ANALYTICAL AND NUMERICAL-SOLUTIONS OF FIXED AND MOVING BOUNDARY-PROBLEMS FOR LIQUID SURFACTANT MEMBRANE PROCESSES

A mathematical model describing a batch liquid-surfactant-membrane (LS M) process is developed. In addition to unsteady-state diffusion, reac tion, and external mass transfer, consumption of the reagent by the re action is also considered. Thus, depending on the Thiele modulus and t he initial amount of the reagent, the model either remains to be a fix ed boundary problem throughout the whole process or changes to a movin g boundary problem after some initial period of fixed boundary. For th e fixed boundary problem a general analytic solution is obtained in an eigenfunction expansion by a self-adjoint formalism in linear operato r theory. In contrast to the known perturbation solution, the solution is exact and straightforward to use. For typical operations the serie s solution converges rapidly within a few terms, providing a useful to ol in design and analysis of the LSM process. Also the solution readil y enables determination of the time of transition from fixed to moving boundary. For the moving boundary model, a simple numerical method ba sed on a fixed-grid finite difference method is constructed for soluti on. In this method the position of the moving boundary is approximated to a grid point at which depletion of the reagent occurred most recen tly. 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.