The author studies steady-state solutions of the heat equation in a domain \(\Omega \subset \mathbb{R}^3\), that is, solutions of the equation \(- \Delta u = f(x)\) in \(\Omega\). He derives an integral identity for solutions of the form \(\int_{\partial P} {- \nu \cdot \text{grad} u \over u} ds = F + G\), where \(P \subset \Omega\) and \(F\) and \(G\) are interpreted as flux of entropy and rate of entropy generation, respectively. The main result of the paper is that \(G\) is unbounded and \(F\) is bounded from above by \(C \cdot \text{diam} (\text{supp} f)\).