2. Thermoelectric effects

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 α 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 [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 ntype 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

290 Bringing Thermoelectricity into Reality

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 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:

$$V\_{ab} = \int\_{1}^{2} (\alpha\_b - \alpha\_a) dT \tag{1}$$

where the parameters α<sup>a</sup> and α<sup>b</sup> are the Seebeck coefficients for semiconductor materials a and b.

Figure 1. Single thermocouple model for a TEC system.

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

$$
\alpha\_{ab} = \frac{V}{\Delta T} \tag{2}
$$

3. Thermoelectric refrigeration in nonequilibrium thermodynamics

charge transport relations, consistent with the Onsager theory [17], are

el is the electric current density, j

∇ ! � κ ∇ ! T <sup>þ</sup> <sup>j</sup>

<sup>κ</sup>ð Þ <sup>T</sup> <sup>∂</sup><sup>2</sup> T ∂x<sup>2</sup> þ

∇ � j

j

j

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

Thermoelectric Cooling: The Thomson Effect in Hybrid Two-Stage Thermoelectric Cooler Systems with Different…

where, α is the Seebeck coefficient, T is the temperature, κ is the thermal conductivity, E is the

Equation (9) is the essential equation for thermoelectric phenomena. The governing equations

<sup>q</sup> ¼ j

el ¼ 0 and ∇ � j

2 <sup>r</sup> � <sup>T</sup> <sup>d</sup><sup>α</sup> dT <sup>J</sup> ! � ∇ !

where r is the electrical resistivity ð Þ r ¼ 1=σ and J 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 [18]. Now, from Equation (11),

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 α 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

� jT <sup>d</sup><sup>α</sup> dT ∂T <sup>∂</sup><sup>x</sup> ¼ � <sup>j</sup>

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

the equation that governs the system for one-dimensional steady state is given by:

∂T ∂x <sup>2</sup>

dκ dT

el ¼ σE � σα∇T (8)

<sup>q</sup> ¼ αTjel � κ∇T (9)

2

<sup>q</sup> is the heat flux and σ is the electric conductivity.

el � E (10)

http://dx.doi.org/10.5772/intechopen.75440

293

T ¼ 0 (11)

<sup>σ</sup>ð Þ <sup>T</sup> (12)

framework

and

are

electric field, j

3.1. Cooling power

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 ptype semiconductors, α is positive and in n-type semiconductors, α is 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

$$Q = \pi\_{ab}I \tag{3}$$

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

$$\frac{dQ}{d\mathbf{x}} = \pi I \frac{dT}{d\mathbf{x}}\tag{4}$$

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

$$
\tau\_a - \tau\_b = T \frac{d\alpha\_{ab}}{dT} \tag{5}
$$

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

$$
\pi\_{ab} = \alpha\_{ab} T \tag{6}
$$

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

$$
\pi = T \frac{d\alpha}{dT} \tag{7}
$$

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.
