Unknown constant parameter, problems

Parameter estimation problems result when we attempt to match a model of known form to experimental data by an optimal determination of unknown model parameters. The exact nature of the parameter estimation problem will depend on the mathematical model. An important distinction has to be made at this point. A model will contain both state variables (concentrations, temperatures, pressures, etc.) and parameters (rate constants, dispersion coefficients, activation energies, etc.). 

Equation 3.32 provides an expression for the flame spread velocity based only on the parameters of the problem and on the experimental conditions, but still relies on the presence of an unknown constant that needs to be determined experimentally. 

The computing problem is concerned with calculating the maximum number of unknown parameters of a proposed reaction system from available experimental data. This data can be any combination of values for constant parameters (rate and equilibrium constants) and variable parameters (concentration versus time data). Moreover, data for different variable parameters need not have the same time scale. When the unknown parameters are calculated, it is important that the mathematical validity of the proposed model be determined in terms of the experimental accuracy of the data. Also, if it is impossible to solve for all unknown parameters, then the model must be automatically reduced to a form that contains only solvable parameters. Thus, the input to CRAMS consists of 1) a description of a proposed reaction system model and, 2) experimental data for those parameters that were measured or previously determined. The output of CRAMS is 1) information concerning the mathematical validity of the model and 2) values for the maximum number of computable unknown parameters and, if possible, the associated reliabilities. The system checks for model validity only in those reactions with unknown rate constants. Thus a simulation-only problem does not invoke any model validation procedures. 

The identification problem described above is somewhat different and more complex as compared to the implicit isothermal problem described in Sect. 3.6, where Np rate constants are estimated from data measured at constant temperature. In fact, in this case, for each reduced model, the unknown parameters are (/ = 1,..., AL) ... 

In theory, by feeding the MWD and experimental rate data into a mathematical model containing a variety of polymerization mechanisms, it should be possible to find the mechanism which explains all the experimental phenomena and to evaluate any unknown rate constants. As pointed out by Zeman (58), as long as there are more independent experimental observations than rate parameters, the solution should, in principle, be unique. This approach involves critical problems in choice of experiments and in experimental as well as computational techniques. We are not aware of its having yet been successfully employed. The converse— namely, predicting MWD from different reactor types on the basis of mathematical models and kinetic data—has been successfully demonstrated, however, as discussed above. The recent series of interesting papers by Hamielec et al. is a case in point. 

However, chemical ionization is sensitive to source parameters and matrix effects, and these problems are exacerbated by the direct introduction of a complex mixture into the source. The effects can be ccnpensated to seme degree by the use of an isotopically labelled internal standard for quantitative work. In the analysis of unknowns in canplex mixtures, the nature of the source chemistry should be a constant concern. 

A variety of techniques is nowadays available for the solution of inverse problems [26,27], However, one common approach relies on the minimization of an objective function that generally involves the squared difference between measured and estimated variables, like the least-squares norm, as well as some kind of regularization term. Despite the fact that the minimization of the least-squares norm is indiscriminately used, it only yields maximum likelihood estimates if the following statistical hypotheses are valid the errors in the measured variables are additive, uncorrelated, normally distributed, with zero mean and known constant standard-deviation only the measured variables appearing in the objective function contain errors and there is no prior information regarding the values and uncertainties of the unknown parameters. 

A practical problem in applying the Thiele modulus is that the values of the kinetic constants Vmax and Km are frequently unknown. Therefore an observable Thiele modulus was introduced which depends on the measurable overall rate, but not on kinetic parameters ... 

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