Rectangular symmetry, effectiveness factors

Catalysts with rectangular symmetry (see Table 18-1). When the intrapellet Damkohler number is larger, (a) the molar density of reactant A at the center of the catalyst is smaller, (b) the concentration gradient at the external surface increases, and (c) the effectiveness factor decreases. These trends are universal. 

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. 

As illustrated below, the gradient of the dimensionless reactant molar density profile is a function of the intrapeUet Damkohler number, so the effectiveness factor is only a function of A and geometry. Numerical values of a are 1, 2, or 3 for catalysts with rectangular, cyhndrical, or spherical symmetry, respectively. 

The dimensionless molar density profile of reactant A is symmetric with respect to T] about the symmetry plane (i.e., z = 0, r) = 0). Consequently, it is only necessary to integrate equation (20-37) from the symmetry plane at the center of the wafer to the external surface, and multiply by 2. The final expression for the effectiveness factor in rectangular coordinates is... 

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). 

Figure 20-3 Dimensionless correlations between the effectiveness factor and the intrapellet Damkohler number for one-dimensional diffusion and nth-order irreversible chemical kinetics in porous catalysts with rectangular symmetry (i.e., n = 0, 1, 2). The quantity on the horizontal axis is A, not A. One-half of the thickness of these porous wafer-like catalysts is the characteristic length in the definition of A.

However, the void area fraction is equivalent to the void volume fraction, based on equation (21-76) and the definition of intrapellet porosity Sp at the bottom of p. 555. Effectiveness factor calculations in catalytic pellets require an analysis of one-dimensional pseudo-homogeneous diffusion and chemical reaction in a coordinate system that exploits the symmetry of the macroscopic boundary of a single pellet. For catalysts with rectangular symmetry as described above, one needs an expression for the average diffusional flux of reactants in the thinnest dimension, which corresponds to the x direction. Hence, the quantity of interest at the local level of description is which represents the local... 

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