Consider a dialysis cartridge in which blood flows through the inner tubes and buffer

flows through the outer "shell". In order to be transported from the inner tube to the

outer shell, the urea in the blood $($a$.$k$.$a$.$ Blood Urea Nitrogen or BUN$)$ must overcome

not only the diffusive resistance of the membrane, but also the convectional mass

transfer resistance on both the tube and the shell sides of the membrane.

The membrane itself can be considered as flat surface due to it being thin compared to

the radius of the tube. The diffusivity of the existing membrane material to urea is $1$

${10}^{-10}\frac{m^2}{s},$ and the partition coefficient of urea for the membrane material with respect

to aqueous solution is $2.$ Through materials research, there is the potential to develop a

new membrane material in which the diffusivity of urea will be double that of the

existing membrane. However, the partition coefficient and thickness is expected to

remain unchanged. If the thickness of the existing membrane is $25m,$ and the flux of

BUN through it is $0\mathrm{.}001g$ of urea $\frac{?}{m^{2}s}$ when its concentration in the blood is $25m\frac{g}{d}L$

$(25m\frac{g}{100}mL)$ and in the dialyzing fluid is zero,

$($a$)$ Calculate by what factor $($ratio of new to old membrane$)$ the flux would change if the

diffusivity were doubled. Assume that the concentrations of BUN in the blood and in the

dialyzing fluid stay the same, as do the convective mass transfer resistances.

$($b$)$ What could you do to further increase the flux of BUN through the membrane?