Solutions of the mass continuity equation in hollow fibers for fully developed flow with some notes on the Lévêque correlation

In this study a historical perspective of the mass continuity equation for fully developed laminar flow in the lumen side of a hollow-fiber membrane contactor is presented with respect to the lumen-wall boundary condition (BC). It is shown that the constant wall concentration case (Dirichlet boundary condition) imposed by the Graetz-Lévêque postulations is a sub-case of the mixed Neumann-Dirichlet linear BC largely overestimating the performance of such contactors. For the linear BC the analytical solution derived by the separation of variables method is revisited proving that it is very accurate and practical even in the region very close to the entrance of the computational domain. The analysis is extended by incorporating and solving nonlinear lumen-wall BCs with the method-of- lines approach by discretizing the radial domain using the Gauss-Jacobi orthogonal collocation and integrating the resulting initial-value differential-algebraic system. The analytical solution, the derivation of the collocation matrices and the numerical solution are presented with the aid of the open-source SageMath and the commercial package Maple.