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 $25\mathrm{.}0\backslash$deg C$.$

Data: Density of solvent $($water$)$ is $\backslash$rho $=0\mathrm{.}997$ kg$/$L$.$

Density $($kg$/$L$)$ of dilute aqueous sucrose solutions is $\backslash$rho $=0\mathrm{.}997+0\mathrm{.}4$x where x is weight fraction sucrose. At low sucrose weight fraction, osmotic pressure $($in atm$)$ at T$=25\backslash$deg C can be estimated as $\backslash$pi $=59\mathrm{.}895$x 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, Jsolv$=1\mathrm{.}5\backslash$times $103[$g$/($cm$2$ s$)].$ An experiment in a highly stirred system $($M$=1\mathrm{.}0)$ with a dilute sucrose solution gives a rejection Ro$=0\mathrm{.}997$ for an inlet weight fraction of $0\mathrm{.}050,$ and cut $\backslash$theta $=0\mathrm{.}45($in weight units$).$ Find Ksolv$/$tms and selectivity $\backslash$alpha $.$ 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 $\backslash$theta $=2/3.$ Feed rate will be $5\mathrm{.}0$ kg$/$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 xp and xr$=$xout. This is a completely nonlinear $($except for linearizing osmotic pressure dependence on sucrose concentration$)$ result. Then find J$$solv and the membrane area.