2. Theoretical assessment of a thermoelectric cooler

#### 2.1. The general heat diffusion equation

Consider a non-uniformly heated thermoelectric material having isotropic properties (the same transport properties in all directions) crossed by a constant current density J ! [2, 14– 16]. The continuity equation is

$$
\overrightarrow{\nabla} \cdot \overrightarrow{\J} = 0 \tag{9}
$$

where J ! is the current density vector and ∇ ! is the gradient vector, a differential operator with respect to the three orthogonal directions:

$$
\vec{\nabla} = \vec{i}\,\,\frac{\partial}{\partial x} + \vec{j}\,\,\frac{\partial}{\partial y} + \vec{k}\,\,\frac{\partial}{\partial z} \tag{10}
$$

A temperature gradient generates an electric field E ! . This electric field depends on the temperature gradient ∇ ! T and on the current density J ! and is expressed as

$$
\overrightarrow{E} = \underbrace{\rho \cdot \overrightarrow{J}}\_{\text{Ohm's law}} + \underbrace{\alpha \cdot \overrightarrow{\nabla}}\_{\text{Sebeck voltage}} \tag{11}
$$

with the electrical resistivity <sup>r</sup> in <sup>Ω</sup>�m and the Seebeck coefficient <sup>α</sup> in V�K�<sup>1</sup> .

The heat flow rate q ! also depends on the temperature gradient ∇ ! T and is expressed as

$$\overrightarrow{q} = \underbrace{-k \cdot \overrightarrow{\nabla}}\_{\text{Fourier's law}} + \underbrace{\alpha \cdot T \cdot \overrightarrow{J}}\_{\text{Pelltier heat}} \tag{12}$$

where <sup>k</sup> is the thermal conductivity in W�(m�K)�<sup>1</sup> .

The main components of a refrigeration unit (Figure 3) are [7, 9, 10]:

between the fins is collected by means of a fan [11–13];

same external power supply that powers the TEC;

ing, it is possible to directly control the cooling capacity.

less than 5 l the thickness is 4 cm);

Figure 3. Schematic of a TEM used for the refrigeration unit.

volume;

226 Bringing Thermoelectricity into Reality

• the insulated refrigerator cabinet with thermoelectric technology having variable dimensions (e.g. for a capacity from 5 to 40 l, the thickness is from 5 to 10 cm, and for a capacity

• the cooling thermoelectric system with semiconductors (TECs) useful to cool the insulated

• the heat sink, considered as a heat exchanger useful to facilitate the heat transfer from the hot side of TEC to the environment. The TEC can operate in a definite operating range of its temperature difference. To keep this temperature difference inside the specific operating range, a TEC is compulsory to have a heat sink at the hot end to dissipate heat from the TEC to environment. Sometimes, another heat sink with fins is fixed inside the compartment to improve the heat transfer from the insulated volume which is cooled (fluid, solid) to the cold side of the TEC. In this case, the heat sink is cooled at a temperature lower than the insulated volume, and the heat flowing

• one or many fans which transfer heat through convection and allow the dissipation of heated or cooled air in order to avoid operational problems; the fans are powered by the

• the control system useful for an accurate temperature control; as the current value is chang-

The general heat diffusion equation for transient state [14] is

$$-\overrightarrow{\nabla} \cdot \overrightarrow{q} + \dot{\mathbf{q}}\_{\text{vol}} = \rho \cdot c\_{\text{p}} \cdot \frac{\partial T}{\partial t} \tag{13}$$

where r is the electrical resistivity in Ω�m, c<sup>p</sup> is the specific heat capacity at constant pressure in <sup>J</sup>�(kg�K)�<sup>1</sup> and <sup>q</sup>\_ vol is the volumetric heat generation, in W�m�<sup>3</sup> .

The volumetric heat generation is also given by

$$\dot{\mathbf{q}}\_{\text{vol}} = \overrightarrow{E} \cdot \overrightarrow{J} = \left(\rho \cdot \overrightarrow{J} + \alpha \cdot \overrightarrow{\nabla} \, T\right) \cdot \overrightarrow{J} = \rho \cdot \overrightarrow{J}^2 + \overrightarrow{J} \cdot \alpha \cdot \overrightarrow{\nabla} \, T \tag{14}$$

Based on Thomson's relationship and Osanger's relationship, the heat flow rate vector is written as [16]

$$
\overrightarrow{q} = \alpha \cdot T \cdot \overrightarrow{J} \quad -k \cdot \overrightarrow{\nabla} \ T \tag{15}
$$

Substituting Eq. (14) and Eq. (15) into Eq. (13), with successive elaborations, yields

$$
\rho \mathbf{J}^2 + \overrightarrow{J} \cdot \boldsymbol{\alpha} \cdot \overrightarrow{\nabla} \ \mathbf{T} = \overrightarrow{\nabla} \cdot \overrightarrow{q} + \rho \cdot \mathbf{c\_{P}} \cdot \frac{\partial T}{\partial t}
$$

$$
\rho \mathbf{J}^2 + \overrightarrow{J} \cdot \boldsymbol{\alpha} \cdot \overrightarrow{\nabla} \ \mathbf{T} = \overrightarrow{\nabla} \cdot \left( -\mathbf{k} \cdot \overrightarrow{\nabla} \ \mathbf{T} + \boldsymbol{\alpha} \cdot \mathbf{T} \cdot \overrightarrow{J} \right) + \rho \cdot \mathbf{c\_{P}} \cdot \frac{\partial T}{\partial t}
$$

$$
\rho \mathbf{J}^2 + \overrightarrow{J} \cdot \boldsymbol{\alpha} \cdot \overrightarrow{\nabla} \ \mathbf{T} = -\overrightarrow{\nabla} \cdot \left( \mathbf{k} \cdot \overrightarrow{\nabla} \ \mathbf{T} \right) + \underbrace{\overrightarrow{\nabla} \cdot \left( \boldsymbol{\alpha} \cdot \mathbf{T} \cdot \overrightarrow{J} \right)}\_{T \cdot \overrightarrow{J} \not\approx \overrightarrow{\nabla} T + \overrightarrow{J} \cdot \boldsymbol{\alpha} \cdot \overrightarrow{\nabla} T} + \rho \cdot \mathbf{c\_{P}} \cdot \frac{\partial T}{\partial t} \tag{16}
$$

Considering that <sup>μ</sup> <sup>¼</sup> <sup>T</sup><sup>∙</sup> <sup>d</sup><sup>α</sup> <sup>d</sup><sup>T</sup> is the Thomson coefficient, the heat diffusion equation is

$$
\underbrace{\vec{\nabla} \cdot \left(k \cdot \vec{\nabla} \, T\right)}\_{\text{thermal conduction}} + \underbrace{\rho \cdot \mathbf{J}^2}\_{\text{Joule heating}} - \underbrace{\mu \cdot \vec{J} \cdot \vec{\nabla} \, T}\_{\text{Thomas effect}} = \underbrace{\rho \cdot c\_{\text{p}}}\_{\text{transient}} \frac{\partial T}{\partial t} \tag{17}
$$

#### 2.2. Steady-state and transient approaches

#### 2.2.1. The limits of steady-state analysis

Steady-state analysis for a TEC is typically carried out by resorting to a set of approximations. The simplest model is based on the following assumptions: the Seebeck effect does not depend on temperature, there are no thermal or electrical contact resistances, there are no heat losses, and the Thomson coefficient is zero (μ ¼ 0Þ, so that the Thomson heat is absent [2, 14, 16]. In these conditions, there is no heat transfer from or to the external environment, so that the heat flows occur only between the source and the sink. On these assumptions, Eq. (17) becomes

$$\vec{\nabla} \cdot \left( k \cdot \vec{\nabla} \, T \right) + \rho \cdot \mathbf{J}^2 = \mathbf{0} \tag{18}$$

and the heat flow rate at x = l and T = T<sup>h</sup> is

<sup>x</sup>¼<sup>l</sup> <sup>¼</sup> <sup>α</sup>∙I∙T<sup>h</sup> � <sup>k</sup>∙<sup>S</sup>

<sup>c</sup> <sup>¼</sup> ð Þ <sup>α</sup><sup>P</sup> � <sup>α</sup><sup>N</sup> <sup>∙</sup>I∙Tc � <sup>1</sup>

<sup>h</sup> ¼ ð Þ α<sup>P</sup> � α<sup>N</sup> ∙I∙Th þ

dT dx x¼l

Figure 4. Schematic of a TEC (geometric elements and material properties).

Q\_ <sup>c</sup> <sup>¼</sup> <sup>Q</sup>\_

<sup>∙</sup> <sup>r</sup>P∙l<sup>P</sup>

<sup>∙</sup> <sup>r</sup>P∙l<sup>P</sup>

2

Likewise, the heat flow rate at the hot junction is determined for x = l:

1 2 ) <sup>Q</sup>\_

The heat flow rate at the cold junction is obtained by summing up the contributions of the

<sup>S</sup> <sup>þ</sup> <sup>r</sup>N∙l<sup>N</sup> S <sup>∙</sup><sup>I</sup>

<sup>Q</sup>\_ <sup>h</sup> <sup>¼</sup> <sup>Q</sup>\_ <sup>x</sup>¼l,<sup>P</sup> <sup>þ</sup> <sup>Q</sup>\_

<sup>S</sup> <sup>þ</sup> <sup>r</sup>N∙l<sup>N</sup> S <sup>∙</sup><sup>I</sup>

The total electrical resistance R of the thermoelement pair in series, the total thermal conductance K of the thermoelements in parallel, the Seebeck coefficient αNP of the thermoelectric

<sup>x</sup>¼<sup>l</sup> <sup>¼</sup> <sup>α</sup>∙I∙T<sup>h</sup> <sup>þ</sup> <sup>r</sup>∙<sup>I</sup>

2

<sup>2</sup> � <sup>k</sup>P∙S<sup>P</sup> lP þ kN∙S<sup>N</sup> lN

<sup>2</sup> � <sup>k</sup>P∙S<sup>P</sup> lP þ kN∙S<sup>N</sup> lN

<sup>2</sup><sup>S</sup> <sup>∙</sup><sup>l</sup> � ð Þ <sup>T</sup><sup>h</sup> � <sup>T</sup><sup>c</sup> <sup>∙</sup>

<sup>x</sup>¼0,<sup>P</sup> <sup>þ</sup> <sup>Q</sup>\_ <sup>x</sup>¼0,<sup>N</sup> (24)

<sup>∙</sup>ð Þ Tc � Th (25)

<sup>x</sup>¼l,<sup>N</sup> (26)

<sup>∙</sup>ð Þ <sup>T</sup><sup>h</sup> � <sup>T</sup><sup>c</sup> (27)

k∙S

Thermoelectric Refrigeration Principles http://dx.doi.org/10.5772/intechopen.75439 229

<sup>l</sup> (23)

Q\_

Q\_

Q\_

N-type and P-type elements at x = 0:

By replacing in Eq. (18) the current density <sup>J</sup> <sup>¼</sup> <sup>I</sup> <sup>S</sup> and the temperature Laplacian <sup>∇</sup><sup>2</sup><sup>T</sup> <sup>¼</sup><sup>∇</sup> ! ∙ ∇ ! T ¼ i ! ∂ <sup>∂</sup><sup>x</sup> ∙ i ! ∂T <sup>∂</sup><sup>x</sup> <sup>¼</sup> d2<sup>T</sup> <sup>d</sup>x<sup>2</sup> , the one-dimensional differential equation is [2, 14–16]

$$k \cdot \mathbf{S} \cdot \frac{\mathbf{d}^2 T}{\mathbf{d} x^2} + \rho \cdot \frac{I^2}{S} = 0 \Rightarrow k \cdot \mathbf{S} \cdot \mathbf{d} \left(\frac{\mathbf{d} T}{\mathbf{d} x}\right) = -\rho \cdot \frac{I^2}{S} \mathbf{d} x \tag{19}$$

Let us consider the boundary conditions between the following limits (Figure 4):

$$
\infty = 0 \Rightarrow T = T\_c \tag{20}
$$

$$
\infty = l \Rightarrow T = T\_h \tag{21}
$$

The heat flow rate at x = 0 and T = T<sup>c</sup> is expressed as

$$\dot{Q}\_{x=0} = \alpha \cdot I \cdot T\_{\text{c}} - k \cdot S \frac{dT}{d\mathbf{x}}\Big|\_{\mathbf{x}=0} \Rightarrow \dot{Q}\_{x=0} = \alpha \cdot I \cdot T\_{\text{c}} - \frac{\rho \cdot I^2}{2S} \cdot l - (T\_{\text{h}} - T\_{\text{c}}) \cdot \frac{k \cdot S}{l} \tag{22}$$

Figure 4. Schematic of a TEC (geometric elements and material properties).

and the heat flow rate at x = l and T = T<sup>h</sup> is

r∙J <sup>2</sup> <sup>þ</sup> <sup>J</sup> ! ∙α∙ ∇ ! T ¼∇ ! ∙ q ! <sup>þ</sup> <sup>r</sup>∙cp<sup>∙</sup>

T ¼ � ∇ ! ∙ k∙ ∇ ! T � �

> þ r∙J 2 |{z} Joule heating

∇ ! ∙ k∙ ∇ ! T � �

> I 2

Let us consider the boundary conditions between the following limits (Figure 4):

) <sup>Q</sup>\_

By replacing in Eq. (18) the current density <sup>J</sup> <sup>¼</sup> <sup>I</sup>

k∙S∙ d2 T <sup>d</sup>x<sup>2</sup> <sup>þ</sup> <sup>r</sup><sup>∙</sup>

The heat flow rate at x = 0 and T = T<sup>c</sup> is expressed as

dT dx � � � � x¼0

<sup>x</sup>¼<sup>0</sup> <sup>¼</sup> <sup>α</sup>∙I∙T<sup>c</sup> � <sup>k</sup>∙<sup>S</sup>

Q\_

∙ �k∙ ∇ !

r∙J <sup>2</sup> <sup>þ</sup> <sup>J</sup> ! ∙α∙ ∇ ! T ¼∇ !

∇ ! ∙ k∙ ∇ ! T � �

2.2. Steady-state and transient approaches

2.2.1. The limits of steady-state analysis


r∙J <sup>2</sup> <sup>þ</sup> <sup>J</sup> ! ∙α∙ ∇ !

Considering that <sup>μ</sup> <sup>¼</sup> <sup>T</sup><sup>∙</sup> <sup>d</sup><sup>α</sup>

228 Bringing Thermoelectricity into Reality

<sup>∇</sup><sup>2</sup><sup>T</sup> <sup>¼</sup><sup>∇</sup> ! ∙ ∇ ! T ¼ i ! ∂ <sup>∂</sup><sup>x</sup> ∙ i ! ∂T <sup>∂</sup><sup>x</sup> <sup>¼</sup> d2<sup>T</sup> ∂T ∂t

þ r∙cp∙

¼ r∙cp∙

∂T ∂t

þ r∙cp∙

∂T ∂t |fflfflfflffl{zfflfflfflffl} transient

<sup>2</sup> <sup>¼</sup> <sup>0</sup> (18)

<sup>S</sup> and the temperature Laplacian

<sup>S</sup> <sup>d</sup><sup>x</sup> (19)

k∙S

<sup>l</sup> (22)

∂T

<sup>∂</sup><sup>t</sup> (16)

(17)

T þ α∙T∙ J � �!

> ∙ α∙T∙ J � �!


þ ∇ !

> T∙ J ! ∙ dα <sup>d</sup>T∙∇ ! Tþ J ! ∙α∙∇ ! T

<sup>d</sup><sup>T</sup> is the Thomson coefficient, the heat diffusion equation is


� μ∙ J ! ∙ ∇ ! T

Steady-state analysis for a TEC is typically carried out by resorting to a set of approximations. The simplest model is based on the following assumptions: the Seebeck effect does not depend on temperature, there are no thermal or electrical contact resistances, there are no heat losses, and the Thomson coefficient is zero (μ ¼ 0Þ, so that the Thomson heat is absent [2, 14, 16]. In these conditions, there is no heat transfer from or to the external environment, so that the heat flows occur only between the source and the sink. On these assumptions, Eq. (17) becomes

þ r∙J

<sup>S</sup> <sup>¼</sup> <sup>0</sup> ) <sup>k</sup>∙S∙<sup>d</sup> <sup>d</sup><sup>T</sup>

<sup>d</sup>x<sup>2</sup> , the one-dimensional differential equation is [2, 14–16]

¼ �r∙ I 2

2

x ¼ 0 ) T ¼ Tc (20)

x ¼ l ) T ¼ Th (21)

<sup>2</sup><sup>S</sup> <sup>∙</sup><sup>l</sup> � ð Þ <sup>T</sup><sup>h</sup> � <sup>T</sup><sup>c</sup> <sup>∙</sup>

dx � �

<sup>x</sup>¼<sup>0</sup> <sup>¼</sup> <sup>α</sup>∙I∙T<sup>c</sup> � <sup>r</sup>∙<sup>I</sup>

$$\dot{Q}\_{\rm x=l} = a \cdot I \cdot T\_{\rm h} - k \cdot S \frac{dT}{d\mathbf{x}}\bigg|\_{\mathbf{x}=l} \Rightarrow \dot{Q}\_{\rm x=l} = a \cdot I \cdot T\_{\rm h} + \frac{\rho \cdot I^2}{2S} \cdot l - (T\_{\rm h} - T\_{\rm c}) \cdot \frac{k \cdot S}{l} \tag{23}$$

The heat flow rate at the cold junction is obtained by summing up the contributions of the N-type and P-type elements at x = 0:

$$
\dot{Q}\_c = \dot{Q}\_{x=0,\text{P}} + \dot{Q}\_{x=0,\text{N}} \tag{24}
$$

$$\dot{Q}\_c = (a\_\text{P} - a\_\text{N}) \cdot \text{I} \cdot T\_c - \frac{1}{2} \cdot \left(\frac{\rho\_\text{P} \cdot l\_\text{P}}{\text{S}} + \frac{\rho\_\text{N} \cdot l\_\text{N}}{\text{S}}\right) \cdot \text{I}^2 - \left(\frac{k\_\text{P} \cdot \text{S}\_\text{P}}{l\_\text{P}} + \frac{k\_\text{N} \cdot \text{S}\_\text{N}}{l\_\text{N}}\right) \cdot (T\_c - T\_h) \tag{25}$$

Likewise, the heat flow rate at the hot junction is determined for x = l:

$$
\dot{Q}\_h = \dot{Q}\_{x=l,\mathcal{P}} + \dot{Q}\_{x=l,\mathcal{N}} \tag{26}
$$

$$\dot{Q}\_{\text{h}} = (a\_{\text{P}} - a\_{\text{N}}) \cdot I \cdot T\_{\text{h}} + \frac{1}{2} \cdot \left(\frac{\rho\_{\text{P}} \cdot l\_{\text{P}}}{\text{S}} + \frac{\rho\_{\text{N}} \cdot l\_{\text{N}}}{\text{S}}\right) \cdot I^{2} - \left(\frac{k\_{\text{P}} \cdot \text{S}\_{\text{P}}}{l\_{\text{P}}} + \frac{k\_{\text{N}} \cdot \text{S}\_{\text{N}}}{l\_{\text{N}}}\right) \cdot (T\_{\text{h}} - T\_{\text{c}}) \tag{27}$$

The total electrical resistance R of the thermoelement pair in series, the total thermal conductance K of the thermoelements in parallel, the Seebeck coefficient αNP of the thermoelectric

couple and the temperature difference ΔT between the hot surface temperature T<sup>h</sup> and the cold surface temperature T<sup>c</sup> are written as [9]

$$R = \frac{\rho\_{\rm P} \cdot l\_{\rm P}}{S} + \frac{\rho\_{\rm N} \cdot l\_{\rm N}}{S} \tag{28}$$

$$K = \frac{k\_\text{P} \cdot \text{S}\_\text{P}}{l\_\text{P}} + \frac{k\_\text{N} \cdot \text{S}\_\text{N}}{l\_\text{N}} \tag{29}$$

$$
\alpha\_{\rm NP} = \alpha\_{\rm P} - \alpha\_{\rm N} \tag{30}
$$

$$
\Delta T = T\_\text{h} - T\_\text{c} \tag{31}
$$

on the composition of the whole system and cannot be determined for the individual components. Thereby, the temperatures at the TEC terminals can be determined by using a dedicated model of the interconnected components. These temperatures are calculated from the solution of the overall system equations, in which all the temperature-dependent thermoelectric effects

Thermoelectric Refrigeration Principles http://dx.doi.org/10.5772/intechopen.75439 231

Thermoelectric refrigerators are controlled devices that operate in transient conditions. Thereby, it is important to formulate a detailed model taking into account all the thermoelectric

Eq. (17) is written in generic transient conditions. The solution of this equation has been obtained in [18] by constructing an electrothermal equivalent model with resistances and capacities (in which the thermoelectric modules are represented through a multi-node structure and the other

(Peltier, Seebeck, Thomson and Joule) are taken into account [17].

Figure 5. Schematic of temperature profile in a thermoelectric refrigeration system.

effects and the dependence of the model parameters on temperature.

2.2.2. Transient analysis

Considering that N-type and P-type thermocouples are identical (with the same length), the total electrical resistance is <sup>R</sup> <sup>¼</sup> <sup>r</sup>∙l∙ð Þ <sup>S</sup> �<sup>1</sup> with <sup>r</sup> <sup>¼</sup> <sup>r</sup><sup>P</sup> <sup>þ</sup> <sup>r</sup>N, the total thermal conductance of the thermoelements is <sup>K</sup> <sup>¼</sup> <sup>k</sup>∙S∙ð Þ<sup>l</sup> �<sup>1</sup> with <sup>k</sup> <sup>¼</sup> <sup>k</sup><sup>P</sup> <sup>þ</sup> <sup>k</sup><sup>N</sup> the thermal conductivity corresponding to the N and P thermoelement legs in W∙ð Þ <sup>m</sup>∙<sup>K</sup> �<sup>1</sup> , and σ is the electrical conductivity corresponding to the N and P thermoelement legs with <sup>σ</sup> <sup>¼</sup> <sup>σ</sup><sup>P</sup> <sup>þ</sup> <sup>σ</sup><sup>N</sup> in S∙m�1. Then, Eq. (24) and Eq. (26) give the cooling capacity (or the rate of heat absorbed at the cold junction) Q\_ <sup>c</sup> and the rate of heat rejection Q\_ <sup>h</sup> in W:

$$
\dot{Q}\_{\rm c} = \alpha\_{\rm NP} \cdot I \cdot T\_{\rm c} - \frac{1}{2} \cdot \mathbb{R} \cdot I^2 - K \cdot \Delta T \Rightarrow \dot{Q}\_{\rm c} = \dot{Q}\_{\rm hp} - \frac{1}{2} \dot{Q}\_{\rm l} - \dot{Q}\_{\rm cd} \tag{32}
$$

$$
\dot{Q}\_{\text{h}} = \alpha\_{\text{NP}} \cdot I \cdot T\_{\text{h}} + \frac{1}{2} \cdot R \cdot I^2 - K \cdot \Delta T \Rightarrow \dot{Q}\_{\text{h}} = \dot{Q}\_{\text{hp}} + \frac{1}{2} \dot{Q}\_{\text{J}} - \dot{Q}\_{\text{cd}} \tag{33}
$$

where Q\_ hp is the thermoelectric heat pumping at the cold junction, Q\_ <sup>J</sup> is the Joule heat and Q\_ cd is the heat flow conducted from the hot junction to the cold junction.

However, this model can be used only at first approximation for the selection of thermocouple materials [9]. In practice, the semiconductor properties depend on temperature, the contact resistances cannot be avoided, and the Thomson effect cannot be neglected. Moreover, in the steady-state model, the temperatures T<sup>h</sup> and T<sup>c</sup> are input values that have to be determined accurately. If the object to be cooled is directly in contact with the TEC cold surface, the object temperature has the same value as the temperature of the TEC cold surface Tc. However, if the object to be cooled is not directly in contact with the TEC cold surface, e.g. in a refrigerator compartment, a heat exchanger is required on the TEC cold surface. In this case, the cold surface of the TEC has to be some degrees colder than the desired temperature in the refrigerator compartment, and the temperature T<sup>c</sup> is unknown. With a similar reasoning, if a heat exchanger is placed at the hot side, the known value is the ambient temperature, and the temperature T<sup>h</sup> is unknown. The temperature distribution of a complex system (refrigerator) with TEC is depicted in Figure 5.

Therefore, in practical applications in which the TEC is connected to other components (e.g. heat exchangers), (i) the temperatures T<sup>c</sup> and T<sup>h</sup> are unknown, (ii) only the external temperatures can be measured accurately, and (iii) the temperature at each point of the system depends

Figure 5. Schematic of temperature profile in a thermoelectric refrigeration system.

on the composition of the whole system and cannot be determined for the individual components. Thereby, the temperatures at the TEC terminals can be determined by using a dedicated model of the interconnected components. These temperatures are calculated from the solution of the overall system equations, in which all the temperature-dependent thermoelectric effects (Peltier, Seebeck, Thomson and Joule) are taken into account [17].

## 2.2.2. Transient analysis

couple and the temperature difference ΔT between the hot surface temperature T<sup>h</sup> and the

Considering that N-type and P-type thermocouples are identical (with the same length), the total electrical resistance is <sup>R</sup> <sup>¼</sup> <sup>r</sup>∙l∙ð Þ <sup>S</sup> �<sup>1</sup> with <sup>r</sup> <sup>¼</sup> <sup>r</sup><sup>P</sup> <sup>þ</sup> <sup>r</sup>N, the total thermal conductance of the thermoelements is <sup>K</sup> <sup>¼</sup> <sup>k</sup>∙S∙ð Þ<sup>l</sup> �<sup>1</sup> with <sup>k</sup> <sup>¼</sup> <sup>k</sup><sup>P</sup> <sup>þ</sup> <sup>k</sup><sup>N</sup> the thermal conductivity corresponding to

corresponding to the N and P thermoelement legs with <sup>σ</sup> <sup>¼</sup> <sup>σ</sup><sup>P</sup> <sup>þ</sup> <sup>σ</sup><sup>N</sup> in S∙m�1. Then, Eq. (24) and Eq. (26) give the cooling capacity (or the rate of heat absorbed at the cold junction) Q\_

<sup>2</sup> � <sup>K</sup>∙Δ<sup>T</sup> ) <sup>Q</sup>\_

<sup>2</sup> � <sup>K</sup>∙Δ<sup>T</sup> ) <sup>Q</sup>\_

However, this model can be used only at first approximation for the selection of thermocouple materials [9]. In practice, the semiconductor properties depend on temperature, the contact resistances cannot be avoided, and the Thomson effect cannot be neglected. Moreover, in the steady-state model, the temperatures T<sup>h</sup> and T<sup>c</sup> are input values that have to be determined accurately. If the object to be cooled is directly in contact with the TEC cold surface, the object temperature has the same value as the temperature of the TEC cold surface Tc. However, if the object to be cooled is not directly in contact with the TEC cold surface, e.g. in a refrigerator compartment, a heat exchanger is required on the TEC cold surface. In this case, the cold surface of the TEC has to be some degrees colder than the desired temperature in the refrigerator compartment, and the temperature T<sup>c</sup> is unknown. With a similar reasoning, if a heat exchanger is placed at the hot side, the known value is the ambient temperature, and the temperature T<sup>h</sup> is unknown. The temperature distribution of a complex system (refrigerator)

Therefore, in practical applications in which the TEC is connected to other components (e.g. heat exchangers), (i) the temperatures T<sup>c</sup> and T<sup>h</sup> are unknown, (ii) only the external temperatures can be measured accurately, and (iii) the temperature at each point of the system depends

<sup>c</sup> <sup>¼</sup> <sup>Q</sup>\_

<sup>h</sup> <sup>¼</sup> <sup>Q</sup>\_

hp � <sup>1</sup> 2 <sup>Q</sup>\_ <sup>J</sup> � <sup>Q</sup>\_

hp þ 1 2 Q\_

<sup>S</sup> <sup>þ</sup> <sup>r</sup>N∙l<sup>N</sup>

<sup>S</sup> (28)

, and σ is the electrical conductivity

αNP ¼ α<sup>P</sup> � α<sup>N</sup> (30)

ΔT ¼ T<sup>h</sup> � T<sup>c</sup> (31)

(29)

<sup>c</sup> and

cd

cd (32)

<sup>J</sup> � <sup>Q</sup>\_ cd (33)

<sup>J</sup> is the Joule heat and Q\_

<sup>R</sup> <sup>¼</sup> <sup>r</sup>P∙l<sup>P</sup>

<sup>K</sup> <sup>¼</sup> <sup>k</sup>P∙S<sup>P</sup> lP þ kN∙S<sup>N</sup> lN

cold surface temperature T<sup>c</sup> are written as [9]

230 Bringing Thermoelectricity into Reality

the N and P thermoelement legs in W∙ð Þ <sup>m</sup>∙<sup>K</sup> �<sup>1</sup>

<sup>h</sup> in W:

2 ∙R∙I

1 2 ∙R∙I

is the heat flow conducted from the hot junction to the cold junction.

hp is the thermoelectric heat pumping at the cold junction, Q\_

<sup>c</sup> <sup>¼</sup> <sup>α</sup>NP∙I∙T<sup>c</sup> � <sup>1</sup>

<sup>h</sup> ¼ αNP∙I∙T<sup>h</sup> þ

the rate of heat rejection Q\_

where Q\_

Q\_

Q\_

with TEC is depicted in Figure 5.

Thermoelectric refrigerators are controlled devices that operate in transient conditions. Thereby, it is important to formulate a detailed model taking into account all the thermoelectric effects and the dependence of the model parameters on temperature.

Eq. (17) is written in generic transient conditions. The solution of this equation has been obtained in [18] by constructing an electrothermal equivalent model with resistances and capacities (in which the thermoelectric modules are represented through a multi-node structure and the other components are represented by a single node). The implicit finite difference method has been used to solve the equations. In this model, the input data are the number of modules, the geometric parameters (lengths and cross areas), the structural characteristics of the components, the heat flow rate produced by the heat source, the voltage supply from the electrical system and the environment temperature. The structural characteristics can be given as constant values (density, specific heat, surface electrical resistivity of the thermoelectric elements) or can be expressed as functions of the temperatures (Seebeck coefficient, electrical resistivity and thermal conductivity). Since the model is non-linear, the solution requires an iterative process, so that the initialization of the temperatures at each node of the model has to be provided as well. The outputs of the method (with their evolutions in time) are the temperatures at all the nodes, the heat flow rates in each component, the power produced by the modules and consumed by the fan and the efficiencies of the modules and of the system. This formulation is consistent with an experimental application, such as the one presented in [17].

#### 2.3. Energy indicators for TEC performance

The energy indicators useful for the design and the performance of TEC are the cooling capacity, the rate of heat rejection, the input electrical power, the dimensionless figure of merit ZT and the coefficient of performance (COP).

The temperature difference ΔT created when a current flows through the TEC generates a raising voltage [9]. This voltage depends on the voltage referring to the Seebeck effect αNP∙ΔT and the voltage at the thermoelectric couple αNP∙ΔT:

$$V = \alpha\_{\rm NP} \cdot \Delta T + R \cdot I \tag{34}$$

thermoelements. In this case, a thermoelectric semiconductor with a higher figure of merit is advantageous because it gives a superior cooling power. To obtain a higher figure of merit, a thermoelectric material optimization is required. This means to optimize the ZT dimensionless parameter by a maximization of the power factor, which depends on material properties like electrical conductivity and Seebeck coefficient, as well as a minimization of the thermal con-

The best materials with high ZT are high doped semiconductors. Metals have relatively small Seebeck coefficients, and insulators have low electrical conductivity. The thermoelectric cooling materials are alloys which contain bismuth telluride (Bi2Te3) with antimony telluride (Sb2Te3) (like p-type Bi0.5Sb1.5Te3 composites) [22] and Bi2Te3 with bismuth selenide Bi2Se3 (like n-type Bi2Te2.7Se0.3) [23], each having ZT ffi1 at room temperature [21]. The thermoelectric materials with good electrical properties and low thermal conductivities are bulk materials and nanostructured materials considering the dimensionless figure of merit ZT ≥ 1 [24, 25].

The figure of merit of thermoelectric modules rises with the Seebeck coefficient, while the cooling capacity of the heat sink becomes narrow [26]. Much more, the figure of merit of a thermoelectric element limits the temperature differential achieved between the sides of the module, while the

The number of thermoelements in a thermoelectric module mainly depends on the required cooling capacity and the maximum electric current [9]. An expression for the cooling capacity shows that it also depends on thermal and electrical contact resistances (at both sides of the

<sup>c</sup> <sup>¼</sup> <sup>k</sup>∙ð Þ <sup>Δ</sup>Tmax � <sup>Δ</sup><sup>T</sup>

where l<sup>c</sup> is the thickness of the contact layers, r is the thermal contact parameter (which is the ratio between the thermal conductivity of the thermoelements and the thermal conductivity of the contact layers), COP is the coefficient of performance and ΔTmax the maximum tempera-

An insignificant effect of the contact resistances on the cooling capacity is observed for the thermoelements with long lengths, while significant changes of the cooling capacity are obtained when the contact resistances are improved, and this is for the case of short thermoel-

design to find the operating conditions [28]. The maximum temperature difference ΔTmax

The maximum current represents the current which gives the maximum possible temperature

� T<sup>h</sup> þ

1 Z � �<sup>2</sup>

" #<sup>1</sup>

� <sup>T</sup><sup>2</sup> h 2

<sup>c</sup> ¼ 0 [4]. Practically, operating under the maximum

1 Z � �

obtainable between the hot and cold sides always occurs at Imax, Vmax and Q\_

ΔTmax ¼ T<sup>h</sup> þ

<sup>l</sup> <sup>þ</sup> <sup>2</sup>∙r∙l<sup>c</sup> <sup>þ</sup> <sup>r</sup>∙lc∙COP�<sup>1</sup> (38)

<sup>c</sup>max and the maximum COPmax are used in

<sup>c</sup> ¼ 0:

Thermoelectric Refrigeration Principles http://dx.doi.org/10.5772/intechopen.75439 233

(39)

length-to-surface ratio for the thermoelements defines the cooling capacity [3].

thermoelectric module), as well as the thermoelement length of the module [20]:

Q\_

ements [27]. The maximum cooling capacity Q\_

difference ΔTmax which takes place when Q\_

ductivity [1].

ture difference.

The input electrical power Pel or electrical power consumption [19] is

$$P\_{\rm el} = \dot{Q}\_h - \dot{Q}\_c = \alpha\_{\rm NP} \cdot I \cdot \Delta T + R \cdot I^2 \tag{35}$$

An important physical property of the TEM is the figure of merit Z. It depends on the transport parameters (Seebeck coefficient of the thermoelectric couple, total electrical resistivity and total thermal conductivity):

$$Z = \frac{\alpha\_{\rm NP}^2}{\rho \cdot k} \tag{36}$$

The thermal performance of a thermoelectric cooler is given by dimensionless figure of merit ZT. The absolute temperature T is the mean device temperature T between the hot side and cold sides of the TEC [20, 21]:

$$\overline{T} = \frac{T\_{\text{c}} + T\_{\text{h}}}{2} \tag{37}$$

Generally the expression ZT is written without indicating the averaging symbol for T. The parameter ZT represents the efficiency of the semiconductor materials of N-type and P-type thermoelements. In this case, a thermoelectric semiconductor with a higher figure of merit is advantageous because it gives a superior cooling power. To obtain a higher figure of merit, a thermoelectric material optimization is required. This means to optimize the ZT dimensionless parameter by a maximization of the power factor, which depends on material properties like electrical conductivity and Seebeck coefficient, as well as a minimization of the thermal conductivity [1].

components are represented by a single node). The implicit finite difference method has been used to solve the equations. In this model, the input data are the number of modules, the geometric parameters (lengths and cross areas), the structural characteristics of the components, the heat flow rate produced by the heat source, the voltage supply from the electrical system and the environment temperature. The structural characteristics can be given as constant values (density, specific heat, surface electrical resistivity of the thermoelectric elements) or can be expressed as functions of the temperatures (Seebeck coefficient, electrical resistivity and thermal conductivity). Since the model is non-linear, the solution requires an iterative process, so that the initialization of the temperatures at each node of the model has to be provided as well. The outputs of the method (with their evolutions in time) are the temperatures at all the nodes, the heat flow rates in each component, the power produced by the modules and consumed by the fan and the efficiencies of the modules and of the system. This formulation is consistent with an

The energy indicators useful for the design and the performance of TEC are the cooling capacity, the rate of heat rejection, the input electrical power, the dimensionless figure of merit

The temperature difference ΔT created when a current flows through the TEC generates a raising voltage [9]. This voltage depends on the voltage referring to the Seebeck effect αNP∙ΔT

<sup>h</sup> � <sup>Q</sup>\_ <sup>c</sup> <sup>¼</sup> <sup>α</sup>NP∙I∙Δ<sup>T</sup> <sup>þ</sup> <sup>R</sup>∙<sup>I</sup>

An important physical property of the TEM is the figure of merit Z. It depends on the transport parameters (Seebeck coefficient of the thermoelectric couple, total electrical resistivity and total

> <sup>Z</sup> <sup>¼</sup> <sup>α</sup><sup>2</sup> NP

The thermal performance of a thermoelectric cooler is given by dimensionless figure of merit ZT. The absolute temperature T is the mean device temperature T between the hot side and

<sup>T</sup> <sup>¼</sup> <sup>T</sup><sup>c</sup> <sup>þ</sup> <sup>T</sup><sup>h</sup>

Generally the expression ZT is written without indicating the averaging symbol for T. The parameter ZT represents the efficiency of the semiconductor materials of N-type and P-type

V ¼ αNP∙ΔT þ R∙I (34)

<sup>r</sup>∙<sup>k</sup> (36)

<sup>2</sup> (37)

<sup>2</sup> (35)

experimental application, such as the one presented in [17].

2.3. Energy indicators for TEC performance

232 Bringing Thermoelectricity into Reality

ZT and the coefficient of performance (COP).

thermal conductivity):

cold sides of the TEC [20, 21]:

and the voltage at the thermoelectric couple αNP∙ΔT:

The input electrical power Pel or electrical power consumption [19] is

<sup>P</sup>el <sup>¼</sup> <sup>Q</sup>\_

The best materials with high ZT are high doped semiconductors. Metals have relatively small Seebeck coefficients, and insulators have low electrical conductivity. The thermoelectric cooling materials are alloys which contain bismuth telluride (Bi2Te3) with antimony telluride (Sb2Te3) (like p-type Bi0.5Sb1.5Te3 composites) [22] and Bi2Te3 with bismuth selenide Bi2Se3 (like n-type Bi2Te2.7Se0.3) [23], each having ZT ffi1 at room temperature [21]. The thermoelectric materials with good electrical properties and low thermal conductivities are bulk materials and nanostructured materials considering the dimensionless figure of merit ZT ≥ 1 [24, 25].

The figure of merit of thermoelectric modules rises with the Seebeck coefficient, while the cooling capacity of the heat sink becomes narrow [26]. Much more, the figure of merit of a thermoelectric element limits the temperature differential achieved between the sides of the module, while the length-to-surface ratio for the thermoelements defines the cooling capacity [3].

The number of thermoelements in a thermoelectric module mainly depends on the required cooling capacity and the maximum electric current [9]. An expression for the cooling capacity shows that it also depends on thermal and electrical contact resistances (at both sides of the thermoelectric module), as well as the thermoelement length of the module [20]:

$$\dot{Q}\_{\text{c}} = \frac{k \cdot (\Delta T\_{\text{max}} - \Delta T)}{l + 2 \cdot r \cdot l\_{\text{c}} + r \cdot l\_{\text{c}} \cdot \text{COP}^{-1}} \tag{38}$$

where l<sup>c</sup> is the thickness of the contact layers, r is the thermal contact parameter (which is the ratio between the thermal conductivity of the thermoelements and the thermal conductivity of the contact layers), COP is the coefficient of performance and ΔTmax the maximum temperature difference.

An insignificant effect of the contact resistances on the cooling capacity is observed for the thermoelements with long lengths, while significant changes of the cooling capacity are obtained when the contact resistances are improved, and this is for the case of short thermoelements [27]. The maximum cooling capacity Q\_ <sup>c</sup>max and the maximum COPmax are used in design to find the operating conditions [28]. The maximum temperature difference ΔTmax obtainable between the hot and cold sides always occurs at Imax, Vmax and Q\_ <sup>c</sup> ¼ 0:

$$
\Delta T\_{\text{max}} = \left( T\_{\text{h}} + \frac{1}{Z} \right) - \left[ \left( T\_{\text{h}} + \frac{1}{Z} \right)^2 - T\_{\text{h}}^2 \right]^{\frac{1}{2}} \tag{39}
$$

The maximum current represents the current which gives the maximum possible temperature difference ΔTmax which takes place when Q\_ <sup>c</sup> ¼ 0 [4]. Practically, operating under the maximum current, there is insufficient current to obtain ΔTmax. Working above the maximum current, the power dissipation inside the TEC starts to rise the device temperature and to decrease ΔT. The maximum current is almost constant which is the operating range of the device:

$$I\_{\text{max}} = \alpha\_{\text{NP}} \cdot \left( T\_h - \Delta T\_{\text{max}} \right) \cdot \text{R}^{-1} \tag{40}$$

The COPmax depends on the current and does not depend on the number of TEC pairs

The COPmax and the maximum temperature difference ΔTmax are affected by the figure of

<sup>1</sup> <sup>þ</sup> ZT <sup>p</sup> � Th∙Tc

where COPC is the (ideal) Carnot COP and COPr is the relative COP. The COP depends on the temperature difference. Mainly, the COP rises with the reduction of the temperature difference ΔT. For household applications, to obtain an adequate cooling effect, the temperature differential between the sides of TEC is considered to be about ΔT = 25 ÷ 30 K. In this case, the values COP = 0.5 ÷ 0.7 represent about 50% of the COP of a vapour compressor refrigerator [34].

�<sup>1</sup> � �<sup>∙</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi


<sup>∙</sup> <sup>α</sup>NP∙I∙T<sup>c</sup> � <sup>k</sup>∙Δ<sup>T</sup> � <sup>R</sup>∙<sup>I</sup>

<sup>1</sup> <sup>þ</sup> ZT <sup>p</sup> <sup>þ</sup> <sup>1</sup> � ��<sup>1</sup>

2 2 � � (44)

Thermoelectric Refrigeration Principles http://dx.doi.org/10.5772/intechopen.75439

(45)

235

<sup>2</sup> <sup>þ</sup> <sup>α</sup>NP∙I∙Δ<sup>T</sup> � ��<sup>1</sup>

∙ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

3. Methods to enhance the TEC performance in refrigeration units

• Optimization of the internal temperature controller of the insulated compartment

A thermoelectric refrigerator unit operates with COP typically less 0.5 due to the limited cooling temperature to ΔTmax ffi 20 K under the ambient temperature [20]. Figure 6 shows a comparison of the theoretical COP of a TEC with respect to household refrigerators [36]. Refrigerators with thermoelectric modules with materials based on alloys of Bi2Te3 have a COP about 1 [9] which is low enough to be competitive to the vapour-compression systems with COP = 2 ÷ 4 [37–39]. The low COP values of TECs are not considered a drawback. These systems are more suitable for a niche market sector (below 25 W) such as military and medical industries, in applications such as temperature stabilization of semiconductor lasers and vaccine cooling. Furthermore, they are also suitable for the civil market (e.g., portable refrigeration, car-seat cooler, high-quality beverage conservators). For these applications, the thermoelectric elements have the advantages that

COPmax ¼ R∙I

COPmax <sup>¼</sup> Tc∙ΔT�<sup>1</sup>

merit. The COPmax is reached at low Th, high T<sup>c</sup> and high Z [33]:


Some methods to enhance the TEC performance are [35]:

do not suffer vibrations and shocks [21, 40–42].

• Development of thermoelectric materials with high performance

3.1. Development of thermoelectric materials with high performance

[32, 27]:

• TEC design

• Thermal design

The maximum cooling capacity Q\_ cmax for a TEC is the maximum thermal load obtained when ΔT = 0 and I=Imax:

$$\dot{Q}\_{\text{c}\_{\text{max}}} = \alpha\_{\text{NP}} \,^2 \left( T\_{\text{h}}^2 - \Delta T\_{\text{max}}^2 \right) \cdot \left( 2R \right)^{-1} \tag{41}$$

The maximum voltage represents the DC voltage which gives ΔTmax at I = Imax. In this case COP has a minimum value. At maximum voltage the power dissipation inside the TEC starts to rise the device temperature and to decrease ΔT. The maximum voltage depends on the temperature:

$$V\_{\text{max}} = \alpha\_{\text{NP}} \cdot T\_h \tag{42}$$

The coefficient of performance (COP) represents the heat absorbed at the cold junction or cooling capacity, divided by the input electrical power:

$$\text{COP} = \frac{\dot{Q}\_{\text{c}}}{P\_{\text{el}}} \tag{43}$$

Various papers explain the COP dependence of the characteristics of the materials on the Thomson effect and on temperature. The TEC performance is improved by raising the figure of merit of the thermoelectric elements and considering the Thomson effect [29]. The validity of the Thomson effect is taken into account in the relationships of the cooling capacity and input electrical power and implicitly in the COP relationship, if the dependence on temperature of Seebeck coefficient is considered [30]. In this case the Thomson effect gives a reduction with about 7.1% for the input electrical power and with about 7% for the cooling capacity considering the positive values of the Thomson coefficient; instead, an improvement in both the input electrical power and the cooling capacity is observed for negative values of the Thomson coefficient [31].

The COP is also influenced by the thermal and electrical contact resistances. The COP of the thermoelectric module can be improved up to 60% by decreasing the electrical contact resistances [27]. Furthermore, the COP depends on the thermoelement length. The COP rises with the increment of the thermoelement length. For a thermoelement with a shorter length, the contact resistance becomes closer to the resistance of the thermoelement, notably affecting this indicator [27].

The maximum COP (indicated as COPmax) of a TEC is used for its sizing [9, 20]. The COPmax has the benefit of minimum input electrical power, therefore, minimum total heat to be rejected by the heat sink: Q\_ <sup>h</sup> <sup>¼</sup> <sup>Q</sup>\_ <sup>c</sup> þ Pel:

The COPmax depends on the current and does not depend on the number of TEC pairs [32, 27]:

$$\text{COP}\_{\text{max}} = \left(\mathbf{R} \cdot \mathbf{I}^2 + \alpha\_{\text{NP}} \cdot \mathbf{I} \cdot \Delta T\right)^{-1} \cdot \left(\alpha\_{\text{NP}} \cdot \mathbf{I} \cdot \mathbf{T}\_{\text{c}} - k \cdot \Delta T - \frac{\mathbf{R} \cdot \mathbf{I}^2}{2}\right) \tag{44}$$

The COPmax and the maximum temperature difference ΔTmax are affected by the figure of merit. The COPmax is reached at low Th, high T<sup>c</sup> and high Z [33]:

$$\text{COP}\_{\text{max}} = \underbrace{T\_{\mathcal{C}} \cdot \Delta T^{-1}}\_{\text{COP}\_{\mathcal{C}}} \cdot \underbrace{\left(\sqrt{1 + ZT} - T\_{h'}T\_{c}^{-1}\right) \cdot \left(\sqrt{1 + ZT} + 1\right)^{-1}}\_{\text{COP}\_{\text{r}}} \tag{45}$$

where COPC is the (ideal) Carnot COP and COPr is the relative COP. The COP depends on the temperature difference. Mainly, the COP rises with the reduction of the temperature difference ΔT. For household applications, to obtain an adequate cooling effect, the temperature differential between the sides of TEC is considered to be about ΔT = 25 ÷ 30 K. In this case, the values COP = 0.5 ÷ 0.7 represent about 50% of the COP of a vapour compressor refrigerator [34].
