Performance optimization of a thermoelectric generator element with linear, spatial material profiles in a one-dimensional setup

Abstract

Graded and segmented thermoelectric elements are studied in order to improve the performance of thermogenerators that are exposed to a large temperature difference. The linear thermodynamics of irreversible processes is extended by assuming spatially dependent material parameters like the Seebeck coefficient, the electrical and thermal conductivities. For the particular case in which these transport coefficients exhibit a constant gradient, we present an analytical solution of the one-dimensional thermal energy balance in terms of Bessel functions. Given linear spatial material profiles, we discuss the optimization of performance parameters like the electrical power P_el and the efficiency η of a graded thermogenerator element of fixed length and fixed boundary temperatures. The results are compared with the constant properties model, i.e., physically and chemically homogeneous material, as a suitable reference for performance evaluation.

APPENDIX A: TRANSFORMATION OF THE THERMAL ENERGY BALANCE AND ITS SOLUTION EXPRESSED IN SPECIAL FUNCTIONS

The thermal energy balance Eg. (2b) can be written in another notation where it shows

((A1))

with the abbreviations

As one easily sees the b gives the appropriate slope, e.g., κ′(x) = b(κ). There is another form of writing the material profiles with the chosen abbreviations, e.g., κ = a(κ)ξ_κ + b(κ)x. To derive the Bessel equation from the (homogeneous part of the) thermal energy balance [see Eq. (A1)], we used the following substitution:

$$z = - 2\sqrt {{{ - jb\left( S \right)} \over {{b^2}\left( {\rm{\kappa }} \right)}}} \sqrt {{\rm{\kappa }}\left( x \right)} \equiv A\sqrt {{\rm{\kappa }}\left( x \right)} \quad,$$

((A2))

leading to T″(z) + z^–1T′(z) + T(z) = 0 for the homogeneous part of Eq. (A1) which is equivalent to a Bessel equation of order 0 (after multiplying with z²).

The complete inhomogeneous differential equation can be calculated to

((A3))

Some properties of Bessel functions are helpful in calculating the performance parameters,⁹⁷ f.i. J′₀(z) =–J₁(z), which leads in our case to

$${d \over {dx}}\left[ {{{\rm{J}}_0}\left( {A\sqrt {{\rm{\kappa }}\left( x \right)} } \right)} \right] = - {{Ab\left( {\rm{\kappa }} \right)} \over {2\sqrt {{\rm{\kappa }}\left( x \right)} }}{{\rm{J}}_1}\left( {A\sqrt {{\rm{\kappa }}\left( x \right)} } \right)\quad.$$

((A4))

The integral of the Bessel function gives

$$\int {{{\rm{J}}_0}\left( {A\sqrt {{\rm{\kappa }}\left( x \right)} } \right)dx = {{2\sqrt {{\rm{\kappa }}\left( x \right)} } \over {Ab\left( {\rm{\kappa }} \right)}}{{\rm{J}}_1}\left( {A\sqrt {{\rm{\kappa }}\left( x \right)} } \right)} \quad.$$

((A5))

Analogous relations are found for the Bessel function of the second kind Y₀(z).

Here, Si(x) and Ci(x) are the respective sine and cosine integral98, 99

((A6))

with Euler’s constant γ.

The function Li₂(z) is the dilogarithm function100 defined as

$$\matrix{ {{\rm{L}}{{\rm{i}}_2}\left( z \right) = \sum\limits_{k = 1}^\infty {{{{z^k}} \over {{k^2}}}} } & {{\rm{or}}} & {{\rm{L}}{{\rm{i}}_2}\left( z \right) \equiv \int\limits_z^0 {{{\ln \left( {1 - t} \right)} \over t}dt\quad.} } \cr } $$

((A7))

A more detailed discussion and a number of relations can be found in Refs. 56–60 and 97–101 and references therein.