A paint company is concentrating latex particles in an aqueous suspension with a UF module. The module is perfectly mixed on the retentate side and operates with \(p_{p}=1.0\) bar and \(p_{r}=2.2\) bar. Because of extensive stirring, concentration polarization modulus is fairly low, \(\mathrm{M}=1.2\), and no gel forms \(\left(\mathrm{x}_{\text {wall }}<\mathrm{x}_{\text {gel }}=0.5\right)\). Osmotic pressure of latex particles is negligible. Density of solutions can be assumed to equal density of pure water \(=0.997 \mathrm{~kg} / \mathrm{L}\). All latex particles are retained by the membrane \(\left(\mathrm{R}^{\mathrm{o}}=1.0\right)\). Feed to UF module is \(\mathrm{x}_{\mathrm{r}, \text { in }}=\) 0.10 weight fraction latex (for all parts). We operate with a cut \(\theta^{\prime}=0.20\). Feed rate of suspension, \(\mathrm{F}_{\mathrm{r}, \mathrm{in}}^{\prime}\), is \(100.0 \mathrm{~kg} / \mathrm{h}\). Flux rate of the membrane with pure water is \(2500.0 \mathrm{~L} /\left(\mathrm{m}^{2}\right.\) day \()\).

a.* Find \(\mathrm{F}_{\mathrm{r}, \text { out }}^{\prime}, \mathrm{x}_{\mathrm{p}}\), and \(\mathrm{x}_{\mathrm{r}, \text { out }}\).

b. What membrane area is required?

c. If we decrease stirring and \(M\) increases, at what value of \(M\) will gel formation occur \(\left(\theta^{\prime}=0.20\right)\) ?

d. At \(\mathrm{M}=1.2\), what value of the cut \(\theta^{\prime}\) will cause gel formation to occur?

e. If \(M=1.2\) and \(\theta^{\prime}=0.20\), at what feed weight fraction will gel formation occur? What is \(\mathrm{x}_{\mathrm{r}, \mathrm{out}}\) ?

f. We are doing an experiment of slowly decreasing the amount of stirring. If gel formation first occurs with \(\mathrm{x}_{\mathrm{F}}=\) 0.20 and \(\theta^{\prime}=0.25\), what are the values of \(\mathrm{M}\) and mass transfer coefficient \(\mathrm{k}\) with this amount of stirring? \(\mathrm{J}_{\text {solv }}=\) 2500.0 , but the area has changed.

g. With the same amount of stirring as in part \(\mathrm{f}\) but otherwise new conditions, we do an experiment with \(\mathrm{x}_{\mathrm{F}}=0.20\), \(\mathrm{p}_{\mathrm{r}}=3.4 \mathrm{bar}, \mathrm{p}_{\mathrm{p}}=1.0 \mathrm{bar}\), and \(\theta^{\prime}=0.20\). Does a gel form? What is the solvent flux in \(\mathrm{L} /\left(\mathrm{m}^{2}\right.\) day \()\) ?

h. We do an experiment with the same conditions as part \(\mathrm{g}\), except \(\theta^{\prime}=0.26\). Does a gel form? What is the solvent flux in \(L /\left(\mathrm{m}^{2}\right.\) day \()\) ?