Optimal heat transfer and optimal exit times

TL;DR

This paper investigates optimal flow strategies in heat exchangers to maximize heat transfer efficiency by minimizing the mean exit time of particles, revealing flow configurations that are effective regardless of diffusivity at high energy levels.

Contribution

It introduces a novel optimization framework for fluid flow in heat exchangers based on minimizing particle exit times, with analytical and numerical solutions.

Findings

Optimal velocity fields can cool the domain efficiently.

Exit times become independent of diffusivity at high energy flows.

Analytical solutions are obtained in certain limits.

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.