Dimensionless mathematical model of a thermoelectric cooler: $\Delta T _{\operatorname{max}}$ mode

The thermal resistances on the cold and hot sides substantially affect the output characteristics of thermoelectric devices. A dimensionless mathematical model of a thermoelectric cooler, which makes it possible to calculate device parameters, such as the optimal thermal resistance ratio on the cold and hot sides as well as the optimal current taking into account the influence of thermal resistances, is presented. The maximal temperature difference $\Delta T_{\operatorname{max}}$ mode is considered. It is shown that the optimal cooler parameters are different for implementation of the $\Delta T_{\operatorname{max}}$ and $Q_{\operatorname{max}}$ modes. The determining factor for the $\Delta T_{\operatorname{max}}$ mode is the influence of the thermal resistance on the hot side, and the optimal current is 0.4–0.7 of the maximal current in most cases for the material with $ZT$ = 1. It is shown that an additional increase in $\Delta T_{\operatorname{max}}$ of a cooler is attained with a decrease in the thermal conductivity of the thermoelectric material due to a decrease in the influence of the thermal resistance on the hot side besides the effect from an increase in $ZT$. An increase in the length of thermoelectric legs has the same positive effect of an increase in $\Delta T_{\operatorname{max}}$ of a cooler, while a decrease in the leg length negatively affects $\Delta T_{\operatorname{max}}$.