Identification of Model Parameters

Jacquez and Perry [37] developed the program ID ENT to investigate the identification of model parameters. In most cases, problems of identification can be detected by inspection of the standard errors of the model parameters (the standard error of a model parameter is a measure of the credibility of the parameter value, which is provided by the most fitting programs). A high standard error (for example, more than 50% of the parameter value) indicates that the parameter value cannot be assessed from the data, most likely due to an identification problem. In that case, the parameter value itself is meaningless, and thus the parameter set should be discarded (see Section 13.2.8.5). 

Figure 1 shows a computational framework, representing many years of Braun s research and development efforts in pyrolysis technology. Input to the system is a data base including pilot, commercial and literature sources. The data form the basis of a pyrolysis reactor model consistent with both theoretical and practical considerations. Modern computational techniques are used in the identification of model parameters. The model is then incorporated into a computer system capable of handling a wide range of industrial problems. Some of the applications are reactor design, economic and flexibility studies and process optimization and control. 

Advanced control strategies require a model that accurately represents the behavior of the process. Model identification involves determining an appropriate model structure, performing experiments, collecting data that allow identification of model parameters, and estimating the parameters. There are several ways to model crystallization processes, but a review of parameter estimation is beyond the scope of this chapter. A discussion of the most relevant methods of model identification for continuous crystallizers is given below. 

Cook, 1990] Cook, P. R. (1990). Identification of Control Parameters in an Articulatory Vocal Tract Model, with Applications to the Synthesis of Singing. PhD thesis, Elec. Eng. Dept., Stanford University. 

The rather complex issue of chemical kinetics has been discussed in a quantitative way, in order to stress out two main ideas, namely, the necessity of resorting to simplified kinetic models and the need of adequate methods of data analysis to estimate the kinetic parameters. These results introduce Chap. 3, in which basic concepts and up-to-date methods of identification of kinetic parameters are presented. 

Theories are not used directly, as in the discussion presented in Sect. 3.1, but allow building a mathematical model that describes an experiment in the unambiguous language of mathematics, in terms of variables, constants, and parameters. As an example, when considering the identification of kinetic parameters of chemical reactions from isothermal experiments performed in batch reactors, the relevant equations of mass conservation (presented in Sect. 2.3.1) give a set of ordinary differential equations in the general form... 

The description of small scale turbulent fields in confined spaces by fundamental approaches, based on statistical methods or on the concept of deterministic chaos, is a very promising and interesting research task nevertheless, at the authors knowledge, no fundamental approach is at the moment available for the modeling of large-scale confined systems, so that it is necessary to introduce semi-empirical models to express the tensor of turbulent stresses as a function of measurable quantities, such as geometry and velocity. Therefore, even in this case, a few parameters must be adjusted on the basis of independent measures of the fluid dynamic behavior. In any case, it must be underlined that these models are very complex and, therefore, well suited for simulation of complex systems but neither for identification of chemical parameters nor for online control and diagnosis [5, 6],... 

A new approach is the application of chemometrics (and neural networks) in modeling [73]. This should allow identification of the parameters of influence in solvent-resistant nanofiltration, which may help in further development of equations. Development of a more systematic model for description and prediction of solute transport in nonaqueous nanofiltration, which is applicable on a wide range of membranes, solvents and solutes, is the next step to be taken. The Maxwell-Stefan approach [74] is one of the most direct methods to attain this. 

The proposal of an adequate flow model of the reactor and the identification of the parameters are the main requirements of this application. Solving this type of problem involves two distinct actions first the selection of the flow model and second the computations involved in identifying the parameters. 

The notion of model parameters defines one or more numerical values that are contained as symbolic notations in the mathematical model of a process. These numerical values cannot be obtained without any experimental research. In reality, the most important part of experimental research is dedicated to the identification of the models parameters. Generally, all the experimental works, laboratory... 

Identification of the Parameters of a Model by the Steepest Slope Method... 

To begin the identification of the parameters with the Gauss-Newton method, the mathematical model of the process must be available. This model allows computation of the values of the temperature at the centre and the surface of the cylinder. At the same time, to estimate the starting vector of parameters (Pq), the method needs a first evaluation of the thermal conductivity 2.q and of the heat transfer coefficient Uq. 

The identification of the parameters of a process can be examined from two completely different viewpoints. The former is given by laboratory researchers, who consider the identification of parameters together with a deep experimental analysis it is then frequently difficult to criticize the experimental working methods, the quality and quantity of the experimental data. The latter is given by researchers specialized in mathematical modelling and simulation. These researchers consider that the mathematical aspects in the identification of parameters are prevailing. Nevertheless, this last consideration has some limits because, in all cases, a similar number of parameters and independent experimental data are necessary for a correct identification. 

The level of coalescence between particles, the size of the particles, and the packing arrangement dictate the size of air cavities and, thus, the size of the bubble initially formed in the melt. Once formed, the bubbles remain stationary in the melt. A relatively small bubble diameter, combined with the high viscosity of the melt, prevents the movement of the bubbles into the melt. The bubble removal is known to be a diffusion-controlled process. The identification of key parameters in the dissolution of bubbles formed in the melt has been done using a theoretical model that describes this process. The disappearance of the air bubble formed into the melt was modeled based on... 

The model allows the identification of the parameter that controls the relative efficiency of pulsatile stimuli of different periods. Indeed, numerical simulations indicate that the main process governing the response of the system to such stimuli is the rate of dephosphorylation, which determines the rapidity at which the receptor resensitizes between successive stimuli (this point is elaborated further in section 8.5). In D. discoideum amoebae, the rate of resensitization is thus governed by the activity of a phosphatase the kinetic constant and the concentration of that enzyme are such that dephosphorylation takes place... 

Trying to set up a physicochemically exact kinetic model for all simultaneously proceeding reactions with identification of all parameters would be a task so extremely time-consuming tiiat it could not be justified economically. Even modem computer programs, which use non-linear optimization techniques for the parameter adjustment in complex models, require an amount of analytical information on all substances participating in the process which is not to be underestimated [46]. 

Econometrics or macroeconomic models are primarily applied to economic description and forecasting problems. They are based on both theory and data. Emphasis is placed an specification of structural relations, based upon economic theory, and the identification of unknown parameters, using available data, in the behavioral equations. The method requires expertise in economics, statistics, and computer use. It can be quite expensive and time consuming. Macroeconomic models have been widely used for short- to medium-term economic analysis and forecasting. 

Comet JF, Diassap CG, Cluzel P, Dubertret G A stmctured model for simulation of cultures of the cyanobacterium Spirulina platensis in photobioreactors II. Identification of kinetic parameters under light and mineral limitations, Biotechnol Bioeng 40(7) 826—834, 1992. 

Zhong, L. Youchao, S. 2007. Research on Maintainability Evaluation Model Based on Fuzzy Theory. Chinese Journal of Aeronautics 20 402—407 Zio, E. etal. 2003. The analytic hierarchy process as a systematic approach to the identification of important parameters for the reliability assessment of passive systems. Nuclear Engineering and Design 226 311 336... 

Certain techniques for the application of thermodynamics in separation technology are introduced in Chapter 11, for example, the concept of residue curve maps, a general procedure for the choice of suitable solvents for the separation of azeotropic systems, the verification of model parameters prior to process simulation and the identification of separation problems. 

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