One-dimensional modeling of a Peltier element

For dimensioning and optimum control of Peltier coolers and heat pumps, an accurate numerical description of the performance parameters under various operation conditions is required. Here, the situation for homogeneous bismuth antimony telluride based Peltier cooler material is discussed, using representative values of constant material parameters in comparison to real experimental data of the temperature dependence of the thermoelectric properties. For the case of a constant pellet cross section, given pellet length and neglecting heat transfer aside, the problem can be treated as one-dimensional. The relation between electric current density j, temperature difference ▵T, and absorbed cooling power or COP, respectively, along a single Peltier element have been considered by ab-initio calculations and plotted for the entire two-dimensional range (over j and ▵T) of relevant operating conditions, assuming constant material properties. Accordingly, the spatial temperature distribution inside the thermoelectric material has been calculated. Differential equations governing thermoelectric transports have been analytically and numerically solved by the software tool MATHEMATICA. This instrument is capable of solving the inhomogeneous second order differential equation for the temperature profile even for nonconstant coefficients with mixed boundary conditions provided. Thus, exact temperature profiles along the pellet can be easily calculated taking into account the correct temperature dependence of the material properties. The maximum temperature difference has been determined for arbitrarily given cooling power or COP, respectively, including zero temperature difference and adiabatic cold side (ΔT _max) cases. Evidence is provided for a very good quantitative agreement between ΔT_max values calculated using real temperature dependent material properties or volume averaged constant values