Diffusivities equation

Two applications of the flucUiathig diffusion equation are made here to illustrate tlie additional infonnation the flucUiations provide over and beyond the detenninistic behaviour. Consider an infinite volume with an initial concentration, c, that is constant, Cq, everywhere. The solution to the averaged diffusion equation is then simply (c) = Cq for all t. However, the two-time correlation fiinction may be shown [26] to be... 

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... 

If tire diffusion coefficient is independent of tire concentration, equation (C2.1.22) reduces to tire usual fonn of Pick s second law. Analytical solutions to diffusion equations for several types of boundary conditions have been derived [M]- In tlie particular situation of a steady state, tire flux is constant. Using Henry s law (c = kp) to relate tire concentration on both sides of tire membrane to tire partial pressure, tire constant flux can be written as... 

The diffusion layer widtli is very much dependent on tire degree of agitation of tire electrolyte. Thus, via tire parameter 5, tire hydrodynamics of tire solution can be considered. Experimentally, defined hydrodynamic conditions are achieved by a rotating cylinder, disc or ring-disc electrodes, for which analytical solutions for tire diffusion equation are available [37, 4T, 42 and 43]. 

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... 

Thus far we have considered systems where stirring ensured homogeneity witliin tire medium. If molecular diffusion is tire only mechanism for mixing tire chemical species tlien one must adopt a local description where time-dependent concentrations, c r,f), are defined at each point r in space and tire evolution of tliese local concentrations is given by a reaction-diffusion equation... 

This complex Ginzburg-Landau equation describes the space and time variations of the amplitude A on long distance and time scales detennined by the parameter distance from the Hopf bifurcation point. The parameters a and (5 can be detennined from a knowledge of the parameter set p and the diffusion coefficients of the reaction-diffusion equation. For example, for the FitzHugh-Nagumo equation we have a = (D - P... 

In order to investigate such front instabilities quantitatively one may derive an equation for the profile ( i(y, t) of the front directly from the reaction-diffusion equation. This Kuramoto-Sivashinsky equation 1691... 

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. 

The starting point for developing the model is the set of diffusion equations for a gas mixture in the presence of temperature, pressure and composition gradients, and under the influence of external forces." These take the following form... 

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 finite element results obtained for various values of (3 are compared with the analytical solution in Figure 2.27. As can be seen using a value of /3 = 0.5 a stable numerical solution is obtained. However, this solution is over-damped and inaccurate. Therefore the main problem is to find a value of upwinding parameter that eliminates oscillations without generating over-damped results. To illustrate this concept let us consider the following convection-diffusion equation... 

Nguen, N. and Reynen, J., 1984. A space-time least-squares finite element scheme for advection-diffusion equations. Cornput. Methods Appl Mech. Eng. 42, 331- 342. 

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. 

Sorption Rates in Batch Systems. Direct measurement of the uptake rate by gravimetric, volumetric, or pie2ometric methods is widely used as a means of measuring intraparticle diffusivities. Diffusive transport within a particle may be represented by the Fickian diffusion equation, which, in spherical coordinates, takes the form... 

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. 

Mass Transport. An expression for the diffusive transport of the light component of a binary gas mixture in the radial direction in the gas centrifuge can be obtained directly from the general diffusion equation and an expression for the radial pressure gradient in the centrifuge. For diffusion in a binary system in the absence of temperature gradients and external forces, the general diffusion equation retains only the pressure diffusion and ordinary diffusion effects and takes the form... 

Under conditions of limiting current, the system can be analyzed using the traditional convective-diffusion equations. For example, the correlation for flow between two flat plates is... 

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). 

Hyperbolic Equations The most common situation yielding hyperbohc equations involves unsteady phenomena with convection. Two typical equations are the convective diffusive equation... 

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... 

Another method often used for hyperbohc equations is the Mac-Cormack method. This method has two steps, and it is written here for the convective diffusion equation. 

Diffusion-controlled mass transfer is assumed when the vapor or liquid flow conforms to Tick s second law of diffusion. This is stated in the unsteady-state-diffusion equation using mass-transfer notation as... 

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... 

Diffusion equations mav also be used to study vapor diffusion in porous materi s. It should be clear that aU estimates based on relationships that assume constant diffusivity are approximations. Liquid diffusivity in sohds usually decreases with moisture concentration. Liquid and vapor diffusivity also change, and material shrinks during diying. 

Salt flux across a membrane is due to effects coupled to water transport, usually negligible, and diffusion across the membrane. Eq. (22-60) describes the basic diffusion equation for solute passage. It is independent of pressure, so as AP — AH 0, rejection 0. This important factor is due to the kinetic nature of the separation. Salt passage through the membrane is concentration dependent. Water passage is dependent on P — H. Therefore, when the membrane is operating near the osmotic pressure of the feed, the salt passage is not diluted by much permeate water. 

A situation which is frequently encountered in tire 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 tire cavity will be mainly by diffusion. If the mainstream gas velocity is high, there exists the possibility of turbulent flow at tire mouth of tire cavity, but since this is rare in vapour deposition processes, the assumption that the gas widrin dre cavity is stagnant is a good approximation. The appropriate solution of dre diffusion equation for the steady-state transport of material tlrrough the stagnant layer in dre cavity is... 

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