2D companion to the 3D membrane-module scene: the same solution-diffusion separation, computed with the identical Weller & Steiner (1950) cross-flow equations, shown here as a schematic cross-section plus a purity-vs-recovery trade-off curve instead of a rendered 3D module.
N₁ = P₁·(p_hi·x₁ − p_lo·y₁) (CO₂ flux)
N₂ = P₂·(p_hi·x₂ − p_lo·y₂) (CH₄ flux)
α = P₁ / P₂ (membrane selectivity)
φ = p_lo / p_hi (pressure ratio)
φ(1−α)·y₁² + [1 + (α−1)(x₁+φ)]·y₁ − α·x₁ = 0 → solve for y₁
- Cross-section (top) — CH₄ (amber) and CO₂ (blue-grey) molecules drift down the retentate channel while CO₂ preferentially crosses the dashed membrane wall into the permeate space above and below it, at a rate set by the local driving force.
- Trade-off curve (bottom) — sweeps membrane area from minimum to maximum at the current selectivity, pressure ratio and feed composition, plotting CH₄ purity against CH₄ recovery. The white dot marks where the Membrane area slider currently sits on that curve — every real membrane skid lives somewhere on this same curve and can only trade purity for recovery, not escape it.
- CH₄ recovery — the fraction of feed CH₄ that stays in the retentate rather than slipping through the membrane.
- Methane slip — CH₄ lost into the permeate, a greenhouse-gas and revenue penalty real plants are penalized for.
- Stage cut θ — the fraction of feed gas that permeates; larger membrane area raises θ, which raises purity but costs recovery.
Higher selectivity α (better membrane material) and a lower pressure ratio φ (bigger vacuum/compression duty) both push the whole trade-off curve toward higher purity at a given recovery — watch the curve reshape as you move those sliders. The 2-stage cascade recompresses the first-stage permeate and re-separates it, recovering CH₄ that a single stage would lose — the standard industrial fix, at the cost of extra compression energy.