Optimal heat transfer and optimal exit times

Abstract

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

16 pages, 5 figures. SIAM LaTeX style with custom margins. Fixed a few typos

References in corpus (4)

• Heat Transport by Coherent Rayleigh-Bénard Convection
• Optimal wall-to-wall transport by incompressible flows
• Optimizing the Source Distribution in Fluid Mixing
• Exit times of diffusions with incompressible drift

Cited by in corpus (8)

• Dissipation Enhancement by Mixing
• Shape matters: A Brownian microswimmer in a channel
• Wall-to-wall optimal transport in two dimensions
• On the optimal design of wall-to-wall heat transport
• Bounds on heat transfer by incompressible flows between balanced sources and sinks
• Optimal cooling of an internally heated disc
• Bounds on the heat transfer rate via passive advection
• Enhancing wall-to-wall heat transport with unsteady flow perturbations