Question

Consider the surge drum in the figure. Initially, the system is at steady state with a hold-up \( \bar{V} = 5 \, \text{m}^3 \), which is \( 50\% \) of full tank capacity, \( V_{\text{full}} \), and volumetric flow rates \( F_{\text{in}} = F_{\text{out}} = 1 \, \text{m}^3/\text{h} \). The high hold-up alarm limit \( V_{\text{high}} = 0.8 V_{\text{full}} \) while the low hold-up alarm limit \( V_{\text{low}} = 0.2 V_{\text{full}} \). A proportional (P-only) controller manipulates the outflow to regulate the hold-up as \( F_{\text{out}} = K_c (V - \bar{V}) + F_{\text{out} \). At \( t = 0 \), \( F_{\text{in}} \) increases as a step from \( 1 \, \text{m}^3/\text{h} \) to \( 2 \, \text{m}^3/\text{h} \). Assume linear control valves and instantaneous valve dynamics. Let \( K_c^{\text{min}} \) be the minimum controller gain that ensures \( V \) never exceeds \( V_{\text{high}} \). The value of \( K_c^{\text{min}} \), in \( \text{h}^{-1} \), rounded off to 2 decimal places, is:} \includegraphics[width=0.5\linewidth]{q63 CE.PNG}

Hint:

In surge drum problems with controllers, calculate the gain by analyzing the maximum allowable deviation in the hold-up.

Answer 1

Step 1: Dynamics of the system. The rate of change of hold-up \( V \) in the surge drum is: \[ \frac{dV}{dt} = F_{\text{in}} - F_{\text{out}}, \] where: \[ F_{\text{out}} = K_c (V - \bar{V}) + F_{\text{out}}. \] At \( t = 0 \), \( F_{\text{in}} \) increases from \( 1 \, \text{m}^3/\text{h} \) to \( 2 \, \text{m}^3/\text{h} \), causing a change in \( V \). Step 2: Steady-state condition. Initially, \( F_{\text{in}} = F_{\text{out}} = 1 \, \text{m}^3/\text{h} \), and \( V = \bar{V} = 5 \, \text{m}^3 \). Step 3: Determine \( K_c^{\text{min}} \). To prevent \( V \) from exceeding \( V_{\text{high}} = 0.8 V_{\text{full}} \), the proportional controller gain \( K_c \) must be sufficient to counteract the inflow increase. The maximum deviation \( \Delta V \) is given by: \[ \Delta V = \frac{\Delta F_{\text{in}}}{K_c}, \] where \( \Delta F_{\text{in}} = F_{\text{in,new}} - F_{\text{in,old}} = 2 - 1 = 1 \, \text{m}^3/\text{h} \). Substitute \( \Delta V \leq V_{\text{high}} - \bar{V} = 0.8 V_{\text{full}} - 0.5 V_{\text{full}} = 0.3 V_{\text{full}} \): \[ \frac{1}{K_c} \leq 0.3 V_{\text{full}}. \] Rearrange for \( K_c \): \[ K_c \geq \frac{1}{0.3 V_{\text{full}}}. \] With \( V_{\text{full}} = 10 \, \text{m}^3 \): \[ K_c \geq \frac{1}{0.3 \cdot 10} = 0.33 \, \text{h}^{-1}. \] Step 4: Conclusion. The minimum controller gain \( K_c^{\text{min}} \) is \( 0.33 \, \text{h}^{-1} \).