Mass transfer intrapellet

The effect of intrapellet mass transfer is to reduce the rate below what it would be if there were no internal-concentration gradient. The effect of the temperature gradient is to increase the rate for an exothermic reaction. This is because intrapellet temperatures will be greater than surface values. For endothermic reactions temperature and concentration gradients both reduce the rate below that evaluated at outer-surface conditions. 

Surface migration is pertinent to a study of intrapellet mass transfer if its contribution is significant with respect to diffusion in the pore space. When multimolecular-layer adsorption occurs, surface diffusion has been explained as a flow of the outer layers as a condensed phase. However, surface transport of interest in relation to reaction occurs in the monomolecular layer. It is more appropriate to consider, as proposed by deBoer," that such transport is an activated process, dependent on surface characteristics as well as those of the adsorbed molecules. Imagine that a molecule in the gas phase strikes the pore wall and is adsorbed. Then two alternatives are possible desorption into the gas (Knudsen diffusion) or movement to an adjacent active site on the pore wall (surface diffusion). If desorption occurs,... 

The Effect of Intrapellet Mass Transfer on Observed Rate... 

In Sec. 10-1 we saw that neglecting external resistances could lead to misleading conclusions about reaction order and activation energy. Similar errors may occur when intrapellet mass transfer is neglected. Consider the situation where <1> > 5. In this region intrapellet transport has a strong effecion the rate Fig. 11-7 shows, that 77 is less than about 0.2. From Eqs. (11-54) and (11-50) for a first-order reaction... 

SECTION 11-9 THE EFFECT OF INTRAPELLET MASS TRANSFER ON OBSERVED RATE... 

The intrapellet mass transfer resistance for component i is directly proportional to the molar density difference on the left side of (30-8). It should be emphasized that the stoichiometric relations given by (30-5) and (30-8) are applicable throughout the entire volume of the pellet when one chemical reaction governs the rate of conversion of reactants to products. 

In a porous catalyst, as reactants diffuse in the radial direction toward the center of the particle, reaction occurs on the pore walls, releasing or absorbing heat as required by the reaction. The interior surface of a porous catalyst is not as effective as its exterior because each point on the inner surface is exposed to a lower reactant concentration than that of the exterior (Cas). The net effect of intrapellet mass transfer resistance is to reduce the global rate beneath the rate evaluated at surface conditions. The net effect of intrapellet heat transfer resistance, on the other hand, depends on the exothermicity-endothermicity of the surface reaction and on the relative significance of... 

FIGURE 7.24 Mass transfer in adsorption processes (a) fixed beds and (b) intrapellet mass transfer. (Reproduced from C. Tien. Butterworth-Heinemann Series in Chemical Engineering. Butterworth-Heinemann, Boston, 1994. With permission.)... 

Intramolecular chain transfer, 20 220 Intramolecular cycloacylations, 72 177 Intramolecular self assembly, 20 482 Intramolecular stretching modes, 74 236 Intraoperative auto transfusion, 3 719 Intraparticle mass transfer, 75 729-730 Intrapellet Damkohler number, 25 294,... 

The starting point of a number of theoretical studies of packed catalytic reactors, where an exothermic reaction is carried out, is an analysis of heat and mass transfer in a single porous catalyst since such system is obviously more conductive to reasonable, analytical or numerical treatment. As can be expected the mutual interaction of transport effects and chemical kinetics may give rise to multiple steady states and oscillatory behavior as well. Research on multiplicity in catalysis has been strongly influenced by the classic paper by Weisz and Hicks (5) predicting occurrence of multiple steady states caused by intrapellet heat and mass intrusions alone. The literature abounds with theoretical analysis of various aspects of this phenomenon however, there is a dearth of reported experiments in this area. Later the possiblity of oscillatory activity has been reported (6). 

The effects of diffusional restrictions on the activity and selectivity of FT synthesis processes have been widely studied (32,52,56-60). Intrapellet diffusion limitations are common in packed-bed reactors because heat transfer and pressure-drop considerations require the use of relatively large particles. Bubble columns typically use much smaller pellets, and FT synthesis rates and selectivity are more likely to be influenced by the rate of mass transfer across the gas-liquid interface as a gas bubble traverses the reactor (59,61,62). 

The effective thermal conductivities of catalyst pellets are surprisingly low. Therefore significant intrapellet temperature gradients can exist, and the global rate may be influenced by thermal effects. The effective conductivity is the energy transferred per unit of total area of pellet (perpendicular to the direction of heat transfer). The defining equation, analogous to Eq. (11-18) for mass transfer, may be written... 

For an endothermic reaction there is a decrease in temperature and rate into the pellet. Hence 17 is always less than unity. Since the rate decreases with drop in temperature, the effect of heat-transfer resistance is diminished. Therefore the curves for various are closer together for the endothermic case. In fact, the decrease in rate going into the pellet for endothermic reactions means that mass transfer is of little importance. It has been shown that in many endothermic cases it is satisfactory to use a thermal effectiveness factor. Such thermal 17 neglects intrapellet mass transport that is, ri is obtained by solution of Eq. (11-72), taking C = Q. 

A limiting case of intrapellet transport resistances is that of the thermal effectiveness factor In this situation of zero mass-transfer resistance, the resistance to intrapellet heat transfer alone establishes the effectiveness of the pellet. Assume that the temperature effect on the rate can be represented by the Arrhenius function, so that the rate at any location is given by r = A... 

Most important, heterogeneous surface-catalyzed chemical reaction rates are written in pseudo-homogeneous (i.e., volumetric) form and they are included in the mass transfer equation instead of the boundary conditions. Details of the porosity and tortuosity of a catalytic pellet are included in the effective diffusion coefficient used to calculate the intrapellet Damkohler number. The parameters (i.e., internal surface area per unit mass of catalyst) and Papp (i.e., apparent pellet density, which includes the internal void volume), whose product has units of inverse length, allow one to express the kinetic rate laws in pseudo-volumetric form, as required by the mass transfer equation. Hence, the mass balance for homogeneous diffusion and multiple pseudo-volumetric chemical reactions in one catalytic pellet is... 

This second-order ordinary differential equation given by (16-4), which represents the mass balance for one-dimensional diffusion and chemical reaction, is very simple to integrate. The reactant molar density is a quadratic function of the spatial coordinate rj. Conceptual difficulty arises for zeroth-order kinetics because it is necessary to introduce a critical dimensionless spatial coordinate, ilcriticai. which has the following physically realistic definition. When jcriticai which is a function of the intrapellet Damkohler number, takes on values between 0 and 1, regions within the central core of the catalyst are inaccessible to reactants because the rate of chemical reaction is much faster than the rate of intrapellet diffusion. The thickness of the dimensionless mass transfer boundary layer for reactant A, measured inward from the external surface of the catalyst,... 

Table 16-2 illustrates the functional dependence of ]criticai on the intrapellet Damkohler number, A. Notice that the numerical results for ]criticai = /(A) are identical for spheres and cylinders when A > 15. For all catalyst shapes, licriticai 1 in the diffusion-limited regime when A oo, and the mass transfer boundary layer thickness measured inward from the external surface of the catalyst becomes infinitesimally small. If equation (16-25), which defines / critical, is solved for A instead of ]criticab then ... 

Hence, it is not possible to redefine the characteristic length such that the critical value of the intrapellet Damkohler number is the same for all catalyst geometries when the kinetics can be described by a zeroth-order rate law. However, if the characteristic length scale is chosen to be V cataiyst/ extemai, then the effectiveness factor is approximately A for any catalyst shape and rate law in the diffusion-limited regime (A oo). This claim is based on the fact that reactants don t penetrate very deeply into the catalytic pores at large intrapellet Damkohler numbers and the mass transfer/chemical reaction problem is well described by a boundary layer solution in a very thin region near the external surface. Curvature is not important when reactants exist only in a thin shell near T] = I, and consequently, a locally flat description of the problem is appropriate for any geometry. These comments apply equally well to other types of kinetic rate laws. 

Hence, at the center of spherical catalytic pellets, the first term on the right side of the mass transfer equation with diffusion and chemical reaction depends on the intrapellet Damkohler number and adopts a value between zero and — when the... 

In this problem, we explore the dimensionless mass transfer correlation between the effectiveness factor and the intrapellet Damkohler number for one-dimensional diffusion and Langmuir-Hinshelwood surface-catalyzed chemical reactions within the internal pores of flat-slab catalysts under isothermal conditions. Perform simulations for vs. A which correspond to the following chemical reaction that occurs within the internal pores of catalysts that have rectangular symmetry. 

Problem. Consider zeroth-order chemical kinetics in pellets with rectangular, cylindrical and spherical symmetry. Dimensionless molar density profiles have been developed in Chapter 16 for each catalyst geometry. Calculate the effectiveness factor when the intrapellet Damkohler number is greater than its critical value by invoking mass transfer of reactant A into the pellet across the external surface. Compare your answers with those given by equations (20-50). 

Catalysts with Spherical Symmetry. This analysis is based on the mass transfer equation with diffusion and chemical reaction in spherical catalysts. For zeroth-order kinetics, the molar density of reactant A is equated to zero at the critical value of the dimensionless radial coordinate, iciiticai = /(A). The relation between the critical value of the dimensionless radial coordinate and the intrapellet Damkohler number is obtained by solving the following nonlinear algebraic equation ... 

Consider one-dimensional diffusion and zeroth-order chemical reaction in a flat-slab porous wafer-type catalyst. The conditions are approximately isothermal and the inirapellet Damkohler number of reactant A is Aa. intrapellet = VS. The mass transfer equation is solved numerically, not analytically. 

What is the defining expression for the isothermal effectiveness factor in spherical catalysts Reactant A is consumed by three independent first-order irreversible chemical reactions on the interior catalytic surface. Your final expression should be based on mass transfer via diffusion and include the reactant concentration gradient at the external surface of the catalyst, where t) = 1. Define the intrapellet Damkohler number in your final answer. 

The effectiveness factor E is evaluated for the appropriate kinetic rate law and catalyst geometry at the corresponding value of the intrapellet Damkohler number of reactant A. When the resistance to mass transfer within the boundary layer external to the catalytic pellet is very small relative to intrapellet resistances, the dimensionless molar density of component i near the external surface of the catalyst (4, surface) IS Very similar to the dimensionless molar density of component i in the bulk gas stream that moves through the reactor ( I, ). Under these conditions, the kinetic rate law is evaluated at bulk gas-phase molar densities, 4, . This is convenient because the convective mass transfer term on the left side of the plug-flow differential design equation d p /di ) is based on the bulk gas-phase molar density of reactant A. The one-dimensional mass transfer equation which includes the effectiveness factor. 

When the intrapellet Damkohler number for reactant A is large enough and the catalyst operates in the diffusion-limited regime, the effectiveness factor is inversely proportional to the Damkohler number (i.e., Aa, intrapeiiet)- Under these conditions, together with a large mass transfer Peclet number which minimizes effects due to interpellet axial dispersion, the following scaling law is valid ... 

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