Diffusion and reaction in a slab

To illustrate the salient features of the method, we first consider the following boundary value problem in an abstract form, and then later attempt an elementary example of diffusion and reaction in a slab of catalyst material. We shall assume there exists an operator of the type discussed in Chapter 2 (Section 2.5) so that in compact form, we can write... 

We illuminate these attractive features by considering the difficult problems of diffusion and reaction in a slab catalyst sustaining highly nonlinear Hinshelwood kinetics. The mass balance equations written in nondimensional form are taken to be... 

Internal Pore Diffusion and Reaction in a Slab-Shaped Catalyst Pellet... 

Figure 7.3 Diffusion and reaction in a semi-infinite slab of porous catalyst.

Porous catalysts are used in a variety of shapes, including spheres, cylinders, rings, irregular particles, and thin coatings on tubes or flat surfaces. Diffusion plus reaction in a flat slab is a simple case to analyze, since the area for diffusion does not change with distance from the external surface. The equation is similar to that for diffusion and reaction in a straight... 

Note that Eq. (7.25) has the same form as Eq. (4.75) for pore diffusion and reaction in a flat slab, but the boundary conditions are different, since the gradient for A is not zero at the edge of the liquid film. 

Figure 2.24 (a,b) Diffusion and reaction in a semi-infinite flat slab. (Adapted from Ref. [15], Figure 4.11 Copyright 2012, Wiley-VCFI GmbFI Co. KGaA.)...

Cardoso SSS, Rodrigues AE. Diffusion and reaction in a porous catalyst slab Perturbation solutions. AIChE Journal 2006 52 3924-3932. 

While we have contact with this problem, however, we will look at the asymptotics of reactions in a slab of catalyst. In Chapter 3 we used the special form of the equations for diffusion and reaction to get rather general expressions for the Thiele modulus and effectiveness in terms of the center concentration v... 

Communications on the theory of diffusion and reaction-I A complete parametric study of the first-order, irreversible exothermic reaction in a flat slab of catalyst (with D.W. Drott). Chem. Eng. ScL 24, 541-551 (1969). 

Some of the first considerations of the problem of diffusion and reaction in porous catalysts were reported independently by Thiele [E.W. Thiele, Ind. Eng. Chem., 31, 916 (1939)] Damkohler [G. Damkohler, Der Chemie-Ingenieur, 3, 430 (1937)] and Zeldovich [Ya.B. Zeldovich, Acta Phys.-Chim. USSR, 10, 583 (1939)] although the first solution to the mathematical problem was given by Jiittner in 1909 [F. Jiittner, Z. Phys. Chem., 65, 595 (1909)]. Consider the porous catalyst in the form of a flat slab of semi-infinite dimension on the surface, and of half-thickness W as shown in Figure 7.3. The first-order, irreversible reaction A B is catalyzed within the porous matrix with an intrinsic rate (—r). We assume that the mass-transport process is in one direction though the porous structure and may be represented by a normal diffusion-type expression, that there is no net eonveetive transport eontribution, and that the medium is isotropic. For this case, a steady-state mass balance over the differential volume element dz (for unit surface area) (Figure 7.3), yields... 

We illustrate the above five variations of weighted residuals with the following example of diffusion and first order chemical reaction in a slab catalyst (Fig. 8.1). We choose the first order reaction here to illustrate the five methods of weighted residual. In principle, these techniques can apply equally well to nonlinear problems, however, with the exception of the collocation method, the integration of the form (8.7) may need to be done numerically. 

In laminar flow stirred tanks, the packet diffusion model is replaced by a slab-diffusion model. The diffusion and reaction calculations are similar to those for the turbulent flow case. Again, the conclusion is that perfect mixing is almost always a good approximation. 

For first order reaction in a porous slab this problem is solved in P7.03.16. Three dimensionless groups are involved in the representation of behavior when both external and internal diffusion are present, namely, the Thiele number, a Damkohler nunmber and a Biot number. Problem P7.03.16 also relates r)t to the common effectiveness based on the surface concentration,... 

Rajagopalan and Luss (1979) developed a theoretical model to predict the influence of pore properties on the demetallation activity and on the deactivation behavior. In this model the change in restricted diffusion with decreasing pore size was included. Catalysts with slab and spherical geometry composed of nonintersecting pores with uniform radius but variable pore lengths were assumed. The conservation equation for diffusion and first-order reaction in a single pore of radius rp is... 

When a chemical reaction is fast enough to become complete within the diffusion film, the chemical rate and diffusion rates are coupled differently. Fig. 5.4 shows the basis for derivation of the rate expression for a reaction in a two-phase organic reactant/water system when the reaction is first order in solute with rate constant k, the diffusion coefficient of the organic species in water is D and the saturation solubility of the organic reactant in water is Csat. We consider the system at steady state and take a mass balance across a slab... 

Figure 9 shows the approximate solutions of dimensionless potential and concentration with different terms for a second order reaction in a porous slab electrode, and shows the comparisons between the approximate and numerical solutions. The potential and concentration profiles are obtained by using the coupled equation model with diffusion. 

According to the above definitions, the effectiveness factor for any of the above shapes can adequately describe simultaneous reaction and diffusion in a catalyst particle. The equation for the effectiveness factor in a slab is the simplest in Table 6.3.1 and will be used for all pellet shapes with the appropriate Thiele modulus ... 

Metal deposition occurs with sharp gradients within a catalyst pellet, usually concentrated on the outside of catalyst pellets forming a U-shaped distribution. Sato et at [3] related this metal deposition with simultaneous diffusion and reaction, and suggested a value of 8 for the Thiele modulus in a slab geometry, Tamm [4] suggested that this distribution can be characterized by a theta factor defined in a ( Undricat geometry as... 

The prototype problem is heat conduction in a slab or reaction and diffusion in a flat layer, as described in Chapter 9. Here the methods are demonstrated on a problem with reaction and diffusion in a flat layer with a first-order chemical reaction. 

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. 

As an alternative to the previous example, we can also solve the problems with inhomogeneous boundary conditions by direct application of the finite integral transform, without the necessity of homogenizing the boundary conditions. To demonstrate this, we consider the following transient diffusion and reaction problem for a catalyst particle of either slab, cylindrical, or spherical shape. The dimensionless mass balance equations in a catalyst particle with a first order... 

The problem of pore diffusion is only limited to immobilized enzyme catalysts, and not enzyme catalyzed reactions in which the enzyme is used in the native or soluble form. Immobilized enzymes are supported catalysts in which the enzyme is supported or immobilized on a suitable inert support such as alumina, kiesulguhr, silica, or microencapsulated in a suitable polymer matrix. The shape of the immobilized enzyme pellet may be spherical, cylindrical, or rectangular (as in a slab). If the reaction follows Michaelis-Menten kinetics discussed previously, then a shell balance around a spherical enzyme pellet results in the following second order differential equation ... 

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