A 1-D model of a plate-and-frame module: feed flows along a slit with the membrane on one wall and an impermeable plastic filler on the other. CO₂ permeates faster than the feed can resupply it, so a CO₂-starved boundary layer forms (concentration polarization), and the retentate is progressively depleted down the channel. The model integrates the Wijmans–Baker film model cell-by-cell along the length and lets you compare co-current against counter-current. Every molecule in the animation is conserved — a CO₂ that leaves the feed is the same one that enters the permeate.
Mole fraction vs position. Gap between retentate (bulk) and surface = polarization. Hollow circles = your measured values.
Solid bars = model at the current settings; hollow bars = your measurements (enter them in the panel →).
Reading it. Red = CO₂ (fast), purple = N₂, green = O₂ (slow). Two losses stack up, and they respond to different knobs.
1 · Polarization. The membrane only “sees” the depleted surface — the bulk–surface gap in the plot. It worsens with high permeance, high pressure and a wider gap. In this one-sided slit the laminar flow sits on its fully-developed limit (Sh ≈ 4.86), so k ≈ 4.86·D/(2·gap): the cure here is the gap, not the feed rate — halving the gap doubles k, while raising the flow barely moves it. Not a better membrane, either: a faster material only deepens the boundary layer.
2 · Axial depletion. The retentate loses CO₂ down the channel, and in co-current the permeate enriches alongside the feed so the driving force collapses toward the outlet — counter-current keeps a larger average driving force and usually wins. This loss is the flow-sensitive one: a faster feed cuts the stage cut and keeps the retentate rich.
A 1-D plate-and-frame module discretised into cells along the length. In each cell the feed-side film model (Wijmans, Athayde & Baker) is solved with the local solution–diffusion flux, then feed and permeate molar flows are marched along the channel (forward for co-current; iterated as a two-point problem for counter-current). For each species i:
The mass-transfer coefficient k comes from a slit-channel Sherwood correlation with one active wall (membrane) and one inert wall (the plastic filler): fully-developed laminar limit Sh ≈ 4.86 on hydraulic diameter dh = 2·gap, blended with the Lévêque entrance solution, and a turbulent branch for Re > 2300.
Regime note: with gases (Sc ≈ 1) in a millimetre slit the fully-developed floor dominates over practically the whole slider range — Lévêque only takes over above Re·Sc·dh/L ≈ 27, and Re stays below ~1300 at 5 bar, so the turbulent branch is never reached. Hence k is set by the gap and is nearly independent of feed velocity here. A different channel geometry, a feed spacer or a turbulence promoter would change that, and would need its own correlation. Gas diffusivity and viscosity scale as T1.7/p. The flow reading is converted to a molar feed rate (NL/min assumes 0 °C, 1 atm — the usual MFC reference; “actual” takes the volume at feed p,T), and the corresponding actual volumetric flow through the width × gap cross-section sets the channel velocity that drives k. The active membrane area can be smaller than the channel wall (length × width) to account for gaskets/sealing: flux acts only on that fraction of each cell while the hydrodynamics keep the full channel. Each species' permeance (CO₂, N₂, O₂) is set independently — type measured values directly — and the selectivities shown are simply their ratios. Defaults (1000 / 25 / 75 GPU) are illustrative.
Sources: Wijmans, Athayde, Daniels, Ly, Kamaruddin (1996) J. Membr. Sci. 109:135 (film model). R.W. Baker, Membrane Technology and Applications (Wiley), ch. 4–5. One-wall slit Sherwood asymptote: Shah & London, Laminar Flow Forced Convection in Ducts. Recent CP-in-gas-separation impact studies: high-performance zeolite membranes (Membranes/MDPI 2024); biogas upgrading (Chem. Eng. Res. Des. 2024).