Spherical catalyst pellet

Figure 1.6.1 Comparison of asymptotic and exact solutions for a first order, non-isothermal reaction in a spherical catalyst pellet. ...

One must understand the physical mechanisms by which mass transfer takes place in catalyst pores to comprehend the development of mathematical models that can be used in engineering design calculations to estimate what fraction of the catalyst surface is effective in promoting reaction. There are several factors that complicate efforts to analyze mass transfer within such systems. They include the facts that (1) the pore geometry is extremely complex, and not subject to realistic modeling in terms of a small number of parameters, and that (2) different molecular phenomena are responsible for the mass transfer. Consequently, it is often useful to characterize the mass transfer process in terms of an effective diffusivity, i.e., a transport coefficient that pertains to a porous material in which the calculations are based on total area (void plus solid) normal to the direction of transport. For example, in a spherical catalyst pellet, the appropriate area to use in characterizing diffusion in the radial direction is 47ir2. 

Consider the spherical catalyst pellet of radius R shown in Figure 12.4. The effective diffusivity approach presumes that diffusion of all types can be represented in terms of Fick s first law and an overall effective diffusion coefficient that can be taken as a constant. That is, the appropriate flux representation is... 

Schematic diagram of porous spherical catalyst pellet.

Effectiveness factor ratios for first-order kinetics on spherical catalyst pellets. 

Analysis of the same reaction carried out using spherical catalyst pellets leads to similar results and conclusions, since the first-order rate constant is again replaced by the group k K + 1 )/K. 

Using this definition of the Thiele modulus, the reaction rate measurements for finely divided catalyst particles noted below, and the additional property values cited below, determine the effectiveness factor for 0.5 in. spherical catalyst pellets fabricated from these particles. Comment on the reasons for the discrepancy between the calculated value of rj and the ratio of the observed rate for 0.5 in. pellets to that for fine particles. 

This relation is plotted as curve Bin Figure 12.11. Smith (66) has shown that the same limiting forms for are observed using the concept of effective dififusivities and spherical catalyst pellets. Curve B indicates that, for fast reactions on catalyst surfaces where the poisoned sites are uniformly distributed over the pore surface, the apparent activity of the catalyst declines much less rapidly than for the case where catalyst effectiveness factors approach unity. Under these circumstances, the catalyst effectiveness factors are considerably less than unity, and the effects of the portion of the poison adsorbed near the closed end of the pore are not as apparent as in the earlier case for small hr. With poisoning, the Thiele modulus hp decreases, and the reaction merely penetrates deeper into the pore. 

The ortho-para conversion of molecular hydrogen is catalyzed by NiO. A supported catalyst is available with a specific surface area of 305 m2/g and a void volume of 0.484 cm3/g. A spherical catalyst pellet has an apparent density of 1.33 g/cm3 and a diameter of 0.5 cm. If the system is not far from equilibrium, an apparent first-order rate constant (kr) can be defined in the following manner. 

A first-order chemical reaction occurs isothermally in a reactor packed with spherical catalyst pellets of radius R. If there is a resistance to mass transfer from the main fluid stream to the surface of the particle in addition to a resistance within the particle, show that the effectiveness factor for the pellet is given by ... 

What size of spherical catalyst pellets = 10 cm /cm cat s) would ensure that pore resistance effects do not intrude to slow the rate of reaction ... 

In problems such as the drying of droplets or diffusion through films around spherical catalyst pellets, it is more convenient to use Eqs. (40b) and (49) in spherical coordinates. Then for steady state diffusion in the radial direction alone, one has in the absence of chemical reactions... 

Villadsen, J. and Michelsen, M. L. (1972) Diffusion and reaction in spherical catalyst pellets steady state and local stability analysis. Chem. Engng Sci. 27, 751. 

Derive an expression for the effectiveness factor of a spherical catalyst pellet in which a first-order isothermal reaction occurs. 

When the right-hand side of (5.53) is linear, the DE can be solved analytically. The second-order differential equation for the spherical catalyst pellet is... 

Here we consider a spherical catalyst pellet with negligible intraparticle mass- and negligible heat-transfer resistances. Such a pellet is nonporous with a high thermal conductivity and with external mass and heat transfer resistances only between the surface of the pellet and the bulk fluid. Thus only the external heat- and mass-transfer resistances are considered in developing the pellet equations that calculate the effectiveness factor rj at every point along the length of the reactor. 

The solution method, the Adomian Decomposition Method (ADM), is mechanized for solving the nonlinear models according to the principle of Parameter Decomposition .2,3 A Mathematica code of the ADM,12 for general order reactions in planar or spherical catalyst pellets, is given in more detail in the Appendix. Thus, the algebraic expressions of the approximate solutions and the computed data of results can all be easily obtained. 

Figure 6 presents the effectiveness factors of a spherical catalyst pellet for reaction orders of 1.0, 2.0 ( Figure 6a) and 0.5 (up curve in Figure 6b). A reasonable agreement between the four-term decomposition solutions and the finite difference method is achieved over most of the range of Thiele modulus.

The following results refer to a bed 0.91 m deep containing spherical catalyst pellets of diameter 1.52 mm, with porosity 0.4 due to pores of diameter 75 A and tortuosity factor 3.5. 

The well known Thiele modulus of the reaction. This is defined as the ratio of the intrinsic chemical rate, calculated at bulk fluid phase conditions, to the maximum rate of effective diffusion at the external pellet surface. For spherical catalyst pellets, the Thiele modulus is given by... 

Equation 51 has been derived for a spherical catalyst pellet. The corresponding solution for the cylinder is... 

Equation 56 can be used only for spherical catalyst pellets and first order, irreversible reactions. However, for convenience, and in analogy to the Thiele modulus, a generalized modulus ij/pn can be defined as well which applies to arbitrary pellet shape and arbitrary reaction order. This is defined as... 

In Fig. 13, typical curves for the effectiveness factor as a function of the Thiele modulus arc given for a first order, irreversible reaction in a spherical catalyst pellet. These curves have been obtained numerically by Weisz and Hicks [110], for the case of negligible intcr-... 

Assuming a spherical catalyst pellet and that interphase heat and mass transfer resistances can be neglected, substitution of eq 51 into eq 123 gives, upon rearrangement,... 

So far, we have obtained an explicit expression for the normalized concentration of reactant Ai inside the pellet. Analogously to eq 42, which relates the effectiveness factor of a spherical catalyst pellet to the nor-... 

A reaction of the order Vi is carried out in a spherical catalyst pellet. For the given surface temperature and concentration the product ksCAjmii equals 0.2 s 1. The effective diffusion coefficient has been determined as 2 x Iff7 m2 s"1. The diameter of the sphere is 6 mm. Fur-... 

Pommersheim and Dixit (7 ) have developed models for poisoning occurring in the pores of flat plate and spherical catalyst pellets. 

Schematic of the shell baltmce on a spherical catalyst pellet.

Figure 6.3.7 illustrates the effect of the Thiele modulus on the concentration profile within a spherical catalyst pellet. 

Figure 6.3.8 illustrates the relationship between the effectiveness factor and the Thiele modulus for a spherical catalyst pellet. 

Assuming a spherical catalyst pellet with radius Rp, the Thiele modulus is ... 

Substituting the necessary terms into the expression for the Thiele modulus yields the radius of the spherical 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... 

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. 

The irreversible, first-order reaction of gaseous A lo B occurs in spherical catalyst pellets with a radius of 2 mm. For this problem, the molecular diffusivity of A is 1.2 X 10" cm s and the Knudsen diffusivity is 9 X 10 " cm s. The intrinsic first-order rate constant determined from detailed laboratory measurements was found to be 5.0 s . The concentration of A in the surrounding gas is 0.01 mol L . Assume the porosity and the tortuosity of the pellets are 0.5 and 4, respectively. 

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