Optimal design of a model energy conversion device

Fuel cells, batteries, thermochemical and other energy conversion devices involve the transport of a number of (electro-)chemical species through distinct materials so that they can meet and react at specified multi-material interfaces. Therefore, morphology or arrangement of these different materials can be critical in the performance of an energy conversion device. In this paper, we study a model problem motivated by a solar-driven thermochemical conversion device that splits water into hydrogen and oxygen. We formulate the problem as a system of coupled multi-material reaction-diffusion equations where each species diffuses selectively through a given material and where the reaction occurs at multi-material interfaces. We express the problem of optimal design of the material arrangement as a saddle point problem and obtain an effective functional which shows that regions with very fine phase mixtures of the material arise naturally. To explore this further, we introduce a phase-field formulation of the optimal design problem, and numerically study selected examples.