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Membrane Gas Separations
Membranes can be used for separating gas mixtures where they act as a permeable barrier through which different compounds move across at different rates or not move at all. The membranes can be nanoporous, polymer, etc. and the gas molecules penetrate according to their size, diffusivity or solubility.

Fundamentals of Membrane Gas Separation
There are 3 main diffusion mechanisms. Molecular sieving is referred to the case where pores of the membrane are too small to let one component pass, which is not so practical in gas applications because the molecules are too small. Knudsen diffusion holds at very low pressures where lighter molecules move across faster in the stable larger pores. In these cases the movement of molecules is best described by pressure-driven convective flow through capillaries, which is quantified by Darcy’s Law. However, the more general model in gas applications is the solution-diffusion where particles are first dissolved onto the membrane and then diffuse through it both at different rates. This model is employed when the pores in the polymer membrane appear and disappear faster relative to the movement of the particles.

In a typical membrane system the incoming feed stream is separated into two components: permeate and retentate. Permeate is the gas that travels across the membrane and the retentate is what is left of the feed. On both sides of the membrane a gradient of chemical potential is maintained, by a pressure difference, which is the driving force for the gas molecules to pass through. The ease of transport of each species is quantified by the permeability, P. With the assumptions of ideal mixing on both sides of the membrane, ideal gas law, constant diffusion coefficient and Henry’s Law, the flux of a species can be related to the pressure difference by Fick’s Law :

$$j=P(p_R-p_P )=P^'/L (p_R-p_P )=DH/L (p_R-p_P )$$

where, Permeability, which traditionally has the units of barrer, is therefore a combination of the diffusivity and the solubility (expressed by H) of the gas.
 * j is the molar flux
 * L is membrane thickness
 * P’ is permeance
 * D is diffusivity
 * H is the Henry coefficient
 * pR and pP refer to the pressure on the retentate and the permeate sides.

Selectivity or separation factor is another important characteristic of the membranes. For a system of two components (A and B) it is defined as the ratios of the mole fractions in the permeate side divided by the ratio on the retentate side. If permeate pressure is assumed to be very close to vacuum an ideal separation factor can be defined as the ratios of the permeabilities of the components :

$$\alpha\{A,B\}=(X\{A,P\}/X\{B,P\})/(X\{A,R\}/X\{B,R\} )=P_A/P_B $$

Another term used in the literature is stage cut θ defined as the portion of the feed that travel through the membrane. So, θ = jP/jF. A basic mass balance over the membrane gives another equation defining the system :

$$j_F X\{A,F\}=j_R X\{A,R\}+j_P X\{A,P\}=j_F ((1-\theta)X\{A,R\}+X\{A,P\}\theta)$$

The same equation can easily be written for the other component. This analysis assumes that the solubility and diffusion properties of the components in the mixture are the same as the pure components. This assumption is better at lower pressures where components don’t interact with each other much.

The flux and the composition of the feed are generally set by the specific application of the gas separation. Pressures of both sides of the membrane are maintained at steady state by pumps that can operate at pump ratios that are economically and technologically optimal. This sets a limit to the pressure gradient that can be maintained. The parameters that can easily change are the thickness and the area of the membrane and the compositions on the permeate and retentate sides.

In some applications there may be a theoretical upper limit to the extent of separation. The pressure driving force across the membrane must be positive to maintain a net flux to the permeate side. For a single component this would set up the maximum achievable mole fraction in the permeate :

$$p_F X_F-p_P X_P\geqslant0$$

$$X_P max=(p_F X_F)/p_P $$