Intrinsic reaction rates

In addition to the influence of the complexation equilibrium constant K, the observed reaction rate of arenediazonium salts in the presence of guest complexing reagents is influenced by the intrinsic reaction rate of the complexed arenediazonium ion. This system of reactions can be rationalized as in Scheme 11-1. Here we are specifically interested in the numerical value of the intrinsic rate constant k3 of the complexed diazonium ion relative to the rate constant k2 of the free diazonium ion. 

Diffusion is important in reactors with unmixed feed streams since the initial mixing of reactants must occur inside the reactor under reacting conditions. Diffusion can be a slow process, and the reaction rate will often be limited by diffusion rather than by the intrinsic reaction rate that would prevail if the reactants were premixed. Thus, diffusion can be expected to be important in tubular reactors with unmixed feed streams. Its effects are difficult to calculate, and normal design practice is to use premixed feeds whenever possible. 

The numerator of the right side of this equation is equal to the chemical reaction rate that would prevail if there were no diffusional limitations on the reaction rate. In this situation, the reactant concentration is uniform throughout the pore and equal to its value at the pore mouth. The denominator may be regarded as the product of a hypothetical diffusive flux and a cross-sectional area for flow. The hypothetical flux corresponds to the case where there is a linear concentration gradient over the pore length equal to C0/L. The Thiele modulus is thus characteristic of the ratio of an intrinsic reaction rate in the absence of mass transfer limitations to the rate of diffusion into the pore under specified conditions. 

When the effectiveness factors for both reactions approach unity, the selectivity for two independent simultaneous reactions is the ratio of the two intrinsic reaction-rate constants. However, at low values of both effectiveness factors, the selectivity of a porous catalyst may be greater than or less than that for a plane-catalyst surface. For a porous spherical catalyst at large values of the Thiele modulus s, the effectiveness factor becomes inversely proportional to (j>S9 as indicated by equation 12.3.68. In this situation, equation 12.3.133 becomes... 

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. 

Modem catalysts have to be very active and very (100%) selective, that is, they have to catalyze the desired reaction in the temperature window, where the equilibrium conversion is the highest possible and the reaction rate is high enough to permit suitable process economics. To engineer the reaction, one has to obtain first the intrinsic reaction rate, free of heat- and mass-transfer limitations. In many cases this is very difficult, because in the core of the catalytic process there are several physical and chemical steps that must occur and which may preclude the reaction running in the kinetic regime. These steps are as follows ... 

The catalyst activity depends not only on the chemical composition but also on the diffusion properties of the catalyst material and on the size and shape of the catalyst pellets because transport limitations through the gas boundary layer around the pellets and through the porous material reduce the overall reaction rate. The influence of gas film restrictions, which depends on the pellet size and gas velocity, is usually low in sulphuric acid converters. The effective diffusivity in the catalyst depends on the porosity, the pore size distribution, and the tortuosity of the pore system. It may be improved in the design of the carrier by e.g. increasing the porosity or the pore size, but usually such improvements will also lead to a reduction of mechanical strength. The effect of transport restrictions is normally expressed as an effectiveness factor q defined as the ratio between observed reaction rate for a catalyst pellet and the intrinsic reaction rate, i.e. the hypothetical reaction rate if bulk or surface conditions (temperature, pressure, concentrations) prevailed throughout the pellet [11], For particles with the same intrinsic reaction rate and the same pore system, the surface effectiveness factor only depends on an equivalent particle diameter given by... 

For a triphasic reaction to work, reactants from a solid phase and two immiscible liquid phases must come together. The rates of reactions conducted under triphasic conditions are therefore very sensitive to mass transport effects. Fast mixing reduces the thickness of the thin, slow moving liquid layer at the surface of the solid (known as the quiet film or Nemst layer), so there is little difference in the concentration between the bulk liquid and the catalyst surface. When the intrinsic reaction rate is so high (or diffusion so slow) that the reaction is mass transport limited, the reaction will occur only at the catalyst surface, and the rate of diffusion into the polymeric matrix becomes irrelevant. Figure 5.17 shows schematic representations of the effect of mixing on the substrate concentration. 

If, however, both reactions were influenced by intraparticle diffusion effects, the rate of reaction of a particular component would be given by the product of the intrinsic reaction rate, fecg, and the effectiveness factor, Tj. Substituting eqn. (6) for the effectiveness factor gives (for a first-order isothermal reaction) the overall rate as 0tanh< >. As is often the case, the molecular weights of the diffusing reactants are similar and can be... 

At one extreme diffusivity may be so low that chemical reaction takes place only at suface active sites. In that case p is equal to the fraction of active sites on the surface of the catalyst. Such a polymer-supported phase transfer catalyst would have extremely low activity. At the other extreme when diffusion is much faster than chemical reaction p = 1. In that case the observed reaction rate equals the intrinsic reaction rate. Between the extremes a combination of intraparticle diffusion rates and intrinsic rates controls the observed reaction rates as shown in Fig. 2, which profiles the reactant concentration as a function of distance from the center of a spherical catalyst particle located at the right axis, When both diffusion and intrinsic reactivity control overall reaction rates, there is a gradient of reactant concentration from CAS at the surface, to a lower concentration at the center of the particle. The reactant is consumed as it diffuses into the particle. With diffusional limitations the active sites nearest the surface have the highest turnover numbers. The overall process of simultaneous diffusion and chemical reaction in a spherical particle has been described mathematically for the cases of ion exchange catalysis,63 65) and catalysis by enzymes immobilized in gels 66-67). Many experimental parameters influence the balance between intraparticle diffusional and intrinsic reactivity control of reaction rates with polymer-supported phase transfer catalysts, as shown in Fig. 1. 

Following Carbeny (1976), in this book the term rate coefficient is used for the proportionality coefficients kt in the typical rate expression of the form r] = kJ(C). To simplify the following analysis, a first-order elementary reaction is considered. Then the intrinsic reaction rate can be... 

On the other hand, if kg —> °° or kg ks, the resistance to die overall rate is owing to the intrinsic reaction rate and CG — C s. The rate-controlling step is the reaction rate and the overall rate is... 

For example, consider a second order reaction. In this case, the intrinsic reaction rate is... 

Reaction-limited second-order reaction For a second-order reaction of the form 2A —> products, or in the more general case of a reaction of the form a A + bB -> products of second order with respect to A, the intrinsic reaction rate is... 

If kf —> oo or kf km, then by the rate-controlling step concept, the resistance to the overall rate is due to the intrinsic reaction rate and thus... 

Reactor model for a first-order reaction To illustrate the effect of pressure drop, consider an isothermal two-phase fixed-bed operation (gas-solid system). In terms of a reactant, the intrinsic reaction rate is... 

It is obvious that this form of reaction rate becomes very complicated to work with when the surface concentration is introduced. In this case, we can use the concept of the ratecontrolling step. In our case, we assume that kt -> x or kt km, the resistance to the overall rate is owing to the intrinsic reaction rate. Then, due to the equality of the individual rates, if kt -> co then (CA- CAS) -> 0 or CA -> CAS, and so the individual rates are finite and equal to the overall rate. The rate in terms of the fluid bulk-phase concentration (overall rate per unit volume of reactor) is... 

Here, the simple case of a first-order reaction is presented. The intrinsic reaction rate in each phase is... 

For liquid-phase catalytic or enzymatic reactions, catalysts or enzymes are used as homogeneous solutes in the hquid, or as sohd particles suspended in the hquid phase. In the latter case, (i) the particles per se may be catalysts (ii) the catalysts or enzymes are uniformly distributed within inert particles or (hi) the catalysts or enzymes exist at the surface of pores, inside the particles. In such heterogeneous catalytic or enzymatic systems, a variety of factors that include the mass transfer of reactants and products, heat effects accompanying the reactions, and/or some surface phenomena, may affect the apparent reaction rates. For example, in situation (iii) above, the reactants must move to the catalytic reaction sites within catalyst particles by various mechanisms of diffusion through the pores. In general, the apparent rates of reactions with catalyst or enzymatic particles are lower than the intrinsic reaction rates this is due to the various mass transfer resistances, as is discussed below. 

The reaction is carried out over a silver gauze or low surface supported catalyst at 600—700°C, indicating a very fast chemical reaction. This implies that determination of the intrinsic reaction rate in laboratory reactors is complicated by the interference of heat and mass transfer limitations. To avoid this problem, studies have been made at much lower temperatures, which in turn run the risk of being non-representative. 

The second factor to consider is the catalyst conditions and the catalyst-particles, environment. From the point of view of the effectiveness factor, the size, shape, porosity, metal-loading, and other internal characteristics, must be the same. Otherwise, the intrinsic reaction rate and selectivity will be compromised. In practice there will be a trend to reduce costs by using less metal. This will change the process markedly, and other adjustments will be required. 

Hydrodemetallation reactions require the diffusion of multiringed aromatic molecules into the pore structure of the catalyst prior to initiation of the sequential conversion mechanism. The observed diffusion rate may be influenced by adsorption interactions with the surface and a contribution from surface diffusion. Experiments with nickel and vanadyl porphyrins at typical hydroprocessing conditions have shown that the reaction rates are independent of particle diameter only for catalysts on the order of 100 /im and smaller (R < 50/im). Thus the kinetic-controlled regime, that is, where the diffusion rate DeU/R2 is larger than the intrinsic reaction rate k, is limited to small particles. This necessitates an understanding of the molecular diffusion process in porous material to interpret the diffusion-disguised kinetics observed with full-size (i -in.) commercial catalysts. 

The experimentally-determined effectiveness factor is determined as the ratio of the experimental macro reaction rate to the intrinsic reaction rate under the same interface (bulk) composition and temperature. Based on the experimental conditions of the macrokinetics, the predicted effectiveness factors of the methanation reaction and the WGSR are obtained by solving the above non-isothermal one-dimensional and two-dimensional reaction-diffusion models for the key components. Table 1 shows the calculated effectiveness factors and the experimental values. By... 

For Ha > 0.02, there is a considerable scope for process intensification. If a reaction is intrinsically fast (a large reaction rate constant) the design aim is to provide sufficiently intense mixing to move it into the slow reaction regime (Ha < 0.02) such that the reaction is limited by the intrinsic reaction rate rather than the mass transfer rate. 

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