Raw biogas from an anaerobic digester is roughly 55–65% CH₄ and 35–45% CO₂. A polymer membrane module upgrades it to pipeline/vehicle-grade biomethane (>95% CH₄) using the solution-diffusion mechanism: CO₂ dissolves into and diffuses through the polymer far faster than CH₄, so it preferentially crosses the membrane wall into a low-pressure permeate stream while CH₄ is left behind, concentrated, in the high-pressure retentate.
At every point along the membrane the local permeate composition y₁ (CO₂) is fixed by the cross-flow permeation balance (Weller & Steiner, 1950), solved here in closed form at each of 80 integration steps along the 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
Each step removes dN₁ + dN₂ from the retentate flow over a slice of membrane area dA, updates the retentate composition x₁, and repeats. Integrating over the whole module gives:
- CH₄ purity — the retentate composition at the exit (the biomethane product spec).
- 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 — the fundamental purity/recovery trade-off of every real membrane skid.
Higher selectivity α (better membrane material) and a lower pressure ratio φ (bigger vacuum/compression duty) both push toward higher purity at a given recovery. 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.