Question

The outlet concentration \(C_A\) of a plug flow reactor (PFR) is controlled by manipulating the inlet concentration \(C_{A0}\). The transfer function is \[ \frac{C_A(s)}{C_{A0}(s)}=\exp\!\left[-\left(\frac{V}{F}\right)(k+s)\right], \] with \(V=1\ \text{m}^3\), \(F=0.1\ \text{m}^3\text{min}^{-1}\), \(k=0.5\ \text{min}^{-1}\). Measurement and valve transfer functions are 1. Find the ultimate gain (proportional controller gain for sustained oscillations).

Hint:

For a pure delay process, the ultimate gain is simply \(K_u=1/|K_p|\); the delay sets the oscillation frequency \(\omega=(2n+1)\pi/\theta\) but not \(K_u\).

Answer 1

Correct Answer: 148

Step 1: Rewrite the process as a pure dead-time with static gain: \[ \frac{C_A}{C_{A0}}=K_p\,e^{-\theta s},\quad K_p=e^{-(V/F)k}=e^{-5},\qquad \theta=\frac{V}{F}=10\ \text{min}. \] Step 2: For proportional control of a dead-time process, the marginal (sustained) oscillation condition from \(1+K_cK_p e^{-\theta s}=0\) at \(s=j\omega\) gives \[ |K_cK_p|=1 \;\Rightarrow\; K_u=\frac{1}{|K_p|}=e^{5}=148.413\dots \] Rounded to one decimal place: \(\boxed{148.4}\).