Low-order dynamic models

A. Kienle, Low-order dynamic models for ideal multicomponent distillation processes using nonlinear wave propagation theory. Chem. 

Steps 4 and 5 Computing and (s). These are computed following the procedure in the section on Generating Low-Order Dynamic Models, which gives linearized models for the high-pressure column in the LSF configuration ... 

From a practical perspective, this is the model that should be used to design a (multivariable) controller that manipulates the inputs us to fulfill the control objectives ys. It is important to note that the availability of a low-order ODE model of the process-level dynamics affords significant flexibility in designing the supervisory control system, since any of the available inversion- or optimization-based (e.g., Kravaris and Kantor 1990, Mayne et al. 2000, Zavala... 

Since most processes have low-order dynamics and the overall transfer function, M, is likely to have a large order, plenty of cancellations must occur between Gy and M. However, many of these cancellations will not occur if the model estimates are even slightly off. Thus, there is a potential of creating a very large order model, even if it is not warranted. Furthermore, this method can only be used for identifying data obtained when the process has external excitation. 

Verbitsky, M. Ya. and B. Saltzman, 1994. Heinrich type glacial surges in a low-order dynamical climate model. Climate Dynamics, 10, 39-47. 

The basic strategy of modal reduction approaches is to retain only certain modes of the high-order model in the low-order model. Wilson et al. (1974) summarized these techniques and showed that many of the published modal approaches are equivalent since they produce identical reduced models. Bonvin (1980) also provides a comparison of the various modal techniques with respect to their steady-state and dynamic accuracies as well as to the dependence of the reduced models on the retained state variables. 

When more satisfactory forms of diffusion coefficient for the hydro-dynamic repulsion effect become available, these should be incorporated into the diffusion equation analysis. The effect of competitive reaction processes on the overall rate of reaction only becomes important when the concentration of both reactants is so large that it would require exceptional means to generate such concentrations of reactants and a solvent of extremely low diffusion coefficient to observe such effects. This effect has been the subject of much rather repetitive effort recently (see Chap. 9, Sect. 5.5). By contrast, the recent numerical studies of reactions between uncharged species is a most welcome study of the effect of this competition in various small clusters of reactants (see Chap. 7, Sect. 4.4). It is to be hoped that this work can be extended to reactions between ions in order to model spur decay processes in solvents less polar than water. One other area where research on the diffusion equation analysis of reaction rates would be very welcome is in the application of the variational principle (see Chap. 10). 

The order of the postulated model is a very important factor. For a well-known process such as a stirred-tank heater, it is not a problem. On the other hand, it is not obvious what order of dynamics we should assume for a fluid catalytic cracker (see Example 4.15). Also, it is not obvious what type of low-order model we should use to approximate the high-order models of even simple distillation columns (see Example 4.16). As a general starting point one could employ first- or second-order models with or without dead time. There exist a surprisingly large number of processes which could be effectively described by such low-order models. 

For proper dynamics modeling it is necessary to estimate the composition and nature of VOCs. GC-MS analysis of exhausted carbon samples shows the presence of more titan 11 main components. Decane, dodecane, 1,2,4-trimethylbenzene and xylene and their derivatives arc present at the highest concentration [131]. All these species have high boiling temperature (from 413 K up to 443 K), density around 0.7-O.9 g cm and low saturation vapor pressure at ambient conditions. TA analysis showed that VOC desorption from activated carbons occurs at temperature between 473 K and 773 K. Total initial concentration of VOCs estimated from the amount adsorbed and column operation time is in the order of 1.4 ppm. 

Rathnasingham and Breuer [4] developed the first low-order model of a synthetic jet using a control volume model for the flow and an empirical model for the stmctural dynamics of the diaphragm. A lumped element model of a piezoelectric-driven synthetic jet actuator was derived in [5]. 

The capability to calculate low-order equivalent linear systems from time domain simulations of SOFC models using the ARX algorithm has been established. After the SOFC model was created, it was reduced to transfer fimctions using the ARX algorithm thus, the transfer function (reduced-order model) exhibited the same dynamic response as the original SOFC model. 

Stochastic identification techniques, in principle, provide a more reliable method of determining the process transfer function. Most workers have used the Box and Jenkins [59] time-series analysis techniques to develop dynamic models. An introduction to these methods is given by Davies [60]. In stochastic identification, a low amplitude sequence (usually a pseudorandom binary sequence, PRBS) is used to perturb the setting of the manipulated variable. The sequence generally has an implementation period smaller than the process response time. By evaiuating the auto- and cross-correlations of the input series and the corresponding output data, a quantitative model can be constructed. The parameters of the model can be determined by using a least squares analysis on the input and output sequences. Because this identification technique can handle many more parameters than simple first-order plus dead-time models, the process and its related noise can be modeled more accurately. 

Armero, F. Romero, I. 2001a. On the formulation of high-frequency dissipative timestepping algorithms for nonlinear dynamics. Part I Low order methods for two model problems and nonlinear elastodynamics . Comp. Meth. Appl. Mech. Eng., 190 2603-2649. 

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