Cross-Flow Model for Gas Separation by Membranes

A detailed flow diagram for the cross-flow model derived by Weller and Steiner (W3, W4) is shown in Fig. 13.6-1. In this case the longitudinal velocity of the high-pressure or reject stream is large enough so that this gas stream is in plug flow and flows parallel to the membrane. On the low-pressure side the permeate stream is almost pulled into vacuum, so that the flow is essentially perpendicular to the membrane. 

This model assumes no mixing in the permeate side and also no mixing on the high-pressure side. Hence, the permeate composition at any point along the membrane is determined by the relative rates of permeation of the feed components at that point. This cross-flow pattern approximates that in an actual spiral-wound membrane 

Referring to Fig. 13.6-1, the local permeation rate over a differential membrane area dA at any point in the stage is 

This equation relates the permeate composition y to the reject composition x at a point along the path. It is similar to Eq. (13.4-5) for complete mixing. Hwang and Kammermeyer (HI) give a computer program for the solution of the above system of differential equations by numerical methods. 

Weller and Steiner (W3, W4) used some ingenious transformations and were able to obtain an analytical solution to the three equations as follows  

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