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This result was also given in Eq. (7.10). Letting I = (B, - Y)/B, as was done in step 2, we obtain from Eq. (19.10)
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Assume that Ti > T2. As t i approaches 03, the second term on the right side of Eq. (19.11) becomes much smaller than the first term and we can write as an approximation to Eq. (19.11) for large t 1
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= pe- dTl - pe--tilTi T1
Tl - T2 -
(19.12)
where the term I, is the approximation of I at large values of time and P is the value of I, at tl = 0. When tIlTI = 1, or tl = T1, we obtain from Eq. (19.12) I, = Pe- = 0.368P (19.13)
This proves step 4 of the graphical procedure. Now let A = I, - I. From Eqs. (19.11) and (19.12) we obtain A=-. & T1 - T2
-tllT2 - Re-t IT2
(19.14)
This relation plots as a straight line on semi-log paper. When tllT2 = 1, or tl = T2, we obtain from Eq. (19.14) A = Re- = 0.368R This proves Step 5. To appreciate the nature of this graphical construction, the reader is encouraged to solve the problems requiring its use at the end of the chapter.
Fkequency Testing
We have shown in the section on frequency response that a process having a transfer function G(s) can be represented by a frequency response diagram (or Bode plot) by taking the magnitude and phase angle of G(jo). This procedure can be reversed to obtain G(s) from an experimentally determined frequency response diagram. The procedure requires that a device be available to produce a sinusoidal signal over a range of frequencies. We describe such a device as a sine wave generator. In frequency testing of an industrial process, a sinusoidal variation in pressure is applied to the top of the control valve so that the manipulated variable can be varied sinusoidally over a range of frequencies. The block diagram that applies during frequency testing is the same as the one of Fig. 19.3 with the step input (M/s) replaced by a sinusoidal signal. The sine wave generator used to test electronic devices operates at frequencies that are too high for many slow moving chemical processes. For frequency testing of chemical processes, special low-frequency generators must be built that can produce a sinusoidal variation in pressure to a control valve. To preserve the sinusoidal signal in the flow of manipulated variable through the valve, the valve must be linear. In the 1960s when frequency response methods were first introduced to chemical engineers as a means for process identification, several chemical and
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petroleum companies constructed mobile units containing low-frequency sine wave generators and recorders that could be moved to processing units in a plant for the purpose of frequency testing. The great disadvantage of frequency testing is that it takes a long time to collect frequency response data over a range of frequencies that can be used to construct frequency response plots. The time is especially long for chemical processes, often having long time constants measured in minutes or even hours. The frequency test at a given frequency must last long enough to make sure that the transients have disappeared and only the ultimate periodic response is represented by the data. Frequency testing usually ties up plant equipment too long to be recommended as a means of process identification. Step testing and pulse testing take much less time and can usually provide satisfactory process identification.
Pulse Rsting
Pulse testing is similar to step testing; the only difference in the experimental procedure is that a pulse disturbance is used in place of a step disturbance. The pulse is introduced as a variation in valve top pressure as was done for step testing (see Fig. 19.3). In applying the pulse, the open-loop system is allowed to reach steady state, after which the valve top pressure is displaced from its steady-sqte value for a short time and then returned to its original value. The response is recorded at the output of the measuring element (B in Fig. 19.3). An arbitrary pulse and a typical response are shown in Fig. 19.14. Usually the pulse shape is rectangular in experimental work, but other well defined shapes are also used. The input-output data obtained in a pulse test are converted to a frequency response diagram, which can be used to tune a controller. The transfer function of the valve, process, and measuring element (referred to as the process transfer function, for convenience) is given by:
where Y(s) = Laplace transform of the function representing the recorded output response X(s) = Laplace transform of the function representing the pulse input Applying the definition of the Laplace transform [Eq. (2. l)] to the numerator and denominator of Eq. (19.15) and replacing s by jo gives
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