First-order reactions pellets

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. 

The analytical result for a first-order reaction in a spherical pellet is ... 

FIGURE 10.3 Nonisothermal effectiveness factors for first-order reactions in spherical pellets. (Adapted from Weisz, P. B. and Hicks, J. S., Chem. Eng. Sci., 17, 265 (1962).)... 

Suppose that catalyst pellets in the shape of right-circular cylinders have a measured effectiveness factor of r] when used in a packed-bed reactor for a first-order reaction. In an effort to increase catalyst activity, it is proposed to use a pellet with a central hole of radius i /, < Rp. Determine the best value for RhjRp based on an effective diffusivity model similar to Equation (10.33). Assume isothermal operation ignore any diffusion limitations in the central hole, and assume that the ends of the cylinder are sealed to diffusion. You may assume that k, Rp, and eff are known. 

In order to verify that the fixed bed and the micro-channel reactor are equivalent concerning chemical conversion, an irreversible first-order reaction A —) B with kinetic constant was considered. For simplicity, the reaction was assumed to occur at the channel surface or at the surface of the catalyst pellets, respectively. Diffusive mass transfer to the surface of the catalyst pellets was described by a correlation given by Villermaux [115]. 

The Effectiveness Factor Analysis in Terms of Effective Diffusivities First-Order Reactions on Spherical Pellets. Useful expressions for catalyst effectiveness factors may also be developed in terms of the concept of effective diffusivities. This approach permits one to write an expression for the mass transfer within the pellet in terms of a form of Fick s first law based on the superficial cross-sectional area of a porous medium. We thereby circumvent the necessity of developing a detailed mathematical model of the pore geometry and size distribution. This subsection is devoted to an analysis of simultaneous mass transfer and chemical reaction in porous catalyst pellets in terms of the effective diffusivity. In order to use the analysis with confidence, the effective diffusivity should be determined experimentally, since it is difficult to obtain accurate estimates of this parameter on an a priori basis. 

Thus a zero-order reaction appears to be 1/2 order and a second-order reaction appears to be 3/2 order when dealing with a fast reaction taking place in porous catalyst pellets. First-order reactions do not appear to undergo a shift in reaction order in going from high to low effectiveness factors. These statements presume that the combined diffusivity lies in the Knudsen range, so that this parameter is pressure independent. 

Effectiveness factor chart for first-order reaction in spherical pellets for y = 20. [From P. B. Weisz and J. S. Hicks, Chemical Engineering Science, 17 (265), 1962. Copyright 1962. Reprinted with permission of Pergamon Press.]... 

An enzyme is immobilized by adsorption on porous pellets of a carrier. The differential equation for the concentration of a reactant in a porous spherical pellet is derived in problem P7.03.01 and integrated for a first order reaction, rc = kC, in problem P7.03.06. An expression is derived for the effectiveness of the adsorbed enzyme for first order reaction as... 

The conclusion, therefore, is that when parallel competing first-order reactions occur in isothermal pellets with large pores the intrinsic selectivity is unaffected. However, in Izurge pellets with small pores, the selectivity reduces to the square root of the value for the unimpeded reaction. Thus for large 0... 

In practice, of course, it is rare that the catalytic reactor employed for a particular process operates isothermally. More often than not, heat is generated by exothermic reactions (or absorbed by endothermic reactions) within the reactor. Consequently, it is necessary to consider what effect non-isothermal conditions have on catalytic selectivity. The influence which the simultaneous transfer of heat and mass has on the selectivity of catalytic reactions can be assessed from a mathematical model in which diffusion and chemical reactions of each component within the porous catalyst are represented by differential equations and in which heat released or absorbed by reaction is described by a heat balance equation. The boundary conditions ascribed to the problem depend on whether interparticle heat and mass transfer are considered important. To illustrate how the model is constructed, the case of two concurrent first-order reactions is considered. As pointed out in the last section, if conditions were isothermal, selectivity would not be affected by any change in diffusivity within the catalyst pellet. However, non-isothermal conditions do affect selectivity even when both competing reactions are of the same kinetic order. The conservation equations for each component are described by... 

The kinetics of a homogeneous liquid reaction are studied in a flow reactor, and to approximate plug flow the 48-cm long reactor is packed with 5-mm nonporous pellets. If the conversion is 99% for a mean residence time of 1 sec, calculate the rate constant for the first-order reaction... 

A certain spherical porous catalyst with a pellet diameter of 1/8 in. has a Thiele modulus of 0.5 for a first-order reaction and gives 90% conversion in a packed bed reactor. It is proposed to... 

First-order reactions without internal mass transfer limitations A number of reactions carried out at high temperatures are potentially mass-transfer limited. The surface reaction is so fast that the global rate is limited by the transfer of the reactants from the bulk to the exterior surface of the catalyst. Moreover, the reactants do not have the chance to travel within catalyst particles due to the use of nonporous catalysts or veiy fast reaction on the exterior surface of catalyst pellets. Consider a first-order reaction A - B or a general reaction of the form a A - bB - products, which is of first order with respect to A. For the following analysis, a zero expansion factor and an effectiveness factor equal to 1 are considered. 

Show that the selectivity of two concurrent first-order reactions occurring in flat-shaped porous catalyst pellets is independent of the effect of either heat or mass transfer if the activation energies of both reactions are equal. 

Newson (1975) was among the first to develop a pore plugging model of demetallation to predict catalyst life. By using the pore structure model of Wheeler (1951), the pellet was assumed to have N pores of identical length but with a specified distribution of pore radii. Metal deposition was assumed to be a first-order reaction over an outer fraction of the pore length and to have a uniform thickness. This model showed that the broadness of the size distribution had little effect on the catalyst life for the same average radii, but that increasing the radii from 45 to 65 A more than doubled the catalyst life. The restricted form of the diffusivity (see Section IV,B,5) was not employed in this model. 

The model formulated by Ahn and Smith (1984) considered partial surface poisoning for HDS and pore mouth plugging for HDM reactions. The conservation equations with first-order reactions for metal-bearing and sulfur-bearing species were based on spherical pellet geometry rather than on single pores. Hence, a restricted effective diffusivity was employed... 

By using the theory of multicomponent first-order reaction and diffusion (Wei, 1962), the specific conservation equation for A and B in the catalyst pellet is expressed as... 

For the nonisothermal catalyst pellet with negligible external mass and heat transfer resistances, i.e., with Sh —> 00 and Nu —> 00 and for a first-order reaction, the dimensionless concentration and temperature are governed by the following couple of boundary value differential equations... 

For single, irreversible reactions obeying simple, integer order power rate laws, this problem can generally be solved analytically. In the case of a first-order reaction in a spherical pellet, the following mass balance is found ... 

Consider, by way of example, an irreversible first-order reaction, A B, occurring in a pellet with slab geometry. As a result of the concentration profile, the reaction rate depends on the position within the catalyst pellet. Hence, a mass balance for the reactant has to be taken over an infinitesimal slice of the slab. At the steady state this leads to ... 

Assuming a steady state, for first-order reaction-diffusion system A -> B under nonisothermal catalyst pellet conditions, the mass and energy balances are... 

Figure 9.3 is a plot of Eq. (9.28), which shows that if r/> -> 0 then rj -> 1, which means there is no considerable diffusion resistance. As diffusion resistance increases, we have (j> - oo and hence rj -> 0. The latter can occur not for small diffusivity, for large pellet size L, or for very fast reaction rate, or for all three factors. This regime where the diffusion strongly affects the rate of reaction is called strong pore resistance. For a first-order reaction, a general criterion of... 

In the case that the reactants are already in the phase surrounding the catalyst pellet, the components only have to be transported through this single phase, being either liquid or gas. Reactant A is assumed to be converted according to a first-order reaction. For convenience a chemical reaction rate R is defined per unit of external surface ... 

Nonisothermal Catalyst Pellets and First-order Reactions... 

This is equivalent to the differential equation describing the concentration profile inside a catalyst pellet for first-order reactions with reaction rate... 

For a first-order reaction taking place in a ring-shaped catalyst pellet, the effectiveness factor was already calculated by Gunn [1], That solution, however, is very complex because the mathematical techniques used are not the most suitable with which to solve the equations that arise. Therefore, the following solution derives another expression for the effectiveness factor. 

Cast in the form of Equation 8, the problem is equivalent to the classical chemical engineering problem of determining the effectiveness of reactant penetration into a porous catalyst pellet which is the site of a distributed pseudo-homogeneous first-order reaction. We can then quote well known results (12,13). Inside the cloud c i is governed by ... 

Develop expressions for the Thiele modulus and the concentration profile of A for the following reversible first-order reaction that takes place in a flat plate catalyst pellet ... 

Since the equations are nonlinear, a numerical solution method is required. Weisz and Hicks calculated the effectiveness factor for a first-order reaction in a spherical catalyst pellet as a function of the Thiele modulus for various values of the Prater number [P. B. Weisz and J. S. Hicks, Chem. Eng. Sci., 17 (1962) 265]. Figure 6.3.12 summarizes the results for an Arrhenius number equal to 30. Since the Arrhenius number is directly proportional to the activation energy, a higher value of y corresponds to a greater sensitivity to temperature. The most important conclusion to draw from Figure 6.3.12 is that effectiveness factors for exothermic reactions (positive values of j8) can exceed unity, depending on the characteristics of the pellet and the reaction. In the narrow range of the Thiele modulus between about 0.1 and 1, three different values of the effectiveness factor can be found (but only two represent stable steady states). The ultimate reaction rate that is achieved in the pellet... 

Consider a first-order reaction occurring on a nonporous flat plate catalyst pellet. In Section 6.2, it was shown that the concentration of reactant A on the external surface of the catalyst is related to both the mass transfer coefficient, k, and the surface rate constant, ks. ... 

Now consider the first-order reaction in a porous flat plate catalyst pellet so that both external (interphase) and internal (intraphase) transport limitations are encountered. At steady state, the flux of A to the surface of the pellet is equal to the flux entering the pellet ... 

A lack of significant intraphase diffusion effects (i.e., 17 > 0.95) on an irreversible, isothermal, first-order reaction in a spherical catalyst pellet can be assessed by the Weisz-Prater criterion [P. B. Weisz and C. D. Prater, Adv. Catal., 6 (1954) 143] ... 

The isothermal, first-order reaction of gaseous A occurs within the pores of a spherical catalyst pellet. The reactant concentration halfway between the external surface and the center of the pellet is equal to one-fourth the concentration at the external surface. 

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