Properties of thermoelectric (TE) elements.
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.
- ideal equation (IE)
- Thomson effect
- two-stage micro-cooler
- Peltier effect
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 , 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 () values positives and negatives respectively, joined by a conductor metal with low value; 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 . 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 (
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
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 . 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
where the parameters and are the Seebeck coefficients for semiconductor materials
The differential Seebeck coefficient, under open-circuit conditions, is defined as the ratio of the voltage,
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, is positive and in n-type semiconductors, is negative . Peltier coefficient is equal to the rate of heating or cooling, , ratio at each junction to the electric current, . The rate of heat exchange at the junction is
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
When Seebeck coefficient is considered independent of temperature, Thomson coefficient will not be taken into account in calculations, is zero.
2.1. Thomson relations
Seebeck effect is a combination of the Peltier and Thomson effects . 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, , given by
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 , are
where, is the Seebeck coefficient, is the temperature, is the thermal conductivity, is the electric field, is the electric current density, is the heat flux and is the electric conductivity. Equation (9) is the essential equation for thermoelectric phenomena. The governing equations are
For one-dimensional model, from Equations (8) and (9), we get for the heat flux
where is the electrical resistivity and is 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 . Now, from Equation (11), the equation that governs the system for one-dimensional steady state is given by:
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 , and the metal that connects the p-type with the n-type leg has a low value, 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
where is the cross-sectional area, is the thermal conductivity, and is the temperature gradient. Heat is removed from the source at the rate
The rate of generation of heat per unit length from the Joule effect is . This heat generation implies that there is a non-constant thermal gradient
Using next boundary conditions: at and at , we get
where the subscripts
where . The thermal conductance of the two legs in parallel is
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 
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, , flows from the positive to the negative terminal [22, 23].
, , and are, respectively, cold junction side temperature, amount of heat that can be absorb and amount of heat rejected from stage 1 to 2 of TEC. , and are, 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 is the heat flow from stage 1 to stage 2, that is, , and is the average temperature in the system. For calculations, we use TDPM model  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. and are, respectively, the temperatures at the cold side junction for p-type and n-type legs in stage 1. and are, respectively, the temperatures at the hot side junction for p-type and n-type legs in stage 2 . Solving with next boundary conditions: and , we have for the first stage
and for the second stage, with and
where , , , for and when . 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 ,
with , for and , for ; ; and for . A general solution for the heat fluxes in two-stage system is found in  where Thomson effect is studied. The system’s coefficient of performance,
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, between stages, on performance. Now solving for , knowing that , from Equations (25) and (26)
where , , , and for . Once again we must notice the relationship that exists between both stages according to average temperature , which also depends on the Thomson effect.
4.2.1. Influence of Thomson effect on performance (COP) and cooling power ()
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 for material and and for material .
Figure 3 shows the
Similar behaviour, to what happens with the performance
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 is the electric current flow in stage 1 and is the electric current flow in stage 2.
According to the continuity of the heat flow between both stages, , from Equations (32) and (33), we solve for the average temperature,
The system’s coefficient of performance,
In the previous section, it is shown that
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 and , in accordance with Figure 2, we define the dimensionless temperature, , and the parameter as,
Dimensionless differential equation corresponding to Equation (12) is given by:
that is, is the relation between Thomson heat with thermal conduction. From Equation (38), if , we get the ideal equation (IE) when Thomson effect not considered. Dimensionless parameter, , is the relation between Joule heating to the thermal conduction, and the parameter , which is the ratio of temperature difference to the high junction temperature, defined as:
5.1. Cooling power: the ideal equation and Thomson effect ()
If we consider Seebeck coefficient independent of temperature, Thomson coefficient is negligible (), 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
The resulting equation considering the Thomson effect is given by:
5.2. Geometric parameter between stages: area-length ratio (W =
Figure 6 shows a simple thermocouple with length,
for the first and second stage, respectively.
We define the geometric parameter, , which allows us to determine the optimal geometric parameters of the stages, which is expressed as,
In terms of the geometric parameters, and , we get:
We have for the cooling power, in terms of the geometric parameters,and
For ideal equation, , and Thomson effect, , we have
Finally, we introduce the ratio,
The total number of thermocouples, , for both stages is given by,
5.3. Material properties considerations: CPM and TDPM models
The two different semiconductor materials and their properties are given in Table 3: Material , which is obtained from commercial module of laird and its properties were provided by the manufacturer , and material , , , where and 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
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:
Now, according to optimal electric current values, determined in the previous section, we show the effect of the semiconductor geometric parameters on the
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 . 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 and are used in the first and the second stage, respectively; and the inverse system (b) materials and are used in the first and the second stage, respectively.
5.5.1. Average system temperature,
A two-stage TEMC consists of and thermocouples in the first and second stages, respectively. The heat flux at the cold side, , and the heat flux at the hot side, , in the first stage, and for the second stage, and , respectively. Thus, heat flux equations in the first stage are ,
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 and element length of , with a total number of thermocouples of in 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 , 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 , 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, , 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, , as a function of the length, , 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 is 19.05% better than . 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
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, , 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 . 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.
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.