Seawater RO Control Room

Solution-diffusion membrane

Water crosses the membrane at Jw = A (ΔP - Δπ) where Δπ = π(Cw) - π(Cp); salt crosses at Js = B (Cw - Cp) so Cp = B Cw / (Jw + B). Permeability rises about 3 % per degree (A) and 6 % per degree (B). Osmotic pressure follows van't Hoff, π = 0.771 S T/298 K bar for S in g/kg, which gives 27 bar for 35 g/kg at 25 C.

Along the vessel

Each of the seven elements is solved in turn: Cw = C eJw/k (concentration polarisation, k about 3e-5 m/s), a bisection solves the implicit flux, then salt and water balances give the next element's feed. Channel pressure drop is 0.45 (q/10)1.7 bar per element. The lead element carries the highest flux; the tail is limited by the rising concentrate osmotic pressure.

Recovery and the pressure the pump must supply

With auto control the pump pressure is found by bisection so that recovery Y = Qp/Qf hits the target; the real pump follows with a 10 s lag. The brine leaves at roughly 1-2 bar below the feed pressure, so about half of the pump hydraulic power is still in the concentrate.

Energy recovery

Without recovery the pump supplies the full feed: SEC = Qf Pf / (η Qp). A turbine returns 0.85 Qb Pb. A rotary pressure exchanger passes the brine pressure directly to fresh feed (efficiency 96 %, a few percent salt mixing), so the main pump moves only the permeate flow and a small booster makes up the pressure loss. The minimum possible work of separation is Wmin = -π0 ln(1-Y)/Y, about 1 kWh/m3 at 50 % recovery; no real plant can go below it.

Fouling

Flux factor falls as d(f)/dt = -4e-7 SDI (J/15)2 per second and salt passage grows as the membrane fouls; a clean in place restores it (minus a small permanent loss).

Honest simplifications

One identical pressure vessel is solved and multiplied, constant channel properties, no scaling chemistry or boron, no second pass, no intake or post-treatment energy (add roughly 0.2-0.4 kWh/m3 in practice). Constants are typical datasheet values, not a design for a specific membrane or plant.