A New Solution for Flow In Porous Tubes and Channels Incorporating Darcy's Law

Pressure-driven fluid flow through a porous tube occurs in applications such as filtration, membrane reactors, reverse osmosis for desalination, and medical devices.  Typically, a suspension or solution is pumped axially through a tubular membrane, purified filtrate is extracted through the membrane as a result of a transmembrane pressure difference, and concentrate exits downstream. In spite of how common this flow is, an exact solution for this flow has not been found.  Such a generalized solution for pressure-driven, incompressible, Newtonian flow in a porous tubular membrane is challenging due to the coupling between the transmembrane pressure and velocity. To date, all analytical solutions require simplifications such as neglecting the coupling between the transmembrane pressure and velocity, assuming the form of the velocity fields, or expanding in powers of parameters involving the tube length.  Moreover, previous solutions have not been validated with comparison to direct numerical simulation. In this research, we comprehensively revisit the problem to develop a robust analytical solution incorporating Darcy’s law on the membrane. We make no assumptions about the tube length or form of the velocity fields. We also present a solution for porous channels.

Our analytic solution is validated with detailed comparisons to direct numerical simulations, including cases of axial flow exhaustion and cross flow reversal. We explore the validity of typical assumptions used in modeling porous tube flow for conditions typical of ultra- and micro-filtration. In addition to being more accurate than previous solutions, the implementation of the solution in a practical situation is quite straightforward. The entire velocity and pressure fields inside the porous tube can be found analytically based on only the flow rate and transmembrane pressure at a particular location, as well as the membrane and fluid properties.  The solution can also be easily rewritten in terms of the transmembrane pressure at two different axial locations, such as the inlet and outlet, or the flow rate at two different locations.  The equations for the solution, while somewhat long and complicated, do not require any computational or numerical methods for solution.  Hence, they can provide the basis for more sophisticated analyses that incorporate solute transport, concentration polarization, or the effect of osmotic pressure on transmembrane flow.