Approximate transfer functions

Once the test has been performed and the ultimate gain and ultimate frequency have been determined, we may simply use it to calculate Ziegler-Nichols settings. Alternatively, it is possible to use this information, along with other easily determined data, to calculate approximate transfer functions. The idea is to pick some simple forms of transfer functions (gains, deadtime, first- or second-order lags) and find the parameter values that fit the ATV results. 

The other data needed are the steadystate gain and the deadtime. 

This kind of calculation is called a rating calculation (equipment size is fixed), as opposed to a design calculation in which equipment is sized. 

DEADTIME. The deadtime D in the transfer function can be easily read off the initial part of the ATV test. It is simply the time it takes x to start responding to the initial change in m. 

Now we are ready to find an approximate model. We assume one of the forms given in Eqs. (14.41) and (14.42) given below. 

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We put in a step disturbance m, and record the output variable x, as a function of time, as illustrated in Fig. 14.1. The quick-and-dirty engineering approach is to simply look at the shape of the x, curve and find some approximate transfer function Gjj, that would give the same type of step response. 

If a transfer-function model is desired, approximate transfer functions can be fitted to the experimental curves. First the log modulus Bode plot is used. The low-frequency asymptote gives the steadystate gain. The time constants can be found from the breakpoint frequency and the slope of the high-frequency asymptote. The damping coefficient can be found from the resonant peak. 

Once the log modulus curve has been adequately fitted by an approximate transfer function G(J ), the phase angle of G( a) is compared with the experimental phase-angle curve. The difference is usually the contribution of deadtime. The procedure is illustrated in Fig. 14.2. 

J. The frequency-response data given below were obtained from direct sine-wave tests of a chemical plant. Fit an approximate transfer function to these data. 

A distillation column has an approximate transfer function between overhead composition Xp and reflux flow rate R of... 

Fitting an approximate transfer function to experimental frequency response data. 

Once the log modulus curve has been adequately fitted by an approximate transfer function the phase angle of G is compared with the experimental phase... 

It is usually important to get an accurate fit of the model frequency response to the experimental frequency response near the critical region where the phase angle is between 135 and — 180 . It doesn t matter how well or how poorly the approximate transfer function fits the data once the phase angle has dropped below — 180 . So the fitting of the approximate transfer function should weight heavily the differences between the model and the data over this frequency range. 

These simple approximate transfer function models are particularly useful in multivariable systems. For example, to use the BLT method discussed in Chapter 13, we need all the N X N transfer functions relating the N inputs to the N outputs. The ATV method provides a quick and fairly accurate way to obtain all these transfer functions. 

The space velocity, often used in the technical literature, is the total volumetric feed rate under normal conditions, F o(Nm /hr) per unit catalyst volume (m X that is, PbF o/W. It is related to the inverse of the space time W/F g used in this text (with W in kg cat. and F q in kmol A/hr). It is seen that, for the nominal space velocity of 13,800 (m /m cat. hr) and inlet temperatures between 224 and 274 C, two top temperatures correspond to one inlet temperature. Below 224 C no autothermal operation is possible. This is the blowout temperature. By the same reasoning used in relation with Fig. 11.5.e-2 it can be seen that points on the left branch of the curve correspond to the unstable, those on the right branch to the upper stable steady state. The optimum top temperature (425°C), leading to a maximum conversion for the given amount of catalyst, is marked with a cross. The difference between the optimum operating top temperature and the blowout temperature is only 5°C, so that severe control of perturbations is required. Baddour et al. also studied the dynamic behavior, starting from the transient continuity and energy equations [26]. The dynamic behavior was shown to be linear for perturbations in the inlet temperature smaller than 5°C, around the conditions of maximum production. Use of approximate transfer functions was very successful in the description of the dynamic behavior. 

Overall, then, we obtain the approximate transfer function for an -tray column ... 

The normalized step responses of the original and approximate transfer functions are shown in Fig. 6.11. The second-order model provides an excellent approximation, because the neglected time constants are much smaller than the retained time constants. The first-order-plus-time-delay model is not as accurate, but it does provide a suitable approximation of the actual Mnse. 

A process (including valve and sensor-transmitter) has the approximate transfer function, G s) = 2e l s + 1) with time constant and time delay in minutes. Determine PI controller settings and the corresponding gain... 

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