Reaction rate pore diffusion effect

Thus we can say that if the Thiele modulus is much less than unity, there is no pore diffusion effect on the reaction rate because the reactant concentration remains at Cas down the pore, but if the Thiele modulus is much greater than unity, then the rate is proportional to l/[Pg.287]

The conversion of cyclohexanes to aromatics is a highly endothermic reaction (AH 50 kcal./mole) and occurs very readily over platinum-alumina catalyst at temperatures above about 350°C. At temperatures in the range 450-500°C., common in catalytic reforming, it is extremely difficult to avoid diffusional limitations and to maintain isothermal conditions. The importance of pore diffusion effects in the dehydrogenation of cyclohexane to benzene at temperatures above about 372°C. has been shown by Barnett et al. (B2). However, at temperatures below 372°C. these investigators concluded that pore diffusion did not limit the rate when using in, catalyst pellets. 

The pure compound rate constants were measured with 20-28 mesh catalyst particles and reflect intrinsic rates (—i.e., rates free from diffusion effects). Estimated pore diffusion thresholds are shown for 1/8-inch and 1/16-inch catalyst sizes. These curves show the approximate reaction rate constants above which pore diffusion effects may be observed for these two catalyst sizes. These thresholds were calculated using pore diffusion theory for first-order reactions (18). Effective diffusivities were estimated using the Wilke-Chang correlation (19) and applying a tortuosity of 4.0. The pure compound data were obtained by G. E. Langlois and co-workers in our laboratories. Product yields and suggested reaction mechanisms for hydrocracking many of these compounds have been published elsewhere (20-25). 

If the reactions were not influenced by in-pore diffusion effects, the intrinsic kinetic selectivity would be kjk2(= S). When mass transfer is important, the rate of reaction of both A and X must be calculated with this in mind. From equation 3.9, the rate of reaction for the slab model is ... 

A model Is presented for char gasification with simultaneous capture of sulfur In the ash minerals as CaS. This model encompasses the physicochemical rate processes In the boundary layer, In the porous char, and around the mineral matter. A description of the widening of the pores and the eventual collapse of the char structure Is Included. The modeling equations are solved analytically for two limiting cases. The results demonstrate that pore diffusion effects make It possible to capture sulfur as CaS In the pores of the char even when CaS formation Is not feasible at bulk gas conditions. The model predictions show good agreement with experimentally determined sulfur capture levels and reaction times necessary to complete gasification. 

A measure of the absence of internal (pore diffusion) mass transfer limitations is provided by the internal effectiveness factor, t, which is defined as the ratio of the actual overall rate of reaction to the rate that would be observed if the entire interior surface were exposed to the reactant concentration and temperature existing at the exterior of the catalyst pellet. A value of 1 for rj implies that all of the sites are being utilized to their potential, while a value below, say, 0.5, signals that mass transfer is limiting performance. The value of rj can be related to that of the Thiele modulus, 4>, which is an important dimensionless parameter that roughly expresses a ratio of surface reaction rate to diffusion rate. For the specific case of an nth order irreversible reaction occurring in a porous sphere,... 

At impeller speeds above the second or third term on the right-hand side of Equation (CS11.3) becomes controlling i.e., the overall rate is controlled by intrinsic chemical reaction or pore diffusion. Under such conditions, if experiments are performed at different particle sizes, then the effects of internal diffusion and chemical reaction can be elucidated. 

Bodrov et al. (1964), investigated the steam reforming of methane on nickel foil to eliminate pore diffusion effects. A differential reactor was used at atmospheric pressure and in the temperature range 800-900 °C. The results demonstrated that the reaction rate is first order with respect to methane at 900 C, and at a lower temperature (800°C), H2O, H2 and CO, have an influence that can be described by the following expression ... 

It is observed that when Thiele modulus i is low (order of magnitude 0.5), the effectiveness factor rj is almost equal to 1 indicating that the observed reaction rate is not diffusion limited. On the other hand, the effectiveness factor varies linearly with Thiele modulus for i values that range from 2 to 10, indicating that the observed reaction rate is diffusion limited. For i values over 10, the effectiveness factor r] has a more significant deviation, indicating a strong diffusion effect inside pores of the spherical particle. 

Figure 4.5.25 Arrhenius plot of effective rate constant for the three regimes of control by reaction, interplay of reaction and pore diffusion, and control by external diffusion.

To conclude, an overall summary of calculations based on the above results indicates that the usual order of events is to have first chemical reaction control throughout the pellet. Next, with higher intrinsic rates of reaction, internal pore diffusion begins to have an effect, followed by external heat transfer resistance. Finally, for extremely rapid reactions there is the possibility of external mass transfer resistance and temperature gradients of some significance. Only for unrealistic situations is it likely that particle instabilities might occur, and even then only for narrow ranges of temperature. 

Equation (5.28) also incorporates the mass fraction of the combustible surface species, which is carbon wc here, and an effectiveness feictor rj, which is defined as the area of the reacting surface divided by the external (equivalent-sphere) surfece area of the particle [7] the latter is the basis for the reaction rate terms. The effectiveness factor and the carbon mass fraction determine the accessible surface of the particle, allowing the empirical adoption of a pore diffusion restriction as indicated in Equation (5.17). In the frequent case of the reaction order being unity, expression (5.28) can be simplified to... 

Catalyst Effectiveness. Even at steady-state, isothermal conditions, consideration must be given to the possible loss in catalyst activity resulting from gradients. The loss is usually calculated based on the effectiveness factor, which is the diffusion-limited reaction rate within catalyst pores divided by the reaction rate at catalyst surface conditions (50). The effectiveness factor E, in turn, is related to the Thiele modulus,

first-order rate constant, a the internal surface area, and the effective diffusivity. It is desirable for E to be as close as possible to its maximum value of unity. Various formulas have been developed for E, which are particularly usehil for analyzing reactors that are potentially subject to thermal instabilities, such as hot spots and temperature mnaways (1,48,51). 

As a reactant molecule from the fluid phase surrounding the particle enters the pore stmcture, it can either react on the surface or continue diffusing toward the center of the particle. A quantitative model of the process is developed by writing a differential equation for the conservation of mass of the reactant diffusing into the particle. At steady state, the rate of diffusion of the reactant into a shell of infinitesimal thickness minus the rate of diffusion out of the shell is equal to the rate of consumption of the reactant in the shell by chemical reaction. Solving the equation leads to a result that shows how the rate of the catalytic reaction is influenced by the interplay of the transport, which is characterized by the effective diffusion coefficient of the reactant in the pores, and the reaction, which is characterized by the first-order reaction rate constant. 

Figure 10 shows that Tj is a unique function of the Thiele modulus. When the modulus ( ) is small (- SdSl), the effectiveness factor is unity, which means that there is no effect of mass transport on the rate of the catalytic reaction. When ( ) is greater than about 1, the effectiveness factor is less than unity and the reaction rate is influenced by mass transport in the pores. When the modulus is large (- 10), the effectiveness factor is inversely proportional to the modulus, and the reaction rate (eq. 19) is proportional to k ( ), which, from the definition of ( ), implies that the rate and the observed reaction rate constant are proportional to (1 /R)(f9This result shows that both the rate constant, ie, a measure of the intrinsic activity of the catalyst, and the effective diffusion coefficient, ie, a measure of the resistance to transport of the reactant offered by the pore stmcture, influence the rate. It is not appropriate to say that the reaction is diffusion controlled it depends on both the diffusion and the chemical kinetics. In contrast, as shown by equation 3, a reaction in solution can be diffusion controlled, depending on D but not on k. 

Although the point values of the rate diminish with p, in the steady state the rate of reaction equals the rate of diffusion at the mouth of the pores. The effectiveness of the catalyst is a ratio... 

Inert gas pressure, temperature, and conversion were selected as these are the critical variables that disclose the nature of the basic rate controlling process. The effect of temperature gives an estimate for the energy of activation. For a catalytic process, this is expected to be about 90 to 100 kJ/mol or 20-25 kcal/mol. It is higher for higher temperature processes, so a better estimate is that of the Arrhenius number, y = E/RT which is about 20. If it is more, a homogeneous reaction can interfere. If it is significantly less, pore diffusion can interact. 

Few fixed-bed reactors operate in a region where the intrinsic kinetics are applicable. The particles are usually large to minimize pressure drop, and this means that diffusion within the pores. Steps 3 and 7, can limit the reaction rate. Also, the superficial fluid velocity may be low enough that the external film resistances of Steps 2 and 8 become important. A method is needed to estimate actual reaction rates given the intrinsic kinetics and operating conditions within the reactor. The usual approach is to define the effectiveness factor as... 

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. ... 

Equation (10.29) is the appropriate reaction rate to use in global models such as Equation (10.1). The reaction rate would be —ka if there were no mass transfer resistance. The effectiveness factor rj accounts for pore diffusion and him resistance so that the effective rate is — 7ka. 

If diffusion of reactants to the active sites in pores is slower than the chemical reaction, internal mass transfer is at least partly limiting and the reactant concentration decreases along the pores. This reduces the reaction rate compared to the rate at external surface conditions. A measure of the reaction rate decrease is the effectiveness factor, r, which has been defined as ... 

The occurrence of pore diffusion can usually be determined by simply grinding the catalyst into smaller and smaller particles. If the rate per gram of catalyst increases as the particles become smaller and smaller, then pore diffusion is likely to be occurring. This effect is due to the fact that the pore lengths are decreased by the catalyst particles being ground into smaller and smaller pieces. Eventually, the pores become short enough that the reactants can readily diffuse in and out of them faster than the chemical reaction occurs on the surface. 

This situation is termed pore-mouth poisoning. As poisoning proceeds the inactive shell thickens and, under extreme conditions, the rate of the catalytic reaction may become limited by the rate of diffusion past the poisoned pore mouths. The apparent activation energy of the reaction under these extreme conditions will be typical of the temperature dependence of diffusion coefficients. If the catalyst and reaction conditions in question are characterized by a low effectiveness factor, one may find that poisoning only a small fraction of the surface gives rise to a disproportionate drop in activity. In a sense one observes a form of selective poisoning. 

Now consider the other extreme condition where diffusion is rapid relative to chemical reaction [i.e., hT( 1 — a) is small]. In this situation the effectiveness factor will approach unity for both the poisoned and unpoisoned reactions, and we must retain the hyperbolic tangent terms in equation 12.3.124 to properly evaluate Curve C in Figure 12.11 is calculated for a value of hT = 5. It is apparent that in this instance the activity decline is not nearly as sharp at low values of a as it was at the other extreme, but it is obviously more than a linear effect. The reason for this result is that the regions of the catalyst pore exposed to the highest reactant concentrations do not contribute proportionately to the overall reaction rate because they have suffered a disproportionate loss of activity when pore-mouth poisoning takes place. 

As a measure of how much the effective rate is lowered by the resistance to pore diffusion, the effectiveness factor qpore is used. This factor is defined as the ratio of the actual mean reaction rate within the pore to the maximum rate if not... 

---