Transfer in a Spherical Pellet

Consider mass transfer in a spherical pellet.[l] The governing equation in dimensionless form is 

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Effective Thermal Conductivities of Porous Catalysts. The effective thermal conductivity of a porous catalyst plays a key role in determining whether or not appreciable temperature gradients will exist within a given catalyst pellet. By the term effective thermal conductivity , we imply that it is a parameter characteristic of the porous solid structure that is based on the gross geometric area of the pellet perpendicular to the direction of heat transfer. For example, if one considers the radial heat flux in a spherical pellet one can say that... 

The effectiveness factor is obtained by the resolution of the mass transfer equations [12] in a spherical pellet with simple assumptions ... 

The external temperature difference can be related to the reaction rate and the heat generated in a pellet using the appropriate correlation to predict the heat transfer coefficient for the gas film. At steady state, the rate of heat generation is equal to the rate of heat removal. For a first-order reaction in a spherical pellet, the heat balance is... 

The above results are general for mass transfer from a spherical pellet and do not presuppose a specific distribution of solute within the pellet. The consequences of the shrinking core model are now examined (equations 7.62-7.68). According to this model the solute-rich phase (liquid oil in the present example) is concentrated within a core, the radius (re) of which gradually diminishes with time. 

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. 

Problem 4-3. Reaction-Diffusion in a Spherical Catalyst Pellet. Consider a spherical catalyst pellet. Assume that transport of product is by diffusion, with diffusivity Deff. As in the problem discussed in Section F, we assume that the transport of reactant within the pellet is decoupled from the transport of product. Mass transfer in the gas is sufficiently rapid so that the reactant concentration at the catalyst surface is maintained at c,x,. 

With no resistance to mass transfer, the concentration is Cm throughout the whole spherical pellet, and the reaction rate, which must be equal to the mass transfer rate in a steady-state process, is ... 

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. 

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. 

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 ... 

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. 

To begin our discussion on the diffusion of reactants from the bulk fluid to the external smface of a catalyst, we shall focus attention on the flow past a single catalyst pellet. Reaction takes place only on the catalyst and not in the fluid surroimding it. The fluid velocity in the vicinity of the spherical pellet will vaiy with position aroimd the sphere. The hydrodynamic boundary layer is usually defined as the distance from a solid object to where the fluid velocity is 99% of the bulk velocity U. Similarly, the mass transfer boundary layer thickness, 8, is defined as the distance from a solid object to where the concentration of the diffusing species reaches 99% of the bulk concentration. 

Figure 12-5 (a) Effectiveness factor plot for nth-order kinetics spherical catalyst particles (from Mass Transfer in Heterogeneous Catalysis, hy C. N. Satterfield, 1970 reprint edition Robert E. Krieger Publishing Co., 1981 reprinted by permission of the author), (b) First-order reaction in different pellet geometries (from R. Aris, Introduction to the Analysis of Chemical Reactors, 1965, p. 131 reprinted by permission of Prentice-Hall, Englewood Cliffs, NJ)... 

Intraparticle Diffusion and External Mass-Transfer Resistance For typical industrial conditions, external mass transfer is important only if there is substantial intraparticle diffusion resistance. This subject has been discussed by Luss, Diffusion-Reaction Interactions in Catalyst Pellets, in Carberry and Varma (eds.), Chemical Reaction and Reactor Engineering, Dekker, 1987. This, however, may not be the case for laboratory conditions, and care must be exerted in including the proper data interpretation. For instance, for a spherical particle with both external and internal mass-transfer limitations and first-order reaction, an overall effectiveness factor r, can be derived, indicating the series-of-resistances nature of external mass transfer followed by intraparticle diffusion-reaction ... 

This is a different kind of heterogeneous reaction—a gas-solid noncatalytic one. Let us examine the process at initial conditions (t 0), so that there has been no opportunity for a layer of UF4.(5) to be formed around the UO pellet The process is much like that for gas-solid catalytic reactions. Hydrogen fluoride gas is transferred from the bulk gas to the surface of the UO2 pellets and reacts at the pellet-gas interface, and H2O diffuses out into the bulk gas. If the pellet is nonporous, all the reaction occurs at the outer surface of the UO2 pellet, and only an external transport process is possible. Costa studied this system by suspending spherical pellets 2 cm in diameter in a stirred-tank reactor. In one run, at a bulk-gas temperature of 377°C, the surface temperature was 462°C and the observed rate was — Tuo = 6.9 x 10 mole U02/(sec) (cm reaction surface). At these conditions the concentrations of... 

This type of plot has been used widely to correlate experimental mass-transfer data. De Acetis and Thodos have summarized the data available up to 1960 in a single rve of vs Reynolds number, as shown in Fig. 10-2. For spherical pellets /)sis the diameter for other shapes dp can be taken as the diameter of a sphere with the same external area. Other investigations and correlations of mass-transfer datg include those of Carberry, Yeh, Bradshaw and Bennett, and Thoenes and Kramers. ... 

The 4-step UT-3 tbermochemical cycle with bromine-calcium-iron developed at Tokyo University is considered in Japan to be superior to many other thermochemical cycles. It has been successfully transferred into a bench-scale continuous model plant MASCOT. The system (Fig. 4-9) consists of four reactor furnaces containing the solid reactants CaBr2, CaO, Fe203, and FeBr2, respectively, which are manufactured as spherical pellets. Only gases are passed through the reactors which eases material flow control. Eleven cycles have been completed with a yield of 0.2 - 0.3 1 of hydrogen per cycle [59, 90]. 

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