Modeling dynamic

The steady-state modeling is not sufficient for batch and semibatch RD processes or if one tries to optimize the start-up and shut-down phases of the process. In this case, a knowledge of dynamic process behavior is necessary. Further areas where the dynamic information is crucial are the process control as well as safety issues and training. 

In the dynamic rate-based stage model, molar holdup terms have to be considered in the mass balance equations, whereas the change of both the specific molar component holdup and the total molar holdup are taken into account. For the liquid phase, these equations are as follows  

The vapor holdup can often be neglected due to the low vapor-phase density, and the component balance equation reduces to Eq. (10.4) (see also [85]). 

The dynamic formulation of the model equations requires a careful analysis of the whole system in order to prevent high-index problems during the numerical solution [86]. As a consequence, a consistent set of initial conditions for the dynamic simulations and a suitable description of the hydrodynamics have to be introduced. For instance, pressure drop and liquid holdup must be correlated with the vapor and liquid flows. 

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Figure 1. Dynamic model of testing of the material as a black box , where IN - loads and actions OUT - response of the material as measuring characteristics N - noise (combination of uncertain factors) ...

It is very important to make classification of dynamic models and choose an appropriate one to provide similarity between model behavior and real characteristics of the material. The following general classification of the models is proposed for consideration deterministic, stochastic or their combination, linear, nonlinear, stationary or non-stationary, ergodic or non-ergodic. 

Where Ui denotes input number i and there is an implied summation over all the inputs in the expression above A, Bj, C, D, and F are polynomials in the shift operator (z or q). The general structure is defined by giving the time delays nk and the orders of the polynomials (i.e., the number of poles and zeros of the dynamic models trom u to y, as well as of the noise model from e to y). Note that A(q) corresponds to poles that are common between the dynamic model and the noise model (useful if noise enters system close to the input). Likewise Fj(q) determines the poles that are unique for the dynamics from input number i and D(q) the poles that are unique for the noise N(t). 

An idea of investigation of AE response of the material to different types of loads and actions seems to be useful for building up a dynamic model of the material. In this ease AE is representing OUT data, and it is possible to take various AE parameters for this purpose. It is possible to consider a single AE pulse in time or frequency domain or AE pulses sequence as... 

Dynamic models for ionic lattices recognize explicitly the force constants between ions and their polarization. In shell models, the ions are represented as a shell and a core, coupled by a spring (see Refs. 57-59), and parameters are evaluated by matching bulk elastic and dielectric properties. Application of these models to the surface region has allowed calculation of surface vibrational modes [60] and LEED patterns [61-63] (see Section VIII-2). 

We now proceed to some examples of this Fourier transfonn view of optical spectroscopy. Consider, for example, the UV absorption spectnun of CO2, shown in figure Al.6.11. The spectnuu is seen to have a long progression of vibrational features, each with fairly unifonu shape and width. Wliat is the physical interpretation of tliis vibrational progression and what is the origin of the width of the features The goal is to come up with a dynamical model that leads to a wavepacket autocorrelation fiinction whose Fourier transfonn... 

For very fast reactions, as they are accessible to investigation by pico- and femtosecond laser spectroscopy, the separation of time scales into slow motion along the reaction path and fast relaxation of other degrees of freedom in most cases is no longer possible and it is necessary to consider dynamical models, which are not the topic of this section. But often the temperature, solvent or pressure dependence of reaction rate... 

Luntz A C and Harris J 1991 CH dissociation on metals—a quantum dynamics model Surf. Sc/. 258 397... 

Table Bl.13.1 Selected dynamic models used to calculate spectral densities.

The approach is ideally suited to the study of IVR on fast timescales, which is the most important primary process in imimolecular reactions. The application of high-resolution rovibrational overtone spectroscopy to this problem has been extensively demonstrated. Effective Hamiltonian analyses alone are insufficient, as has been demonstrated by explicit quantum dynamical models based on ab initio theory [95]. The fast IVR characteristic of the CH cliromophore in various molecular environments is probably the most comprehensively studied example of the kind [96] (see chapter A3.13). The importance of this question to chemical kinetics can perhaps best be illustrated with the following examples. The atom recombination reaction... 

I. Theory, and application to a quantum-dynamics model J. Chem. Phys. 102 8011... 

Bala, P., Grochowsky, R, Lesyng, B., McCammon, J.A. Quantum-classical molecular dynamics. Models and applications. In Quantum mechanical simulation methods for studying biological systems, D. Bicout and M. Field, eds. Springer, Berlin (1996) 119-156. 

Van Vlimmeren, B.A.C., Fraaije, J.G.E.M. Calculation of noise distribution in mesoscopic dynamics models for phase-separation of multicomponent complex fluids. Comput. Phys. Comm. 99 (1996) 21-28. 

Approximation Properties and Limits of the Quantum-Classical Molecular Dynamics Model... 

P. Bala, P. Grochowski, B. Lesyng, and J. A. McCammon Quantum-classical molecular dynamics. Models and applications. In Quantum Mechanical Simulation Methods for Studying Biological Systems (M. Fields, ed.). Les Houches, France (1995)... 

Bornemann, F. A., Schiitte, Ch. On the Singular Limit of the Quantum-Classical Molecular Dynamics Model. Preprint SC 97-07 (1997) Konrad-Zuse-Zentrum Berlin. SIAM J. Appl. Math, (submitted)... 

Abstract. The overall Hamiltonian structure of the Quantum-Classical Molecular Dynamics model makes - analogously to classical molecular dynamics - symplectic integration schemes the methods of choice for long-term simulations. This has already been demonstrated by the symplectic PICKABACK method [19]. However, this method requires a relatively small step-size due to the high-frequency quantum modes. Therefore, following related ideas from classical molecular dynamics, we investigate symplectic multiple-time-stepping methods and indicate various possibilities to overcome the step-size limitation of PICKABACK. 

F.A. Bornemann and Ch. Schiitte. On the singular limit of the quantum-classical molecular dynamics model. Preprint SC 97-07, ZIB Berlin, 1997. Submitted to SIAM J. Appl. Math. 

Suffice it to say that a dynamic model of this system was proposed that allowed the estimation of kinetic parameters and gave reasonable agreement with the experimental observations in the bioreactor [22]. 

Ogunnaike, B. A., and W. H. Ray. Frocess Dynamics, Modeling, and Con-trol, Oxford University Press (1994). 

Simulation of Dynamic Models Linear dynamic models are particularly useful for analyzing control-system behavior. The insight gained through linear analysis is invaluable. However, accurate dynamic process models can involve large sets of nonlinear equations. Analytical solution of these models is not possible. Thus, in these cases, one must turn to simulation approaches to study process dynamics and the effect of process control. Equation (8-3) will be used to illustrate the simulation of nonhnear processes. If dcjdi on the left-hand side of Eq. (8-3) is replaced with its finite difference approximation, one gets ... 

Fitting Dynamic Models to E erimental Data In developing empirical transfer functions, it is necessary to identify model parameters from experimental data. There are a number of approaches to process identification that have been pubhshed. The simplest approach involves introducing a step test into the process and recording the response of the process, as illustrated in Fig. 8-21. The i s in the figure represent the recorded data. For purposes of illustration, the process under study will be assumed to be first order with deadtime and have the transfer func tion ... 

The above FF controller can be implemented using analog elements or more commonly by a digital computer. Figure 8-33 compares typical responses for PID FB control, steady-state FF control (.s = 0), dynamic FF control, and combined FF/FB control. In practice, the engineer can tune K, and Tl in the field to improve the performance oTthe FF controller. The feedforward controller can also be simplified to provide steady-state feedforward control. This is done by setting. s = 0 in Gj. s). This might be appropriate if there is uncertainty in the dynamic models for Gl and Gp. 

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