Variable diffusivity

When steady-state conditions prevail and D varies with position (e.g., D = D(r ), the diffusion equation can readily be integrated. Equation 4.2 then takes the form 

---

The Morse function which is given above was obtained from a study of bonding in gaseous systems, and dris part of Swalin s derivation should probably be replaced with a Lennard-Jones potential as a better approximation. The general idea of a variable diffusion step in liquids which is more nearly akin to diffusion in gases than the earlier treatment, which was based on the notion of vacant sites as in solids, remains as a valuable suggestion. 

Diffuser vanes are used to decelerate a high velocity flow to create a pressure rise. They are usually at the periphery of each impeller. The variable diffuser vane system may be controlled manually by a handwheel or automatically by a hydraulic or air-operated positioner. See Figure 12-44B. 

This section introduces the method of Boltzmann transformation to solve onedimensional diffusion equation in infinite or semi-infinite medium with constant diffusivity. For such media, if some conditions are satisfied, Boltzmann transformation converts the two-variable diffusion equation (partial differential equation) into a one-variable ordinary differential equation. 

A double-zeta (DZ) basis in which twice as many STOs or CGTOs are used as there are core and valence atomic orbitals. The use of more basis functions is motivated by a desire to provide additional variational flexibility to the LCAO-MO process. This flexibility allows the LCAO-MO process to generate molecular orbitals of variable diffuseness as the local electronegativity of the atom varies. Typically, double-zeta bases include pairs of functions with one member of each pair having a smaller exponent (C, or a value) than in the minimal basis and the other member having a larger exponent. 

This relationship is captured in differential-equation form as Eq. 7.60. Since the momentum and energy equations (Eqs. 7.59 and 7.62) explicitly involve r2, the radial coordinate has become a dependent variable, not an independent variable. A consequence of the Von Mises transformation is that the radial velocity v is removed as a dependent variable and the radial convective terms are eliminated, which is a bit of a simplification. However, the fact that the group of dependent variables pur2 appear within the diffusion terms is a bit of a complication. The factor pur2 plays the role of an apparent variable diffusion coefficient. ... 

In the solution of mass transport problems, several dimensionless groups are used in order to reduce the number of variables. Diffusion layer thicknesses etc. are expressed in much of the literature in terms of these dimensionless variables. 

Method of Separation of Variables Diffusion on a Finite Domain... 

This model is significant because 1) a variable diffusivity as a function of coke content is incorporated, 2) coke content profiles both within a pellet and the reactor bed are predicted with time and space, 3) catalyst activity is related to coke content, thus with time and space also, and 4) the model is supported by experimental data. 

The above reasoning shows that the stretched exponential function (4.14), or Weibull function as it is known, may be considered as an approximate solution of the diffusion equation with a variable diffusion coefficient due to the presence of particle interactions. Of course, it can be used to model release results even when no interaction is present (since this is just a limiting case of particles that are weakly interacting). 

Buckingham (B4), Gardner (G2, G3), and Wilsdon (W5) attempted the difficult problem of applying variable diffusivities to the diffusion equations by employing a capillary potential instead of a concentration potential. 

Ceaglske and Hougen (C4) showed that the movement of moisture in sand is controlled entirely by capillarity and gravity, and that diffusion is not involved. Diffusion equations cannot be made to apply by using variable diffusivity values. 

Process Variable Diffusion Control Permeation Control... 

Heat, mass or momentum transfer in solids is typically represented by boundary value problems (boundary value problems). Variable diffusivity or thermal conductivity, nonlinear source terms or nonlinear boundary conditions make the boundary value... 

Solve the following diffusion problem with a variable diffusivity (appropriate for some polymers) ... 

Helfferich, F.G., and D. Petruzzelli. 1985b. Diffusion with variable diffusion coefficients. Rep. no. r/107, Consiglio Nazionale Ricerche, Rome. 

Using Eq. (40), the mass balance for either electrons or ions can be written as a usual diffusion equation with a variable diffusivity. 

The results of numerous investigations on the kinetics of sorption of pure substances in zeolites have since then appeared in the literature and the field has been reviewed recently by Walker et al. 42). The total uptake or loss of sorbate in a large number of crystallites is commonly observed, and it is generally assumed that the rate of these processes is controlled by diffusion in the solid. Variable diffusion coefficients were sometimes observed by this method, and it appears possible that other processes than diffusion in the solid had some influence on the rate in these cases. The apparent diffusivity will depend only on concentration (besides temperature) if the migration of sorbate particles in the solid is rate controlling. A simple criterion whether this condition exists can be obtained by measuring sorption or desorption rates repeatedly for various initial concentrations and boundary conditions, as described by Diinwald and Wagner 43). 

---