Geodesic Lower Bound of the Energy Consumption to Achieve Membrane Separation within Finite Time

Abstract

With its promise for low energy consumption, membrane technology has stimulated efficient approaches to separate mixtures. Technological progress has been made to achieve better purification with lower energy cost and higher productivity. Separation with high productivity is typically accomplished within a limited operation time, which unavoidably results in an excess energy consumption due to the fundamental laws of thermodynamics. Reduction of the energy consumption is of practical importance in the application of membrane separation. However, little is known about the fundamental limit of the least excess energy consumption in a finite operation time. We derive such a limit for the separation of binary mixed gases and show its proportionality to the square of the geometric distance between the initial state and the final state and its inverse proportionality to the operation time. The result shows that optimizing the separation protocol is equivalent to finding the geodesic curve in a geometric space. We predict that for the symmetric protocols, the complete separation of 1 mol of a binary mixture of equally mixed ideal single-atom gases at environmental temperature T_{0} within operation time τ requires at least (12−8sqrt[2])N_{A}k_{B}T_{0}τ_{p}/τ for a particle-transport-dominated process and 2(ln⁡2)^{2}N_{A}k_{B}T_{0}τ_{h}/3τ for a heat-exchange-dominated process, where N_{A} is the Avogadro number and τ_{p} and τ_{h} are the timescales for particle transport and heat exchange, respectively. Interestingly, for a symmetric system, the minimum excess energy consumption is achieved by symmetry-breaking protocols. Our geometric approach may inspire the optimization of industrial membrane separation protocols.