Pellets effective diffusivity

In contrast to the cell experiments of Gibilaro et al., it is now seen from equation (10.45) that measurement of the delay time gives no information about diffusion within the pellets this can be obtained only through equation (10.46) from measurements of the second moment. As in the case of the cell experiment, the results can also be Interpreted in terms of an "effective diffusion coefficient" associated with a Fick equation for the... 

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). 

Physical properties of catalysts also may need to be checked periodically, includiug pellet size, specific surface, porosity, pore size and size distribution, and effective diffusivity. The effectiveness of a porous catalyst is found by measuring conversions with successively smaller pellets until no further change occurs. These topics are touched on by Satterfield (Heterogeneous Cataly.sls in Jndustiial Practice, McGraw-Hill, 1991). 

Diffusivity and tortuosity affect resistance to diffusion caused by collision with other molecules (bulk diffusion) or by collision with the walls of the pore (Knudsen diffusion). Actual diffusivity in common porous catalysts is intermediate between the two types. Measurements and correlations of diffusivities of both types are Known. Diffusion is expressed per unit cross section and unit thickness of the pellet. Diffusion rate through the pellet then depends on the porosity d and a tortuosity faclor 1 that accounts for increased resistance of crooked and varied-diameter pores. Effective diffusion coefficient is D ff = Empirical porosities range from 0.3 to 0.7, tortuosities from 2 to 7. In the absence of other information, Satterfield Heterogeneous Catalysis in Practice, McGraw-HiU, 1991) recommends taking d = 0.5 and T = 4. In this area, clearly, precision is not a feature. 

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 first-order chemical reaction takes place in a reactor in which the catalyst pellets are platelets of thickness 5 mm. The effective diffusivity De for the reactants in the catalyst particle is I0"5 m2/s and the first-order rate constant k is 14.4 s . 

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. 

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. 

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. 

Barrer (19) has developed another widely used nonsteady-state technique for measuring effective diffusivities in porous catalysts. In this approach, an apparatus configuration similar to the steady-state apparatus is used. One side of the pellet is first evacuated and then the increase in the downstream pressure is recorded as a function of time, the upstream pressure being held constant. The pressure drop across the pellet during the experiment is also held relatively constant. There is a time lag before a steady-state flux develops, and effective diffusion coefficients can be determined from either the transient or steady-state data. For the transient analysis, one must allow for accumulation or depletion of material by adsorption if this occurs. 

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. 

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... 

ILLUSTRATION 12.3 DETERMINATION OF CATALYST EFFECTIVENESS FACTOR FOR THE CUMENE CRACKING REACTION USING THE EFFECTIVE DIFFUSIVITY APPROACH Use the effective diffusivity approach to evaluate the effectiveness factor for the silica-alumina catalyst pellets considered in Illustration 12.2. 

The physical properties of the catalyst (specific surface area, porosity, effective thermal conductivity, effective diffusivity, pellet density, etc.). 

It may be assumed that the accumulation of hydrogen within the pellet is negligible and that it may be treated as being in a quasi-steady-state condition. The finite difference form of Fick s first law may be used to determine the flow rate of hydrogen through the pellet. The diffusion constant appearing in this equation may be considered as an effective Knudsen diffusion coefficient. 

Transport Properties. Because the feed is primarily air and because substantial amounts of N2 and 02 are present in the effluent stream, we will assume that the fluid viscosity is that of air for purposes of pressure drop calculations. For the temperature range of interest, the fluid viscosity may be taken as equal to 320 micropoise. The pressure range of interest does not extend to levels where variations of viscosity with pressure need be considered. The effective diffusivities of naphthalene and phthalic anhydride in the catalyst pellet may be evaluated using the techniques developed in Section 12.2. 

The catalyst activity depends not only on the chemical composition but also on the diffusion properties of the catalyst material and on the size and shape of the catalyst pellets because transport limitations through the gas boundary layer around the pellets and through the porous material reduce the overall reaction rate. The influence of gas film restrictions, which depends on the pellet size and gas velocity, is usually low in sulphuric acid converters. The effective diffusivity in the catalyst depends on the porosity, the pore size distribution, and the tortuosity of the pore system. It may be improved in the design of the carrier by e.g. increasing the porosity or the pore size, but usually such improvements will also lead to a reduction of mechanical strength. The effect of transport restrictions is normally expressed as an effectiveness factor q defined as the ratio between observed reaction rate for a catalyst pellet and the intrinsic reaction rate, i.e. the hypothetical reaction rate if bulk or surface conditions (temperature, pressure, concentrations) prevailed throughout the pellet [11], For particles with the same intrinsic reaction rate and the same pore system, the surface effectiveness factor only depends on an equivalent particle diameter given by... 

The term in the square bracket is an effective diffusion coefficient DAB. In principle, this may be used together with a material balance to predict changes in concentration within a pellet. Algebraic solutions are more easily obtained when the effective diffusivity is constant. The conservation of counter-ions diffusing into a sphere may be expressed in terms of resin phase concentration Csr, which is a function of radius and time. 

Wheeler s treatment of the intraparticle diffusion problem invokes reaction in single pores and may be applied to relatively simple porous structures (such as a straight non-intersecting cylindrical pore model) with moderate success. An alternative approach is to assume that the porous structure is characterised by means of the effective diffusivity. (referred to in Sect. 2.1) which can be measured for a given gaseous component. In order to develop the principles relating to the effects of diffusion on reaction selectivity, selectivity in isothermal catalyst pellets will be discussed. 

The pellets of the commercial catalyst were crushed to grain size from 0.5 to 1 mm. A calculation on the basis of the measurements of the effective diffusion coefficient showed that the reaction proceeded in the kinetic region. Bed density of the catalyst was 1.23 g/cm3, specific surface after kinetic experiments was 36 m2/g. In the temperature range of 150-225°C reaction (342) is practically irreversible. The experiments proved (348) to be valid thus, the kinetics on low- and high-temperature catalysts is the same. 

Both Knudsen and molecular diffusion can be described adequately for homogeneous media. However, a porous mass of solid usually contains pores of non-uniform cross-section which pursue a very tortuous path through the particle and which may intersect with many other pores. Thus the flux predicted by an equation for normal bulk diffusion (or for Knudsen diffusion) should be multiplied by a geometric factor which takes into account the tortuosity and the fact that the flow will be impeded by that fraction of the total pellet volume which is solid. It is therefore expedient to define an effective diffusivity De in such a way that the flux of material may be thought of as flowing through an equivalent homogeneous medium. We may then write ... 

Thiele(I4>, who predicted how in-pore diffusion would influence chemical reaction rates, employed a geometric model with isotropic properties. Both the effective diffusivity and the effective thermal conductivity are independent of position for such a model. Although idealised geometric shapes are used to depict the situation within a particle such models, as we shall see later, are quite good approximations to practical catalyst pellets. 

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... 

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