  Given the growing concern of reducing CO2 emissions, it is desirable to
identify, for a given separation carried out through a membrane cascade, the
optimum design that yields the lowest energy consumption. Nevertheless,
designing a membrane cascade is challenging since, there are often multiple
feasible configurations that differ in their energy consumption and cost. In
this work, we develop a Mixed Integer Non-linear Program (MINLP) that, for a
given binary separation, which may be either liquid or gaseous, finds the
cascade and its operating conditions that minimize energy consumption. To model
the separation at each membrane in the cascade, we utilize the analytical
solution of a system of differential and algebraic equations derived from the
crossflow model and the solution-diffusion theory. We provide numerical
evidence which shows that our single-stage membrane model accurately predicts
experimental data. Unfortunately, the resulting membrane model is non-convex
and, even state-of-the-art solvers struggle to prove global optimality of the
cascades and the operating conditions identified. In this paper, we derive
various cuts that help with relaxation quality and, consequently, accelerate
convergence of branch-and-bound based solvers. More specifically, we
demonstrate, on various examples, that our cuts help branch-and-bound solvers
converge within 5\% optimality gap in a reasonable amount of time and such a
tolerance level was not achieved by a simple formulation of the membrane model.
The proposed optimization model is an easy-to-use tool for practitioners and
researchers to design energy efficient membrane cascades.
