Laplacian processes

The differential equation of Laplace (V c = 0) describes the formation of fractal distribution of solid matter that results from very different processes (Figure 7.5). Diffusion-limited aggregation (c is the concentration), electrogalvanic deposition (c is the electric potential), and viscous invasion (c is the local pressure) are three Laplacian processes that produce similar fractal distributions. They all imply a strong positive feedback and have the same mathematics. The first two are of significance for materials synthesis. 

The disadvantage of Laplacian probability is its use is limited to calculating the probability of processes for which all outcomes are known and equally probable. This eliminates the use of Laplacian probability for determining the probability of process system failure. 

Actually, this is not really diffusion-XimiiQd, but rather Laplacian growth, since the macroscopic equation describing the process, apart from fluctuations, is not a diffusion equation but a Laplacian equation. There are some crucial differences, which will become clearer below. In some sense DLA is diffusion-limited aggregation in the limit of zero concentration of the concentration field at infinity. 

Because the conditional scalar Laplacian is approximated in the FP model by a non-linear diffusion process (6.91), (6.145) will not agree exactly with CMC. Nevertheless, since transported PDF methods can be easily extended to inhomogeneous flows,113 which are problematic for the CMC, the FP model offers distinct advantages. 

Here, the first term on the right-hand side gives the net diffusive inflow of species A into the volume element. We have assumed that the diffusive process follows Fick s law and that the diffusion coefficient does not vary with position. The spatial derivative term V2a is the Laplacian operator, defined for a general three-dimensional body in x, y, z coordinates by... 

In classical, continuum theories of diffusion-reaction processes based on a Fickian parabolic partial differential equation of the form, Eq. (4.1), specification of the Laplacian operator is required. Although this specification is immediate for spaces of integral dimension, it is less straightforward for spaces of intermediate or fractal dimension [47,55,56]. As examples of problems in chemical kinetics where the relevance of an approach based on Eq. (4.1) is open to question, one can cite the avalanche of work reported over the past two decades on diffusion-reaction processes in microheterogeneous media, as exemplified by the compartmentalized systems such as zeolites, clays and organized molecular assemblies such as micelles and vesicles (see below). In these systems, the (local) dimension of the diffusion space is often not clearly defined. 

Divide the entire mass transfer equation by the scahng factor for diffusion (i.e ,mixCAo/i )- This is an arbitrary but convenient choice. Any of the r + 2 dimensional scaling factors can be chosen for this purpose. When the scaling factor for the diffusion term in the dimensional mass transfer equation is divided by ,mixCAo/T, the Laplacian of the molar density contains a coefficient of unity. When the remaining r - -1 scaling factors in the dimensional mass transfer equation are divided by i,mixCAo/T, the dimensionless mass transfer equation is obtained. Most important, r -h 1 dimensionless transport numbers appear in this equation as coefficients of each of the dimensionless mass transfer rate processes, except diffusion. Remember that the same dimensionless number appears as a coefficient for the accumulation and convective mass transfer rate processes on the left-hand side of the equation. 

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