Nonlinear Models in Parameters. Single Reaction

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. 

Inadequate stoichiometry and poor calibration of the analytical device are interconnected problems. The kinetic model itself follows the stoichiometric rules, but an inadequate calibration of the analytical instrument causes systematic deviations. This can be illustrated with a simple example. Assume diat a bimolecular reaction, A + B P, is carried out in a liquid-phase batch reactor. The density of the reaction mixture is assumed to be constant. The reaction is started with A and B, and no P is present in the initial mixture. The concentrations are related by cp=CoA-Cj=Cob -Cb, i e. produced product, P, equals with consumed reactant. If the concentration of the component B has a calibration error, we get instead of the correct concentration cb an erroneous one, c n ncs, which does not fulfil the stoichiometric relation. If the error is large for a single component, it is easy to recognize, but the situation can be much worse calibration errors are present in several components and all of their effects are spread during nonlinear regression, in the estimation of the model parameters. This is reflected by the fact that the total mass balance is not fulfilled by the experimental data. A way to check the analytical data is to use some fonns of total balances, e.g. atom balances or total molar amounts or concentrations. For example, for the model reaction, A + B P, we have the relation ca+cb+cp -c()a+c0 -constant (again c0p=0). 

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