Hollow fiber membranes made from polymers that allow the transport of one

species $($but not others$)$ can be used for gas separation. Consider a cylindrical

fiber as shown in the figure, below. A gas mixture $($A & B$)$ flows down the center

of the hollow fiber. We will assume that species A will pass through the

membrane wall, but species B will not. Outside of the fiber, we can assume that

another gas sweeps away all of the A so that the concentration of $A$ is

essentially zero. To simplify our analysis, we will also assume that in the core,

the concentration of $A$ is uniform and constant, $c_A=c_{A1},$ and that the solubility

of A in the membrane is given as $c_A^{\text{m}\text{e}\text{m}\text{b}\text{r}\text{a}\text{n}\text{e}}=m{c}_A^{\text{g}\text{a}\text{s}}.$ Our focus is the transport

$($diffusion$)$ of $A$ in the membrane. The process operates at steady$-$state and we

can neglect any variation in the $z-$direction $($the direction of flow inside the

hollow core$).$

:$r_2$

:$r_1$

:$0$

Set up the differential equation for the concentration of $A$ in the membrane with

the appropriate boundary conditions.

Solve the differential equation from problem $4$ to obtain $c_A$ as a function of

radial distance, $r,$ and show that the flux through the membrane at the outside

wall is

$N_{Ar}{|}_r=r_2=\frac{m{D}_{AM}{c}_{A1}}{r_{2}ln(\frac{r_2}{r_1})}$