1. Fickian Solution-Diffusion Permeation Model
Gas permeation across a dense non-porous polymeric membrane follows the classic solution-diffusion mechanism, wherein gas molecules dissolve into the upstream polymer face, diffuse across the active selective skin, and desorb into the low-pressure permeate channel. The molar flux \(J_i\) of component \(i\) is governed by Fick's first law:
\[ J_i = \frac{P_i}{l} \left( p_{f,i} - p_{p,i} \right) = Q_i \left( x_{f,i} P_{\text{feed}} - y_{p,i} P_{\text{perm}} \right) \]
where \(P_i\) is intrinsic membrane permeability, \(l\) is the effective selective layer thickness (\(0.1\text{--}1.0\ \mu\text{m}\)), \(Q_i = P_i/l\) is permeance expressed in Gas Permeation Units (\(1\ \text{GPU} = 10^{-6}\ \text{cm}^3(\text{STP}) / (\text{cm}^2 \cdot \text{s} \cdot \text{cmHg}) = 3.348 \times 10^{-10}\ \text{mol}/(\text{m}^2 \cdot \text{s} \cdot \text{Pa})\)), and \(p_{f,i}, p_{p,i}\) are feed and permeate partial pressures.
2. Membrane Ideal Selectivity & Separation Factor
The inherent selectivity \(\alpha_{i/j}\) between fast-permeating acid gas (\(\text{CO}_2\)) and slow-permeating methane (\(\text{CH}_4\)) is the ratio of their permeabilities:
\[ \alpha_{\text{CO}_2/\text{CH}_4} = \frac{P_{\text{CO}_2}}{P_{\text{CH}_4}} = \frac{Q_{\text{CO}_2}}{Q_{\text{CH}_4}} \]
Commercial polyimide hollow fibers exhibit ideal selectivities \(\alpha \approx 25\text{--}35\), whereas cellulose acetate modules achieve \(\alpha \approx 20\text{--}28\) under typical natural gas pipeline conditions.
3. Log Mean Partial Pressure Driving Force (LMPD)
Because the feed gas becomes progressively depleted in \(\text{CO}_2\) as it travels from module inlet to retentate outlet while permeate partial pressure builds along the bore side, the effective driving force is calculated via the logarithmic mean partial pressure difference:
\[ \Delta p_{\text{LMPD}} = \frac{\Delta p_{\text{in}} - \Delta p_{\text{out}}}{\ln(\Delta p_{\text{in}} / \Delta p_{\text{out}})} \]
where \(\Delta p_{\text{in}} = (z_{\text{CO}_2} P_{\text{feed}} - y_{\text{CO}_2,\text{perm}} P_{\text{perm}})\) and \(\Delta p_{\text{out}} = (x_{\text{CO}_2,\text{ret}} P_{\text{feed}} - y_{\text{CO}_2,\text{perm,local}} P_{\text{perm}})\).
4. Active Membrane Area Sizing & Commercial Module Packaging
The total net membrane surface area \(A_{\text{mem}}\) required to achieve the target sales gas \(\text{CO}_2\) specification is derived from the molar permeation balance:
\[ A_{\text{mem}} = \frac{\dot{n}_{\text{CO}_2,\text{perm}}}{Q_{\text{CO}_2} \cdot \Delta p_{\text{LMPD}}} \quad [\text{m}^2] \]
Commercial hollow-fiber skid assemblies package between \(250\ \text{m}^2\) and \(500\ \text{m}^2\) of active membrane area into each cylindrical module vessel (\(8\text{--}12\ \text{inches}\) in diameter, \(3\text{--}4\ \text{meters}\) long). The total module count is:
\[ N_{\text{modules}} = \left\lceil \frac{A_{\text{mem}}}{A_{\text{module}}} \right\rceil \]
5. Two-Stage Recycle Recompression Power Balance
In high-recovery two-stage configurations, permeate from Stage 1 is recompressed by an electric-motor-driven reciprocating or centrifugal compressor and fed to Stage 2, whose retentate recycles to the feed inlet. The compression work required is:
\[ W_{\text{comp}} = \frac{\dot{m}_{\text{recycle}}}{\eta_c} \left( \frac{\gamma}{\gamma - 1} \right) Z_{\text{avg}} R_{\text{gas}} T_{\text{suct}} \left[ \left( \frac{P_{\text{feed}}}{P_{\text{perm}}} \right)^{\frac{\gamma - 1}{\gamma}} - 1 \right] \quad [\text{kW}] \]