The following presentation pertains to an equilibrium chemical conversion that is bimolecular in both directions

but may be adjusted for monomolecular behavior in either direction. The reactants are assumed to be at chemical equilibrium and remain at equilibrium during the membrane permeation of product S, which shifts the equilibrium.(Other components could also be permeated, of course, complicating the presentation. Even more so, the chemical conversion rate could be accommodated as well, further complicating the representation.)

The initial moles of reacting components (both reactants and products)are, by definition, specified on the basis of a mole of feed. That is, the initial moles of each, when added up, equal 1 mole of feed. The degree of conversion is denoted by the symbol X. The accompanying membrane transfer of component S, on the same basis, is denoted by Y.

A succession of arbitrary values are assumed for Y, starting at Y =0, corresponding to the initial equilibrium condition. There is an ensuing relationship for X in terms of Y, as determined from the reaction equilibrium constant. This in turn enables the calculation of the simultaneously existing mole fraction x S of component S, which is also that of the membrane reject stream.

There is a limiting value for x S, however, by virtue of the integral that establishes the corresponding membrane area:

where y _S = 1 for a permeate of component S only. Therefore, it is necessary that

Accordingly, the permeation pressures may be adjusted to accommodate this restriction, which becomes...