Different Forms of the Diffusion Equation

Thus in this set of experiments, the molar and volume average velocities are zero for ideal gases and the volume and mass average velocities are close to zero for liquids. The mass average velocity is often inappropriate for gases, and the molar average velocity is rarely used for liquids. The volume average velocity is appropriate most frequently, and so it will be emphasized in this book. 

The five most common forms of diffusion equations are given in Table 3.2-1. Each of these forms uses a different way to separate diffusion and convection. Of course. 

Choice Total flux (diffusion -1-convection) Diffusion equation Reference velocity Where best used 

Mass i =yr+Pi y7=Pi(vi-v) = —DpScoi V = COiV] + CO2V2 pv = 1+ 2 Constant-density liquids coupled mass and momentum transport 

Volume 1 =Jl+ClV Ji = ci(vi -/) = -DVci V = Cl FiV] - - C2V2V2 = Fini -1- F2 2 Best overall good for constant-density liquids and for ideal gases may use either mass or mole concentration 

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The latter expression is simply a finite difference form of the diffusion equation with the diffusion coefficient given by D =... 

Note the differences between the form of the diffusion term that appears in this Stratonovich form of the diffusion equation and those that appear in the Ito (or forward Kolmogorov) form of Eq. (2.222) and in the physical diffusion equation of Eq. (2.78). 

The tracer diffusion data shown in Fig. 1.55 correlate well with this equation. The thick curves in this figure, (a) and (b), are calculated ones using fitting parameters listed in Table 1.5. The difference between eqn (1.183) and eqn (1.197) is in the number of exponential terms. The deviation from eqn (1.183) indicates that the correct form of the diffusion equation must contain more than one exponential term. 

Application of the Crank-Nicholson method based on the spatial difference scheme (5.39) results in the following discretized form of the diffusion equation ... 

Rate equations 28 and 30 combine the advantages of concentration-independent mass transfer coefficients, even in situations of multicomponent diffusion, and a familiar mathematical form involving concentration driving forces. The main inconvenience is the use of an effective diffusivity which may itself depend somewhat on the mixture composition and in certain cases even on the diffusion rates. This advantage can be eliminated by working with a different form of the MaxweU-Stefan equation (30—32). One thus obtains a set of rate equations of an unconventional form having concentration-independent mass transfer coefficients that are defined for each binary pair directiy based on the MaxweU-Stefan diffusivities. 

The effect of using upstream derivatives is to add artificial or numerical diffusion to the model. This can be ascertained by rearranging the finite difference form of the convective diffusion equation... 

Fick s second law (Eq. 18-14) is a second-order linear partial differential equation. Generally, its solutions are exponential functions or integrals of exponential functions such as the error function. They depend on the boundary conditions and on the initial conditions, that is, the concentration at a given time which is conveniently chosen as t = 0. The boundary conditions come in different forms. For instance, the concentration may be kept fixed at a wall located atx0. Alternatively, the wall may be impermeable for the substance, thus the flux at x0 is zero. According to Eq. 18-6, this is equivalent to keeping dC/dx = 0 at x0. Often it is assumed that the system is unbounded (i.e., that it extends from x = - °o to + °°). For this case we have to make sure that the solution C(x,t) remains finite when x -a °°. In many cases, solutions are found only by numerical approximations. For simple boundary conditions, the mathematical techniques for the solution of the diffusion equation (such as the Laplace transformation) are extensively discussed in Crank (1975) and Carslaw and Jaeger (1959). 

However, for (r0 — R) < a, Noyes [269] developed a different form of j30. Since such motion must be on a molecular scale a < R) or even less, the author is doubtful that such approximations are more valid than that of the diffusion equation. To estimate a, it may be noted that Northrup and Hynes [103] have found that... 

The equations were transformed into dimensionless form and solved by numerical methods. Solutions of the diffusion equations (7 or 13) were obtained by the Crank-Nicholson method (9) while Equation 2 was solved by a forward finite difference scheme. The theoretical breakthrough curves were obtained in terms of the following dimensionless variables... 

In Chapter 3, several different types of diffusivity were introduced for diffusion in a chemically homogeneous system or for interdiffusion in a solution. In each case, Fick s law applies, but the appropriate diffusivity depends on the particular system. The development of the diffusion equation in this chapter depends only on the form of Fick s law, J = -DVc. D is a placeholder for the appropriate diffusivity, just as J and c are placeholders for the type of component that diffuses. 

In Chapter 4 we described many of the general features of the diffusion equation and several methods of solving it when D varies in different ways. We now address in more detail methods to solve the diffusion equation for a variety of initial and boundary conditions when D is constant and therefore has the relatively simple form of Eq. 4.18 that is,... 

Equation 4.62 is a finite-difference form of the expression above with the convenient definition of a positive diffusive flux toward an adsorbent surface. 

A number of solutions exist by integration of the diffusion equation (7-12) that are dependent on the so-called initial and boundary conditions of special applications. It is not the goal of this section to describe the complete mathematical solution of these applications or to make a list of the most well-known solutions. It is much more useful for the user to gain insight into how the solutions are arrived at, their simplifications and the errors stemming from them. The complicated solutions are usually in the form of infinite series from which only the first or first few members are used. In order to understand the literature on the subject it is necessary to know how the most important solutions are arrived at, so that the different assumptions affecting the derivation of the solutions can be critically evaluated. 

Undeniably, the speed vector, by its size and directional character, masks the effect of small displacements of the particle. Another difference comes from the different definition of the diffusion coefficient, which, in the case of the property transport, is attached to a concentration gradient of the property it means that there is a difference in speed between the mobile species of the medium. A second difference comes from the dimensional point of view because the property concentration is dimensional. When both equations are used in the investigation of a process, it is absolutely necessary to transform them into dimensionless forms [4.6, 4.7, 4.37, 4.44]. 

By examining these characteristic dimensionless numbers, it is possible to appreciate possible interactions of different processes (convection, diffusion, reaction and so on) and to simplify the governing equations accordingly. A typical dimensionless form of the governing equation can be written (for a general variable, (p) ... 

Carslaw, H. S and Jaeger, J. C. (1959) Cottduciion of Heat in Solids, Oxford University Press. Oxford, 2nd ed. This work includes a compilation of solutions to the equation of unsteady heat conduction in the absence of flow for many different geometries, initial and boundaiy conditions. The basic equation is of the same form as the diffusion equation with the thermal diffusivity. K/pC, in place of the diffusion coefficient. (Here k, p, and C are the thermal conductivity, density, and specific heatof the continuous fluid.) Like D, the thcnnal diffusivity has cgs- dimensions of cm"/sec,... 

There are two forms of phenomenological equations for describing Brownian motion the Smoluchowski equation and the Langevin equation. These two equations, essentially the same, look very different in form. The Smoluchowski equation is derived from the generalization of the diffusion equation and has a clear relation to the thermodynamics of irreversible processes. In Chapters 6 and 7, its application to the elastic dumbbell model and the Rouse model to obtain the rheological constitutive equations will be discussed. In contrast, the Langevin equation, while having no direct relation to thermodynamics, can be applied to wider classes of stochastic processes. In this chapter, it will be used to obtain the time-correlation function of the end-to-end vector of a Rouse chain. 

They showed that such a model could reproduce the kind of behaviour observed experimentally in samples of polypropylene. The question of the kind of behaviour to be expected from such a system had previously been addressed mathematically by Cheung [80], but the results he had obtained were not accessible except in the two limits mentioned above. Although the form of the mathematical equation involved was already well known from the study of conduction of heat in solids [15], the work of Packer et al, [79] was the first approach that allowed the relationship between the observed relaxation behaviour and the intrinsic relaxation and spin diffusion properties of the different regions to be explored between the two limits of slow and... 

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