Characteristics of Real Processes in Visual Studio .NET

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Characteristics of Real Processes
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FIG 2.3. The step response of a multicapacity process can be reduced to dead time plus a single capacity.
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effective time constant equals the total lag in the process:
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n2 + n + 71 = T 2
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(2.4)
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Equation (2.4) requires that the step response of any number of equal interacting lags reach 63.2 percent at time .(nz + n)/2, which is corroborated by Fig. 2.2. Ziegler and Nichols2 noted t hat the period of oscillation will be four times the effective dead time, whether the process is int eracting or not. SO t he technique of dealing with single capacity plus dead time takes on added value in being applicable to these examples of complex dynamics. This is an important insight-without it, numerical methods must be rejected for use on any process containing more than two dynamic element s. And when on-the-spot analysis must be made, the shortcut numerical method is invaluable. Fortunately, a single-capacity plus dead-time process can be made to represent any degree of difficulty from one extreme to the other, simply by varying the ratio TJT~. Thus its application is universal, if approximate. As an example, the lo-capacity interacting process of Fig. 2.2 has an effective dead time of about 0.15 of the total lag. Since the balance is the dominant time constant, the ratio ~,JT~ = 0.15/0.85 = 0.18. The proportional band needed for s/4-amplitude damping of this process can be found by referring to Fig. 1.26. Figure 2.4 is a correlation of the ratio of effective dead time to effective lag, against the number of interacting stages. Data from tests on systems of 2 to 10 capacities fall in a straight line on semilogarithmic coordi-
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FIG 2.4. The ratio of elfective dead time to effective lag of n equal interacting capacities varies with the logarithm of n.
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1 Ud erstanding Feedback Control n
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rtates. This relationship is extremely useful in predicting the dynamic behavior of any process with a discrete number of interacting stages. Diffusive and distributed processes ought to consist of an infinite number of interacting stages. Their response does not correspond to n = 00 in Fig. 2.4, however, probably because their interaction is incomplete. Transmission lines typically exhibit ratios of 7d/r1 in the range from O;l to o.3.3
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Predicting the Behavior of a Loop
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The appearance of a piece of processing equipment often reveals the nature of its dynamic characteristics. If all the dimensions are similar, as in a cylindrical tank where t he height is of the same magnitude as the diameter, capacity will predominate-dead time, if any, being short. But, if the vessel has one dimension much greater than the others, dead time may be dominant, though not without some capacity. Thus a shell and tube heat exchanger will exhibit considerable dead time, compared to a heated tank whose principal elements would be lags. Just the appearance of a tower, whether it be distillation, absorption, or whatever, indicates the presence of dead time. One could almost generalize to t,he extent of relating controllability to dimension:
Of course such an expression could only he writ,ten to apply within a specific system, because many more fact,ors are involved. Nonetheless, if dead time is related to length, t,he natural period is similarly related to length, as with a simple pendulum.
GAIN AND ITS DEPENDENCE
The damping of a feedback loop is a function of the gain product of all the elements in the loop, both dynamic arid steady-state. Xormally only one of these elements is adjustable-the controller. All others are fixed by the design of the procsess. Icor a given damping, the eont~roller adjust mcnts are a function of the gain of the fixed elcmcnts. Up to this point, only dynamic gain has been considered. But any clemeut whose output differs from its input has a gain contribution:
Element V a l v e . P r o c e s s , Transmitter, Input Signal Flow Measurement output Flow Ylcasurcrrlcnt Signal
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