Consider a hollow$-$fiber membrane used as an artificial kidney to remove urea and other contaminants from blood. The fiber has an inside diameter $D_i,$ an outside diameter $D_0,$ and length $L.$ Blood enters the tube side of the membrane with a urea concentration $C_{Ai}.$ On the outside $($shell side$)$ is a saline solution to sweep away urea. The diffusivity of urea within the membrane wall $($which is mostly water within a sponge$-$like polymer$)$ is $D_{AB}.$ The blood is$/$not pressurized, so convective flow through the fiber wall may be neglected.

A$.$ Assuming steady state and diffusion only in the radial direction through the tube wall, perform a shell balance to derive ordinary differential equations goveming the flux and concentration in the membrane wall. Solve for concentration $C_A(r)$ and flux $N_{Ar}(r),$ subject to the boundary conditions $C_A=C_{Ai}$ at $r=\frac{D_i}{2}$ and $C_A=C_{A0}$ at $r=\frac{D_o}{2}.$ Note that these boundary conditions are based on the diffusion resistance in the wall dominating over the film resistances on the inside and outside of the membrane.

B$.$ Using $D_i=0\mathrm{.}5mm,D_0=1\mathrm{.}0mm,C_{Ai}=10mM,C_{A0}=0($fast flow of saline without urea$)$ and $D_{AB}=1\mathrm{.}5{10}^{-5}c\frac{m^2}{s},$ calculate the flux of urea through the membrane in units of $mo\frac{l}{c}{m}^2-s$ based on the inside area of the tube. Then, noting that the typical blood volume in an adult

$2$

human is $5L,$ how many hollow fibers of length $L=20cm$ are required for an artificial kidney to remove $50\%$ of the urea from the blood in a $3-$hr dialysis session, using the calculated flux?