**2.5 Thermogenerator composite parameters**

The effective composite materials for thermoelectric generator include heat source, the thermocouple, the heat sink and the load. The composite parameters of the generator are the total Seebeck coefficient of the junction S, the total internal resistance r, and the total thermal conductivity *k*. The Seebeck coefficient can have either a positive or a negative sign. A material that has a negative S is referred to as an n-type material while a material having positive S is referred to as a p-type material.

Assuming that all the composite parameters are independent of temperature, and the Seebeck coefficients of the legs of thermocouple are Sn and Sp, and the electrical resistivity of each leg is ρn and ρp, and the thermal conductivities of each leg is *k*n and *k*p, then the parameters are defined by the following equations.

$$\mathbf{S} = \mathbf{S}\_{\text{P}} - \mathbf{S}\_{\text{n}} = \begin{vmatrix} \mathbf{S}\_{\text{p}} \end{vmatrix} + \begin{vmatrix} \mathbf{S}\_{\text{n}} \end{vmatrix} \tag{24}$$

$$\mathbf{r} = \frac{\rho\_\mathrm{n} l\_\mathrm{n}}{\mathbf{A}\_\mathrm{n}} + \frac{\rho\_\mathrm{p} l\_\mathrm{p}}{\mathbf{A}\_\mathrm{p}} \tag{25}$$

$$\mathbf{k} = \frac{\mathbf{k}\_\mathbf{n} \mathbf{A}\_\mathbf{n}}{l\_\mathbf{n}} + \frac{\rho\_\mathbf{p} \mathbf{A}\_\mathbf{p}}{l\_\mathbf{p}} \tag{26}$$

$$\mathfrak{h} = \mathfrak{p}\_{\text{out}} / \mathbb{Q}\_{\text{in}} = \, l^2 \mathbb{R} / \mathbb{Q}\_{\text{in}} \tag{27}$$

$$\mathbf{Q\_{ln}} = \mathbf{S} \mathbf{T\_n} l + \mathbf{k(T\_n - T\_c)} - \frac{l^\* r}{2} \tag{28}$$

$$\mathbf{V\_o = S(T\_n - T\_c)}\tag{29}$$

$$\mathbf{I} = \begin{array}{c} \mathbf{v}\_{\mathbf{o}} \\ \hline \mathbf{R} + \mathbf{r} \end{array} \tag{30}$$

$$\eta = \frac{\mathbf{T\_n} - \mathbf{T\_c}}{\mathbf{T\_n}} \left\{ \frac{\mathbf{M} - \mathbf{1}}{\mathbf{M} + \mathbf{^T \mathbf{c}}/\_{\mathbf{T\_n}}} \right\} \tag{31}$$

$$\mathbf{M} = \|\mathbf{m}\|\_{\mathbf{d}\_\mathbf{h}/\mathbf{d}\_\mathbf{m=o}} = \sqrt{\mathbf{Z}(\mathbf{T}\_\mathbf{h} - \mathbf{T}\_\mathbf{2})/2} \\ \mathbf{M} = \|\mathbf{m}\|\_{\mathbf{d}\_\mathbf{h}/\mathbf{d}\_\mathbf{m=o}} = \sqrt{\mathbf{Z}(\mathbf{T}\_\mathbf{h} - \mathbf{T}\_\mathbf{2})/2} \tag{32}$$

$$
\eta\_{\rm c} = \frac{\mathbf{T\_n} - \mathbf{T\_c}}{\mathbf{T\_n}} \tag{33}
$$

### **2.7 Figure of merit**

This is a measure of the ability of a given thermoelectric material in power generation, heating or cooling at a given temperature T. ZT is given by the equation.

$$ZT = S^2 \sigma T / k\tag{35}$$

Where

470 Solar Radiation

Where *l*n, An, and lp, Ap, refers to the length and area of the n-type and p-type materials respectively, the heat source has a temperature Tn, and the heat sink has temperature Tc.

R is the electrical load resistance. Heat input Qin consists of the Peltier heat STnI plus the conduction heat *K* (Tn-Tc) less one-half of the Joule heat I2r librated in the thermocouple legs,

Q�� = ST����(T� − T�) <sup>−</sup> ���

Losses in maintaining the temperature Tn are not considered by this efficiency and thus it is

I = ��

����� ��

Using these quantities, and selecting m to give optimum loading, the optimized efficiency,

This efficiency for an optimum load consists of a Carnot efficiency ηc and device efficiency

η� <sup>=</sup> ����� ��

η� <sup>=</sup> ���

The device efficiency ηd will be a maximum for the largest value of M, for a fixed Tn and Tc; this requires a maximum value of Z. For most good thermoelectrics, Z (Tn+Tc)/2 ≈1, so for

Tn/Tc ≈1, the efficiency is about 20% of the thermodynamic limit.

� ���

M = m|��⁄���� = √Z(T� − T�) 2⁄ ) M = m|��⁄���� = √Z(T� − T�) 2⁄ ) (32)

The ratio of load resistance R to internal resistance r is defined as m = R/r

η =

η=p���⁄Q�� = ��R Q⁄ �� (27)

V� = S(T� − T�) (29)

��� (30)

���� �� � � (31)

(33)

���� �� � (34)

� (28)

The efficiency η of generator is the power output I2R divided by the heat input Qin.

**2.6 Thermogenerator efficiency** 

The open-circuit voltage,

the efficiency expression becomes

Z is the figure of merit.

The current,

Where

ηd thus

not a total efficiency including heat source losses.

i.e.

S = the thermo power of the material = The electrical conductivity *K* = The thermal conductivity

The largest values of ZT are attained in semimetals and highly doped semiconductors, which are the materials normally used in practical thermoelectric devices. Figure of merit for single materials and thermocouples formed from two such materials varies hence one thermocouple can be better than another at one temperature but less effective at a second temperature.

Z depends upon the material parameters Sp, *K*p, ρp Sn, *K*n, ρn and the dimensions of the two legs Ap, ℓn, ℓp ,An . Maximizing Z with respect to the area-to-length ratio of the legs gives

$$\|Z\|\_{\mathbf{d}\_{\mathbf{n}}/\mathbf{d}\_{\mathbf{x}=\mathbf{0}}} = \frac{(\mathbf{S}\_{\mathbf{p}} - \mathbf{S}\_{\mathbf{n}})^2}{[(\mathbf{k}\_{\mathbf{p}} l\_{\mathbf{p}})^{1/2} + (\mathbf{k}\_{\mathbf{n}} l\_{\mathbf{n}})^{1/2}]^2} \tag{36}$$

When equation

$$
\left[\frac{k\_\mathrm{n}l\_\mathrm{n}}{k\_\mathrm{p}l\_\mathrm{p}}\right]^{1/2} = \left[\frac{A\_\mathrm{p}l\_\mathrm{n}}{A\_\mathrm{n}l\_\mathrm{p}}\right] \tag{37}
$$

For the optimum area-to-length ratio Z depends only upon the specific properties of the thermoelectric material. Generally, the parameters S, *K*, and ρ are not independent of temperature, and in fact the temperature dependence of the n and p legs may differ radically. The most widely used generator materials are lead telluride, which has a maximum figure of merit of approximately 1.5x10-3 K-1. It can be doped to produce both ptype and n-type material and has a useful temperature range of about 300-700K (80-800oF). In material development, existing thermoelectric p and n materials operates from 300 to 1300K (80 to 1900oF) and yield an overall theoretical thermal efficiency of 18%.

To maximize power output, it is necessary to produce the largest possible voltage, thus Seebeck coefficient S should be made large, and hence proper selection of materials are required. Materials should have low electrical resistance in the generator. The legs should also have low thermal conductivities *K* since heat energy is carried away by thermal conduction. Hence the requirements for materials to be used in thermoelectric power generators are high S, low ρ and *K* and high figures of merit Z. Since the figures of merit Z for single materials vary with temperature, so do the figures of merit for thermocouples formed from two materials.

### **3. The thermocouple system**

Thermocouples are differential temperature-measurement devices. They are constructed with two wires of dissimilar metals. One wire is pre-designated as the positive side (Copper, Iron, Chromel) and the other as the negative (Constantan, Alumel). Basic system suitable for the application of thermoelectricity in power generation is that of several thermocouples connected in series to form a thermopile (a device with increased output relative to a single thermocouple). The junctions forming one end of the thermocouple are at the same low temperature TL and the other junctions at the hot temperature TH.

The thermopile is connected to a device in which the temperature TL is fixed when connected to a heat sink. The temperature TH is determined by the output of the heat source and the thermal output of the thermopile. The load is run by the charges generated. With a thermopile, the multiplication of thermocouple involves a corresponding increase of resistance, hence it follows that one thermocouple can be better than another at one temperature but less effective at a second temperature. In order to take maximum advantage of the different materials, the thermocouples are cascaded, producing power in stages and increase power output.
