Equation of Diffusion

A fluid, which is not flowing, contains small particles in Brownian motion. Gradients exist in the concentration of particles, all of which are of the same size. Concentrations are small, however,, so that any small flow.s that accompany diffusion can be neglected. A balance can be can ied out on the number of particles in an eiementnl volume of fluid fixed in space (Fig, 2.1). as follows The rale at which particles enter the elemental volume across the face ABCO is 

The nel rate o( iransptm inio the element for these two faces is obtained by summing the two previous expressions to give 

The rate of change of the number of particles in the elemental volume SxSy z is given by 

The relationship between the flux and the concentration gradient depends on an experimental observation A one-dimensional gradient in the particle concentration is set up in a fluid by fixing the concentration at two parallel planes. The fluid is isothermal and stationary. It is observed that the rate at which particles are transported from the high to the low concentration (particlcs/cm- sec) is proportional to the local concentration gradient, Bn/dx  

If the properties of the fluid are the same in all directions, it is said to be isotropic. This is the usual case, and D then has the same value for diffusion in any direction. 

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Using the Stokes-Einstein equation of diffusion coefficient ... 

Referring to the Pick Equations of diffusion, let us now reexamine 1/T, the jump frequency, so as to relate diffusion processes to lattice vibration processes. The reciinrocal of the time of stay is the jump frequency and is related to the diffusion coefficient by ... 

Burns and Curtiss (1972) and Burns et al. (1984) have used the Facsimile program developed at AERE, Harwell to obtain a numerical solution of simultaneous partial differential equations of diffusion kinetics (see Eq. 7.1). In this procedure, the changes in the number of reactant species in concentric shells (spherical or cylindrical) by diffusion and reaction are calculated by a march of steps method. A very similar procedure has been adopted by Pimblott and La Verne (1990 La Verne and Pimblott, 1991). Later, Pimblott et al. (1996) analyzed carefully the relationship between the electron scavenging yield and the time dependence of eh yield through the Laplace transform, an idea first suggested by Balkas et al. (1970). These authors corrected for the artifactual effects of the experiments on eh decay and took into account the more recent data of Chernovitz and Jonah (1988). Their analysis raises the yield of eh at 100 ps to 4.8, in conformity with the value of Sumiyoshi et al. (1985). They also conclude that the time dependence of the eh yield and the yield of electron scavenging conform to each other through Laplace transform, but that neither is predicted correctly by the diffusion-kinetic model of water radiolysis. 

The cage effect was also analyzed for the model of diffusion of two particles (radical pair) in viscous continuum using the diffusion equation [106], Due to initiator decomposition, two radicals R formed are separated by the distance r( at / = 0. The acceptor of free radicals Q is introduced into the solvent it reacts with radicals with the rate constant k i. Two radicals recombine with the rate constant kc when they come into contact at a distance 2rR, where rR is the radius of the radical R Solvent is treated as continuum with viscosity 17. The distribution of radical pairs (n) as a function of the distance x between them obeys the equation of diffusion ... 

By changing the variables, the partial differential equation of diffusion has been turned into an ordinary differential equation... 

There has been very much effort devoted to the solution of the diffusion equation of motion for a reactant particle executing Brownian motion. The Euler equation of diffusion... 

Master equations of diffusion type were characterized by the property that the lowest non-vanishing term in their -expansion is not a macroscopic deterministic equation but a Fokker-Planck equation. One may ask whether it is still possible to obtain an approximation in the form of a deterministic equation, although Q is no longer available as an expansion parameter. The naive device of omitting from the Fokker-Planck equation the term involving the second order derivatives is, of course, wrong the result would depend on which of the various equivalent forms (4.1), (4.7), (4.17), (4.18) one chooses to mutilate in this way. 

Master equations of diffusion type are characterized by y0 = p0. In that case (5.8) and (5.9) reduce to... 

THE GENERAL EQUATIONS OF DIFFUSION AND FLOW IN A STRAIGHT TUBE... 

In the absence of catalysis on the surface, similarity of the concentration and temperature fields is achieved precisely at the ignition limit if the coefficients of diffusion and thermal diffusivity are equal, since in this case both the diffusion gradient and the temperature gradient at the igniting surface are equal to zero, and the equations of diffusion and thermal conductivity with the chemical reaction may be reduced to the form of an identity (see our work on flame propagation [3]). 

The vector qa gives the direction of flow and its magnitude in grams per cm2 per second at a given point. The general equation of diffusion has the form... 

This similarity was established in [2] by consideration of the second-order differential equations of diffusion and heat conduction. Under the assumptions made about the coefficient of diffusion and thermal diffusivity, similarity of the fields, and therefore constant enthalpy, in the case of gas combustion occur throughout the space this is the case not only in the steady problem, but in any non-steady problem as well. It is only necessary that there not be any heat loss by radiation or heat transfer to the vessel walls and that there be no additional (other than the chemical reaction) sources of energy. These conditions relate to the combustion of powders and EM as well, and were tacitly accounted for by us when we wrote the equations where the corresponding terms were absent. 

After having integrated equation (F.12), one can obtain the equation of diffusion... 

The latter form is the basic equation of diffusion generally identified as Fick s first law, formulated in 1855 [13]. Fick s first law, of course, can be deduced from the postulates of irreversible thermodynamics (Section 3.2), in which fluxes are linearly related to gradients. It is historically an experimental law, justified by countless laboratory measurements. The convergence of all these approaches to the same basic law gives us confidence in the correctness of that law. However, the approach used here gives us something more. 

Diffusion Pattern from a Continuous Point Source—The distribution of particles from a point source in a moving fluid can be determined provided we assume that the concentration gradients in the direction of fluid motion are small compared to those at right angles to it. If C, is defined as the concentration of particles over a unit area of a plane horizontal surface downstream and to one side of the mean path of the diffusing stream from a point source, then the equation of diffusion at any point x downstream and at a distance y from the mean path is... 

Noncompartmental models were introduced as models that allow for transport of material through regions of the body that are not necessarily well mixed or of uniform concentration [248]. For substances that are transported relatively slowly to their site of degradation, transformation, or excretion, so that the rate of diffusion limits their rate of removal from the system, the noncompartmental model may involve diffusion or other random walk processes, leading to the solution in terms of the partial differential equation of diffusion or in terms of probability distributions. A number of noncompartmental models deal with plasma time-concentration curves that are best described by power functions of time. 

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. 

Comparison of the computerized results obtained for the kinetic curves (Fig. 2a) reveals a very interesting feature of the IE process under discussion. Kinetic curves F (Fo) for variants I and III, resolved when a favorable isotherm of the B ion and the relation D >Dg apply, are described by the kinetic equation of diffusion into a spherical bead with constant diffiisivity D. In other words the kinetic curves F(Fo) coincide with the isotope exchange kinetic curves if a sharp profile appears in the bead and do not coincide if the exchange zone is greatly spread. The remaining kinetic curves in Fig. 2 formally correspond to the exchange process where varies in value. This is especially evident when comparing kinetic curves II and I.e (Fig. 2a). 

As a consequence, the telegraph equations (8.17) become the equations of diffusion... 

The usual differential equation of diffusion processes is due to Pick... 

This process governs the rate of deposition of the molecules of nonvolatile compounds on the surface of gas ducts, and contributes to broadening of the chromatographic zones. Being of the order of 0.1 pm at STP, the mean free path of molecules, which is inversely proportional to pressure, reaches 1 cm only at about 0.01 mmHg. In dense enough gas, in the absence of convective flow, the macroscopic picture of migration of molecules (as well as of aerosol particulates) is described by the equations of diffusion. The mean squared diffusional displacement z2D of molecules, the time of diffusion t and the mutual diffusion coefficient >i 2 are related by ... 

The 15 chapters fall into three parts. Part I (Chapters 1-6) deals with the basic equations of diffusion in multicomponent systems. Chapters 7-11 (Part II) describe various models of mass and energy transfer. Part III (Chapters 12-15) covers applications of multicomponent mass transfer models to process design. 

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