Consider a Membrane Reactor, as illustrated below, for the reversible elementary reaction A à B + C

Then we can apply the conservation of mass to each species (A, B, C) to obtain differential equations governing FA(z), FB(z) and FC(z) within the core-side of the membrane reactor, recognizing that TWO process are taking place, and each can be modeled with a rate expression, as follows:

(1) Reaction of A à B + C,

-rA = k{ CA – CBCC/Keq}, obtained assuming both forward and reverse reactions are elementary, and recognizing that overall rate of reaction is the sum of forward and reverse rates.

(2) Permeation of B through the membrane into the sweep channel,

-rP = PB*(CB – CB,sweep), assuming that permeation of species B through the membrane reactor wall is directly related to the concentration difference between the core fluid and the annular fluid. This form of the permeation rate expression is common for porous membranes which separate products based upon molecule size.

Then our set of equations are as follows:

dFa/dz = -rA

dFb/dz = +rA – rP

dFc/dz = +rA

A numerical solution of this set of equations is easily obtained using MATLAB. Please download and run the following pair of matlab files,

supp_4.m, main program

supp_4_eqns.m, file containing the differential equations

Type ‘supp_4’ at the command prompt to run the files. The results should look like this:

We can see that as the removal rate of B by permeation increases, conversion of A is enhanced by shifting the equilibrium to the right. This is very important for equilibrium reactions, such as the water-gas-shift reaction for hydrogen generation.

We can also “shut off” the reverse reaction by setting the equilibrium coefficient to a sufficiently high value (say, 1x1010 or so…) and see what happens for an irreversible reaction in a membrane reactor:

Well, the molar flowrate of B is quite different, since it is being extracted via the membrane as a pure gas product; thus we are still combining production of product B with separation of product B from by-product C. It is interesting to note that the rate of reaction is mildly enhanced via reduction in gas volume by removal of product. This is in itself important, e.g. for generating “clean” hydrogen gas for a fuel cell.