Optimal kinetic parameter estimates

Table 4.4 Optimal kinetic parameter estimates (A, in M s and Ea, in kj/mol) and uncertainties (1 standard deviation)...

Kinetic Parameter Estimation. Since the values of ki and k were already estimated to be 6.7x10 and 9.6x10 3 mol/g-Ag.min.atm. respectively, five parameters k2, kj, k9, k9 and ki, should be estimated by a parameter optimization technique, using a digital computer. 

There are several principal sets of problem layouts in chemical reaction engineering the calculation of reactor performance, the sizing of a reactor, the optimization of a reactor, and the estimation of kinetic parameters from the experimental data. The primary problem is, however, a performance calculation that delivers the concentrations, molar amounts, temperature, and pressure in the reactor. Successful solutions of the remaining problems—sizing, optimization, and parameter estimation—require knowledge of performance calculations. The performance of a chemical reactor can be prognosticated by mass and energy balances, provided that the outlet conditions and the kinetic and thermodynamic parameters are known. 

The kinetic parameters associated with the synthesis of norbomene are determined by using the experimental data obtained at elevated temperatures and pressures. The reaction orders with respect to cyclopentadiene and ethylene are estimated to be 0.96 and 0.94, respectively. According to the simulation results, the conversion increases with both temperature and pressure but the selectivity to norbomene decreases due to the formation of DMON. Therefore, the optimal reaction conditions must be selected by considering these features. When a CSTR is used, the appropriate reaction conditions are found to be around 320°C and 1200 psig with 4 1 mole ratio of ethylene to DCPD in the feed stream. Also, it is desirable to have a Pe larger than 50 for a dispersed PFR and keep the residence time low for a PFR with recycle stream. 

Based on the experimental data kinetic parameters (reaction orders, activation energies, and preexponential factors) as well as heats of reaction can be estimated. As the kinetic models might not be strictly related to the true reaction mechanism, an optimum found will probably not be the same as the real optimum. Therefore, an iterative procedure, i.e. optimization-model updating-optimization, is used, which lets us approach the real process optimum reasonably well. To provide the initial set of data, two-level factorial design can be used. 

For a fixed molar ratio (ns/riAh equal to 0.05887, the temperature as applied in experiment E4, and a batch time of 347.8 dimensionless units, the feed rate of B (and thus the feed time) was optimized by computation to find tj = 323.19 dimensionless units. A run was carried out at these conditions. The data collected from this experiment were then used for re-estimation of the kinetic parameters. The new kinetic model was used to evaluate the new optimum feed rate for the same total amount of B. The optimum batch time reduced to 275.36 and the feed time to 242.75 units. Table 5.4-19 summarizes the results for three successive optimizations and re-estimations. Evidently, even a very simplified kinetic model can be successfully used in search for an optimum provided that kinetic parameters are updated based on every subsequent run carried out at the optimum conditions evaluated from the preceding set of kinetic parameters. 

For example let us consider the estimation of the two kinetic parameters in the Bodenstein-Linder model for the homogeneous gas phase reaction of NO with 02 (first presented in Section 6.5.1). In Figure 8.4 we see that the use of direct search (LJ optimization) can increase the overall size of the region of convergence by at least two orders of magnitude. 

These four procedures are all recommended to be performed in the order shown to achieve optimal parameter estimation followed by a final validation of the gravity sewer process model (Figure 7.7). In the case of design of a new sewer system, procedure number 4 is, of course, not relevant and kinetic parameters for the sewer biofilm must be evaluated and selected based on information from comparative systems. 

Those based on strictly empirical descriptions Mathematical models based on physical and cnemical laws (e.g., mass and energy balances, thermodynamics, chemical reaction kinetics) are frequently employed in optimization applications. These models are conceptually attractive because a general model for any system size can be developed before the system is constructed. On the other hand, an empirical model can be devised that simply correlates input/output data without any physiochemical analysis of the process. For these models, optimization is often used to fit a model to process data, using a procedure called parameter estimation. The well-known least squares curve-fitting procedure is based on optimization theory, assuming that the model parameters are contained linearly in the model. One example is the yield matrix, where the percentage yield of each product in a unit operation is estimated for each feed component... 

The kinetic and deactivation models were fitted by non-linear regression analysis against the experimental data using the Modest software, especially designed for the various tasks -simulations, parameter estimation, sensitivity analysis, optimal design of experiments, performance optimization - encountered in mathematical modelling [6], The main interest was to describe the epoxide conversion. The kinetic model could explain the data as can be seen in Fig. 1 and 2, which represent the data sets obtained at 70 °C and 75°C, respectively. The model could also explain the data for hydrogenated alkyltetrahydroanthraquinone. 

Rawlings and co-workers proposed to carry out parameter estimation using Newton s method, where the gradient can be cast in terms of the sensitivity of the mean (Haseltine, 2005). Estimation of one parameter in kinetic, well-mixed models showed that convergence was attained within a few iterations. As expected, the parameter values fluctuate around some average values once convergence has been reached. Finally, since control problems can also be formulated as minimization of a cost function over a control horizon, it was also suggested to use Newton s method with relatively smooth sensitivities to accomplish this task. The proposed method results in short computational times, and if local optimization is desired, it could be very useful. 

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. 

A recommended approach for conducting toxicokinetic studies generally involves three steps. Step 1 is a preliminary study, which uses a minimum number of animals to estimate the range of blood/tissue concentrations, the required quantitation limit for the analytical method, and the optimal sampling times for the definitive toxicokinetic studies. Step 2 is the definitive study and generates blood and/or tissue concentration data for calculating the toxicokinetic parameters. Step 3 is the toxicokinetic study conducted in conjunction with the toxicology study to determine the internal dose and the effects of age and continuous exposure on kinetic parameters. 

Numerical identifiability also becomes a problem with a poorly or inadequately designed experiment. For example, a drug may exhibit multi-exponential kinetics but due to analytical assay limitations or a sampling schedule that stops sampling too early, one or more later phases may not be identifiable. Alternatively, if sampling is started too late, a rapid distribution phase may be missed after bolus administration. In these cases, the model is identifiable but the data are such that all the model components cannot be estimated. Attempting to fit the more complex model to data that do not support such a model may result in optimization problems that either do not truly optimize or result in parameter estimates that are unstable and highly variable. 

In this work, the proposed optimization procedure is based on the combination of different optimization techniques, to know real-coded GA (RGA), Placket-Burman (PB) design, and QN. The approach is applicable when the stmeture of a kinetic model has been set up and the kinetic parameters should be estimated. 

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