It is desired to control the exit temperature $T_2$ of the heat exchanger by adjusting the steam flow rate $W_s.$ Unmeasured disturbances occur in inlet temperature $T_1.$ The dynamic behavior of the heat exchanger can be approximated by the transfer functions

$\frac{T_2^{\text{'}}(s)}{W_s^{\text{'}}(s)}=\frac{2\mathrm{.}5e^{-2s}}{12s+1},\frac{C}{k\frac{g}{s}}$

$\frac{T_2^{\text{'}}(s)}{T_1^{\text{'}}(s)}=\frac{1\mathrm{.}3e^{-3s}}{5s+1}$ dimensionl

dimensionless

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

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

$w_s=0\mathrm{.}5\sqrt[\phantom{2}]{p-4},$

where $p$ is the controller output expressed in $m$ A$.$ At the nominal operating condition, $p=12mA.$

After a sudden change in the controller output, $w_5$ reaches a new steady$-$state value in $25s($assumed to take five time constants$).$

The temperature transmitter has time constant $1s$ and is designed so that its output signal varies linearly from $4$ to $20mA$ as $T_2$ varies from $50C$ to $90C.$

A$.$ Draw a block diagram.

B$.$ Write a characteristic equation.

C$.$ If a proportional$-$only feedback controller is used, what is $K_{cm}?$ What is the frequency of the resulting oscillation when $K_c=K_{cm}?$ Use the direct$-$substitution method and nonlinear solver $($Matlab$,$ Excel$).$

D$.$ Estimate $Kcm$ using the Routh test.

E$.$ Determine the offset for $Kc=40.$