Variable diffusion coefficient

Diffusion equation 3-10 is for constant diffusivity. When diffusivity varies for one-dimensional diffusion, then Equation 3-9 must be used. Diffusivity may vary as a function of t (e.g., when temperature varies with time), or as a function of C (diffusivity in general depends on composition), and less often, as a function of X. 

Because D increases with increasing temperature (the Arrhenius equation 1-73), time-dependent D is often encountered in geology because an igneous rock may have cooled down from a high temperature, or metamorphic rock may have experienced a complicated thermal history. If the initial and boundary conditions are simple and if D depends only on time, the diffusion problem is easy to deal with. Because D is independent of x. Equation 3-9 can be written as 

The above equation is equivalent to Equation 3-10 by making a equivalent to t and D equal 1. Hence, solutions obtained for constant D may be applied to time-dependent D. For example, by analogy to Equation 3-38, the solution to the diffusion-couple problem for time-dependent D is 

Similarly, by analogy to Equation 3-40b, the solution for semi-infinite medium with uniform initial concentration and constant surface concentration is 

These solutions are used often in treating diffusion during cooling. 

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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. ... 

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). 

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

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). 

Most kinetic studies of diffusion into keratin fibers employ equations derived from this form of Pick s law and provide approximate diffusion coefficients, assumed to be constant throughout the diffusion reaction. Pfowever, Crank [82] has provided equations for evaluating diffusion data under a wide variety of circumstances, including a variable diffusion coefficient described later in this chapter in the section entitled The Case of a Variable Diffusion Coefficient. ... 

The diffusion equations described in the previous section have been derived from Pick s second law for unidirectional diffusion with the assumption that the diffusion coefficient is constant throughout the reaction. Crank [82] has also derived equations for evaluating diffusion data for systems with a variable diffusion coefficient that can be used to test one s data. 

In general, the penetration of solvents (that promote swelhng) into polymers may be described as processes with a variable diffusion coefficient. For more comprehensive treatment of this subject, see the books by Alexander et al. [93] and Crank [82],... 

Scatter in the data shown in Fig. 1 could be due to several factors, such as variable diffusion coefficients, sample intervals too widely spaced to accurately indicate gradients, poorly constrained sedimentation rates, failure to recover seafloor sediments in cores and consequent depth inaccuracies, an oxidizing (bioturbated) zone near the seafloor, advective supply of sulfate at depth and intense anaerobic methane oxidation. 

Things can be different when the Laplace equation is used to solve current distributions coupled with diffusion in stagnant electrolytes (e.g. in the diffusion layer) with variable diffusion coefficients but these problems are beyond the scope of this work. 

Some simulation results of the first- and second-order FRFs for the nonisothermal micropore diffusion model with variable diffusion coefficient are given in Figure 11.16. They correspond to literature data for adsorption of CO2 on silicalite-1 [34], Ps= 10 kPa and Tg = 298 K, and to moderate heat transfer resistances [57], The functions H2,pp(co, —co), ff2,Tx(<. —co), and //2,px( , —co), which are identically equal to zero, are not shown. In Rgurc 11.16a we also give the FRFs corresponding to isothermal case (the parameter very large). Notice that for that case the Fp set of FRFs describes the system completely. 

Integration of Eq. (61.1) for the desired geometry and boundary conditions yields the total rate of permeation of the penetrant gas through the polymer membrane. Integration of Eq. (61.2) yields information on the temporal evolution of the penetrant concentration profile in the polymer. Equation (61.2) requires the specification of the initial and boundary conditions of interest. The above relations apply to homogeneous and isotropic polymers. Crank [3] has described various techniques of solving Pick s equations for different membrane geometries and botmdary conditions, for constant and variable diffusion coefficients, and for both transient and steady-state transport. 

Note that the above theoretical treatment is highly approximate the modern approach to data analysis would be via digital simulation allowing for depletion of Z and for variable diffusion coefficients, as well as for providing a rigorous (rather than approximate) solution of the coupled dilfusion-kinetic equations involved. 

Fig. 3.21 Simulation of polymerisation within a droplet with a variable diffusion coefficient according to Eq. (3.25)...

Scientists developed a two-stage model, which takes into account water-vapor-sorption kinetics of wool fibers and can be used to describe the coupled heat and moisture transfer in wool fabrics. The predictions fi om the model showed good agreement with experimental observations obtained from a sorption-cell experiment. More recently, Scientists further improved the method of mathematical simulation of the coupled diffusion of the moisture and heat in wool fabric by using a direct numerical solution of the moisture-diffusion equation in the fibers with two sets of variable diffusion coefficients. These researeh publieations were focused on fabrics made fi om one type of fiber. The features and differences in the physical mechanisms of coupled moisture and heat diffusion into fabrics made fi om different fibers have not been systematieally investigated. 

These equations are valid for variable mass or molar density, p or C, and variable diffusion coefficient Bab- Their generality can be reduced in certain cases as is shown below. 

For a constant diffusion coefficient and boundary conditions of constant current (galvanostatic operation) or constant surface concentration (je.g., for a potential step experiment), this equation can be integrated directly [58]. For nonconstant boundary conditions but constant diffusion coefficient, the equation can be solved using DuhameTs superposition integral [59]. With an arbitrarily variable diffusion coefficient, the equation must be solved numerically. 

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