Q$5(20$ pts$).$ We have solved the question below in the classroom $($Note: The question and

its solution is provided below$)$:

In patients with severe kidney disease, urea must be removed from the blood with a "hemodialyzer."

In that device, the blood passes by special membranes through which urea can pass. A salt solution

$("$dialysate$")$ flows on the other side of the membrane to collect the urea and to maintain the de$-$

sired concentrations of vital salts in the blood. One geometry for hemodialyzer design is with flat

membranes in a rectangular system. For such a geometry, consider the following typical values:

Blood side:

mass$-$transfer coefficient for the urea

average urea concentration within the dialyzer

Dialysate side:

mass$-$transfer coefficient for the urea

average urea concentration within the dialyzer

Membrane:

thickness

diffusivity of urea in the membrane

total membrane area

porosity

$0\mathrm{.}0019c\frac{m}{s}$

$0\mathrm{.}020$gmo$\frac{l}{L}$

$0\mathrm{.}0011c\frac{m}{s}$

$0\mathrm{.}003$gmo$\frac{l}{L}$

$0\mathrm{.}0016cm$

$1\mathrm{.}8{10}^{-5}c\frac{m^2}{s}$

$1\mathrm{.}2m^2$

$20\%$

Based on these values, what is the initial removal rate of urea?

Solution

$\{\begin{array}{c}\text{:}\frac{1(m)}{100(cm)}\frac{60(s)}{\text{m}\text{i}\text{n}}=0\mathrm{.}0065\frac{\text{g}\text{m}\text{o}\text{l}}{\text{m}\text{i}\text{n}}\end{array}$

Your company manufactures hemodialyzers that have the characteristics described in the

above example. A colleague in the company has proposed replacing the membranes with

better ones, which have the same thickness, area, and porosity but for which the urea diffusivity

in the membrane is $2\mathrm{.}7{10}^{-5}c\frac{m^2}{s}.$

Assuming that the average concentrations of urea in the blood and dialysate are the same as

with the old membranes, by what percentage would the new membranes increase the urea

removal rate? In terms of resistances, explain why this turns out to be such a small

improvement.