Single slab geometry

One of the most accurate approaches to solve the LDF equations for the single slab geometry is the full-potential linearized augmented plane wave (FLAPW) method (10). Here, we highlight only the essential characteristics of this approach for further details the reader is referred to a recent review article (11). 

Figure 2. The polarization energy Wi of a single charge of magnitude e as a function of the distance of the charge from the slab for different slab widths. The dielectric coefficients of the slab geometry are ei = 80 2 = 2 3 = 80. The polarization energy is normalized by kT where T = 300 K. The ICC curves as obtained from different approaches (PC/PC, SC/PC, and SC/SC the explanation of the abbreviations can be found in the main text) are compared to the analytical solution [66],...

Figure 5 Tjrpical periodic boundary conditions used for computer simulations of metal-water interfaces (a) and (b) geometries that are periodic in the directions parallel to the metal surface (c) geometry that is periodic in all three dimensions. The ome-tries illustrated correspond to (a) a single slab of water sandwiched between metal surfaces, (b) a slab of water that is on top of a metal slab and has a free surface, and (c) a system with an infinite number of parallel, alternating, metal and water slabs.

Note the very good accuracy of the Double-P method in slab geometry. More extensive tests do not however single out this method, or any other, as decisively superior in accuracy for a fixed n. 

Gottifredi JC, Gonzo EE, Quiroga OD. Isothermal effectiveness factor—I Analytical expression for single reaction with arbitrary kinetics. Slab geometry. Chemical Engineering Science 1981 36 713-719. 

Six fuel boxes make up the simplest form of the reactor. The individual fuel boxes are separated by graphite wedges. The inner and outer reactor core reflector is graphite, whereas the upper and lower is light water. Various core geometries varying from a single slab (6 boxes) to an annular core (24 boxes) can be readily constructed. 

The single cylindrical pore is of course not the geometry we are interested in for porouS catalysts, which may be spheres, cylinders, slabs, or flakes. Let us consider first a honeycomb catalyst of thickness It with equal-sized pores of diameter cfp, as shown in Figure 7-14. The centers of the pores may be either open or closed because by symmetry there is no net flux across the center of the slab. (If the end of the pore were catalytically active, the rate would of course be sHghtly different, but we will ignore this case.) Thus the porous slab is just a collection of many cylindrical pores so the solution is exactly the same as we have just worked out for a single pore. 

The behaviour of the simpler autocatalytic models in each of these three class A geometries seems to be qualitatively very similar, so we will concentrate mainly on the infinite slab, j = 0. For the single step process in eqn (9.3) the two reaction-diffusion equations, for the two species concentrations, have the form... 

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... 

Notice that if we use Eq. (6.4.11) for A and Eq. (6.4.10) for pi Sg we recover the same Thiele modulus Eq. (6.4.9) for the straight tube. The geometry of the situation for a slab is identical with that for a single pore, and both give the same expression for the effectiveness factor. The assumption that the edges are sealed forces the problem into a one-dimensional form by making it possible to take the concentration constant over any plane parallel to the face. The graph of j versus h is shown as the left-hand curve in Fig. 6.7. 

FIGURE 10.6 Computational models of Au surfaces. Models are shown with optimized O2 geometries. For Au(m) and Au(211), a single unit cell of the periodic slab is shown [34]. Reprinted with permission from [34]. Copyright (2005) American Chemical Society. 

In his first studies of the Fourier equation for temperature distribution in a reactive system, Frank-Kamenetskii restricted his attention to three shapes, the infinite slab, the infinite cylinder, and the sphere. For these three geometries (class A) the Laplacian operator can be expressed in terms of a single co-ordinate, and the steady-state problem is reduced to solving the ordinary differential equation... 

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