Chapter 1.2, Problem 35P

A chemical of fixed concentration $c_i$ flows into a continuously stirred tank reactor at a constant volume flow rate $r_i$ and flows out at the same rate. While in the reactor, the chemical undergoes a simple reaction in which it disappears at a rate proportional to the concentration:

$\frac{dc}{dt}=\frac{r_i}{V}{c}_i-r_i\frac{c}{V}-kc$, (i)

Where $V$ is the volume of the reactor and $k$ is the rate of reaction.

Use the dimensionless variables

$C=\frac{c}{c_i},\tau =\frac{t}{V/r_i}$

to express Eq. (i) in dimensionless form

$\frac{dC}{d\tau }=1-C-\alpha C$, (ii)

Where $\alpha =\frac{kV}{r_i}$.

Determine the equilibrium solution of Eq. (ii), draw a phase line diagram, and determine whether the equilibrium solution is asymptotically stable or unstable. Then sketch a phase portrait with a representative set of solution curves.