Diffusion, generally isotropic

Solution. We will determine the Du by the general method used to obtain Eq. 8.11. According to Eq. 4.66, the diffusion is isotropic in directions perpendicular to X3. We shall therefore determine the net flux, Jnet, parallel to xi across the CD plane illustrated in Fig. 8.23. 

Prager [302] examined diffusion in concentrated suspensions using the variational approach. (A discussion of the basic principles in variational theory is given in Ref. 6.) Prager s result is applicable to a very general class of isotropic porous media. Prager s solution for a limiting case of a dilute suspension of particles was... 

It can be noted that in general this result predicts that the ratio of the dispersion coefficient to the free-solution diffusion coefficient is different from the ratio of the effective mobility to the free-solution mobility. In the case of gel electrophoresis, where it is expected that the (3 phase is impermeable (i.e., the gel fibers), the medium is isotropic, and the a phase is the space between fibers, the transport coefficients reduce to... 

Small-step rotational diffusion is the model universally used for characterizing the overall molecular reorientation. If the molecule is of spherical symmetry (or approximately this is generally the case for molecules of important size), a single rotational diffusion coefficient is needed and the molecular tumbling is said isotropic. According to this model, correlation functions obey a diffusion type equation and we can write... 

The general case of mass transfer includes both diffusion and convection. Hence, there are both diffusive flux and convective flux for a component. Therefore, the total flux is the sum of the two fluxes. For a given component in a binary and isotropic system, the total flux is... 

Expand Fick s law and Gauss theorem (Eq. 18-12) to three dimensions and derive Fick s second law for the general situation that the diffusivities Dx, Dy, and Dz are not equal (anisotropic diffusion) and vary in space. Show that the result can then be reduced to Eq. (1) of Box 18.3 provided that D is isotropic (Dx = Dy=Dz) and spatially constant. 

While molecular diffusivity is commonly independent of direction (isotropic, to use the correct expression), turbulent diffusivity in the horizontal direction is usually much larger than vertical diffusion. One reason is the involved spatial scales. In the troposphere (the lower part of the atmosphere) and in surface waters, the vertical distances that are available for the development of turbulent structures, that is, of eddies, are generally smaller than the horizontal distances. Thus, for pure geometrical reasons the eddies are like flat pancakes. Needless to say, they are more effective in turbulent mixing along their larger axes than along their smaller vertical extension. 

The smoothing of a rough isotropic surface such as illustrated in Fig. 3.7 due to vacancy flow follows from Eq. 3.69 and the boundary conditions imposed on the vacancy concentration at the surface.12 In general, the surface acts as an efficient source or sink for vacancies and the equilibrium vacancy concentration will be maintained in its vicinity. The boundary condition on cy at the surface will therefore correspond to the local equilibrium concentration. Alternatively, if cy, and therefore Xy, do not vary significantly throughout the crystal, smoothing can be modeled using the diffusion potential and Eq. 3.72 subject to the boundary conditions on a at the surface and in the bulk.13... 

In general, the properties of crystals and other types of materials, such as composites, vary with direction (i.e., macroscopic materials properties such as mass diffusivity and electrical conductivity will generally be anisotropic). It is possible to generalize the isotropic relations between driving forces and fluxes to account for... 

Equation 7.52 is of central importance for atomistic models for the macroscopic diffusivity in three dimensions (see Chapter 8). For isotropic diffusion in a system of dimensionality, d, the generalized form of Eq. 7.52 is... 

A complete theory of turbulence is still lacking, so we must restrict our discussions to two general cases of interest to us (a) the diffusion of particles from point or line sources where the turbulence may be said to be isotropic, and (b) the behavior of particles near large land surfaces— as for example, dust storms. We shall begin our discussion with an explanation of the meaning of eddy-diffusion, which is characteristic of the conditions to be more fully discussed later. 

In all of the above descriptions and equations, it was assumed that molecular tumbling is isotropic in the solution. This is an idealized case that can be approximated by real proteins only in fortunate instances. In general, proteins exhibit totally anisotropic tumbling7 but in special cases, the simplification of only an axially symmetric motion is used, for which two of the molecular axes are assumed to be identical and thus the corresponding rotational diffusion tensor can be simplified. 

Fast and satisfactory mass transfer calculations are necessary since we may have to repeat such calculations many times for a rate-based distillation column model or two-phase flow with mass transfer between the phases in the design and simulation process. The generalized matrix method may be used for multicomponent mass transfer calculations. The generalized matrix method utilizes the Maxwell-Stefan model with the linearized film model for diffusion flux, assuming a constant diffusion coefficient matrix and total concentration in the diffusion region. In an isotropic medium, Fick s law may describe the multicomponent molecular mass transfer at a specified temperature and pressure, assuming independent diffusion of the species in a fluid mixture. Such independent diffusion, however, is only an approximation in the following cases (i) diffusion of a dilute component in a solvent, (ii) diffusion of various components with identical diffusion properties, and (iii) diffusion in a binary mixture. 

In general, the diffusion equation depends on all the microscopic parameters. The microscopic parameters of van Kampen s model are the local values of the effective trap density cr, which is density times cross-section and work function . The traditional diffusion relation of Eq. (6.305) is valid only for isotropic diffusion and under the restrictive conditions that cr cx exp( homogeneous system with nontrivial geometry. Equation (6.306) is valid when the effective trap concentration is constant, which is more realistic for liquids. 

However, we remark here that the simplifications of the expressions in the Cartesian coordinates system have been accepted as in the case of an isotropic and nonproperty dependent diffusion coefficient of a property. Indeed, the independence of the general diffusion coefficient with respect to the all-internal or external solicitations of the transport medium appears unrealistic in some situations. 

The liquid phase of molecular matter is usually isotropic at equilibrium but becomes birefringent in response to an externally applied torque. The computer can be used to simulate (1) the development of this birefringence —the rise transient (2) the properties of the liquid at equilibrium under the influence of an arbitrarily strong torque and (3) the return to equilibrium when the torques are removed instantaneously—the fall transient. Evans initially considered the general case of the asymmetric top (C2 symmetry) diffusing in three-dimensional space and made no assumptions about the nature of the rotational and translational motion other than those inherent in the simulation technique itself. A sample of 108 such molecules was taken, each molecule s orientation described by three unit vectors, e, Cg, and parallel to its principal moment-of-inertia axes. 

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