$11\mathrm{.}18$ It is desired to control the exit temperature T$2$ of the heat exchanger shown in Fig. E$11\mathrm{.}18$ by adjusting the steam flow rate ws$.$ Unmeasured disturbances occur in inlet temperature T$1.$ The dynamic behavior of the heat exchanger can be approximated by the transfer functions:$11\mathrm{.}18It$ is desired to control the exit temperature $T_2$ of the

heat exchanger shown in Fig. E$11\mathrm{.}18$ by adjusting the steam

flow rate $w_s.$ Unmeasured disturbances occur in inlet temper$-$

ature $T_1.$ The dynamic behavior of the heat exchanger can be

approximated by the transfer functions

$\frac{T_2^{\text{'}}(s)}{W_s^{\text{'}}(s)}=\frac{2\mathrm{.}5e^{-s}}{10s+1}[=]\frac{F}{l\frac{b}{s}}$

$\frac{T_2^{\text{'}}(s)}{T_1^{\text{'}}(s)}=\frac{0\mathrm{.}9e^{-2s}}{5s+1}[=]$ dimensionless

where the time constants and time delays have units of seconds.

The control valve has the following steady$-$state characteristic:

$w_s=0\mathrm{.}6\sqrt[\phantom{2}]{p-4}$

where $p$ is the controller output expressed in $mA.$ At the nom$-$

inal operating condition, $p=12mA.$ After a sudden change in

the controller output, $w,$ reaches a new steady$-$state value in

$20s($assumed to take five time constants$).$ The temperature

transmitter has negligible dynamics and is designed so that its

Figure E$11\mathrm{.}18$

output signal varies linearly from $4$ to $20mA$ as $T_2$ varies from

$120$ to $160F.$

$($a$)$ If a proportional$-$only feedback controller is used, what is

$K_{cm}?$ What is the frequency of the resulting oscillation when

$K_c=K_{cm}?($Hint: Use the direct$-$substitution method and

Euler's identity.$)$

$($b$)$ Estimate $K_{on}$ using direct substitution and a $\frac{1}{1}$ Pad$$

approximation for the time$-$delay term. Does this analysis

provide a satisfactory approximation?

where the time constants and time delays have units of seconds.

The control valve has the following steady$-$state characteristic:

where p is the controller output expressed in mA$.$ At the nominal operating condition, p $=12$ mA$.$ After a sudden change in the controller output, ws reaches a new steady$-$state value in $20$ s $($assumed to take five time constants$).$ The temperature transmitter has negligible dynamics and is designed so that its output signal varies linearly from $4$ to $20$ mA as T$2$ varies from $120$ to $160$F$.$

$($a$)$ If a proportional$-$only feedback controller is used, what is Kcm$?$ What is the frequency of the resulting oscillation when Kc $=$ Kcm$?($Hint: Use the direct$-$substitution method and Euler$$s identity.$)$

$($b$)$ Estimate Kcm using direct substitution and a $1/1$ Pad$$ approximation for the time$-$delay term. Does this analysis provide a satisfactory approximation?