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As already described in the application of the Z-N tuning method, a step change in the input to a process produces a response, which is called the process reaction curve. For many processes in the chemical industry, the process reaction curve is an S-shaped curve as shown in Fig. 19.4. It is important that no disturbances other than the test step enter the system during the test, otherwise the transient will be corrupted by these uncontrolled disturbances and will be unsuitable for use in deriving a process model. For systems that produce an S-shaped process reaction curve, a general model that can be fitted to the transient is the following second-order with transport lag model:
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K e-TdS G,(s) =
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P Y(s) =-
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(TIS + l)(T2s + 1)
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X(s)
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(19.7)
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This model is an extension of the one used in the C-C tuning method, in which there was only one first-order term. We shall now describe a graphical procedure for obtaining the transfer function of Eq. (19.7) from a process reaction curve.
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SEMI-LOG PLOT FOR MODELING. The transfer function given by Eq. (19.7)
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can be obtained from a process reaction curve by a graphical method in which the logarithm of the incomplete response is plotted against time. In principle, this method can extract from the process reaction curve the two time constants in Eq. (19.7). The method, referred to as the semi-log plot method, is outlined below. The method applies for Ti > T2. 1. Determine (if transport lag is present) the time at which the process reaction curve of Fig. 19.12 first departs from the time axis; this time is taken as the transport lag Td . 2. From the process reaction curve of Fig. 19.12, plot I versus t 1 on semi-log paper as shown in Fig. 19.13 where Z is the fractional incomplete response and tl is the shifted time starting at Td (i.e., t 1 = t - Td). I is defined by
where B, is the ultimate value of Y. 3. Extend a tangent line through the data points at large values of f 1 (see Fig. 19.13). Refer to this tangent line as I, and let the intersection of the tangent line with the vertical axis at t 1 = 0 be called Z 4. To find the time constant T1 , read from the graph in Fig. 19.13 the time at which I, = 0.368P. This time is TI. 5. Plot A versus t i where A = I, - I. If the data points (A, t i) fall on a straight line, the system can be modeled as a second-order transfer function
----Bu --__-t T----Y
Ior F I G U R E t
9 - 1 2 Process reaction curve used in the semi-log plot method of modeling.
PROCESS
APPLICATIONS
T2 T I
(arithmetic scale)
FIGURE 19-13 Graphical construction for use in modeling by semi-log plot method.
with transport lag as given by Eq. (19.7) with time constants T1 and T2. The value of -Tz is the time at which A = 0.368R where R is the intersection of the line A with the vertical axis at t i = 0. If one does not get a straight line when A is plotted against t i, the procedure can be extended to get more first-order time constants, T3, T4, and so on; however, the data must be very accurate for this method to be successful in identifying more than two time constants. Usually the data scatter, especially at large values of time, and one must be satisfied in drawing straight lines through the scattered points. 6. The process gain is simply
Kp = B,IM
PROOF OF SEMI-LOG METHOD. By shifting the time axis from t to t i in Fig.
19.12 we have accounted for the effect of Td in Eq. (19.7) and the transient to be considered (Y vs. t 1) is described by the transfer function X(s) Introducing X = MIS and K, = B,/M into Eq. (19.8) gives 1 Y(s) - = s(Tls + 1)(T2s + 1) BU The time response of this expression is given by
Bu KP Gp(s) = (Tls + 1)(T2s + 1) Y(s) =-
(19.8)
(19.9)
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