GATE 2023 CH – Question 61
The outlet concentration $C_A$ of a plug flow reactor (PFR) is controlled by manipulating the inlet concentration $C_{A0}$. The following transfer function describes the dynamics of this PFR. $\frac{C_A(s)}{C_{A0}(s)} = \exp\left[-\left(\frac{V}{F}\right)(k + s)\right]$
In the above equation, $V = 1$ m$^3$, $F = 0.1$ m$^3$min$^{-1}$ and $k = 0.5$ min$^{-1}$. The measurement and valve transfer functions are both equal to 1. The ultimate gain, defined as the proportional controller gain that produces sustained oscillations, for this system is ______ (rounded off to one decimal place).
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Correct answer: 147.66 to 149.14
Explanation
The space time is $\frac{V}{F} = 10$ min, so $G(s) = e^{-10k}e^{-10s} = e^{-5}e^{-10s}$, a pure delay with gain $e^{-5} = 0.006738$. Sustained oscillation needs a phase lag of 180°: $10\omega = \pi$, so $\omega = 0.314$ rad/min. The magnitude is constant, $|G| = e^{-5}$, so the ultimate gain is $K_u = \frac{1}{e^{-5}} = e^5 = 148.4$.