Your company's well water, which contains \(190.0 \mathrm{ppm}\) (weight) of salt, is fine for most purposes but not for a particular manufacturing process that needs purer water. A vendor is trying to sell you a spiral-wound RO unit that has an inherent salt rejection coefficient of \(R^{\circ}=0.98\), a pure water flux of \(J^{\prime}{ }_{\text {water }}=102.0 \mathrm{~g} /\left(\mathrm{m}^{2} \mathrm{~s}\right)\) at \(\Delta p=\) \(35.0 \mathrm{bar}\), and approximately \(600.0 \mathrm{~m}^{2}\) of area per \(\mathrm{m}^{3}\) volume. Modules are available with volumes of 0.5 \(\left(300.0 \mathrm{~m}^{2}\right), 1.0\left(600.0 \mathrm{~m}^{2}\right)\), and \(2.0 \mathrm{~m}^{3}\left(1200.0 \mathrm{~m}^{2}\right)\). Modules can be put in parallel or in series. Your company wants to produce \(20,000 \mathrm{~m}^{3} /\) day of water that contains \(12.0 \mathrm{ppm}\) or less of salt. The retentate, at \(480.0 \mathrm{ppm}\) and 1.4 bar, will be used elsewhere in the plant. Inlet pressure of the water is 28.0 bar and pressure cannot be safely increased without a major capital expense. Operation will be at \(25.0^{\circ} \mathrm{C}\). The salesperson wants to do the design for the company (at a price) and will not release the vendor's other data. However, they estimate that two large and one medium module will provide the water your company wants.

a. Using available data, estimate the area of the membrane and the operating conditions required to produce the desired permeate and retentate products. Use mass transfer data for spiral-wound systems, but do calculations for a perfectly mixed system.

b. How did the salesperson calculate the number?

Data: Typical spiral-wound unit: channels 0.03 to \(0.1 \mathrm{~cm}\), velocity \(=0.5\) to \(1.5 \mathrm{~m} / \mathrm{s}, \mathrm{Re}=500.0\) to 1000.0 (Wang and Zhou, 2013).