$3.(30)$ Recall the mass balance on solute in a continuous stirred tank reactor $($CSTR$)$ derived in class,

$lonV\frac{dc}{dt}=H(\frac{c}{F}-c)-(1-lon)V\frac{dq}{dt}$

where $lon$ is the fluid fraction of the tank working volume, $V$;$c$ is the solute concentration in the fluid, $c_F$ is the feed concentration in the fluid; and $q$ is the adsorbate $($adsorbed solute$)$ concentration on the adsorbent.

$($a$)$ State any assumptions that were made in deriving the mass balance above. We will consider an idealized case of instantaneous desorption, so modify the balance above for the situation where no solute is being fed. In addition, assume a linear isotherm, $q=Kc,$ and that equilibrium is achieved very quickly in the tank at any instant. Further modify the equation above appropriately, and make the mass balance equation dimensionless using the following definitions

definitions $\frac{c}{c_F},T=\frac{t}{[lon+(1-lon)K]\frac{V}{H}},Q=\frac{q}{c_F}$

$d$

$\frac{dc}{dt}*\frac{dH}{dL}=\frac{dc}{d}$

You should obtain the following simple differential equation,

$\frac{dC}{d}=-C$

For the initial condition, consider the special case where the adsorbent previously had come to equilibrium with the feed concentration, $C_F,$ but now instantaneously releases all bound solute at time zero. With this initial condition, solve the differential equation to obtain $C$ as a fulilction of $T.$

$($b$)$ Determine the initial slope of $C$ versus $$ and the value of $T$ where $C=0\mathrm{.}01.$

$($c$)$ Sketch a plot of $C$ versus $T$ and make any appropriate marks on the axes that are relevant to your plot.

$($d$)$ Consider the case where a second CSTR of exactly the same design as the first is added downstream so that there are two identical tanks in series. In this arrangement, the effluent from the first tank is the feed to the second tank. The solute concentration in the second tank is given by $C_2$ and the corresponding dimensionless concentration, $C_2,$ equals $\frac{c_2}{C_F}.$ Consider the case where instantaneous desorption also is occurring in the second tank and derive a dimensionless mass balance for the solute in the second tank similar to that in part a for the first tank.

$($e$)$ Wityout solving the differential equation obtained in part d$,$ determine the initial slope of $C_2$ ver us at $T=0),$ and sketch a plot of $C_2$ versus $T$ making sure to note key features of the plot including any appropriate marks on the axes. Add an additional representative sketch to the plot of the expected dimensionless concentration versus $T$ for the effluent from the last in a series of many identical CSTRs where desorption also is taking place. Compare the features of the two sketches on this plot and state clearly why and in what way one may expect them to be different.