This simulation shows the concentration profiles on the feed and permeate sides of a membrane resulting from a diffusion boundary layer and the concentration profile inside the membrane. It also calculates the molar flux through the membrane. Use the sliders to change the membrane diffusion constant, the equilibrium constant, and the membrane thickness. The feed concentration C₁ and the desired permeate concentration C₂ are set. Place your cursor over a black dot to see its concentration displayed on the right.

The molar flux at steady state is calculated using the analog of series resistances. The flux through each layer is:

feed: N_A = k_c, 1 (C₁ − C_1, i) ,

membrane: N_A = D_AL (C_1, i^★ − C_2, i^★) ,

C_1, i^★ = K C_1, i ,  C_2, i^★ = K C_2, i ,

permeate: N_A = k_c, 2 (C_2, i − C₂) ,

where k_c is the mass transfer coefficient, C₁ and C₂ are the feed and permeate concentrations, C_1, i and C_2, i are the concentrations at the interface, C_1, i^★ and C_2, i^★ are the concentrations at the surfaces, D_A is diffusivity in the membrane, L is membrane thickness and K is the equilibrium constant.

The flux through the membrane can be written in terms of permeance P:

P = D_A KL ,  N_A = P (C_1, i − C_2, i) .

Because the system is at steady state, N_A is the same through all three layers:

N_A = k_c, 1 (C₁ − C_1, i) = P (C_1, i − C_2, i) = k_c, 2 (C_2, i − C₂) .

Solving each N_A equation for the concentration differences:

C₁ − C_1, i = N_Ak_c, 1 ,  C_1, i − C_2, i = N_AP ,  C_2, i − C₂ = N_Ak_c, 2 .

Adding these equations together and solving for N_A gives the flux in terms of series resistances:

N_A = (C₁ − C₂) ( 1k_c, 1 + 1P + 1k_c, 2 )⁻¹ .

This simulation was created in the Department of Chemical and Biological Engineering at University of Colorado Boulder for LearnChemE.com by John L. Falconer using Claude AI. It is a JavaScript/HTML5 implementation of a Mathematica simulation by Rachael L. Baumann. It was prepared with financial support from the National Science Foundation (DUE 2336987 and 2336988) in collaboration with Washington State University. Address any questions or comments to LearnChemE@gmail.com.