Simulation of Time-Lag Permeation Experiments Using Finite Differences

Abstract

Membrane-based pressure driven processes are used in an increasing number of applications. To properly design membrane applications, it is necessary to have a good estimate of membrane properties. To characterize membrane permeation properties, the time-lag method is commonly used. A study has been undertaken to gain a deeper understanding on the accuracy of the time-lag method under realistic boundary conditions using numerical methods. Numerical simulations offer the opportunity to obtain a solution to the Fick’s diffusion equation under various boundary conditions and for nonlinear sorption behaviour for which analytical solutions are difficult or impossible to obtain. This paper is mainly concerned with the selection of the optimal finite difference scheme for solving the Fick’s diffusion equation that leads to the accurate determination of the membrane time lag. Pressure responses in the upstream and downstream reservoirs at both membrane interfaces are determined from the concentration gradients. The concentration gradient at the upstream side of the membrane is initially very steep and to accurately extract membrane properties, it is important to predict it very accurately. Simulation results for the prediction of concentration profiles and gradients at both interfaces are compared with known benchmark analytical equations to assess the precision of numerous numerical schemes where the effect of mesh size and time step is quantified. Results show that a variable mesh size is required to predict accurately the concentration gradient at the upstream interface. The choice of a variable mesh size scheme is important as a compromise must be struck between the smallest mesh size and the time step as it greatly impacts on the computation time. Results also showed that both the implicit and explicit finite difference schemes gave very similar results.