Diffusion examples

The Merrill and Hamrin criterion was derived for a first-order reaction. It should apply reasonably well to other simple reactions, but reactions exist that are quite sensitive to diffusion. Examples include the decomposition of free-radical initiators where a few initial events can cause a large number of propagation reactions, and coupling or cross-linking reactions where a few events can have a large effect on product properties. 

The tube models consider the oceanic mass of water as subdivided into columns. Mass transfer between columns takes place by advection and diffusion. Examples of tube models may be found in Munk (1966) and Bieri et al. (1966), to whom we refer readers for further clarification. 

Fig. 13.1. The boundary conditions for the diffusion example before diffusion process...

Write a small finite difference program to solve the one dimensional convection-diffusion example (8.59). 

Fig, 4. Methane concentration profile through the rock salt seal obtained for case A with diffusion example for an extremely prestation-ary flow. 

C. Theodoropoulos, Y. Qian, and I. G. Kevrekidis (2000) Coarse stability and bifurcation analysis using time-steppers A reaction-diffusion example. Proc. Natl. Acad. Sci. USA 97, pp. 9840-9843... 

Our present task is to build on the foundations laid in chap. 7, but now with special reference to the diffusive processes that take place at extended defects. The basic argument will be that by virtue of the more open atomic-level environments near extended defects, the activation energy both for point defect formation and migration will often be reduced relative to bulk values. We will build our case around a fundamental case study through the consideration of diffusion at surfaces. The surface diffusion example will illustrate not only how diffusive processes are amended at extended defects, but will also illustrate the shortcomings of the transition state formalism when the detailed atomic-level mechanisms are not known a priori. 

The porosils have appeared only recently upon the scene [lOj. They include the silica end-products of synthesis of zeolites richer and richer in silica, such as silicalites I and II, and also other species which do not always have a zeolite counterpart. They can be sub-divided into clathrasils in which openings between intracrystalline cavities are too small for molecule migration and zeosils in which these openings are adequate for molecule diffusion. Examples of each sub-division are ... 

The following examples illustrate the variety of geochemical problems treated with the numerical simulator. We first consider geochemical processes in porous media nitrification (Example 1), validation of the nonequilibrium model (Example 2) and TCE transformation (Example 3). After that we treat the same problems in fractured porous media matrix diffusion (Example 4), two-member decay chain (Example 5). Finally, Example 5 is extended to demonstrate the influence of parameter variation for the concentration distribution that occurs either under equilibrium or nonequilibriiun conditions (Example 6). 

Figure 7.7-2. Concentrations for numerical method for unsteady-state diffusion. Example 7.7-1.

Fig. 3.1. Error of the standard finite difference method (SFD) and compact differences (CFD) for the reaction-diffusion example (3.1), BC=boundary condition...

Chemical reactions produce extra terms in the diffusion example, a following reaction scheme (EC) like... 

An estimation of the minimal value of >jex is given in Example 4.5.1. Note that for an exothermic (and thus non-isothermal) reaction jex may be >1 even for a first-order reaction (see end of Section 4.5.3.2). Figure 4.5.8 shows ri x as a function of kA/fi according to Eq. (4.5.21) but also for reaction orders m that differ from unity. For a negative reaction order, the rate of the chemical reaction increases with decreasing concentration, and therefore we get the rare case that tjex increases for a limitation of the rate by external diffusion (Example 4.5.2). 

Note that the activation energy is different if the rate is defined based on partial pressures and not on molar concentrations (Topic 6.13.1). Also note that we have assumed that the reaction rates are not influence by external or internal diffusion (Example 6.13.2). 

When O2 is available in a specific environment, this is called an aerobic environment when no O2 is available, the conditions are anaerobic. Several anaerobic environments exist, especially in places where oxygen is consumed or depleted more rapidly than it is replaced by diffusion. Examples include the bottoms of rivers, canals and lakes with a lot of organic debris, landfills, the rumen of herbivores and so on. Besides these natural examples anaerobic conditions also exist in several man-controlled environments such as septic tanks, anaerobic wastewater treatment plants, sludge digesters or solid waste biogasification plants. These anaerobic environments show a high biological activity that can be quite different from aerobic conditions. 

Although the macroscopic curvature of the surface can have a small effect on the equilibrium properties of the adsorbed polymer layer, it will have a strong effeet if the kinetics of polymer adsorption are dominated by diffusion. Examples of frequently applied well-defined geometries were presented in flie previous section. Wang et al. [34] have studied exchange kinetics in a spherical geometry. It was shown that the influence of bulk diffusion is reduced with decreasing radius of the adsorbent particle. For slow processes the macroscopic curvature will not be very important, and powders with poorly defined macroscopic curvatures ean still be used. 

The case of two-sided finite initial boundary conditions is discussed more precisely below. If we apply a diffusion source on one side of a very thin, long diffusion channel (corresponding to one-sided finite or semiinfinite boundary conditions), Eq. (6.68) applies once again (comparatively small times), provided So is replaced by double the value. Figure 6.24 shows a tracer diffusion example. 

Figure 12.7. Diffusion profiles for nonlinear diffusion example.

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