Transfer functions model

Transfer function models are linear in nature, but chemical processes are known to exhibit nonhnear behavior. One could use the same type of optimization objective as given in Eq. (8-26) to determine parameters in nonlinear first-principle models, such as Eq. (8-3) presented earlier. Also, nonhnear empirical models, such as neural network models, have recently been proposed for process applications. The key to the use of these nonlinear empirical models is naving high-quality process data, which allows the important nonhnearities to be identified. 

RGA Example In order to illustrate use of the RGA method, consider the following steady-state version of a transfer function model for a pilot-scale, methanol-water distillation column (Wood and Berry, Terminal Composition Control of a Binaiy Distillation Column, Chem. Eng. Sci, 28, 1707, 1973) Ku = 12.8, K = -18.9, K. i = 6.6, and Koo = —19.4. It follows that A = 2 and... 

Our question is to formulate this model under two circumstances (1) when we only vary the dilution rate, and (2) when we vary both the dilution rate and the amount of glucose input. Derive also the transfer function model in the second case. In both cases, C, and C2 are the two outputs. 

Jury, W.A. (1982). Simulation of solute transport using a transfer function model. Water Resources Research 18(2), pp. 363-368. 

A number of techniques have been proposed. We will discuss only the more conventional methods that are widely used in the chemical and petroleum industries. Only the identification of linear transfer-function models will be discussed. Nonlinear identification is beyond the scope of this book. 

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. 

The important feature of the ATV method is that it gives transfer function models that fit the frequency-response data very well near the important frequencies of zero (steadystate gains) and the ultimate frequency (which determines closedloop stability). 

Figure 17.3 gives some comparisons of the performance of the multivariable DMC stmctuie with the diagonal stmcture. Three linear transfer-function models are presented, varying from the 2 x 2 Wood and Berry column to the 4 x 4 sidestream column/stripper complex configuration. The DMC tuning constants used for these three examples are NP = 40 and NC = 15. See Chap. 8, Sec. 8.9. 

Sections 3.2.1—3.2.3 have referred specifically to the system illustrated in Fig. 6. However, the approach in these sections is quite general and can therefore be used in situations where the system transfer function G(s) is other than that given by eqn. (7). For the case of the ideal PFR responses, G(s) is exp(— st) and impulse, step and frequency responses are simply these respective input functions delayed by a length of time equal to r. The non-ideal transfer function models of Sect. 5 may be used to produce families of predicted responses which depend on chosen model parameters. 

Process Transfer Function Models In continuous time, the dynamic behaviour of an ideal continuous flow stirred-tank reactor can be modelled (after linearization of any nonlinear kinetic expressions about a steady-state) by a first order ordinary differential equation of the form... 

To elucidate the details of the OEC reaction mechanism, bioinorganic chemists have made many new complexes to mimic the proposed biological mechanisms. Work on small molecules has clarified many issues relevant to the OEC such as the necessity of multiple metals and the importance of proton transfer. Functional model systems are still very much less active than the OEC, but current mechanistic models show activity very analogous to mechanisms proposed for the OEC [13,28, 30-34],... 

As Figure 1 shows, the main stages of the proposed methodology are the preliminary data analysis, the identification of all process variables (outputs, inputs and disturbance variables), the transfer function model (i.e. the mathematical model that describes the... 

After the preliminary data analysis, a satisfactory Box-Jenkins transfer function model was developed to describe, as much as possible, the dynamic behaviour of the bleached... 

Jury, W.A., G. Sposito, and R.E. White. 1986. A transfer function model of solute transport through soil I. Fundamental concepts. Water Resour. Res. 22 243-247. 

Following the MVC framework [102, 148[, consider a process described by a linear discrete-time transfer function model ... 

Dyson, J.S., and R.E. White. 1987. A comparison of the convective dispersion equation and transfer function model for predicting chloride leaching through undisturbed structured clay soil. J. Soil. Sci. 38 157-172. 

White, R.E., Thomas, G.W., and Smith, M.S. Modelling water flow through undisturbed soil cores using a transfer function model derived from 3HOH and Cl transport. Journal of soil science 35,159-168. 1984. 

To illustrate the topics of this section we will use as example a distillation column for the high purity separation of a benzene/toluene mixture. The following transfer function model in scaled variables represents the column s dynamics (Dimian, Bildea Kiss, 2001) ... 

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. 

Whatever the form of the input, the basic idea is to use a difference equation model for the process in which the current output y is related to previous values of the output yn- >yn-2> ) 3nd present and past values of the input (m , Wn-i,.. . ) hi the simple model structures, the relationship is linear, so classical least-squares can be used to solve for the best values of the unknown coefficients. These difference equation models occur naturally in sampled-data systems (see Chapter 15) and can be easily converted to Laplace-domain transfer function models. 

Hydrodynamic Transfer function model (Laplace transform s))... 

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