This 2D view plots the same pressure-retarded osmosis (PRO) equations as a live characteristic curve instead of a 3D chamber scene: the water flux line and the power-density parabola are drawn directly as functions of the applied backpressure ΔP, exactly like an engineer would read them off a plant datasheet. A flat membrane-flow schematic underneath shows the same flux visually as crossing dots.
The driving force is the van't Hoff osmotic pressure difference between the two solutions:
Δπ = i · M · R · T
i = 2 (NaCl dissociates into Na⁺ + Cl⁻)
M = draw-side molar salt concentration (mol/L)
R = 0.08314 L·bar/(mol·K)
T = absolute temperature (K)
Net water flux through the membrane follows the solution-diffusion model, opposed by the applied hydraulic backpressure ΔP — this is a straight line in ΔP with slope −A:
Jw(ΔP) = A · (Δπ − ΔP) [L/(m²·h)]
Power(ΔP) = ΔP · Jw(ΔP) (parabola, zero at ΔP=0 and ΔP=Δπ)
- ΔP < Δπ — PRO regime: the curve marker sits on the rising/falling arc of the power parabola, generating power. The parabola's vertex — its maximum — sits exactly at ΔP = Δπ/2, where dPower/dΔP = A·(Δπ − 2ΔP) = 0.
- ΔP = Δπ — the flux line crosses zero; the power parabola returns to zero at this second root.
- ΔP > Δπ — flux and power both go negative (reverse-osmosis regime): the plant consumes power instead of generating it.
Real-world relevance: this is the mechanism behind salinity-gradient ("blue energy") power plants proposed at river mouths, where the Gibbs free energy of mixing fresh and salt water — theoretically about 1.4 MJ per cubic metre of fresh water — is harvested instead of being lost to uncontrolled mixing.