Lumen mass transfer in hollow-fiber membrane processes with nonlinear boundary conditions

Hollow-fiber membrane separation or reaction processes, with a nonlinear boundary condition at the lumen interface, are analyzed numerically Sixteen different nonlinear boundary conditions in the literature were simplified to four dimensionless boundary conditions, in which only two parameters are needed to incorporate the effect on mass transport of membrane and shell resistances, lumen inlet concentration, shell concentration, phase and/or chemical equilibria, and interface reaction kinetics. The lumen mass-transfer coefficients were obtained by solving the continuity mass-conservation equation with the corresponding boundary conditions. The calculated results demonstrate that the lumen mass-transfer coefficient is a function of lumen-inlet concentration and shell concentration, as opposed to the case when a linear boundary condition occurs at the lumen interface. By using the empirical correlations of the lumen local mass-transfer coefficient, an ordinary differential equation substituting a partial differential equation can be applied to obtain the lumen dimensionless mixed-cup concentration with an absolute error of less than 0.005 in most cases, thus making it possible to design and evaluate the hollow-fiber membrane separation or reaction processes.