Intrapellet Damkohler number

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,... 

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... 

The dimensionless spatial coordinate rj is measured in the thinnest dimension of rectangular catalysts. For cylindrical and spherical catalysts, r] is measured in the radial direction. The characteristic length L which appears in the intrapellet Damkohler number and is required to make the spatial coordinate dimensionless (i.e., rj = spatial coordinate/L) is one-half the thickness of catalysts with rectangular symmetry, measured in the thinnest dimension the radius of long cylindrical catalysts or the radius of spherical catalysts. q A is the molar density of reactant A divided by its value in the vicinity of the external surface of the catalyst, CAsurf- Hence, by definition, q A = 1 at r = 1. 

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,... 

In other words, reactants exist everywhere within the pores of the catalyst when the chemical reaction rate is slow enough relative to intrapellet diffusion, and the intrapellet Damkohler number is less than, or equal to, its critical value. These conditions lead to an effectiveness factor of unity for zerofli-order kinetics. When the intrapellet Damkohler number is greater than Acnticai, the central core of the catalyst is reactant starved because criticai is between 0 and 1, and the effectiveness factor decreases below unity because only the outer shell of the pellet is used to convert reactants to products. In fact, the dimensionless correlation between the effectiveness factor and the intrapeUet Damkohler number for zeroth-order kinetics exhibits an abrupt change in slope when A = Acriticai- Critical spatial coordinates and critical intrapeUet Damkohler numbers are not required to analyze homogeneous diffusion and chemical reaction problems in catalytic pellets when the reaction order is different from zeroth-order. When the molar density appears explicitly in the rate law for nth-order chemical kinetics (i.e., n > 0), the rate of reaction antomaticaUy becomes extremely small when the reactants vanish. Furthermore, the dimensionless correlation between the effectiveness factor and the intrapeUet Damkohler nnmber does not exhibit an abrupt change in slope when the rate of reaction is different from zeroth-order. 

Critical Dimensionless Spatial Coordinates and Intrapellet Damkohler Numbers... 

Notice that the molar density of reactant A does not decrease to zero at the center of the catalyst, where t] = 0, when the intrapellet Damkohler number is below its critical value. In fact. 

It is only necessary to solve this equation for values of tjcnticai between 0 and 1. Obviously, ciiticai is a function of the intrapellet Damkohler number, but an explicit analytical function for ( critical = /(A) is not possible. If A or A is incremented from its critical value to extremely large values in the diffusion-limited regime, then a Newton-Raphson root-finding approach can be implemented to find the reahstic root for ( critical at each value of the intrapellet Damkohler number (see Table 16-1). 

TABLE 16-1 Effect of the Intrapellet Damkohler Number on the Critical Dimensionless Radius for Radial Diffusion and Pseudo-Homogeneous Zeroth-Order Chemical Kinetics in Porous Catalysts with Cylindrical Symmetry... 

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 ... 

When the intrapellet Damkohler number is less than its critical value (i.e., /6), the critical dimensionless spatial coordinate Jjcnticai is negative, and boundary condition 2b must be employed instead of 2a. Under these conditions, the dimensionless molar density profile for reactant A within the catalytic pores is adopted from equation (16-24) by setting //criucai to zero. Hence,... 

REDEFINING THE INTRAPELLET DAMKOHLER NUMBER SO THAT ITS CRITICAL VALUE MIGHT BE THE SAME FOR ALL PELLET GEOMETRIES... 

This is an interesting challange from the standpoint of developing geometry-insensitive universal correlations for all catalyst shapes. As illustrated above, the critical value of the intrapellet Damkohler number is... 

REDEFINITION OF THE INTRAPELLET DAMKOHLER NUMBER When the characteristic length L is defined as follows ... 

Calculate the intrapellet Damkohler number when ijcnticai = 0, which corresponds to the largest value of A that is consistent with the presence of reactant A throughout the catalyst. This is the definition of the critical intrapellet Damkohler number, Acnticai- At higher values of A, reactant A... 

A summary of the final results follows. When L = Vcataiyst/5 extemab critical values of the intrapellet Damkohler number are as follows ... 

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. 

Figure 16-1 Effect of the intrapellet Damkohler number on dimensionless reactant concentration profiles for one-dimensional diffusion and pseudo-homogeneous zeroth-order chemical kinetics in porous catalyst with rectangular symmetry.

It is not necessary to introduce a critical spatial coordinate because the rate of disappearance of reactant A is extremely small when its molar density approaches zero in the central core of the catalyst at large values of the intrapellet Damkohler number. One-dimensional diffusion and first-order irreversible chemical reaction in rectangular coordinates is described mathematically by a frequently occurring... 

What is the analytical expression for the effectiveness factor vs. the intrapellet Damkohler number that corresponds to one-dimensional diffusion and first-order irreversible chemical reaction in catalytic pellets with cylindrical symmetry The radius of the cylinder is used as the characteristic length in the definition of the intrapellet Damkohler number. 

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... 

Long cylindrical catalysts (see Table 18-1). Effectiveness factors in cylindrical pellets are larger than their counterparts in catalysts with rectangnlar symmetry, at the same value of the intrapellet Damkohler number (see additional comments below on the relative magnirnde of E for catalysts of various geometries). 

At the same value of the intrapellet Damkohler number (i.e., A ), the following trend in effectiveness factors is universal for the three catalyst geometries discussed above ... 

If L = R for spherical pellets, as defined in Section 18-1, then the intrapellet Damkohler number (i.e., A ) is nine-fold larger than Ay g, and the analytical and numerical solutions for radial diffusion and first-order irreversible chemical reaction proceed as follows. Now, the dimensionless independent spatial variable r) ranges from 10 near the center of the catalyst to 1 at the external surface, and the geometric factor a is 3 for spheres. 

DIMENSIONLESS CORRELATION FOR THE EFFECTIVENESS FACTOR IN TERMS OF THE INTRAPELLET DAMKOHLER NUMBER... 

Effectiveness factor calculations summarized in Tables 19-1 to 19-5 are consistent with Langmuir-Hinshelwood kinetics, as discussed in this chapter. E is larger and approaches 1 asymptotically in the reaction-controlled regime where the intrapellet Damkohler number is small, and E decreases in the diffusion-controlled regime at large values of A a- These trends are verified by simulations provided in Table 19-1. 

Effectiveness factor is given vs. the intrapellet Damkohler number. 

Reactant equilibrium constants Kp and affect the forward kinetic rate constant, and all Ki s affect die adsorption terms in the denominator of the Hougen-Watson rate law via the 0, parameters defined on page 493. However, the forward kinetic rate constant does not appear explicitly in the dimensionless simulations because it is accounted for in Ihe numerator of the Damkohler number, and is chosen independently to initiate the calculations. Hence, simulations performed at larger adsorption/desorption equilibrium constants and the same intrapellet Damkohler number implicitly require that the forward kinetic rate constant must decrease to offset the increase in reactant equilibrium constants. The vacant-site fraction on the internal catalytic surface decreases when adsorption/desorption equilibrium constants increase. The forward rate of reaction for the triple-site reaction-controlled Langmuir-Hinshelwood mechanism described on page 491 is proportional to the third power of the vacant-site fraction. Consequently, larger T, s at lower temperature decrease the rate of reactant consumption and could produce reaction-controlled conditions. This is evident in Table 19-3, because the... 

Effectiveness factor is given vs. the intrapellet Damkohler number for different stoichiometric imbalances between reactants A2 and B, denoted by I b, surf-... 

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