Problem $3.$ D$5.$ You are working on a low$-$temperature and energy$-$efficient method of

concentrating syrup for a sugar company by using a cellulose acetate RO membrane to concentrate

a dilute sucrose mixture. Operation is at $25\mathrm{.}0C.$

Data: Density of solvent $($water$)$ is $=0\mathrm{.}997k\frac{g}{L}.$

Density $(k\frac{g}{L})$ of dilute aqueous sucrose solutions is $=0\mathrm{.}997+0\mathrm{.}4x$ where $x$ is weight fraction

sucrose.

At low sucrose weight fraction, osmotic pressure $($in atm$)$ at $T=25C$ can be estimated as

$=59\mathrm{.}895x$ where $x$ is the weight fraction sucrose.

Molecular weight of water is $18\mathrm{.}016.$ Molecular weight of sucrose is $342\mathrm{.}3.$

$($Note: Some of this data may not be needed to solve this problem.$)$

a$.$ We measure pure water flux $($no sucrose present$)$ and find at a pressure drop of $102\mathrm{.}0$atm across

the membrane, $J_{\text{s}\text{o}\text{l}\text{v}}=1\mathrm{.}5{10}^{-3}[\frac{g}{c{m}^2(s)}].$ An experiment in a highly stirred system $(M=1\mathrm{.}0)$ with

a dilute sucrose solution gives a rejection $R=0\mathrm{.}997$ for an inlet weight fraction of $0\mathrm{.}050,$ and cut

${}^{\text{'}}=0\mathrm{.}45($in weight units$).$ Find $\frac{K_{\text{s}\text{o}\text{l}\text{v}}^{\text{'}}}{t_{ms}}$ and selectivity ${}^{\text{'}}.$

b$.$ We will design a perfectly mixed RO system to operate at conditions where $M=3\mathrm{.}0.$ Feed is $2\mathrm{.}0$

wt $\%$ sucrose. We want a cut of ${}^{\text{'}}=\frac{2}{3}.$ Feed rate will be $5\mathrm{.}0k\frac{g}{s}.$ Pressure drop across the

membrane is $78\mathrm{.}0$atm. Plot the RT curve, Eq$.(19-19$a$),$ and find the intersection with the operating

line, Eq$.(19-22),$ to find $x_p$ and $x_r=x_{\text{o}\text{u}\text{t}}.$ This is a completely nonlinear $($except for linearizing

osmotic pressure dependence on sucrose concentration$)$ result. Then find $J_{\text{'}\text{s}\text{o}\text{l}\text{v}}$ and the membrane

area.