Properties of thermoelectric (TE) elements.

## 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 (*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 *COP* and 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 *a* and *b*, the voltage is given by:

where the parameters *a* and *b*.

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

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,

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

where *x* is a spatial coordinate and *T* the 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

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

### 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

These last effects have a relation to the Thomson coefficient,

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

and

where,

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

where

### 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 *COP* does 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

where

The rate of generation of heat per unit length from the Joule effect is

Using next boundary conditions:

where the subscripts *n* and *p* are for the n-type and p-type elements, respectively. From Equation (10), we find for the cooling power at the cold side

where

and the electrical resistance of the two legs in series is

### 3.2. Coefficient of performance

The total power consumption in the TEC system is

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

## 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,

### 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.

and for the second stage, with

where

with *COP,* is determined by

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,

where

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

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

With

Figure 3 shows the *COP* and the cooling power *COP* behaviour is influenced directly by the Thomson effect ratio of both stages. *COP* values increase when there is an increase in the ratio *I.* We must notice that for lower values of *COP* values are very closely one with another, with a maximum difference of 17% as compared an electric current value of 1 *A* with an electric current of 4 *A,* when *COP* values increase for all the different electric current values.

Similar behaviour, to what happens with the performance *COP*, happens for the cooling power *A* is 11 % higher compared with electric current values of 4 *A,* when

### 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

where

According to the continuity of the heat flow between both stages,

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

In the previous section, it is shown that *COP* increases for higher values of Thomson coefficient ratio between both stages. The behaviour of the *COP* for the case where two different electric currents flow in the system, shown in Figure 5, is now analysed. Three different values of Thomson coefficients, *COP* in function of the electric current ratio between both stages, *COP* value is obtained for higher values of the ratio *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

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

where

that is,

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

If we consider Seebeck coefficient independent of temperature, Thomson coefficient is negligible (

The resulting equation considering the Thomson effect is given by:

### 5.2. Geometric parameter between stages: area-length ratio (W = *w*1/*w*2)

Figure 6 shows a simple thermocouple with length, *L* and 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,

for the first and second stage, respectively.

We define the geometric parameter,

In terms of the geometric parameters,

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

For ideal equation,

Finally, we introduce the ratio, *M*, of the number of thermocouples in the first stage,

The total number of thermocouples,

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

The two different semiconductor materials and their properties are given in Table 3: Material

### 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: *COP* and *COP* and *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 *COP* and *COP* than material *COP* for material

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 *COP* and *COP* of material _{c} value in material

### 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

#### 5.5.1. Average system temperature, T m

A two-stage TEMC consists of

and for the second stage,

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

#### 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

#### 5.5.3. Analysis and coefficient of performance and cooling power (Q_{c})

Figure 10 shows COP and Q_{c} 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

#### 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 *COP* and *W* on *COP* and *COP*(*w*) and *w*), which turns out to be case (a) where material *w*. *COP* and *m* to be a constant value. Figure 11 (b) shows *COP* increases by 8.9% and

## 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,

## 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.