General Approach for Composite Thermoelectric Systems with Thermal Coupling: The Case of a Dual Thermoelectric Cooler

Abstract

In this work, we show a general approach for inhomogeneous composite thermoelectric systems, and as an illustrative case, we consider a dual thermoelectric cooler. This composite cooler consists of two thermoelectric modules (TEMs) connected thermally in parallel and electrically in series. Each TEM has different thermoelectric (TE) properties, namely thermal conductance, electrical resistance and the Seebeck coefficient. The system is coupled by thermal conductances to heat reservoirs. The proposed approach consists of derivation of the dimensionless thermoelectric properties for the whole system. Thus, we obtain an equivalent figure of merit whose impact and meaning is discussed. We make use of dimensionless equations to study the impact of the thermal conductance matching on the cooling capacity and the coefficient of the performance of the system. The equivalent thermoelectric properties derived with our formalism include the external conductances and all intrinsic thermoelectric properties of each component of the system. Our proposed approach permits us changing the thermoelectric parameters of the TEMs and the working conditions of the composite system. Furthermore, our analysis shows the effect of the number of thermocouples on the system. These considerations are very useful for the design of thermoelectric composite systems. We reproduce the qualitative behavior of a commercial composite TEM connected electrically in series.

1. Introduction

Composite thermoelectric systems, based on thermoelectric effects, such as Seebeck effect and Peltier effect, have a variety of uses nowadays. A composite thermoelectric cooling system (composite TECS) has many advantages in comparison with the traditional cooling systems, such as the lack of moving parts, low weight, does not need maintenance and the fact that it is environmentally friendly due to lack of cooling substances. Disadvantage of the TECS are the low cooling capacity Q_c and coefficient of performance (COP). New techniques and thermoelectric materials have been proposed for designing new thermoelectric specific systems that allow the improvement of the device performance through the optimization of the thermal and electrical transport properties [1–3]. These TECS, known as solid-state devices, are used in many different applications, ranging from controlling the temperature of laser diodes, infrared detectors, superconductor applications, aerospace applications, electronic devices and food storage [4].

It is well known that the thermal conductance of ceramic plates plays a vital role in the performance of TECS. Yamanashi [5] has considered the effects of the thermal resistance of heat exchangers on the performance of a TEC, with a constant thermocouple number. He has obtained the design parameters through the dimensionless heat balance equations. On the other hand, Xuan [6], through introducing equivalent impedances to take into account the thermal contact effect of a single stage, have derived the maximum temperature difference, cooling capacity and COP. Recently, Pearson and Lents [7] have studied a thermal network with an integrated TEC and performed a dimensionless analysis. They concluded that dimensionless parameters reduce the complexity of the results, enabling the evaluation of the system without knowing the detailed information of the geometries or the materials. These proposals consider homogeneous TECS, i.e., TECS with thermoelectric modules (TEMs) that have the same thermoelectric properties, namely the Seebeck coefficient, electrical resistance and thermal conductance. However, the behavior of inhomogeneous composite TECS, i.e., composite TECS with TEMs that have different thermoelectric properties, can not be predicted.

Apertet et al. [8] have considered an inhomogeneous thermoelectric generator system (TEGS) composed of two different thermoelectric modules, electrically and thermally connected in parallel, using linear irreversible thermodynamics, and they have proposed equivalent parameters for an equivalent thermoelectric generator system, including realistic thermal coupling [9]. Using this linear approach, others authors [10,11] have considered different configurations for composite TEGS.

Moreover, a widely-considered approach in the analysis for both TECS and TEGS is Ioffe’s approach [12]. This approach includes the Joulean heat loss [5,6]. Thus, we are interested in the analysis of inhomogeneous composite TECS, and we propose a general approach, based in Ioffe’s approach, which includes all thermoelectric properties of the TEMs constituting the TECS. We illustrate this general approach for a dual thermoelectric cooling system that consists of two TEMs with different thermoelectric properties and thermal coupling. This approach can be useful for designing a composite solid-state device with thermal coupling for switching between low power consumption or high heat pumping performance [13].

The paper is organized as follows: In Section 2, we present the configuration of the dual thermoelectric cooler system. We derive the equivalent parameters for our system and transform it into its dimensionless form, in Section 3. In Section 4, we analyze our numerical results for COP and for cooling capacity, Q_C, and their behavior as a function of the thermal impedances. Finally, in Section 5, we present our conclusions, and future work is discussed briefly.

2. Dual Thermoelectric Cooling System

Our composite TEC system is composed of two thermoelectric modules thermally connected in parallel and electrically connected in series, as depicted in Figure 1. The system is coupled to two heat reservoirs by thermal exchangers, and the temperatures of the heat reservoirs are T_c and T_h with T_h > T_c. K_c and K_h are the thermal conductances of the heat exchangers at both sides, cold and hot, respectively. Each TEM is characterized by a Seebeck coefficient α_i, thermal conductance K_i and an electrical resistanceR_i, where i can be one or two as appropriate. The TEM₁ and TEM₂ are formed by m and n thermocouples number, respectively. I is the electrical current through the TEMs.

3. Heat Balance Equations

We assume that the heat flow between the TEMs and the surroundings is ignored, except for the cold and hot end of each TEM; also, the properties of the N- and P -type elements are assumed independent of temperature. In Figure 1, Q_h is the heat rejected from the TEC system to the heat reservoir and Q_c is the cooling capacity or the heat absorbed from the cooled object. Assuming that the heat flow between each stage and the heat reservoirs obey Newton’s law, we have:

$$Q_h=K_h(T_1-T_h)$$

(1)

$$Q_c=K_c(T_c-T_2)$$

(2)

The heat flux flowing through the dual TEC system is the sum of two heat fluxes flowing through TEM₁ and TEM₂,

$$Q_h=Q_{h1}+Q_{h2}$$

(3)

$$Q_c=Q_{c1}+Q_{c2}$$

(4)

The heat fluxes, Q_ci and Q_hi (i = 1, 2), for each TEM are given by:

$$Q_{h1}=m[{\alpha }_{1}I{T}_1+\frac{1}{2}{I}^2{R}_1-K_1(T_1-T_2)]$$

(5)

$$Q_{c1}=m[{\alpha }_{1}I{T}_2-\frac{1}{2}{I}^2{R}_1-K_1(T_1-T_2)]$$

(6)

$$Q_{h2}=n[{\alpha }_{2}I{T}_1+\frac{1}{2}{I}^2{R}_2-K_2(T_1-T_2)]$$

(7)

$$Q_{c2}=n[{\alpha }_{2}I{T}_2-\frac{1}{2}{I}^2{R}_2-K_2(T_1-T_2)]$$

(8)

where m and n are the number of thermocouples for TEM₁ and TEM₂, respectively. Equations (5)–(8) contain three terms, namely αIT, Peltier heat, I²R, the internal heat generated by the Joulean loss and K(T₁ − T₂), conduction heat loss. Clearly, the properties of different thermoelectric semiconductor materials of each TEM are included in the above equations.

3.1. Equivalent Dual TEC System

In this section, we derive the equivalent thermoelectric parameters of the dual TEC system as a whole. Combining Equations (1)–(3), (5) and (7), we obtain for the heat rejected, Q_h,

$$Q_h={\alpha }_{eq}{}_{H}I{T}_h+\frac{1}{2}{I}^2{R}_{eqH}-K_{eqH}(T_h-T_c)$$

(9)

where,

$${\alpha }_{eqH}=\frac{(m{\alpha }_1+n{\alpha }_2)(1+\frac{m{\alpha }_{1}I+n{\alpha }_{2}I}{K_c})}{1+\frac{(m{K}_1+n{K}_2)(K_c+K_h)}{K_h{K}_c}+\frac{(m{\alpha }_{1}I+n{\alpha }_{2}I)(K_h-K_c)}{K_h{K}_c}-\frac{{(m{\alpha }_{1}I+n{\alpha }_{2}I)}^2}{K_h{K}_c}}$$

(10)

$$R_{eqH}=\frac{(m{R}_1+n{R}_2)(1+\frac{m{\alpha }_{1}I+n{\alpha }_{2}I}{K_c}+2\frac{m{K}_1+n{K}_2}{K_c})}{1+\frac{(m{K}_1+n{K}_2)(K_c+K_h)}{K_h{K}_c}+\frac{(m{\alpha }_{1}I+n{\alpha }_{2}I)(K_h-K_c)}{K_h{K}_c}-\frac{{(m{\alpha }_{1}I+n{\alpha }_{2}I)}^2}{K_h{K}_c}}$$

(11)

$$K_{eqH}=\frac{(m{K}_1+n{K}_2)}{1+\frac{(m{K}_1+n{K}_2)(K_c+K_h)}{K_h{K}_c}+\frac{(m{\alpha }_{1}I+n{\alpha }_{2}I)(K_h-K_c)}{K_h{K}_c}-\frac{{(m{\alpha }_{1}I+n{\alpha }_{2}I)}^2}{K_h{K}_c}}$$

(12)

Similarly, combining Equations (1), (2), (4), (6) and (8), we have for absorbed heat Q_c,

$$Q_c={\alpha }_{eqC}I{T}_c-\frac{1}{2}{I}^2{R}_{eqC}-K_{eqC}(T_h-T_c)$$

(13)

where:

$${\alpha }_{eqC}=\frac{(m{\alpha }_1+n{\alpha }_2)(1-\frac{m{\alpha }_{1}I+n{\alpha }_{2}I}{K_h})}{1+\frac{(m{K}_1+n{K}_2)(K_c+K_h)}{K_h{K}_c}+\frac{(m{\alpha }_{1}I+n{\alpha }_{2}I)(K_h-K_c)}{K_h{K}_c}-\frac{{(m{\alpha }_{1}I+n{\alpha }_{2}I)}^2}{K_h{K}_c}}$$

(14)

$$R_{eqC}=\frac{(m{R}_1+n{R}_2)(1-\frac{m{\alpha }_{1}I+n{\alpha }_{2}I}{K_h}+2\frac{m{K}_1+n{K}_2}{K_h})}{1+\frac{(m{K}_1+n{K}_2)(K_c+K_h)}{K_h{K}_c}+\frac{(m{\alpha }_{1}I+n{\alpha }_{2}I)(K_h-K_c)}{K_h{K}_c}-\frac{{(m{\alpha }_{1}I+n{\alpha }_{2}I)}^2}{K_h{K}_c}}$$

(15)

$$K_{eqC}=\frac{(m{K}_1+n{K}_2)}{1+\frac{(m{K}_1+n{K}_2)(K_c+K_h)}{K_h{K}_c}+\frac{(m{\alpha }_{1}I+n{\alpha }_{2}I)(K_h-K_c)}{K_h{K}_c}-\frac{{(m{\alpha }_{1}I+n{\alpha }_{2}I)}^2}{K_h{K}_c}}$$

(16)

The equivalent thermoelectric parameters of our dual TEC system are given by Equations (10)–(12) and (14)–(16). These equations generalize the previously obtained results for homogeneous thermoelectric cooling systems considered by other authors [6]. For example, the parameters α_eqH and α_eqC are the effective Seebeck coefficients for the hot and cold side, respectively. These equivalent parameters not only combine all intrinsic thermoelectric properties of both modules, but also they are significantly influenced by external thermal conductances. In Section 4.4, we will show that the numerical behavior of the figure of merit for the whole system satisfies Bergman’s theorem. Notice that the equivalent heat balance equations of the system (9) and (13), depend only on the temperatures T_c and T_h. Furthermore, we will use these results and recover, as limit cases, the design parameters for TEC system, which have been previously studied by other authors; see Section 4.5.

3.2. Dimensionless Equivalent Heat Balance Equations

When a TEC system is optimized, it is convenient to rewrite the heat balance Equations (9) and (13), into the dimensionless form as follows [5],

$$q_h=\phi \frac{1+\phi \delta }{\mu }+\frac{1}{2}{\phi }^2\frac{1}{Z_{eq}{T}_h}\frac{1+\phi \delta +2\delta }{\mu }-\frac{1-\theta }{\mu }$$

(17)

$$q_c=\phi \frac{1-\phi \rho }{\mu }-\frac{1}{2}{\phi }^2\frac{1}{Z_{eq}{T}_c}\frac{1-\phi \rho +2\rho }{\mu }-\frac{\frac{1}{\theta }-1}{\mu }$$

(18)

where dimensionless quantities are defined as,

$$\theta =\frac{T_c}{T_h}$$

(19)

$$\delta =\frac{(m{K}_1+n{K}_2)}{K_c}$$

(20)

$$\rho =\frac{(m{K}_1+n{K}_2)}{K_h}$$

(21)

$$\phi =I\frac{(m{\alpha }_1+n{\alpha }_2)}{(m{K}_1+n{K}_2)}$$

(22)

$$Z_{eq}=\frac{{(m{\alpha }_1+n{\alpha }_2)}^2}{(m{R}_1+n{R}_2)(m{K}_1+n{K}_2)}$$

(23)

$$q_h=\frac{Q_h}{(m{K}_1+n{K}_2)T_h}$$

(24)

$$q_c=\frac{Q_c}{(m{K}_1+n{K}_2)T_c}$$

(25)

$$\mu =1+\delta +\rho +(\delta -\rho )\phi -{\phi }^2\delta \rho$$

(26)

In equations (17) and (18), the dimensionless current is represented by φ. The parameters δ and ρ are dimensionless thermal conductances normalized by the cold and hot external thermal conductances, respectively. θ is the temperature ratio of heat reservoirs. Z_eq is the equivalent figure of merit and µ is a coupling factor. Notice that we recover the dimensionless quantities q_h and q_c, which have been considered as the entropy flow normalized by the intrinsic thermal conductances, mK₁ + nK₂, of the inhomogeneous TEC system.

The equivalent figure of merit Z_eq, Equation (23), can be written in terms of the number of pairs of each TEM as:

$$Z_{eq}=\frac{{\alpha }_1^2{(D+\sqrt{Z_r})}^2}{R_1{K}_1{(D+Z_r)}^2}=Z_1\frac{{(D+\sqrt{Z_r})}^2}{{(D+Z_r)}^2}$$

(27)

where the number of pairs ratio, D, and the figure of merit ratio, Z_r, are given by:

$$D=\frac{m}{n}$$

(28)

$$Z_r=\frac{Z_1}{Z_2}$$

(29)

Now, the well-known COP for a cooling system,

$$COP=\frac{Q_c}{Q_h-Q_c}$$

(30)

can be written in terms of dimensionless heat quantities,

$$COP=\frac{q_c}{\frac{1}{\theta }{q}_h-q_c}$$

(31)

We point out that this approach can be used in the analysis of a thermoelectric heat pump.

4. Results and Discussion

In this section, we study the effect of the thermal conductance matching on the cooling capacity q_c and the COP of the TEC system. In our calculations, we use values T_c = 286K and T_h = 296K for the temperatures of the cold and hot reservoirs, respectively.

4.1. Cooling Capacity q_c: External Conductances Match

The behavior of the cooling capacity and the COP as a function of the dimensionless current, φ, for different external conductance ratios,

$$K_{ch}=\frac{K_c}{K_h}$$

(32)

are shown by Figures 2 and 6, respectively (see also Figures 11, 13 in Appendix).

Figure 2 shows that a limit value of q_c is reached as the ratio K_ch decreases for a given value of φ when K_c < K_h. As the external conductance ratio decreases, the change in maximal values of the cooling capacity are determined by the electrical current φ. Moreover, the limit values of q_c are determined by the ratio K_ch. Thus, the condition K_c < K_h is bounded, because q_c will reach limit values. Notice that it is possible to reach the same limit value of q_c, using different values of electrical current, φ.

From the results obtained in Section 3, our approach permits us to calculate the cooling capacity, q_c, in terms of any two thermoelectric parameters (φ, K_ch, Z_r) of the equivalent TEC system. For example, Figure 3 shows the behavior of q_c as a function of the figure of merit ratio Z_r and external conductances ratio, K_ch, for different values of electrical current, φ.

Notice that the behavior of q_c includes several effects that are contained in different planes of Figure 3. For example, maximum values for cooling capacity q_c are shown in the plane q_c vs. Z_r, at different values of electrical current φ, for small figure of merit ratios, Z_r, and thermal conductances ratios, K_ch. This result is more clearly shown in Figure 4.

Our approach permits us to know the effect of (1) thermal conductances, K_ch, and (2) the figure of merit ratio, Z_r, on the cooling capacity q_c at different working conditions. These effects are shown in Figures 11 and 12 of the Appendix.

Furthermore, we show the effect on q_c of both φ and Z_r, in Figure 5, for a given value of K_ch.

In fact, Figure 5 shows the optimal values of q_c in terms of φ and Z_r for constant values of other parameters, namely Z₁, T_h, θ, K_ch and D, which determine the working conditions of the composite TEC system.

We point out that our proposed approach permits us to change the thermoelectric parameters of the TEMs and the working conditions of the composite system. This fact is very useful for the design of thermoelectric composite systems.

4.2. Coefficient of Performance (COP): External Conductances Match

Analogously, Figures 6, 7 and 8 show the corresponding behavior for COP of the system as a function of K_ch, Z_r and φ, respectively.

Again, we obtain several results for COP of the system, which are included in different planes of Figures 7 and 8. See Figures 13, 14 and 15 in the Appendix.

For example, Figures 6 and 13 show that the maximal values of the COP tend to a limit maximum value as the ratio K_ch changes. This fact shows that it is not possible to optimize the system by just fixing the K_ch, but also a limit for the external conductances K_ch exists.

Table 1. COP for different values K_ch.

Kch	COP	∆COP
0	1.9866	0.045
0.05	1.9416	0.0437
0.10	1.8979	0.0424
0.15	1.8555	0.0412
0.20	1.8143	0.0400
0.25	1.7743	0.0389
0.30	1.7354	0.0378
0.35	1.6976	0.0368
0.40	1.6608	0.0358
0.45	1.6250	0.0349
0.50	1.5901	0.0339
0.55	1.5562	0.0331
0.60	1.5231	0.0322
0.65	1.4909	0.0314
0.70	1.4595	0.0306
0.75	1.4289	0.0298
0.80	1.3991	0.0292
0.85	1.3699	0.0284
0.90	1.3415	0.0278
0.95	1.3137	0.0271

shows the behavior of limit values for COP as we change the ratio K_ch.

As is well known, the optimal working conditions are different for both maximum values of q_c and COP. A maximum value for q_c is obtained for high values of φ; meanwhile, the maximum values of COP are obtained in low ranges of φ. In Section 4.3, we obtain the qualitative behavior of COP for a commercial composite TEM connected in series.

4.3. Numerical Validation

Firstly,

Table 2. Values of ρ and δ for given values of K_ch.

Kch	0.3	0.5	1	2
δ	0.3522	0.3522	0.3522	0.3522
ρ	0.10566	0.1761	0.3522	0.7044

shows some values of K_ch, δ and ρ used in our calculations, only for completeness reasons.

Nowadays, it is possible to fabricate composite TEMs connected in different configurations [13,14]. We use the thermoelectric parameters, α₁ = α₂ = α_Luo, R₁ = R₂ = R_Luo and K₁ = K₂ = K_Luo, for a composite TEM connected in series proposed by Luo [15] for obtaining Figure 9. The behavior of the COP, shown in Figure 9, is very near to that reported in the datasheet for the SP − 254 − 1.0−2.5 (series) TEM connected electrically in series, with the same thermoelectric properties for each component of the TEM [16].

4.4. Role of the Equivalent Figure of Merit

In the previous section, we studied the effect of the external conductance matching on the performance and the cooling capacity of our system. In this section, we analyze the role of the equivalent figure of merit, Z_eq, in the q_c and COP parameters in terms of the number of pairs, n and m, and the figure of merit of each component TEM, Z₁ and Z₂.

Using Equation (23) or Equation (27) and the numerical values of the thermoelectric properties of TEM 2, shown in

Table 3. Numerical values of the thermoelectric properties of TEM 2 when Z₁ = 0.0027.

Zr	α2	R2	K2	Z2
0.1	0.0170	0.341	0.0313	0.0273
0.5	0.0381	1.705	0.1565	0.0054
1	0.054	3.41	0.313	0.0027
2	0.0766	6.82	0.626	0.0013
3	0.0935	10.23	0.939	0.0009

, for a fixed value of Z₁ = 0.0027, we obtain Figure 10, which shows the equivalent figure of merit of the system as a function of the number pairs of each TEM.

Our results show that if the thermoelectric properties of both modules have the same value, Z_r = 1, the equivalent figure of merit Z_eq becomes independent of the number of pairs. On the other hand, if Z_r ≠ 1, the equivalent figure of merit approaches Z_r = 1, as the number of pairs increases. Meanwhile, if the number of pairs ratio decreases, the value of Z_eq tends to the figure of merit that satisfies the Z_r. Notice that the value of Z_eq is not greater that the values of the figure of merit for each TEM of the system, independently of the number, n or m, of the number of pairs. Thus, the result is according to the theorem of Bergman [17], which says that the equivalent figure of merit of the TEM, can only be lower than the highest Z of the more efficient TEM. The value of Z_eq is affected by the figure of merit ratio Z_r. When Z_r is lower than the unit, or Z₂ > Z₁, it is possible to increase the values of the equivalent figure of merit by having a lower number of pairs ratio n > m, but it is not possible to get a higher value than the Z₂ that satisfies the Z_r. Our analysis is consistent with the obtained results for thermoelectric generator systems [8] because of the generality of Bergman Theorem.

4.5. Simplified Approaches: Previously Considered Cases

Finally, in this section, we derive some results previously obtained for more simple models. From Equation (18), by setting q_c = 0, it is possible to get the full solution for the temperature ratio θ,

$$\theta =\frac{1}{\phi (1-\phi \rho )-\frac{1}{2}{\phi }^2\frac{1-\phi \rho +2\rho }{Z_{eq}{T}_c}+1}$$

(33)

In order to proceed with the approaches, it is necessary to define a new temperature ratio θ_h:

$${\theta }_h=\frac{1}{\theta }=\phi (1-\phi \rho )-\frac{1}{2}{\phi }^2\frac{1-\phi \rho +2\rho }{Z_{eq}{T}_c}+1.$$

(34)

Now, solving $\frac{d{\theta }_h}{d\phi }=0$, φ_max can be approximated through a power series expansion, leading to:

$${\phi }_{\mathit{max}}=Z_{eq}{T}_c-(\frac{1}{2}{Z}^2{T}_c^2+2Z_{eq}{T}_c)\rho +O({\rho }^2)$$

(35)

and finally, substituting φ_max into Equation (34) yields:

$${\theta }_h{}_{(\mathit{max})}={\phi }_{\mathit{max}}(1-{\phi }_{\mathit{max}}\rho )-\frac{1}{2}{\phi }_{\mathit{max}}^2\frac{(1-{\phi }_{\mathit{max}}\rho +2\rho )}{Z_{eq}{T}_c}+1$$

(36)

In the limit ρ → 0 or when K_c >> (mK₁ + nK₂), the maximum dimensionless current and temperature ratio are respectively reduced to:

$${\phi }_{\mathit{max}}=Z_{eq}{T}_c$$

(37)

$${\theta }_{h(\mathit{max})}=\frac{1}{2}{Z}_{eq}{T}_c+1$$

(38)

We highlight the fact that if the thermoelectric properties of the two modules are the same, i.e., α₁ = α₂ = α, R₁ = R₂ = R and K₁ = K₂ = K, then the equivalent figure of merit becomes the ordinary one, and from Equations (37) and (38) we obtain,

$${\phi }_{\mathit{max}}=Z{T}_c$$

(39)

$${\theta }_{h(\mathit{max})}=\frac{1}{2}Z{T}_c+1$$

(40)

Equations (39) and (40) have been previously obtained by many authors [5,6].

5. Conclusions

Using our proposed approach to analyze a composite TEC system, formed by two TEMs, with different thermoelectric properties each, connected thermally in parallel and electrically in series, equivalent thermoelectric properties have been derived. These equivalent properties depend on all thermoelectric parameters of the composite TEC system, namely Seebeck coefficients, α₍₁,₂₎, thermal conductivity, K₍₁,₂₎, the electrical resistance, R₍₁,₂₎, of each TEM and external thermal conductances K_(c,h). The corresponding dimensionless heat balance equations may be very useful for the design of composite thermoelectric systems, because they show the relation of the thermoelectric parameters of the TEM components of the system. The main parameters of the TEC system are external thermal conductances of heat exchangers, K_ch, and the figures of merit ratio, Z_r. The obtained results from this approach show the effect of two or more thermoelectric parameters on the COP and q_c of the system. In this work, it is shown that the maximum values for COP and q_c are determined by the external thermal conductances ratio K_ch, with the condition K_c < K_h. In general, we have shown (see Section 4) that our approach permits us to determine the optimal values of q_c and COP for different working conditions determined. The obtained results in this work are useful for designing composite TEC systems with thermal coupling, K_ch. We show the consistency of our approach obtaining the COP as a function of the input voltage for a commercial thermoelectric module connected electrically in series. Furthermore, we have derived results previously obtained by many authors as limit cases of our approach. Finally, we point out that the proposed approach in this work can be easily extended to include many TEMs with different thermoelectric properties, each one connected thermally in parallel and electrically in series.