The diffusion equation

Having specified the initial distribution of B reactants about A reactants in eqn. (3), it is necessary to find out what the distribution will be at subsequent times, subject to the two boundary conditions of eqns. (4) and (5). The region where p(r, t) is of interest is shown in Fig. 2. Once p(r,t) has been determined, the flux and hence the particle current of B towards A can be calculated. The rate of reaction is the number of B reactants diffusing towards each A reactant multiplied by the concentration of A reactants. From this, the rate coefficient for reaction may be evaluated. 

In order to calculate the density of reactant B about A, it is necessary to know by what means the reactants migrate in solution. Under most circumstances, diffusion is a very adequate description (the limitations of and complications to diffusion are discussed in Sect. 6, Chap. 8 Sect. 2 and Chap. 11). In this simple analysis of diffusion, Fick s laws will be used with little further justification, save to note that Fick s second law is identical to the equation satisfied by a random walk function. Hardly a surprising result, because diffusion is a random walk with no retention of information about where the diffusing species was before its current location. In Chap. 3 Sect. 1, the diffusion equation is derived from thermodynamic considerations and the continuity equation (law of conservation of mass). 

Because all the molecules in a solution are diffusing throughout the entire volume, any concentration differences tend to be smoothed out because there is a net migration from regions of high concentration to low concentration. Net diffusion occurs because of concentration gradients, such as dc/dr, where c = [B] for notational convenience. The flux 

The flux of particles is in the opposite sense to the direction of the concentration gradient increase. Equation (6) is Fick s first law, which has been experimentally confirmed by many workers. D is the mutual diffusion coefficient (units of m2 s 1), equal to the sum of diffusion coefficients for both reactants, and for mobile solvents D 10 9 m2 s D = DA + jDb. The diffusion coefficient is approximately inversely dependent upon viscosity and is discussed in Sect. 6.9. As spherical symmetry is appropriate for the diffusion of B towards a spherically symmetric A reactant, the flux of B crossing a spherical surface of radius r is given by eqn. (6) where r is the radial coordinate. The total number of reactant B molecules crossing this surface, of area 4jrr2, per second is the particle current I 

The current of B towards A crossing the surface of radius r, less the current crossing a surface of radius r + dr 

Having specified the initial distribution of B reactants about A reactants in eqn. (3), it is necessary to find out what the distribution will be at subsequent times, subject to the two boundary conditions of eqns. (4) and 

The Diffusion Equation Describes How Concentration Gradients Change over Time and Space 

The total accumulation of particles can also be computed in a different way, as (volume) x (midpoint concentration) at two different times, t and t + At, or AAx[c(x - - Ax/2, t At) - c(x -t Ax/2, t)]. Combining these two calculations of the same property gives 

Substituting Equation (18.2) into Equation (18.8) gives Fick s second law, also called the diffusion equation. 

318 Chapter 18. Physical Kinetics Diffusion, Permeation Flow 

The diffusion equation is a partial differential equation. You solve it to find c x,y,z,t), the particle concentration as a function of spatial position and time. To solve it, you need to know two boundary conditions and one initial condition, that is, two pieces of information about the concentration at particular points in space and one piece of information about the concentration distribution at some particular time. Example 18.1 shows how to solve the diffusion equation for the permeation of particles through a planar film or membrane. 

From now on, we assume that the diffusion ( = molecular) transport is not negligible, so we need some expression relating the diffusion flux to measurable quantities, e.g., 

The first term on the right-hand side can be expanded as 

The right-hand side is the sum of three terms describing diffusion, advection, and chemical reaction, respectively. 

For the one-dimensional equation with x as the space variable, the diffusion equation is a partial differential equation of the first order in time and the second order in x. It therefore requires concentration to be known everywhere at a given time (in general =0) and, at any time t 0, concentration, flux, or a combination of both, to be known in two points (boundary conditions). In the most general case, the diffusion equation is a partial differential equation of the first order in time and the second order in the three space coordinates x, y, z. Concentration or flux conditions valid at any time 0 must then be given along the entire boundary. 

Taking the one-dimensional problem with x as the space variable as an example, boundary conditions can be of three forms  

The diffusion of particles can be described by the second-order differential equation (Pick s second law)  

Equation (2) implies that the concentration C x,y,z) in a certain point x,y,z) can change in time only when the second derivatives of C with respect to X, y or z are all not zero. Equation (2) cannot be used as such for the description of self-diffusion, since self-diffusion even occurs when the macroscopic concentration is the same everywhere in the system. 

Note that the differentiation in the right hand part of Eq. (3) is with respect to the 7 co-ordinates. P(7 7, t) is not only a solution of Eq. (3) but, since it is a probability function, it must also fulfil the normalization equation  

The diffusion equation (3) can be written in a more concise form as  

In this chapter, we will review various solution techniques for the diffusion equation, which is generally dehned as the mass transport equation with diffusive terms. These techniques will be applied to chemical transport solutions in sediments. There are also a number of applications to chemical transport in biohlms. There are many other applications of the diffusion equation, including most of the topics of this text, but they require more background with regard to the physics of mixing processes, which will be addressed in later chapters. 

What is mass (or chemical) transport It is the transport of a solute (the dissolved chemical) in a solvent (everything else). The solute is the dissolvee and the solvent is the dissolver. There are liquids that are generally classified as solvents because they typically play that role in industry. Some examples would be degreasing and dry cleaning solvents, such as trichloroethylene. In environmental applications, these solvents are the solutes, and water or air is usually the solvent. In fact, when neither water nor air is the solvent, the general term nonaqueous phase liquid is applied. A nonaqueous phase liquid is defined as a liquid that is not water, which could be composed of any number of compounds. 

The substance being transported can be either dissolved (part of the same phase as the water) or particulate substances. We will develop the diffusion equation by considering mass conservation in a fixed control volume. The mass conservation equation can be written as 

Flux rate - Flux rate + Source - Sink = Accumulation IN OUT rate rate 

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Piela L, Kostrowicki J and Scheraga H A 1989 The multiple-minima problem in the conformational analysis of molecules. Deformation of the potential energy hypersurface by the diffusion equation method J. Phys. Chem. 93 3339... 

Most of our ideas about carrier transport in semiconductors are based on tire assumption of diffusive motion. Wlren tire electron concentration in a semiconductor is not unifonn, tire electrons move diffuse) under tire influence of concentration gradients, giving rise to an additional contribution to tire current. In tliis motion, electrons also undergo collisions and tlieir temporal and spatial distributions are described by the diffusion equation. The... 

J. Kostrowicki and H.A. Scheraga, Application of the diffusion equation method for global optimization to oligopeptides, J. Phys. Chem. 96 (1992), 7442-7449. M. Levitt, A simplified representation of protein confomations for rapid simulation of protein folding, J. Mol. Biol. 104 (1976), 59-107. 

Equipped with a proper boundary condition and a complete solution for the mass mean velocity, let us now turn attention to the diffusion equations (4.1) which must be satisfied everywhere. Since all the vectors must... 

The rate of chemical diffusion in a nonfiowing medium can be predicted. This is usually done with an equation, derived from the diffusion equation, that incorporates an empirical correction parameter. These correction factors are often based on molar volume. Molecular dynamics simulations can also be used. 

Diffusion and Mass Transfer During Leaching. Rates of extraction from individual particles are difficult to assess because it is impossible to define the shapes of the pores or channels through which mass transfer (qv) has to take place. However, the nature of the diffusional process in a porous soHd could be illustrated by considering the diffusion of solute through a pore. This is described mathematically by the diffusion equation, the solutions of which indicate that the concentration in the pore would be expected to decrease according to an exponential decay function. 

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,... 

Example Consider the diffusion equation, with boundary and initial conditions. 

We denote by C the value of c(x , t) at any time. Thus, C is a function of time, and differential equations in C are ordinary differential equations. By evaluating the diffusion equation at the ith node and replacing the derivative with a finite difference equation, the following working equation is derived for each node i, i = 2,. . . , n (see Fig. 3-52). 

The diffusion equation for the falling-rate drying period for a slab can be derived from the diffusion equation if one assumes that the surface is diy or at an equilibrium moisture content and that the initial moisture distribution is uniform. For these conditions, the following equation is obtained ... 

Equation (12-31) assumes that Df is constant however, Df is rarely constant but varies with moisture content, temperature, and humidity. For long diying times, Eq. (12-31) simphfies to a limiting form of the diffusion equation as... 

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

In the kinetics of formation of carbides by reaction of the metal widr CH4, the diffusion equation is solved for the general case where carbon is dissolved into tire metal forming a solid solution, until the concentration at the surface reaches saturation, when a solid carbide phase begins to develop on the free surface. If tire carbide has a tirickness at a given instant and the diffusion coefficient of carbon is D in the metal and D in the carbide. Pick s 2nd law may be written in the form (Figure 8.1)... 

The odor threshold for most atmospheric pollutants may be found in the literature (1). By properly applying the diffusion equations, one can calculate the height of a stack necessary to reduce the odor to less than its threshold at the ground or at a nearby structure. A safety factor of two orders of magnitude is suggested if the odorant is particularly objectionable. 

To be specific, we consider the two-dimensional growth of a pure substance from its undercooled melt in about its simplest form, where the growth is controlled by the diffusion of the latent heat of freezing. It obeys the diffusion equation and appropriate boundary conditions [95]... 

At small driving forces a completely fiat interface cannot move at a constant speed. This is basically a result of the inherent scaling property of the diffusion equation, which scales lengths proportional to the square-root of time, so an advancing interface would slow down with time. 

For an ideal gas, the total molar concentration Cj is constant at a given total pressure P and temperature T. This approximation holds quite well for real gases and vapours, except at high pressures. For a liquid however, CT may show considerable variations as the concentrations of the components change and, in practice, the total mass concentration (density p of the mixture) is much more nearly constant. Thus for a mixture of ethanol and water for example, the mass density will range from about 790 to 1000 kg/m3 whereas the molar density will range from about 17 to 56 kmol/m3. For this reason the diffusion equations are frequently written in the form of a mass flux JA (mass/area x time) and the concentration gradients in terms of mass concentrations, such as cA. 

Liquid phase diffusivities are strongly dependent on the concentration of the diffusing component which is in strong contrast to gas phase diffusivities which are substantially independent of concentration. Values of liquid phase diffusivities which are normally quoted apply to very dilute concentrations of the diffusing component, the only condition under which analytical solutions can be produced for the diffusion equations. For this reason, only dilute solutions are considered here, and in these circumstances no serious error is involved in using Fick s first and second laws expressed in molar units. 

As noted previously, for equimolecular counterdiffusion, the film transfer coefficients, and hence the corresponding HTUs, may be expressed in terms of the physical properties of the system and the assumed film thickness or exposure time, using the two-film, the penetration, or the film-penetration theories. For conditions where bulk flow is important, however, the transfer rate of constituent A is increased by the factor Cr/Cgm and the diffusion equations can be solved only on the basis of the two-film theory. In the design of equipment it is usual to work in terms of transfer coefficients or HTUs and not to endeavour to evaluate them in terms of properties of the system. 

In the presence of a large amount of supporting electrolyte, the diffusion equation in a static solution without any convection is expressed as... 

The basic equation and boundary conditions for the symmetrical fluctuations are the same as those for the asymmetrical fluctuations except for the superscript s. The diffusion equation is written in the form... 

Then the diffusion equation for the fluctuation of the metal ion concentration is given by Eq. (68), and the mass balance at the film/solution interface is expressed by Eq. (69). These fluctuation equations are also solved with the same boundary condition as shown in Eq. (70). 

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. 

Introducing a new variable of 0 equal to, the diffusion equation (Eq. (2)) can be transformed into... 

The two BCs of the TAP reactor model (1) the reactor inlet BC of the idealization of the pulse input to tiie delta function and (2) the assumption of an infinitely large pumping speed at the reactor outlet BC, are discussed. Gleaves et al. [1] first gave a TAP reactor model for extracting rate parameters, which was extended by Zou et al. [6] and Constales et al. [7]. The reactor equation used here is an equivalent form fi om Wang et al. [8] that is written to be also applicable to reactors with a variable cross-sectional area and diffusivity. The reactor model is based on Knudsen flow in a tube, and the reactor equation is the diffusion equation ... 

A thin slab of solid material dries first by evaporation from the top surface and then by diffusion from the interior of the solid. The water movement is approximated by the diffusion equation... 

For analytical solutions, it is more convenient to work with nondimensional forms of the diffusion equations. We choose the following nondimensional substitutions. The time coordinate / is replaced by the nondimensional parameter /, and a is the root radius ... 

Therefore if the carbon substrate is present at sufficiently high concentration anywhere in the rhizosphere (i.e., p p, ax), the microbial biomass will increase exponentially. Most models have considered the microbes to be immobile and so Eq. (33) can be solved independently for each position in the rhizosphere provided the substrate concentration is known. This, in turn, is simulated by treating substrate-carbon as the diffusing solute in Eq. (32). The substrate consumption by microorganisms is considered as a sink term in the diffusion equation, Eq. (8). 

In order to calculate the tip current response, the diffusion equations must be solved subject to the boundary and initial conditions of the system. Prior to the potential step. 

In addition to the fact that MPC dynamics is both simple and efficient to simulate, one of its main advantages is that the transport properties that characterize the behavior of the macroscopic laws may be computed. Furthermore, the macroscopic evolution equations can be derived from the full phase space Markov chain formulation. Such derivations have been carried out to obtain the full set of hydrodynamic equations for a one-component fluid [15, 18] and the reaction-diffusion equation for a reacting mixture [17]. In order to simplify the presentation and yet illustrate the methods that are used to carry out such derivations, we restrict our considerations to the simpler case of the derivation of the diffusion equation for a test particle in the fluid. The methods used to derive this equation and obtain the autocorrelation function expression for the diffusion coefficient are easily generalized to the full set of hydrodynamic equations. 

The diffusion equation describes the evolution of the mean test particle density h(r,t) at point r in the fluid at time t. Denoting the Fourier transform of the local density field by hk(t), in Fourier space the diffusion equation takes the form... 

To derive the diffusion equation we return to Eq. (30), which we write as... 

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