Diffusion problems equation

This exactly reduces to the stagnant layer one-dimensional pure diffusion problem (Equation (9.56)). 

APPENDIX 12 SOLUTION OF THE TRANSIENT GAS-PHASE DIFFUSION PROBLEM EQUATIONS (12.4M12.7)... 

For each set o(Il), the concentration profile is calculated by solving the direct diffusion problem (Equation 12.78). The set of coefficients a(Il) that yields the concentr discrepancy... 

In test examples, we calculated the experimental profile by solving direct diffusion problems (Equation 12.78) with a diffusion coefficient (Equation 12.81) for preset parameters (l)°-a(3) . In order to do so, we used the numerical procedure of solving nonhnear parabolic equations described in [68]. After that, the calculated profile was distorted by the addition of a random concentration proportional to... 

Magnitudes of n have been empirically established for those kinetic expressions which have found most extensive application e.g. values of n for diffusion-limited equations are usually between 0.53 and 0.58, for the contracting area and volume relations are 1.08 and 1.04, respectively and for the Avrami—Erofe ev equation [eqn. (6)] are 2.00, 3.00 etc. The most significant problem in the use of this approach is in making an accurate allowance for any error in the measured induction period since variations in t [i.e. (f + f0)] can introduce large influences upon the initial shape of the plot. Care is needed in estimating the time required for the sample to reach reaction temperature, particularly in deceleratory reactions, and in considering the influences of an induction period and/or an initial preliminary reaction. 

Vabishchevich, P. (1994) Monotone difference schemes for convection-diffusion problems. Differential Equations, 30, 503-531 (in Russian). 

Neal and Nader [260] considered diffusion in homogeneous isotropic medium composed of randomly placed impermeable spherical particles. They solved steady-state diffusion problems in a unit cell consisting of a spherical particle placed in a concentric shell and the exterior of the unit cell modeled as a homogeneous media characterized by one parameter, the porosity. By equating the fluxes in the unit cell and at the exterior and applying the definition of porosity, they obtained... 

The concentrations of the reactants and reaction prodncts are determined in general by the solution of the transport diffusion-migration equations. If the ionic distribution is not disturbed by the electrochemical reaction, the problem simplifies and the concentrations can be found through equilibrium statistical mechanics. The main task of the microscopic theory of electrochemical reactions is the description of the mechanism of the elementary reaction act and calculation of the corresponding transition probabilities. 

The mathematical formulations of the diffusion problems for a micropippette and metal microdisk electrodes are quite similar when the CT process is governed by essentially spherical diffusion in the outer solution. The voltammograms in this case follow the well-known equation of the reversible steady-state wave [Eq. (2)]. However, the peakshaped, non-steady-state voltammograms are obtained when the overall CT rate is controlled by linear diffusion inside the pipette (Fig. 4) [3]. 

Nonequilibrium thermodynamics provides a second approach to combined convection and diffusion problems. The Kedem-Katchalsky equations, originally developed to describe combined convection and diffusion in membranes, form the basis of this approach [6,7] ... 

Fick s first law is a concise mathematical statement however, it is not directly applicable to solutions of most pharmaceutical problems. Fick s second law presents a more general and useful equation in resolving most diffusion problems. Fick s second law can be derived from Fick s first law. 

Membrane diffusion illustrates the uses of Fick s first and second laws. We discussed steady diffusion across a film, a membrane with and without aqueous diffusion layers, and the skin. We also discussed the unsteady diffusion across a membrane with and without reaction. The solutions to these diffusion problems should be useful in practical situations encountered in pharmaceutical sciences, such as the development of membrane-based controlled-release dosage forms, selection of packaging materials, and experimental evaluation of absorption potential of new compounds. Diffusion in a cylinder and dissolution of a sphere show the solutions of the differential equations describing diffusion in cylindrical and spherical systems. Convection was discussed in the section on intrinsic dissolution. Thus, this chapter covered fundamental mass transfer equations and their applications in practical situations. 

The differential equation is evaluated at certain collocation points. The collocation points are the roots to an orthogonal polynomial, as first used by Lanczos [Lanczos, C.,/. Math. Phys. 17 123-199 (1938) and Lanczos, C., Applied Analysis, Prentice-Hall (1956)]. A major improvement was proposed by Villadsen and Stewart [Villadsen, J. V., and W. E. Stewart, Chem. Eng. Sci. 22 1483-1501 (1967)], who proposed that the entire solution process be done in terms of the solution at the collocation points rather than the coefficients in the expansion. This method is especially useful for reaction-diffusion problems that frequently arise when modeling chemical reactors. It is highly efficient when the solution is smooth, but the finite difference method is preferred when the solution changes steeply in some region of space. The error decreases very rapidly as N is increased since it is proportional to [1/(1 - N)]N 1. See Finlayson (2003) and Villadsen, J. V., and M. Michelsen, Solution of Differential Equation Models by Polynomial Approximations, Prentice-Hall (1978). 

With the expressions for the velocities, i.e. equations (24) and (25), at hand, one can turn to the diffusion problem and seek to solve equation (23). Apart from dropping the terms with derivatives in the z-direction, it can be shown that the diffusive term 02c,-/0y2 is also relatively small in comparison with 02c,/0x2 (see p. 87 in [25]). Thus, equation (23) boils down to ... 

Solve the following equation, which appears in connection with some diffusion problems, in C(t) from t = 0 to f = 0.5... 

This kinetic-diffusion problem in the steady state can be described by the coupled second-order differential equations ... 

The key physics of our model (see Eqs. (9) and (10)) is contained in the nonlocal diffusion kernels which occur after integrating over the atomic processes which produce step fluctuations. We have calculated these kernels for a variety of physically interesting cases (see Appendix C) and have related the parameters in those kernels to atomic energy barriers (see Appendix B). The model used here is close in spirit to the work of Pimpinelli et al. [13], who developed a scaling analysis based on diffusion ideas. The theory of Einstein and co-workers and Bales and Zangwill is based on an equihbrated gas of atoms on each terrace. The concentration of this gas of atoms obeys Laplace s equation just as our probability P does. To make complete contact between the two methods however, we would need to treat the effect of a gas of atoms on the diffusion probabilities we have studied. Actually there are two effects that could be included. (1) The effect of step roughness on P(J) - we checked this numerically and foimd it to be quite small and (2) The effect of atom interactions on the terrace - This leads to the tracer diffusion problem. It is known that in the presence of interactions, Laplace s equation still holds for the calculation of P(t), but there is a concentration... 

Mathematically, studies of diffusion often require solving a diffusion equation, which is a partial differential equation. The book of Crank (1975), The Mathematics of Diffusion, provides solutions to various diffusion problems. The book of Carslaw and Jaeger (1959), Conduction of Heat in Solids, provides solutions to various heat conduction problems. Because the heat conduction equation and the diffusion equation are mathematically identical, solutions to heat conduction problems can be adapted for diffusion problems. For even more complicated problems, including many geological problems, numerical solution using a computer is the only or best approach. The solutions are important and some will be discussed in detail, but the emphasis will be placed on the concepts, on how to transform a geological problem into a mathematical problem, how to study diffusion by experiments, and how to interpret experimental data. 

Equation 3-10 is the most basic diffusion equation to be solved, and has been solved analytically for many different initial and boundary conditions. Many other more complicated diffusion problems (such as three-dimensional diffusion with spherical symmetry, diffusion for time-dependent diffusivity, and... 

To verify the equation is the solution to the diffusion problem, you may verify that the expression satisfies the diffusion equation, the initial condition and the boundary conditions. 

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 important and very useful because many solutions obtained for the simple one-dimensional diffusion equation can be applied to the spherical diffusion problem. Below is an example. 

This equation is the same as the diffusion equation in isotropic media with D = 1. Hence, theoretically, solutions in isotropic media can be applied to diffusion problems in anisotropic media after these transformations. However, it must be realized that the transformed coordinates may correspond to strange and unin-... 

Many diffusion problems cannot be solved anal3dically, such as concentration-dependent D, complicated initial and boundary conditions, and irregular boundary shape. In these cases, numerical methods can be used to solve the diffusion equation (Press et al., 1992). There are many different numerical algorithms to solve a diffusion equation. This section gives a very brief introduction to the finite difference method. In this method, the differentials are replaced by the finite differences ... 

One-dimensional crystal growth at constant growth rate The crystal growth rate may be controlled by factors other than the diffusion process itself. In such a case, the growth rate may be constant. Assume constant D and uniform initial melt. The diffusion problem can be described by the following set of equations ... 

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