Kinetic parameter error model

For the separate calorimetric investigation, the quality of the fit to the model was similar to that for the combined investigation discussed above (A q = 173 W2, A A = 2.685). In this determination, the error for the infrared measurements, A A, was obtained by using the kinetic parameters obtained from fitting the calorimetric data to calculate concentration-time... 

Once the kinetic parameters have been estimated, (3.60) becomes linear in the unknown parameters A Hr . Therefore, the errors between the total heat of reaction, computed via the detailed model, and the total heat, computed via each reduced model, can be minimized by resorting to the least squares solution of a linear regression problem, discussed in Sect. 3.4. The molar heats of reaction, included in the vector of parameters... 

The previous analysis shown that the initial values of most of the kinetic parameters obtained from DFT calculations provide a good description of the reaction kinetics data collected over a wide range of conditions. The principal difference between the values of the final kinetic parameters used in the model and the initial values obtained from DFT calculations is that the fitted enthalpy changes for the formation of C2Ha transition states involved in cleavage of the C-C bond are lower than the initial values predicted from DFT calculations. This difference may be explained by the structure sensitivity of the system and/or by the inherent error of the DFT calculations. 

When once a mechanism has been built up, techniques of sensitivity analysis (see Sect. 2.5.4) and of parametric estimation (see Sect. 5) allow a determination both of the numerical values of a few kinetic parameters (or of combinations thereof) and the degree of confidence which can be placed in these estimates (assuming, as usual, that there are no systematic errors both in experimental results and in reaction and reactor models). 

Numerous reports are available [19,229-248] on the development and analysis of the different procedures of estimating the reactivity ratio from the experimental data obtained over a wide range of conversions. These procedures employ different modifications of the integrated form of the copolymerization equation. For example, intersection [19,229,231,235], (KT) [236,240], (YBR) [235], and other [242] linear least-squares procedures have been developed for the treatment of initial polymer composition data. Naturally, the application of the non-linear procedures allows one to obtain more accurate estimates of the reactivity ratios. However, majority of the calculation procedures suffers from the fact that the measurement errors of the independent variable (the monomer feed composition) are not considered. This simplification can lead in certain cases to significant errors in the estimated kinetic parameters [239]. Special methods [238, 239, 241, 247] were developed to avoid these difficulties. One of them called error-in-variables method (EVM) [239, 241, 247] seems to be the best. EVM implies a statistical approach to the general problem of estimating parameters in mathematical models when the errors in all measured variables are taken into account. Though this method requires more information than do ordinary non-linear least-squares procedures, it provides more reliable estimates of rt and r2 as well as their confidence limits. 

Comparison between One and Two-Dimensional Models. In a recent article ( 2), the differences between the responses of the one and two-dimensional versions of the same type of model were compared. It was found that they depend to a greater extent on kinetic parameters (j3 and y) than on physical parameters (Re and dp/dt). In order to explain this behaviour an analysis of the error in the evaluation of the reaction rate for a one-dimensional model is carried out. Since the reaction rate at radial mean concentration and temperature generally differs from the radial mean reaction rate, we can define ... 

Parameter Estimates, Standard Errors, Root Mean Square Errors, and Correlation Coefficients for First-Order Kinetic and Irreversible Models for Various Columns... 

Figure 4. Conversion vs. nondimensional reaction rate constant, K. The two limiting cases of one phase (Orcutt-Davidson) PM and PF models are the solid lines. Zone A is the limit of operation allowing for a 10% error in the kinetic parameters of Quach et al. Zone B is the experimental limit of operation.

Min and Ray (1978) have compared the predictions of the population balance model wiA the experimental results obtained by Gerrens (1959) on poly(methyl methacrylate) latexes. The correspondence between theory and experiment is qualitatively acceptable, and the same kinetic parameters model the kinetic behavior of the polymerization process (see Fig. 5). Again, the PSD was measured at the conclusion of the polymerization reaction, raising once more the problem of cancelling errors in the theory. Min and Ray (1978) used literature values for all but two of the rate coefficients... 

To illustrate the validity of the models presented in the previous section, results of validation experiments using lab-scale BSR modules are taken from Ref. 7. For those experiments, the selective catalytic reduction (SCR) of nitric oxide with excess ammonia served as the test reaction, using a BSR filled with strings of a commercial deNO catalyst shaped as hollow extrudates (particle diameter 1.6 or 3.2 mm). The lab-scale BSR modules had square cross sections of 35 or 70 mm. The kinetics of the model reaction had been studied separately in a recycle reactor. All parameters in the BSR models were based on theory or independent experiments on pressure drop, mass transfer, or kinetics none of the models was later fitted to the validation experiments. The PDFs of the various models were solved using a finite-difference method, with centered differencing discretization in the lateral direction and backward differencing in the axial direction the ODEs were solved mostly with a Runge-Kutta method [16]. The numerical error of the solutions was... 

Uncertainty and disturbances can be described in terms of mathematical constraints defining a finite set of hounded regions for the allowable values of the uncertain parameters of the model and the parameters defining the disturbances. If uncertainty or disturbances were unbounded, it would not make sense to try to ensure satisfaction of performance requirements for all possible plant parameters and disturbances. If the uncertainty cannot be related mathematically to model parameters, the model cannot adequately predict the effect of uncertainty on performance. The simplest form of description arises when the model is developed so that the uncertainty and disturbances can be mapped to independent, bounded variations on model parameters. This last stage is not essential to the method, but it does fit many process engineering problems and allows particularly efficient optimization methods to be deployed. Some parameter variations are naturally bounded e.g.. feed properties and measurement errors should be bounded by the quality specification of the supplier. Other parameter variations require a mixture of judgment and experiment to define, e.g., kinetic parameters. 

The kinetic parameters for the n order kinetic model have been obtained using these definitions of reactivity for the pure steam gasification experiments of birch. All the activation energies lie between 228-238 kJ/mol and the reaction orders between 0.54 and 0.58, apart from definition 3. The frequency factors are somewhat more scattered, lying between 5-10 and 3-10 . Regarding the uncertainty of the calculation, definitions 2, 5 and 4 seem to give more precise results and it is interesting to notice that the error of the reaction order calculation does not depend on how a representative reactivity value is defined. 

Where S, G, X, E and Enz are respectively the starch, glucose, cells, ethanol and enzyme concentrations inside the reactor, Si is the starch concentration on the feed, F is the feed flow rate, V is the volume of hquid in the fermentor and (pi, (p2, (ps represent the reaction rates for starch degradation, cells growth and ethanol production, respectively. The unstructured model presented in (Ochoa et al., 2007) is used here as the real plant. The ki (for i=l to 4) kinetic parameters of the model for control were identified by an optimization procedure given in Mazouni et al. (2004), using as error index the mean square error between the state variables of the unstructured model and the model for control. 

A number of factors limit the accuracy with which parameters needed for the design of commercial equipment can be determined. The kinetic parameters may be affected by inaccurate accounting for laboratory reactor heat and mass transport, and hydrodynamics correlations for these are typically determined under nonreacting conditions at ambient temperature and pressure and with nonreactive model fluids and may not be applicable or accurate at reaction conditions. Experimental uncertainty including errors in analysis, measurement,... 

The limitations of QSAR for enzymes are related to the fact that the experimental measurement of kinetic parameters is inherently prone to errors. Kinetic constants for the same compound vary substantially among studies, depending on the enzyme source (recombinant enzyme, purified enzyme, subcellular fraction, etc.) or experimental conditions. Reported Vmax values for the same compound can vary by 2 to 3 orders of magnitude, seriously impacting regression-based QSAR modeling. Therefore much larger, consistent datasets for each enzyme will be required to increase the predictive scope of such models. 

Computational modeling is a powerful tool to predict toxicity of drugs and environmental toxins. However, all the in silico models, from the chemical structure-related QSAR method to the systemic PBPK models, would beneht from a second system to improve and validate their predictions. The accuracy of PBPK modeling, for example, depends on precise description of physiological mechanisms and kinetic parameters applied to the model. The PBPK method has primary limitations that it can only predict responses based on assumed mechanisms, without considerations on secondary and unexpected effects. Incomplete understanding of the biological mechanism and inappropriate simplification of the model can easily introduce errors into the PBPK predictions. In addition values of parameters required for the model are often unavailable, especially those for new drugs and environmental toxins. Thus a second validation system is critical to complement computational simulations and to provide a rational basis to improve mathematical models. 

An alternative approach implies that all kinetic parameters used in the model are determined independently from the modeling itself. The probability of serious errors is still high in this case too, although for many parameters important for alkane oxidation and combustion such errors are evaluated (see, for instance, Tsang and Hampson, 1986 Warnatz, 1984). Despite that, it seems that in this case we are dealing with the lesser of two evils. Moreover,—and this is of principal importance—every value included into the scheme can be independently corrected as soon as newer and more reliable data appear. This does not require any modification of any other kinetic parameter in the model. 

In the following discussion of the main important catalyst pellet characteristics using Skrzypek el al. (1985) kinetic model, the kinetic parameters have been corrected by Elnashaie, and reported by Skrzypek (1988), where it was found that there is an error of 10 in the pre-exponential factors of the reaction rate constants. 

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