Parameter estimation single reactions

Parameter Estimation and Statistical Testing of Models and Parameters in Single Reactions... 

The techniques referred to above (Sects. 1—3) may be operated for a sample heated in a constant temperature environment or under conditions of programmed temperature change. Very similar equipment can often be used differences normally reside in the temperature control of the reactant cell. Non-isothermal measurements of mass loss are termed thermogravimetry (TG), absorption or evolution of heat is differential scanning calorimetry (DSC), and measurement of the temperature difference between the sample and an inert reference substance is termed differential thermal analysis (DTA). These techniques can be used singly [33,76,174] or in combination and may include provision for EGA. Applications of non-isothermal measurements have ranged from the rapid qualitative estimation of reaction temperature to the quantitative determination of kinetic parameters [175—177]. The evaluation of kinetic parameters from non-isothermal data is dealt with in detail in Chap. 3.6. 

Thermal methods in kinetic modelling. Methods for the estimation of thermokinetic parameters based on experiments in a reaction calorimeter will be discussed below. As mentioned in section 5.4.4.3, instantaneous heat evolved due to a single reaction is directly proportional to the reaction rate. Assume that the reaction is of first order. Then for isothermal operation ... 

The next two steps after the development of a mathematical process model and before its implementation to "real life" applications, are to handle the numerical solution of the model s ode s and to estimate some unknown parameters. The computer program which handles the numerical solution of the present model has been written in a very general way. After inputing concentrations, flowrate data and reaction operating conditions, the user has the options to select from a variety of different modes of reactor operation (batch, semi-batch, single continuous, continuous train, CSTR-tube) or reactor startup conditions (seeded, unseeded, full or half-full of water or emulsion recipe and empty). Then, IMSL subroutine DCEAR handles the numerical integration of the ode s. Parameter estimation of the only two unknown parameters e and Dw has been described and is further discussed in (32). 

Fowle and Fein (1999) measured the sorption of Cd, Cu, and Pb by B. subtilis and B. licheniformis using the batch technique with single or mixed metals and one or both bacterial species. The sorption parameters estimated from the model were in excellent agreement with those measured experimentally, indicating that chemical equilibrium modeling of aqueous metal sorption by bacterial surfaces could accurately predict the distribution of metals in complex multicomponent systems. Fein and Delea (1999) also tested the applicability of a chemical equilibrium approach to describing aqueous and surface complexation reactions in a Cd-EDTA-Z . subtilis system. The experimental values were consistent with those derived from chemical modeling. 

Note, however, that, in the case of fundamental models, there is not always a need to discriminate among rival models since, often, only a single model has been built up. Furthermore, the best criterion of the quality of a model is the consistency of fundamental parameter estimates with other values obtained by means of several methods under a large range of experimental conditions. Let us not be misled about the principle enemy the systematic errors both in experiments and in reaction and reactor models. 

As another example of the first interaction, a potential parameter in the analysis of the CSTR is estimating the actual reactor volume. CSTR shown in Fig. 30-7. The steady-state material balance for this CSTR having a single reaction can be represented as ... 

Another possible correlation between coal structure and pyrolysis behavior is indicated by the temperature dependence of the evolution of pyrolytic water being strikingly different for the two coals. Figure 5 shows pyrolytic water evolution data for experiments in which the sample was heated at 1000°C/sec to the peak temperature indicated on the abscissa and then immediately allowed to cool at around 200°C/sec. The smooth curves are based on a single reaction, first-order decomposition model (7,8) and on the stated temperature-time history. Parameters used for the lignite have been published (8) while for the bituminous coal the Arrhenius frequency factor and activation energy were taken as 1013 sec"1 and 35 kcal/mol, respectively, with the yield of pyrolytic water ultimately attainable estimated from experimental measurements as 4.6 wt % of the coal (as-received). 

Nonlinear Models in Parameters, Single Reaction In practice, the parameters appear often in nonlinear form in the rate expressions, requiring nonlinear regression. Nonlinear regression does not guarantee optimal parameter estimates even if the kinetic model adequately represents the true kinetics and the data width is adequate. Further, the statistical tests of model adequacy apply rigorously only to models linear in parameters, and can only be considered approximate for nonlinear models. 

Network of Reactions The statistical parameter estimation for multiple reactions is more complex than for a single reaction. As indicated before, a single reaction can be represented by a single con-... 

Although the LO curves lump together the chemical and physical processes, and the experimental conditions, they can be used to estimate the rate parameters provided that a reliable model accounting for all the nonchemical characteristics is available. We consider a simple mixture of reactive species subject to a single reaction ... 

Now we turn to the single most important parameter estimation problem in chemical reactor modeling determining reaction-rate constants given dynamic concentration measurements. We devote the rest of the chapter to developing methods for this problem. 

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