Model number


When flow of solute in a solution under pressure is retarded so that its flux across a membrane is slower than the solvent flux, it accumulates on the upstream side of the membrane. This "polarization" retards solvent flow osmotically.


  Concentration polarization: From a solution of constant concentration C0 there 
  is flow of the solution through a tube towards a filtering membrane at 
  distance L from the inlet. The tube diameter doesn't matter and we consider 
  only fluid velocity toward the membrane and diffusion of solute, both being 
  one-dimensional normal to the membrane surface. The solute does NOT permeate, 
  and therefore become concentrated above the membrane, creating a gradient
  in concentration, dCdx or C:x. The solute activity coefficient
  is assumed to remain constant at 1. The buildup in C(x) will retard
  solvent flow osmotically.unless pressure is raised to maintain flow.	

fig 1


The basic equation for this problem in the steady state balances the diffusion at any point upstream of the membrane with the convection of the solute downstream towards the membrane.


where C is the concentration of the diffusing species, D is the rate of diffusion of the species in the axial direction and vel is the velocity of the solute/solvent mix entering the tube. Two boundary conditions are applied at the tube entrance:



where C0 is the concentration of the solute at the entrance to the tube. The analytical solution to this problem is given by:


The equations for this model may be viewed by running the JSim model applet and clicking on the Source tab at the bottom left of JSim's Run Time graphical user interface. The equations are written in JSim's Mathematical Modeling Language (MML). See the Introduction to MML and the MML Reference Manual. Additional documentation for MML can be found by using the search option at the Physiome home page.

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  Bassingthwaighte, JB. Transport in Biological Systems, Springer Verlag, New York, 2007.
Key terms
Osmotic pressure
Transport Phsyiology

Please cite in any publication for which this software is used and send one reprint to the address given below:
The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.

Model development and archiving support at provided by the following grants: NIH U01HL122199 Analyzing the Cardiac Power Grid, 09/15/2015 - 05/31/2020, NIH/NIBIB BE08407 Software Integration, JSim and SBW 6/1/09-5/31/13; NIH/NHLBI T15 HL88516-01 Modeling for Heart, Lung and Blood: From Cell to Organ, 4/1/07-3/31/11; NSF BES-0506477 Adaptive Multi-Scale Model Simulation, 8/15/05-7/31/08; NIH/NHLBI R01 HL073598 Core 3: 3D Imaging and Computer Modeling of the Respiratory Tract, 9/1/04-8/31/09; as well as prior support from NIH/NCRR P41 RR01243 Simulation Resource in Circulatory Mass Transport and Exchange, 12/1/1980-11/30/01 and NIH/NIBIB R01 EB001973 JSim: A Simulation Analysis Platform, 3/1/02-2/28/07.