Desalination Membrane Flow: Osmotic Pressure & Solution-Diffusion Transport (2D)

Feed & Pressure

Applied pressure (ΔP) 55 bar

Feed salinity (C_f) 35.0 g/L

Membrane & Conditions

Water permeability (A) 2.0 L/m²h·bar

Feed temperature (T) 20 °C

Live Readout

Water flux J_w

0.0 L/m²h

Salt rejection

0.0%

Osmotic π (wall)

0.0 bar

Net driving ΔP−Δπ

0.0 bar

Polarization β

1.00×

Recovery

0.0%

Status

—

Batch time

0.0 h

How it works

Reverse osmosis pushes water backward across a semi-permeable membrane, against its natural direction (osmosis), by applying hydraulic pressure that exceeds the solution's osmotic pressure. Osmotic pressure follows the van't Hoff equation:

π = i · M · R · T
  i = 2 for NaCl (dissociates into Na⁺ + Cl⁻)
  M = molar concentration (mol/L)
  R = 0.08314 L·bar/(mol·K)

Transport follows the solution-diffusion model: water flux is driven by the net pressure left over after subtracting the osmotic pressure difference, and salt leaks through separately, driven only by its own concentration difference:

Jw = A · (ΔP − Δπ)     [water flux, solution-diffusion]
Js = B · (C_wall − C_permeate)   [salt flux]

Right at the membrane surface, rejected salt accumulates faster than it can diffuse back into the bulk feed — concentration polarization — raising the effective concentration (and osmotic pressure) the membrane actually has to fight above the bulk feed value. The polarization factor β = C_wall / C_bulk builds up over a few seconds toward its steady value exp(Jw/k) as the boundary layer forms.

Because the feed reservoir is a finite batch, every liter of permeate removed leaves the remaining feed saltier — recovery rises, osmotic pressure climbs, and flux falls over time, exactly like a real batch RO cell. If applied pressure ever drops below the wall osmotic pressure, net driving pressure goes negative and flux clamps to zero — no amount of membrane area fixes that; only more pressure does.

Feed (saline) Permeate (fresh) Rejected salt