Performance modeling of thermoelectric devices by perturbation method

A nonlinear differential heat conduction equation for a leg of a thermoelement with temperature-dependent thermoelectric material properties is reduced to an integral equation. Its solution is obtained by a perturbation method in the form of a series in terms of powers of parameters proportional to Thomson and Joule heat. The first six coefficients of the series and the heat balance equations, i.e. the dependence of heat fluxes at the junctions on its temperatures and the electric current are found. The calculation results of main energy characteristics of thermoelectric devices (thermoelectric generators and thermoelectric coolers) is compared by five methods: exact solution, solution at constant thermoelectric properties taken at a mean temperature of hot and cold junctions, method of average parameters (thermoelectric material properties averaged over the relevant temperature range by integration), linear and quadratic approximations of the perturbation theory. It is shown that the accuracy acceptable in real-world applications (error less than 1%) is obtained for thermoelectric generators by the method of average parameters and for thermoelectric coolers - by quadratic approximation of the perturbation method.