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The Peltier effect is heat transfer associated with an electric current crossing a junction between materials with different Peltier coefficients. At fixed junction temperature, its contribution changes sign when the conventional current reverses. A junction can absorb heat from its surroundings for one direction and release it for the other. Joule heating, proportional to the square of the electric current, accompanies this transport but does not reverse sign. Consequently, observing a warm device alone does not identify the Peltier effect.

This makes the Peltier effect useful as an electrically controlled heat pump: reversing the supply can switch an object's thermal role without changing any plumbing. In thermoelectric cooling, however, the useful cooling is the junction's absorbed heat minus returning heat conduction and its share of Joule heating. Increasing electric current eventually makes cooling worse. The distinction between transported heat and net cooling is the central practical lesson of the thermoelectric cooler energy balance.

At fixed face temperatures, differentiation of the thermoelectric cooler energy balance givesThus current chosen for maximum cooling power is not automatically current for maximum coefficient of performance. Cooling exists only when $(\alpha T_c{)}^2>2RK\Delta T$, and then only between the two positive rootsAt equality the parabola merely touches $Q_c=0$. For the toy module the interval is approximately $0.767<I<10.433\mathrm{A}$, with $I_{\ast }=5.60\mathrm{A}$ and $Q_{c,\max }=23.36\mathrm{W}$. At $12\mathrm{A}$, the object is heated even in the nominal cooling polarity: $Q_c=-17.6\mathrm{W}$. This mathematical interval supplies no permission to exceed an actual device's current or temperature ratings.

The fixed-face constraint matters. With fixed $T_h$ but a freely cooling object, $T_c$ must be solved from its heat load and leaks rather than held at $280\mathrm{K}$ while claiming an arbitrary temperature drop. For zero load and optimal current, the same model givesHere $Z$ has units ${\mathrm{K}}^{-1}$ and is a module parameter. The toy model gives $Z=0.0020{\mathrm{K}}^{-1}$ and $T_{c,\min }\approx 241.62\mathrm{K}$ when $T_h=300\mathrm{K}$. This is an ideal zero-load limit with stipulated constant properties; cooling a real load requires $Q_c$ above zero and reduces the attainable drop. Contact thermal resistance, a warming heat sink, external leaks and temperature-dependent coefficients further change the result. Electrical control is convenient, but conservation of energy still requires a thermal path that removes the extracted heat plus the electrical input.