Rate parameters estimation

Notably, with a single set of rate parameter estimates, the present model can also correctly describe the effects of all the investigated process conditions on product distribution. Figure 16.10 compares experimental and calculated ASF product distributions in five of the investigated process conditions. It is worth noticing also that the model predicts the hydrocarbons selectivity up to n = 49,... 

This empirical rate expression considers the active sites of the catalyst as only a fraction of the total adsorption sites for ammonia and is consistent vfith the presence of a reservoir of ammonia adsorbed species which can take part in the reaction. The ammonia reservoir is likely associated vfith poorly active but abundant W and Ti surface sites, which can strongly adsorb ammonia in fact, nhs roughly corresponds to the surface coverage of V. Once the ammonia gas-phase concentration is decreased, the desorption of ammonia species originally adsorbed at the W and Ti sites can occur followed by fast readsorption. When readsorption occurs at the reactive V sites, ammonia takes part in the reaction. Also, the analysis of the rate parameter estimates indicates that at steady state the rate of ammonia adsorption is comparable to the rate of its surface reaction with NO, whereas NH3 desorption is much slower. Accordingly, the assumption of equilibrated ammonia adsorption, which is customarily assumed in steady-state kinetics, may be incorrect, as also suggested by other authors [55]. 

Parameter estimation, objective junctions, and maximum likelihood. A common problem is to find the set of values of the parameters (e.g., rate coefficients, equilibrium constants, etc.) that best fit the observed values of a measurable variable (e.g., a rate). Parameter estimation entails the calculation of the most likely value from a set of experimental values which are subject to random error. This is achieved by minimizing the value of an objective function that expresses the discrepancy between observed values (yj)obs of the variable and the values (yj)calc calculated with sets of values of the model s parameters. The best set is that which gives the minimum value of the objective function. 

Baldwin, A. C., J. R. Barker, D. M. Golden, and D. G. Hendry (1977). Photochemical smog rate parameter estimates and computer simulations. J. Phys. Chem. 81, 2483-2492. 

This allows us to write a quantitative expression for the rate of catalyst decay in this system. With this information we can purge both of the experiments we already have, removing decay influences so that we obtain all the rate data already established in the two experiments in a form suitable for fitting a decay-free rate expression for the reaction. Having two experiments presents an opportunity for extensive crosscheck of our rate expressions. We also note that we have a set of zero decay rates from the intercepts r(0)(x,T). This set is usually too small to be used for rate parameter estimation by itself but can be used as a further cross check, at conditions of special interest to theoreticians, on the parameters of the rate expression for the purged rates from the two experiments. 

The above analyses of species concentrations and net reaction rates clearly indicate which reactions and which chemical species are most important in this reaction mechanism, under the particular conditions considered. However, for purposes of refining a reaction mechanism by eliminating unimportant reactions and species and by improving rate parameter estimates and thermochemical property estimates for the most important reactions and species, it would be helpful to have a quantitative measure of how important each reaction is in determining the concentration of each species. This measure is obtained by sensitivity analysis. In this approach, we define sensitivity coefficients as the partial derivative of each of the concentrations with respect to each of the rate parameters. We can write an initial value problem like that given by equation (35) in the general form... 

The aim of the parameter estimation is to deduee the growth rate G, nuelea-tion rate agglomeration kernel /3aggi and disruption kernel /3disr from the experimental CSD. The CSD is deseribed mathematieally by the population balanee (Randolph and Larson, 1988)... 

When estimates of k°, k, k", Ky, and K2 have been obtained, a calculated pH-rate curve is developed with Eq. (6-80). If the experimental points follow closely the calculated curve, it may be concluded that the data are consistent with the assumed rate equation. The constants may be considered adjustable parameters that are modified to achieve the best possible fit, and one approach is to use these initial parameter estimates in an iterative nonlinear regression program. The dissociation constants K and K2 derived from kinetic data should be in reasonable agreement with the dissociation constants obtained (under the same experimental conditions) by other means. 

We determined the reaction parameters using the optimal parameter estimation technique with the experimentally obtained copolymer yield and norbomene composition data. Based on the literature report, we assume that k = 3 [5]. Fig. 1 shows that the estimated rate constant values depend on the norbomene block length. Note that the reaction rate constant... 

Parameter estimation. Integral reactor behavior was used for the interpretation of the experimental data, using N2O conversion levels up to 70%. The temperature dependency of the rate parameters was expressed in the Arrhenius form. The apparent rate parameters have been estimated by nonlinear least-squares methods, minimizing the sum of squares of the residual N2O conversion. Transport limitations could be neglected. 

Hydrogenation of lactose to lactitol on sponge itickel and mtheitium catalysts was studied experimentally in a laboratory-scale slurry reactor to reveal the true reaction paths. Parameter estimation was carried out with rival and the final results suggest that sorbitol and galactitol are primarily formed from lactitol. The conversion of the reactant (lactose), as well as the yields of the main (lactitol) and by-products were described very well by the kinetic model developed. The model includes the effects of concentrations, hydrogen pressure and temperature on reaction rates and product distribution. The model can be used for optinuzation of the process conditions to obtain highest possible yields of lactitol and suppressing the amounts of by-products. 

This expression was used in the parameter estimation, i.e., just the concentrations cp and cmb were used in the data fitting, and Ca was calculated from Eq. (17). The rate constants included in the model were described by the modified Arrhenius equation... 

Chemical kinetics is an area that received perhaps most of the attention of chemical engineers from a parameter estimation point of view. Chemical engineers need mathematical expressions for the intrinsic rate of chemical reactions... 

In biochemical engineering we are often faced with the problem of estimating average apparent growth or uptake/secretion rates. Such estimates are particularly useful when we compare the productivity of a culture under different operating conditions or modes of operation. Such computations are routinely done by analysts well before any attempt is made to estimate true kinetics parameters like those appearing in the Monod growth model for example. 

Using the initial rate data given above do the following (a) Determine the parameters, kR, kH and KA for model-A and model-B and their 95% confidence intervals and (b) Using the parameter estimates calculate the initial rate and compare it with the data. Shah (1965) reported the parameter estimates given in Table 16.14. 

Subsequently, Watts performed a parameter estimation by using the data from all temperatures simultaneously and by employing the formulation of the rate constants as in Equation 16.19. The parameter values that they found as well as their standard errors are reported in Table 16.18. It is noted that they found that the residuals from the fit were well behaved except for two at 375°C. These residuals were found to account for 40% of the residual sum of squares of deviations between experimental data and calculated values. 

Watts (1994) dealt with the issue of confidence interval estimation when estimating parameters in nonlinear models. He proceeded with the reformulation of Equation 16.19 because the pre-exponential parameter estimates "behaved highly nonlinearly." The rate constants were formulated as follows... 

An advantage over the line source method and other parameter estimation techniques is that the estimate can be made directly on the measured return temperature. Using the derived average heat extraction or injection rate may... 

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