$4(15).$ For conventional filtration, we derived in class the expression below that describes the change in total volume of filtrate, $V,$ with time if the cake is incompressible. We discussed the possibilities of running a conventional filtration operation at constant pressure drop, $p$; constant volumetric flow rate, Q; and constant pump power.

$($a$)$ If we wish to operate at constant $Q,$ derive the expression for $p$ as a function of time $($also given below$)$ from the expression for $d\frac{V}{d}t.$ You may ignore the membrane resistance, $R_M.$

$($b$)$ If $300f{t}^3$ of suspension are to be filtered in $1hr$ at a constant volumetric flow rate, calculate the $p($in psi$)$ at the conclusion of the filtration process. Use the following data: $A=100f{t}^2$;$=2\mathrm{.}73{10}^{11}f\frac{t}{l}{b}_m$;$=6{10}^{-4}l\frac{b}{\frac{?}{?}}(fts)$;${}_o=1\mathrm{.}5l\frac{b}{\frac{?}{?}}f{t}^3.$

$\frac{1}{A}\frac{dV}{(d)t}=\frac{p}{[{}_o(\frac{V}{A})+R_M]},p={}_o(\frac{Q}{A}{)}^{2}t$