2. Review of numerical methods applied to thermoelectricity

Despite the fact that there exists a large amount of works that use simplified analytical models [8–10] and semi-analytical techniques such as Laplace transforms [11], numerical modeling of thermoelectricity is a fundamental aspect in order to design and optimize sophisticated devices made out of thermoelectric materials. In this connection, there are at least five numerical methods applied to solve the set of two coupled partial differential equations (PDE) of thermoelectricity. For more details, readers are referred to [12], in which some of these numerical methods are compared and discussed.

• Network or circuit simulator method, which can be considered as a representation of the natural convection and uses the Kirchhoff's theorems. The main characteristic of this method is the discretization of the spatial coordinates while time is assumed to be a continuous variable. For instance, the method developed in [13] considers the Thomson effect, which is often neglected in the models for simplicity, and also the temperaturedependency of all material parameters.


2. Review of numerical methods applied to thermoelectricity

ical methods are compared and discussed.

Symbol Description

r ¼ r f <sup>q</sup>; r<sup>m</sup>

J ¼ j; j s

ga <sup>¼</sup> <sup>V</sup><sup>~</sup> <sup>a</sup>; <sup>T</sup>~a; ~\_

Ta

272 Bringing Thermoelectricity into Reality

Table 1. Table of symbols.

Despite the fact that there exists a large amount of works that use simplified analytical models [8–10] and semi-analytical techniques such as Laplace transforms [11], numerical modeling of thermoelectricity is a fundamental aspect in order to design and optimize sophisticated devices made out of thermoelectric materials. In this connection, there are at least five numerical methods applied to solve the set of two coupled partial differential equations (PDE) of thermoelectricity. For more details, readers are referred to [12], in which some of these numer-

Theoretical formulation

c, α, κ, γ Heat capacity, Seebeck coefficient, and thermal and electric conductivities Finite element formulation

Ω, Ω<sup>∞</sup>, Γ, n Domain, surrounding, boundary and outward normal

n o Free electric charge and mass density

Ωe, Γ<sup>e</sup> Subdomain and boundary of any finite element R, K, C Residual, tangent stiffness and capacity matrices

Applications

� � Electric and entropy fluxes

d, δ Exact differential and variation t, s, T, V time, entropy, temperature and voltage q, x, Heat flux and Cartesian coordinates

n o Degrees of freedom

a, b Global numbering of elements k Newton-Raphson counter c1, c<sup>2</sup> Time integration parameters

Tc, Th Cold and hot temperatures

Qc , Qh Extracted heat at cold and hot ends I, Z Intensity and figure-of-merit

Ξ ¼ Σ<sup>s</sup> Entropy production

• Network or circuit simulator method, which can be considered as a representation of the natural convection and uses the Kirchhoff's theorems. The main characteristic of this method is the discretization of the spatial coordinates while time is assumed to be a • FE method [19] is the most advanced technique for solving PDE in science and engineering. Furthermore, this method is optimal for applying to nonlinear coupled problems as thermoelectricity. In this connection, the authors of the present chapter have extensive experience in developing thermodynamic consistent FE formulations, which are implemented in the research code FEAP [20], see [21–25] and [26]. Regarding commercial FE codes, the authors in [27, 28] use ANSYS to perform a comprehensive numerical analysis focusing on the cooling performance of miniaturized thermoelectric coolers for microelectronics applications. In particular, they study the effects of parameters such as the load current, geometric size. Finally, the software COMSOL is used in [29] to study the thermal stresses in Peltier cells.

In short, the main disadvantage of the boundary element is, as commented, its intrinsic difficulty to solve nonlinear problems. On the contrary, all the other methods can be applied to solve nonlinear thermoelectricity. The FE and finite volume methods require of complex mathematical developments and they are commonly used by the continuum mechanics community. In contrast, the finite difference method is the most direct approach to numerically solve PDE. Finally, the circuit method is preferably used by the electric engineering community.

Joule who formed the basis of the first law of thermodynamics, since he established that the various forms of energy (mechanical, electrical and heat) were in essence the same and could be changed. Other relevant contribution emerged in 1852 when Joule and Thomson discovered the Joule-Thomson effect, which is commonly exploited in thermal machines and played a crucial role in the development of the

Computational Thermoelectricity Applied to Cooling Devices

http://dx.doi.org/10.5772/intechopen.75473

275

The term continuum physics refers to several branches of physics such electromagnetism and nonequilibrium thermodynamic for which the matter consists of material points that are localized by the Euclidean position vector x. This fact is mathematically grounded on the continuum hypothesis [30], which allows to work with balance equations that state the conservation, production or annihilation of certain quantities such electric charge and specific

Consider a closed system of domain Ω, boundary Γ with outward normal n and its surrounding Ω<sup>∞</sup>. A general expression of a balance equation for the quantity ϒ at time t and in any

where r, J and Ξ denote the ϒ-density, ϒ-current density and ϒ-production, respectively. This

• Inflow quantity from Ω<sup>∞</sup> to Ω through the boundary Γ, first term on the right-hand side

In thermoelectricity, both electric and thermal fields are present and, consequently, there are two ϒ quantities as shown in Table 2. As observed, the quantities are the free electric charge r

and the specific entropy s; the current densities are the electric j and entropy j<sup>s</sup> fluxes. Regarding the production terms, there is not production for the electric field due to the fact that the

Quantity ϒ Current density J Production Ξ

Specific entropy rms Entropy flux j<sup>s</sup> Entropy production Σ<sup>s</sup>

<sup>q</sup> Electric flux j – –

∇ � Jð Þ ϒ; x; t dΩ þ

ð Ω

Ξð Þ ϒ; x; t dΩ , (1)

f q

Ω

• Production/annihilation of ϒ inside Ω, second term on the right-hand side.

refrigeration industry in the nineteenth century.

<sup>r</sup>ð Þ <sup>ϒ</sup>; <sup>x</sup>; <sup>t</sup> <sup>d</sup><sup>Ω</sup> ¼ �<sup>ð</sup>

• The quantity of ϒ inside Ω, term on the left-hand side.

(after application of divergence theorem).

f

Table 2. Quantities to be balanced in thermoelectricity.

equation balances the quantity ϒ since it contains information on:

3.1. Balance of electric charges and entropy

entropy.

material point x is given by: d d t ð Ω

Free electric charge r

Despite the difficulty of the FE method, it is used in the present chapter since it presents two main advantages: i) It is a very general an efficient method, and ii) it is easy to improve the accuracy of the solutions by two approaches: mesh refinement and/or increasing the order of the interpolation functions.
