Generalized formulas to express maximum temperature difference for two-stage thermoelectric cooling modules

Application of the two-stage modules has been widely considered to obtain a large temperature difference, required for devices such as a freezer, by thermoelectric cooling. Due to the complexity of the formulas for maximum parameters, however, design of the application devices in practical use has been limited. Generalized formulas have been derived to express the maximum temperature difference and maximum current at ΔT_max-mode for two-stage thermoelectric modules connected electrically in series or in parallel: ΔT_max=T_k(a/c)(M´-1/M´+1), I_max=2T _h(a/b)(e₁/R₁)(1/M´+1), and M´=(1+2T_k(ac/b)((Z₁)/(K₂/K₁ )))^1/2. Corresponding to these expressions, ΔT _max=T_h-(M´-1)/(Z₁)=T_h (M´-1)/(M´+1), I_max=(K₁)/(e₁)(M´-1)=2T_h(e ₁)/(R₁)(1)/(M´+1), and M´=(1+2·Z₁·T_k)^1/2 for one-stage modules