Solution of the Diffusion Equations

For interactions between packaging and product the above descriptions of both material transport processes by diffusion and convection as well as the simultaneous chemical reactions come into consideration. The general transport equation (8.6) is the starting point for solutions of all specific cases occurring in practice. Material loss through poorly sealed regions in the package can be considered as convection currents and/or treated as diffusion in the gas phase. 

The diffusion coefficient D can be assumed to be constant or practically constant for most cases of practical interest. In addition, simplified solutions for diffusion along the x-axis can be used instead of the general solution, except for some particular cases. This greatly simplifies presentation of the problem and the resulting equation for diffusion is  

The simplest case to solve is when the concentration stays constant over time in the polymer. Under such steady state conditions, dc/dt = 0, as in the case of permeation, the equation of diffusion (8.4) turns to Tick s first law. 

This particular case exists, for example, in the diffusion of a substance 

Concentration is variable with time, Pick s second law Most interactions involving mass transfer between the packaging and food behave under non-steady state conditions and are referred to as migration. A number of solutions exist by integration of the diffusion equation 8.7 that are dependent on the so-called initial and boundary conditions of special applications. Many solutions are taken from analogous solutions of the heat conductance equation that has been known for many years  

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Example The equation 3c/3f = D(3 c/3a. ) represents the diffusion in a semi-infinite medium, a. > 0. Under the boundary conditions c(0, t) = Cq, c(x, 0) = 0 find a solution of the diffusion equation. By taking the Laplace transform of both sides with respect to t,... 

If the total amount of radioactivity transfened from one cylinder to another is measured the solution of the diffusion equation is... 

The first and second integrals have their coordinate systems centered on the catalytic C and noncatalytic N spheres, respectively. The local nonequilibrium average microscopic density field for species a is pa(r) = [Y = 5(r - ( )) The solution of the diffusion equation can be used to estimate this nonequilibrium density, and thus the velocity of the nanodimer can be computed. The simple model yields results in qualitative accord with the MPC dynamics simulations and shows how the nonequilibrium density field produced by reaction, in combination with the different interactions of the B particles with the noncatalytic sphere, leads to directed motion [117],... 

The last two assumptions are the most critical and are probably violated under field conditions. Smith et al. (3) found that at least a half-hour was required to achieve adsorption equilibrium between a chemical in the soil water and on the soil solids. Solution of the diffusion equation has shown that many volatile compounds have theoretical diffusion half-lives in the soil of several hours. Under actual field conditions, the time required to achieve adsorption equilibrium will retard diffusion, and diffusion half-lives in the soil will be longer than predicted. Numerous studies have reported material bound irreversibly to soils, which would cause apparent diffusion half-lives in the field to be longer than predicted. 

With the advent of picosecond-pulse radiolysis and laser technologies, it has been possible to study geminate-ion recombination (Jonah et al, 1979 Sauer and Jonah, 1980 Tagawa et al 1982a, b) and subsequently electron-ion recombination (Katsumura et al, 1982 Tagawa et al, 1983 Jonah, 1983) in hydrocarbon liquids. Using cyclohexane solutions of 9,10-diphenylanthracene (DPA) and p-terphenyl (PT), Jonah et al. (1979) observed light emission from the first excited state of the solutes, interpreted in terms of solute cation-anion recombination. In the early work of Sauer and Jonah (1980), the kinetics of solute excited state formation was studied in cyclohexane solutions of DPA and PT, and some inconsistency with respect to the solution of the diffusion equation was noted.1... 

For small colloidal particles, which are subject to random Brownian motion, a stochastic approach is more appropriate. These methods are based on the formulation and solution of the diffusion equation in a force field, in the presence of convection... 

A situation which is frequently encountered in the production of microelectronic devices is when vapour deposition must be made into a re-entrant cavity in an otherwise planar surface. Clearly, the gas velocity of the major transporting gas must be reduced in the gas phase entering the cavity, and transport down the cavity will be mainly by diffusion. If the mainstream gas velocity is high, there exists the possibility of turbulent flow at the mouth of the cavity, but since this is rare in vapour deposition processes, the assumption that the gas within the cavity is stagnant is a good approximation. The appropriate solution of the diffusion equation for the steady-state transport of material through the stagnant layer in the cavity is... 

The trouble is now that the source term does not include the sum of sines, so we will use a trick resting on the Leibniz s rule for differentiating integrals. A particular solution of the diffusion equation with radiogenic accumulation is... 

Butler and Pillingf) calculated an exact numerical solution of the diffusion equation. They showed that the interpolation formula proposed by Gosele et al.e) reproduces the numerical solution with high precision. 

Finally, as described in Box 4.1 of Chapter 4, an exact numerical solution of the diffusion equation (based on Fick s second law with an added sink term that falls off as r-6) was calculated by Butler and Pilling (1979). These authors showed that, even for high values of Ro ( 60 A), large errors are made when using the Forster equation for diffusion coefficients > 10 s cm2 s 1. Equation (9.34) proposed by Gosele et al. provides an excellent approximation. 

A solution of the diffusion equation for an electrode reaction for repetitive stepwise changes in potential can be obtained by numerical integration [44]. For a stationary planar diffusion model of a simple, fast, and reversible electrode reaction (1.1), the following differential equations and boundary conditions can be formulated ... 

In Eq. (14) we took care of the fact that the differentiation in the diffusion Eq. (3) was with respect to the primed co-ordinates. Because X is an arbitrary constant, the general one-dimensional solution of the diffusion equation is ... 

In many treatments of free diffusion the propagator is immediately written as a Gaussian function with the argument that it fulfils the diffusion equation. Equation (27) shows the relation with the normal mode solution of the diffusion equation. For diffusion in a bounded region the propagator is no... 

The general solution of the diffusion equation for a system with spherical symmetry can therefore be written as ... 

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. 

The DICTRA programme is based on a numerical solution of multi-component diffusion equations assuming that thermodynamic equilibrium is locally maintained at phase interfaces. Essentially the programme is broken down into four modules which involve (1) the solution of the diffusion equations, (2) the calculation of... 

Kirkaldy J.S., Weichert D., and Haq Z.U. (1963) Diffusion in multicomponent metallic systems, VI some thermodynamic properties of the D matrix and the corresponding solutions of the diffusion equations. Can. f. Phys. 41, 2166-2173. 

Sorption relates to a compound sticking to the surface of a particle. Adsorption relates to the process of compound attachment to a particle surface, and desorption relates to the process of detachment. Example 2.2 was on a soluble, nonsorptive spiU that occurred into the ground and eventually entered the groundwater. We will now review sorption processes because there are many compounds that are sorptive and subject to spills. Then, we can examine the solutions of the diffusion equation as they apply to highly sorptive compounds. 

We will now turn our attention from one-dimensional solutions of the diffusion equation to two-dimensional solutions. 

Infinite fluid volume and solid diffusion control Practically, infinite solution volume condition (w 1) amounts to constant liquid-phase concentration. For a constant diffu-sivity and an infinite fluid volume, the solution of the diffusion equations is (Helfferich, 1962 Ruthven, 1984)... 

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... 

In a transient method one creates an initial distribution of labels in the plane of the membrane that is nonuniform, c(x, y, 0), where c is the mole % of the spin-label lipid (e.g., at point x, y in the membrane, at time t = 0), For times f O the lateral distribution of label c(x, y, r) is then determined from a solution of the diffusion equation... 

IV. Solutions of the Diffusion Equations of Interest in Chemica. Engineering.. 198... 

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