$9\mathrm{.}5.$ The diagram for a reactor temperature controller cascaded to a coolant flow controller is shown in Fig. P$9-6.$ The control valve is linear with constant pressure drop and

sized for a maximum flow of $500gpm($gallons per minute$)$; its time constant is $0\mathrm{.}2$min. The flow transmitter $($FT$)$ has range of $0$ to $500gpm$ and negligible time constant. The

flow controller $($FC$)$ is proportional integral $($PI$)$ with a gain of $1\mathrm{.}0\%C\frac{O}{\%}TO$ and the integral time set equal to the valve time constant. The transfer function of the reactor temperature to the coolant flow is first$-$order with a gain of $-2\frac{F}{g}pm$ and a time constant of $5\mathrm{.}0$min. The temperature transmitter $($TT$)$ has a range of $160$ to $200F$ and a time constant of $0\mathrm{.}5$min.

$($a$)$ Draw the block diagram for the cascade control loop showing the transfer function of each device. Pay particular attention to the valve fail position and the action $($direct or reverse$)$ of the flow and temperature controllers, and show the appropriate signs for each of their transfer functions.

$($b$)$ Calculate the ultimate gain and period of the temperature control loop.

$($c$)$ Repeat parts $($a$)$ and $($b$)$ assuming the flow controller is removed and the temperature controller directly manipulates the valve. Caution: the controller action may change.