This paper documents the fundamental problem of designing a porous flow architecture that meets the requirements of facilitating flow access while storing and releasing heat to a flowing fluid. Examples of such designs are regenerators that operate cyclically in various types of heating or reheating furnaces. The main geometrical scales are determined for parallel flow channels in a fixed regenerator volume with a fixed porosity, by matching the time scales of convection along the channels and thermal diffusion. In accord with the constructal law, the route to better architectures for maximum heat transfer and minimum pressure losses is the morphing of the regenerator architecture from parallel channels to dendritic channels. Copyright (c) 2012 John Wiley & Sons, Ltd.