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Equation diffusion

Measurement of the diffusion of an atom through a perfect crystal will give a concentration profile. The diffusion coefficient is extracted from the concentration profile by solution of one of two diffusion equations. For one-dimensional diffusion, along x, they are [Pg.476]

This is known as Fick s second law of diffusion or more commonly as the diffusion equation. In these equations, J is called the flux of the diffusing species, with units of [amount of substance (atoms or equivalent units) m2 s-1], c is the concentration of the diffusing species, with units of [amount of substance (atoms or equivalent units) m-3] at position x (m) after time t (s) D is the diffusion coefficient, units (m2 s 1). [Pg.476]


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... [Pg.704]

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... [Pg.2359]

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... [Pg.2536]

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]. [Pg.2721]

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... [Pg.2883]

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... [Pg.3064]

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... [Pg.3066]

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... [Pg.3071]

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. [Pg.223]

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... [Pg.19]

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... [Pg.30]

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... [Pg.61]

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. [Pg.109]

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. [Pg.115]

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... [Pg.259]

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. [Pg.87]

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... [Pg.92]

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... [Pg.66]

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,... [Pg.458]

Example Consider the diffusion equation, with boundary and initial conditions. [Pg.479]

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). [Pg.479]

Hyperbolic Equations The most common situation yielding hyperbohc equations involves unsteady phenomena with convection. Two typical equations are the convective diffusive equation... [Pg.481]

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... [Pg.481]

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. [Pg.481]

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... [Pg.1181]

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 ... [Pg.1181]

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... [Pg.1181]

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. [Pg.1181]

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. [Pg.2035]

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... [Pg.108]


See other pages where Equation diffusion is mentioned: [Pg.400]    [Pg.703]    [Pg.1540]    [Pg.3065]    [Pg.3066]    [Pg.3068]    [Pg.3071]    [Pg.261]    [Pg.414]    [Pg.156]    [Pg.349]    [Pg.434]    [Pg.380]    [Pg.385]    [Pg.481]    [Pg.481]    [Pg.1179]    [Pg.1179]    [Pg.1516]    [Pg.2183]    [Pg.211]   
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A critique of the diffusion equation and molecular pair treatments

Adsorption diffusion equation

Advection-reaction-diffusion equation

Advective-diffusion equation

Advective-diffusion equation, sediment

Analytical Solutions of the diffusion equation

Apparent diffusion coefficient, equation

Apparent diffusion coefficient, equation describing

Approximate solutions equations with diffusion

Arrhenius equation diffusion

Atmospheric diffusion equation

Atmospheric diffusion equation boundary conditions

Atmospheric diffusion equation constraints

Atmospheric diffusion equation errors

Atmospheric diffusion equation parameters

Atmospheric diffusion equation solution

Atmospheric diffusion equation validity

Averaged diffusion equation

Boundary value problems diffusion equation

Brownian Motion, Levy Flight, and the Diffusion Equations

Brownian motion diffusion equation with fractional

Brownian motion fractional diffusion equations

Cahn-Hilliard diffusion equation

Channel convective-diffusion equations

Chemical separation equations with diffusion

Classical finite difference equations, diffusion

Constitutive equations, diffusion

Constrained Brownian motion diffusion equation

Convection convective-diffusion equation

Convection-Diffusion Equation with Reactions

Convection-diffusion equation

Convection-diffusion-reaction equation

Convective Diffusion Equation in Two and Three Dimensions

Convective diffusion equation

Convective diffusion equation boundary layer

Convective diffusion equation for

Convective diffusion equation species

Convective diffusion equation with chemical reactions

Cooperative diffusion coefficient equation

Derivatives fractional diffusion equations

Different Forms of the Diffusion Equation

Differential Equations of Diffusion

Differential equation diffusion impedance

Differential equations diffusion

Differential equations diffusion modeling, boundary value problems

Differential equations for diffusion

Diffuse double layer, equation

Diffuse double layer, equation state

Diffuse reflectance Kubelka-Munk equation

Diffusion Basic equations

Diffusion Darken’s equation

Diffusion Equation Approach

Diffusion Equation for Two-component Gas Mixture (Without and With a Potential Field)

Diffusion Equation in Cylindrical and Spherical Coordinates

Diffusion Equation in Miscellaneous Coordinate Systems

Diffusion Fokker-Planck equation

Diffusion Form of LG Model Equations

Diffusion From Equation of Motion

Diffusion Reference equation

Diffusion Smoluchowski equation

Diffusion Stefan-Maxwell equation

Diffusion Stokes-Einstein equation

Diffusion boundary layer equation

Diffusion coefficient Einstein equation

Diffusion coefficient Fokker-Planck equation

Diffusion coefficient, equation

Diffusion coefficients Hayduk-Laudie equation

Diffusion coefficients Stokes-Einstein equation

Diffusion coefficients Wilke-Chang equation

Diffusion control rate equations

Diffusion dispersion equation

Diffusion equation 366 index

Diffusion equation Einstein

Diffusion equation Einstein-Smoluchowski

Diffusion equation Laplace transforms

Diffusion equation anisotropic

Diffusion equation backward

Diffusion equation boundary conditions

Diffusion equation concentration-dependent diffusivities

Diffusion equation concentrations

Diffusion equation constant diffusivities

Diffusion equation cylindrical, steady-state

Diffusion equation diffusivity diagonalization method

Diffusion equation dimensionless

Diffusion equation direct solution

Diffusion equation finite slab

Diffusion equation general formulation

Diffusion equation geometrical interpretation

Diffusion equation line source

Diffusion equation linearization

Diffusion equation localized sources

Diffusion equation method

Diffusion equation multi-component systems

Diffusion equation point source

Diffusion equation region

Diffusion equation regularization

Diffusion equation scaling

Diffusion equation short chains

Diffusion equation solutions

Diffusion equation space-fractional

Diffusion equation species

Diffusion equation spherical, steady-state

Diffusion equation steady-state

Diffusion equation superposition

Diffusion equation time-dependent

Diffusion equation time-dependent boundary conditions

Diffusion equation variable diffusivity, steady-state

Diffusion equation variational interpretation

Diffusion equation, chemical species

Diffusion equation, constrained Brownian

Diffusion equation, forced rotational

Diffusion equation, interpretation

Diffusion equation, thermal

Diffusion equations binary

Diffusion equations droplet phase reactions

Diffusion equations for

Diffusion equations in three dimensions

Diffusion equations multicomponent, derivation

Diffusion equations simplified

Diffusion equations simulation

Diffusion equations, liquid phase chemical

Diffusion equations, liquid phase chemical dynamics

Diffusion equations, redox properties

Diffusion flux concentrated sources, equations

Diffusion general dynamic equation

Diffusion layer equation

Diffusion mass, equation

Diffusion migration equation

Diffusion nonlinear equations

Diffusion overpotential equations

Diffusion permeability equations

Diffusion problems equation

Diffusion rate, equation

Diffusion separation equations with

Diffusion system equation

Diffusion transport equation through membrane

Diffusion-advection equation

Diffusion-annihilation equation

Diffusion-current equation

Diffusion-limited reactions, Smoluchowski equation

Diffusion/reaction mass transfer equation

Diffusive mixing Dynamical equation

Diffusive/advective transport /reaction equation

Diffusive/advective transport equation

Diffusivities equation

Diffusivities equations for

Dimensionless Form of the Generalized Mass Transfer Equation with Unsteady-State Convection, Diffusion, and Chemical Reaction

Disordered systems fractional diffusion equations

Drift-diffusion equation

Effective diffusion coefficient relations from different equations

Effects of Rotational Diffusion on Fluorescence Anisotropies The Perrin Equation

Electrons diffusion equation

Elliptic equations diffusion modeling

Energy diffusion equation, generalization

Equation fractional diffusion

Equation of Convective Diffusion

Equation of Diffusion

Equation with diffusion, approximate

Equations for the diffusive flux (Ficks law)

Equations of diffusion and reaction

Equations parabolic diffusion

Examples of kinetic equations without diffusion

Fick diffusion equation

Fick s diffusion equation

Fickian diffusion equation

Ficks Laws and the Diffusion Equations

Flow-diffusion equation

Flux Equations for Multicomponent Diffusion

Flux equations facilitated diffusion

Fokker Planck equation anomalous diffusion

Fokker-Planck equation fractional rotational diffusion

Fractional Klein Kramers equation diffusion

From the diffusion equation to escape and survival probabilities

Fundamental Equations of Turbulent Diffusion

Fundamental equations atmospheric diffusion equation

Fundamental equations eddy diffusivity

Gaussian diffusion equation

Generalization of the Nonlinear Phase Diffusion Equation

Generalized diffusion equation

Global distribution 263 Diffusion equation

Goldman diffusion equation

HA Diffusion Equation with Higher Order Derivatives

Harmonic amplitudes perturbative equations, joint diffusion

Heat diffusion equation

Heat transfer concentrated diffusion flux equations

Homogeneity, fractional diffusion equations

Hydrocarbon diffusivity equation

Hydrodynamic methods convective-diffusion equation

Hydrodynamic methods convective-diffusion equation, solution

Hydrodynamic repulsion and the diffusion equation

Hyperbolic reaction-diffusion equations

Investigation diffusion equation, fractal

Kinetic equations for reactions with diffusion

Kinetic equations of reactions without diffusion

Kinetic rate equations, diffusion control

Kramers diffusion equation

Langevin-diffusion equation

Laplacian diffusion equation

Lattice diffusion equation

Laws Laplacian diffusion equation

Laws reaction-diffusion equations

Laws time-dependent diffusion equation

Linear diffusion equation

Linear diffusion equations, solution

Long-range transfer and the diffusion equation

Macropore diffusion equation

Mass Transport in Binary Mixtures and the Diffusion Equation

Mass transfer diffusion equation

Master diffusion equation approximation

Master equation and diffusion

Molecules characterization, diffusion equation

Monomer diffusion equation

Multicomponent diffusion equation

Multicomponent diffusion equations solution

Multicomponent diffusion flux equations

Multigroup diffusion equations

Multigroup diffusion equations multi-dimensional

Multigroup diffusion equations time-dependent

Multivariate diffusion equation

Nemst-Planck diffusion equations, applied

Nemst-Planck diffusion equations, applied membranes

Nemst-Planck equation diffusive term

Nernst-Einstein diffusion equation

Nernst-Einstein equation molar conductivity-diffusion coefficient

Nondimensionalization convective diffusion equation

Numerical Solution of the diffusion equation

One-velocity Diffusion Equation

Overall parabolic diffusion equation

Oxidation-enhanced diffusion equation

Oxygen diffusion equation

Parameters in the Atmospheric Diffusion Equation

Partial differential equations diffusion modeling

Phase diffusion equation

Phase diffusion equation coefficient

Phase diffusion equation linearized

Phase diffusion equation nonlinear

Phenomenological Derivation of the Reaction-Diffusion Equation

Physical derivation of the multicomponent diffusion equation

Polymer diffusion coefficient equation

Position-dependent diffusion equation

Quasilinear diffusion equation

Reaction kinetics diffusion equation

Reaction-diffusion equation

Reaction-diffusion equation , model

Reaction-diffusion equation , model system

Reaction-diffusion equation boundaries

Reaction-diffusion equation general form

Reaction-diffusion equation generalized

Reaction-diffusion equation with electric field

Reaction-diffusion master equation

Reactor, batch equations with diffusion

Relaxation fractional derivatives diffusion equation

Rotary diffusion equation

Rotating disk electrode convective-diffusion equation, solution

Rotational diffusion equation

Rotational diffusion equation dielectric relaxation

Rotational diffusion equation, forces

Scaling of the Diffusion Equation

Similarity solutions linear diffusion equation

Similarity solutions nonlinear diffusion equation

Simple Diffusion Equation

Smoluchowski Equation and Diffusion Tensor

Smoluchowski equation including both translational and rotational diffusion

Smoluchowski equation, diffusion controlled reactions

Smoluchowsky equation for diffusion

Solid-state diffusion continuity equation

Solution diffusion model transport equation through membrane

Solution of Diffusion Equation Near an Interface

Solution of Parabolic Partial Differential Equations for Diffusion

Solution of a diffusion equation

Solution of the Atmospheric Diffusion Equation for an Instantaneous Source

Solution of the Multicomponent Diffusion Equations

Solution of the Transient Gas-Phase Diffusion Problem Equations

Solution of the Transient Gas-Phase Diffusion Problem Equations (11.4) to

Solution of the diffusion equation when Le

Solution of the reaction-diffusion equations

Solution to the Diffusion Equation with a Step in Concentration

Solutions of the Radial Diffusion Equation

Solutions of the Steady-State Atmospheric Diffusion Equation

Solutions of the diffusion equation

Solutions of the diffusion equation parallel flux

Solutions to the Diffusion Equation

Solutions to the diffusion equation with no solute elimination or generation

Solutions to the diffusion equation with solute binding and elimination

Solving the Diffusion Equations

Some Cases for which there is no Solution of the Diffusion Equation

Some Properties of the Nonlinear Phase Diffusion Equation

Some simple solutions to the diffusion equation at steady state

Stationary medium, particle diffusion equations

Stefan-Maxwell equations molecular diffusion

Stokes-Einstein equation Brownian diffusion coefficient

Stokes-Einstein equation for diffusion

Stokes-Einstein equation size-dependent diffusion

The Binary Diffusion Equations

The Combined Slowing-down and Diffusion Equation

The Convective Diffusion Equation

The Discretized Diffusion Equation

The General Equations of Diffusion and Flow in a Straight Tube

The Multicomponent Diffusion Equations

The Rate Equation for Film Diffusion

The Time-Dependent Diffusion Equation

The differential equations of diffusion

The diffusion equation

The diffusion equation assumptions and applications

Towards the diffusion equation analysis

Translational diffusion coefficient equation

Transport Equation with Turbulent Diffusion Coefficients

Transport equations diffusive flow

Turbulent flux of a scalar quantity averaged diffusion equation

Turing Instabilities in Hyperbolic Reaction-Diffusion Equations

Unsteady-state diffusion equation

Variable Reaction-Diffusion Equations

Variational Interpretation of the Diffusion Equation

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