Consider the problem of designing a dialysis membrane $($the individual tubes in a hollow fiber dialyzer, as discussed in Section $12\mathrm{.}1\mathrm{.}4).$ A schematic for transport of urea from the blood through a tube wall is shown in Figure $12\mathrm{.}23$ where the tube wall can be considered as a flat surface due to it being thin compared to the radius of the tube.

Convection occurs on both sides of the wall. Through materials research, there is the potential to come up with a membrane material that has twice the diffusivity. With this new membrane material, the thickness can remain the same as the existing membrane material at $25um.$ The distribution coefficient of the existing membrane material for a urea solution is $2$ and is also expected to remain unchanged. The concentrations of the dialyzing fluid in blood and urea will stay the same for both membranes as well. The diffusivity of the existing membrane material to urea is $1{10}^{-10}\frac{m^2}{s},$ and its flux value is $0\mathrm{.}001g$ of urea $\frac{?}{m^2}.$s$.$ The concentration of urea in the blood is $0\mathrm{.}0002\frac{g}{c}{m}^3.1)$ Through steady$-$state mass transfer analysis of transport through the membrane, show by what factor the flux would change $($ratio of new flux to old flux$)$ if the diffusivity were doubled. $2)$ Explain why the flux does not double even if the diffusivity is doubled. $3)$ If this new material were not available $($diffusivity of the membrane could not be increased$)$ and the membrane could not be made any thinner, how would you try to increase the rate of urea transport across the membrane?

$**$THE ANSWER TO #$1$ IS $1\mathrm{.}45.$ PLEASE DO NOT SUBMIT A SOLUTION IF YOU DO NOT GET THIS ANSWER