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A heat exchanger can be modeled as a closed domain containing anincompressible fluid. The moving fluid has a temperature distribution obeyingthe advection-diffusion equation, with zero temperature boundary conditions atthe walls. Starting from a positive initial temperature distribution in theinterior, the goal is to flux the heat through the walls as efficiently aspossible. Here we consider a distinct but closely related problem, that of theintegrated mean exit time of Brownian particles starting inside the domain.Since flows favorable to rapid heat exchange should lower exit times, weminimize a norm of the exit time. This is a time-independent optimizationproblem that we solve analytically in some limits, and numerically otherwise.We find an (at least locally) optimal velocity field that cools the domain on amechanical time scale, in the sense that the integrated mean exit time isindependent on molecular diffusivity in the limit of large-energy flows.

Optimal heat transfer and optimal exit times