Global rates/kinetics

The global rate of the process is r = rj + r2. Of all the authors who studied the whole reaction only Fang et al.15 took into account the changes in dielectric constant and in viscosity and the contribution of hydrolysis. Flory s results fit very well with the relation obtained by integration of the rate equation. However, this relation contains parameters of which apparently only 3 are determined experimentally independent of the kinetic study. The other parameters are adjusted in order to obtain a straight line. Such a method obviously makes the linearization easier. 

The model considers the noble-metal catalyzed oxidation reactions of CO, two hydrocarbons of differing reactivities and H2, and the reaction kinetics was described by the global rate expressions of the dual-site Langmuir-Hinshelwood type [2]. 

Equations 12.6.2 to 12.6.4 and the relation between s, y, and are sufficient to calculate the global rate at specified values of TB and CB. Unfortunately, information on the last relation is rather limited. The curves presented in Figure 12.10 and reference 61 give the desired relation for first-order kinetics, but numerical solutions for other reaction orders are not available to this extent we will presume that numerical solutions may be generated if needed for design purposes. 

The global rates of heat generation and gas evolution must be known quite accurately for inherently safe design.. These rates depend on reaction kinetics, which are functions of variables such as temperature, reactant concentrations, reaction order, addition rates, catalyst concentrations, and mass transfer. The kinetics are often determined at different scales, e.g., during product development in laboratory tests in combination with chemical analysis or during pilot plant trials. These tests provide relevant information regarding requirements... 

The outer-sphere electron-transfer initiation mechanism cannot account for the observed kinetics, the half-reaction time being more than 100 times greater than that observed. The chain process considerably enhances the global rate of the reaction (without a chain process, the half-reaction time would be three centuries). 

Vatcha reports that the rate expression given by Eq. (1) describes the global rate, thus allowing gas phase concentrations to be used in the reaction analysis. Global reaction kinetics will be used in the analysis to follow. Consequently, these kinetics must account for microscopic processes such as adsorption/desorption on the catalyst surface and intraparticle diffusion. Since most available kinetic information is based on steady-state data, a major... 

The value of reaction rate Eq. (43) can be negative when N02 present in the mixture is transformed to NO via backward reaction, typically at higher temperatures. A comparison of measured and simulated outlet N02 concentrations in dependence on temperature can be seen for two different space velocities in Fig. 13. The pre-exponential factor k j and activation energy Ej of the kinetic constant no/no2 in the global rate law were evaluated by the weighted least squares method, Eq. (35). 

This type of rate law is employed in the global DOC kinetic model given in Table II (cf. reaction R5). A typical evolution of the outlet NOx concentration in the course of a slow temperature ramp is shown in Fig. 14. From this type of experiment, the selectivity and inhibition constants A(7) are evaluated, considering exponential temperature dependence, Eq. (36). Again, simpler HC + 02 + NO reaction mixtures with single hydrocarbon are examined first, followed by more complex inlet gas compositions. 

In Eqn. 5.3-1, rj is the effectiveness factor of the catalyst with respect to the dissolved gaseous reactant and the temperature of the outer surface. The rate of reaction within the catalyst pores is comprised in rj. R is the reaction rate expressed in moles of gaseous reactant, A, per unit of bubble-free liquid, per unit of time. Reaction is irreversible. In equation (1) it has not been assumed that the gas is pure gas A, its concentration in the bubbles being Cg. Also, Henry s law for the gas is assumed and written as in Eqn. 5,3-4. Using Henry s law, Eqn. 5.3-4, the intermediate concentrations (Cs, CL) can be eliminated using the above system of equations. This provides an expression of the global rate in terms of an apparent constant, ko, that contains the various kinetic and mass transfer steps. Therefore, the observed rate can be written as ... 

This expression of the current-potential relationship is totally general. For each particular situation, the expressions of the rate constants (through a given kinetic model) and of the limiting currents and mass transport coefficients should be provided to analyze the influence of the different factors that can control the global rate. 

The observed diffusion and reaction rate coefficients can be obtained from specific experiments. To quantify the rate coefficients on the right-hand side of Eq. (5.23), kinetic experiments could be conducted such that the global rate is preferably determined by FD, PD, or CR. In the laboratory these steps can be simulated separately by conducting experiments using static, stirred, or vortex batch adsorption systems (Ogwada and Sparks, 1986b). Therefore, to these systems one can assign additive resistance relations as follows ... 

To have a quantitative idea of the higher unit surface area activity of the Monolith catalyst, rate constants based on surface area were considered essential to know. To accomplish this, the global reaction kinetics of desulfurization and denitrogen-ation were determined. For the desulfurization the following three kinetic models, as suggested in the literature, were tested to determine which best represented the data of this study. 

One has to be careful when using these global rate laws, as there may be more than one solution. The overall rate expression may provide some information on the kinetics however, individual parameters cannot be used separately to predict their effect on the rate. Kinetic lumping is another method that is often used by scientists to derive a simple rate formula avoiding the use of elementary reactions [85]. 

Edelman, R. B. Fortune, 0. F. "A Quasi-Global Chemical Kinetic Model for the Finite-Rate Combustion of Hydrocarbon Fuels With Application to Turbulent Burning and Mixing in Hypersonic Engines and Nozzles" AIAA Paper 69-86, AIAA,... 

This nudeation kinetic mechanism is based on the activation of reaction sites, followed by growth of the product nuclei (B4C, in this case) through chemical reaction. The global rate constant, k, describes either of these two rate determining steps for the reaction mechanism. The values of m corresponds to... 

Equations (10-6) and (10-7) show that for the intermediate case the observed rate is a function of both the rate-of-reaction constant, ic and.. the mass-transfer coefficient k. In a design problem k and k would be known, so that Eqs. (10-6) and (10-7) give the global rate in terms of Cj. Alternately, in interpreting laboratory kinetic data k would be measured. If k is known, k can be calculated from Eq. (10-7). In the event that the reaction is not first order Eqs. (10-1) and (10-2) cannot be combined easily to eliminate C. The preferred approach is to utilize the mass-transfer coefficient to evaluate Q and then apply Eq. (10-2) to determine the order of the reaction n and the numerical value of k. One example of this approach is described by Olson et al. ... 

Example 11-7 The rate of isomerization of o-butane with a silica-alumina catalyst is measured at 5 atm and 50°C in a laboratory reactor with high turbulence in the gas phase surrounding the catalyst pellets. Turbulence ensures that external-diffusion resistances are negligible, and so Q = Q. Kinetic studies indicate that the rate is first order and reversible. At 50°C the equilibrium conversion is 85%. The effective diffusivity is 0.08 cm /sec at reaction conditions, and the density of the catalyst pellets is 1.0 g/cm, regardless of size. The measured, global rates when pure n-butane surrounds the pellets are as follows ... 

Example 11-7 illustrates one of the problems in scale-up of catalytic reactors. The results showed that for all but -in. pellets intrapellet diffusion significantly reduced the global rate of reaction. If this reduction were not considered, erroneous design could result. For example, suppose the laboratory kinetic studies to determine a rate equation were made with f-in. pellets. Then suppose it was decide tojise f-ih. pellets in the commercial reactor to reduce the pressure drop through the bed. If the rate equation were used for the -in. pellets without modification, the rate would be erroneously high. At the conditions of part b) of Example 11-7 the correct would be only 0.68/0.93, or 73% of the rate measured with -in. pellets. 

In the laboratory either integral or differential (see Sec. 4-3) tubular units or stirred-tank reactors may be used. There are advantages in using stirred-tank reactors for kinetic studies. Steady-state operation with well-defined residence-time conditions and uniform concentrations in the fluid and on the solid catalyst are achieved. Isothermal behavior in the fluid phase is attainable. Stirred tanks have long been used for homogeneous liquid-phase reactors and slurry reactors, and recently reactors of this type have been developed for large catalyst pellets. Some of these are described in Sec. 12-3. When either a stirred-tank or a differential reactor is employed, the global rate is obtained directly, and the analysis procedure described above can be initiated immediately. 

Also, because the reaction is first order, could be easily eliminated between Eqs. (10-1) and (11-43), and the global rate could be expressed explicitly in terms of Cj, [Eq. (I) of Example 12-1]. For other kinetics a trial solution of Eqs. (10-1), (11-43), the defining equation for 0, and the O-vs-j relationship would be simplest. 

Finally, the first-order kinetics allowed a direct display of the relative importance of the diffusion resistances on the global rate. The quantities 0.92 and 6.15 in the denominator of the previous equation measure the resistances of external diffusion and internal diffusion plus reaction. The value of rj divides the latter into internal-diffusion resistance and the resistance of the intrinsic reaction on the interior catalyst site . 

This expression still includes the effect of longitudinal dispersion. It is identical to Eq. (6-41), except that the rate for a homogeneous reaction has been replaced with the global rate XpPs per unit volume for a heterogeneous catal)dic reaction. In Sec. 6-9 Eq. (6-41) was solved analytically for first-order kinetics to give Eq. (6-45). Hence that result can be adapted for fixed-bed catalytic reactors. The first-order global rate would be... 

To utilize Eqs. (6-45) and (13-15), or solutions of Eq. (13-14) for other kinetics, requires the longitudinal diffusivity as well as the global rate. 

Example 13-5 Using the one-dimensional method, compute curves for temperature and conversion vs catalyst-bed depth for comparison with the experimental data shown in Figs. 13-10 and 13-14 for the oxidation of sulfur dioxide. The reactor consisted of a cylindrical tube, 2.06 in. ID. The superficial gas mass velocity was 350 lb/(hr)(ft ), and its inlet composition was 6.5 mole % SO2 and 93.5 mole % dry air. The catalyst was prepared from -in. cylindrical pellets of alumina and contained a surface coating of platinum (0.2 wt % of the pellet). The measured global rates in this case were not fitted to a kinetic equation, but are shown as a function of temperature and conversion in Table 13-4 and Fig. 13-13. Since a fixed inlet gas composition was used, independent variations of the partial pressures of oxygen, sulfur dioxide, and sulfur trioxide were not possible. Instead these pressures are all related to one variable, the extent of conversion. Hence the rate data shown in Table 13-4 as a function of conversion are sufficient for the calculations. The total pressure was essentially constant at 790 mm Hg. The heat of reaction was nearly constant over a considerable temperature range and was equal to — 22,700 cal/g mole of sulfur dioxide reacted. The gas mixture was predominantly air, so that its specific heat may be taken equal to that of air. The bulk density of the catalyst as packed in the reactor was 64 Ib/ft. ... 

The global reaction rate depends on three factors (i) chemical kinetics (the intrinsic reaction rate), (ii) the rates that chemical species are transported (transport hmitations), and (hi) the rnterfacial surface per imit volume. Therefore, even when a kinetic-transport model is carefully constructed (using the concepts described above), it is necessary to determine the interfacial surface per unit volume. The interfacial surface depends on the way the two phases are contacted (droplet, bubble, or particle size) and the holdup of each phase in the reactor. All those factors depend on the flow patterns (hydrodynamics) in the reactor, and those are not known a priori. Estimating the global rate expression is one of the most challenging tasks in chemical reaction engineering. 

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