Predictive kinetics rate parameters estimation

Laboratory data collected over honeycomb catalyst samples of various lengths and under a variety of experimental conditions were described satisfactorily by the model on a purely predictive basis. Indeed, the effective diffusivities of NO and NH3 were estimated from the pore size distribution measurements and the intrinsic rate parameters were obtained from independent kinetic data collected over the same catalyst ground to very fine particles [27], so that the model did not include any adaptive parameters. 

During the selectivity kinetic parameter estimation, the relationship for x in terms of C5 - is determined from Eq. (12). For an assumed set of rate constants K, x is calculated for each composition data point such that the experimentally measured C5- equals that estimated from Eq. (12). Selectivity composition profiles as a function of C5- are generated in this manner. The proper selectivity matrix K will be that which minimizes the deviation between experimental and predicted profiles for the hydrocarbons other than C5-, as illustrated in Fig. 10. 

The rate parameters of importance in the multicomponent rate model are the mass transfer coefficients and surface diffusion coefficients for each solute species. For accurate description of the multicomponent rate kinetics, it is necessary that accurate values are used for these parameters. It was shown by Mathews and Weber (14), that a deviation of 20% in mass transfer coefficients can have significant effects on the predicted adsorption rate profiles. Several mass transfer correlation studies were examined for estimating the mass transfer coefficients (15, jL6,17,18,19). The correlation of Calderbank and Moo-Young (16) based on Kolmogaroff s theory of local isotropic turbulence has a standard deviation of 66%. The slip velocity method of Harriott (17) provides correlation with an average deviation of 39%. Brian and Hales (15) could not obtain super-imposable curves from heat and mass transfer studies, and the mass transfer data was not in agreement with that of Harriott for high Schmidt number values. 

The QSAR models can be used to estimate the treatability of organic pollutants by SCWO. For two chemical classes such as aliphatic and aromatic compounds, the best correlation exists between the kinetic rate constants and EHOMO descriptor. The QSAR models are compiled in Table 10.13. By analyzing the behavior of the kinetic parameters on molecular descriptors, it is possible to establish a QSAR model for predicting degradation rate constants by the SCWO for organic compounds with similar molecular structure. This analysis may provide an insight into the kinetic mechanism that occurs with this technology. 

Landrum et al. (1992) developed a kinetic bioaccumulation model for PAHs in the amphipod Diporeia, employing first-order kinetic rate constants for uptake of dissolved chemical from the overlying water, uptake by ingestion of sediment, and elimination of chemical via the gills and feces. In this model, diet is restricted to sediment, and chemical metabolism is considered negligable. The model and its parameters, as Table 9.3 summarizes, treat steady-state and time-variable conditions. Empirically derived regression equations (Landrum and Poore, 1988 and Landrum, 1989) are used to estimate the uptake and elimination rate constants. A field study in Lake Michigan revealed substantial differences between predicted and observed concentrations of PAHs in the amphipod Diporeia. Until more robust kinetic rate constant data are available for a variety of benthic invertebrates and chemicals, this model is unlikely to provide accurate estimates of chemical concentrations in benthic invertebrates under field conditions. 

In Figure 5 the predictions of the second model are compared against the experimental data published in (2) and obtained in a small, well stirred, vessel reactor with 1 It total volume. The various initiators were tested under conditions representative of polymerization in commercial units, that is with 20 60 seconds residence time and an operating pressure between 1278 and 2352 atm. For the sake of convenience we will use here the same nomenclature and dimensions as in (2). The kinetic parameters used were those given in Table I. The relative size of the two small volumes and the recirculation rates were estimated once and for all cases from equations (13) and (15). The other parameter values, determined independently, were not changed in order to obtain a better fit with the data. As can be seen, the imperfectly mixed model is in excellent agreement with the experimental data, and accurately accounts for the effect of initiator type (Figure 5). 

Figure 30.6 shows a prediction of the plasma concentration of ARA-C and total radioactivity (ARA-C plus ARA-U) following administration of two separate bolus intravenous injections of 1.2 mg/kg to a 70-kg woman. All compartment sizes and blood flow rates were estimated a -priori, and all enzyme kinetic parameters were determined from published in vitro studies. None of the parameters was selected specifically for this patient only the dose per body weight was used in the simulation. The prediction has the correct general shape and magnitude. It can be made quantitative by relatively minor changes in model parameters with no requirement to adjust the parameters describing metabolism. 

As outlined in 12.3.7.2, there are close parallels in the kinetic parameters for corresponding electrochemical and homogeneous reactions, at least for outer-sphere pathways. Aside from the fundamental virtues of testing theoretically based relationships between electrochemical and homogeneous reactivities, such correlations can have predictive value, especially for estimating electrochemical reactivities from the much larger body of homogeneous rate data. 

When one begins to construct a chemical kinetic model, there are several different types of required input information, Fig. 4. Obviously, one needs some specification of the initial concentrations of the reactants, and of the reaction conditions (e.g. T, P, timescale) of interest. Normally one wants to numerically solve the kinetic model to predict species and/or temperature profiles, so the inputs must also include some specification of numerical tolerances on these outputs, and options for the differential equation solver. The most complicated input information required to construct a kinetic model is the chemistry what species, reactions, or reaction types will be considered How will all the thermochemical and rate parameters be estimated ... 

On a practical level, the heuristic approach includes first collecting all the possible data during the experiments as a function of the parameters which are deemed to be important, i.e. concentrations, temperature, pressures, pH, catalyst concentration, volume, etc. Then the rate constants are estimated by regression analysis and the adequacy of the model is judged based on some criteria (like residual sums and parameter significance, which will be discussed further). If a researcher is not satisfied, then additional experiments are performed, followed by parameter estimation and sometimes simulations outside the studied parameter domain. The latter procedure provides the possibility to test the predictive power of a kinetic model. The kinetic model is then gradually improved and the experimental plan is modified, if needed. This process continues until the researcher is satisfied with the kinetic model. 

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... 

Simulation Model Results Initially, the assumption was tested that succinic acid can act as its own catalyst in the esterification reaction. In Figure 4.6, the experimental results on the esterification of PPSu are compared to the theoretical model predictions using kinetic rate constant that are either acid catalyzed (dashed and dotted lines) or not (solid line). As can be seen, the simulation of the experimental data by the theoretical model is very good when the kinetic rate constants used are not acid catalyzed. However, when the kinetic rate constants are assumed to be acid catalyzed, using Equations 4.30 and 4.31, the experimental data are not predicted equally well. Using values to accurately predict the initial rate data, the final data are underestimated. In contrast, when such values are used to predict the final experimental data, the initial data are overestimated. Thus, it was concluded that in the synthesis of the poly(alkylene succinates) studied here, the presence of the metal catalyst tetrabutoxy titanium (TBT) leads to a poor activity of self-catalyzed acid. This was also observed for PBSu by Park et al. [42]. Therefore, Equations 4.30 and 4.31 were not used and only parameters and Arg need to be estimated. The values of these parameters were calculated for every different system studied from fitting to the experimental data. The final values are reported in Table 4.2. Notice that these values are correct only for the specific catalyst type. 

Computer modeling of the kinetics of a reaction by solving rate equations is useful in the determination of mechanism and the estimation of rate parameters. Such analysis of kinetic data represents a higher-level approach to the problem of modeling than the molecular modeling discussed above. Here, we assume knowledge of the bulk properties of the system (the kinetic equations) and proceed to model the system comparing predictions to experimental measurements. 

---