An analytical expression for the facilitation factor of component A across a liquid membrane is derived in case of an instantaneous reaction A(g)+B(l)AB(l) inside the liquid membrane. The present expression has been derived based on the analytical results of Olander (A.I.Ch.E. J. 6(2) (1960) 233) obtained for the enhancement factor for G–L systems with bulk. The analytical expression for the facilitation factor allows for arbitrary diffusivities of all species involved and does not contain any simplification or approximations. The facilitation factor starts from the value of unity, goes through a maximum and then reduces back to unity as the equilibrium constant is increased. The maximum facilitation factor occurs at higher values of the equilibrium constant as the ratio of the permeate-complex over carrier diffusivity is reduced whereas the maximum facilitation factor occurs at the same value of the equilibrium constant for all values of DA/DB (ratio of the permeate over carrier diffusivity). A similar behavior is seen for the flux of A as a function of the equilibrium constant. The facilitation factor remains constant with changes in the film thickness whereas the flux of A reduces with an increase in the thickness of the film. A linear increase of the facilitation factor and flux of A are seen with increasing initial carrier concentration.
Al-marzouqi, M., Hogendoorn, K., & Versteeg, G. (2002). Analytical Solution for facilitated transport across a membrane. Chemical engineering science, 57(22-23), 4817-4829. https://doi.org/10.1016/S0009-2509(02)00285-3