Unified theory for inhomogeneous thermoelectric generators and coolers including multistage devices

Abstract

A novel generalized Lagrange multiplier method for functional optimization with inclusion of subsidiary conditions is presented and applied to the optimization of material distributions in thermoelectric converters. Multistaged devices are considered within the same formalism by inclusion of position-dependent electric current in the legs leading to a modified thermoelectric equation. Previous analytical solutions for maximized efficiencies for generators and coolers obtained by Sherman [J. Appl. Phys. 31, 1 (1960)], Snyder [Phys. Rev. B 86, 045202 (2012)], and Seifert et al. [Phys. Status Solidi A 207, 760 (2010)] by a method of local optimization of reduced efficiencies are recovered by independent proof. The outstanding maximization problems for generated electric power and cooling power can be solved swiftly numerically by solution of a differential equation-system obtained within the new formalism. As far as suitable materials are available, the inhomogeneous TE converters can have increased performance by use of purely temperature-dependent material properties in the thermoelectric legs or by use of purely spatial variation of material properties or by a combination of both. It turns out that the optimization domain is larger for the second kind of device which can, thus, outperform the first kind of device.

Optimizing the electrical power output of a thermogenerator with the Gerstenmaier/Wachutka approach

“…It is found when the first variation of the integral disappears: where is the Lagrange function depending on the degrees of freedom of the problem. Contrary to the standard variational formulation, Gerstenmaier and Wachutka have used a generalized Euler–Lagrange formulation with the heat equation as side condition. For further calculations we write Eq.…”

“…Following Ref. , it is also advantageous to rewrite the heat equation in terms of the new variable …”

“…Since only two equations are available, optimization can only be performed for two functions out of jα ( x ), T ( x ), κ ( x ), z ( x ); just as in Ref. we choose whereas z ( x ), κ ( x ) can, on principle, be arbitrarily fixed . However, for reaching maximum power, κ ( x ) resp.…”

“…The TE material properties α , σ , and κ are in general temperature and position dependent quantities and measurable under certain constraints. Gerstenmaier and Wachutka have focused on decoupled dependencies, i.e ., either a temperature or spatial dependence of the material coefficients. Consequently they have considered optimization of the performance parameters in the “ x ‐domain” and in the “ T ‐domain”.…”

Inhomogeneous thermal conductivity enhances thermoelectric cooling

“…Indeed, the devices made of functionally graded TE materials (FGTM) with inhomogeneous transport properties was first proposed by Ioffe 13 in 1960 and then be widely studied by many researchers to enhance the device performance. [14][15][16][17][18][19] For example, Bian et al 20 found that an enhancement of (∆T) max can be achieved in FGTM with spatial-dependent Seebeck coefficient.…”