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## Q. 15.3

It is desired to reduce the concentration of $CO _{2}$ in the flue gas from a coal-fired power plant, in order to reduce greenhouse gas emissions. The effluent flue gas is sent to an ammonia scrubber, where most of the $CO _{2}$ is absorbed in a liquid ammonia solution, as shown in Fig. E15.3. A feedforward control system will be used to control the $CO _{2}$ concentration in the flue gas stream leaving the scrubber $C _{ CO _{2}}$ which cannot be measured on-line. The flow rate of the ammonia solution entering the scrubber $Q_{A}$ can be manipulated via a control valve. The inlet flue gas flow rate $Q_{F}$ is a measured disturbance variable.

(a) Draw a block diagram of the feedforward control system. (It is $n o t$ necessary to derive transfer functions.)

(b) Design a feedforward control system to reduce $CO _{2}$ emissions based on a steady-state design (Eq. 15-40).

$G_{f}=K_{f}=\frac{-K_{d}}{K_{v} K_{t} K_{p}}$           (15-40)

Available Information:

(i) The flow sensor-transmitter and the control valve have negligible dynamics.

(ii) The flow sensor-transmitter has a steady-state gain of $0.08 mA /( L / min ) .$

(iii) The control valve has a steady-state gain of 4 $( gal / min ) / mA .$

(iv) The following steady-state data are available for a series of changes in $Q_{A}$ :

 $Q_{A}( gal / min )$ $C_{C O_{2}}( ppm )$ 15 125 30 90 45 62

(v) The following steady-state data are available for a series of changes in $Q_{F}$,

 $Q_{F}( L / min )$ $C_{C O_{2}}( ppm )$ 100 75 200 96 300 122

## Verified Solution

(a) Block diagram of the feedforward control system

(b) Feedforward design based on a steady-state analysis

The starting point in feedforward controller design is Eq. 15-21. For a design based on a steady-state analysis, the transfer functions in $(15-21)$ are replaced by their corresponding steady-state gains:

$G_{f}=-\frac{G_{d}}{G_{t} G_{v} G_{p}}$              (15-21)

$G_{F}(s)=-\frac{K_{d}}{K_{t} K_{v} K_{p}}$           (1)

From the given information,

\begin{aligned}K_{t} &=0.08 \frac{ mA }{ L / min } \\K_{v} &=4 \frac{ gal / min }{ mA }\end{aligned}

Next, calculate $K_{p}$ and $K_{d}$ from the given data. Linear regression gives:

\begin{aligned}K_{p} &=-2.1 \frac{ ppm }{ gal / min } \\K_{d} &=0.235 \frac{ ppm }{ L / min }\end{aligned}

Substitute these gains into (1) to get:

\begin{aligned}&G_{F}(s)=-\frac{0.235 \frac{ ppm }{ L / min }}{\left(0.08 \frac{ mA }{ L / min }\right)\left(4 \frac{ gal / min }{ mA }\right)\left(-2.1 \frac{ ppm }{ gal / min }\right)} \\&G_{F}(s)=0.35\end{aligned}