In some applications, exceptionally tight level control, through enforcement of a residence time or a material balance for a unit operation, is needed for best product quality. The quantity and quality of product for continuous reactors and crystallizers depend on residence times. For fed-batch operations, there may be an optimum batch level. The variability in column temperature that is an inference of product concentration in a direct material balance control scheme depends on the tightness of the overhead receiver level control. Since these overhead receivers are often horizontal tanks, a small change in level can represent a huge change in inventory and manipulated reflux flow.
In other applications, level control can be challenging due to shrink and swell (e.g. boiler drums and column sumps) or because of the need for the level to float to avoid upsetting the feed to downstream units (e.g. surge tanks). If the level controller gain is decreased to reduce the reaction to inverse response from shrink and swell or to allow the level to float within alarm limits, the reset time must be increased to prevent slow oscillations.
Adaptive level controllers can not only account for the effect of vessel geometry, but also deal with the changes in process gain from changes in fluid density and nonlinear valves. Even if these nonlinearities are not significant, the adaptive level control with proper tuning rules removes the confusion of the allowable gain window, and prevents the situation of level loops being tuned with not enough gain and too much reset action.
Specific Dynamics for Conical Tank Level
Conical tanks with gravity discharge flow are used as an inexpensive way to feed slurries and solids such as lime, bark and coal to unit operations. The conical shape prevents the accumulation of solids on the bottom of the tank. The Madras Institute of Technology (MIT) at Anna University in Chennai, India, has a liquid conical tank controlled by a distributed control system (DCS) per the latest international standards for the process industry as shown in Figure 1. The use of a DCS in a university lab offers the opportunity for students to become proficient in industrial terminology, standards, interfaces and tools. The DCS allows graduate students and professors to explore the use of industry’s state-of-the-art advanced control tools. Less recognized is the opportunity to use the DCS for rapid prototyping and deployment of leading edge advances developed from university research.
The conical tank with gravity flow introduces a severe nonlinearity from the extreme changes in area. The dependence of discharge flow on the square root of the static head creates another nonlinearity and negative feedback. The process no longer has a true integrating response. In Appendix A online (www.controlglobal.com/1002_LevelAppA.html), the equations for the process time constant (τp) and process gain (Kp) are developed from a material balance applicable to liquids or solids. The equations are approximations because the head term (h) was not isolated. Since the radius (r) of the cross-sectional area at the surface is proportional to the height of the level as depicted in Figure 2, it is expected that the decrease in process time constant is much larger than the decrease in process gain with a decrease in level.
Controller Tuning Rules
The lambda controller tuning rules allow the user to provide a closed-loop time constant or arrest time from a lambda factor (λf) for self-regulating and integrating processes, respectively. The upper and lower controller gain limits are a simple fall out of the equations and can be readily enforced as part of the tuning rules in an adaptive controller.
For a self-regulating process the controller gain (Kc) and reset time (Ti) are computed as follows from the process gain (Kp), process time constant and process dead time (θp):
The upper gain limit to prevent fast oscillations occurs when the closed loop time constant equals to the dead time.
For an integrating process the controller gain (Kc) and reset time (Ti) are computed as follows from the integrating process gain (Ki) and process deadtime (θp):
The upper gain limit to prevent fast oscillations occurs when the closed loop arrest time equals the dead time:
The lower gain limit to prevent slow oscillations occurs when the product of the controller gain and reset time is too small.