Figure 2 illustrates the response in level to an upset to the material balance. When the material balance is closed (imbalance is zero), vessel level is constant. In Figure 2, this is the case prior to time 0. At that point the discharge valve opening is reduced by 10%, which decreases discharge flow and causes level to increase.
All examples we'll discuss pertain to a straight-walled vessel containing a constant density liquid, hence the ramp has a constant slope as in Figure 2. We'll express the level as a percentage of the level measurement span. The response in Figure 2 is for a 12,000-L vessel. The average flow through the vessel is 200 L/min, giving a residence time of 60 min or 1 hr.
A simple characterization of a level process relies on two parameters whose value can be readily obtained from the response in Figure 2:
Process gain, KF. This is the effect of a 10% change in the controller output on the slope of the ramp. From Figure 2, a 10% reduction in the controller output causes the slope of the ramp to change from zero to 0.49%/min. So:
KF = (0.49 %/min)/10% = 0.049 (%/min)/%
A decrease leads to an increase in level, so the process is reverse acting.
Process lag, θ. The material balance suggests the ramp should commence immediately, as indicated by the dashed line in Figure 2. Instead, the slope changes gradually from zero to 0.49 %/min. By the time the slope reaches 0.49 %/min, the actual response lags by 0.4 min.
The process lag shown by the ramp in Figure 2 includes the following:
Control valve lag. A digital system can change its output by 10% very quickly but all final control elements exhibit some lag in responding to a change in their input signal. Rarely are the response characteristics of the final control element well known.
Measurement device lag. This depends on the measurement technology employed and, sometimes, on how the device is installed. Rarely is this lag quantified.
Smoothing of the process variable. When smoothing is applied either within the measurement device or the control system, quantitative values are available. However, with some level measurement technologies, smoothing can be applied externally.
The lag observed in Figure 2 is roughly the sum of these lags. The combined effect often is approximated by a transportation lag or dead time. In the simple approximations of the dynamics of a level process, the process lag θ is considered to be entirely transportation lag.
Unfortunately, for many level loops, conducting a test such as in Figure 2 is impractical due to the presence of noise on the measured level value and variability in the feed flow to the vessel.
Testing procedures are available to determine KF and θ in face of both measurement noise and flow upsets. One approach is to use a pseudo-random binary signal (PRBS) for the output to the control element. Model predictive control technology relies on such tests to find process characteristics. However, such tests are long-duration (days) and difficult to justify for level control applications.
When a proportional-integral (PI) controller is used, the relationships between the tuning coefficients and the process characteristics are:
Controller gain, KC: inversely proportional to the process gain KF; inversely proportional to the lag θ.
Reset time, TI: not affected by KF; directly proportional to θ.
Especially for large tanks with long residence times, tuning equations often suggest unreasonably large values for KC. Most tuning relationships link the product KFKC (the loop gain) to the process dynamics. The large value for KC results from two factors:
1. For responsive processes, the tuning equations suggest a large value for KFKC. For a vessel with a residence time of 1 hr, a lag of 0.4 min is trivial.
2. For large vessels, the process sensitivity KF is small.
Using the Ziegler-Nichols tuning equations, the suggested values for the tuning coefficients are: