Analytical Solutions of the diffusion equation

When the fast reactions occurring in the system have stoichiometries different from the simple one shown by Eq. (5.78), analytical solutions of the diffusion equations are difficult to obtain. Nevertheless, numerical solutions can be obtained by iterative routines, and the results are conceptually similar to those described. The additional complications introduced by non-steady-state diffusion and nonlinear concentration gradients can be similarly handled. 

Klinger (1974, 1980) used the Green s function to obtain an exact analytical solution of the diffusion equation with convection terms without making any special assumptions concerning the velocity field. The absence of a detailed knowledge of the convection field makes this approach of limited use as well. 

Numerical and some analytical solutions of the diffusion/reaction equations are represented closely by an empirical curve/fit,... 

Hong and Noolandi (1978a) first gave an analytical solution for the diffusion equation of an e-ion pair in the absence of an external field—that is, of Eq. (7.30) with F = -eVer2, where is the dielectric constant of the medium. They then extended their solution in the presence of an external field of arbitrary strength (Hong and Noolandi, 1978b). Since the method involves fairly complex mathematical manipulations, we will only present its outline and some important conclusions. 

The root time method of data analysis for diffusion coefficient determination was developed by Mohamed and Yong [142] and Mohamed et al. [153]. The procedure used for computing the diffusion coefficient utilizes the analytical solution of the differential equation of solute transport in soil-solids (i.e., the diffusion-dispersion equation) ... 

Chapter 2 The Diffusion Equation. The diffusion equation provides the mathematical foundation for chemical transport and fate. There are analytical solutions to the diffusion equation that have been developed over the years that we will use to our advantage. The applications in this chapter are to groundwater, sediment, and biofihn transport and fate of chemicals. This chapter, however, is very important to the remainder of the applications in the text, because the foundation for solving the diffusion equation in environmental systems will be built. 

Equation (2.38) has a first-order sink and a zero-order source, which meets our criteria for an analytical solution to the diffusion equation. Ce is the concentration of C at equilibrium for the reaction. This technique of assuming that multiple reactions are zero-order and first-order reactions will be utilized in Example 2.9. 

The analytical solution of the Smoluchowski equation for a Coulomb potential has been found by Hong and Noolandi [13]. Their results of the pair survival probability, obtained for the boundary condition (11a) with R = 0, are presented in Fig. 2. The solid lines show W t) calculated for two different values of Yq. The horizontal axis has a unit of r /D, which characterizes the timescale of the kinetics of geminate recombination in a particular system For example, in nonpolar liquids at room temperature r /Z) 10 sec. Unfortunately, the analytical treatment presented by Hong and Noolandi [13] is rather complicated and inconvenient for practical use. Tabulated values of W t) can be found in Ref. 14. The pair survival probability of geminate ion pairs can also be calculated numerically [15]. In some cases, numerical methods may be a more convenient approach to calculate W f), especially when the reaction cannot be assumed as totally diffusion-controlled. 

These equations, for the case of solid diffusion-controlled kinetics, are solved by arithmetic methods resulting in some analytical approximate expressions. One common and useful solution is the so-called Nernst-Plank approximation. This equation holds for the case of complete conversion of the solid phase to A-form. The complete conversion of solid phase to A-form, i.e. the complete saturation of the solid phase with the A ion, requires an excess of liquid volume, and thus w 1. Consequently, in practice, the restriction of complete conversion is equivalent to the infinite solution volume condition. The solution of the diffusion equation is... 

Despite the large number of analytical solutions available for the diffusion equation, their usefulness is restricted to simple geometries and constant diffusion coefficients. However, there are many cases of practical interest where the simplifying assumptions introduced when deriving analytical solutions are unacceptable. Such a case, for example, is the diffusion in polymer systems characterized by concentration-dependent diffusion coefficients.This chapter gives an overview of the most powerful numerical methods used at present for solutions of the diffusion equation. Indeed the application of these methods in practice needs the use of adequate computer programs (software). 

Solution of the diffusion equations leads to a result in the Laplace domain that cannot be inverted analytically, numerical inversion being necessary. The final result, after inversion, can be expressed in the form... 

In cases where comparisons have been made, theoretical data obtained by digital simulations are always in agreement with those from analytical solutions of the diffusion-kinetic equations within the limit of experimental error of quantities which can be measured. A definite advantage of simulation over the other calculation techniques is that it does not require a strong mathematical background in order to learn and to use the technique. A very useful guide for the beginner has recently appeared (Britz, 1981). 

For Dp Dp and /fpp < 1 we obtain eqn 8.24. If the diffusion coefficients in the packaging and in the food are approximately equal, the partition coefficient, /fpp determines transport through the system. The packaging determines the rate of the whole process. If the migrant dissolves much better in the food than in the packaging, that means /fpp < 1 and the food determines the rate of the whole process. But if the migrant dissolves much better in the packaging than in the food, A pp 1. If Dp < Dp the mass transport is determined by the diffusion coefficient in the food. Dp and the partition coefficient, A"pp. This leads to the build up of a concentration profile in the foodstuff. An exact analytical solution of the differential equation that takes into consideration the diffusion in food and finite values for Vp and Vp is not available and in consequence the application of numerical methods is necessary. 

Assuming a constant surface area, dissolution at a solution-solid interface (Case I) results in linear kinetics in which the rate of mass transfer is constant with time (equation 1). Analytical solutions to the diffusion equation result in parabolic rates of mass transfer (, 16) (equation 2). This result is obtained whether the boundary conditions are defined so diffusion occurs across a progressively thickening, leached layer within the silicate phase (Case II), or across a growing precipitate layer forming on the silicate surface (Case III). Another case of linear kinetics (equation 1) may occur when the rate of formation of a metastable product or leached layer at the fresh silicate surface becomes equal to the rate at which this layer is destroyed at the aqueous... 

In the chapters that follow, we will examine the solution of the diffusion equations under a variety of conditions. The analytical mathematical methods for attacking these problems are discussed briefly in Appendix A. Numerical methods, including digital simulations (Appendix B), are also frequently employed. 

In the next sections, we describe the solution of the diffusion equations with the appropriate boundary conditions for electrode reactions with heterogeneous rate constants spanning a wide range, and we discuss the observed responses. An analytical approach based on an integral equation is used here, because it has been widely applied to these types of problems and shows directly how the current is affected by different experimen-... 

The diffusion layer width is very much dependent on the degree of agitation of the eleetrolyte. Thus, via the parameter 5, the hydrodynamics of the solution ean be eonsidered. Experimentally, defined hydrodynamie conditions are aehieved by a rotating cylinder, dise or ring-disc electrodes, for whieh analytical solutions for the diffusion equation are available [37, 41,42 and 43]. 

An exact analytic solution to the diffusion equation with constant D and s was given in 1938 by W.J. Archibald for the problem posed (Fujita 1975). However, it is sufficiently complex that its use in application to sedimentation experiments is difficult. To simplify the form of the solution, H. Faxen, as far back as 1929, had introduced the approximation of considering the sector cell to be infinitely extended, corresponding to We shall outline the solution... 

Equation 12 is linear and can be solved analytically under fairly general conditions. It appears in several other contexts such as in the solution of the diffusion equation and in wave propagation problems with evanescent waves. If... 

The superscript FT denotes the Fourier rransform of the corresponding quantity. The coefficients are obtained by solution of the diffusion equation for Qi as functions of rg, Yb and the periodicity of the microdomains. The sums in Eq. (188) can be analytically evaluated if is assumed to be independent of the chain length of the individual species. This assumption may be compared to that made in Sect. 3.3, i.e. the interaction energies per segment do not depend on rg, and allows the effective potentials ([Pg.106]

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