Concentration Polarization at the RO Membrane

Feed Conditions

Applied pressure ΔP 55 bar

Crossflow velocity 0.20 m/s

Feed salinity 35 g/L

Membrane fouling none

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Permeate flux J_w

– LMH

Polarization factor β

–

Salt rejection

– %

Wall salinity C_w

– g/L

Net driving pressure (ΔP − Δπ)

– bar

How it works

Water crosses the membrane by the solution-diffusion model: it dissolves into the polymer and diffuses down a pressure-driven chemical-potential gradient, while dissolved salt leaks through far more slowly.

J_w = A · (ΔP − σ·Δπ)         water flux
J_s = B · (C_w − C_p)         salt flux
π   = i·R·T·C                 van 't Hoff osmotic pressure

As water is pulled through, salt is left behind faster than turbulence can carry it back into the bulk feed — a thin concentration boundary layer forms at the wall. Film theory gives its strength:

C_w / C_b = exp(J_w / k)        concentration polarization
k = Sh·D / d_h,  Sh = 0.065·Re^0.875·Sc^0.25   (spacer channel)

Because Δπ depends on C_w, and C_w depends on J_w, the two are solved together each frame by fixed-point iteration — exactly as a process engineer sizes a real RO train.

Applied pressure — raises J_w, which thickens the wall-side salt layer (higher β) and raises the osmotic pressure it must overcome.
Crossflow velocity — more turbulence raises the mass-transfer coefficient k, sweeping salt away and thinning the boundary layer (lower β).
Feed salinity — sets the bulk osmotic pressure the pump has to beat before any net flux appears.
Membrane fouling — a cake/scale layer adds hydraulic resistance, cutting effective permeability A and starving the flux at a given pressure.

Ion motion in the channel is a real drift–diffusion (Fokker–Planck) simulation: each particle drifts downstream, drifts toward the wall at a rate set by J_w, jitters randomly at a rate set by k, and — on reaching the membrane — is rejected back into the feed with probability equal to the instantaneous rejection, or removed as permeate otherwise. The resulting particle density near the wall is a direct, physically driven picture of C_w/C_b.

Bulk feed ions Wall boundary layer Membrane surface

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