Analytical and numerical solutions of fixed and moving boundary problems for liquid surfactant membrane processes

A mathematical model describing a batch liquid-surfactant-membrane (LSM) process is developed. In addition to unsteady-state diffusion, reaction, and external mass transfer, consumption of the reagent by the reaction is also considered. Thus, depending on the Thiele modulus and the initial amount of the reagent, the model either remains to be a fixed boundary problem throughout the whole process or changes to a moving boundary problem after some initial period of fixed boundary. For the fixed boundary problem a general analytic solution is obtained in an eigenfunction expansion by a self-adjoint formalism in linear operator theory. In contrast to the known perturbation solution, the solution is exact and straightforward to use. For typical operations the series solution converges rapidly within a few terms, providing a useful tool in design and analysis of the LSM process. Also the solution readily enables determination of the time of transition from fixed to moving boundary. For the moving boundary model, a simple numerical method based on a fixed-grid finite difference method is constructed for solution. In this method the position of the moving boundary is approximated to a grid point at which depletion of the reagent occurred most recently. The calculated position of the moving boundary has been found to be accurate to the size of the grid. The method has also been found to be stable and reliable for all practical ranges of its application.