Kinetic modeling parameter estimation

L. L. Oliveira and E. Biscaia Catalytic cracking kinetic models. Parameter estimation and model evaluation. Industrial and Engineering Chemistry Research, 28, 264-271 (1989). 

Oliveira, L.L. and Biscaia, E.C., "Catalytic Cracking Kinetic Models. Parameter Estimation and Model Evaluation", Ind. Eng. Chem. Res., 28, 264-271 (1989). 

Discrimination of the Kinetic Models and Estimation of their Parameters... 

Thus far, the transport model, after incorporation of equilibrium or kinetic retention, was used in a calibration mode where, along with nonlinear least-squares approximation, a best fit of the model to the experimental BTC was attempted. This resulted in a set of model parameter estimates that provided the best fit of the BTC for a... 

It should be pointed out that not all software programs lead to the same model structural model parameter estimates and variance components. Roe (1997) compared the simulated pharmacokinetics of a drug having monoexponential kinetics where clearance was a function of saturable protein binding and renal function and volume of distribution was a function of saturable protein binding only. The basis for the simulated concentrations was a population analysis of 361 quinidine concentration-time measurements from 136 male patients who had experienced cardiac arrhythmia (Verme et al., 1992). The same distribution of simulated observations (e.g., 46 patients had only one sample collected, 33 patients had two samples collected) was used as in the actual study. She and many other participants on the project analyzed the dataset with seven different... 

In the field of pharmacokinetics, there has been much recent work on developing methods for estimating interindividual variation in kinetic model parameters, particularly in sparse data situations where there are... 

Todic et al. [14] developed a comprehensive micro-kinetic model based on the carbide mechanism that predicts FT product distribution up to carbon number 15. This model explains the non-ASF product distribution using a carbon number dependent olefin formation rate (e term). The rate equations for the olefins and paraffins used in the model are shown in Figure 2. The derivation of the rate equations and physical meaning of the kinetic parameters, as well as their fitted values, can be found in Todic et al. [14]. In the current study, a MATLAB code which uses the Genetic Algorithm Toolbox has been developed, following the method of Todic et al. [14], to estimate the kinetic model parameters. In order to validate our code, model output from Todic et al. [14] was used as the input data to our code, and the kinetic parameter values were back-calculated and compared to the values fi om [14], as shown in Table 1. The model has 19 kinetic parameters that are to be estimated. The objective function to be minimized was defined as... 

The seventy one kinetic parameters of the proposed kinetic model were estimated using the experimental information obtained at different temperatures and WHSV. For each reaction step, a kinetic expression was formulated as a function of product yields and kinetic constants. 

In the example that follows, a possible mechanism is presented to illustrate the procedure followed to obtain an acceptable kinetic model and estimates of the kinetic parameters. 

One of the main differences between the polymerization kinetics with coordination catalysts and free-radical initiators is that the former depends on the characteristics of the active site as well as on monomer type, while the latter is almost exclusively regulated by monomer type. As we will see, even though this may not constitute a problem for establishing an operative mechanism for coordination polymerization, it creates a significant challenge for model parameter estimation. 

Verneuil et al. (Verneuil, V.S., P. Yan, and F. Madron, Banish Bad Plant Data, Chemical Engineering Progress, October 1992, 45-51) emphasize the importance of proper model development. Systematic errors result not only from the measurements but also from the model used to analyze the measurements. Advanced methods of measurement processing will not substitute for accurate measurements. If highly nonlinear models (e.g., Cropley s kinetic model or typical distillation models) are used to analyze unit measurements and estimate parameters, the Hkelihood for arriving at erroneous models increases. Consequently, resultant models should be treated as approximations. 

Cropley made general recommendations to develop kinetic models for compUcated rate expressions. His approach includes first formulating a hyperbolic non-linear model in dimensionless form by linear statistical methods. This way, essential terms are identified and others are rejected, to reduce the number of unknown parameters. Only toward the end when model is reduced to the essential parts is non-linear estimation of parameters involved. His ten steps are summarized below. Their basis is a set of rate data measured in a recycle reactor using a sixteen experiment fractional factorial experimental design at two levels in five variables, with additional three repeated centerpoints. To these are added two outlier... 

Tire simplest model for describing binary copolyinerization of two monomers, Ma and Mr, is the terminal model. The model has been applied to a vast number of systems and, in most cases, appears to give an adequate description of the overall copolymer composition at least for low conversions. The limitations of the terminal model generally only become obvious when attempting to describe the monomer sequence distribution or the polymerization kinetics. Even though the terminal model does not always provide an accurate description of the copolymerization process, it remains useful for making qualitative predictions, as a starting point for parameter estimation and it is simple to apply. 

Obtaining Kinetic Samples for Reactive Extrusion. To develop and test kinetic models, homogeneous samples with a well defined temperature-time history are required. Temperature history does not necessarily need to be isothermal. In fact, well defined nonisothermal histories can provide very good test data for models. However, isothermal data is very desirable at the initial stages of model building to simplify both model selection and parameter estimation problems. 

De Gaetano A, Mingrone G, Castagneto M. NONMEM improves group parameter estimation for the minimal model of glucose kinetics. Am J Physiol 1996 271 E932-7. 

In Fig. 1, a comparison can be observed for the prediction by the honeycomb reactor model developed with the parameters directly obtained from the kinetic study over the packed-bed flow reactor [6] and from the extruded honeycomb reactor for the 10 and 100 CPSI honeycomb reactors. The model with both parameters well describes the performance of both reactors although the parameters estimated from the honeycomb reactor more closely predict the experiment data than the parameters estimated from the kinetic study over the packed-bed reactor. The model with the parameters from the packed-bed reactor predicts slightly higher conversion of NO and lower emission of NHj as the reaction temperature decreases. The discrepancy also varies with respect to the reactor space velocity. 

The kinetic parameters estimated by the experimental data obtained frmn the honeycomb reactor along with the packed bed flow reactor as listed in Table 1 reveal that all the kinetic parameters estimated from both reactors are similar to each other. This indicates that the honeycomb reactor model developed in the present study can directly employ intrinsic kinetic parameters estimated from the kinetic study over the packed-bed flow reactor. It will significantly reduce the efibrt for predicting the performance of monolith and estimating the parameters for the design of the commercial SCR reactor along with the reaction kinetics. 

Based on surface science and methods such as TPD, most of the kinetic parameters of the elementary steps that constitute a catalytic process can be obtained. However, short-lived intermediates cannot be studied spectroscopically, and then one has to rely on either computational chemistry or estimated parameters. Alternatively, one can try to derive kinetic parameters by fitting kinetic models to overall rates, as demonstrated below. 

To illustrate the utility of the MWBD method, a series of commercial polyvinyl acetates and low density polyethylenes are analyzed. Either kinetic models or 13c nuclear magnetic resonance results are used to estimate the branching structural parameter. 

Estimation of parameters. Model parameters in the selected model are then estimated. If available, some model parameters (e.g. thermodynamic properties, heat- and mass-transfer coefficient, etc.) are taken from literature. This is usually not possible for kinetic parameters. These should be estimated based on data obtained from laboratory expieriments, if possible carried out isothermal ly and not falsified by heat- and mass-transport phenomena. The methods for parameter estimation, also the kinetic parameters in complex organic systems, and for discrimination between models are discussed in more detail in Section 5.4.4. More information on parameter estimation the reader will find in review papers by Kittrell (1970), or Froment and Hosten (1981) or in the book by Froment and Bischoff (1990). 

A survey of the mathematical models for typical chemical reactors and reactions shows that several hydrodynamic and transfer coefficients (model parameters) must be known to simulate reactor behaviour. These model parameters are listed in Table 5.4-6 (see also Table 5.4-1 in Section 5.4.1). Regions of interfacial surface area for various gas-liquid reactors are shown in Fig. 5.4-15. Many correlations for transfer coefficients have been published in the literature (see the list of books and review papers at the beginning of this section). The coefficients can be evaluated from those correlations within an average accuracy of about 25%. This is usually sufficient for modelling of chemical reactors. Mathematical models of reactors arc often more sensitive to kinetic parameters. Experimental methods and procedures for parameters estimation are discussed in the subsequent section. 

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