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.

On the Optimal Design of Wall‐to‐Wall Heat Transport

“…The energy-and enstrophy-constrained wall-to-wall optimal transport problems (1.4) and (1.5) were introduced in [19] and studied further in [39] by a combination of asymptotic and numerical methods. Similar methods have since been applied to study other related optimal transport problems [1,25,28]. A key question left unresolved by these works is whether the local maximizers constructed therein actually achieve heat transport comparable to that of global optimizers.…”

Wall-to-wall optimal transport in two dimensions

“…Heat transfer via fluid advection is a critical component of atmospheric, oceanographic, geophysical, and astrophysical dynamics, as well as being the basis of cooling systems in engineering applications. Numerous studies on how to design systems that achieve enhanced heat transfer by either manipulation of domain geometry or through the discovery of suitable flow structures have recently appeared in the literature Toppaladoddi et al (2017); Alben (2017); Marcotte et al (2018); Motoki et al (2018a,b). A particularly fruitful approach to discovering flow structures was first introduced by Hassanzadeh et al (2014) where it was formulated via an optimal control approach.…”

Shape matters: a Brownian microswimmer in a channel

“…A standard problem in drift-diffusion processes is to compute the mean exit time (MET) of a particle to some exit, also called a first-passage time (Ward & Keller 1993;Redner 2001;Holcman & Schuss 2014;Kurella et al 2015;Grebenkov 2016;Marcotte et al 2018). Associated with the reduced drift-diffusion equation (4.17) is a reduced equation for the mean exit time τ (θ) μ(θ…”