You are working on a low-temperature and energy-efficient method of concentrating syrup for a sugar company. Initial experiments are done with aqueous sucrose solutions and an RO system with a cellulose acetate membrane at \(25.0^{\circ} \mathrm{C}\).

Data: Density of solvent (water) is \(ho=0.997 \mathrm{~kg} / \mathrm{L}\).

Density ( \(\mathrm{kg} / \mathrm{L}\) ) of dilute aqueous sucrose solutions is \(ho=0.997+0.4 \mathrm{x}\) where \(\mathrm{x}\) is weight fraction sucrose.

At low sucrose weight fraction, osmotic pressure (in atm) at \(\mathrm{T}=25^{\circ} \mathrm{C}\) can be estimated as \(\pi=59.895 \mathrm{x}\) where \(\mathrm{x}\) is the weight fraction sucrose.

Molecular weight of water is 18.016 . Molecular weight of sucrose is 342.3 .

Experiment A. This experiment is done in a well-mixed stirred tank. At \(1000.0 \mathrm{rpm}\) with a \(3.0 \mathrm{wt} \%\) solution of sucrose in water using \(p_{r}=75.0 \mathrm{~atm}\) and \(p_{p}=2.0 \mathrm{~atm}\), we obtain \(J_{\text {solv }}^{\prime}=4.625 \mathrm{~g} /\left(\mathrm{m}^{2} \mathrm{~s}\right)\). Mass transfer coefficient \(\mathrm{k}=6.94 \times 10^{-5} \mathrm{~m} / \mathrm{s}\). We measure \(\mathrm{x}_{\mathrm{r}}=\mathrm{x}_{\text {out }}=0.054\) and \(\mathrm{x}_{\mathrm{p}}=3.6 \times 10^{-4}\).

a. Calculate concentration polarization modulus \(\mathrm{M}\).

b. Calculate selectivity \(\alpha^{\prime}, \mathrm{K}_{\text {solv }}^{\prime} / \mathrm{t}_{\mathrm{ms}}\), and \(\mathrm{K}_{\mathrm{A}}^{\prime} / \mathrm{t}_{\text {ms. }}\).

c. Estimate mass transfer coefficient \(\mathrm{k}\) if rpm is increased to 2000.0 .