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Substituting for ul, Ml, and Mz in Eq. (3.7) yields
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Level in the measuring chamber, h 2, is related to average level h by h - hl = (hz - h) 2
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Including this in Eq. (3.8) yields the response of measured level hz to average level h:
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But velocity uz is the rate of change of level, dhz/dt. ential equation can be written eliminating ~2:
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Therefore a differ-
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L1 + Lzd2h 2g dtz
(3.9)
This differential equation is descriptive of a second-order undamped system. The U tube resonates at a natural period established by the
Analysis of Some Common Loops
square root of the coefficient of the differential:
Notice that the period is unaffected by density, area, or any property other than the distance L, + Lz between the bounded surfaces. Compare it t o that of a pendulum, also a function of length and gravity only. Liquid in a vessel may also oscillatje without the benefit of a U tube. The period of oscillation of the surface of diameter I, is: (3.11) Rectangular vessels can oscillate at two different periods. Vessels with an att)ached measuring chamber can oscillate with at least two different periods. The natural period of any control loop containing a resonant element cannot exceed t,hat of the resonant element. The phase shift of a resonant element is exactly -90 at its natural period, no matter how heavily damped it may be. Since the integration of flow into average level represents an inherent phase shift of -9O , the process, from flow to measured level, will exhibit -180 at the natural period of the vessel. To damp the measuring chamber by throttling its connecting valves will not change this period, but will only reduce the amplitude of the resonance.
example 3.3 As an example of a liquid-level control problem, consider a vessel with a measuring chamber of the following description: Volume V: 100 gal Maximum flow F: 50 g-pm Diameter L: 2.0 ft
Normal level L1: 3.6 ft
Chamber Le: 4.4 ft
The liquid can oscillate on the surface and in the U tube. Rut since the largest resonant period is alwars the limiting one, only the period of the L tube is important To = 3.6 ft + 4.4 ft $4 = 2.2 set 2T [2(32.2 ft/se ) (2.2/60) (6.28)(l !%o)
The dynamic gain of the int.egrator is G = 2T;,F = = 0.003
This control problem can be accommodated with a proportional band of ZOOG = 6 percent.
1 Understanding
Feedback
Control
Since dimensions of process vessels generally fall bet ween 2 and 200 ft, liquid resonance lies principally in the region from l- to lo-set period. Hence it is only of serious consequence, from the standpoint of controlloop stability, in vessels with time constants of less than 1 min.
Liquid-level Noise
Measurement of liquid level is usually noisy, because of splashing and turbulence of fluids entering the vessel. As we have seen, loops that resonate respond to random disturbances by oscillat ing at their natural period. As a result, level measurements are rarely quiet, often fluctuating 20 or 30 percent of scale. This is particularly true in vessels containing boiling liquids, where turbulence is high. Although a narrow proportional band, like the one determined in the example, may be sufficient, for control-loop stability, random fluctuations of only a few percent will drive the control valve to its limits. This may be unobjectionable in some cases, but too severe in others. Often the liquid level in a tank is used to control flow into another part of the process. It is certain that wide fluctuations in feed rate are not tolerated in most operations. To provide steady flow in these instances, the proportional band is widened and reset is relied upon t o maintain control. In many applications, exact regulation of liquid level is not important. In fact, a surge tank does not fulfill its purpose if t ight control is imposed on it. As a result, cont rol adjustments are often relaxed, and the process is sometimes left to be operated manually, if its time constant is long enough. In some applications, a special controller whose proportional band changes with deviation is warranted. This type of csontroller is devised to deliver smooth flow while level is normal, but to change flow radically in the event that high or low limits are approached. 5 discusses more details of this function.
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