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## Q. 15.13

$It$ is desired to design a feedforward control scheme in order to control the exit composition $x_{4}$ of the two-tank blending system shown in Fig. E15.13. Flow rate $q_{2}$ can be manipulated, while disturbance variables, $q_{5}$ and $x_{5}$, can be measured. Assume that controlled variable $x_{4}$ cannot be measured and that each process stream has the same density. Also, assume that the volume of liquid in each tank is kept constant by using an overflow line. The transmitters and control valve have negligible dynamics.

(a) Using the steady-state data given below, design an ideal feedforward control law based on steady-state considerations. State any additional assumptions that you make.

(b) Do you recommend that dynamic compensation be used in conjunction with this feedforward controller? Justify your answer.

 Steady-State Data Stream Flow (gpm) Mass Fraction 1 1900 0.000 2 1000 0.990 3 2400 0.167 4 3400 0.409 5 500 0.800

## Verified Solution

\begin{aligned}&0=\bar{q}_{5}+\bar{q}_{1}-\bar{q}_{3} (1) \\&0=\bar{q}_{3}+\bar{q}_{2}-\bar{q}_{4} (2) \\&0=\bar{x}_{5} \bar{q}_{5}+\bar{x}_{1} \bar{q}_{1} ^{\nearrow ^{0}} -\bar{x}_{3} \bar{q}_{3} (3) \\&0=\bar{x}_{3} \bar{q}_{3}+\bar{x}_{2} \bar{q}_{2}-\bar{x}_{4} \bar{q}_{4} (4)\end{aligned}

Solve (4) for $\bar{x}_{3} \bar{q}_{3}$ and substitute into (3),

$0=\bar{x}_{5} \bar{q}_{5}+\bar{x}_{2} \bar{q}_{2}-\bar{x}_{4} \bar{q}_{4}$            (5)

Rearrange,

$\bar{q}_{2}=\frac{\bar{x}_{4} \bar{q}_{4}-\bar{x}_{5} \bar{q}_{5}}{\bar{x}_{2}}$               (6)

In order to derive the feedforward control law, let

$\bar{x}_{4} \rightarrow x_{4 s p} \quad \bar{x}_{2} \rightarrow x_{2}(t) \quad \bar{x}_{5} \rightarrow x_{5}(t) \quad \text { and } \quad \bar{q}_{2} \rightarrow q_{2}(t)$

Thus,

$q_{2}(t)=\frac{x_{4 s p} \bar{q}_{4}-x_{5}(t) q_{5}(t)}{\bar{x}_{2}}$             (7)

Substitute numerical values:

$q_{2}(t)=\frac{(3400) x_{4 s p}-x_{5}(t) q_{5}(t)}{0.990}$                  (8)

or

$q_{2}(t)=3434 x_{4 s p}-1.01 x_{5}(t) q_{5}(t)$           (9)

Note: If the transmitter and control valve gains are available, then an expression relating the feedforward controller output signal, $p(t)$, to the measurements, $x_{5 m}(t)$ and $q_{5 m}(t)$, can be developed.

Dynamic compensation: It will be required because of the extra dynamic lag introduced by the tank on the left hand side. The stream 5 disturbance affects $x_{3}$ while $q_{3}$ does not.