Laminar flow with injection through a long dead‐end cylindrical porous tube: Application to a hollow fiber membrane

Abstract

Effects of membrane length and hydraulic resistance on the steady-state laminar flow of a fluid with injection in a dead-end cylindrical porous tube have been investigated using the perturbation approach of the Navier-Stokes and continuity equations. Analytic solutions of the dynamic equations, reduced to non linear differential equations, were obtained and applied to submerged, dead-end hollow fiber membranes of large resistances and small wall Reynolds numbers. The velocity and pressure profiles were solved using the method of separation of variables as with physical properties of the membrane and fluid and the wall velocity at the dead end. The constant flux approximation was found to be valid only for a short membrane with a large hydraulic resistance. The constant permeability approximation used in this study is universal for a membrane of an arbitrary length, through which the fluid velocities and pressure increase exponentially from the dead end to the open end.

Flow behavior in weakly permeable micro-tube with varying viscosity near the wall

“…In order to incorporate the effect of variation in viscosity (either due to concentration polarization or due to temperature gradient) in a weakly permeable or impermeable tube, one can use the concept of effective viscosity being introduced in the present work through equation (12). For the case where integration of viscosity term is diffi cult, numerical integration using equation (13) can be performed.…”

“…(10) as follows: Substituting Eq. (7) and rearranging we get (12) In a practical situation, however, the viscosity of a fl uid depends on the local conditions of temperature and composition; therefore, a suitable function representing the variation of viscosity in terms of r that can be used for integrating Eq. (12) may not be known.…”

A filtration model for prediction of local flux distribution and optimization of submerged hollow fiber membrane module

“…Revised manuscript AIChE-15-16842R2 for submission to AIChE Journal 16 repeated three times to confirm the reproducibility of the experimental data.…”

“…Especially, Kim et al 16 described a perturbation approach of the Navier-Stokes and continuity equations to investigate the effect of membrane length and hydraulic resistance on the steady-state laminar flow of the fluid in the fiber lumen. It indicated that the flow velocity in fiber lumen represents the exponential variation along the axial direction.…”

“…To achieve maximum energy balance coupled with the application of exponential model to predict the permeate flux of hollow fiber ultrafiltration membrane. The simulated results showed that the effect of operating pressure on permeate flux is more significant than that of feed concentration.Especially, Kim et al 16 described a perturbation approach of the Navier-Stokes and continuity equations to investigate the effect of membrane length and hydraulic resistance on the steady-state laminar flow of the fluid in the fiber lumen. It indicated that the flow velocity in fiber lumen represents the exponential variation along the axial direction.…”