for programed simulation. The following APDL codes have taken temperature variation of materials properties, thermal contact and thermal radiation (although its influence is very weak) into consideration. According to the practical requirements, the readers could use the code more concisely by neglecting certain physical effects. The unit referring to length is meter and the temperature unit is Celsius.

*! defining the TEG cell dimensions ln=1.6e-3 ! n-type thermoelement thickness lp=1.6e-3 ! p-type thermoelement thickness wn=1.4e-3 ! p-type thermoelement width wp=1.4e-3 ! p-type thermoelement width d=1.0e-3 ! Distance between the thermoelements hs=0.2e-3 ! copper strap thickness hc=1e-3 !substrate thickness ! definition of several physical parameters rsvx=1.8e-8 ! copper electrical resistivity kx=200 ! copper thermal conductivity kxs=24 !substrate thermal conductivity T1=250 ! temperature of heat source T0=30 ! temperature of heat sink Toffst=273 ! temperature offset ! defining TEG output parameters and the load \*dim,P0,array,1 ! defining P0 as the output power \*dim,R0,array,1 ! defining R0 as the load \*dim,Qh,array,1 ! defining Qh as the heat flow into the TEG cell \*dim,I,array,1 ! defining I as the current \*dim,enta,array,1 ! defining enta as the energy efficiency*

*\*vfill,R0(1),ramp,0.025 ! setting the load (Ohm)*

*! pre-processing before calculation, defining element type, building the structure and meshing /PREP7*

*toffst,Toffst ! set temperature offset*

*et,1,226,110 ! 20-node thermoelectric brick element*

*et,2,shell57 ! shell57 element for radiation simulation*

*et,3,conta174 ! conta174 element for contact simulation*

*et,4,targe170 ! target170 element for contact simulation*

*keyopt,3,1,4 ! taking temperature and voltage as the degree of freedom*

*keyopt,3,9,0*

Contact properties of the leg-strap junction are implemented with element pairs CON-TACT174/TARGET170. Detailed finite element formulations in ANSYS are introduced in [4], and the range of contact thermal conductivity and electrical resistivity is explicated in [11].

ANSYS Parametric Design Language (APDL) is widely used for programed simulation. The following APDL codes have taken temperature variation of materials properties, thermal contact and thermal radiation (although its influence is very weak) into consideration. According to the practical requirements, the readers could use the code more concisely by neglecting certain physical effects. The unit referring to length is meter and the temperature

**4.2. APDL codes for TEG simulation**

494 Thermoelectrics for Power Generation - A Look at Trends in the Technology

*! defining the TEG cell dimensions*

*ln=1.6e-3 ! n-type thermoelement thickness lp=1.6e-3 ! p-type thermoelement thickness*

*wn=1.4e-3 ! p-type thermoelement width wp=1.4e-3 ! p-type thermoelement width*

*! definition of several physical parameters*

*rsvx=1.8e-8 ! copper electrical resistivity*

*kx=200 ! copper thermal conductivity*

*kxs=24 !substrate thermal conductivity*

*! defining TEG output parameters and the load*

*\*dim,R0,array,1 ! defining R0 as the load*

*\*dim,I,array,1 ! defining I as the current*

*\*dim,P0,array,1 ! defining P0 as the output power*

*\*dim,Qh,array,1 ! defining Qh as the heat flow into the TEG cell*

*\*dim,enta,array,1 ! defining enta as the energy efficiency*

*T1=250 ! temperature of heat source*

*T0=30 ! temperature of heat sink*

*Toffst=273 ! temperature offset*

*hs=0.2e-3 ! copper strap thickness*

*hc=1e-3 !substrate thickness*

*d=1.0e-3 ! Distance between the thermoelements*

unit is Celsius.

*keyopt,3,10,1*

*keyopt,4,2,0*

*keyopt,4,3,0*

*! Temperature data points*

*mptemp,1,25,50,75,100,125,150*

*mptemp,7,175,200,225,250,275,300*

*mptemp,13,325,350*

*! Seebeck coefficient of the n-type material (V·K−1)*

*mpdata,sbkx,1,1,-160e-6,-168e-6,-174e-6,-180e-6,-184e-6,-187e-6*

*mpdata,sbkx,1,7,-189e-6,-190e-6,-189e-6,-186.5e-6,-183e-6,-177e-6*

*mpdata,sbkx,1,13,-169e-6,-160e-6*

*! electrical resistivity of the n-type material (Ohm\*m)*

*mpdata,rsvx,1,1,1.03e-5,1.06e-5,1.1e-5,1.15e-5,1.2e-5,1.28e-5*

*mpdata,rsvx,1,7,1.37e-5,1.49e-5,1.59e-5,1.67e-5,1.74e-5,1.78e-5*

*mpdata,rsvx,1,13,1.8e-5,1.78e-5*

*! thermal conductivity of the n-type material (m\* K−1)*

*mpdata,kxx,1,1,1.183,1.22,1.245,1.265,1.265,1.25*

*mpdata,kxx,1,7,1.22,1.19,1.16,1.14,1.115,1.09*

*mpdata,kxx,1,13,1.06,1.03*

*! Seebeck coefficient of the p-type material (V·K−1)*

*mpdata,sbkx,2,1,200e-6,202e-6,208e-6,214e-6,220e-6,223e-6 mpdata,sbkx,2,7,218e-6,200e-6,180e-6,156e-6,140e-6,120e-6 mpdata,sbkx,2,13,101e-6,90e-6 ! electrical resistivity of the p-type material (Ohm\*m) mpdata,rsvx,2,1,1.0e-5,1.08e-5,1.18e-5,1.35e-5,1.51e-5,1.7e-5 mpdata,rsvx,2,7,1.85e-5,1.98e-5,2.07e-5,2.143e-5,2.15e-5,2.1e-5 mpdata,rsvx,2,13,2.05e-5,2.0e-5 ! thermal conductivity of the p-type material (m\* K−1) mpdata,kxx,2,1,1.08,1.135,1.2,1.25,1.257,1.22 mpdata,kxx,2,7,1.116,1.135,1.13,1.09,1.12,1.25 mpdata,kxx,2,13,1.5,2.025 ! material property for cooper strap mp,rsvx,3,rsvx mp,kxx,3,kx ! material property for the substrate mp,kxx,4,kxs !radiation property for the p-n materials mp,emis,5 ! contact friction coefficient mp,mu,6,0 ! build the TEG cell structure block,d/2,wn+d/2,-ln,0,,t block,-(wp+d/2),-d/2,-lp,0,,t block,d/2,wn+d/2,,hs,,t block,-(wp+d/2),-d/2,,hs,,t block,-d/2,d/2,,hs,,t block,-(wp+d/2),-d/2,-lp,-(lp+hs),,t block,d/2,wn+d/2,-ln,-(ln+hs),,t block,-(wp+d/2),wn+d/2,hs,hs+hc,,t block,-(wp+d/2),wn+d/2,-(lp+hs),-(lp+hs+hc),,t*

*! glue the copper strap and the substrate vsel,s,loc,y,0,hs vsel,a,loc,y,hs,hc+hs vglue,all allsel vsel,s,loc,y,-lp-hs,-lp vsel,a,loc,y,-lp-hs-hc,-lp-hs vglue,all allsel ! meshing the TEG cell structure numcmp,all mshape,0,3d mshkey,1 type,1 mat,3 lsel,s,loc,x,-d/2,d/2 lsel,r,loc,y,0 lsel,r,loc,z,t lesize,all,d/3 vsel,s,loc,x,-d/2,d/2 vsel,r,loc,y,0,hs vsweep,all allsel esize,ww/3 type,1 mat,3 vsel,s,loc,y,0,hs vsel,u,loc,x,-d/2,d/2 vsweep,all vsel,s,loc,y,-lp-hs,-lp*

*mpdata,sbkx,2,1,200e-6,202e-6,208e-6,214e-6,220e-6,223e-6 mpdata,sbkx,2,7,218e-6,200e-6,180e-6,156e-6,140e-6,120e-6*

496 Thermoelectrics for Power Generation - A Look at Trends in the Technology

*mpdata,rsvx,2,1,1.0e-5,1.08e-5,1.18e-5,1.35e-5,1.51e-5,1.7e-5 mpdata,rsvx,2,7,1.85e-5,1.98e-5,2.07e-5,2.143e-5,2.15e-5,2.1e-5*

*! electrical resistivity of the p-type material (Ohm\*m)*

*! thermal conductivity of the p-type material (m\* K−1)*

*mpdata,kxx,2,1,1.08,1.135,1.2,1.25,1.257,1.22 mpdata,kxx,2,7,1.116,1.135,1.13,1.09,1.12,1.25*

*mpdata,sbkx,2,13,101e-6,90e-6*

*mpdata,rsvx,2,13,2.05e-5,2.0e-5*

*mpdata,kxx,2,13,1.5,2.025*

*mp,rsvx,3,rsvx mp,kxx,3,kx*

*mp,kxx,4,kxs*

*mp,emis,5*

*mp,mu,6,0*

*! material property for cooper strap*

*! material property for the substrate*

*! contact friction coefficient*

*! build the TEG cell structure*

*block,d/2,wn+d/2,-ln,0,,t*

*block,d/2,wn+d/2,,hs,,t*

*block,-d/2,d/2,,hs,,t*

*block,-(wp+d/2),-d/2,,hs,,t*

*block,-(wp+d/2),-d/2,-lp,-(lp+hs),,t block,d/2,wn+d/2,-ln,-(ln+hs),,t*

*block,-(wp+d/2),wn+d/2,hs,hs+hc,,t*

*block,-(wp+d/2),wn+d/2,-(lp+hs),-(lp+hs+hc),,t*

*block,-(wp+d/2),-d/2,-lp,0,,t*

*!radiation property for the p-n materials*

*vsweep,all type,1 mat,1 vsel,s,loc,x,d/2,d/2+wn vsel,r,loc,y,-ln,0 vmesh,all mat,2 vsel,s,loc,x,-(wp+d/2),-d/2 vsel,r,loc,y,-lp,0 vmesh,all type,1 mat,4 vsel,s,loc,y,hs,hs+hc vsel,a,loc,y,-lp-hs-hc,-lp-hs vsweep,all allsel ! defining the contact parameters r,5 ! selecting the thermal contact conductivity and resistivity RMORE, rmore,,7e5 ! setting the thermal contact conductivity rmore,0.67e8,0.5 ! setting the thermal contact resistivity ! defining the contact layer between p-leg and upper copper strap vsel,s,loc,y,0,hs asel,s,ext asel,r,loc,y,0 nsla,s,1 nsel,r,loc,x,-(wp+d/2),-d/2 type,3 mat,6 real,5*

*esurf allsel ! defining the target layer between p-leg and upper copper strap vsel,s,mat,,2 asel,s,ext asel,r,loc,y,0 nsla,s,1 type,4 mat,6 esurf allsel ! defining the contact layer between n-leg and upper copper strap vsel,s,loc,y,0,hs asel,s,ext asel,r,loc,y,0 nsla,s,1 nsel,r,loc,x,d/2,d/2+wn type,3 mat,6 real,5 esurf allsel ! defining the target layer between n-leg and upper copper strap vsel,s,mat,,1 asel,s,ext asel,r,loc,y,0 nsla,s,1 type,4 mat,6 esurf*

*vsweep,all*

*vsel,s,loc,x,d/2,d/2+wn*

*vsel,s,loc,x,-(wp+d/2),-d/2*

*vsel,r,loc,y,-ln,0*

*vsel,r,loc,y,-lp,0*

*vsel,s,loc,y,hs,hs+hc*

*vsel,a,loc,y,-lp-hs-hc,-lp-hs*

*! defining the contact parameters*

*r,5 ! selecting the thermal contact conductivity and resistivity*

498 Thermoelectrics for Power Generation - A Look at Trends in the Technology

*rmore,,7e5 ! setting the thermal contact conductivity*

*rmore,0.67e8,0.5 ! setting the thermal contact resistivity*

*! defining the contact layer between p-leg and upper copper strap*

*vmesh,all*

*vmesh,all*

*vsweep,all*

*RMORE,*

*vsel,s,loc,y,0,hs*

*asel,s,ext*

*nsla,s,1*

*type,3 mat,6 real,5*

*asel,r,loc,y,0*

*nsel,r,loc,x,-(wp+d/2),-d/2*

*allsel*

*type,1 mat,4*

*mat,2*

*type,1 mat,1*

*allsel ! defining the contact layer between p-leg and bottom copper strap vsel,s,loc,y,-hs-lp,-lp vsel,r,loc,x,-wp-d/2,-d/2 asel,s,ext asel,r,loc,y,-lp nsla,s,1 type,3 mat,6 real,5 esurf allsel ! defining the target layer between p-leg and bottom copper strap vsel,s,mat,,2 asel,s,ext asel,r,loc,y,-lp nsla,s,1 type,4 mat,6 esurf allsel ! defining the contact layer between n-leg and bottom copper strap vsel,s,loc,y,-hs-ln,-ln vsel,r,loc,x,d/2,d/2+wn asel,s,ext asel,r,loc,y,-ln nsla,s,1 type,3 mat,6 real,5*

*esurf allsel ! defining the target layer between n-leg and bottom copper strap vsel,s,mat,,1 asel,s,ext asel,r,loc,y,-ln nsla,s,1 type,4 mat,6 esurf allsel ! defining the shell element for radiation simulation, outputting radiation matrix ! defining the shell element for copper strap type,2 aatt,3,,2 asel,s,loc,x,-(wp+d/2),wn+d/2 asel,r,loc,y,0,hs asel,u,loc,y,0 asel,u,loc,y,hs amesh,all allsel asel,s,loc,x,-d/2,d/2 asel,r,loc,y,0 amesh,all allsel aatt,3,,2 asel,s,loc,x,-(wp+d/2),wn+d/2 asel,r,loc,y,-lp-hs,-lp asel,u,loc,y,-lp asel,u,loc,y,-lp-hs*

*allsel*

*asel,s,ext*

*nsla,s,1 type,3 mat,6 real,5 esurf allsel*

*asel,r,loc,y,-lp*

*vsel,s,mat,,2*

*asel,r,loc,y,-lp*

*vsel,s,loc,y,-hs-ln,-ln vsel,r,loc,x,d/2,d/2+wn*

*asel,s,ext*

*nsla,s,1 type,3 mat,6 real,5*

*asel,r,loc,y,-ln*

*asel,s,ext*

*nsla,s,1 type,4 mat,6 esurf allsel*

*vsel,s,loc,y,-hs-lp,-lp*

*vsel,r,loc,x,-wp-d/2,-d/2*

*! defining the contact layer between p-leg and bottom copper strap*

500 Thermoelectrics for Power Generation - A Look at Trends in the Technology

*! defining the target layer between p-leg and bottom copper strap*

*! defining the contact layer between n-leg and bottom copper strap*

*amesh,all allsel aatt,4,,2 asel,s,loc,x,-d/2,d/2 asel,r,loc,y,-lp-hs amesh,all ! defining the shell element for p-n thermoelements allsel aatt,5,,2 asel,s,loc,x,-(wp+d/2),wn+d/2 asel,r,loc,y,-lp,0 asel,u,loc,y,-lp asel,u,loc,y,0 amesh,all ! defining the space node for radiation simulation n,10000,0,0,3e-3 fini ! using radiation matrix method /aux12 emis,3,1 ! setting the emissivity emis,4,1 emis,5,1 allsel geom,0 stef,5.68e-8 ! setting the Stefan-Boltzmann constant vtype,hidden space,10000 write,teg,sub ! outputting the radiation super element fini /prep7*

*! deleting the shell elements and the corresponding mesh allsel asel,s,type,,2 aclear,al etdele,2 allsel et,5,matrix50,1 ! defining radiation matrix element ! defining boundary conditions and the load nsel,s,loc,y,hs+hc ! TEG cell hot side cp,1,temp,all ! coupling of temperature degree of freedom nh=ndnext(0) ! getting the master node d,nh,temp,Th ! setting the temperature constraint to the hot side nsel,all nsel,s,loc,y,-(ln+hs+hc) ! selecting the TEG cell cold side d,all,temp,Tc ! setting the temperature constraint to the cold side nsel,s,loc,y,-(ln+hs),-ln nsel,r,loc,x,d/2+wn cp,3,volt,all ! electrical coupling nn=ndnext(0) ! getting the master node d,nn,volt,0 ! setting the ground connection node nsel,all nsel,s,loc,y,-(lp+hs),-lp nsel,r,loc,x,-(wp+d/2) cp,4,volt,all ! ! electrical coupling np=ndnext(0) ! getting the master node nsel,all type,5 allsel d,10000,temp,300 ! setting the temperature of the space node se,teg,sub ! reading the radiation super element*

*amesh,all*

*amesh,all*

*allsel aatt,5,,2*

*asel,s,loc,x,-d/2,d/2 asel,r,loc,y,-lp-hs*

*asel,s,loc,x,-(wp+d/2),wn+d/2*

*! using radiation matrix method*

*emis,3,1 ! setting the emissivity*

*asel,r,loc,y,-lp,0 asel,u,loc,y,-lp asel,u,loc,y,0*

*n,10000,0,0,3e-3*

*amesh,all*

*fini*

*/aux12*

*emis,4,1 emis,5,1*

*allsel geom,0*

*fini /prep7*

*vtype,hidden space,10000*

*! defining the shell element for p-n thermoelements*

502 Thermoelectrics for Power Generation - A Look at Trends in the Technology

*! defining the space node for radiation simulation*

*stef,5.68e-8 ! setting the Stefan-Boltzmann constant*

*write,teg,sub ! outputting the radiation super element*

*allsel aatt,4,,2*

```
et,6,CIRCU124,0 ! setting the load resistor element
fini
/prep7
! setting the load value and property
r,1,R0(1)
type,6
real,1
numcmp,all
e,np,nn
esel,s,type,,6
circu_num=elnext(0) !getting circuit element number
allsel
fini
! starting the calculation
/SOLU
antype,static ! solution type
cnvtol,heat,1,1.e-3 ! setting the converging value for heat condition
cnvtol,amps,1,1.e-3 ! setting the converging value for the current
neqit,50 ! calculation iteration step
solve ! starting solving
fini
*get,P0(1),elem,circu_num,nmisc,1 ! getting the output power of the TEG cell
*get,Qh(1),node,nh,rf,heat ! getting the heat flow into the TEG cell
*get,I(1),elem,circu_num,smisc,2 ! getting the current
*voper,enta,P0,div,Qh ! calculating the energy efficiency of the TEG cell
```
## **5. Conclusions**

The built one-dimensional model, which is validated by test results, can calculate TEG output power and energy efficiency accurately. By simplifying this model, it is convenient to analyze influences of different thermal and electrical parameters on TEG performance. And basic factors to enhance TEG output power and energy efficiency are extracted. At last, ANSYS simulation considering thermal contact and radiation effects for TEGs is introduced briefly, and basic APDL codes are shared.

## **Author details**

*et,6,CIRCU124,0 ! setting the load resistor element*

504 Thermoelectrics for Power Generation - A Look at Trends in the Technology

*circu\_num=elnext(0) !getting circuit element number*

*cnvtol,heat,1,1.e-3 ! setting the converging value for heat condition cnvtol,amps,1,1.e-3 ! setting the converging value for the current*

*\*get,Qh(1),node,nh,rf,heat ! getting the heat flow into the TEG cell*

*\*voper,enta,P0,div,Qh ! calculating the energy efficiency of the TEG cell*

*\*get,I(1),elem,circu\_num,smisc,2 ! getting the current*

*\*get,P0(1),elem,circu\_num,nmisc,1 ! getting the output power of the TEG cell*

The built one-dimensional model, which is validated by test results, can calculate TEG output power and energy efficiency accurately. By simplifying this model, it is convenient to analyze

*! setting the load value and property*

*fini*

*/prep7*

*r,1,R0(1)*

*numcmp,all*

*esel,s,type,,6*

*! starting the calculation*

*antype,static ! solution type*

*solve ! starting solving*

**5. Conclusions**

*neqit,50 ! calculation iteration step*

*e,np,nn*

*allsel fini*

*/SOLU*

*fini*

*type,6 real,1*

Fuqiang Cheng

Address all correspondence to: chengfq101@aliyun.com

1 Key Laboratory of Fault Diagnosis and Maintenance of In-orbit Spacecraft, Xi'an Satellite Control Center, Xi'an, China

2 Equipment Academy, Beijing, China

## **References**


Provisional chapter

## **Performance Analysis of Composite Thermoelectric Generators** Performance Analysis of Composite Thermoelectric Generators

Alexander Vargas Almeida,

[9] Cheng F., Hong Y., Zhu C.: A physical model for thermoelectric generators with and without Thomson heat. Journal of Energy Resources Technology. 2014;136(4):2280–

[10] Tatarinov D., Wallig D., Bastian G.: Optimized characterization of thermoelectric generators for automotive application. Journal of Electronic Materials. 2012;41(96):

[11] Ziolkowski P., Poinas P., Leszczynski J., Karpinski G., Muller E.: Estimation of thermoelectric generator performance by finite element modeling. Journal of Electronic

2285.

1706–1712.

Materials. 2010;39(9):1934–1943.

506 Thermoelectrics for Power Generation - A Look at Trends in the Technology

Miguel Angel Olivares‐Robles and Henni Ouerdane Alexander Vargas Almeida, Miguel Angel Olivares-Robles and Henni Ouerdane

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/66143

#### Abstract

Composite thermoelectric generators (CTEGs) are thermoelectric systems composed of different modules arranged under various thermal and electrical configurations (series and/or parallel). The interest for CTEGs stems from the possibility to improve device performance by optimization of configuration and working conditions. Actual modeling of CTEGs rests on a detailed understanding of the nonequilibrium thermodynamic processes at the heart of coupled transport and thermoelectric conversion. In this chapter, we provide an overview of the linear out-of-equilibrium thermodynamics of the electron gas, which serves as the working fluid in CTEGs. The force-flux formalism yields phenomenological linear, coupled equations at the macroscopic level, which describe the behavior of CTEGs under different configurations. The relevant equivalent quantities—figure of merit, efficiency, and output power—are formulated and calculated for two different configurations. Our results show, that system performance in each of these configurations is influenced by combination of different materials and their ordering, that is, position in the arrangement structure. The primary objective of our study is to contribute new design guidelines for development of composite thermoelectric devices that combine different materials, taking advantage of the performance of each in proper temperature range and type of configuration.

Keywords: thermoelectric energy conversion, thermoelectric devices, thermodynamic constraints on energy production, thermoelectric figure of merit, thermoelectric optimization, efficiency

## 1. Introduction

Thermoelectric devices are heat engines, which may operate as generators under thermal bias or as heat pumps. For waste energy harvesting and conversion, thermoelectricity offers quite

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

appropriate solutions, when temperature difference between heat source and heat sink is not too large. The physics underlying this type of energy conversion is based on the fundamental coupling between electric charge and energy that each mobile electron carries. The coupling strength is given by the so-called Seebeck coefficient or thermoelectric power [1]. The performance of thermoelectric system is usually assessed against the so-called figure of merit [2]: a dimensionless quantity denoted ZT, which combines the system's thermal and electrical transport properties, as well as their coupling at temperature T.

To qualify as a good thermoelectric, a material (semiconductor or strongly correlated) must boast the following characteristics: small thermal conductivity and large electrical conductivity on the one hand, so that, it behaves as a phonon glass—electron crystal system [2], and large thermoelectric power on the other hand. All these properties, which can be optimized, are temperature-dependent, so they may take interesting values only in a particular temperature range. Improvement of thermoelectric devices in terms of performance and range of applications is highly desired, as their conversion efficiency is not size-dependent, and the typical device does not contain moving parts. Much progress in the field of thermoelectricity has been achieved since the early days, which saw the pioneering works of Seebeck [3] and Peltier [4], but decisive improvement of the energy conversion efficiency, typically 10% of the efficiency of ideal Carnot thermodynamic cycle, is still in order.

In a general manner, transport phenomena are irreversible processes: the generation of fluxes within the system, upon which external constraints are applied, are accompanied by energy dissipation and entropy production [5]. Therefore, thermoelectric effects may be viewed as the result of the mutual interaction of two irreversible processes, electrical transport, and heat transport, as they take place [6]. Not too far from equilibrium, these transport phenomena obey linear phenomenological laws; so, general macroscopic description of thermoelectric systems is, in essence, phenomenological. Linear nonequilibrium thermodynamics provides the most convenient framework to characterize the device properties and the working conditions to achieve various operation modes.

A thermoelectric generator (TEG) is under the influence of two potentials: electrochemical (μe) and thermal (T); for each of which there is a flux and a force (as shown in examples of Table 1). If force is capable of getting the system to state close to equilibrium after perturbation, then the linear regime may characterize the situation, and approximation in this case is the linear response theory (LRT). In this chapter, we will review and discuss these issues considering thermoelectric system composed of different modules: we are particularly interested in the performance analysis of composite thermoelectric generator (CTEG). For this purpose, we will use a framework based on LRT, which allows to derive a set of linear coupled equations, which contain the system's thermoelectric properties: Seebeck coefficient (α), thermal conductivity (κ), and electrical resistivity (ρ), which are combined to form the effective transport parameters of CTEG in different thermal and electrical arrangements.

The present chapter is organized as follows: as thermoelectric conversion results primarily from nonequilibrium thermodynamic processes, a brief overview of some of the basic concepts and tools developed by Onsager [7, 8] and Callen [6] is very instructive, and we will see, that the force-flux formalism is perfectly suited for a description of thermoelectric processes [9]. Then, we will turn our attention to the physical model of composite thermoelectric generators, deriving and analyzing the figure of merit, the conversion efficiency and maximum output power. The chapter ends with a discussion and concluding remarks.


Table 1. Linear thermodynamic phenomenological laws—illustrative examples of forces and fluxes.

## 2. Basic notions of linear nonequilibrium thermodynamics

#### 2.1. Instantaneous entropy

appropriate solutions, when temperature difference between heat source and heat sink is not too large. The physics underlying this type of energy conversion is based on the fundamental coupling between electric charge and energy that each mobile electron carries. The coupling strength is given by the so-called Seebeck coefficient or thermoelectric power [1]. The performance of thermoelectric system is usually assessed against the so-called figure of merit [2]: a dimensionless quantity denoted ZT, which combines the system's thermal and electrical trans-

To qualify as a good thermoelectric, a material (semiconductor or strongly correlated) must boast the following characteristics: small thermal conductivity and large electrical conductivity on the one hand, so that, it behaves as a phonon glass—electron crystal system [2], and large thermoelectric power on the other hand. All these properties, which can be optimized, are temperature-dependent, so they may take interesting values only in a particular temperature range. Improvement of thermoelectric devices in terms of performance and range of applications is highly desired, as their conversion efficiency is not size-dependent, and the typical device does not contain moving parts. Much progress in the field of thermoelectricity has been achieved since the early days, which saw the pioneering works of Seebeck [3] and Peltier [4], but decisive improvement of the energy conversion efficiency, typically 10% of the efficiency of

In a general manner, transport phenomena are irreversible processes: the generation of fluxes within the system, upon which external constraints are applied, are accompanied by energy dissipation and entropy production [5]. Therefore, thermoelectric effects may be viewed as the result of the mutual interaction of two irreversible processes, electrical transport, and heat transport, as they take place [6]. Not too far from equilibrium, these transport phenomena obey linear phenomenological laws; so, general macroscopic description of thermoelectric systems is, in essence, phenomenological. Linear nonequilibrium thermodynamics provides the most convenient framework to characterize the device properties and the working condi-

A thermoelectric generator (TEG) is under the influence of two potentials: electrochemical (μe) and thermal (T); for each of which there is a flux and a force (as shown in examples of Table 1). If force is capable of getting the system to state close to equilibrium after perturbation, then the linear regime may characterize the situation, and approximation in this case is the linear response theory (LRT). In this chapter, we will review and discuss these issues considering thermoelectric system composed of different modules: we are particularly interested in the performance analysis of composite thermoelectric generator (CTEG). For this purpose, we will use a framework based on LRT, which allows to derive a set of linear coupled equations, which contain the system's thermoelectric properties: Seebeck coefficient (α), thermal conductivity (κ), and electrical resistivity (ρ), which are combined to form the effective transport parameters

The present chapter is organized as follows: as thermoelectric conversion results primarily from nonequilibrium thermodynamic processes, a brief overview of some of the basic concepts and tools developed by Onsager [7, 8] and Callen [6] is very instructive, and we will see, that

port properties, as well as their coupling at temperature T.

508 Thermoelectrics for Power Generation - A Look at Trends in the Technology

ideal Carnot thermodynamic cycle, is still in order.

tions to achieve various operation modes.

of CTEG in different thermal and electrical arrangements.

The thermodynamic formulation presented here is that of Callen [10]. To each set of extensive variables associated to a thermodynamic system, there is a counterpart, that is, a set of intensive variables. The thermodynamic potentials are constructed from these variables. At the macroscopic scale, the equilibrium states of a system may be characterized by a number of extensive variables Xi macroscopic by nature. As one may assume that a macroscopic system is made of several subsystems, which may exchange matter and/or energy among themselves, the values taken by the variables Xi correspond to these exchanges, which occur as constraints are imposed and lifted. When constraints are lifted, relaxation processes take place until the system reaches a thermodynamic equilibrium state, for which a positive and continuous function S differentiable with respect to the variables Xi can be defined as follows:

$$S: X\_i \mapsto S(X\_i). \tag{1}$$

The function S, called entropy, is extensive; its maximum characterizes equilibrium as it coincides with the values that the variables Xi finally assume after the relaxation of constraints. Note, that extensive variables Xi differ from microscopic variables because of typical time scales, over which they evolve: the relaxation time of microscopic variables is extremely fast, while the variables Xi are slow in comparison. To put it simply, relaxation time toward local equilibrium τrelax is much smaller than the time necessary for the evolution toward the macroscopic equilibrium τeq. Hence, one may define an instantaneous entropy, SðXiÞ, at each step of the relaxation of the variables Xi. The differential of the function S is as follows:

$$\mathbf{d}\mathbf{S} = \sum\_{i} \frac{\partial \mathbf{S}}{\partial \mathbf{X}\_{i}} \mathbf{dX}\_{i} = \sum\_{i} \mathbf{F}\_{i} \mathbf{dX}\_{i},\tag{2}$$

where each quantity Fi is the intensive variable conjugate of the extensive variable Xi.

#### 2.2. Thermodynamic forces and fluxes

Examples of well-known linear phenomenological laws are given in Table 1. These laws establish a proportionality relationship between forces, which derive from potentials, and fluxes. Proportionality factors are transport coefficients, as fluxes are the manifestation of transport phenomena. Indeed, the system's response to externally applied constraints is transport, and when these are lifted, the system relaxes toward an equilibrium state.

Following the introductory discussion of this section, we now see in more detail how these forces and fluxes appear. The notions, which follow, are easily introduced considering the case of a discrete system like, for instance, two separate homogeneous systems initially prepared at two different temperatures and then put in thermal contact through a thin diathermal wall. The thermalization process triggers a flow of energy from one system to the other. So, assume now an isolated system composed of two weakly coupled subsystems, to which an extensive variable taking the values Xi and Xi ′, is associated. One has Xi þ Xi ′ <sup>¼</sup> <sup>X</sup><sup>ð</sup>0<sup>Þ</sup> <sup>i</sup> ¼ constant and SðXiÞ þ SðXi ′Þ ¼ <sup>S</sup>ðX<sup>ð</sup>0<sup>Þ</sup> <sup>i</sup> Þ. Then, the equilibrium condition reads:

$$\frac{\partial \mathbf{S}^{(0)}}{\partial X\_i}|\_{X\_i^{(0)}} = \frac{\partial (\mathbf{S} + \mathbf{S}^{'})}{\partial X\_i} \mathbf{d}X\_i|\_{X\_i^{(0)}} = \frac{\partial \mathbf{S}}{\partial X\_i} \mathbf{-} \frac{\partial \mathbf{S}^{'}}{\partial X\_i} = F\_i - F\_i^{'} = \mathbf{0},\tag{3}$$

as it maximizes the total entropy. Therefore, if the difference F<sup>i</sup> =Fi – F′<sup>i</sup> is equal to zero, the system is in equilibrium; otherwise, irreversible process takes place and drives the system to equilibrium. The quantity F<sup>i</sup> is the affinity or generalized force allowing the evolution of the system toward equilibrium. Further, we also introduce the variation rate of the extensive variable Xi, as it characterizes the response of the system to the applied force:

$$I\_i = \frac{\mathbf{dX}\_i}{\mathbf{dt}}.\tag{4}$$

The relationship between affinities and fluxes characterizes the changes due to irreversible processes: non-zero affinity yields non-zero conjugated flux, and a given flux cancels, if its conjugate affinity cancels.

In local equilibrium, fluxes depend on their conjugate affinity, but also on the other affinities; so, we see, that there are direct effects and indirect effects. Therefore, the mathematical expression for the flux Ii, at a given point in space and time ðr;tÞ, shows a dependence on the force Fi, but also on the other forces Fj≠i:

$$I\_i(\mathbf{r}, t) \mathbb{1}\_i(\mathcal{F}\_1, \mathcal{F}\_2, \dots). \tag{5}$$

Close to equilibrium Iiðr;tÞ can be written as Taylor expansion:

$$I\_k(\mathbf{r}, t) = \sum\_j \frac{\partial I\_k}{\partial \mathcal{F}\_j} \mathbf{F}\_j + \frac{1}{2!} \sum\_{i,j} \frac{\partial^2 I\_k}{\partial \mathcal{F}\_i \mathcal{F}\_j} \mathcal{F}\_i \mathcal{F}\_j + \dots \\ = \sum\_k L\_{jk} \mathcal{F}\_k + \frac{1}{2} \sum\_{i,j} L\_{ijk} \mathcal{F}\_i \mathcal{F}\_j + \dots \. \tag{6}$$

The quantities Ljk are the first-order kinetic coefficients; they are given by the equilibrium values of intensive variables Fi. The matrix ½L� of kinetic coefficients characterizes the linear response of the system. Onsager put forth the idea that there are symmetry and antisymmetry relations between kinetic coefficients [6, 7]: the so-called reciprocal relations must exist in all thermodynamic systems, for which transport and relaxation phenomena are well described by linear laws. The main results can be summarized as follows [5]: (1) Onsager's relation: Lik ¼ Lki; (2) Onsager-Casimir relation: Lik ¼ εiεkLki; (3) generalized relations: LikðH; ΩÞ ¼ εiεkLkið−H;−ΩÞ, where H and Ω denote, respectively, the magnetic field and angular velocity associated with Coriolis field; the parameters ε<sup>i</sup> denote the parity with respect to time reversal: if the quantity studied is invariant under time reversal transformation, it has parity þ1; otherwise, this quantity changes sign, and it has parity −1.

## 3. Thermoelectric forces and fluxes

2.2. Thermodynamic forces and fluxes

510 Thermoelectrics for Power Generation - A Look at Trends in the Technology

variable taking the values Xi and Xi

∂Sð0<sup>Þ</sup> ∂Xi j Xð0<sup>Þ</sup> i

′Þ ¼ <sup>S</sup>ðX<sup>ð</sup>0<sup>Þ</sup>

conjugate affinity cancels.

but also on the other forces Fj≠i:

Ikðr;tÞ ¼ ∑ j ∂Ik ∂F<sup>j</sup> F<sup>j</sup> þ 1 2! ∑ i;j

SðXiÞ þ SðXi

Examples of well-known linear phenomenological laws are given in Table 1. These laws establish a proportionality relationship between forces, which derive from potentials, and fluxes. Proportionality factors are transport coefficients, as fluxes are the manifestation of transport phenomena. Indeed, the system's response to externally applied constraints is trans-

Following the introductory discussion of this section, we now see in more detail how these forces and fluxes appear. The notions, which follow, are easily introduced considering the case of a discrete system like, for instance, two separate homogeneous systems initially prepared at two different temperatures and then put in thermal contact through a thin diathermal wall. The thermalization process triggers a flow of energy from one system to the other. So, assume now an isolated system composed of two weakly coupled subsystems, to which an extensive

′, is associated. One has Xi þ Xi

<sup>−</sup> <sup>∂</sup>S′ ∂Xi ′

<sup>¼</sup> Fi−F′

<sup>¼</sup> <sup>∂</sup><sup>S</sup> ∂Xi ′ <sup>¼</sup> <sup>X</sup><sup>ð</sup>0<sup>Þ</sup>

<sup>d</sup><sup>t</sup> : (4)

Iiðr;tÞ≡IiðF1; F2;…Þ: (5)

1 2 ∑ i;j

<sup>i</sup> ¼ constant and

<sup>i</sup> ¼ 0, (3)

LijkFiF<sup>j</sup> þ … : (6)

port, and when these are lifted, the system relaxes toward an equilibrium state.

<sup>i</sup> Þ. Then, the equilibrium condition reads:

dXij Xð0<sup>Þ</sup> i

as it maximizes the total entropy. Therefore, if the difference F<sup>i</sup> =Fi – F′<sup>i</sup> is equal to zero, the system is in equilibrium; otherwise, irreversible process takes place and drives the system to equilibrium. The quantity F<sup>i</sup> is the affinity or generalized force allowing the evolution of the system toward equilibrium. Further, we also introduce the variation rate of the extensive

Ii <sup>¼</sup> <sup>d</sup>Xi

The relationship between affinities and fluxes characterizes the changes due to irreversible processes: non-zero affinity yields non-zero conjugated flux, and a given flux cancels, if its

In local equilibrium, fluxes depend on their conjugate affinity, but also on the other affinities; so, we see, that there are direct effects and indirect effects. Therefore, the mathematical expression for the flux Ii, at a given point in space and time ðr;tÞ, shows a dependence on the force Fi,

FiF<sup>j</sup> þ … ¼ ∑

The quantities Ljk are the first-order kinetic coefficients; they are given by the equilibrium values of intensive variables Fi. The matrix ½L� of kinetic coefficients characterizes the linear

k

LjkF<sup>k</sup> þ

Þ ∂Xi

variable Xi, as it characterizes the response of the system to the applied force:

Close to equilibrium Iiðr;tÞ can be written as Taylor expansion:

∂<sup>2</sup>Ik ∂FiF<sup>j</sup>

<sup>¼</sup> <sup>∂</sup>ð<sup>S</sup> <sup>þ</sup> <sup>S</sup>′

#### 3.1. Coupled fluxes of heat and electrical charges

The thermoelectric effect results from the mutual interference of two irreversible processes occurring simultaneously in the system, namely heat transport and charge carriers transport. The Onsager force-flux derivation is obtained from the laws of conservation of energy and matter:

$$\mathbf{I}\_E = \mathbf{I}\_Q + \mu\_\mathbf{e} \mathbf{I}\_N,\tag{7}$$

where I<sup>E</sup> is energy flux, I<sup>Q</sup> is heat flux, and I<sup>N</sup> is particle flux. Each flux is the conjugate variable of its potential gradient. Considering the electron gas, correct potentials for particles and energy are μe=T and 1=T, and related forces are as follows: F<sup>E</sup> ¼ ∇ð1=TÞ and F<sup>N</sup> ¼ ∇ð−μe=TÞ, where μ<sup>e</sup> is the electrochemical potential [1]. Then, the linear coupling between forces and fluxes may simply be described by a linear set of coupled equations involving the so-called kinetic coefficient matrix ½L�:

$$
\begin{pmatrix} \mathbf{I}\_N \\ \mathbf{I}\_E \end{pmatrix} = \begin{pmatrix} L\_{NN} & L\_{NE} \\ L\_{EN} & L\_{EE} \end{pmatrix} \begin{pmatrix} \nabla(-\mu\_e/T) \\ \nabla(\mathbf{1}/T) \end{pmatrix}, \tag{8}
$$

where LNE ¼ LEN. Now, to treat properly heat flow and electrical current, it is more convenient to consider I<sup>Q</sup> instead of IE. Using I<sup>E</sup> ¼ I<sup>Q</sup> þ μeIN, we obtain:

$$
\begin{pmatrix} \mathbf{I}\_N \\ \mathbf{I}\_Q \end{pmatrix} = \begin{pmatrix} L\_{11} & L\_{12} \\ L\_{21} & L\_{22} \end{pmatrix} \begin{pmatrix} -\nabla(\mu\_\mathbf{e}/T) \\ \nabla(1/T) \end{pmatrix} \tag{9}
$$

with L<sup>12</sup> ¼ L21. Since ∇ð−μe=TÞ ¼ −μe∇ð1=TÞ−1=T∇ðμeÞ, then heat flow and electrical current read:

$$
\begin{pmatrix} \mathbf{I}\_N \\ \mathbf{I}\_Q \end{pmatrix} = \begin{pmatrix} L\_{NN} & L\_{NE} - \mu\_\mathbf{e} L\_{NN} \\ L\_{NE} - \mu\_\mathbf{e} L\_{NN} & -2L\_{NE} \mu\_\mathbf{e} + L\_{EE} + \mu\_\mathbf{e}^2 L\_{NN} \end{pmatrix} \begin{pmatrix} \nabla (-\mu\_\mathbf{e}/T) \\ \nabla (1/T) \end{pmatrix} \tag{10}
$$

with the following relationship between kinetic coefficients:

$$L\_{11} = L\_{\text{NN}},\tag{11}$$

$$L\_{12} = L\_{NE} - \mu\_{\text{e}} L\_{\text{NN}},\tag{12}$$

$$L\_{22} = L\_{EE} - 2\mu\_{\rm e}L\_{EN} + \mu\_{\rm e}^2 L\_{NN}.\tag{13}$$

Note, that since electric field derives from electrochemical potential, we also obtain:

$$\mathcal{E} = -\frac{1}{\mathcal{e}} \nabla \mu\_{\text{e}}.\tag{14}$$

#### 3.2. Thermoelectric transport coefficients

The thermoelectric transport coefficients can be derived from the expressions of electron and heat flux densities depending on applied thermodynamic constraints: isothermal, adiabatic, electrically open or closed circuit conditions. Under isothermal conditions, electrical current may be written in the form:

$$\mathbf{I}\_N = \frac{-L\_{11}}{T} \nabla(\mu\_\mathbf{e}).\tag{15}$$

This is expression of Ohm's law, since with I ¼ eI<sup>N</sup> we obtain the following relationship between electrical current density and electric field:

$$e\mathbf{I}\_N = \mathbf{I} = e\frac{-L\_{11}}{T}\nabla(\mu\_\mathbf{e}) = \sigma\_T \left(-\frac{\nabla(\mu\_\mathbf{e})}{e}\right) = \sigma\_T \mathcal{E},\tag{16}$$

which contains the definition for isothermal electrical conductivity expressed as follows:

<sup>σ</sup><sup>T</sup> <sup>¼</sup> <sup>e</sup><sup>2</sup> <sup>T</sup> <sup>L</sup>11: (17)

Now, if we consider the heat flux density in the absence of any particle transport or, in other words, under zero electrical current, we get:

$$\mathbf{I}\_N = \mathbf{0} = -L\_{11}\left(\frac{1}{T}\nabla(\mu\_\mathbf{e})\right) + L\_{12}\nabla(\frac{1}{T}),\tag{18}$$

so that, the heat flux density under zero electrical current, IQI¼<sup>0</sup> , reads:

$$\mathbf{I}\_{Q\_{l=0}} = \frac{1}{T^2} \left[ \frac{L\_{21}L\_{12} - L\_{11}L\_{22}}{L\_{11}} \right] \nabla(T). \tag{19}$$

This is Fourier's law, with thermal conductivity under zero electrical current given by:

$$\kappa\_{I} = \frac{1}{T^{2}} \left[ \frac{L\_{11}L\_{22} - L\_{21}L\_{12}}{L\_{11}} \right]. \tag{20}$$

We can also define the thermal conductivity κ<sup>E</sup> under zero electrochemical gradient, that is, under closed circuit conditions:

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$$\mathbf{I}\_{\rm Q\_{E=0}} = \frac{L\_{22}}{T^2} \nabla(T) = \kappa\_E \nabla(T). \tag{21}$$

It follows, that thermal conductivities κ<sup>E</sup> and κ<sup>I</sup> are simply related through:

$$
\kappa\_E = T \alpha^2 \sigma r + \kappa\_I. \tag{22}
$$

As thermal and electric processes are coupled, the actual strength of the coupling is given by Seebeck coefficient:

$$\alpha \equiv \frac{-\frac{1}{e}\nabla(\mu\_e)}{\nabla(T)} = \frac{1}{eT}\frac{L\_{12}}{L\_{11}},\tag{23}$$

defined as the ratio of two forces that derive from electrochemical potential for one and from temperature for the other.

The analysis and calculations developed above allow to establish complete correspondence between kinetic coefficients and transport parameters:

$$L\_{11} = \frac{\sigma\_T}{\varepsilon^2} T,\tag{24}$$

$$L\_{12} = \frac{\sigma\_T \mathbf{S}\_I T^2}{\mathcal{e}^2},\tag{25}$$

$$L\_{22} = \frac{T^3}{\varepsilon^2} \sigma\_T \mathbf{S}\_I^2 + T^2 \kappa\_I,\tag{26}$$

so that, expressions for electronic current and heat flow may take their final forms:

$$\mathbf{I}\_N = \frac{\sigma\_T}{c^2} T \left( -\frac{\nabla(\mu\_e)}{T} \right) + \frac{\sigma\_T \mathbf{S}\_I T^2}{c^2} \left( \nabla(\frac{1}{T}) \right), \tag{27}$$

$$\mathbf{I}\_{Q} = \frac{\sigma\_{T}\mathbf{S}\_{I}}{\varepsilon^{2}}T^{2}\left(-\frac{\nabla(\mu\_{\rm e})}{T}\right) + \left[\frac{T^{3}}{\varepsilon^{2}}\sigma rS\_{I}^{2} + T^{2}\kappa\_{I}\right]\left(\nabla(\frac{1}{T})\right). \tag{28}$$

Since I ¼ eIN, it follows that:

L<sup>11</sup> ¼ LNN, (11)

<sup>e</sup>LNN: (13)

∇μe: (14)

<sup>T</sup> <sup>∇</sup>ðμeÞ: (15)

<sup>T</sup> <sup>L</sup>11: (17)

¼ σTE; (16)

Þ, (18)

∇ðTÞ: (19)

: (20)

L<sup>12</sup> ¼ LNE−μeLNN, (12)

<sup>L</sup><sup>22</sup> <sup>¼</sup> LEE−2μeLEN <sup>þ</sup> <sup>μ</sup><sup>2</sup>

Note, that since electric field derives from electrochemical potential, we also obtain:

3.2. Thermoelectric transport coefficients

512 Thermoelectrics for Power Generation - A Look at Trends in the Technology

between electrical current density and electric field:

words, under zero electrical current, we get:

under closed circuit conditions:

eI<sup>N</sup> ¼ I ¼ e

−L<sup>11</sup>

I<sup>N</sup> ¼ 0 ¼ −L<sup>11</sup>

so that, the heat flux density under zero electrical current, IQI¼<sup>0</sup> , reads:

<sup>I</sup>QI¼<sup>0</sup> <sup>¼</sup> <sup>1</sup> T2

> <sup>κ</sup><sup>I</sup> <sup>¼</sup> <sup>1</sup> T2

may be written in the form:

E ¼ − 1 e

<sup>I</sup><sup>N</sup> <sup>¼</sup> <sup>−</sup>L<sup>11</sup>

The thermoelectric transport coefficients can be derived from the expressions of electron and heat flux densities depending on applied thermodynamic constraints: isothermal, adiabatic, electrically open or closed circuit conditions. Under isothermal conditions, electrical current

This is expression of Ohm's law, since with I ¼ eI<sup>N</sup> we obtain the following relationship

∇ðμeÞ e 

þ L12∇ð

1 T

<sup>T</sup> <sup>∇</sup>ðμeÞ ¼ <sup>σ</sup><sup>T</sup> <sup>−</sup>

which contains the definition for isothermal electrical conductivity expressed as follows:

<sup>σ</sup><sup>T</sup> <sup>¼</sup> <sup>e</sup><sup>2</sup>

1 <sup>T</sup> <sup>∇</sup>ðμe<sup>Þ</sup> 

This is Fourier's law, with thermal conductivity under zero electrical current given by:

Now, if we consider the heat flux density in the absence of any particle transport or, in other

L21L12−L11L<sup>22</sup> L<sup>11</sup> 

L11L22−L21L<sup>12</sup> L<sup>11</sup> 

We can also define the thermal conductivity κ<sup>E</sup> under zero electrochemical gradient, that is,

$$\mathbf{I} = \sigma\_T \mathbf{E} - \frac{\sigma\_T \mathbf{S}\_I}{\varepsilon} \nabla(T),\tag{29}$$

from which we obtain:

$$\mathbf{E} = \rho\_T \mathbf{I} + a \nabla(T),\tag{30}$$

where ρ<sup>T</sup> is the isothermal conductivity. This is a general expression of Ohm's law.

#### 4. Formulation of physical model for thermoelectric generators

For TEG performance analysis, we have applied the model given by [11, 12], associating thermal circuit for heat transport and electrical circuit for charge carriers transport, see Figure 1.

Electrical current and heat flow, Ii and IQi , are functions of generalized forces [11], related to differences in voltage, ΔVi, and temperature, ΔTi, of thermoelectric generator:

$$
\begin{pmatrix} I\_i \\ I\_{Q\_i} \end{pmatrix} = \begin{pmatrix} 1/\mathsf{R}\_i & \alpha\_i(\mathsf{1}/\mathsf{R}\_i) \\ \alpha\_i(\mathsf{1}/\mathsf{R}\_i)T & \alpha\_i^2(\mathsf{1}/\mathsf{R}\_i)T + \mathsf{K}\_i \end{pmatrix} \begin{pmatrix} \Delta V\_i \\ \Delta T\_i \end{pmatrix},\tag{31}
$$

where T is average temperature.

In this model, TEG is characterized by its internal electrical resistance, R, thermal conductance under open electrical circuit condition, K, and Seebeck coefficient, α. Physical conditions assumed for this model are as follows: (i) thermoelectric properties are independent on temperature, (ii) the only electrical resistance taken into account is that of the legs, (iii) there is no thermal contact resistance between the ends of the legs and heat source, and (iv) in this model, doping of the legs (p- or n-type) is not taken into account, so that, TEG can be seen as only one leg.

Figure 1. Circuit model for thermoelectric generator, red (thermal circuit), blue (electrical circuit), where ΔV, voltage; R, electrical resistance; K, thermal conductance; Tcold, temperature of the cold side; Thot, temperature of the hot side; ΔT, temperature difference; α, Seebeck coefficient; and T, average temperature.

#### 5. Heat balance equation

The heat balance in TEG is governed by the following equations; basically, there are two extreme points: one in contact with the heat source (incoming point):

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$$Q\_{\rm in} = \alpha T\_h I - \frac{1}{2} R\_{\rm in} I^2 + K(T\_h - T\_c),\tag{32}$$

the other point is point, where heat is rejected:

4. Formulation of physical model for thermoelectric generators

differences in voltage, ΔVi, and temperature, ΔTi, of thermoelectric generator:

<sup>¼</sup> <sup>1</sup>=Ri <sup>α</sup>ið1=Ri<sup>Þ</sup> <sup>α</sup>ið1=RiÞ<sup>T</sup> <sup>α</sup><sup>2</sup>

In this model, TEG is characterized by its internal electrical resistance, R, thermal conductance under open electrical circuit condition, K, and Seebeck coefficient, α. Physical conditions assumed for this model are as follows: (i) thermoelectric properties are independent on temperature, (ii) the only electrical resistance taken into account is that of the legs, (iii) there is no thermal contact resistance between the ends of the legs and heat source, and (iv) in this model, doping of the legs (p- or n-type) is not taken into account, so that, TEG can be seen as only one

The heat balance in TEG is governed by the following equations; basically, there are two

Figure 1. Circuit model for thermoelectric generator, red (thermal circuit), blue (electrical circuit), where ΔV, voltage; R, electrical resistance; K, thermal conductance; Tcold, temperature of the cold side; Thot, temperature of the hot side; ΔT,

extreme points: one in contact with the heat source (incoming point):

temperature difference; α, Seebeck coefficient; and T, average temperature.

<sup>i</sup> ð1=RiÞT þ Ki

ΔVi

Figure 1.

leg.

Electrical current and heat flow, Ii and IQi

where T is average temperature.

5. Heat balance equation

Ii IQi 

514 Thermoelectrics for Power Generation - A Look at Trends in the Technology

For TEG performance analysis, we have applied the model given by [11, 12], associating thermal circuit for heat transport and electrical circuit for charge carriers transport, see

, are functions of generalized forces [11], related to

ΔTi 

, (31)

$$Q\_{re} = \alpha T\_c I + \frac{1}{2} R\_{in} I^2 + K(T\_h - T\_c),\tag{33}$$

where αTiI is Seebeck heat, <sup>1</sup> <sup>2</sup> RinI <sup>2</sup> is Joule heat, and <sup>K</sup>ðTh−Tc<sup>Þ</sup> is thermal conduction heat; in terms of these quantities, electrical power is defined as:

$$P\_{electrical} = Q\_{in} - Q\_{rt} = \alpha I (T\_h - T\_c) - R\bar{I}^2. \tag{34}$$

#### 6. Composite thermoelectric generator (CTEG)

We consider a composite thermoelectric generator, which is composed of three thermoelectric elements (TEGs) in different configurations, each TEG is made of a different thermoelectric material, see Figure 2. The configurations considered are as follows: (A) two-stage thermally and electrically connected in series (TES-CTEG); (B) segmented TEG, conventional TEG, thermally and electrically connected in parallel (PSC-CTEG). Also, we consider the effect of the arrangement of the materials on the performance of the composite system. Thus, for each of the systems (A, B), we have the following arrangements:


In the following sections, we analyze and show results for CTEG by applying the conditions listed above in order to contribute to development of new design guidelines for thermoelectric systems with news architectures and even to provide some clues to the search for new physical conditions in the area of science and engineering of thermoelectric materials.

#### 6.1. Formulation of equivalent figure of merit for CTEG

To analyze CTEG performance, equivalent quantities are defined, which contain the overall contribution of individual properties of each TEG building up composite system. These quantities are as follows: equivalent Seebeck coefficient (αeq), equivalent electrical resistance (Req), and equivalent thermal conductance (Keq), in terms of which it is possible to have equivalent figure of merit (Zeq). We show the impact of the configuration of the system on Zeq for each of configuration (A, B) listed in Section 6, and we suggest the optimum configuration. In order to justify the effectiveness of the equivalent figure of merit, the corresponding efficiency has been calculated for each configuration.

Figure 2. Composite thermoelectric generator (CTEG) (components are three TEGs, each made of different material).

#### 6.1.1. Two-stage thermally and electrically connected in series

Schematic view of this system is shown in Figure 3. The first stage (bottom stage) consists of two different thermoelectric modules (TEG), while the top stage consists of only one TEG. Each of components is characterized by proper thermoelectric properties ðαi;Ri;KiÞ [13].

Using Eq. (31), the heat flux within any segment in TEGs is:

$$I\_{Q\_i} = \alpha\_i T I\_i + \mathcal{K}\_i \Delta T\_i. \tag{35}$$

By continuity of the heat flux through the interface between stages of TES-CTEG:

$$I\_{Q1} = I\_{Q2} + I\_{Q3}$$

$$K\_1(T\_{hot} - T\_i) + \alpha\_1 TI = K\_2(T\_i - T\_{cold}) + \alpha\_2 TI + K\_3(T\_i - T\_{cold}) + \alpha\_3 TI,\tag{36}$$

from which we obtain the average temperature at the interface between stages [12]:

$$T\_i = \frac{K\_1 T\_{hot} + (K\_2 + K\_3) T\_{cold} + (\alpha\_1 - \alpha\_2 - \alpha\_3) T I}{K\_1 + K\_2 + K\_3}.\tag{37}$$

Since all components are electrically connected in series, the total voltage is given by:

$$
\Delta V = -\alpha\_1 (T\_{hvt} - T\_i) - \alpha\_2 (T\_i - T\_{cold}) - \alpha\_3 (T\_i - T\_{cold}) + (R\_1 + R\_2 + R\_3)I,\tag{38}
$$

substituting the value of Ti in the last equation, we have:

#### Performance Analysis of Composite Thermoelectric Generators http://dx.doi.org/10.5772/66143 517

$$\begin{array}{ll} \Delta V &= \left[ \frac{-\left(\alpha\_2 + \alpha\_3\right)K\_1 - \alpha\_1 K\_2 - \alpha\_1 K\_3}{K\_1 + K\_2 + K\_3} \right] \left[ T\_{hot} - T\_{cold} \right] + \\ &+ \left[ \frac{\left(\alpha\_1 - \alpha\_2 - \alpha\_3\right)^2 T}{K\_1 + K\_2 + K\_3} + \left(R\_1 + R\_2 + R\_3\right) \right] I. \end{array} \tag{39}$$

Figure 3. Schematic representation of thermoelectric system composed of two stages thermally and electrically connected in series (TES-CTEG). (a) Equivalent circuit for TES-CTEG, where ΔV is the voltage, Ri is the electrical resistance, Ki is the thermal conductance, Tcold is the temperature of the cold side, Thot is the temperature of the hot side, ΔT is the temperature difference, α<sup>i</sup> is the Seebeck coefficient, T is the average temperature, Rload is the load; (b) practical device related to TES-CTEG, where ni is the ith n-type material, pi is the ith p-type material.

From Eq. (39), we identified the equivalent series Seebeck coefficient, αeq−TES, and equivalent series electrical resistance, Req<sup>−</sup>TES, as follows:

$$\alpha\_{eq-TES} = \frac{-(\alpha\_2 + \alpha\_3)K\_1 - \alpha\_1 K\_2 - \alpha\_1 K\_3}{K\_1 + K\_2 + K\_3},\tag{40}$$

$$R\_{\text{eq}-TES} = R\_1 + R\_2 + R\_3 + R\_{\text{relax}},\tag{41}$$

where

6.1.1. Two-stage thermally and electrically connected in series

516 Thermoelectrics for Power Generation - A Look at Trends in the Technology

Using Eq. (31), the heat flux within any segment in TEGs is:

substituting the value of Ti in the last equation, we have:

Schematic view of this system is shown in Figure 3. The first stage (bottom stage) consists of two different thermoelectric modules (TEG), while the top stage consists of only one TEG. Each

Figure 2. Composite thermoelectric generator (CTEG) (components are three TEGs, each made of different material).

IQ<sup>1</sup> ¼ IQ<sup>2</sup> þ IQ<sup>3</sup>

Ti <sup>¼</sup> <sup>K</sup>1Thot þ ðK<sup>2</sup> <sup>þ</sup> <sup>K</sup>3ÞTcold þ ðα1−α2−α3ÞTI K<sup>1</sup> þ K<sup>2</sup> þ K<sup>3</sup>

K1ðThot−TiÞ þ α1TI ¼ K2ðTi−TcoldÞ þ α2TI þ K3ðTi−TcoldÞ þ α3TI, (36)

ΔV ¼ −α1ðThot−TiÞ−α2ðTi−TcoldÞ−α3ðTi−TcoldÞþðR<sup>1</sup> þ R<sup>2</sup> þ R3ÞI, (38)

IQi ¼ αiTIi þ KiΔTi: (35)

: (37)

of components is characterized by proper thermoelectric properties ðαi;Ri;KiÞ [13].

By continuity of the heat flux through the interface between stages of TES-CTEG:

from which we obtain the average temperature at the interface between stages [12]:

Since all components are electrically connected in series, the total voltage is given by:

$$R\_{\text{relax}} = \frac{\left(\alpha\_1 - \alpha\_2 - \alpha\_3\right)^2 T}{K\_1 + K\_2 + K\_3}.\tag{42}$$

Considering open circuit condition for the system, I ¼ 0, we find, that equivalent thermal conductance for the whole system:

$$K\_{eq-TES} = \frac{K\_1(K\_2 + K\_3)}{K\_1 + K\_2 + K\_3}.\tag{43}$$

We define the figure of merit in terms of equivalent quantities [12]:

$$Z\_{\rm eq} = \frac{\alpha\_{\rm eq}^2}{R\_{\rm eq} K\_{\rm eq}}.\tag{44}$$

By replacing the results obtained in Eqs. (40)–(43), we have:

$$Z\_{\text{eq-TES}} = \frac{\left[\frac{-(\alpha\_2 + \alpha\_3)K\_1 - \alpha\_1 K\_2 - \alpha\_1 K\_3}{K\_1 + K\_2 + K\_3}\right]^2}{\left[\frac{(\alpha\_1 - \alpha\_2 - \alpha\_3)^2 T}{K\_1 + K\_2 + K\_3} + (R\_1 + R\_2 + R\_3)\right] \left[\frac{K\_1 (K\_2 + K\_3)}{K\_1 + K\_2 + K\_3}\right]}.\tag{45}$$

#### 6.1.2. Segmented TEG-conventional TEG thermally and electrically connected in parallel

In this section, we consider CTEG system, which is composed by segmented TEG and conventional TEG. These TEGs are thermally and electrically connected in parallel (PSC-CTEG), as is shown in Figure 4.

In the composite system, there are two currents, Is for TEG 1 and TEG 2, Ic for TEG 3. If the electrical current is conserved, then [13]:

$$I\_{eq} = I\_s + I\_c.\tag{46}$$

The heat flux through the whole system is the sum of the heat flux flowing through segmented generator and the heat flux in conventional generator. Thus:

$$I\_{Q-eq} = I\_{Q\_\*} + I\_{Q\_\*}.\tag{47}$$

Figure 4. Schematic representation of (PSC-CTEG). (a) Thermal-electrical circuit, where ΔV is the voltage, Ri is the electrical resistance, Ki is the thermal conductance, Tcold is the temperature of the cold side, Thot is the temperature of the hot side, ΔT is the temperature difference, α<sup>i</sup> is Seebeck coefficient, T is the average temperature, Rload is the load resistance, TM is the intermediate temperature; (b) structure design, where ni is the i th n-type material, pi is the i th p-type material.

To obtain the equivalent electrical resistance, Req<sup>−</sup>PSC, using Eq. (45), the isothermal condition, ΔT ¼ 0, is required. Under this condition, we recover the usual expression of equivalent electrical resistance for an ohmic circuit. Thus, we get:

$$R\_{\text{eq-PSC}} = \frac{R\_s R\_c}{R\_s + R\_c} \,\text{,}\tag{48}$$

where Rc is the internal electrical resistance of conventional TEG and Rs is the electrical resistance of the segmented TEG:

$$R\_8 = R\_1 + R\_2 + R\_{relax} \tag{49}$$

and

<sup>Z</sup>e<sup>q</sup> <sup>¼</sup> <sup>α</sup><sup>2</sup>

By replacing the results obtained in Eqs. (40)–(43), we have:

generator and the heat flux in conventional generator. Thus:

ðα1−α2−α3Þ 2 T

6.1.2. Segmented TEG-conventional TEG thermally and electrically connected in parallel

Zeq−TES ¼

518 Thermoelectrics for Power Generation - A Look at Trends in the Technology

electrical current is conserved, then [13]:

shown in Figure 4.

material.

eq ReqKe<sup>q</sup>

−ðα2þα3ÞK1−α1K2−α1K<sup>3</sup> K1þK2þK<sup>3</sup> h i<sup>2</sup>

<sup>K</sup>1þK2þK<sup>3</sup> þ ðR<sup>1</sup> <sup>þ</sup> <sup>R</sup><sup>2</sup> <sup>þ</sup> <sup>R</sup>3<sup>Þ</sup> h i <sup>K</sup>1ðK2þK3<sup>Þ</sup>

In this section, we consider CTEG system, which is composed by segmented TEG and conventional TEG. These TEGs are thermally and electrically connected in parallel (PSC-CTEG), as is

In the composite system, there are two currents, Is for TEG 1 and TEG 2, Ic for TEG 3. If the

The heat flux through the whole system is the sum of the heat flux flowing through segmented

Figure 4. Schematic representation of (PSC-CTEG). (a) Thermal-electrical circuit, where ΔV is the voltage, Ri is the electrical resistance, Ki is the thermal conductance, Tcold is the temperature of the cold side, Thot is the temperature of the hot side, ΔT is the temperature difference, α<sup>i</sup> is Seebeck coefficient, T is the average temperature, Rload is the load

resistance, TM is the intermediate temperature; (b) structure design, where ni is the i

: (44)

h i : (45)

K1þK2þK<sup>3</sup>

Ieq ¼ Is þ Ic: (46)

IQ<sup>−</sup>eq ¼ IQs þ IQc : (47)

th n-type material, pi is the i

th p-type

$$R\_{relax} = \frac{\left(\alpha\_1 - \alpha\_2\right)^2 T}{K\_1 + K\_2}.\tag{50}$$

Assuming the condition of closed circuit, ΔV ¼ 0, and applying Eq. (45), we have for equivalent Seebeck coefficient [13]:

$$
\alpha\_{\alpha \eta - PSC} = \frac{R\_{\epsilon} \alpha\_{s} + R\_{s} \alpha\_{\epsilon}}{R\_{s} + R\_{\epsilon}},
\tag{51}
$$

where

$$\alpha\_s = \frac{K\_2 \alpha\_1 + K\_1 \alpha\_2}{K\_1 + K\_2}.\tag{52}$$

To determine equivalent thermal conductance, Keq, we use the open circuit condition, Ieq ¼ 0, which is satisfied when Is ¼ −Ic ¼ I, and, due to preservation of heat flow:

$$K\_{eq-PSC} = K\_s + K\_c + \frac{(\alpha\_s - \alpha\_c)TI}{\Delta T},\tag{53}$$

where

$$K\_s = \frac{K\_2 K\_1}{K\_1 + K\_2}.\tag{54}$$

Under open circuit condition, Ieq ¼ 0, so that, ΔV ¼ −αeqΔT. Applying this result, we have for I:

$$I = \frac{1}{R\_s + R\_c} (\alpha\_s - \alpha\_c) \Delta T. \tag{55}$$

Using this last result in Eq. (53), we have:

$$K\_{\text{eq-PSC}} = K\_s + K\_c + \left(\alpha\_s - \alpha\_c\right)^2 T \, \frac{1}{R\_s + R\_c} \,. \tag{56}$$

Now, we can write the figure of merit for this PSC-CTEG system:

$$Z\_{\text{eq-PSC}} = \frac{\alpha\_{\text{eq-PSC}}^2}{R\_{\text{eq-PSC}} K\_{\text{eq-PSC}}}.\tag{57}$$

Using the results obtained in Eqs. (48), (51), and (56), we have:

$$Z\_{\text{eq-PSC}} = \frac{(\frac{R\_c\alpha\_c + R\_s\alpha\_c}{R\_s + R\_c})^2}{\left[\frac{R\_sR\_c}{R\_s + R\_s}\right]\left[K\_s + K\_c + \left(\alpha\_s - \alpha\_c\right)^2 \ T \ \frac{1}{R\_s + R\_c}\right]} \tag{58}$$

#### 6.1.3. Analysis of equivalent figure of merit for composite systems

Equivalent figure of merit (Zeq) is calculated in this section for TES and PSC systems. For performing calculations, the best known thermoelectric materials for commercial applications have been selected: BiTe, PbTe, and SiGe (experimental data taken from Refs. [14–16] have been used as numerical values of thermoelectric parameters). It has also been calculated equivalent maximum efficiency ðηeq−maxÞ.

It is important to emphasize, that in this study we analyzed also the behavior of Zeq and ηeq, when ordering of materials in the composite system changes (i.e., change its position).

Table 2 shows, that performance of composite system is affected by the type of thermal and electrical connection, as well as ordering of materials. For example, PSC case reaches the highest value of Zeq and ηeq with the ordering TEG 1 = PbTe, TEG 2 = SiGe, TEG 3 = BiTe.

To analyze the performance of the composite system, with each of the different orderings, we have built plots (Figure 5a, b), that show variation of equivalent figure of merit with Seebeck coefficients ratio αj=αi.

#### 6.2. Maximum efficiency

The figure of merit measures the performance of materials in thermoelectric device, but, if we measure the performance when the TEG is operating under a temperature difference, then the value called thermal efficiency quantifies the ability of TEG to utilize the supplied heat effectively.


Table 2. Numerical values of Zeq and ηeq in each equivalent thermoelectric system for different arrangements of the TE materials.

From thermodynamics, Carnot cycle thermal efficiency is known as:

Now, we can write the figure of merit for this PSC-CTEG system:

520 Thermoelectrics for Power Generation - A Look at Trends in the Technology

Using the results obtained in Eqs. (48), (51), and (56), we have:

6.1.3. Analysis of equivalent figure of merit for composite systems

equivalent maximum efficiency ðηeq−maxÞ.

coefficients ratio αj=αi.

6.2. Maximum efficiency

effectively.

materials.

Zeq<sup>−</sup>PSC <sup>¼</sup> <sup>ð</sup>

RsRc RcþRs h i

<sup>Z</sup>eq−PSC <sup>¼</sup> <sup>α</sup><sup>2</sup>

eq−PSC Req<sup>−</sup>PSCKeq−PSC

> RcαsþRsα<sup>c</sup> RsþRc Þ 2

Ks þ Kc þ ðαs−αcÞ

Equivalent figure of merit (Zeq) is calculated in this section for TES and PSC systems. For performing calculations, the best known thermoelectric materials for commercial applications have been selected: BiTe, PbTe, and SiGe (experimental data taken from Refs. [14–16] have been used as numerical values of thermoelectric parameters). It has also been calculated

It is important to emphasize, that in this study we analyzed also the behavior of Zeq and ηeq,

Table 2 shows, that performance of composite system is affected by the type of thermal and electrical connection, as well as ordering of materials. For example, PSC case reaches the highest value of Zeq and ηeq with the ordering TEG 1 = PbTe, TEG 2 = SiGe, TEG 3 = BiTe.

To analyze the performance of the composite system, with each of the different orderings, we have built plots (Figure 5a, b), that show variation of equivalent figure of merit with Seebeck

The figure of merit measures the performance of materials in thermoelectric device, but, if we measure the performance when the TEG is operating under a temperature difference, then the value called thermal efficiency quantifies the ability of TEG to utilize the supplied heat

TEG 1 TEG 2 TEG 3 Zeq<sup>−</sup>TES Zeq<sup>−</sup>PSC ηeq−TES ηeq−PSC BiTe PbTe SiGe 0.000433 0.000463 0.079936 0.084392 PbTe SiGe BiTe 0.000508 0.001905 0.091045 0.224724 SiGe BiTe PbTe 0.000574 0.000622 0.100217 0.106658

Table 2. Numerical values of Zeq and ηeq in each equivalent thermoelectric system for different arrangements of the TE

when ordering of materials in the composite system changes (i.e., change its position).

<sup>2</sup> T <sup>1</sup> RsþRc h i : (58)

: (57)

Figure 5. (a) Zeq<sup>−</sup>TES vs. ratio α3=α2, maintaining α<sup>1</sup> and α<sup>2</sup> constant; (b) Zeq<sup>−</sup>PSC vs. ratio, α2=α1, maintaining α<sup>1</sup> and α<sup>3</sup> constant.

In terms of ηCarnot and Zeq, the maximum efficiency of thermoelectric device is defined by the next equation (with thermoelectric properties ðα;R;κÞ constant with respect to temperature) [2]:

$$
\eta\_{\text{max-}j} = \frac{\Delta T}{T\_{\text{hot}}} \cdot \frac{\sqrt{1 + Z\_{eq-j}T} - 1}{\sqrt{1 + Z\_{eq-j}T} + \frac{T\_{\text{cold}}}{T\_{\text{hot}}}},\tag{60}
$$

where Zeq<sup>−</sup><sup>j</sup> with j ¼ TES; PSC is given by Eqs. (45) and (58), respectively. Thus, we have for the maximum efficiency of TES-CTEG system:

$$\eta\_{eq-TES} = \frac{\Delta T}{T\_{hot}} \cdot \frac{\sqrt{1 + Z\_{eq-TES}T} - 1}{\sqrt{1 + Z\_{eq-TES}T} + \frac{T\_{cold}}{T\_{hot}}}.\tag{61}$$

For the maximum efficiency of PSC-CTEG system:

$$\eta\_{eq-PSC} = \frac{\Delta T}{T\_{hot}} \cdot \frac{\sqrt{1 + Z\_{eq-PSC}T} - 1}{\sqrt{1 + Z\_{eq-PSC}T} + \frac{T\_{cold}}{T\_{hot}}}.\tag{62}$$

Our results are shown in Figure 6.

Plots in Figure 6 show typical dependences of CTEGs efficiency on the properties of component materials. The presented results of maximum efficiency reached by the thermoelectric device approach the limit established by Bergman's theorem for composite materials [17]: the efficiency of composite thermoelectric system cannot be greater than the module's component with highest efficiency.

Figure 6. (a) ηmax−TES vs. ratio α3=α2. (b) ηmax−PSC vs. ratio α2=α1.

The maximum efficiencies achieved by studied CTEGs, see plots in Figure 6, are of similar order of magnitude as CTEG systems investigated in some works, e.g. [18], where reported efficiencies from 17 to 20%.

#### 6.3. CTEG: maximum output power

We analyze also the maximum output power of the studied CTEG system, again, assuming configurations and physical conditions shown in Section 6. The obtained results have been compared with some analytical work and numerical simulations.

For the case of thermoelectric generator connected to load resistor Rload (Figure 7), the power delivered to Rload is given by the following equation [19]:

$$P\_{\text{out-m}} = \frac{[\alpha(T\_H - T\_C)]^2 m}{\left(m + 1\right)^2 R},\tag{63}$$

The strategy consists of defining the optimal ratio m ¼ Rload=R and then by applying the method of maximizing variable to obtain the value of the load resistance, which maximizes power. It yields Rload ¼ R, and in this case, the maximum output power is:

$$P^{\text{max}} = \frac{\alpha^2 \left(T\_H - T\_C\right)^2}{4\mathcal{R}}.\tag{64}$$

Figure 7. Thermal-electrical circuit for TEG delivering power to the load, where ΔV is the voltage, Ri is the electrical resistance, Ki is the thermal conductance, Tcold is the temperature of the cold side, Thot is the temperature of the hot side, ΔT is the temperature difference, α<sup>i</sup> is Seebeck coefficient, Rload is the load resistance.

#### 6.3.1. Formulation of output power for CTEG

device approach the limit established by Bergman's theorem for composite materials [17]: the efficiency of composite thermoelectric system cannot be greater than the module's component

The maximum efficiencies achieved by studied CTEGs, see plots in Figure 6, are of similar order of magnitude as CTEG systems investigated in some works, e.g. [18], where reported

We analyze also the maximum output power of the studied CTEG system, again, assuming configurations and physical conditions shown in Section 6. The obtained results have been

For the case of thermoelectric generator connected to load resistor Rload (Figure 7), the power

m

2

<sup>R</sup> , (63)

<sup>4</sup><sup>R</sup> : (64)

Pout<sup>−</sup><sup>m</sup> <sup>¼</sup> <sup>½</sup>αðTH−TCÞ�<sup>2</sup>

The strategy consists of defining the optimal ratio m ¼ Rload=R and then by applying the method of maximizing variable to obtain the value of the load resistance, which maximizes

Pmax <sup>¼</sup> <sup>α</sup><sup>2</sup>ðTH−TC<sup>Þ</sup>

ðm þ 1Þ 2

compared with some analytical work and numerical simulations.

power. It yields Rload ¼ R, and in this case, the maximum output power is:

delivered to Rload is given by the following equation [19]:

with highest efficiency.

efficiencies from 17 to 20%.

6.3. CTEG: maximum output power

Figure 6. (a) ηmax−TES vs. ratio α3=α2. (b) ηmax−PSC vs. ratio α2=α1.

522 Thermoelectrics for Power Generation - A Look at Trends in the Technology

Here, in similar way as in previous sections, formulating of output power will be considered using thermoelectric equivalent quantities, see Sections 6.1.1, 6.1.2 [20]. Thus, using Eqs. (62, 63) in terms of αeq and Req, we can write:

$$P\_{out-eq-m} = \frac{\left[\alpha\_{eq}(T\_H - T\_C)\right]^2}{R\_{eq}} \frac{m}{\left(m+1\right)^2},\tag{65}$$

$$P\_{eq}^{\text{max}} = \frac{\alpha\_{eq}^2 (T\_H - T\_C)^2}{4R\_{eq}}.\tag{66}$$

Application of the formalism described above Eqs. (64, 65) give the output power for each configuration as follows.

Two-stage thermoelectric system connected in series:

$$P\_{out-eq-(TES-CTG)-m} = \frac{\left(\left[\frac{-(\alpha\_2 + \alpha\_3)K\_1 - \alpha\_1 K\_2 - \alpha\_1 K\_3}{K\_1 + K\_2 + K\_3}\right] (T\_H - T\_C)\right)^2}{\left[R\_1 + R\_2 + R\_3 + \frac{(\alpha\_1 - \alpha\_2 - \alpha\_3)^2 T}{K\_1 + K\_2 + K\_3}\right]} \frac{m}{(m+1)^2} \tag{67}$$

and the maximum power is given by:

$$P\_{eq-(TES-CTEG)}^{\text{max}} = \frac{\left(\left[\frac{-(a\_2 + a\_3)K\_1 - a\_1K\_2 - a\_1K\_3}{K\_1 + K\_2 + K\_3}\right](T\_H - T\_C)\right)^2}{4\left[R\_1 + R\_2 + R\_3 + \frac{(a\_1 - a\_2 - a\_3)^2 \overline{T}}{K\_1 + K\_2 + K\_3}\right]}\tag{68}$$

Segmented-conventional thermoelectric system in parallel (PSC-CTEG):

$$P\_{out-eq-(PSC)-m} = \frac{\left(R\_c \left[\frac{K\_2\alpha\_1 + K\_1\alpha\_2}{K\_1 + K\_2}\right] + \left[R\_1 + R\_2 + \left[\frac{(\alpha\_1 - \alpha\_2)^2 T}{K\_1 + K\_2}\right]\right] \alpha\_c\right)^2 \left(T\_H - T\_C\right)^2}{\left[\left(R\_1 + R\_2 + \frac{(\alpha\_1 - \alpha\_2)^2 T}{K\_1 + K\_2}\right) R\_c \left(R\_s + R\_c\right)\right]} \tag{69}$$

and using Eqs. (51, 48) and Eq. (66), the maximum power of this system obtained is:

$$P\_{eq-(\text{PSC})}^{\text{max}} = \frac{1}{4} \frac{\left(R\_c \left[\frac{K\_2 \alpha\_1 + K\_1 \alpha\_2}{K\_1 + K\_2}\right] + \left[R\_1 + R\_2 + \left[\frac{(\alpha\_1 - \alpha\_2)^2 \overline{T}}{K\_1 + K\_2}\right]\right] \alpha\_c\right)^2 \left(T\_H - T\_C\right)^2}{\left[\left(R\_1 + R\_2 + \frac{(\alpha\_1 - \alpha\_2)^2 \overline{T}}{K\_1 + K\_2}\right) R\_c (R\_s + R\_c)\right]}.\tag{70}$$

#### 6.3.2. Analysis of output power

We show the behavior of the electrical output power delivered in each CTEG configuration using the data of Section 6.1.3. Figure 8, panels (a) and (b), shows the output power as a function of the ratio between the electrical resistance of the load and the electrical resistance of the thermoelectric system <sup>m</sup> <sup>¼</sup> Rload R .

Figure 8. (a) Plot for output power delivered by TES-CTEG system as function of ratio Rload=R; combination, producing the highest output power, is (TEM 1=SiGe, TEM 2=BiTe, TEM 3=PbTe); (b) plot for output power delivered by the PSC-CTEG system as function of ratio Rload=R; combination, producing the highest output power, is (TEM 1=PbTe, TEM 2= SiGe, TEM 3=BiTe).

Plots in Figure 8 show, that similarly to the equivalent figure of merit and equivalent efficiency (Sections 6.1.3 and 6.2), the output power of a composite system is also influenced by the type of thermal-electrical connection and ordering of materials, and again, PSC-CTEG case shows the highest performance quantified by generated output power. This result is consistent with the results obtained by Vargas-Almeida et al. [20], and the behavior of the output power for each array of equivalent TES-CTEG is consistent with the results obtained by Apertet et al. [11]. Table 3 shows the comparison of maximum output power values for different types of connections and possible arrangements.


Table 3. Numerical values of maximum output power, in terms of equivalent amounts of each compound of CTEG, evaluated for each order of building TEGs.

To confirm the validity of our results, we have built plots for CTEG output power using ΔT values of some work: [21] (experimental) and [22, 23] (analytical). Plots in Figure 9 were produced using the temperature difference of Ref. [21].

The results for comparisons with [22, 23] are shown in [24].

Segmented-conventional thermoelectric system in parallel (PSC-CTEG):

Rc <sup>K</sup>2α1þK1α<sup>2</sup> K1þK<sup>2</sup> h i

R .

<sup>þ</sup> <sup>R</sup><sup>1</sup> <sup>þ</sup> <sup>R</sup><sup>2</sup> <sup>þ</sup> <sup>ð</sup>α1−α2<sup>Þ</sup>

� �<sup>2</sup>

<sup>R</sup><sup>1</sup> <sup>þ</sup> <sup>R</sup><sup>2</sup> <sup>þ</sup> <sup>ð</sup>α1−α2<sup>Þ</sup>

h i

and using Eqs. (51, 48) and Eq. (66), the maximum power of this system obtained is:

h i h i

2 T K1þK<sup>2</sup>

<sup>þ</sup> <sup>R</sup><sup>1</sup> <sup>þ</sup> <sup>R</sup><sup>2</sup> <sup>þ</sup> <sup>ð</sup>α1−α2<sup>Þ</sup>

h i h i

2 T K1þK<sup>2</sup>

h i

� �<sup>2</sup>

<sup>R</sup><sup>1</sup> <sup>þ</sup> <sup>R</sup><sup>2</sup> <sup>þ</sup> <sup>ð</sup>α1−α2<sup>Þ</sup>

h i

We show the behavior of the electrical output power delivered in each CTEG configuration using the data of Section 6.1.3. Figure 8, panels (a) and (b), shows the output power as a function of the ratio between the electrical resistance of the load and the electrical resistance

Plots in Figure 8 show, that similarly to the equivalent figure of merit and equivalent efficiency (Sections 6.1.3 and 6.2), the output power of a composite system is also influenced by the type of thermal-electrical connection and ordering of materials, and again, PSC-CTEG case shows the highest performance quantified by generated output power. This result is consistent with the results obtained by Vargas-Almeida et al. [20], and the behavior of the output power for

Figure 8. (a) Plot for output power delivered by TES-CTEG system as function of ratio Rload=R; combination, producing the highest output power, is (TEM 1=SiGe, TEM 2=BiTe, TEM 3=PbTe); (b) plot for output power delivered by the PSC-CTEG system as function of ratio Rload=R; combination, producing the highest output power, is (TEM 1=PbTe, TEM 2=

2 T K1þK<sup>2</sup>

RcðRs þ RcÞ

αc

2 T K1þK<sup>2</sup>

RcðRs þ RcÞ h i : (70)

αc

ðTH−TCÞ

2

ðTH−TCÞ

m ðm þ 1Þ

2

<sup>2</sup> , (69)

Rc <sup>K</sup>2α1þK1α<sup>2</sup> K1þK<sup>2</sup> h i

524 Thermoelectrics for Power Generation - A Look at Trends in the Technology

4

Pout<sup>−</sup>eq−ðPSCÞ−<sup>m</sup> ¼

Pmax eq−ðPSC<sup>Þ</sup> <sup>¼</sup> <sup>1</sup>

6.3.2. Analysis of output power

SiGe, TEM 3=BiTe).

of the thermoelectric system <sup>m</sup> <sup>¼</sup> Rload

Figure 9. Output power POut<sup>−</sup>eq−PSC delivered by composed PSC system vs ratio Rload=R. At temperature difference ΔT = 20K, curves behave similarly to the plots shown in Ref. [21]. This figure is consistent with the result obtained by Abdelkefi [21]. Our results have also been compared to other published works [22, 23].

## 7. Opportunity analysis to improve CTEG design by varying configuration

In this section, we generalize results shown in previous sections by formulating corollary and including some results with realistic approaches, for example, consideration of contact thermal conductance. To achieve this goal, we combine physical conditions imposed in Section 6 with the next options: (1) the whole system is formed of the same thermoelectric material (α1; K1; R<sup>1</sup> ¼ α2; K2; R<sup>2</sup> ¼ α3; K3; R3); (2) the whole system is constituted by only two different thermoelectric materials (αi; Ki; Ri ¼ αj; Kj; Rj ≠ αl; Kl; Rl), where i; j; l can be 1, 2 or 3, [25].

#### 7.1. Case A: homogeneous thermoelectric properties, configuration effect

We consider configurations of CTEG with the same thermoelectric material, <sup>ð</sup>α1; <sup>K</sup>1; <sup>R</sup>1Þ¼ðα2; <sup>K</sup>2; <sup>R</sup>2Þ¼ðα3; <sup>K</sup>3; <sup>R</sup>3Þ. In this case, equivalent figure of merit <sup>Z</sup><sup>h</sup> eq is as follows,

for homogeneous TES-CTEG:

$$Z\_{eq-TES}^h = \frac{\left(\frac{-4\alpha\_i}{3}\right)^2}{\left(\frac{(-\alpha\_i)^2 T}{3K\_i} + 3R\_i\right)\left(\frac{2K\_i}{3}\right)},\tag{71}$$

for homogeneous PSC-CTEG:

$$Z\_{eq-PSC}^h = \frac{\left(\alpha\_i\right)^2}{\left(\frac{2R\_i}{3}\right)\left(\frac{3K\_i}{2}\right)},\tag{72}$$

where i ¼ ðBiTe; PbTe , SiGeÞ.

Table 4 shows numerical values of equivalent figure of merit Zh eq obtained by us for CTEG with considered configurations.


Table 4. Numerical values of Z<sup>h</sup> eq, for each of three configurations with different materials.

It is important to note, that fulfillment condition TEG 1=TEG 2=TEG 3 evidences the fact, that although composite system is made of single material, the figure of merit reaches different values depending on type of connection.

#### 7.2. Case B: two different materials in CTEG

CTEG is made of two same materials and the other one different. Thus, two TEGs include same semiconductor material and the other one different semiconductor material. In this case, equivalent figure of merit Zh eq is as follows, for heterogeneous TES-CTEG:

$$Z\_{eq-TES}^{Inh} = \frac{\left(\frac{-(\alpha\_{\bar{\prime}} + \alpha\_{\bar{\prime}})\mathcal{K}\_{\bar{\prime}} - \alpha\_{\bar{\prime}}(\mathcal{K}\_{\bar{\prime}} + \mathcal{K}\_{\bar{\prime}})}{\mathcal{K}\_{\bar{\prime}} + \mathcal{K}\_{\bar{\prime}} + \mathcal{K}\_{\bar{\prime}}}\right)^2}{\left(\frac{(\alpha\_{\bar{\prime}} - \alpha\_{\bar{\prime}} - \alpha\_{\bar{\prime}})^2 T}{\mathcal{K}\_{\bar{\prime}} + \mathcal{K}\_{\bar{\prime}} + \mathcal{K}\_{\bar{\prime}}} + R\_{\bar{\prime}} + R\_{\bar{\prime}} + R\_{\bar{\prime}}\right) \left(\frac{\mathcal{K}\_{\bar{\prime}}(\mathcal{K}\_{\bar{\prime}} + \mathcal{K}\_{\bar{\prime}})}{\mathcal{K}\_{\bar{\prime}} + \mathcal{K}\_{\bar{\prime}} + \mathcal{K}\_{\bar{\prime}}}\right)},\tag{73}$$

for heterogeneous PSC-CTEG:

7. Opportunity analysis to improve CTEG design by varying configuration

In this section, we generalize results shown in previous sections by formulating corollary and including some results with realistic approaches, for example, consideration of contact thermal conductance. To achieve this goal, we combine physical conditions imposed in Section 6 with the next options: (1) the whole system is formed of the same thermoelectric material (α1; K1; R<sup>1</sup> ¼ α2; K2; R<sup>2</sup> ¼ α3; K3; R3); (2) the whole system is constituted by only two different thermoelectric materials (αi; Ki; Ri ¼ αj; Kj; Rj ≠ αl; Kl; Rl), where i; j; l can be

We consider configurations of CTEG with the same thermoelectric material,

−4α<sup>i</sup> 3 <sup>2</sup>

2

3

, (71)

, (72)

eq−TES Z<sup>h</sup>

eq obtained by us for CTEG with

eq−PSC

eq is as

<sup>ð</sup>α1; <sup>K</sup>1; <sup>R</sup>1Þ¼ðα2; <sup>K</sup>2; <sup>R</sup>2Þ¼ðα3; <sup>K</sup>3; <sup>R</sup>3Þ. In this case, equivalent figure of merit <sup>Z</sup><sup>h</sup>

ð−αiÞ 2 T <sup>3</sup>Ki þ 3Ri <sup>2</sup>Ki

eq−PSC <sup>¼</sup> <sup>ð</sup>αi<sup>Þ</sup>

It is important to note, that fulfillment condition TEG 1=TEG 2=TEG 3 evidences the fact, that although composite system is made of single material, the figure of merit reaches different

eq, for each of three configurations with different materials.

BiTe 0.00212133 0.00305269 PbTe 0.00055109 0.000657238 SiGe 0.000287562 0.00033337

2Ri 3 <sup>3</sup>Ki 2

7.1. Case A: homogeneous thermoelectric properties, configuration effect

526 Thermoelectrics for Power Generation - A Look at Trends in the Technology

Zh eq−TES ¼

Table 4 shows numerical values of equivalent figure of merit Zh

Zh

1, 2 or 3, [25].

follows,

for homogeneous TES-CTEG:

for homogeneous PSC-CTEG:

where i ¼ ðBiTe; PbTe , SiGeÞ.

considered configurations.

Table 4. Numerical values of Z<sup>h</sup>

values depending on type of connection.

Material Z<sup>h</sup>

$$Z\_{\alpha\eta\text{-}PSC}^{\text{Inh}} = \frac{\left(\frac{R\_l\left(\frac{K\_i + R\_i + k\_j}{K\_i + K\_j}\right) + \left(R\_i + R\_j + \frac{(a\_i - w\_j)^2 T}{K\_i + K\_j}\right) a\_l}{R\_l + R\_l + R\_l + \frac{(a\_i - w\_j)^2 T}{K\_i + K\_j}}\right)^2}{\left(\frac{R\_l\left(R\_i + R\_j + \frac{(a\_i - w\_j)^2 T}{K\_i + K\_j}\right)}{R\_l + R\_l + R\_j + \frac{(a\_i - w\_j)^2 T}{K\_i + K\_j}}\right)\left(\frac{K\_i K\_i}{K\_i + K\_j} + K\_l + \left(\frac{(K\_i \alpha + K\_i \alpha\_i)}{K\_i + K\_j} - \alpha\_l\right)^2 \frac{T}{R\_i + R\_j + R\_l + \frac{(a\_i - w\_j)^2 T}{K\_i + K\_j}}\right)}\tag{74}$$

Eqs. (72) and (73) are applied with condition TEGi ¼ TEGj, that is, two TEGs are made of the same thermoelectric material, and third TEGl is made of different thermoelectric material. Thus, we have three possibilities (TEG 1=TEG 2≠TEG3, TEG 1=TEG 3≠TEG 2, TEG 2=TEG 3≠ TEG 1) for each configuration [25]. Note that, each arrangement has six different combinations, if the cyclical order of the material is taken into account.

The behavior of the equivalent figure of merit as a function of the ratio of the thermal conductivities of the two component materials is shown in Figure 10. This step is important, because it shows numerical values, that CTEG maker must meet for both component materials to reach the highest value of Zeq.

Table 5 shows maximum values of equivalent figure of merit of CTEG with material arrangements in every configuration, when TEGi ¼ TEGj≠TEGl.

Table 6 shows each configuration with the most efficient material arrangements for every TEG.

Results show again, that the most efficient system of three configurations is PSC with corresponding material arrangement, namely TEG 1=TEG 2=PbTe≠TEG 3=BiTe; see Figure 11.

Again, it is important to note, that this result proves, that although the performance of composite systems is affected by combination of different materials, it is affected by the position of such materials in the system structure as well.

#### 7.3. Performance analysis with realistic approximations

The results of the previous sections have argued, that application of output power and efficiency as quantities to measure performance of the system is reasonable; however, in this new section, we extend the analysis of these quantities using realistic considerations. Numerical treatment is performed with ZInh eq−PSC.

Figure 10. (a) Equivalent figure of merit for heterogeneous TES-CTEG, under condition TEG 2=TEG 3≠TEG 1, the highest numerical value is corresponding to TEG 2=TEG 3=BiTe≠TEG 1=PbTe; (b) Equivalent figure of merit for heterogeneous PSC-CTEG under condition TEG 1=TEG 2≠TEG 3, the highest numerical value is corresponding to TEG 1=TEG 2=PbTe≠ TEG 3=BiTe.


Table 5. Maximum values of equivalent figure of merit of CTEG with material arrangements in every configuration, when TEMi ¼ TEMj≠TEMl.


Table 6. Most efficient material arrangements TEGi ¼ TEGj≠TEGl for TES and PSC-CTEG systems.

#### 7.3.1. Maximum output power

Figure 10. (a) Equivalent figure of merit for heterogeneous TES-CTEG, under condition TEG 2=TEG 3≠TEG 1, the highest numerical value is corresponding to TEG 2=TEG 3=BiTe≠TEG 1=PbTe; (b) Equivalent figure of merit for heterogeneous PSC-CTEG under condition TEG 1=TEG 2≠TEG 3, the highest numerical value is corresponding to TEG 1=TEG 2=PbTe≠

528 Thermoelectrics for Power Generation - A Look at Trends in the Technology

TEG 3=BiTe.

In the following analysis, we consider thermoelectric modules as isolated units only. Although this is usually considered as an ideal situation, such an approach is useful to study the performance of materials in the composite system. However, for real applications, modules must be coupled to heat exchangers, which produces thermal conductance of contact (Kc) at the coupling points. This affects system performance and reflects in the output power. Here, the maximum output power is calculated using the maximum value of the equivalent figure of merit (ZInh eq−PSC) [23]:

$$P\_{\text{max-PSC}} = \frac{(K\_c \Delta T)^2}{4(K\_{l=0} + K\_c)\overline{T}} \frac{Z\_{eq-\text{PSC}}^{\text{Inh}}\overline{T}}{1 + Z\_{eq-\text{PSC}}^{\text{Inh}}\overline{T} + K\_c/K\_{l=0}}.\tag{75}$$

Figure 12a shows maximum output power values for PSC system as function of ratio KI¼<sup>0</sup>=Kc, that is, in terms of internal thermal conductance KI¼<sup>0</sup> and contact thermal conductance Kc, under condition TEG 1=TEG 2≠TEG 3.

Figure 11. Optimal configuration corresponds to PSC-CTEG with arrangement TEG 1=TEG 2=PbTe≠TEG 3=BiTe.

#### 7.3.2. Efficiency

To calculate the efficiency of PSC systems with TEG 1=TEG 2=PbTe≠TEG 3=BiTe arrangement, we applied the equation:

$$\eta\_{eq-PSC}^{Inh} = \frac{\Delta T}{T\_H} \frac{\sqrt{1 + \overline{Z\_{eq-PSC}^{Inh}} \overline{T}} - 1}{\sqrt{1 + \overline{Z\_{eq-PSC}^{Inh}} \overline{T}} + \frac{T\_C}{\overline{T}\_H}}. \tag{76}$$

Finally, for an ideal TEG, that is, without taking into account heat exchangers, we can analyze TEG efficiency considering intrinsic thermal conductances ratio (K3=K1;2) and electrical resistances ratio (R3=R<sup>1</sup>;2).

Figure 12b shows contour plot for different values of ηInh eq−PSC as function of ratios, K3=K<sup>1</sup>;<sup>2</sup> and R3=R<sup>1</sup>;2. We can see, that the range of optimal values for the best efficiency of PSC—CTEG lies in intervals 0.1–1.0 and 0.1–0.5 for K3=K<sup>1</sup>;<sup>2</sup> and R3=R<sup>1</sup>;2, respectively. It is remarkable, that thermal conductances ratio shows a wider range of good values in comparison with electrical resistances ratio, which shows narrower range.

Performance Analysis of Composite Thermoelectric Generators http://dx.doi.org/10.5772/66143 531

7.3.2. Efficiency

ment, we applied the equation:

tances ratio (R3=R<sup>1</sup>;2).

To calculate the efficiency of PSC systems with TEG 1=TEG 2=PbTe≠TEG 3=BiTe arrange-

Figure 11. Optimal configuration corresponds to PSC-CTEG with arrangement TEG 1=TEG 2=PbTe≠TEG 3=BiTe.

q

Finally, for an ideal TEG, that is, without taking into account heat exchangers, we can analyze TEG efficiency considering intrinsic thermal conductances ratio (K3=K1;2) and electrical resis-

R3=R<sup>1</sup>;2. We can see, that the range of optimal values for the best efficiency of PSC—CTEG lies in intervals 0.1–1.0 and 0.1–0.5 for K3=K<sup>1</sup>;<sup>2</sup> and R3=R<sup>1</sup>;2, respectively. It is remarkable, that thermal conductances ratio shows a wider range of good values in comparison with electrical

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> <sup>Z</sup>Inh

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> ZInh

eq−PSCT

eq−PSCT

−1

<sup>þ</sup> TC TH : (76)

eq−PSC as function of ratios, K3=K<sup>1</sup>;<sup>2</sup> and

ηInh

530 Thermoelectrics for Power Generation - A Look at Trends in the Technology

Figure 12b shows contour plot for different values of ηInh

resistances ratio, which shows narrower range.

eq−PSC <sup>¼</sup> <sup>Δ</sup><sup>T</sup> TH

Figure 12. (a) Maximum power of PSC system under condition TEG 1=TEG 2≠TEG 3, the highest numerical value corresponding to arrangement TEG 1=TEG 2=PbTe≠TEG 3=BiTe. (b) Contour plot: efficiency of PSC system under condition TEG 1=TEG 2≠TEG 3, assuming the maximum value of efficiency ZInh eq−PSC for arrangement TEG 1=TEG 2= PbTe≠TEG 3=BiTe.

## 7.3.3. Corollary: maximum efficiency Zeq for composite thermoelectric generator

Based on the progress presented in this paper, we have been formulated the following corollary: two features of design must be met to ensure the maximum value of Zeq of CTEG:


## 8. Conclusions

The main objective of this chapter was to present new ideas for designing more complex thermoelectric systems taking into account the effects of electrical and thermal connection, combination of different materials and ordering of materials in CTEG. For this purpose, we considered the framework of linear response theory for nonequilibrium thermodynamic processes, and we used the constant parameter model. Through the definition of equivalent parameters αeq, Req, and Keq, we have shown the significant impact of these parameters on the system's properties, which characterize the performance of CTEG, namely Zeq, ηeq, and Peq. The numerical results show, that the optimal configuration for CTEG considered here is the thermal and electrical connection in parallel with arrangement (PbTe, SiGe and BiTe). For completeness, we have shown the effect of contact thermal conductance on the parameter ZInh eq−PSC for the most efficient case—PSC-CTEG system, in terms of both ratio K3=K<sup>1</sup>;<sup>2</sup> (intrinsic thermal conductances) and R3=R<sup>1</sup>;<sup>2</sup> (intrinsic electrical resistance). Although in this study, the composite system is restricted to only three components, the results can be generalized to systems consisting of N modules, either analytically by extension of the mathematical model or through numerical simulations; guidelines for this purpose are provided by the corollary 7.3.3.

## Author details

Alexander Vargas Almeida<sup>1</sup> , Miguel Angel Olivares-Robles<sup>2</sup> \* and Henni Ouerdane3,4

\*Address all correspondence to: molivares67@gmail.com

1 Departamento de Termofluidos, Facultad de Ingenieria, Universidad Nacional Autonoma de Mexico, Mexico

2 Instituto Politecnico Nacional, SEPI-Esime Culhuacan, Coyoacan, Ciudad de Mexico, Mexico

3 Russian Quantum Center, Skolkovo, Moscow Region, Russian Federation

4 UFR Langues Vivantes Etrangeres, Universite de Caen Normandie, Esplanade de la Paix, Caen, France

## References

7.3.3. Corollary: maximum efficiency Zeq for composite thermoelectric generator

a specific type of thermal—electrical connection.

532 Thermoelectrics for Power Generation - A Look at Trends in the Technology

8. Conclusions

Author details

de Mexico, Mexico

Caen, France

Alexander Vargas Almeida<sup>1</sup>

Based on the progress presented in this paper, we have been formulated the following corol-

• If the material is the same in all components, CTEG reaches the maximum value of Zeq with

• When components of TEGs composing CTEG are made of different materials, TEGi≠TEGj≠TEGl where i; j; l can be 1, 2, or 3; then, for a given thermal-electrical connection, there exists an optimal arrangement of thermoelectric materials for which Zeq is maximum.

The main objective of this chapter was to present new ideas for designing more complex thermoelectric systems taking into account the effects of electrical and thermal connection, combination of different materials and ordering of materials in CTEG. For this purpose, we considered the framework of linear response theory for nonequilibrium thermodynamic processes, and we used the constant parameter model. Through the definition of equivalent parameters αeq, Req, and Keq, we have shown the significant impact of these parameters on the system's properties, which characterize the performance of CTEG, namely Zeq, ηeq, and Peq. The numerical results show, that the optimal configuration for CTEG considered here is the thermal and electrical connection in parallel with arrangement (PbTe, SiGe and BiTe). For completeness, we

efficient case—PSC-CTEG system, in terms of both ratio K3=K<sup>1</sup>;<sup>2</sup> (intrinsic thermal conductances) and R3=R<sup>1</sup>;<sup>2</sup> (intrinsic electrical resistance). Although in this study, the composite system is restricted to only three components, the results can be generalized to systems consisting of N modules, either analytically by extension of the mathematical model or through numerical

, Miguel Angel Olivares-Robles<sup>2</sup>

1 Departamento de Termofluidos, Facultad de Ingenieria, Universidad Nacional Autonoma

2 Instituto Politecnico Nacional, SEPI-Esime Culhuacan, Coyoacan, Ciudad de Mexico, Mexico

4 UFR Langues Vivantes Etrangeres, Universite de Caen Normandie, Esplanade de la Paix,

eq−PSC for the most

\* and Henni Ouerdane3,4

have shown the effect of contact thermal conductance on the parameter ZInh

simulations; guidelines for this purpose are provided by the corollary 7.3.3.

3 Russian Quantum Center, Skolkovo, Moscow Region, Russian Federation

\*Address all correspondence to: molivares67@gmail.com

lary: two features of design must be met to ensure the maximum value of Zeq of CTEG:


**Section 6**
