Extracting Intrinsic Rate Constants

The industrial rates obtained earlier from the pseudohomogeneous model actually include diffusional limits and are suitable for the specific reactor with the specific catalyst particle size for which the data was extracted. Such pseudohomogeneous models do not account explicitly for the catalyst packing of the reactor. On the other hand, heterogeneous models account for the catalyst explicitly by considering the diffusion of reactants and of products through the pores of the catalyst pellet. 

In this section we refer to the same industrial reactor as in 7.5.1, with its data given on p. 508. Further reactor specifications and catalyst-bed properties of this plant are as follows. 

The apparent kinetic data as discussed in the previous section is given in terms of the dimensionless frequency factors Ai and the activation energy ), measured in kJ/kmol with 

Frequency factors and activation energies of the apparent rates of reactions and the frequency factors of the intrinsic rates in equation (7.183) 

The set of constants in columns 3 and 4 above are not intrinsic rate constants. They can, however, be used as starting values or as the initial guess in an iterative scheme to obtain kinetics that are suitable for the heterogeneous model. 

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Both the intrinsic rate constant and the effective diffusivity (KD) can be extracted from measurements of the reaction rate with different size fractions of the zeohte crystals. This approach has been demonstrated by Haag et al. [116] for cracking of n-hexane on HZSM5 and by Post et al. [117] for isomerization of 2,2-dimethylbutane over HZSM-5. It is worth commenting that in Haag s analysis the equilibrium constant (or distribution coefficient K) was omitted, leading to erroneously large apparent diffusivity values. 

Cumene is cracked in a recycle reactor over commercial H-ZSM5 extrudates. A Thiele modulus approach is used to determine the diffusion coefficient and the intrinsic rate constant. The results are compared to those obtained from pulse experiments. A linear model for diffusion, adsorption and reaction rate is applied for reactants and products. In contrast to literature it is argued that if the Thiele modulus is greater than five, the system becomes over parameterised. If additionally adsorption dynamics are negligible, only one lumped parameter can be extracted, which is the apparent reaction constant found from steady state experiments. The pulse experiment of cumene is strongly diffusion limited showing no adsorption dynamics of cumene. However, benzene adsorbed strongly on the zeolite and could be used to extract transient model parameters which are compared to steady state parameters. 

The rate constants, efficiencies, and thermodynamic data used to extract intrinsic barriers via the analysis outlined above appear in Table I. Sample input parameters for KRKM calculations have been published elsewhere. Table II and Figure 4 contain the intrinsic barriers for the systems we have examined. 

Finally, a phenomenon called concentration quenching or static quenching can lead to upward curvature of Stern Volmer plots even at moderate quencher concentrations (c q > 0.01 M). Molecules that are located next to a quencher at the time of excitation will be quenched immediately. Therefore, the fluorescence decay curve will be nonexponential initially, exhibiting a very fast initial component. Moreover, the initial depletion of these molecules will result in an inhomogeneous distribution of the remaining excited molecules around the quenchers. As a result, the diffusion coefficient kA is no longer a constant, but becomes a function of time, kd(t), until the statistical distribution of excited molecules is re-established. The impact of these effects has been analysed in detail.231 Intrinsic rates of electron transfer in donor acceptor contact pairs can be extracted from the resulting curvature in Stern Volmer plots.232... 

Generally, as the Thiele modulus of main reaction is smaller than unity, the order of deactivation is equal to unity and, therefore, the apparent deactivation rate constant can he extracted from the slope of -In Xs(t) versus t Krishnaswamy and Kittrell (ref. 10) have shown that the relationship between the intrinsic deactivation rate constant k and the apparent deactivation rate constant kda can be expressed as... 

For an intrinsic rate-limited reaction in the organic phase, if the rate equation can be expressed as rate = ky[catalyst]oj.g[substrate]o g, then for a given substrate the rate of reaction depends mainly on the concentration of the active catalytic species in the organic phase and on the intrinsic rate coefficient, k . The distribution of catalyst in the organic phase can be determined by the extraction constant for the two-phase organic/aqueous system. If the transferred catalyst is in the form of a catalyst-anion pair, then it is important to take the extent of aggregation into account to obtain the effective concentration of the active catalytic species. 

Wang and Wu [70] analyzed the extraction equilibrium of the effects of catalyst, solvent, NaOH/organic substrate ratio, and temperature on the consecutive reaction between 2,2,2-trifluoroethanol with hexachlorocyclotriphosphazene in the presence of aqueous NaOH. Wu and Meng [69] reported the reaction between phenol with hexachlorocyclotriphosphazene. They first obtained the intrinsic reaction-rate constant and overall mass transfer coefficient simultaneously, and reported that the mass transfer resistance of QX from the organic to aqueous phase is larger than that of QY from the aqueous to organic phase. The intrinsic reaction-rate constant and overall mass transfer coefficients were obtained in three ways. 

In the developments just presented, the constants of the MM equation are the intrinsic constants of the IME catalyst which are independent of particle size and liquid flow rate. Another approach is to extract apparent values of the constants for any given flow rate and particle size and correlate them as functions of these parameters. Several methods of extracting these apparent constants have been proposed (e.g., Lilly et al., 1968 Shiraishi, 1993, 1995 Miyakawa and Shiraishi, 1997). It is important to note, however, that even the intrinsic constants have different values for the free and immobilized forms of an enzyme, but this can often be ignored. 

By building in a feedback mechanism which enables the tip-substrate separation to be accurately controlled and known, it is possible to extract intrinsic ET rate constant values, k, as a function of tip position. Intermittent contact mode SG-TC SECM (2-pm-diameter tip) was employed on highly doped polished freestanding MC BDD, of a similar grain size to the studies above (Figure 5.14d (right)) [18]. For the redox couples FcTMA and Ru(NH3)g, two characteristic Atq... 

It should be noted that even if the trajectory lifetime distribution is intrinsically non-RRKM, the enharmonic RRKM rate constant can be extracted from the t = 0 intercept of the trajectory P(t) plot. 

The airflow equations presented above are based on the assumption that the soil is a spatially homogeneous porous medium with constant intrinsic permeability. However, in most sites, the vadose zone is heterogeneous. For this reason, design calculations are rarely based on previous hydraulic conductivity measurements. One of the objectives of preliminary field testing is to collect data for the reliable estimation of permeability in the contaminated zone. The field tests include measurements of air flow rates at the extraction well, which are combined with the vacuum monitoring data at several distances to obtain a more accurate estimation of air permeability at the particular site. 

Once Rp is determined by the EIS method, icorr is evaluated in the same way as with the polarization-resistance method (i.e., with Eq 6.28). Therefore, the Tafel constants still must be experimentally determined. The intrinsic value of the EIS method lies in the fact that extensive information is extracted (i.e., Rp, Rs, and C are all determined) and, ideally, interpreted to not only determine the corrosion rate but also the rate-controlling mechanisms at the material surface and within the electrolyte. 

An intrinsic problem during the investigation of lipid oxidative deterioration is the uncertainty about the rate of initiative reactions. One possible way of overcoming this problem is to introduce into the reaction mixture a compound that decomposes at a constant rate to free radicals (X ) capable of extracting a hydrogen atom from the fatty acid (RH) and consequently initiating the autoxidation process. 

To proceed with this analysis, we have made independent measurements of the biological degradation rate in soil-free systems and coupled this wifii the abiotic mass transfer rates found from the desorption studies. In other words, in an effort to understand the total system, we have separated two of the key factors in the ultimate fate of a contaminant, desorption and biodegradation, and measured each in the absence of the other. The biodegradation rate can be extracted from the intrinsic biokinetic parameters for the active degraders. In our case, we will employ the Michaelis-Menten parameters measured in soil-free systems for the inoculated culture CRE7. For an initial cell concentration of 1.94 X 10 CFU/mL, the biokinetic parameters for CRE7 were as follows r 7.10 X 10 pg/L/day Km=SS3 pg/L. These parameters were coupled widi both the labile and resistant desorption constants fit to the desorption profiles to calculate a bioavailabilty number which is summarized in Table 3. 

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