The previous sections of this chapter have dealt with the tuning of controllers for self-regulat-ing processes such as temperature and flow. Liquid-level control, however, has characteristics distinctly different from those of the previous loops. Some of the differences are these:
• Liquid level is usually a non-self-regulating (integrating) process.
the gain.”) do not apply, and in fact will usually produce results that are the opposite of those expected.
• Liquid-level control loops, once properly tuned, do not usually go out of tune.
Most processes can be described at best by an approximate process model that must often be determined by process testing. On the other hand, most liquid-level control loops readily yield to an analytical approach. A simple process model can be formulated, desired performance parameters can be established, and from this controller tuning parameters can be calculated.
Once this is done, other attributes of the control loop, such as the period of oscillation, can be predicted.
Determining tuning parameters for a liquid-level loop should probably be considered as an engineering activity, rather than being left for field trial-and-error tuning, for two reasons: the counterintuitive nature of liquid-level loops makes tuning by trial-and-error techniques diffi-cult, and liquid-level loops are amenable to an analytical approach. Hence it can be said that liquid-level control loops should be engineered, not tuned.
Many engineering studies start with an ideal model, then incorporate subsequent consider-ations to account for differences found in real situconsider-ations. Engineering design problems are usu-ally based upon some assumed worst-case conditions, even though those conditions may never be experienced in reality. We will follow the same approach in engineering liquid-level con-trol loops.
Idealized Control Loop Model
An idealized liquid-level control system is shown in Figure 6-15.
)&
)7 /7
/&
)LQ
)RXW /
Attributes of this idealized system are these:
• The tank has constant cross-sectional area.
• The level controller is cascaded to a flow controller.
• A valve positioner is installed on the flow control valve.
• All inflow goes to outflow; the tank is merely a buffer storage tank.
• The maximum outflow is the same as the maximum inflow.
• The size of the tank is substantial.
• There is no thermal effect such as the boiling liquid found in boiler-drum level con-trol.
• The level controller operates at constant set point.
The implications for these attributes are as follows:
• The level-control loop constitutes a linear system.
• There is no effect from upstream or downstream pressure, line loss, or pump curve.
• The system is not affected by the size of the valve.
• The dynamics of the flow loop are significantly faster than the dynamics of the vessel, and therefore can be ignored.
• There is no dead time in the loop.
• Response to set point change need not be considered since set point changes are rarely made; instead, the response to a disturbance is of critical importance.
Figure 6-15 shows a common situation in which the level controller manipulates the outflow in response to changes in inflow. There are also cases where the level controller manipulates the inflow in response to varying demands for the outflow. The technology developed here is applicable to both cases, generally by exchanging the words inflow and outflow.
A key parameter required for analysis is the tank holdup time, also called the tank time con-stant. If the tank geometry (diameter, distance between the level taps) and maximum outflow rate (flow rate corresponding to 100 percent output of level controller) are known, then the tank time constant can be calculated from the following:
(6-18)
where: TL = tank time constant
Q = holdup quantity, between upper- and lower-level sensor taps fout = maximum outflow
L fout
T = Q
constant is then expressed in minutes.
A block diagram for the loop with PI control is shown in Figure 6-16.
If the dynamics of the flow loop are negligible compared with the tank dynamics, then it can be assumed that F(s), the closed-loop transfer function of the flow loop, is equal to unity. If the loop operates at constant set point, then we are more interested in the response to a disturbance (i.e., to a change in fin) than to the set point response. However, in addition to the response of the level to a change in fin, we may also be interested in the response of the outflow, fout, to a change in fin. Transfer functions representing these two responses, derived from Figure 6-16, are given by the following two equations:
(6-19)
(6-20)
These equations display the fact that the loop acts as a second-order system. These transfer functions, written with traditional servomechanism terminology, are:
(6-21) Figure 6-16. Block Diagram for Ideal Liquid-level Process Model
(6-22)
where ζ, the damping factor, and , the natural frequency, are given by:
(6-23)
(6-24)
The damping factor is a dimensionless number. The natural frequency is in radians per minute if TI is in minutes (minutes per repeat) and TL is in minutes.
If 0 < ζ < 1, the control loop is said to be underdamped. A step change in disturbance or a set point change will result in a decaying, oscillatory response.
If ζ = 1, the control-loop response is said to be critically damped. A step change in disturbance or a set point change will result in a relatively rapid return to set point without oscillation.
If ζ > 1, the control-loop response is said to be overdamped. A step change in disturbance or a set point change will result in a relatively slow return to set point without oscillation.
Many practicing engineers may be more familiar with the concept of decay ratio, DR, rather than the damping factor, ζ. (Decay ratio is defined for both a set point change and a distur-bance in Figure 6-1.) It can be shown that the damping factor and decay ratio are related by the following relations:
(6-25)
. (6-26)
For example, for the familiar quarter-amplitude decay, ζ has a value of 0.215.
out n n2
In addition to knowing the tank holdup time, TL (see Equation 6-18), analytical determination of tuning parameters requires three additional decisions:
• ∆Fin – the maximum step change in disturbance (inflow) that can be expected, in per-centage of full scale (this is the “worst-case” assumption);
• ∆Lmax – the maximum allowable deviation from set point, in percentage of full scale (choosing this parameter lets us determine whether we want “tight” or “loose” tun-ing);
• DR – the desired decay ratio, in the event of a step inflow disturbance.
Using these values, the value for TL , and Equation 6-25 to convert decay ratio into damping factor, the relations given in Table 6-3, 6-4 and 6-5 can be derived.7 Table 6-3 presents rela-tions for calculating tuning parameters, while Tables 6-4 and 6-5 present relarela-tions for predict-ing attributes of the level and outflow response to a step change in inflow.
The column labeled “Rigorous” demonstrates that the tuning parameters are entirely a func-tion of the four fixed or chosen parameters, TL, ζ (as determined from the chosen decay ratio),
∆Fin and ∆Lmax. The column labeled “Simplified” produces the same results, calculated in terms of some previously calculated quantity.
Table 6-3. Liquid-level Tuning Parameter Relations for Ideal Model
Tuning Parameter
Underdamped
ζ < 1 Critically Damped
ζ = 1
Rigorous Simplified
KC
TI
Note: In the table above,
∆
Once the tuning parameters have been calculated, the predicted behavior for the level as well as for the outflow can be calculated from Tables 6-4 and 6-5.
The level arrest time, TaL, represents the time from the disturbance until the maximum devia-tion from set point. The period, P, is the predicted period of oscilladevia-tion of the level-control loop, if it is underdamped. See Equation 6-1 for the definition of IAE.
represents the maximum change in outflow. The ratio
will always be greater than 1. The outflow arrest time, TaF, represents the time from the distur-bance until the maximum change in outflow. The maximum rate of change of outflow is given since it is this quantity, rather than the size of the outflow change itself, that represents the maximum disturbance to a downstream process unit.
The relations given in Tables 6-3, 6-4, and 6-5 show the development of this tuning technique, but they are not very useful as working relations because of the amount of computation required. For three specific decay ratios, Tables 6-6 and 6-7 present working relations. The three decay ratios chosen are these:
Table 6-4. Predicted Behavior for Level – Ideal Model
Behavior Attribute
Underdamped
ζ < 1 Critically Damped
ζ = 1
Note: In the above table,
∆
• critically damped,
• one-quarter decay,
• one-twentieth decay.
Critically damped is chosen because it represents a recognized extreme form of tuning; one-quarter decay is chosen because of its familiarity. The last, although less familiar, is chosen because it provides both the minimum IAE and the lowest maximum rate of change of out-flow. Figure 6-17 depicts these three forms of response with equal values of maximum devia-tion.
Note: In the following expressions,
∆Fout max−
Figure 6-17. Three Forms of Disturbance Response; Equal Magnitude Maximum Deviation
Table 6-6. Working Equations Liquid-level Tuning Parameters – Ideal Model (If the loop has dead time or is not a cascade loop, use Tables 6-8 and 6-9 rather than 6-6 and 6-7.)
Decay Ratio
Example 2:
Suppose you have a tank with the following specifications:
Tank diameter: 5.0 feet,
Distance between level transmitter taps: 8.0 feet,
Maximum outflow (upper end of outflow transmitter measuring span): 250 gpm.
Calculate the tank holdup time:
Surge volume ,
Surge quantity = Q = ,
Holdup time = .
(If the loop has dead time or is not a cascade loop, use Tables 6-8 and 6-9 rather than 6-6 and 6-7.)
157.3 ft 7.48gal 1176.6 gal
× ft =
1176.6
4.7 min
L 250
T = =
Also, suppose that you anticipate that a worst-case disturbance would be a step inflow change of 10 percent. In the event of this disturbance, you want the level to deviate no more than 5 percent (about five inches). You would like for the system to settle out fairly rapidly, so you choose a 0.05 decay ratio.
∆Fin = 10%,
∆Lmax = 5%.
With this data, use Table 6-6 to calculate tuning parameters:
.
We can use Table 6-7 to predict other properties of the response:
Level arrest time: ,
Period: P .
The period may seem to be excessive, but recall that because of the fast settling behavior selected, the maximum deviation during the second half-cycle will be about 1.1 inches, during the third half-cycle about 0.25 inches, and so on.
End of Example 2.
“Real-world” Considerations
The results presented so far have been based on an idealized process model. Many real appli-cations will fail to meet one or more of the criteria for the idealized model. The following are commonly encountered situations, along with suggested procedures for coping with them.
Irregularly Shaped Vessels
For irregularly shaped vessels, such as horizontal or spherical drums, using the level measure-ment directly as the process variable for the level controller may create highly nonlinear char-acteristics for the behavior of the control loop. In this case, the loop can be linearized using the volumetric holdup in the vessel rather than the level measurement. This may be computed from the vessel geometry and the actual level measurement. The process variable then should be scaled in terms of percentage of maximum volumetric holdup.
No Cascade Loop
In cases where there is no secondary flow controller as shown in Figure 6-15, the holdup time cannot be calculated from Equation 6-14 because the maximum outflow cannot be related to the maximum setting of a secondary flow controller. It would probably be futile to attempt to
0.50 10
ables such as line loss, the effect of the pump curve, head effects in the tank, and so on. In addition, the process response is probably nonlinear so the maximum flow rate with a wide-open valve would probably vary with level in the tank. What is required is the apparent holdup time at the nominal operating point. An additional required parameter is the valve gain, KV. (KV replaces F(s) in Figure 6-16.)
We now present a method by which we can determine the apparent holdup time and the valve gain at the nominal operating point from process tests. Both of these methods start with the supposition that the operation is stable with constant inflow, with constant level at the nominal set point, and with the controller in the automatic mode with its output somewhere between the extreme limits. Also, we assume that the inflow remains constant for the duration of the test.
The valve should have a positioner, or at least be free of stiction and hysteresis.
Put the controller in manual and change the output by a small amount, say ∆m percent. The outflow will change by an amount ∆F, and the level will begin changing (see Figure 6-18).
After a certain period of time, say ∆t, change the controller back to its original position. The level should stop changing. Now determine the change in level, ∆L, during the test. (Use the absolute values in percentage of full scale for ∆L, ∆Fout, and ∆m.) The apparent holdup time can be estimated from:
, (6-27)
and the valve gain from:
. (6-28)
∆ ∆
∆
L Fout t
T = L
∆
∆
V Fout
K = m
$/
$)RXW
$9 $7
7LPH
9DOYH)ORZ2XW/HYHO
Dead Time
The ideal process model for a liquid-level loop contains neither dead time nor lag time. Real processes may contain either dead time or lag. For instance, a cascaded flow-control loop may have a finite response time. For specific forms of response, Tables 6-8 and 6-9, presented later in this section, contain correction factors for Φ, the ratio of dead time to holdup time. In addi-tion, Table 6-8 contains the valve-gain factor, KV. For cascade loops, KV can be set to unity.
Unequal Inflow and Outflow
When considering step changes in feed rate, consider the feed rate as the liquid actually going to the reservoir, whether from the feed stream, from liquid falling from lower trays on a distil-lation tower, or from some other source. Hence, the change in feed rate, ∆Fin, can be the result of any cause, such as actual change in vessel feed rate, change in the percentage of liquid of the feed, change in reboiler heat, or change in liquid load on the trays in the distillation tower.
Flashing Liquid
In some cases, flashing liquid or foam and froth can produce a false indication of level. An example is the “shrink-and-swell” effect in a boiler drum. The shrink-and-swell effect approx-imates that of dead time; hence the dead-time correction factors of Tables 6-8 and 6-9 may apply.
Modified Tables for Tuning Parameters
Tables 6-8 and 6-9 are modifications of Table 6-6 and 6-7 to account for two real-world phe-nomena:
• Noncascade control,
• Dead time in the level-control loop.
These correction factors were determined as a “best fit” to simulation results.
Sinusoidal Disturbance
If an oscillating inflow is anticipated (for instance, as a result of the cycling of a control loop of an upstream process unit), then both the level and the outflow will oscillate with the same frequency. The maximum deviation in level (half the amplitude of its oscillation) may be more than would result from a step change in inflow equal to half the amplitude of inflow oscilla-tion. Also, the level-control loop may act as an amplifier, causing the outflow’s amplitude of oscillation to exceed that of the inflow. Hence, both the amplitude of oscillation for both the level and the outflow should be investigated, with the possibility that the tuning parameters will need to be modified.
If the frequency of oscillation of the inflow is known, then a key parameter is the ratio of this frequency to the undamped natural frequency (or simply “natural frequency”) of the level-con-trol loop, ω/ωn. Table 6-7 or 6-9 gives relations for calculating the predicted period of oscilla-tion, P, of the control loop. The natural frequency, in radians per minute, can be calculated from the following:
Time and Noncascade Control
∆Fin = max. step change in disturbance
∆Lmax = max. allowable deviation of level from set point
TL = holdup time, minutes KV = Valve Gain
(=1.0 if level is cascaded to flow)
Decay Ratio
Table 6-9. Predicted Performance for Step Change in Inflow with Corrections for Dead Time and Noncascade Control
(6-29)
Once the normalized frequency ratio is known, Figure 6-19 can be used to determine
.
If the frequency of oscillation of the inflow is not known, then a worst-case condition would be when
For an assumed amplitude of input oscillation, Fin(ω), if half the (peak-to-peak) amplitude of level variation, L(ω), exceeds ∆Lmax, then recalculate a new value for KC from
. (6-30)
(Note that we are adjusting KC upward from its original value.) Then return to Table 6-7 and calculate a new value of TI. (If you are using Table 6-19, use column 3 to back-calculate a new value for ∆L, resulting from the adjusted value KC. Then use these new values for KC and ∆L in column 4 to calculate TI.)
Example 3.
In Example 2, for ∆Fin = 10% and ∆Lmax = 5%, the following tuning parameters and pre-dicted period of oscillation were calculated:
KC = 1.0,
TI = 3.55 minutes/repeat, P = 28.7 minutes.
The natural frequency is given by:
radians/minute.
If, rather than a step input change of 10 percent, we anticipate that, as a worst case, there will be an oscillating input of the same frequency whose peak-to-peak amplitude is twice this, then from Figure 6-9:
= 1.0 .
or L(ω) = 20.
Figure 6-19. Magnitude Ratio of Changes in Level to Sinusoidal Changes in Inflow
Z
Z
Z
&ULWLFDOO\'DPSHG
'5
'5
1RUPDOL]HG)UHTXHQF\ WWQ .&[0DJQLWXGH5DWLR / W )LQ W
1 0 20
C in
L( ) L( )
K .
F ( )
ω ω
ω
= ×Since half the amplitude exceeds ∆Lmax, then recompute KC as
With this new value for KC, reenter Table 6-7 and calculate TI.
minutes/repeat.
We had assumed that as a worst case, the frequency of oscillation was the same as the natural frequency, whatever value that is. If the frequency of input oscillation were fixed, say at 0.24 radians/minute (the same as the initial natural frequency of the loop), then we would proceed as we did before and calculate new values for KC and TI. With a new value for TI, however, the predicted period of oscillation of the loop would be 14.08 minutes. Consequently, the value for ωn would change to 0.49 radians/minute. At a normalized frequency ratio of 0.24/
0.49, from Figure 6-19,
so that .
Half of this amplitude is equal to ∆Lmaxso our adjusted tuning is satisfactory. If our new amplitude were still greater than 2 × ∆Lmax , we would have had to calculate a further adjust-ment for KC and TI.
End of Example 3.
For a range of values of the normalized frequency ratio, Figure 6-20 depicts the magnitude ratio for the amplitudes of oscillation of outflow and inflow. This table illustrates the fact that if the normalized frequency ratio, ω/ωn, is near to unity, the level-control loop will act as an amplifier for the outflow. If the ratio is considerably less than 1, the outflow and inflow will oscillate at approximately the same amplitude. If the ratio is greater than unity, however, the amplitude of outflow oscillation will be considerably attenuated from that of the inflow. (By inflow, we refer to the net inflow into the liquid pool in the vessel. This may differ from the total liquid inflow into the vessel itself.) As a consequence of this, we deduce that if two level-controlled vessels (such as distillation towers) are in series, it is preferable that the level-con-trol loops be tuned so their natural frequencies are separated. In particular, the natural fre-quency for the second vessel should be greater than the first, to take advantage of the attenuation of inflow oscillation provided by the first vessel. Since the vessel holdup times are
For a range of values of the normalized frequency ratio, Figure 6-20 depicts the magnitude ratio for the amplitudes of oscillation of outflow and inflow. This table illustrates the fact that if the normalized frequency ratio, ω/ωn, is near to unity, the level-control loop will act as an amplifier for the outflow. If the ratio is considerably less than 1, the outflow and inflow will oscillate at approximately the same amplitude. If the ratio is greater than unity, however, the amplitude of outflow oscillation will be considerably attenuated from that of the inflow. (By inflow, we refer to the net inflow into the liquid pool in the vessel. This may differ from the total liquid inflow into the vessel itself.) As a consequence of this, we deduce that if two level-controlled vessels (such as distillation towers) are in series, it is preferable that the level-con-trol loops be tuned so their natural frequencies are separated. In particular, the natural fre-quency for the second vessel should be greater than the first, to take advantage of the attenuation of inflow oscillation provided by the first vessel. Since the vessel holdup times are