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

Abstract

Weakly permeable micro-tubes are employed in many applications involving heat and/or mass transfer. During these processes, either solute concentration builds up (mass transfer) or steep change in temperature (heat transfer) takes place near the permeable wall causing a change in the viscosity of the fl uid. Results of the present work suggest that such change in viscosity leads to a considerable alteration in the fl ow behavior, and the commonly assumed parabolic velocity profi le no longer exists. To solve the problem numerically, the equation of motion was simplifi ed to represent permeation of incompressible, Newtonian fl uid with changing viscosity through a micro-tube. Even after considerable simplifi cation, the accuracy of the results was the same as that obtained by previously reported results for some specifi c cases using rigorous formulation. The algorithm developed in the present work is found to be numerically robust and simple so that it can be easily integrated with other simulations.

Effects of concentration dependent local resistances and viscosity on overall permeation characteristics of a hollow‐fiber membrane: A simulation approach

“…This equation is applicable only in cases when viscosity of the fluid is constant throughout the pipe. However, considering the simplicity of the Hagen–Poiseuille equation and the fact that radial velocity in hollow‐fiber membrane is several orders of magnitude less than the axial velocity, Gupta et al proposed the concept of effective viscosity for short length permeable tubes ( Δz→ 0) in cases when viscosity of the fluid changes in radial direction. They proposed the following modified Hagen–Poiseuille equation: where Q is the overall volumetric flow rate of the fluid and μ eff is the effective (or overall) viscosity at a particular cross section of the tube, given by …”

“…This equation is applicable only in cases when viscosity of the fluid is constant throughout the pipe. However, considering the simplicity of the Hagen-Poiseuille equation and the fact that radial velocity in hollow-fiber membrane is several orders of magnitude less than the axial velocity, Gupta et al 22 proposed the concept of effective viscosity for short length permeable tubes (Δz→0) in cases when viscosity of the fluid changes in radial direction. They proposed the following modified Hagen-Poiseuille equation:…”

“…Another approach for the study of membrane separation is based on the coupling of model equations for viscous flow (equation of motion or Navier–Stokes equation) and permeation through the membrane (such as Darcy's equations) . However, the coupled hydrodynamics and mass transfer formulation leads to inconsistency at the membrane‐solute interface, which is difficult to be resolved . The problem is further aggravated due to concentration polarization near the membrane leading to significant change in osmotic pressure and the rheology of the fluid …”

“…To solve the coupled model equations, different numerical approaches such as a finite element method finite difference scheme, and finite volume method have been used. In all the above numerical investigations, Navier–Stokes equation (i.e., the equation of motion with constant density and constant viscosity) and the equation of continuity were used to predict velocity profile.…”

“…22 The problem is further aggravated due to concentration polarization near the membrane leading to significant change in osmotic pressure and the rheology of the fluid. 23 To solve the coupled model equations, different numerical approaches such as a finite element method [22][23][24] finite difference scheme, 14,[25][26][27][28][29] and finite volume method [30][31][32][33][34] have been used. In all the above numerical investigations, Navier-Stokes equation (i.e., the equation of motion with constant density and constant viscosity) and the equation of continuity were used to predict velocity profile.…”