The analytic solution of interfacial concentration with observed rejection ratio during dead-end membrane filtration

In this study, we revisit the fundamentals of constant-flux dead-end filtration, develop an analytical solution of the interfacial concentration phi(m)(tau,R-o) as a function of dimensionless time R-o and observed rejection R-o, and compare the solution with previous work developed for constant intrinsic rejection, R-i. The excessive concentration, phi(m)(tau,R-o)-1, consists of three nonlinear terms of tau and reaches 4R(o)tau in an asymptotic limit of tau>1/2. We apply the Robin (mixed) and Dirichlet boundary conditions on the membrane surface and at a far feed-entrance, respectively. The mathematical difficulties for the inverse Laplace transform are resolved using a linear combination of the Laplace transform of error and complementary error functions and applying the convolution theorem. We analytically obtain the unsteady variation of the interfacial concentration after the pressure release using the global mass balance and numerically calculate the required time to reduce the interfacial concentration to a specific limit. More importantly, a relationship between observed and intrinsic rejection ratios is found, such as, R-o similar or equal to root R-i, and verified using experimental data from the literature.