Water crosses the membrane by the solution-diffusion model, and the same fixed-point solve as the 3D version of this sim gives the channel-averaged flux, wall salinity and rejection:
J_w = A·(ΔP − σ·Δπ) π = i·R·T·C
C_w / C_b = exp(J_w / k_avg) film theory (channel-averaged)
This 2D version goes one step further and resolves where along the membrane that boundary layer is thin or thick. In the laminar entrance region of a duct, the local mass-transfer coefficient follows the Lévêque solution, k(x) ∝ x−1/3 — sharpest right at the inlet, falling off downstream as the diffusion layer thickens. Its length-average over a channel of length L is exactly 1.5× the local value at the exit, which fixes the constant of proportionality against the same channel-averaged k used above:
k(x) = (k_avg / 1.5) · (L / x)^(1/3)
δ(x) = D / k(x) film thickness at position x
C_w(x) = C_b · exp(J_w / k(x)) local wall concentration
Inside the film itself, salt is carried toward the wall by the permeating water and pushed back by diffusion; balancing those two fluxes at steady state gives a first-order ODE, D·dC/dy + J_w·(C − C_p) = 0, whose exact solution is the exponential profile drawn as the orange haze above the membrane:
C(x,y) = C_p + (C_w(x) − C_p)·exp(−J_w·y / D)
- Applied pressure — raises Jw, thickening the salt layer everywhere and raising the osmotic pressure it must overcome.
- Crossflow velocity — raises k(x) at every position, sweeping salt away and thinning the boundary layer along the whole membrane length.
- 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 and starving the flux at a given pressure.
Ion motion is a drift–diffusion (Fokker–Planck) particle system: each particle drifts downstream on a parabolic channel profile, drifts toward the wall at a rate set by Jw, jitters at a rate set by the local k(x) it currently sits above, and on reaching the membrane is rejected back into the feed with probability equal to the instantaneous rejection, or removed as permeate otherwise — so particles near the inlet see a thin, fast-mixing layer, and particles near the outlet see a thicker one, exactly the spatial development the channel-averaged 3D model does not show.