$8\mathrm{.}1.$ Determine the analytical expression for a step set point change in the fol

lowing processes under P$-$only and PI feedback control. You should select

values for the tuning constant that give acceptable performance.

c$)$ Example $3\mathrm{.}3$ with CA$2$ as the controlled variable, Cao as the manipu

lated variable, and ASP $=0\mathrm{.}05$ mole$/$m$3.$EXAMPLE $3\mathrm{.}3.$ Two isothermal CSTR reactors

A problem similar to the single CSTR in Example $3\mathrm{.}2$ is presented, with the only

difference that two series reactors are included as shown in Figure $3\mathrm{.}6.$ Each tank

is one$-$half the volume of the tank in Example $3\mathrm{.}2.$

Goal. The same as that of Example $3\mathrm{.}2,$ with the important concentration be$-$

ing in the second reactor. Determine the time when this concentration exceeds

$0\mathrm{.}85$mol$\frac{e}{m^3}.$

Information. The two systems are the liquid in each tank. The data is the same

as in Example $3\mathrm{.}2,$ except that $V_1=V_2=1\mathrm{.}05m^3.$

$F=0\mathrm{.}085\frac{m^3}{m}in$;$(C_{A0}{)}_{\text{i}\text{n}\text{i}\text{t}}=0\mathrm{.}925$mol$\frac{e}{m^3}$;$C_{A0}=0\mathrm{.}925$mol$\frac{e}{m^3}.$

The chemical reaction is first$-$order, ${}_A=-k{C}_A$ with $k=0\mathrm{.}040mi{n}^{-1}.$

The reactor is well mixed and isothermal.