Concentration polarization at a membrane surface

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Directions

This simulation shows concentration polarization: the buildup of rejected solute near a membrane surface caused by convection of feed toward the membrane during filtration.

• permeate flux — increase it to push more solute toward the membrane. The wall concentration rises and the profile becomes more sharply curved, but the boundary-layer thickness does not change.
• mass-transfer coefficient — represents cross-flow velocity and turbulence in the feed channel. Increasing it thins the boundary layer (the shaded region), so solute diffuses back into the bulk over a shorter distance and polarization decreases.
• real rejection — the membrane's intrinsic ability to reject solute. Lower it to see more solute pass into the permeate.
• The solute feed concentration is set at 4.0 g/L, changing the feed concentration from 1 to 20 g/L does not change the shape of the profile or the boundary layer thickness. The only change is that the scale on the y axis changes.

Watch how the observed rejection (based on measurable bulk and permeate concentrations) falls below the membrane's real rejection as polarization becomes more severe — this is why concentration polarization degrades apparent membrane performance even though the membrane itself hasn't changed.

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Details

Within the concentration boundary layer of thickness adjacent to the membrane, solute is carried toward the membrane by convection (at the permeate flux J_v) and diffuses back toward the bulk. At steady state, a solute balance across this layer gives:

where C is the local solute concentration, D is the solute diffusivity, and C_p is the permeate concentration. Integrating from the edge of the boundary layer (x = 0, C = C_b, the bulk concentration) to the membrane wall gives the concentration profile:

Evaluating at the membrane wall (x = , C = C_wall) and defining the mass-transfer coefficient k = D/ gives the classic concentration polarization relation:

The membrane's real (intrinsic) rejection relates the wall and permeate concentrations:

so that C_p = (1 − R)C_wall. Combining this with the polarization relation and solving for the wall concentration gives:

The ratio J_v/k is the Peclet number for the boundary layer, , which measures the strength of convection toward the membrane relative to diffusion back into the bulk. Because the permeate concentration is lower than the wall concentration, the rejection an experimenter would compute from bulk and permeate samples — the observed rejection — is:

Observed rejection is always less than or equal to the real rejection, and the gap between them grows as J_v/k increases — i.e., as concentration polarization becomes more severe.

This simulation uses D = 1.0 × 10⁻⁹ m²/s, typical of a small solute in water, so the boundary-layer thickness = D/k ranges from 10 μm (k = 100 μm/s) to 100 μm (k = 10 μm/s). In the film model, is set by the flow along the membrane, so it depends on k but not on the permeate flux.

Symbols

	local solute concentration (g/L)
	bulk (feed) solute concentration (g/L)
	solute concentration at the membrane wall (g/L)
	permeate solute concentration (g/L)
	solute diffusivity in the feed solution (m2/s); 1.0 × 10−9 m2/s in this simulation
	permeate (volumetric) flux through the membrane (μm/s)
	mass-transfer coefficient in the boundary layer, D/ (μm/s)
	Peclet number, Jv/k (dimensionless)
	real (intrinsic) membrane rejection (dimensionless)
	observed rejection (dimensionless)
	distance from the bulk edge of the boundary layer toward the membrane (μm)
	concentration boundary-layer thickness (μm)

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About

This simulation was created in the Department of Chemical and Biological Engineering at University of Colorado Boulder for LearnChemE.com by John L. Falconer using Claude AI. It was prepared with financial support from the National Science Foundation (DUE 2336987 and 2336988). Address any questions or comments to LearnChemE@gmail.com.