Membrane Reactor for an Equilibrium-Limited Reaction

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The elementary reversible reaction A + B ↔ C + D is carried out in a tubular membrane reactor, which combines reaction with separation. Higher conversions are obtained for this equilibrium-limited reaction by selectively removing product D through the membrane, which is assumed to only permeate product D. The permeate pressure of D (the pressure outside the tube) is assumed to be zero. The molar flow rates of reactants and products are divided by the cross-sectional area of the reactor tube in the plot to make the comparison easier when the reactor diameter changes. All values are dimensionless. Use sliders to change the forward reaction rate constant, equilibrium constant, and membrane permeance. When the membrane permeance is low, the reaction is equilibrium limited for large enough rate constants. Increasing the permeance shifts the equilibrium to the right to obtain higher conversion. Increasing the reactor diameter decreases the conversion because the permeation area per reactor volume decreases.

Material balances:
where Fi is the molar flow rate of component i = A, B, C, D (mol/min), r is rate of reaction (mol/[m3 s]), Ax is the reactor cross-sectional area (m2), rperm is the rate of D permeating (mol/min), and δ is the reactor diameter (m).
where k is the reaction rate constant (m3/[mol s]), Keq is the equilibrium constant (unitless), and km is the membrane permeance, based on concentration difference (mol/[m2 s]).
where FT = ΣFi is the total molar flow rate (mol/min), FT,0 is the initial total molar flow rate (mol/min),  FA,0 and FB,0 are the initial molar flow rates of reactants A and B (mol/min), v and v0 are the volumetric flow rates down the reactor and initially (m3/min).