Diffusion, bulk Effectiveness factor

Table II.1 which depends on the pellet size, so the familiar plot of effectiveness factor versus Thiele modulus shows how t varies with pellet radius. A slightly more interesting case arises if it is desired to exhibit the variation of the effectiveness factor with pressure as the mechanism of diffusion changes from Knudsen streaming to bulk diffusion control [66,...

Example 10.6 A commercial process for the dehydrogenation of ethylbenzene uses 3-mm spherical catalyst particles. The rate constant is 15s , and the diffusivity of ethylbenzene in steam is 4x 10 m /s under reaction conditions. Assume that the pore diameter is large enough that this bulk diffusivity applies. Determine a likely lower bound for the isothermal effectiveness factor. 

Many theoretical embellishments have been made to the basic model of pore diffusion as presented here. Effectiveness factors have been derived for reaction orders other than first and for Hougen and Watson kinetics. These require a numerical solution of Equation (10.3). Shape and tortuosity factors have been introduced to treat pores that have geometries other than the idealized cylinders considered here. The Knudsen diffusivity or a combination of Knudsen and bulk diffusivities has been used for very small pores. While these studies have theoretical importance and may help explain some observations, they are not yet developed well enough for predictive use. Our knowledge of the internal structure of a porous catalyst is still rather rudimentary and imposes a basic limitation on theoretical predictions. We will give a brief account of Knudsen diffusion. 

The concentration of gas over the active catalyst surface at location / in a pore is ai [). The pore diffusion model of Section 10.4.1 linked concentrations within the pore to the concentration at the pore mouth, a. The film resistance between the external surface of the catalyst (i.e., at the mouths of the pore) and the concentration in the bulk gas phase is frequently small. Thus, a, and the effectiveness factor depends only on diffusion within the particle. However, situations exist where the film resistance also makes a contribution to rj so that Steps 2 and 8 must be considered. This contribution can be determined using the principle of equal rates i.e., the overall reaction rate equals the rate of mass transfer across the stagnant film at the external surface of the particle. Assume A is consumed by a first-order reaction. The results of the previous section give the overall reaction rate as a function of the concentration at the external surface, a. ... 

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... 

This sieve effect cannot be considered statically as a factor that only determines the amount of accessible acid groups in the resin in such a way that the boundary between the accessible and non-accessible groups would be sharp. It must be treated dynamically, i.e. the rates of the diffusion of reactants into the polymer mass must be taken into account. With the use of the Thiele s concept about the diffusion into catalyst pores, the effectiveness factors, Thiele moduli and effective diffusion coefficients can be determined from the effect of the catalyst particle size. The apparent rates of the methyl and ethyl acetate hydrolysis [490] were corrected for the effect of diffusion in the resin by the use of the effectiveness factors, the difference in ester concentration between swollen resin phase and bulk solution being taken into account. The intrinsic rate coefficients, kintly... 

A controls the overall rate of conversion, equation 3.81 could be used directly as the rate equation for design purposes. If, however, external mass transfer were important the partial pressures in equation 3.81 would be values at the interface and an equation (such as equation 3.66) for each component would be required to express interfacial partial pressures in terms of bulk partial pressures. If internal diffusion were also important, the overall rate equation would be multiplied by an effectiveness factor either estimated experimentally, or alternatively obtained by theoretical considerations similar to those discussed earlier. 

An enzyme which hydrolyzes the cellobiose to glucose, /3-glucosidase is immobilized in a sodium alginate gel sphere (2.5 mm in diameter). Assume that the zero-order reaction occurs at every point within the sphere with k0 = 0.0795 mol/sm3, and cellobiose moves through the sphere by molecular diffusion with Ds = 0.6 x 10 5 cm2 /s (cellobiose in gel). Calculate the effectiveness factor of the immobilized enzyme when the cellobiose concentration in bulk solution is 10 mol/m3. 

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... 

The effectiveness factor Tj is the ratio of the rate of reaction in a porous catalyst to the rate in the absence of diffusion (i.e., under bulk conditions). The theoretical basis for q in a porous catalyst has been discussed in Sec. 7. For example, for an isothermal first-order reaction... 

Surface reaction with diffusion and heat transfer resistance In fast exothermic reactions, in addition to grad c, also grad T (TG Ts) is present in the boundary layer between the gas bulk phase and the catalyst surface. For the outer effectiveness factor qext this means that... 

Steps 2 and 6 are both pore diffusion processes with apparent activation energies between 2 and 10 kcal/mol. This apparent activation energy is stated to be about 1/2 that of the chemical rate activation energy. The concentration of reactants decreases from the outer perimeter towards the center of the catalyst particle for Step 2. In this case some of the interior of the catalyst is being utilized but not fully. Therefore the effectiveness factor is greater than zero but considerably less than one. These reactions are moderately influenced by temperature but to a greater extent than bulk mass transfer. 

It plays the same role as the effectiveness factor in heterogeneous catalysis and is a measure of the film thickness uniformity. It represents the ratio of the total reaction rate on each pair of wafers to that we would obtain if the concentration in the cell formed by the two wafers were equal to the bulk concentration everywhere. Thus, if the surface reaction is the rate controlling step, n = 1, whereas if the diffusion between the wafers controls, n < 1. In the limit of strong diffusion resistance the deposition is confined to a narrow outer band of the wafers and a strongly nonuniform film results. 

The Damkohler number indicates which characteristic first-order process is faster, external diffusion or reaction. For very large values of Da (ks the surface concentration of reactant approaches zero, whereas for very small values of Da ks the surface concentration approaches the bulk fluid concentration. An interphase effectiveness factor, Tj, is defined as the reaction rate based on surface conditions divided by the rate that would be observed in the absence of diffusional limitations ... 

For first-order reactions we can use an overall effectiveness factor to help us analyze diffusion, flow, and reaction in packed beds. We now consider a situation where external and internal resistance to mass transfer to and within the pellet are of the same order of magnitude (Figure 12-9). At steady state, the transport of the reactant(s) from the bulk fluid to the external surface of the catalyst is equal to the net rate of reaction of the reactant within and on the pellet. 

Hence, to express the actual reaction rate, we have to multiply the reaction rate based on the bulk condition by a correction factor, which accounts for the diffusion effects. The effectiveness factor depends on the ratio between the reaction rate and the diffusion rate and is expressed in terms of a modulus (Thiele modulus), 4>, defined by... 

The equations and plots presented in the foregoing sections largely pertain to the diffusion of a single component followed by reaction. There are several other situations of industrial importance on which considerable information is available. They include biomolecular reactions in which the diffusion-reaction problem must be extended to two molecular species, reactions in the liquid phase, reactions in zeolites, reactions in immobilized catalysts, and extension to complex reactions (see Aris, 1975 Doraiswamy, 2001). Several factors influence the effectiveness factor, such as pore shape and constriction, particle size distribution, micro-macro pore structure, flow regime (bulk or Knudsen), transverse diffusion, gross external surface area of catalyst (as distinct from the total pore area), and volume change upon reaction. Table 11.8 lists the major effects of all these situations and factors. 

The term D/ g(d Cj j,/dz ) is used to represent either diffusion and/or dishow to use n and fl persion in the axial direction. Consequently, we shall use the symbol D, for the dispersion coefficient to represent either or both of these cases. We will come back to this form of the diffusion equation when w e discuss dispersion in Chapter 14. The overall reaction rate within the pellet, -r, is the overall rate of reaction within and on the catalyst per unit mass of catalyst. It is a function of the reactant concentration within the catalyst. This overall rate can be related to the rate of reaction of A that would exist if the entire surface were exposed to the bulk concentration through the overall effectiveness factor fi ... 

Wang and Yang (1991b) have proposed a more realistic model for triphase catalysis in a batch reactor, where they consider mass transfer of reactants in the bulk aqueous and organic phases, diffusion of reactants within the pores of the solid catalyst particle, and intrinsic reactivities of the ion-ex-change and organic reactions at the active sites within the solid catalyst. An apparent overall effectiveness factor of the catalyst is obtained in this case by applying the pseudo-... 

Catalyst supports such as silica and alumina have low thermal conductivities so that temperature gradients within catalyst particles are likely in all but the finely ground powders used for infrinsic kinetic studies. There may also be a film resisfance fo heaf fransfer af fhe exfemal surface of the catalyst. Thus the internal temperatures in a catalyst pellet may be substantially different than the bulk gas temperature. The definition of the effectiveness factor, Equation 10.23, is unchanged, but an exothermic reaction can have reaction rates inside the pellet that are higher than would be predicted using the bulk gas temperature. In the absence of a diffusion limitation, rj > 1 would be expected for an exothermic reaction. (The case > 1 is also possible for some isothermal reactions with weird kinetics.) Mass transfer limitations may have a larger... 

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