Chapter 28, Problem 10P

The reaction $A\to B$ takes place in two reactors in series.

There actors are well mixed but are not at steady state. The unsteady-state mass balance for each stirred tank reactor is shown below:

$\begin{array}{l}\frac{dC{A}_1}{dt}=\frac{1}{\tau }\left(C{A}_0-C{A}_1\right)-kC{A}_1\\ \frac{dC{B}_1}{dt}=-\frac{1}{T\tau }C{B}_1+kC{A}_1\\ \frac{dC{A}_2}{dt}=-\frac{1}{\tau }\left(C{A}_1-C{A}_2\right)-kC{A}_2\\ \frac{dC{B}_2}{dt}=\frac{1}{\tau }\left(C{B}_1-C{B}_2\right)+kC{A}_2\end{array}$

where $C{A}_0=$ concentration of A at the inlet of the first reactor, $C{A}_1=$ concentration of A at the outlet of the first reactor (and inlet of the second), $C{A}_2=$ concentration of A at the outlet of the second reactor, $C{B}_1=$ concentration of B at the outlet of the fi rst reactor (and inlet of the second), $C{B}_2=$ concentration of B in the second reactor $\tau =$ residence time for each reactor, and $k=$ the rate constant for reaction of A to produce B. If $C{A}_0$ A 0 is equal to 20, find the concentrations of A and B in both reactors during their first 10 minutes of operation. Use $k=0.12/\text{min and }\tau \text{=5}$ min and assume that the initial conditions of all the dependent variables are zero.