Reaction effectiveness factor

If the maximum metal content occurs at the edge of the pellet, then the distribution parameter is equal to the reaction effectiveness factor. 

The catalyst intraparticle reaction-diffusion process of parallel, equilibrium-restrained reactions for the methanation system was studied. The non-isothermal one-dimensional and two-dimensional reaction-diffusion models for the key components have been established, and solved using an orthogonal collocation method. The simulation values of the effectiveness factors for methanation reaction Ch4 and shift reaction Co2 are fairly in agreement with the experimental values. Ch4 is large, while Co2 is very small. The shift reaction takes place as direct and reverse reaction inside the catalyst pellet because of the interaction of methanation and shift reaction. For parallel, equilibrium-restrained reactions, effectiveness factors are not able to predict the catalyst internal-surface utilization accurately. Therefore, the intraparticle distributions of the temperature, the concentrations of species and so on should be taken into account. 

The effectiveness factor for a wide range of reaction kinetic models differs little from that of the first-order case. For an isothermal particle, the first-order reaction effectiveness factor is... 

Conventional procedures can be used to find the reaction effectiveness factors of the mono- and bidisperse pellets. If diffusion limitations in the macropores are significant the average reaction rates are... 

Fig.3. Reaction effectiveness factor and deactivation effectiveness factor v s. average pore diameter.

The component effectiveness factors / are shown in Figure 5.29 for o-Xylene, maleic anhydride and phthalic anhydride they all show similar shape to those of the reaction effectiveness factor with a wide... 

The main advantage with packed beds is the flow pattern. Conditions approaching a plug flow are advantageous for most reaction kinetics. Diffusion resistance in catalyst particles may sometimes reduce the reaction rates, but for strongly exothermic reactions, effectiveness factors higher than unity (1) can be obtained. Hot spots appear in highly exothermic... 

The differential material balances contain a large number of physical parameters describing the structure of the porous medium, the physical properties of the gaseous mixture diffusing through it, the kinetics of the chemical reaction and the composition and pressure of the reactant mixture outside the pellet. In such circumstances it Is always valuable to assemble the physical parameters into a smaller number of Independent dimensionless groups, and this Is best done by writing the balance equations themselves in dimensionless form. The relevant equations are (11.20), (11.21), (11.22), (11.23), (11.16) and the expression (11.27) for the effectiveness factor. 

Let us compare computations of the effectiveness factor, using each of the three approximations we have described, with exact values from the complete dusty gas model. The calculations are performed for a first order reaction of the form A lOB in a spherical pellet. The stoichiometric coefficient 10 for the product is unrealistically large, but is chosen to emphasize any differences between the different approaches. 

Reactions in porous catalyst pellets are Invariably accompanied by thermal effects associated with the heat of reaction. Particularly In the case of exothermic reactions these may have a marked influence on the solutions, and hence on the effectiveness factor, leading to effectiveness factors greater than unity and, In certain circumstances, multiple steady state solutions with given boundary conditions [78]. These phenomena have attracted a great deal of interest and attention in recent years, and an excellent account of our present state of knowledge has been given by Arls [45]. 

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). 

The result is shown in Figure 10, which is a plot of the dimensionless effectiveness factor as a function of the dimensionless Thiele modulus ( ), which is R.(k/Dwhere R is the radius of the catalyst particle and k is the reaction rate constant. The effectiveness factor is defined as the ratio of the rate of the reaction divided by the rate that would be observed in the absence of a mass transport influence. The effectiveness factor would be unity if the catalyst were nonporous. Therefore, the reaction rate is... 

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. 

Intraparticle mass transport resistance can lead to disguises in selectivity. If a series reaction A — B — C takes place in a porous catalyst particle with a small effectiveness factor, the observed conversion to the intermediate B is less than what would be observed in the absence of a significant mass transport influence. This happens because as the resistance to transport of B in the pores increases, B is more likely to be converted to C rather than to be transported from the catalyst interior to the external surface. This result has important consequences in processes such as selective oxidations, in which the desired product is an intermediate and not the total oxidation product CO2. 

Rates and selectivities of soHd catalyzed reactions can also be influenced by mass transport resistance in the external fluid phase. Most reactions are not influenced by external-phase transport, but the rates of some very fast reactions, eg, ammonia oxidation, are deterrnined solely by the resistance to this transport. As the resistance to mass transport within the catalyst pores is larger than that in the external fluid phase, the effectiveness factor of a porous catalyst is expected to be less than unity whenever the external-phase mass transport resistance is significant, A practical catalyst that is used under such circumstances is the ammonia oxidation catalyst. It is a nonporous metal and consists of layers of wire woven into a mesh. 

Fig. 6. The three ideal zones (I—III) representing the rate of change of reaction for a porous carbon with increasing temperature where a and b are intermediate zones, is activation energy, and -E is tme activation energy. The effectiveness factor, Tj, is a ratio of experimental reaction rate to reaction rate which would be found if the gas concentration were equal to the atmospheric gas concentration (80).

Heat Sensitivity. The heat sensitivity or polymerization tendencies of the materials being distilled influence the economics of distillation. Many materials caimot be distilled at their atmospheric boiling points because of high thermal degradation, polymerization, or other unfavorable reaction effects that are functions of temperature. These systems are distilled under vacuum in order to lower operating temperatures. For such systems, the pressure drop per theoretical stage is frequently the controlling factor in contactor selection. An exceUent discussion of equipment requirements and characteristics of vacuum distillation may be found in Reference 90. 

Treatment of thermal conductivity inside the catalyst can be done similarly to that for pore diffusion. The major difference is that while diffusion can occur in the pore volume only, heat can be conducted in both the fluid and solid phases. For strongly exothermic reactions and catalysts with poor heat conductivity, the internal overheating of the catalyst is a possibility. This can result in an effectiveness factor larger than unity. 

Diffusion effects can be expected in reactions that are very rapid. A great deal of effort has been made to shorten the diffusion path, which increases the efficiency of the catalysts. Pellets are made with all the active ingredients concentrated on a thin peripheral shell and monoliths are made with very thin washcoats containing the noble metals. In order to convert 90% of the CO from the inlet stream at a residence time of no more than 0.01 sec, one needs a first-order kinetic rate constant of about 230 sec-1. When the catalytic activity is distributed uniformly through a porous pellet of 0.15 cm radius with a diffusion coefficient of 0.01 cm2/sec, one obtains a Thiele modulus y> = 22.7. This would yield an effectiveness factor of 0.132 for a spherical geometry, and an apparent kinetic rate constant of 30.3 sec-1 (106). 

Oxidation kinetics over platinum proceeds at a negative first order at high concentrations of CO, and reverts to a first-order dependency at very low concentrations. As the CO concentration falls towards the center of a porous catalyst, the rate of reaction increases in a reciprocal fashion, so that the effectiveness factor may be greater than one. This effectiveness factor has been discussed by Roberts and Satterfield (106), and in a paper to be published by Wei and Becker. A reversal of the conventional wisdom is sometimes warranted. When the reaction kinetics has a negative order, and when the catalyst poisons are deposited in a thin layer near the surface, the optimum distribution of active catalytic material is away from the surface to form an egg yolk catalyst. 

The ratio of the overall rate of reaction to that which would be achieved in the absence of a mass transfer resistance is referred to as the effectiveness factor rj. SCOTT and Dullion(29) describe an apparatus incorporating a diffusion cell in which the effective diffusivity De of a gas in a porous medium may be measured. This approach allows for the combined effects of molecular and Knudsen diffusion, and takes into account the effect of the complex structure of the porous solid, and the influence of tortuosity which affects the path length to be traversed by the molecules. 

The effectiveness factor depends, not only on the reaction rate constant and the effective diffusivity, but also on the size and shape of the catalyst pellets. In the following analysis detailed consideration is given to particles of two regular shapes ... 

A hydrocarbon is cracked using a silica-alumina catalyst in the form of spherical pellets of mean diameter 2.0 mm. When the reactant concentration is 0.011 kmol/m3, the reaction rate is 8.2 x 10"2 kmol/(m3 catalyst) s. If the reaction is of first-order and the effective diffusivity De is 7.5 x 10 s m2/s, calculate the value of the effectiveness factor r). It may be assumed that the effect of mass transfer resistance in the. fluid external Lo the particles may be neglected. 

In section 11.3 we saw how a classical reaction engineering approach45 can been used to model both electrochemical promotion and metal support interactions. The analysis shows that the magnitude of the effect depends on three dimensionless numbers, II, J and Op (Table 11.3) which dictate the actual value of the promotional effectiveness factor. 

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... 

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 overall effectiveness factor for the first-order reaction is defined using the bulk gas concentration a. 

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