The Spherical Catalyst Particle

Take the pellet to be a sphere with r the distance from the center and R the outside radius. If c(r) is the concentration as a function of r (this assumes symmetry), the usual steady state balance gives 

We also want to scale the reaction rate, although we cannot guarantee that it will be less than 1 if auto-catalysis is present. 

A second-order equation demands two boundary conditions, and symmetry supplies the other, namely  

Clearly there are two parameters, the first of which, written as a square in an intelligent anticipation of square roots to come, is the Thiele modulus. It measures the intensity of reaction in terms of the potential rate of diffusion, for it may be written 

In this expression, 3 is a purely geometric factor generated from the ratio of the surface area to the volume of the sphere the first bracket is the maximum total rate of reaction, achieved when the concentration is everywhere at cf the second bracket is the total diffusive flux across the surface when the 

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The effect of transport limitations can conveniently be evaluated by considering the spherical catalyst particle shown in Fig. 5.32. We will introduce a dimensionless quantity called the Thiele diffusion modulus (Og) [W. Thiele Ind. Eng. Chem. 31... 

Fig. 31. Horizontal and vertical planes through the third-stacked WS, showing the temperature fields in the fluid and through the spherical catalyst particles, with 5% activity.

The radial distribution of the reactant concentrations in the spherical catalyst particle is theoretically given as ... 

The mole balance inside the spherical catalyst particle was described by equation 4, in which the catalyst activity, a, was based on equations 5 and 6 ... 

It is convenient to define the Thiele modulus for the spherical catalyst particle, Os, with radius of the particle R as the size parameter ... 

The internal surface area of the spherical catalyst particles is much higher than the external surface area A (m kg ), and the reaction at the external surface is negligible. 

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. 

We start with an ideal, porous, spherical catalyst particle of radius R. The catalyst is isothermal and we consider a reaction involving a single reactant. Diffusion is described macroscopically by the first and second laws of Pick, stating that... 

This study was carried out to simulate the 3D temperature field in and around the large steam reforming catalyst particles at the wall of a reformer tube, under various conditions (Dixon et al., 2003). We wanted to use this study with spherical catalyst particles to find an approach to incorporate thermal effects into the pellets, within reasonable constraints of computational effort and realism. This was our first look at the problem of bringing together CFD and heterogeneously catalyzed reactions. To have included species transport in the particles would have required a 3D diffusion-reaction model for each particle to be included in the flow simulation. The computational burden of this approach would have been very large. For the purposes of this first study, therefore, species transport was not incorporated in the model, and diffusion and mass transfer limitations were not directly represented. 

Derive an expression for the catalyst effectiveness factor (17) for a spherical catalyst particle of... 

The concentration and temperature Tg will, for example, be conditions of reactant concentration and temperature in the bulk gas at some point within a catalytic reactor. Because both c g and Tg will vary with position in a reactor in which there is significant conversion, eqns. (1) and (15) have to be coupled with equations describing the reactor environment (see Sect. 6) for the purpose of commerical reactor design. Because of the nonlinearity of the equations, the problem can only be solved in this form by numerical techniques [5, 6]. However, an approximation may be made which gives an asymptotically exact solution [7] or, alternatively, the exponential function of temperature may be expanded to give equations which can be solved analytically [8, 9]. A convenient solution to the problem may be presented in the form of families of curves for the effectiveness factor as a function of the Thiele modulus. Figure 3 shows these curves for the case of a first-order irreversible reaction occurring in spherical catalyst particles. Two additional independent dimensionless paramters are introduced into the problem and these are defined as... 

It is possible that the pores of wetted catalyst particles eire filled with liquid. Hence, by virtue of the low values of liquid diffusivities (ca. 10 cm s" ), the effectiveness factor will almost certainly be less than unity. A criterion for assessing the importance of mass transfer in the trickling liquid film has been suggested by Satterfield [40] who argued that if liquid film mass transport were important, the rate of reaction could be equated to the rate of mass transfer across the liquid film. For a spherical catalyst particle with diameter dp, the volume of the enveloping liquid fim is 7rdp /6 and the corresponding interfacial area for mass transfer is TTdn. Hence... 

Find the effectiveness factor for a spherical catalyst particle with first-order kinetics. 

A first-order irreversible catalytic reaction r" = k"C/ ) occurs in a slurry reactor (or fluidized bed reactor). Spherical catalyst particles have diameter R, density surface area/mass Sg, and a fraction g of the reactor is occupied by catalyst. The average pore diameter is d. Find an expression for r C ) in terms of these quantities. 

Figure 12-15 Sketch of concentration profiles between a spherical bubble and a solid spherical catalyst particle in a continuous liquid phase (upper) in a gas-liquid sluny reactor or between a bubble and a planar solid wall (lower) in a catalytic w bubble reactor, It is assmned that a reactant A must migrate from the bubble, tirough the drop, md to tiie solid catdyst smface to react. Concentration variations may occur because of mass transfer limitations around both bubble and solid phases.

At one extreme diffusivity may be so low that chemical reaction takes place only at suface active sites. In that case p is equal to the fraction of active sites on the surface of the catalyst. Such a polymer-supported phase transfer catalyst would have extremely low activity. At the other extreme when diffusion is much faster than chemical reaction p = 1. In that case the observed reaction rate equals the intrinsic reaction rate. Between the extremes a combination of intraparticle diffusion rates and intrinsic rates controls the observed reaction rates as shown in Fig. 2, which profiles the reactant concentration as a function of distance from the center of a spherical catalyst particle located at the right axis, When both diffusion and intrinsic reactivity control overall reaction rates, there is a gradient of reactant concentration from CAS at the surface, to a lower concentration at the center of the particle. The reactant is consumed as it diffuses into the particle. With diffusional limitations the active sites nearest the surface have the highest turnover numbers. The overall process of simultaneous diffusion and chemical reaction in a spherical particle has been described mathematically for the cases of ion exchange catalysis,63 65) and catalysis by enzymes immobilized in gels 66-67). Many experimental parameters influence the balance between intraparticle diffusional and intrinsic reactivity control of reaction rates with polymer-supported phase transfer catalysts, as shown in Fig. 1. 

The Thiele modulus tfi for the case of a spherical catalyst particle of radius R (cm), in which a first-order catalytic reaction occurs at every point within the particles, is given as... 

In Figure 7.4 the effectiveness factor is plotted against the Thiele modulus for spherical catalyst particles. For low values of 0, Ef is almost equal to unity, with reactant transfer within the catalyst particles having little effect on the apparent reaction rate. On the other hand, Ef decreases in inverse proportion to 0 for higher values of 0, with reactant diffusion rates limiting the apparent reaction rate. Thus, decreases with increasing reaction rates and the radius of catalyst spheres, and with decreasing effective diffusion coefficients of reactants within the catalyst spheres. 

A reactant in liquid will be converted to a product by an irreversible first-order reaction using spherical catalyst particles that are 0.4cm in diameter. The first-order reaction rate constant and the effective diffusion coefficient of the reactant in catalyst particles are 0.001 s and 1.2 X 10 ( ii s , respectively. The liquid film mass transfer resistance of the particles can be neglected. 

Communications on the theory of diffusion and reaction-VI The effectiveness of spherical catalyst particles in steep external gradients (with I. Copelowitz). Chem. Eng. Sci. 25,885-896 (1970). 

Equation (23) has been solved numerically for spherical catalyst particles with an IBM electronic computer for values of GKm = 10, 1.0, 0.3, 0.10. In this case

rate constant per unit volume. The curves obtained are plotted in Fig. 16. The various curves have different shapes. This becomes more apparent... 

The reactor is of the heat-exchanger type with catalyst and tubes and rising steam outside. In this project we consider spherical catalyst particles of 5 mm diameter and a bed void fraction of 45%, which offers a good trade-off between efficiency and lower pressure drop. The gas inlet pressure is 10 bar. We aim at a pressure drop less than 15% of the operating pressure, namely a maximum of 1.5 bar. 

By use of this technique, it is possible to prepare fine spherical catalyst particles in the 10-100/rm diameter range, as arc required for typical fluidized-bed catalytic processes. In this technique used for large-scale catalyst manufacture, the feed is generally dilute hydrogel or sol that is sprayed from the top of a tower while hot air is blown in a cocurrcnt or countercurrent direction to dry the droplets before they reach the bottom of the tower. The fine droplets arc produced or atomized by pumping the hydrogel or sol under pressure cither... 

In order to determine the relative importance of mass-transport processes in gas-liquid-solid-reactions, it is recommended to measure the global reaction rate as function of catalyst concentration w (keeping all other operating variables constant). With the assumption of spherical catalyst particles, the specific surface of the catalyst can be calculated as... 

The similar, older slurry process uses a less active catalyst. The monomer is dissolved in isooctane, the titanium catalyst and aluminium cocatalyst are added and this mixture is fed to the reactor which is maintained at 70°C. The inorganic corrosive (Cl) residues are removed in a washing step with alcohols. The atactic material is removed by extraction. A third process employs propene as the liquid in combination with a high activity catalyst. The Himont Spheripol process, which uses spherical catalyst particles, gives spherical polymer beads of millimetre size that need no extrusion for certain purposes. A more recent development is the gas-phase polymerization using an agitated bed. All processes are continuous processes, where the product is continuously removed from the reactor. Over the years we have seen a reduction of the number of process steps. The process costs are very low nowadays, propene feed costs amounting to more than 60% of the total cost. 

Effect of Thiele modulus on the normalized concentration profiles in a spherical catalyst particle with first-order reaction. The external surface of the particle is located at /Rp = 1. 

The influence of mass transfer through the film surrounding a spherical catalyst particle can also be examined with a similar expression. Satisfaction of the following inequality demonstrates that interphase mass transfer is not significantly affecting the measured rate ... 

For the TBR, spherical catalyst particles of uniform size with the catalytically active material either uniformly distributed throughout the catalyst or present in a shell were considered. For the MR, channels of square cross section were assumed to have walls covered by the washcoat distributed in such a way that the comers are approximated by the circle-in-square geometry, while the sides are approximated by a planar slab geometry. The volumetric load of catalytic material was a function of the washcoat thickness... 

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