Introduction & Context

Concentration-polarisation (CP) is the accumulation of rejected solute at a membrane surface. Once the wall concentration reaches the gel or solubility limit, a “polarisation layer” forms and the convective flux toward the wall is balanced by the diffusive back-flux of solute into the bulk. The resulting asymptotic permeate flux is called the concentration-polarisation flux limit and is the maximum throughput that can be sustained without fouling or scaling. It is a key design parameter in reverse osmosis, nanofiltration, and ultrafiltration systems.

Methodology & Formulas

Mass balance in the film
For steady-state, one-dimensional transport across a stagnant film of thickness δ, the convective solute flux toward the wall equals the diffusive back-flux: \[ J C = - D \frac{\mathrm{d}C}{\mathrm{d}y} \] Integration across the film with boundary conditions \(C(0)=C_{\mathrm{g}}\) and \(C(\delta)=C_{\mathrm{b}}\) gives: \[ J_{\mathrm{max}} = k_{\mathrm{L}} \ln\left(\frac{C_{\mathrm{g}}}{C_{\mathrm{b}}}\right) \] where \(k_{\mathrm{L}}=D/\delta\) is the mass-transfer coefficient.

Regime of validity
The film model assumes a fully developed concentration boundary layer and negligible axial gradients. The recommended range for the concentration ratio is:

Parameter	Lower limit	Upper limit	Remarks
\(C_{\mathrm{g}}/C_{\mathrm{b}}\)	2	30	Outside this range, the film model may under- or over-predict flux.

Implementation notes
To avoid numerical exceptions, the ratio is lower-bounded: \[ \frac{C_{\mathrm{g}}}{C_{\mathrm{b}}} \rightarrow \max\left(\frac{C_{\mathrm{g}}}{C_{\mathrm{b}}}, 10^{-9}\right) \] The calculated \(J_{\mathrm{max}}\) is returned in the same volumetric units as the supplied \(k_{\mathrm{L}}\).

Concentration polarization is the accumulation of rejected solutes at the membrane surface. As solvent passes through the membrane, retained species form a boundary layer whose concentration rises above the bulk value. This layer increases osmotic pressure and reduces the effective driving force. When the surface concentration reaches the solubility or gel point, the back-diffusion of solute equals the convective transport toward the wall; the flux becomes independent of pressure and reaches a maximum value known as the limiting flux predicted by film theory.

Use the film-model equation: J* = k ln(C_w/C_b)

J* is the limiting solvent flux (m s⁻¹)
k is the mass-transfer coefficient (m s⁻¹)
C_w is the wall concentration at the gel or solubility point (kg m⁻³)
C_b is the bulk concentration (kg m⁻³)

Obtain k from correlations (e.g., Sherwood = a Re^b Sc^c) for your module geometry and flow regime, then solve for J* once C_w is known.

Focus on increasing the mass-transfer coefficient k or lowering the concentration ratio C_w/C_b:

Raise cross-flow velocity (higher Re → higher k)
Use spacers or corrugated channels to promote turbulence
Increase temperature (reduces viscosity and raises diffusivity)
Lower bulk concentration by pretreatment or diafiltration
Adjust pH or add antiscalants to push C_w (solubility limit) higher

Run a variable-pressure test:

Increase TMP in steps and record steady-state flux
If flux plateaus while TMP continues to rise, the flat region indicates a polarization-limited regime
Check that the plateau flux rises when you increase cross-flow or temperature—both raise k and should lift the limit
Sample the concentrate stream; a measured concentration near the theoretical C_w supports the film-model assumption

Worked Example – Estimating the Maximum Permeate Flux in a Brackish-Water NF Loop

A small nanofiltration skid is being commissioned to soften 5 m³ h⁻¹ of brackish water at 25 °C. A spiral-wound module shows a rapid rise in wall concentration, and the plant engineer wants to know the concentration-polarisation-limited flux before installing a larger pump. Film-theory is used to obtain a quick estimate.

Knowns

Mass-transfer coefficient, \(k_L\) = 45 µm s⁻¹
Bulk (feed-side) salt concentration, \(C_b\) = 5.0 kg m⁻³
Gel or limiting wall concentration, \(C_g\) = 25.0 kg m⁻³
Temperature, \(T\) = 25 °C
Applied pressure difference, ΔP = 0.9 bar (used only for reference)
Measured concentration ratio, \(C_g/C_b\) = 5.0

Step-by-Step Calculation

Convert \(k_L\) to m s⁻¹ for consistency: \[ k_L = 45\ \mu m\ s^{-1} = 45 \times 10^{-6}\ m\ s^{-1} = 4.5 \times 10^{-5}\ m\ s^{-1} \]
Compute the logarithmic concentration driving force: \[ \ln\left(\frac{C_g}{C_b}\right) = \ln(5.0) = 1.609 \]
Apply the film-theory flux limit: \[ J_{max} = k_L\ \ln\left(\frac{C_g}{C_b}\right) \] \[ J_{max} = (4.5 \times 10^{-5}\ m\ s^{-1})(1.609) = 7.243 \times 10^{-5}\ m\ s^{-1} \]
Convert to litres per square metre per hour (LMH), the unit commonly used on the plant floor: \[ J_{max} = 7.243 \times 10^{-5}\ \frac{m^3}{m^2\ s} \times \frac{1000\ L}{1\ m^3} \times \frac{3600\ s}{1\ h} \] \[ J_{max} \approx 72.4\ L\ m^{-2}\ h^{-1} \]

Final Answer

The concentration-polarisation-limited (maximum) permeate flux for this module under the stated conditions is 72.4 L m⁻² h⁻¹. Operation above this value will rapidly foul the membrane regardless of the applied pressure.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle