$8\mathrm{.}1.$ 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.

b$)$ Example $3\mathrm{.}7$ with T as the controlled variable, F as the manipulated

variable, and ASP $=3\backslash$deg C$.($Fc is constant.$)$

EXAMPLE $3\mathrm{.}7.$ Stirred$-$tank heat exchanger

To provide another simple example of an energy balance, the stirred$-$tank heat

exchanger in Figure $3\mathrm{.}9$ is considered.

Goal. The dynamic response of the tank temperature to a step change in the

coolant flow is to be determined.

Information. The system is the liquid in the tank.

Assumptions.

The tank is well insulated, so that negligible heat is transferred to the sur$-$

roundings.

The accumulation of energy in the tank walls and cooling coil is negligible

compared with the accumulation in the liquid.

The tank is well mixed.

Physical properties are constant.

The system is initially at steady state.

Data. $F=0\mathrm{.}085\frac{m^3}{m}in$;$V=2\mathrm{.}1m^3$;$T_s=85\mathrm{.}4C$;$={10}^6\frac{g}{m^3}$;$C_p=1ca\frac{l}{gC}$;

$T_0=150C$;$T_{\text{c}\text{i}\text{n}}=25C$;$F_{cs}=0\mathrm{.}50\frac{m^3}{m}in$;$C_{pc}=1ca\frac{l}{gC}$;${}_c={10}^6\frac{g}{m^3}$;

$a=1\mathrm{.}41{10}^{5}ca\frac{l}{m}inC$;$b=0\mathrm{.}50$