Uniform flow in axisymmetric devices through permeability optimization

Porous media enable the intimate contact between a fluid and a functional solid that can accomplish tasks valuable to chemical engineers, such as catalytic reaction, chemical separations, chemical species detection, and filtration. New additive manufacturing technologies facilitate the creation of porous media with precise control of the geometry of each pore, which could enable improved performance and more flexible design of chemical engineering devices. However, new design tools are needed to accomplish this. In this paper, we analyze an optimization problem, constrained by Darcy’s law, to design porous media columns that achieve uniform transit times of fluid particles, despite having nonuniform geometries. The design/control parameter in this setting is a spatially variable scalar permeability function. We prove existence of solutions to our problem, as well as differentiability, which enables the use of rapidly converging, derivative-based optimization methods. We demonstrate our approach on two axisymmetric columns where we achieve a desired velocity field with uniform transit times despite varying device cross sections. When implemented through our separately released source code, our approach provides a convenient and widely applicable means to design additively manufactured porous media and devices that use them.