A nonlinear unsteady state convective diffusion problem that describes the performance of a constant pressure cell is studied. The cell can be used to either determine membrane constants or to carry out batch filtration operations. It consists of a cylinder closed by a semipermeable membrane at one end and a piston at the other.
The nonlinear partial differential equation governing the system was solved by both integral methods and the use of a similarity transformation. The similarity approach formulates the solution in the form of an infinite series and reduces the problem to finding the solution of an infinite system of ordinary differential equations. The series solution can be considered to be exact but its convergence is questionable for large values of time in the event B ≠ 0; when B = 0, the convergence is substantially better.
Approximate solutions obtained by integral methods were examined in detail. It was found that the results obtained by these methods can involve serious errors, under certain circumstances, and these errors seem unpredictable a priori. Consequently, it is concluded that considerable care should be taken in the use of integral methods for solving mass transfer problems in which the velocity field is coupled with the convective diffusion equation and its boundary conditions.
The numerical results obtained in this work are sufficiently comprehensive to be used, in conjunction with experimental data, to determine membrane constants which are required for the design of both continuous and batch membrane separation systems.