From the proportional-mode equation, you can compute an approximate minimum value for Kc as long as you have the following information:
Value for MR. When reset mode is in use, controller output bias isn't constant. However, when TI is long (as it should be), the value will change very slowly. In the previous responses, the value of MR varies between approximately 60% (for low throughputs) and 80% (for high throughputs).
Value for SP. For all previous responses, the set point for vessel level was 40%.
Location of level switches. In Figure 1 the switches are positioned at vessel levels of approximately 5% and 95%.
Let's require the valve to be fully closed at a level of 10% and fully open at a level of 90%. The level controller is direct acting, so E in the proportional-mode equation is computed as PV – SP. Based on a value of 70% for MR, this translates into the following requirements:
High level trip. At a vessel level of 90% E is 50%. To have a controller output of 100%, Kc = (M – MR)/E = (100% - 70%)/50% = 0.6 %/%.
Low level trip. At a vessel level of 10% E is -30%. To have a controller output of 0%, Kc = (M – MR)/E = (0% - 70%)/(-30%) = 2.33 %/%.
There's one complication to these calculations -- flow through a control valve rarely varies linearly with valve opening. The relationship depends on the nature of the flow system (how much of the pressure drop available for fluid flow is taken by the control valve) and the sizing of the control valve. In a properly sized equal-percentage valve, valve openings less than 50% give very small flows -- not zero, but too small to significantly affect the level. If the requirement is changed to a valve opening of 50% at a vessel level of 10%, Kc = (M – MR)/E = (50% - 70%)/(-30%) = 0.67 %/%.
On the other hand, if the equal-percentage valve is significantly oversized (say by a factor of four), the flow at a valve opening of 50% is nearly the maximum available flow -- once the valve is half open, opening it more has little effect on flow. In such cases, MR would be less than 50% and the above calculations could be performed for M = 10% and M = 50%.
Often flow characteristics are apparent qualitatively but rarely are quantitative data available. The alternative of a nonlinear control equation (as will be described in the next article) usually is the easiest and most reliable approach to assure the valve has been driven to a limit before level reaches either trip point.
Smaller Kc values than those computed above often prove acceptable. Figure 4 illustrates the performance of the controller for a Kc of 0.4 %/% and a TI of 24 min. Even with this low sensitivity for the controller, discharge flow varies considerably. On one occasion, the discharge flow changes from approximately 350 liters/min to approximately 100 liters/min in about 1 hr. During this same period, vessel level varies from just over 75% to under 10% (but above 5%, so there's no process trip).