Solve in MathematicaThe total and species mass balance equations for a filtration system used to concentrate fruit

juice can be written as:

$\frac{dV}{dt}=-Q$

$\frac{d}{dt}(VC)=-QSC$

Here $V$ is the tank volume, $Q$ is the constant flow rate through the membrane, $C$ is the

concentration of flavor in the tank, and $S$ is the leakage ratio of flavor through the membrane

$(S=0$ corresponds to no leakage; $S=1$ corresponds to no membrane selectivity$).$

Note that we can simplify the problem by working on a unit volume basis $[$take $V(0)=1],$

working in relative units of concentration $[$take $C(0)=1],$ and measuring time in units of

volume flux $[$take $Q=1].$

To do the problem:

a$)$ Use DSolve$[]$ to obtain $V(t)$ and $C(t).$

b$)$ For $S=\frac{1}{5},$ plot $V(t)$ and $C(t)$ together for $t$ from $t=0$ to $t=1.$

c$)$ By what factor must we reduce $V,$ to double the concentration?

d$)$ What fraction of the flavor is thrown away in the process?