Parameters for the kinetic model

Solution Given equations for the batch reactor fermentation kinetics are difficult to integrate analytically, we looked at numerical integration techniques to solve 

Concentration profiles for the tequila fermentation in a batch reactor 

92e7 2 5.77e5 as 5.88el2 14 098el0 5 5.35e3 ag 2.15e4 

The mathematical model for the production of biodiesel in a batch reactor is governed by the following ordinary differential equations 3.63 to 3.68 derived from the mass balance of the batch reactor [31]. 

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The optimized parameters for the kinetic model used to describe the catalytic reactions of the three oxygenates are listed in Table VII. In this analysis, eight parameters were found to be sensitive, and the 95% confidence limits are given for these parameters in Table VII. The solid curves in Figs. 11-13 represent predictions of the kinetic model under various reaction conditions. Good agreement is achieved between the predictions of the model and the experimental reaction kinetics for all three reactions. 

Parameters for the Kinetic Model of Steam Reforming of Methanol on BASF K3-110, A Cu/Zn0/A12Q3 Catalyst. 

TABLE 5.1. Estimated regression parameters for the kinetic model described by equations (5-16) to (5-18). Photodegradation of 416 /xmole L of phenol over degiissa... 

The second phase of this work includes the detailed product analysis and obtaining of the rates of formation of paraffins and olefins from the FT experimental data presented here. This data would be used as input for the Genetic Algorithm code in order to fit the parameters for the kinetic model. 

Table 3.1 provides the parameters for the kinetic model given above. 

Fig. 10. Evaluation of kinetic parameters for the DOC model—HC adsorption/desorption (reaction R7 in Table II). Comparison of the measured and simulated outlet Ci0H22 concentrations in the course of the adsorption/desorption experiment. Synthetic gas mixture, other gases 6% C02, 6% H20, N2 balance, SV = 30,000 h 1 (Kryl et al., 2005). Reprinted with permission from Ind. Eng. Chem. Res. 44, 9524, 2005 American Chemical Society.

Fig. 13. Evaluation of kinetic parameters for the DOC model—NO oxidation (reaction R5 in Table II). Comparison of measured and simulated outlet NOx concentrations in the course of temperature ramp (2K/min) for two different space velocities (SV= 50,000 and 100,000 h-1). Lab experiment with isothermal monolith sample using synthetic gas mixture (100 ppm CO, 100 ppm C3H6, 500ppm NO, 8% 02, 8% C02, 8% H20, N2 balance).

This example demonstrates how reaction calorimetry in combination with IR-ATR spectroscopy can be used to discriminate between different postulated reaction models, and to determine the kinetic and thermodynamic parameters for the selected model. In practical applications, when different (semi-) empirical models can be postulated, model discrimination is crucial. 

These estimates bracket the NASA-JPL and lUPAC recommendations of 6.5x10 and 7.7 x 10 cm molecule s [9,60]. It is therefore possible fo reconcile fhe thermochemistry proposed here with the observed lO + NO2 recombination kinetics while employing reasonable input parameters for the unimolecular model. Nevertheless it must be stressed, as emphasized earlier [16], that there is considerable uncertainty in some of the input parameters to an RRKM analysis, especially the Frot term. It is of interest to compare the present kinetic calculations with the Multiwell [61] Master Equation calculations on this system by Golden [16]. He used a Morse potential to locate the centrifugal maximum, and from the bond extension Frot 2.1 is derived, about 1/7 of fhaf used here. On the other hand, the higher Eo value yields a density of sfafes larger by a facfor of 6, and fhese two factors largely cancel. 

A program for simulating runs makes it possible to study the expected behavior of single samples and their mixtures for arbitrarily assigned values of the parameters of the kinetic model. The parameters which are primarily responsible for the shapes of the rate vs time degradation curves are k-, the fraction of chains started per sec, and tne fraction of a started chain unzipping per sec. 

The storage of data for hundreds of runs and the calculation of alpha at the maximum rate represent conventional applications of the computer to the handling of large amounts of data and complex calculations. Programs for obtaining best fit values of parameters for several kinetic models and for simulating a-t data represent unique applications of the computer to degradation kinetics and will be described. 

The greatest support for the actual mechanistic steps rests in the concentration dependence of the response (Equation 1) and more importantly the temperature dependence of the electron capture coefficient. The temperature dependence of the capture coefficient, as it pertains to the various mechanisms associated with Reactions I-IV, are discussed below, and agreement with experiment is strong supporting evidence for the kinetic model. The interpretation of the temperature dependence in terms of molecular parameters, such as molecular electron affinities or... 

At this point the search for the best fit concentrates on parameter optimization. Where the difficulty was (and usually is) greatest was in the inability of available parameter optimization methods to zero in automatically on a unique set of parameters for the DAM model itself. After an exhaustive search of parameter space using experimental data from several experiments, a number of sets of parameters could be found to fit the kinetic data satisfactorily as far as the conventional criterion of fit, the SSR, is concerned. Table 11.1 shows two sets of acceptable parameters. 

Table 3.2 Values of Kinetic Parameters for Different Kinetic Models as Derived from the Reaction Scheme...

Table I. Rate Parameters of the Kinetic Model for Nine Chemicals...

This paper discusses research efforts towards the prediction of hydrocarbon product distribution for the Fischer-Tropsch synthesis (FTS) on a cobalt-based catalyst using a micro-kinetic model taken fiom the literature. In the first part of the study, a MATLAB code has been developed which uses the Genetic Algorithm Toolbox to estimate parameter values for the kinetic model. The second part of the study describes an ongoing experimental campaign to validate the model predictions of the fixed-bed reactor FTS product distribution in both conventional (gas phase) and non-conventional (near-critical and supercritical phase) reaction media. 

In phase II, with OTR = 0, Equ. 3.43 may be reduced to Equ. 2.5d, which describes the rate of respiration of the cell mass. From the slope of the curve, the parameters of the kinetic model can be obtained when x is known (for example, o,max from the maximum slope and Kq from the half maximum see Chap. 5). 

The next step is to perform model selection. Models may be rejected for three different reasons (i) because the differences between the experimental data and the data calculated with the fitted model are much larger than the measurement error (the model is then qualified as inadequate ), (ii) because the fit of the model is significantly worse than an alternative model, and (iii) because one or more parameters in the kinetic model cannot be estimated accurately and independently, which usually indicates that the model contains too many parameters. Although there were large differences between the options available in the packages investigated, none of the packages were capable to perform all these checks. 

We wished to predict precipitation times for a wide range of temperature and hardness levels. This could be done in principle by repeating the above analysis for the desired sets of conditions. We chose instead to set up a simplified kinetic model to calculate hydrolysis as a ftmc-tion of time and then to compare these predictions with the cloud-point contours of Fig, 14. The parameters for the kinetic treatment were derived by fitting the data of Fig. 13 to a rate law, under the assumption that the hydrolysis reaction is a classic chemical equilibrium reaction., ... 

Unzueta and Forcada [31] studied the emulsion copolymerization of methyl methacrylate and n-butyl acrylate. It was assumed that both micellar nucle-ation and homogeneous nucleation are operative in this emulsion polymerization system. Based on the experimental data and computer simulation results, the values of the free radical capture efficiency factors for monomer-swollen micelles (f ) and polymer particles (Fj) that serve as adjustable parameters in the kinetic modeling work are approximately 1(T and 10, respectively. The reason for such a difference in the free radical capture efficiency factors is not available yet. Table 4.2 summarizes some representative data regarding the absorption of free radicals by the monomer-swollen micelles and polymer particles obtained from the literature. 

Based on mechanistic and kinetic studies of the higher alcohol synthesis from synthesis gas, it has been shown that the ethanol in the mixed-oxygenate product is produced from intermediates derived from methanol, not CO [103,109]. Kinetic models of the synthesis have been developed that are able to explain the observed product distribution [110,111]. These models are based on a detailed understanding of the reaction mechanism in which two types of reactions dominate aldol condensation, which yields primarily 2-methyl branched alcohols, and Cl coupling reactions, which yield linear alcohols [106,111]. Estimates of the parameters of the kinetic models that quantitatively describe the oxygenate product distributions suggest that the rate of ethanol formation is about an order of magnitude lower than the rate of production of branched alcohols [111,112]. On the Cs/Cu/Zn catalysts, this results in a minimum in yield of ethanol compared with the yields of methanol, 1-propanol, and 2-methyl-1 propanol. Althou methanol conversion to ethanol has been confirmed as part of the hi er alcohol synthesis from synthesis gas, this synthesis does not offer a plausible route for the conversion of methanol to ethanol. Under the reaction conditions methanol rapidly decomposes, even at a pressme of 0.1 MPa [113], to yield an equilibrium mix of methanol, CO, and H2. Furthermore, as shown by the data in T able 7, the yield of ethanol remains low even with methanol in the feed. 

In order to evaluate rate parameters in the kinetic model, the observed concentration-time data were simulated by a trial and error method. The simulation model for the batch reaction under isothermal conditions can be written as... 

Harris, S.D., Elliott, L., Ingham, D.B., Pourkashanian, M., Wilson, C.W. The optimisation of reaction rate parameters for chemical kinetic modelling of combustion using genetic algorithms. Comput. Methods Appl. Mech. Eng. 190, 1065-1090 (2000)... 

The sequential and the branching models describe the fluorescence kinetics about equally well as judged by their x -values and the residuals while the heterogeneity model appears to have a slightly lower % -value. The improvement in the fit parameters for the latter model is not sufficient, however, to exclude the other models. From these data at the present level of S/N ratio (which is quite good) it is not possible to distinguish between these models. We will thus discuss in the following the relative merits of each of these models. 

Larsson et al. (1998) performed DHP kinetic experiments on a Pt-Sn/ Y-AI2O3 catalyst (0.54 wt% Pt and 1.53 wt% Sn) under these experimental conditions temperature 507-547 °C, total pressure 1.5 bar, and flow rate 60ml/min (propane 18.1—27.2 ml/min, propylene 2.7—4.5 ml/min, and H2 4.1-6.7 ml/min). A power-law model (see Table 2.4), and two categories of L-H models with dissociative adsorption of propane and propylene desorption as the rate-determining step (RDS) respectively were proposed. But the authors had difficulties in obtaining some important parameters mainly due to the fact that the experiments were conducted in a relatively limited range of operating condition. Finally, they adopted a power-law equation for the kinetic model. 

TABLE 11.8 Fitted Kinetic Parameters for the RCD Model for the Pyrolysis of Different Polymers... 

Given the variety of feedstock that the FCC unit processes, it is unlikely that a single set of kinetic parameters will provide accurate and industrially useful yield and property predictions. In addition, changes in catalyst may significantly alter the yield distribution. Therefore, it is necessary to calibrate the model to a base scenario. Table 4.6 lists the key calibration parameters for the FCC model. We group them by their effects on the model predictions. 

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