Regression rate parameter

Figure 3. Temporal variations of the effective latent heat of vaporization H/L and the droplet surface regression rate parameter P with extreme internal heat transport rates (45)...

The solution of problems in chemical reactor design and kinetics often requires the use of computer software. In chemical kinetics, a typical objective is to determine kinetics rate parameters from a set of experimental data. In such a case, software capable of parameter estimation by regression analysis is extremely usefiil. In chemical reactor design, or in the analysis of reactor performance, solution of sets of algebraic or differential equations may be required. In some cases, these equations can be solved an-... 

Linear and nonlinear regressions of data for estimation of rate parameters ... 

The rate expressions Rj — Rj(T,ck,6m x) typically contain functional dependencies on reaction conditions (temperature, gas-phase and surface concentrations of reactants and products) as well as on adaptive parameters x (i.e., selected pre-exponential factors k0j, activation energies Ej, inhibition constants K, effective storage capacities i//ec and adsorption capacities T03 1 and Q). Such rate parameters are estimated by multiresponse non-linear regression according to the integral method of kinetic analysis based on classical least-squares principles (Froment and Bischoff, 1979). The objective function to be minimized in the weighted least squares method is... 

Robust, multimethod regression codes are required to optimize the rate parameters, also in view of possible strong correlations. For example, the BURENL routine, specifically developed for regression analysis of kinetic schemes (Donati and Buzzi-Ferraris, 1974 Villa et al., 1985) has been used in the case of SCR modeling activities. The adaptive simplex optimization method Amoeba was used for minimization of the objective function Eq. (35) when evaluating kinetic parameters for NSRC and DOC. 

In order to develop a suitable kinetic model of the full NH3-N0-N02/02 SCR reacting system, first the active reactions depending on N0/N02 feed ratio and temperature were identified then a dedicated study was performed aimed at clarifying the catalytic mechanism of the fast SCR reaction on the basis of such a reaction chemistry a detailed kinetic model was eventually derived, whose intrinsic rate parameters were estimated from global non-linear regression of a large set of experimental transient runs. 

Using absorbance data collected as a function of time, the distribution function H(k,t) was calculated, and plotted versus In(time) to yield a curve providing a maximum for each first-order process in the reaction. The position of each maximum yields an estimate of the rate constant (t = 2/k), and the area under the maximum provides an estimate of the amount of metal dissociating by that process. Due to concerns of using higher-order derivatives to calculate rate constants for distributions of pathways in multiple first-order mechanisms as described in the literature (16-20), the kinetic spectrum method was used only to obtain initial estimates for the appropriate rate equations. The actual rate parameters reported herein were obtained from a simplex non-linear regression (21) of the original experimental data. When dissociation occurred in both fast (ti/2 > 30 s) and slow (t 1/2 < 30 s) time domains, each data set was treated independently. 

An analysis of the rate of CO, CO2 and H2O evolution during TPO of industrial and laboratory coked cracking catalysts has provided information on the mechanism and energetics of coke combustion. The mechanism has been deduced from previously reported studies on amorphous carbon oxidation [8], while rate parameters have been calculated from non-linear regression simulations of the TPO spectra. The rate of water vapour formation has not been analysed due to re-adsorption expected to affect the apparent kinetics. "Soft" and "hard" coke have been identified in the spectra, and oxidation activation energies of each compared. A further outcome of this work is the proposal that coke deposition on cracking catalysts proceeds from "soft" to "hard" coke via a series of dehydrogenation or dehydration steps. 

Equation 20 is nonlinear with respect to two rate parameters, but it is readily transformed into a linear form. Thus a linear regression technique can be used to obtain initial estimates of these two parameters. Equation 21 cannot be linearized. However it possesses only two unknown parameters instead of eight parameters of Equation 19. The estimates obtained from Equations 20 and 21 supply reasonably accurate initial estimates of these four parameters. Hence, the subsequent nonlinear estimation of all eight parameters of Equation 19 can be simplified substantially. 

Generally, adsorption steps were taken as temperature independent, whereas the rate parameters of surface reactions and desorption steps were described by Arrhenius equations. The kinetic rate parameters for CO oxidation (steps 1-10) and the catalyst properties were taken from [24] with minor adaptation as mentioned. The rate parameters, e.g. activation energies and pre-exponential factors, for steps 11-28 were determined by non-linear regression. It was found [25] that the rates for NO reactions on ceria are independent of the oxidation state of ceria, so the rate parameters for the corresponding steps were taken as the same (i.e. steps 11 and 12 for oxygen, steps 25-28 for NO). 

Table 2, Estimates of the kinetic rate parameters obtained by the simultaneous regression of the cyclic feeding experiments at 523, 548, and 573 K, and a forcing frequency of 1/10 Hz for the nitric oxide reduction by carbon...

On the basis of Eq. (9.44), very accurate preliminary estimates for the parameters Ic, Ka, r, and Kg can be obtained by simple linear regression. These parameters can be used with great advantage for nonlinear regression on the basis of the reaction rate equation as such. 

Equation (10.19) implies that tNo becomes essentially independent of the ammonia surface coverage above a critical NH3 coverage identified by 0 nh3-Like for the ER rate expression, a global multiresponse nonlinear regression of all the TRM and TPR runs performed with 2 % O2 provided the estimates of the three rate parameters in Eq. (10.19) (k°No, E o, 0 nh3)-... 

The prevailing reactions identified in the analyzed reacting systems were used to define the kinetic scheme in Table 18.2. The rate parameters of (Eq. 18.18) were independently estimated from NO oxidation tests, while a global multiresponse nonlinear regression on the whole set of runs involving NH3 provided estimates of the remaining rate parameters in (Eqs. 18.16-18.23). 

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