Theory of the Longitudinally Isothermal Ettingshausen Cooler

Abstract

The temperature distribution T(x), maximum temperature difference ΔTmax, and coefficient of performance ε are determined for an Ettingshausen cooler of rectangular cross section when the temperature is constant in the direction of current flow (longitudinal isothermal condition). The calculation is carried out for the two cases (1) temperature-independent transport coefficients and (2) the Nernst and resistivity coefficients proportional to T−1. In both cases the expressions for ΔTmax and ε correspond closely to those for the transversely isothermal Peltier cooler while the temperature distribution differs considerably. In the longitudinally isothermal Ettingshausen cooler the electric field is constant, in the contrast to the transversely isothermal Peltier cooler, in which the current density is constant. Recognition of this fact allows the validity of the results to be established for all possible values of the figure of merit. A value of ΔTmax measured on a Bi(97)Sb(3) cooler is compared with the value determined by substitution of independently measured values of the transport coefficients into the theoretical expression for ΔTmax. Although the temperature dependence of the transport coefficients is not taken into account exactly, the agreement is good. The effect of shaping the cooler (in effect, cascading) is discussed and experimental evidence for enhancement of cooler performance due to shaping is presented.