Open access peer-reviewed chapter

# Thermoelectric Cooling: The Thomson Effect in Hybrid Two- Stage Thermoelectric Cooler Systems with Different Leg Geometric Shapes

By Pablo Eduardo Ruiz-Ortega, Miguel Angel Olivares-Robles and Amado F. Garcia Ruiz

Submitted: October 30th 2017Reviewed: February 14th 2018Published: July 11th 2018

DOI: 10.5772/intechopen.75440

## Abstract

This chapter aims to analyse the performance of hybrid two-stage thermoelectric cooler systems [two-stage thermoelectric cooling devices (TEC)], which are composed of different thermoelectric materials in each stage with different leg geometric shapes. If we consider a temperature gradient inside a two-stage TEC, then, besides Joule heat, also Thomson heat has to be taken into account. We discuss the out-of-equilibrium thermodynamics equations of a one-dimensional model to provide the performance expressions that govern the system. TEC system performance is analysed in function of the Thomson coefficients ratio of both stages. We describe a recent geometric optimization procedure that includes leg geometry parameters such as ratio of cross-sectional area and length of legs for each stage of the two-stage TEC.

### Keywords

• ideal equation (IE)
• Thomson effect
• two-stage micro-cooler
• Peltier effect

## 1. Introduction

Thermoelectric cooling devices are based on the Peltier effect to convert electrical energy into a temperature gradient. Thermoelectric effects, such as Seebeck effect, Peltier effect and Thomson effect, result from the interference of electrical current and heat flow in various semiconductor materials [1], and its interaction allows to use thermoelectric effects to generate electricity from a temperature differential; conversely, cooling phenomena occurs when a voltage is applied across a thermoelectric material. Seebeck and Peltier effects depend on each other, and this dependence was demonstrated by W. Thomson who also showed the existence of a third thermoelectric effect, known as the Thomson effect. Thomson effect describes reversible heating or cooling, in a homogeneous semiconductor material, when there is both a flow of electric current and a temperature gradient [2, 3]. For thermoelectric cooling devices (TECs), a thermocouple consists of a p-type and n-type legs, with Seebeck coefficients (α) values positives and negatives respectively, joined by a conductor metal with low αvalue; in this chapter, we take this value as zero for calculations. Practical devices make use of modules that contain many thermocouples connected electrically in series and thermally in parallel [4]. TECs suffer from low efficiency, therefore, research on system geometry, for design and fabrication of thermoelectric cooling devices, is investigated in recent days [5, 6]. Coefficient of performance (COP) is the most important parameter for a thermoelectric cooling device, which is defined as the heat extracted from the source due an electrical energy applied [7]. Single-stage devices operate between a heat source and sink at a temperature gradient. However, multistage devices provide an alternative for extending the maximum temperature difference for a thermoelectric cooler. Therefore, two-stage coolers should be used to improve the cooling power, Qc, and COP. In recent days, multistage thermoelectric cooling devices have been developed as many as six stages with bismuth telluride-based alloys. Recent works have investigated the ratio of the TE couple number between the stages and the effects of thermocouple physical size and have found that the cooling capacity is closely related to its geometric structure and operating conditions [8, 9]. In this chapter, a thermodynamics analysis and optimization procedure on performance of two-stage thermoelectric cooling devices based on the properties of established materials, system geometry and energy conversion, is analysed. Energy conversion issues in thermoelectric devices can be solved according to material properties: by increasing the magnitude of the differential Seebeck coefficient, by increasing the electrical conductivities of the two branches, and by reducing their thermal conductivities [10]. Several new theoretical and practical methods for the improvement of materials have been put forward and, at last, it seems that significant advances are being made, at least on a laboratory scale. In this work, we consider temperature-dependent properties material (TDPM) systems in calculations to determine the influence of the Thomson effect on performance [11, 12]. Many investigations have been conducted to improve the cooling capacity of two-stage TEC and found that cooling capacity is closely related to geometric structure and operating conditions of TECs. Our analysis to optimize cooling power of a thermoelectric micro-cooler (TEMC) includes a geometric optimization, that is, different cross-sectional areas for the p-type and n-type legs in both stages [13]. We find a novel procedure based on optimal material configurations, using two different semiconductors with different material properties, to improve the performance of a TEMC device with low-cost production.

This chapter is organized as follows: in Section 2, we give an overview of the thermoelectric effects. In Section 3, we apply thermodynamics theory to solve thermoelectric systems, and consequently, a description of the operation of a TEC device is presented. In Section 4, we proposed a two-stage TEC model taken into account Thomson effect for calculations to show its impact on COPand Qc. In Section 5, geometric parameters, cross-sectional area (A), and length (L) of a proposed two-stage TEMC system is analysed. For this purpose, constant properties of materials (CPM) models and TDPM models are compared to show Thomson effect’s impact on performance. We consider two cases: (a) the same materials in both stages (homogeneous system) and (b) different materials in each stage (hybrid system). We establish optimal configuration of materials that must be used in each stage. Finally, in Section 6, we present a discussion and concluding remarks.

## 2. Thermoelectric effects

Thermoelectricity results from the coupling of Ohm’s law and Fourier’s law. Thermoelectric effects in a system occur as the result of the mutual interference of two irreversible processes occurring simultaneously in the system, namely heat transport and charge carrier transport [14]. To define Seebeck and Peltier coefficients, we refer to the basic thermocouple shown in Figure 1, which consists of a closed circuit of two different semiconductors. For a thermocouple composed of two different materials aand b, the voltage is given by:

where the parameters αaand αbare the Seebeck coefficients for semiconductor materials aand b.

The differential Seebeck coefficient, under open-circuit conditions, is defined as the ratio of the voltage, V, to the temperature gradient,ΔT

αab=VΔTE2

Electrons move through the n-type element towards the positive pole, attraction effect, while the negative pole of the voltage source repels them. Likewise, in the p-type semiconductor, the holes move to the negative potential of the voltage source, while positive potential acts as repel of the holes and they move in the contrary direction to the flow of electrons. As a result, in p-type semiconductors, αis positive and in n-type semiconductors, αis negative [15]. Peltier coefficient is equal to the rate of heating or cooling, Q, ratio at each junction to the electric current, I. The rate of heat exchange at the junction is

Q=πabIE3

Peltier coefficient is regarded as positive if the junction at which the current enters is heated and the junction at which it leaves is cooled. When there is both an electric current and a temperature gradient, the gradient of heat flux in the system is given by

dQdx=τIdTdxE4

where xis a spatial coordinate and Tthe temperature. Thomson coefficient, known as the effect of liberate or absorb heat due to an electric current flux through a semiconductor material in which exist a temperature gradient, is given by the Kelvin relation as follows

τaτb=TdαabdTE5

When Seebeck coefficient is considered independent of temperature, Thomson coefficient will not be taken into account in calculations, τis zero.

### 2.1. Thomson relations

Seebeck effect is a combination of the Peltier and Thomson effects [16]. The relationship between temperature, Peltier, and Seebeck coefficient is given by the next Thomson relation

πab=αabTE6

These last effects have a relation to the Thomson coefficient, τ, given by

τ=TdTE7

To develop an irreversible thermodynamics theory, Thomson's theory of thermoelectricity plays a remarkable role, because this theorem is the first attempt to develop such theory.

## 3. Thermoelectric refrigeration in nonequilibrium thermodynamics framework

Theory of thermoelectric cooling is analysed according to out-of-equilibrium thermodynamics. Under isotropic conditions, when an electrical current density flows through the semiconductor material with a temperature gradient and steady-state condition, the heat transport and charge transport relations, consistent with the Onsager theory [17], are

jel=σEσαTE8

and

jq=αTjelκTE9

where, αis the Seebeck coefficient, Tis the temperature, κis the thermal conductivity, Eis the electric field, jelis the electric current density, jqis the heat flux and σis the electric conductivity. Equation (9) is the essential equation for thermoelectric phenomena. The governing equations are

jel=0andjq=jelEE10

For one-dimensional model, from Equations (8) and (9), we get for the heat flux

κT+j2ρTdTJT=0E11

where ρis the electrical resistivity ρ=1/σand Jis the electric current density. In Equation (11), the first term describes the thermal conduction due to the temperature gradient. According to Fourier’s law, the second term is the joule heating and the third term is the Thomson heat, both depending on the electric current density [18]. Now, from Equation (11), the equation that governs the system for one-dimensional steady state is given by:

κT2Tx2+dTTx2jTdTTx=j2σTE12

### 3.1. Cooling power

Thermoelectric coolers make use of the Peltier effect which origin resides in the transport of heat by an electric current. For this analysis, we assume that thermal conductivity, electrical resistivity, and Seebeck coefficient are all independent of temperature, that is, CPM model [19], and the metal that connects the p-type with the n-type leg has a low αvalue, therefore it is considered as zero. We assume that there is zero thermal resistance between the ends of the branches and the heat source and sink. Thus, only electrical resistance is considered for the thermocouple legs, thereby, the thermocouple legs are the only paths to transfer heat between the source and sink, conduction via the ambient, convection, and radiation are ignored. These considerations have been addressed in previous studies showing that the COPdoes not depend on the semiconductors length when the electrical and thermal contact resistances are not considered in calculations [20]. To determine the coefficient of performance (COP), which is defined as the ratio of the heat extracted from the source to the expenditure of electrical energy, a thermocouple model shown in Figure 1 is used. Thus, for the p-type and n-type legs, the heat transported from the source to the sink is

Qp=αpITKpApdTdx;Qn=αnITKnAndTdxE13

where Ais the cross-sectional area, Kis the thermal conductivity, and dT/dxis the temperature gradient. Heat is removed from the source at the rate

Qc=Qp+Qnx=0E14

The rate of generation of heat per unit length from the Joule effect is I2ρ/A. This heat generation implies that there is a non-constant thermal gradient

κpApd2Tdx2=I2ρpAp;κnAnd2Tdx2=I2ρnAnE15

Using next boundary conditions: T=T1at x=0and T=T2at x=L, we get

κn,pAn,pdTdx=I2ρn,pxLn,p/2An,p+κn,pAn,pT2T1Ln,pE16

where the subscripts nand pare for the n-type and p-type elements, respectively. From Equation (10), we find for the cooling power at the cold side x=0

Qc=αpαnIT1KΔT12I2RE17

where ΔT=T2T1. The thermal conductance of the two legs in parallel is

K=κpApLp+κnAnLnE18

and the electrical resistance of the two legs in series is

R=LpρpAp+LnρnAnE19

### 3.2. Coefficient of performance

The total power consumption in the TEC system is

W=αpαnIΔT+I2RE20

then, the coefficient of performance in a TEC system is defined as [21]

COP=QcWE21

## 4. Thomson effect impact on performance of a two-stage TEC

### 4.1. One-dimensional formulation of a physical two-stage TEC

To determine analytical expressions of cooling power and coefficient of performance in a two-stage TE system, we establish one-dimensional representation model, as shown in Figure 2. When a voltage is applied across the device, as a result, an electric current, I, flows from the positive to the negative terminal [22, 23].

T1=Tc, Qc1, and Qh1are, respectively, cold junction side temperature, amount of heat that can be absorb and amount of heat rejected from stage 1 to 2 of TEC. T2=Th, Qh2and Qc2are, respectively, hot junction temperature, amount of heat rejected to the heat source and amount of heat absorbed from stage 1. It should be noted that Qmis the heat flow from stage 1 to stage 2, that is, Qm=Qh1=Qc2, and Tmis the average temperature in the system. For calculations, we use TDPM model [24] in order to show Thomson effect’s role in the system. Arranging pairs of elements in this way allows the heat to be pumped in the same direction.

### 4.2. TEC electrically connected in series

Considering model from Figure 2, we get temperature distributions for p-type and n-type semiconductor legs in each stage. T11and T12are, respectively, the temperatures at the cold side junction for p-type and n-type legs in stage 1. T21and T22are, respectively, the temperatures at the hot side junction for p-type and n-type legs in stage 2 [25]. Solving with next boundary conditions: T110=T120=T1and T11L11=T12L12=Tm, we have for the first stage

T112=T1mA112x+ΔT±A112L1121eω112L1121eω112x,0xL112E22

and for the second stage, with T21L11=T22L12=Tmand T21L21=T22L22=Th

T212=Tm2A212x+ΔT±A212L2121eω212L2121eω212x,L112xL212E23

where ωij=τijIKijLij, Aij=RijIτijLij, Kij=κijSijLij, Rij=LijσijSijfor i=1,2and j=1,2when ij. The subscripts 1 and 2 describe cold temperature and hot temperature in the junctions. According to the theory of non-equilibrium thermodynamics, for the TEMC, we have for the first stage [26],

Qc1=α1cTcIK1TmTcR1+R1I2E24
Qh1=α1mTmIK1TmTcτ1TmTcR1I2E25
Qc2=α2mTmIK2ThTmR2+R2I2E26
Qh2=α2hThIK2ThTmτ2ThTmR2I2E27

with α1k=α12kα11k, for k=c,mand α2l=α22lα21l, for l=m,h; Rj=Rj1+Rj2; τj=τj2τj1and Rj=Rj1+Rj2Rj1+Rj2for j=1,2. A general solution for the heat fluxes in two-stage system is found in [27] where Thomson effect is studied. The system’s coefficient of performance, COP,is determined by Qcand Qhas follows

COP=α2hThIK2ThTmτ2ThTmR2I2K1TmTc+ThTmτ2IK2+α2hThα1cTcI+R1+R1R2I2E28

Performance depends on Thomson coefficients values of both the first stage and the second stage. In our results, we show the role of the ratio values of the Thomson coefficients, τr=τ1/τ2between stages, on performance. Now solving for Tm, knowing that Qm=Qh1=Qc2, from Equations (25) and (26)

Tm=I2R1R2R2TcK1+τ1IK2ThIα1mα2mτ1K1+K2E29

where Kj=Kj1+Kj2, Kj1=τj1I1eωj1Lj1, Kj2=τj2Ieωj2Lj21, Rj1=Rj111eωj1Lj11ωj1Lj1and Rj2=Rj21ωj2Lj21eωj2Lj21for j=1,2. Once again we must notice the relationship that exists between both stages according to average temperature Tm, which also depends on the Thomson effect.

#### 4.2.1. Influence of Thomson effect on performance (COP) and cooling power (Qc)

Two different materials were used for calculations, thermoelectric properties are shown in Table 1, where only Seebeck coefficient is consider that depends on temperature.

### Table 1.

Properties of thermoelectric (TE) elements.

With α1=2×104+2×1021/Tm1/Tfor material Bi2Te3and α2=62675+1610T2.3T2×106and for material Bi0.5Sb0.5Te3[25].

Figure 3 shows the COPand the cooling power Qc, in function of τr, at different electric current values Bi2Te3and Bi0.5Sb0.5Te3[28]. It is clear that COPbehaviour is influenced directly by the Thomson effect ratio of both stages. COPvalues increase when there is an increase in the ratio τrfor lower values of the electric current I.We must notice that for lower values of τr<1, COPvalues are very closely one with another, with a maximum difference of 17% as compared an electric current value of 1 Awith an electric current of 4 A,when τr=1. Moreover, it is observed that from values of τr>2COPvalues increase for all the different electric current values.

Similar behaviour, to what happens with the performance COP, happens for the cooling power Qc, where maximum values are obtained for higher values of τr, as shown in Figure 3. In this case, Qcvalue for an electric current value of 1 Ais 11 % higher compared with electric current values of 4 A,when τr=1.

### 4.3. TECs electrically connected in parallel

Now, we analyse the case in which different electric currents flow in each stage of the system (Figure 4). The ratio of electric currents between each stage is given by

Ir=I1I2E30
Qc=α1cTcI1K1TmTcR1+R1I12E31
Qm1=α1mTmI1K1TmTcτ1TmTcR1I12E32
Qm2=α2mTmI2K2ThTmR2+R2I22E33
Qh=α2hThI2K2ThTmτ2ThTmR2I22E34

where I1is the electric current flow in stage 1 and I2is the electric current flow in stage 2.

According to the continuity of the heat flow between both stages, Qm1=Qm2, from Equations (32) and (33), we solve for the average temperature, Tm

Tm=R1I12τ1TcI1K1TcK2ThR2+R2I22I1α1mτ1K1+K2α2mI2E35

The system’s coefficient of performance, COP, is given by

COP=α2hThI2K2ThTmτ2ThTmR2I22α2hThI2K2ThTmτ2ThTmR2I22α1cTcI1K1TmTcR1+R1I12E36

In the previous section, it is shown that COPincreases for higher values of Thomson coefficient ratio between both stages. The behaviour of the COPfor the case where two different electric currents flow in the system, shown in Figure 5, is now analysed. Three different values of Thomson coefficients, τr, are considered. Table 2 shows maximum values, from Figure 5, for COPin function of the electric current ratio between both stages, Ir. Maximum COPvalue is obtained for higher values of the ratio τr, that is, a higher value of the electric current I2>I1is desirable to be able to achieve better COP.

### Table 2.

Maximum values of COP.

## 5. Dimensionless equations of a two-stage thermoelectric micro-cooler

Once it has been investigated the role of the Thomson heat on TEC performance, now a procedure to improve the performance of the micro-cooler based on optimum geometric parameters, cross-sectional area (A) and length (L), of the semiconductor elements is proposed. To optimal design of a TEMC, theoretical basis on optimal geometric parameters (of the p-type and n-type semiconductor legs) is required. Next analysis of a TEMC includes these optimization parameters. The configuration of a two-stage TE system considered in this work is shown in Figure 2. Each stage is made of different thermoelectric semiconductor materials. In order to make Equation (12) dimensionless using the boundary conditions T0=T1and TL=T2, in accordance with Figure 2, we define the dimensionless temperature, θ, and the ξparameter as,

θ=TT1T2T1 and ξ=xLE37

Dimensionless differential equation corresponding to Equation (12) is given by:

d2θdξ2βθ1ϕ+1+γ=0E38

where

β=IT2dTΔTΔTLE39

that is, βis the relation between Thomson heat with thermal conduction. From Equation (38), if β=0, we get the ideal equation (IE) when Thomson effect not considered. Dimensionless parameter, γ, is the relation between Joule heating to the thermal conduction, and the parameter ϕ, which is the ratio of temperature difference to the high junction temperature, defined as:

γ=I2RΔTLandϕ=ΔTT2E40

### 5.1. Cooling power: the ideal equation and Thomson effect (τ)

If we consider Seebeck coefficient independent of temperature, Thomson coefficient is negligible (β=0), we can obtain the exact result for the cooling power at the cold junction from Equation (14), which is called the ideal equation (IE) for cooling power

Q̇cIE=α¯T1I12I2RAkLT2T1E41

The resulting equation considering the Thomson effect is given by:

Q̇cβ=α¯T1I12I2RAkLT2T1+βAkLT2T1E42

### 5.2. Geometric parameter between stages: area-length ratio (W = w1/w2)

Figure 6 shows a simple thermocouple with length, Land cross-sectional area, A. Previous studies proved that an improvement on performance of TECs can be achieved by optimizing geometric size of the semiconductor legs [29, 30]. A geometric parameter, ω, is defined as the area-length ratio of the legs in the thermocouple in each stage of the TEMC

ω1=A1L1andω2=A2L2E43

for the first and second stage, respectively.

We define the geometric parameter, W, which allows us to determine the optimal geometric parameters of the stages, which is expressed as,

W=ω1ω2E44

In terms of the geometric parameters, ω1and ω2, we get:

R=Rp+Rn=LpσpAp+LnσnAn=1σpω1+1σnω2E45
K=Kp+Kn=ApkpAp+AnknAn=ω1kp+ω2knE46

We have for the cooling power, in terms of the geometric parameters,ω1andω2

Q̇cIE=αTavgT1I12I21σpω1+1σnω2ω1kp+ω2knT2T1E47

For ideal equation, Q̇cIE, and Thomson effect, Q̇cβ, we have

Q̇cβ=αTavgT1I12I21σpω1+1σnω2ω1kp+ω2knT2T1+βω1kp+ω2knT2T1E48

Finally, we introduce the ratio, M, of the number of thermocouples in the first stage, n1, to the number of thermocouples in second stage,n2

M=n1n2E49

The total number of thermocouples, N, for both stages is given by,

N=n1+n2E50

### 5.3. Material properties considerations: CPM and TDPM models

The two different semiconductor materials and their properties are given in Table 3: Material M1, which is obtained from commercial module of laird CP1012705and its properties were provided by the manufacturer [21], and material M2, Bi0.5Sb0.52Te3[17], Tavg=T1+T2/2, where α¯=αTavgand Seebeck coefficients are dependent on temperature while the electrical resistivity and the thermal conductivity are constant. The sign of n-type elements coefficient is negative while the sign of p-type element coefficients is positive for the Seebeck coefficients values. Then, for materials 1 and 2, we have next equations

α1=0.2068T+138.78×106andα2=62675+1610T2.3T2×106E51

### Table 3.

Properties of thermoelectric (TE) elements used in the TEMC device.

### 5.4. Special case: single-stage TEMC performance analysis

In this section, we analyse a single-stage system to compare with two-stage system to show the differences between both systems. Thereby, we calculate the two important parameters: COPand Qcversus electric current; and COPand Qcversus geometric parameter (w), for both materials. CPM models are compared with TDPM model, for this purpose, in all figures are shown results obtained considering Thomson effect (solid lines) and results using the ideal equation (dashed lines). Figure 7 shows the COP1, Qc,1and COP2, Qc,2for materials, M1and M2respectively, as a function of the electric current. The maximum values of COPand Qcare reached when the Thomson effect is considered, better cooling power is obtained with lower values of β. Results show that material M1achieves higher values of cooling power Qcand COPthan material M2. The COPfor material M1is 15.1% more than for material M2and Qcfor material M1is 40.12% more than for materialM2.

Now, according to optimal electric current values, determined in the previous section, we show the effect of the semiconductor geometric parameters on the COP(w) and Qc(w) of the system. Figure 8 shows that, for COPand Qc, material M1has better results in both cases than material M2. The COPof material M1is 21.18% higher than that for material M2and the Qc value in material M1is 14.85% higher than for materialM2.

### 5.5. Hybrid two-stage TEMC system

Now, we analyse a hybrid two-stage TEMC, that is, a system with a different thermoelectric material in each stage. Homogeneous system can also be analysed, this can be achieved by placing the same materials in both stages, as is shown in [27]. We focus on analysing two-stage hybrid systems, where two temperature gradients are generated and, therefore we must analyse which material works better in each stage. Thus, we determine the optimum thermoelectric material arrangement for the best performance of the TEMC system. For this purpose, two configurations of materials in the hybrid two-stage TEMC model are considered: (a) materials M1and M2are used in the first and the second stage, respectively; and the inverse system (b) materials M2and M1are used in the first and the second stage, respectively.

#### 5.5.1. Average system temperature, Tm

A two-stage TEMC consists of n1and n2thermocouples in the first and second stages, respectively. The heat flux at the cold side, Qc1, and the heat flux at the hot side, Qh1, in the first stage, and for the second stage, Qc2and Qh2, respectively. Thus, heat flux equations in the first stage are [31],

Qc1=n1α1ITc1K1TmTc11/2R1I2+τ1ITmTc1E52
Qh1=n1α1ITmK1TmTc1+1/2R1I2τ1ITmTc1E53

and for the second stage,

Qc2=n2α2ITmK2Th2Tm1/2R2I2+τ2ITh2TmE54
Qh2=n2α2ITh2K2Th2Tm+1/2R2I2τ2ITh2TmE55

For a hybrid system (different materials in each stage), from equations (53) and (54), we obtain the temperature between stages, Tm,

Tm=12I2R1n1+R2n2+12Iτ2n1Th2τ1n1Tc1K2Th2n1K1Tc1n1)In1α112τ1+In212τ2α2K1n1K2n2E56

#### 5.5.2. Dimensionless temperature distribution

For the hybrid two-stage TEMC system, the best configuration of semiconductor thermoelectric materials and its optimal geometric parameters is found in this section. For calculations we use a cross-sectional area of Ac=4.9×109m2and element length of L=15μm, with a total number of thermocouples of n1=n2=100in the first and second stages, respectively. Figure 9 shows the dimensionless spatial temperature distributions, for cases (a) and (b) mentioned earlier. An important factor to analyse in the graphic is the maximum values of the temperature distribution in each stage. When the value of the derivative is to be />0, the semiconductor material is able to absorb a certain amount of heat, that is, Thomson heat acts by absorbing heat. For the case when the value of the derivative is to be /<0, a release of heat occurs in the semiconductor, that is, Thomson heat acts by liberating heat. From Figure 9, maximum temperature distribution values in stage 1, θ=1.06, is near to the junction with stage 2, which is desirable because in the first stage, the system must absorb higher amount of heat to later be released in stage 2. Thereby, dimensionless temperature distribution, θ, as a function of the length, ξ, shows that a lower temperature distribution is required in the first stage and that higher values of temperature distribution are required in the second stage; this is achieved by choosing the optimal arrangement of materials between the two stages. According to this last statement, case (a) is the best configuration of materials to improve the TEMC.

#### 5.5.3. Analysis and coefficient of performance and cooling power (Qc)

Figure 10 shows COP and Qc for the TEMC system for cases (a) and (b) described previously. Case (a) reaches best cooling power and coefficient of performance values. Notice that the COPmaxβais 19.05% better than COPmaxβb. It is clear from the graphic that for the same current values, cooling power values for the case (a) are always over those of the case (b).

#### 5.5.4. Optimization analysis according to the geometric parameter W

In this section, we analyse the physical sizes, length and the cross-sectional area of the thermocouples, when the two stages are related each other. We present an optimization procedure of a two-stage TEMC system, on COPand Qc, by introducing a geometric parameter, W=ω1/ω2. The effect of the parameter Won COPand Qcis analysed when (1) ω1=ω2and (2) when ω1ω2. Figure 11 (a) shows best optimal material configuration for COP(w) and Qc(w), which turns out to be case (a) where material M1is placed in the first stage and material M2in the second stage. Results proved that, higher area-length ratio values do not improve Qc, on the contrary, the cooling power improves for lower values of w. COPand Qcincreases by 19 and 10.5%, respectively, from case (a) to case (b). The most relevant case, geometric parameters ω1ω2, is analysed. In this case, we set ω2=3.26×104mto be a constant value. Figure 11 (b) shows COPWand Qcωwhere it is noted that COPincreases by 8.9% and Qcincreases 6.27% in case (a) compared with case (b). From this last result, it is important to note that although the performance of TEC systems is affected by combination of different materials, it is also affected by the material configuration and the system geometry as well. These results offer a novel alternative in the improvement of thermoelectric systems, when they are used as coolers. Results shown in this chapter are based only on theory of thermoelectricity to optimize a TEMC system, according to geometric parameters. However, parameters as length and cross-sectional area of the semiconductor elements are based on studies which validated similar results with experimental data [32, 33]. In micro-refrigeration, an important problem is the fact of handle heat flux in a small space and it has been proved that thermal interface resistance has beneficial or detrimental effects on cooling performance [34]. For calculation, contact resistances between stages are not considered, since it is known that thermal resistances exist in the interfaces, which are large when the cross-sectional areas are very dissimilar in the stages and negligible for similar cross-sectional areas [35, 36]. Present work can be useful as theoretical basis for future research in the experimental area, development and design of thermoelectric multistage coolers.

## 6. Conclusions

In this chapter, Thomson effect and leg geometry parameters on performance in a hybrid two-stage TEC were evaluated. For this purpose, the basics of two-stage thermoelectric cooler devices are analysed according to one dimension out-of-equilibrium thermodynamics using TDPM model. Two different semiconductor materials were used in all calculations. Results show, Thomson effect leads to a slight improvement on the performance and when the ratio of Thomson coefficients between both stages, τr=τ1/τ2, increases, more cooling power can be achieved. We show that it is convenient to analyse optimal configuration of materials that must be used in each stage, showing that the material with a higher value of Seebeck coefficient must be place in the first stage. The main interest is to improve cooling power, thereby, a new procedure based on optimum leg geometric parameters of the semiconductor elements, is presented. Our analysis shows that, hybrid system reaches maximum cooling power, 15.9% greater than the one-stage system, for the case when the geometric parameter is ω1ω2. An important advantage of this work is that result can be confirmed in laboratories, as prototypes are made by mainly using bismuth telluride, which is the basis of the materials we use in all calculations.

## Acknowledgments

This work was financially supported by research grant 20180069 of Instituto Politecnico Nacional, México. Pablo Eduardo Ruiz Ortega was financially supported by CONACyT-Mexico (CVU No. 490910). The authors acknowledge the editorial assistance in improving the manuscript.

## Conflict of interest

The authors declare no conflict of interest.

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© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Pablo Eduardo Ruiz-Ortega, Miguel Angel Olivares-Robles and Amado F. Garcia Ruiz (July 11th 2018). Thermoelectric Cooling: The Thomson Effect in Hybrid Two- Stage Thermoelectric Cooler Systems with Different Leg Geometric Shapes, Bringing Thermoelectricity into Reality, Patricia Aranguren, IntechOpen, DOI: 10.5772/intechopen.75440. Available from:

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