Three-dimensional diffusion General

Perhaps even more important is die fact that LEM does not require a numerical solution to die Navier-Stokes equation. Indeed, even a three-dimensional diffusion equation is generally less computationally demanding than the Poisson equation needed to find die pressure field. 

The above equation is known as the three-dimensional diffusion equation. One can also write the diffusion in terms of the first component. Hence, C2 is replaced by C (concentration of either component 1 or component 2) below. The above equation is general and accounts for the case when D depends on concentration (such as chemical diffusion to be discussed later). 

This result is rather general and applies to three-dimensional diffusion within the catalytic pores as well as one-dimensional diffusion that is consistent with the homogeneous model of interest in this chapter. Integration of (19-25) from the external catalytic surface where and I a = 1 to an arbitrary position... 

Before we show the general solution of the three-dimensional diifusion Eq. (3) or Eq. (6), we first solve the diffusion equation for diffusion along one axis, e.g., the z axis parallel to the pulsed field gradient. The diffusion equation is then ... 

In the general case of three-dimensional multicomponent diffusion in an anisotropic medium (such as Ca-Fe-Mg diffusion in pyroxene), the mathematical description of diffusion is really complicated it requires a diffusion matrix in which every element is a second-rank tensor, and every element in the tensor may depend on composition. Such a diffusion equation has not been solved. Because rigorous and complete treatment of diffusion is often too complicated, and because instrumental analytical errors are often too large to distinguish exact solutions from approximate solutions, one would get nowhere by considering all these real complexities. Hence, simplification based on the question at hand is necessary to make the treatment of diffusion manageable and useful. 

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... 

Compared to rivers and lakes, transport in porous media is generally slow, three-dimensional, and spatially variable due to heterogeneities in the medium. The velocity of transport differs by orders of magnitude among the phases of air, water, colloids, and solids. Due to the small size of the pores, transport is seldom turbulent. Molecular diffusion and dispersion along the flow are the main producers of randomness in the mass flux of chemical compounds. 

The right-hand side of eqn. (9), which is the diffusion equation or Fick s second law, involves two spherically symmetric derivatives of p(r, t). In the general case of three-dimensional space, lacking any symmetry, it can be shown that the Laplacian operator... 

A particle diffusing in a three-dimensional, inhomogeneous anisotropic medium with external force with potential U(r) obeys a generalization of (3.2),... 

In this chapter generalized mathematical models of three dimensional electrodes are developed. The models describe the coupled potential and concentration distributions in porous or packed bed electrodes. Four dimensionless variables that characterize the systems have been derived from modeling a dimensionless conduction modulus ju, a dimensionless diffusion (or lateral dispersion) modulus 5, a dimensionless transfer coefficient a and a dimensionless limiting current density y. The first three are... 

In this Section so far, ADM is used to solve theoretical generalized models in the forms of ordinary differential equations). For diffusion-convection problems, the distributions along the axial direction of the packed bed electrode were neglected in certain cases, and mass transfer in the three dimensional electrodes were characterized by an average coefficient kh... 

Let us now consider a random walker in a three-dimensional cubic lattice. The atom will jump between sites of the normal lattice for a substitutional diffuser, and from interstitial to interstitial site for an interstitial diffuser. In the present case, the Einstein-Smoluchovskii equation for the diffusion coefficient in three dimensions which is a generalization of Equation 5.36, that is,... 

The class of problem which we are presenting here can not be done with a reasonable amount of computer time, using brute force techniques. It is necessary, for example, to solve a two-dimensional problem rather than a three dimensional problem. Another example is the problem of molecular diffusion (4,5,6). The usual binary diffusion approximation is inadequate. At the very least total mass is transported in this approximation. The primary difficulty in a general treatment is that of inverting a large (N. x N ) matrix at each time step and for each grid... 

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