We are doing \(\mathrm{RO}\) of dilute aqueous sucrose solution at \(25.0^{\circ} \mathrm{C}\). A feed that is \(2.20 \mathrm{wt} \%\) sucrose is separated in a very-well-stirred system \((M=1.0)\) with \(p_{r}=60.0 \mathrm{~atm}\) and \(p_{p}=1.1 \mathrm{~atm}\). The flux \(J^{\prime}\) water \(=3.923 \mathrm{~g} /\left(\mathrm{m}^{2} \mathrm{~s}\right)\) when \(\mathrm{x}_{\mathrm{p}}=0.00032\) and \(\mathrm{x}_{\mathrm{r}, \text { out }}=0.056\). Data are in Problem 19.D4.

Problem 19.D4.

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 cut \(\theta^{\prime}\), inherent rejection \(R^{0}\), and sucrose flux \(J^{\prime}\) sucrose.

b. Calculate selectivity \(\alpha_{\text {water-sucrose }}^{\prime}, \mathrm{K}_{\text {water }}^{\prime} / \mathrm{t}_{\mathrm{ms}}, \mathrm{K}_{\text {sucrose }}^{\prime} / \mathrm{t}_{\text {ms }}\), and \(\mathrm{J}_{\text {water. }}^{\prime}\).

c. If we now repeat the experiment with less mixing so that \(\mathrm{M}=2.1\) with \(\mathrm{x}_{\text {feed }}=0.0220, \mathrm{p}_{\mathrm{r}}=60.0 \mathrm{~atm}, \mathrm{p}_{\mathrm{p}}=\) \(1.1 \mathrm{~atm}\), and \(\theta^{\prime}=0.61\), calculate \(\mathrm{x}_{\mathrm{r}}, \mathrm{x}_{\mathrm{p}},{ }^{\prime}{ }^{\prime}\) water, and \(\mathrm{J}^{\prime}\) sucrose.