Intrinsic reaction kinetics, rate expression

Examples (10.1) and (10.2) used the fact that Steps 4, 5, and 6 must all proceed at the same rate. This matching of rates must always be true, and, as illustrated in the foregoing examples, can be used to derive expressions for the intrinsic reaction kinetics. There is another concept with a time-honored tradition in chemical engineering that should be recognized. It is the concept of rate-determining step or rate-controlling step. 

The units on the rate constants reported by DeMaria et al. indicate that they are based on pseudo homogeneous rate expressions (i.e., the product of a catalyst bulk density and a reaction rate per unit mass of catalyst). It may be assumed that these relations pertain to the intrinsic reaction kinetics in the absence of any heat or mass transfer limitations. 

To arrive at a rate expression that describes intrinsic reaction kinetics and is suitable for engineering design calculations, one must be assured that the kinetic data are free from artifacts that mask intrinsic rates. A variety of criteria have been proposed to guide kinetic analysis and these are thoroughly discussed by Means [D. E. Means, Ind. Eng. Chem. Process Des. Develop., 10 (1971) 541]. 

The kinetic parameters are constants that appear in the intrinsic kinetic rate expressions and are required to describe the rate of a reaction or reaction network. For instance, for the simple global nth-order reaction with Arrhenius temperature dependence ... 

The rate expressions obtained by chemical kinetics describe file dependency of the reaction rate on kinetic parameters related to the chemical reactions. These rate expressions are commonly referred to as the intrinsic rate expressions of the chemical reactions (or intrinsic kinetics). However, in many instances, file local species concentrations depend also on the rate that the species are transported in the reacting medium. Consequently, the actual reaction rate (also referred to as the global reaction rate) is affected by the transport rates of the reactants and products. 

Using artificial neural networks (ANN) the reaction system, including intrinsic reaction kinetics but also internal mass transfer resistances, is considered as a black-box and only input-output signals are analysed. With this approach the conversion rate of the i-th reactant into the j-th species can be expressed in a general form as a complex function, being a mathematical superposition of all above mentioned functional dependencies. This function includes also a contribution of the internal diffusion resistances. So each of the rate equations of Eq. 5 can be described with the following function based on the vanables which uniquely define the state of the system ... 

For process modeling proposes the effective chemical reaction rate has to be expressed as a function of the liquid bulk composition x, the local temperature T, and the catalyst properties such as its number of active sites per catalyst volume c, its porosity e, and its tortuosity t. As discussed in Section 5.4.2, the chemical reaction in the catalyst particles can be influenced by internal and external mass transport processes. To separate the influence of these transport resistances from the intrinsic reaction kinetics, a catalyst effectiveness factor p is introduced by... 

The reaction rate of an immobilized enzyme exhibit a behaviour similar to that of enzyme in its native state, but with different kinetic parameters, assuming that the local intrinsic reaction kinetics of the immobilized biocatalyst can be expressed by the Michaelis-Menten equation. Equation [1.1] where ... 

The design of a catalytic reactor requires the knowledge of the reaction rate and product selectivity as a function of the operating conditions. Rate expression based on the intrinsic reaction kinetics allows scaling up of laboratory data to a pilot plant and further to an industrial unit Without reliable kinetics, optimimi reactor design can be difficult or even impossible to achieve. 

A kinetic parameter (kv) and a transport parameter (Z)A,eff) both appear in the expression for the overall rate. Increasing either ky or Da,cS increases the rate per particle, and the rate per unit weight or per unit volume of catalyst. Given this situation, we caimot say that the rate is controlled by internal transport. Both pore diffusion and the intrinsic reaction kinetics affect the reaction rate. For the situation shown in Figure 9-11, it is more appropriate to say that the reaction is influenced by both pore diffusion and the intrinsic kinetics. 

The reaction will then appear to follow first-order kinetics, regardless of the functional form of the intrinsic rate expression and of the effectiveness factor. This first-order dependence is characteristic of reactions that are mass transfer limited. The term diffusion controlled is often applied to reactions that occur under these conditions. 

To illustrate the masking effects that arise from intraparticle and external mass transfer effects, consider a surface reaction whose intrinsic kinetics are second-order in species A. For this rate expression, equation 12.4.20 can be written as... 

The types of systems we deal with are primarily gas-solid (Section 9.1) and gas-liquid (Section 9.2). In these cases, we assume first- or second-order kinetics for the intrinsic reaction rate. This enables analytical expressions to be developed in some situations for the overall rate with transport processes taken into account. Such reaction models are incorporated in reactor models in Chapters 22 and 24. 

For a more detailed analysis of measured transport restrictions and reaction kinetics, a more complex reactor simulation tool developed at Haldor Topsoe was used. The model used for sulphuric acid catalyst assumes plug flow and integrates differential mass and heat balances through the reactor length [16], The bulk effectiveness factor for the catalyst pellets is determined by solution of differential equations for catalytic reaction coupled with mass and heat transport through the porous catalyst pellet and with a film model for external transport restrictions. The model was used both for optimization of particle size and development of intrinsic rate expressions. Even more complex models including radial profiles or dynamic terms may also be used when appropriate. 

For most reaction systems, the intrinsic kinetic rate can be expressed either by a power-law expression or by the Langmuir-Hinshelwood model. The intrinsic kinetics should include both the detailed mechanism of the reaction and the kinetic expression and heat of reaction associated with each step of the mechanism. For catalytic reactions, a knowledge of catalyst deactivation is essential. Film and penetration models for describing the mechanism of gas-liquid and gas-liquid-solid reactions are discussed in Chap. 2. A few models for catalyst deactivation during the hydrodesulfurization process are briefly discussed in Chap. 4. 

The reaction kinetics were studied by many researchers with sometimes different or contradictory results. Reviews are found in [592], [602], [603], [609]. A reason for this is that often diffusional effects were not eliminated, and therefore no real intrinsic reaction rates were obtained [610], [611]. In industrial practice equations are required for dimensioning reaction vessels and catalyst. The mathematical expressions used for this purpose are for the most part not based on theoretical assumptions and research... 

The observed rate will appear to be first-order with respect to the bulk reactant concentration, regardless of the intrinsic rate expression applicable to the surface reaction. This is a clear example of how external diffusion can mask the intrinsic kinetics of a catalytic reaction. In a catalytic reactor operating under mass transfer limitations, the conversion at the reactor outlet can be calculated by incorporating Equation (6.2.20) into the appropriate reactor model. 

The implications of severe diffusional resistance on observed reaction kinetics can be determined by simple analysis of this more general Thiele modulus. The observed rate of reaction can be written in terms of the intrinsic rate expression and the effectiveness factor as ... 

For any industrial reacting system, the relevant parameters appearing in the rate expression (Eq. (5.14)) need to be obtained by carrying out experiments under controlled conditions. It is necessary to ensure that physical processes do not influence the observed rates of chemical reactions. This is especially difficult when chemical reactions are fast. It may sometimes be necessary to employ sophisticated mathematical models to extract the relevant kinetic information from the experimental data. Some references covering the aspects of experimental determination of chemical kinetics are cited in Chapter 1. It must be noted here that in the above development, the intrinsic rate of all chemical reactions is assumed to follow a power law type model. However, in many cases, different types of kinetic model need to be used (for examples of different types of kinetic model, see Levenspiel, 1972 Froment and Bischoff, 1984). It is not possible to represent all the known kinetic forms in a single format. The methods discussed here can be extended to any type of kinetie model. 

The function expressing the Thiele modulus in terms of kinetic parameters and the catalyst properties depends on the intrinsic reaction rate. For first-order reactions, the modulus is... 

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. 

The difficulty in obtaining a global reaction rate expression—an expression that accounts for both intrinsic kinetics and transport effects. 

The more catalyst present per unit volume of reactor, the higher the rate of reaction can become (expressed as amount converted per unit reactor volume). Therefore, intrinsically slow reactions (reactions that are determined by the kinetic regime, and not by mass transfer) are usually best conducted in a reactor with a large volume fraction of catalyst, such as the trickle-bed reactor. With very active catalysts, on the other hand, where mass transfer dictates the rate of the overall process, slurry reactors are more suitable. 

The order of a reaction may not be as simple as first or second order. We often find nonintegral order in what is called "power-law" kinetics. This typically indicates that the "reaction" rate we have measured is not for a single reaction, which is one elementary step, but for several elementary steps taking place simultaneously, the sum of which is the overall reaction that we observe. Normally, we refer to rate expressions such as these as global rates or kinetics (global in the sense of overall or measurable as opposed to intrinsic or fundamental rates and kinetics). Consider the reaction of A to B ... 

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