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Chapter 15

Q. 15.11

A feedforward-only control system is to be designed for the stirred-tank heating system shown in Fig. E15.11. Exit temperature T will be controlled by adjusting coolant flow rate, q_{c}. The chief disturbance variable is the inlet temperature T_{i} which can be measured on-line. Design a feedforward-only control system based on a dynamic model of this process and the following assumptions:

1. The rate of heat transfer Q between the coolant and the liquid in the tank can be approximated by

Q=U\left(1+q_{c}\right) A\left(T-T_{c}\right)

where U, A, and the coolant temperature T_{c} are constant.

2. The tank is well mixed, and the physical properties of the liquid remain constant.

3. Heat losses to the ambient air can be approximated by the expression Q_{L}=U_{L} A_{L}\left(T-T_{a}\right), where T_{a} is the ambient temperature.

4. The control valve on the coolant line and the T_{i} sensor/ transmitter (not shown in Fig. E15.11) exhibit linear behavior. The dynamics of both devices can be neglected, but there is a time delay \theta associated with the T_{i} measurement due to the sensor location.


Verified Solution

Energy Balance:

\rho V C \frac{d T}{d t}=w C\left(T_{i}-T\right)-U\left(1+q_{c}\right) A\left(T-T_{c}\right)-U_{L} A_{L}\left(T-T_{a}\right)               (1)

Expanding the RHS,

\begin{aligned}\rho V C \frac{d T}{d t}=& w C\left(T_{i}-T\right)-U A\left(T-T_{c}\right) \\&-U A q_{c} T+U A q_{c} T_{c}-U_{L} A_{L}\left(T-T_{a}\right)\end{aligned}            (2)

Linearizing the nonlinear term,

q_{c} T \approx \bar{q}_{c} \bar{T}+\bar{q}_{c} T^{\prime}+\bar{T} q_{c}^{\prime}             (3)

Substituting (3) into (2), subtracting the steady-state equation, and introducing deviation variables,

\begin{aligned}\rho V C \frac{d T^{\prime}}{d t}=& w C\left(T_{i}^{\prime}-T^{\prime}\right)-U A T^{\prime}-U A \bar{T} q_{c}^{\prime}-U A \bar{q}_{c} T^{\prime} \\&+U A T_{c} q_{c}^{\prime}-U_{L} A_{L} T^{\prime}\end{aligned}               (4)

Taking the Laplace transform and assuming steady-state at t =0 gives,

\begin{aligned}\rho V C s T^{\prime}(s) &=w C T_{i}^{\prime}(s)+U A\left(T_{c}-T^{\prime}\right) q_{c}^{\prime}(s) \\&-\left(w C+U A+U A \bar{q}_{c}+U_{L} A_{L}\right) T^{\prime}(s)\end{aligned}                (5)


T^{\prime}(s)=G_{L}(s) T_{i}^{\prime}(s)+G_{p}(s) q_{c}^{\prime}(s)                (6)


\begin{aligned}&G_{d}(s)=\frac{K_{d}}{\tau s+1} \\&G_{p}(s)=\frac{K_{p}}{\tau s+1}\end{aligned}

\begin{aligned}K_{d} &=\frac{w C}{K}              (7) \\K_{p} &=\frac{U A\left(T_{c}-\bar{T}\right)}{K} \\\tau &=\frac{\rho V C}{K} \\K &=w C+U A+U A \bar{q}_{c}+U_{L} A_{L}\end{aligned}

The ideal FF controller design equation is given by,

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

But, G_{t}=K_{t} e^{-\theta s} and G_{v}=K_{v}            (8)

Substituting (7) and (8) gives,

G_{F}=\frac{-w C e^{+\theta s}}{K_{t} K_{v} U A\left(T_{c}-\bar{T}\right)}             (9)

In order to have a physically realizable controller, ignore the e ^{+\theta s} term,

G_{F}=\frac{-w C}{K_{t} K_{v} U A\left(T_{c}-\bar{T}\right)}                  (10)