Membrane Transport Theory: The Solution-Diffusion Model, Concentration Polarisation and the Thermodynamic Limit

Pore flow or solution-diffusion? Why the distinction matters

Microfiltration and ultrafiltration membranes have discrete pores; separation is sieving, and transport is described by a Hagen–Poiseuille or Darcy relation in which flux is proportional to pressure and the solute either passes or does not.

Nanofiltration and reverse osmosis membranes have no continuous pores in the same sense. Their active layer is a dense, cross-linked polyamide film of the order of 100 nm thick, in which transport occurs by dissolution into the polymer and diffusion down a chemical potential gradient. Pressure does not push water through a hole; it raises the chemical potential of water on the feed side so that a diffusion gradient exists.

The practical selection boundaries between these regimes are covered in our guides to ultrafiltration, nanofiltration and reverse osmosis system design.

The transport equations

Combining the three gives the governing relationship for rejection:

Three consequences fall out immediately, and they are the whole of RO operations.

• Rejection rises with flux. More water dilutes the same salt passage. This is why permeate quality degrades when a plant is throttled back, and why running an RO at very low flux to be gentle produces poor water.
• Rejection falls as the membrane ages. B rises with oxidative damage and hydrolysis while A changes little, so conductivity creep is the primary end-of-life indicator.
• Temperature cuts both ways. Warmer water raises A (roughly 3% per °C, so flux rises) but raises B more, so permeate conductivity worsens in summer at constant flux.

Osmotic pressure and the driving force that is not available

The net driving pressure is applied pressure minus osmotic pressure difference, so a substantial fraction of the pump work is spent merely reaching the point at which any water moves at all.

Two refinements matter at high salinity. First, the van’t Hoff expression is an ideal-solution limit; above roughly 10,000 mg/L, activity coefficients fall below unity and rigorous work should use an osmotic coefficient or a Pitzer model. Second, osmotic pressure rises along the pressure vessel as water is removed, so the last element of an array sees far higher π than the first.

The concentration factor is the key bookkeeping quantity:

A 2,000 mg/L feed at 75% recovery therefore produces a 8,000 mg/L concentrate with π ≈ 5.6 bar against 1.4 bar at the feed. The tail element must be pressurised for the concentrate, not the feed — and that, not membrane area, is what sets the pump duty.

Concentration polarisation: the wall is not the bulk

Rejected solute accumulates at the membrane surface faster than it can diffuse back into the bulk, so the wall concentration exceeds the bulk. Film theory gives the classical result:

Worked value. At J_w = 20 L/m²·h = 5.56×10⁻⁶ m/s and k = 4×10⁻⁵ m/s, β = exp(0.139) = 1.15. The wall sees 15% more salt than the bulk. Push flux to 35 L/m²·h and β = exp(0.243) = 1.28.

That modest-sounding number has outsized effects, because it acts on three things simultaneously:

• Osmotic pressure at the wall rises by the same factor, reducing net driving pressure and hence flux — a self-limiting feedback.
• Salt passage rises, since J_s is driven by C_w, degrading permeate quality.
• Scaling risk rises, because saturation indices must be evaluated at wall concentration in the tail element, not at feed concentration.

Critical and limiting flux: the fouling boundary

Below a certain flux, a membrane can be operated for long periods with negligible irreversible fouling; above it, fouling develops rapidly. That threshold is the critical flux, and identifying it is the single most useful piece of pilot work in membrane design.

Mechanistically, particles are carried to the wall by permeation drag and carried away by shear-induced back-transport. The critical flux is where the two balance; above it a cake accumulates, and the added resistance appears in series:

The limiting flux is different and is a mass transfer ceiling: as flux rises, wall concentration rises exponentially until it reaches the gel or saturation concentration, at which point J_lim = k ln(C_g/C_b) and further pressure produces no additional water. A plant operating at limiting flux shows the diagnostic signature of rising pressure at constant flow — and the remedy is more crossflow, not more pressure.

Cleaning strategy and fouling diagnosis are developed in our guide to membrane fouling and clean-in-place.

The thermodynamic minimum and where the energy really goes

There is an irreducible thermodynamic cost to separating water from a solution, independent of technology. For a batch separation at recovery Y, the minimum specific work is:

Worked comparison for the 2,000 mg/L brackish feed above, at 75% recovery with π_f = 1.4 bar:

• Thermodynamic minimum: W_min = 1.4 × ln(4)/0.75 = 1.4 × 1.848 = 2.59 bar-equivalent = 2.59×10⁵ J/m³ = 0.072 kWh/m³.
• Real plant: feed pressure 15 bar (allowing for net driving pressure, element losses and fouling margin), pump efficiency 0.80, no energy recovery. SEC = 15×10⁵/(0.75 × 0.80) = 2.5×10⁶ J/m³ = 0.69 kWh/m³.
• Second-law efficiency = 0.072/0.69 = 10.4%.

Brackish RO is thermodynamically inefficient in relative terms and cheap in absolute terms; seawater RO is the reverse, which is why energy recovery devices are universal at 35 g/L and rare at 2 g/L. The design lever in brackish service is not the pump but the pressure margin: every bar of unnecessary feed pressure costs about 0.046 kWh/m³ at these conditions, so operating at a flux and fouling state that permits 12 bar instead of 15 saves nearly 20% of the energy.

The equivalent arithmetic for high-salinity feeds is set out in our seawater desalination guide, and the alternative of osmotically driven separation in the article on forward osmosis.

Recovery limits: what actually stops you going higher

Recovery is the master variable — it sets concentrate volume, disposal cost, osmotic pressure, scaling risk and energy. It is limited by whichever of four constraints binds first.

Barium sulphate deserves particular attention: with a solubility product around 10⁻¹⁰, barium at 0.05 mg/L in a feed can exceed saturation at a concentration factor of four, so a trace-level analysis that looks unimportant determines the maximum recovery of the whole plant. Antiscalant chemistry buys headroom; it does not remove the constraint. The underlying solubility arithmetic is set out in our guide to softening and scale control.

Worked array design: from duty to element count

Design a brackish RO for 50 m³/h permeate, feed 2,000 mg/L TDS, 15 °C, target 75% recovery.

1. Feed flow = 50/0.75 = 66.7 m³/h; concentrate = 16.7 m³/h at ~8,000 mg/L.
2. Design flux. For a well-pretreated brackish groundwater, 20 L/m²·h is a defensible average. Membrane area required = 50,000 L/h ÷ 20 = 2,500 m².
3. Element count. With 8-inch elements of 37 m² each: 2,500/37 = 68 elements, so 12 vessels of 6 elements (72 elements) gives an average flux of 18.8 L/m²·h.
4. Array configuration. A 2:1 array (8 vessels in stage 1, 4 in stage 2) maintains concentrate velocity in the tail. Check stage-2 concentrate flow against the element minimum: 16.7 m³/h across 4 vessels = 4.2 m³/h per vessel — acceptable.
5. Temperature correction. At 15 °C rather than 25 °C, TCF ≈ 0.75, so the pressure required for the same flux rises by roughly a third. Design the pump on the winter case.
6. Scaling check. Evaluate LSI and the sulphate and silica saturation ratios at the tail-element wall concentration, i.e. at CF = 4 multiplied by β ≈ 1.15, giving an effective concentration factor of about 4.6. Dose antiscalant on that basis, not on the bulk concentrate.

Note step 6. Sizing antiscalant on bulk concentrate rather than wall concentration understates the saturation state by 15% or more, and it is a common cause of scaling on plants that appear, on paper, to be operating within limits. Getting that detail right is characteristic of proper membrane system design rather than catalogue selection.