Assuming that the simpler conditions of Problems $4\mathrm{.}2($a$)$ apply and that the objective

is to control the water temperature T$1$ by adjusting T $.$

$($a$)$ Which of the variables indicated in Eq$.($P$4\mathrm{.}1)$ is the input variable, and which is

the disturbance variable?

$($b$)$ By defining appropriate deviation variables $($in terms of T$1,$ T$2,$ and T$0,$ the

steady$-$state values of T$1,$ T$2,$ T$0$ respectively$)$ rewrite Eq$.($P$4\mathrm{.}1)$ in the form:

t$!$x dt $=$ ay $+$ bu $+")$d $($P$4\mathrm{.}2)$

Explicitly state a$,$ b$,$ andy in terms of process parameters p$,$ CP$,$ c$,$ F$,$ and V$.$

$($c$)$ Express this differential equation model in the transform$-$domain transfer

function form. What are g$($s$)$ and gd$($s$)$ in this case?

$($d$)$ Express the differential equation model in the impulse$-$response form either

directly, or using g$($s$)$ and gd$($s$)$ obtained in $($c$)\mathrm{.}4\mathrm{.}2($a$)$ Figure P$4\mathrm{.}2$ shows a system for heating a continuous flow water kettle using a

hot plate. Assuming that the hot plate temperature can be changed instantaneously

by adjusting the hot plate rate of heat input and assuming uniform water kettle

temperature, show that the following is a reasonable model for the process:

$V{C}_p\frac{d{T}_1}{dt}=c(T_2-T_1)+F{C}_p{T}_0-F{C}_{}{T}_1$

where $,C_p$ are, respectively, water density and heat capacity, and

$c=$a constant

$F=$ volumetric flowrate in and out of the kettle.

Figure P$4\mathrm{.}2.$