Example $1$: The whey produced at a cheese plant is to be further

processed by UF at $293K$ to recover the valuable proteins that it

contains. Tubular membranes will be used with a diameter of $2cm,$ a

clean$-$membrane water permeance of $250\frac{L}{m^2}*h*atm,$ and a rejection

of $100\%$ for the whey proteins. The average diffusivity of the whey

proteins at this temperature is $4{10}^{-7}c\frac{m^2}{s}.$ The protein

concentration of the whey at a certain point in the equipment is $30$

$\frac{g}{L},$ and the whey flows through the tube at a velocity of $1\mathrm{.}5\frac{m}{s}.$

The osmotic pressure of whey can be related to its protein

concentration through the equation:

$(atm)=3\mathrm{.}1646c+8\mathrm{.}749c^2+57\mathrm{.}915c^3$ for $c0\mathrm{.}35k\frac{g}{L}$

Mass transfer correlation for turbulent flow inside circular pipes is as

follow: $Sh=0\mathrm{.}0048R{e}^{0\mathrm{.}913}S{c}^{0\mathrm{.}346}($a$)$Calculate the required pressure differential

to produce a water transmembrane volume flux of $25$ L$/$m$2$h when the

membrane is clean. $($b$)$If the membrane permeance is reduced fivefold

by fouling, calculate the water flux if the applied pressure differential

remains the same.

$($a$)1\mathrm{.}24$ atm, $($b$)22$ liter$/$m$2$ hr

$9$