Solving the Diffusion Equations

For instance, of great interest are through or continuous execution schemes available for solving the diffusion equation with discontinuous diffusion coefficients by means of the same formulae (software). No selection of points or lines of discontinuities of the coefficients applies here. This means that the scheme remains unchanged in a neighborhood of discontinuities and the computations at all grid nodes can be carried out by the same formulae without concern of discontinuity or continuity of the diffusion coefficient. 

We first consider the stmcture of the rate constant for low catalyst densities and, for simplicity, suppose the A particles are converted irreversibly to B upon collision with C (see Fig. 18a). The catalytic particles are assumed to be spherical with radius a. The chemical rate law takes the form dnA(t)/dt = —kf(t)ncnA(t), where kf(t) is the time-dependent rate coefficient. For long times, kf(t) reduces to the phenomenological forward rate constant, kf. If the dynamics of the A density field may be described by a diffusion equation, we have the well known partially absorbing sink problem considered by Smoluchowski [32]. To determine the rate constant we must solve the diffusion equation... 

The normal state of affairs during a diffusion experiment is one in which the concentration at any point in the solid changes over time. This situation is called non-steady-state diffusion, and diffusion coefficients are found by solving the diffusion equation [Eq. (S5.2)] ... 

At high frequencies diffusion of the reactants to and from the electrode is not so important, because the currents are small and change sign continuously. Diffusion does, however, contribute significantly at lower frequencies solving the diffusion equation with appropriate boundary conditions shows that the resulting impedance takes the form of the Warburg impedance ... 

Different techniques are commonly used to solve the diffusion equation (Carslaw and Jaeger, 1959). Analytic solutions can be found by variable separation, Fourier transforms or more conveniently Laplace transforms and other special techniques such as point sources or Green functions. Numerical solutions are calculated for the cases which have no simple analytic solution by finite differences (Mitchell, 1969 Fletcher, 1991), which is the simplest technique to implement, but also finite elements, particularly useful for complicated geometry (Zienkiewicz, 1977), and collocation methods (Finlayson, 1972). 

Finally, to solve the diffusion equation, the inherent assumption required is that the geometry or physical dimensions of the sorbent do not change during the course of the experiment due to abrasion, dissolution, swelling, or other process. 

While Heisig et al. solved the diffusion equation numerically using a finite volume method and thus from a macroscopic point of view, Frasch took a mesoscopic approach the diffusion of single molecules was simulated using a random walk [69], A limited number of molecules were allowed moving in a two-dimensional biphasic representation of the stratum corneum. The positions of the molecules were changed with each time step by adding a random number to each of the molecule s coordinates. The displacement was related... 

The second major difference found in vapor-liquid extraction of polymeric solutions is related to the low values of the diffusion coefficients and the strong dependence of these coefficients on the concentration of solvent or monomer in a polymeric solution or melt. Figure 2, which illustrates how the diffusion coefficient can vary with concentration for a polymeric solution, shows a variation of more than three orders of magnitude in the diffusion coefficient when the concentration varies from about 10% to less than 1%. From a mathematical viewpoint the dependence of the diffusion coefficient on concentration can introduce complications in solving the diffusion equations to obtain concentration profiles, particularly when this dependence is nonlinear. On a physiced basis, the low diffiisivities result in low mass-transfer rates, which means larger extraction equipment. 

Before we show the general solution of the three-dimensional diifusion Eq. (3) or Eq. (6), we first solve the diffusion equation for diffusion along one axis, e.g., the z axis parallel to the pulsed field gradient. The diffusion equation is then ... 

Classical studies of the relaxation processes, caused by translational diffusion, have been presented in the early days by Abragam (18), Torrey (136) and Pfeifer (137). Abragam (18) found, by solving the diffusion equation, the following form of the correlation function for the stochastic function Z>o under translational diffusion of two spins 1/2 ... 

Chen el a/. (1991) studied the solidification of two Al-Li-Cu alloys in an attempt to predict /, vs temperature behaviour and the phases formed during solidification. They modified the composition of the liquid in a fashion very similar to Brody and Flemings (1966) but solved the diffusion equations within their programme. Their results produced good predictions for the formation of the secondary phases for both alloys and for f, of primary phase in one alloy (Al-9at%Li-lat%Cu). However, for a Al-9.9at%Li-7.5at%Cu alloy their predicted /, for the primary phase was significantly different from that observed experimentally. 

Meittinen et al. (1992) and Matsumiya et al. (1993) have also attempted to explicitly solve the diffusion equations using a finite difference approach for the diffusion of solute in the solid phase. Meittinen et al. (1992) used different approaches dependent on whether the solute was fast-moving or slow-moving and treated solidification involving 6-ferrite and austenite in different ways. This may give reasonable answers for steels but the programme then loses general applicability to other material types. 

If a gas such as ammonia or CO2 (phase 1) is absorbing into a liquid solvent (phase 2), the resistance R2 is relatively important in controlling the rate of adsorption. This is also true of the desorption of a gas from solution into the gas phase. Usually R2 is of the order 10 or 10 sec. cm. h though the exact value is a function of the hydrodynamics of the system consequently various hydrodynamic conditions give a variety of equations relating R2 to the Reynolds number and other physical variables in the system. For the simplest system where the liquid is infinite in extent and completely stagnant, one can solve the diffusion equation... 

When a solute such as acetic acid is diffusing from water to benzene, the sum of the two R terms for the liquid phases is usually much greater than the interfacial resistance Rj. In the simplest, unstirred system, with both liquids taken as infinite in extent, one must solve the diffusion equations (35)... 

Several general methods are available for solving the diffusion equation, including Boltzmann transformation, principle of superposition, separation of... 

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 ... 

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. 

Figure 2.1. Common control volumes found in engineering texts and (for the latter three) used in solving the diffusion equation.

Solving the diffusion equation in environmental transport can be challenging because only specihc boundary conditions result in an analytical solution. We may want to consider our system of interest as a reactor, with clearly defined mixing, which is more amenable to time dependent boundary conditions. The ability to do this depends on how well the conditions of the system match the assumptions of reactor mixing. In addition, the system is typically assumed as one dimensional. The common reactor mixing assumptions are as follows ... 

In connection with the interphase mass transfer in liquid-liquid and liquid-gas systems, the diffusion equations (and indeed all the equations of change) are valid in both phases. Hence, in principle, diffusion problems in a two-phase system may be solved by solving the diffusion equations in each phase and then choosing the constants of interaction in such a way that the solutions match up at the interface. It is customary to require that the following two conditions be fulfilled at the interface, in a system in which the solute is being transferred from phase I to phase II (1) the flux of mass leaving phase I must equal the flux of mass entering phase II if the diffusion... 

The interfacial diffusion model of Scott, Tung, and Drickamer is somewhat open to criticism in that it does not take into account the finite thickness of the interface. This objection led Auer and Murbach (A4) to consider a three-region model for the diffusion between two immiscible phases, the third region being an interface of finite thickness. These authors have solved the diffusion equations for their model for several special cases their solutions should be of interest in future analysis of interphase mass transfer experiments. 

Fig. 2. Diagram showing the space available to the density distribution of B reactants about A, p(r, t) where R < r < °° and 0 < ( < °°. The initial and boundary conditions are labelled and drawn boldly. The cross-hatched areas within this rectangle is the region where information on p(r, t) can only be obtained by solving the diffusion equation.

Equation (10) is a diffusion equation which applies equally well for matter or heat. The solution of this equation has been studied by many workers. As with differential equations in general, one arbitrary constant is required for each derivative. Since the diffusion equation has partial derivatives, the arbitrary constants have to be functions of the variable which is not involved in the derivative. The diffusion equation requires two functions of time at fixed values of space coordinates (the boundary conditions) and one function of distance at a given time (the initial condition). These have already been established [eqns. (3)—(5)]. It is possible to proceed to solve the diffusion equation for p(r,t) now and then calculate the particle current of B towards each A reactant and so determine the rate of reaction. 

To solve the diffusion equation and obtain the appropriate rate coefficient with these initial distributions is less easy than with the random distribution. As already remarked, the random distribution is a solution of the diffusion equation, while the other distributions are not. The substitution of Z for r(p(r,s) — p(r, 0)/s) is not possible because an inhomogeneous equation results. This requires either the variation of parameters or Green s function methods to be used (they are equivalent). Appendix A discusses these points. The Green s function g(r, t r0) is called the fundamental solution of the diffusion equation and is the solution to the... 

To solve the diffusion equation (9) or (10) for the density p(r, f) with the random initial condition (3), the outer boundary condition (4) and the partially reflecting boundary condition (22) is straightforward. Again, the solution follows from eqn. (12), but the Laplace transform of eqn. (22) is... 

To solve the diffusion equation (141) for motion of an ion pair in their mutual coulomb potential and an applied electric field [the potential energy of eqn. (145)] is quite difficult [321]. Rather than discuss the mathematics in detail, the reader is referred to articles by Hong and Noolandi [72, 323—326] which amplify the mathematical details of Onsager s analyses [321], Here, the results alone are mentioned. The escape probability p(r0, d0) depends on both the initial separation of distance between the ions and the initial orientation of the ion-pair with respect to the applied electric field, d0. 

Exercise. Solve the diffusion equation in an inverted parabolic potential with initial delta distribution,... 

In order to solve the diffusion equations for the species taking part in the electrode reaction, it is first necessary to specify (a) the initial conditions, (b) the boundary conditions and (c) the relationship between C0/CR at the electrode surface and the electrode potential. 

Methods for solving diffusion problems by setting up and solving the diffusion equation under specified boundary conditions are discussed in Chapter 5. 

Methods to solve the diffusion equation for specific boundary and initial conditions are presented in Chapter 5. Many analytic solutions exist for the special case that D is uniform. This is generally not the case for interdiffusivity D (Eq. 3.25). If D does not vary rapidly with composition, it can be replaced by successive approximations of a uniform diffusivity and results in a linearization of the diffusion equation. The... 

The superposition of two solutions therefore also solves the diffusion equation with superposed boundary and initial conditions. 

---