$$

Q$2$:

$(25$ Marks$)$

A mixing process may be described as follows: A stream with solute concentration $C_i($pounds$/$volume$)$ is fed to a perfectly stirred tank at a constant flow rate of $q($volume$/$time$).$ The perfectly mixed product is withdrawn from the tank, also at the flow rate $q$ at the same concentration as the material in the tank

C$.$ The total volume of solution in the tank is constant at

V$.$ Density may be considered to be independent of concentration. A trace of the tank output concentration versus time appears as shown below. Assume that $C_{iisa}$ sine wave.

$($a$)$ Obtain an expression for $C(t)$ as a function of time.

$($b$)$ Plot on this same figure your best guess of the quantitative behavior of the inlet concentration versus time.

Data: tank dimensions: $8ft$ high by $5-$ft diameter, Tank volume V: $700$gal, Flow rate $q$ : $100ga\frac{l}{m}in,$ and Average density: $70l\frac{b}{f}{t}^3$