= Indoor-temperature limit of a Carnot heat pump
{title2=$T_{c,\min}=T_h-\sqrt{\dot W_{\max}T_h/\lambda}$}
A reversible <heat pump> maintaining $T_h$ against an external $T_c<T_h$ has heating <coefficient of performance> $T_h/(T_h-T_c)$. If its maximum work input is $\dot W_{\max}$ and the building loses heat at rate $\lambda(T_h-T_c)$, maintaining equilibrium requires $\lambda(T_h-T_c)^2\leq\dot W_{\max}T_h$. This yields the displayed minimum external temperature, provided it lies in the physical positive-temperature range. If the rating instead specifies maximum delivered heat, the bound is $T_h-T_c\leq\dot Q_{h,\max}/\lambda$; a heat-output rating cannot be substituted for a work-input rating in the Carnot formula.
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