Diffusion equation boundary conditions

Diffusion equations boundary conditions sweep boundary boundary condi-... 

When the Bom, double-layer, and van der Waals forces act over distances that are short compared to the diffusion boundary-layer thickness, and when the e forces form an energy hairier, the adsorption and desorption rates may be calculated by lumping the effect of the interactions into a boundary condition on the usual ccm-vective-diffusion equation. This condition takes the form of a first-order, reversible reaction on the collector s surface. The apparent rate constants and equilibrium collector capacity are explicitly related to the interaction profile and are shown to have the Arrhenius form. They do not depend on the collector geometry or flow pattern. 

It is well known that W(R0,Rn, N) satisfies the usual diffusion equation (boundary and initial conditions included) ... 

This equation must be solved with the diffusion-interception boundary conditions... 

The derivation of (6.2.8) and (6.2.9) employed only the linear diffusion equations, initial conditions, semi-infinite conditions, and the flux balance. No assumption related to electrode kinetics or technique was made hence (6.2.8) and (6.2.9) are general. From these equations and the boundary condition for LSV, (6.2.3), we obtain... 

Fuchs Theory The matching of continuum and free molecule fluxes dates back to Fuchs (1964), who suggested that by matching the two fluxes at r = A + Rp, one may obtain a boundary condition on the continuum diffusion equation. This condition is, assuming unity accommodation coefficient... 

The semi-infinite medium is employed to study the spatiotemporal patterns that the solution of the non-Fick damped wave diffusion and relaxation equation exhibits. This medium has been used in the study of Pick mass diffusion. The boundary conditions can be different kinds, such as constant wall concentration, constant wall flux (CWF), pulse injection, and convective, impervious, and exponential decay. The similarity or Boltzmann transformation worked out well in the case of the parabolic PDF, where an error function solution can be obtained in the transformed variable. The conditions at infinite width and zero time are the same. The conditions at zero distance from the surface and at infinite time are the same. 

A theoretical method based on limited scale power law form of the interfacial roughness power spectrum and the solution of diffusion equation xmder the diffusion-limited boundary conditions on rough interfaces was developed by Kant and Jha (Kant Jha, 2007). The results were compared with experimentally obtained currents for nano- and microscales of roughness and are applicable for all time scales and roughness factors. Moreover, this work unravels the connection between the anomalous intermediate power law regime exponent and the morphological parameters of limited scales of fractality. 

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... 

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... 

This treatment of reaction at the limit of bulk diffusion control is essentially the same as that presented by HugoC 69j. It is attractive computationally, since only a single two-point boundary value problem must be solved, namely that posed by equations (11.15) and conditions (11.16). This must be re-solved each time the size of the pellet is changed, since the pellet radius a appears in the boundary conditions. However, the initial value problem for equations (11.12) need be solved only once as a preliminary to solving (11.15) and (11.16) for any number of different pellet sizes. 

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. 

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]... 

The overall set of partial differential equations that can be considered as a mathematical characterization of the processing system of gas-liquid dispersions should include such environmental parameters as composition, temperature, and velocity, in addition to the equations of bubble-size and residence-time distributions that describe the dependence of bubble nucleation and growth on the bubble environmental factors. A simultaneous solution of this set of differential equations with the appropriate initial and boundary conditions is needed to evaluate the behavior of the system. Subject to the Curie principle, this set of equations should include the possibilities of coupling effects among the various fluxes involved. In dispersions, the possibilities of couplings between fluxes that differ from each other by an odd tensorial rank exist. (An example is the coupling effect between diffusion of surfactants and the hydrodynamics of bubble velocity as treated in Section III.) As yet no analytical solution of the complete set of equations has been found because of the mathematical difficulties involved. To simplify matters, the pertinent transfer equation is usually solved independently, with some simplifying assumptions. 

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). 

These equations are solved under the initial and boundary conditions as follows Since all the fluctuations at 1 0 are produced by electrode reactions, the initial components induced by diffusion are equal to zero. Therefore,... 

This velocity profile is commonly called drag flow. It is used to model the flow of lubricant between sliding metal surfaces or the flow of polymer in extruders. A pressure-driven flow—typically in the opposite direction—is sometimes superimposed on the drag flow, but we will avoid this complication. Equation (8.51) also represents a limiting case of Couette flow (which is flow between coaxial cylinders, one of which is rotating) when the gap width is small. Equation (8.38) continues to govern convective diffusion in the flat-plate geometry, but the boundary conditions are different. The zero-flux condition applies at both walls, but there is no line of symmetry. Calculations must be made over the entire channel width and not just the half-width. 

The most important mass transfer limitation is diffusion in the micropores of the catalyst. A simplified model of pore diffusion treats the pores as long, narrow cylinders of length The narrowness allows radial gradients to be neglected so that concentrations depend only on the distance I from the mouth of the pore. Equation (10.3) governs diffusion within the pore. The boundary condition at the mouth of the pore is... 

Diffusion of the fluid into the bulk. Rates of diffusion are governed by Pick s laws, which involve concentration gradient and are quantified by the diffusion coefficient D these are differential equations that can be integrated to meet many kinds of boundary conditions applying to different diffusive processes. ... 

With these boundary conditions, the differential transient-diffusion equation (11.1) has the solution... 

The boundary conditions determine the form of balance equation for the inlet and outlet sections. These require special consideration as to whether diffusion fluxes can cross the boundaries in any particular physical situation. The physical situation of closed ends is considered here. This would be the case if a smaller pipe were used to transport the fluid in and out of the reactor, as shown in Figs. 4.13 and 4.14. 

In practice, the Peclet number can always be ignored in the diffusion-convection equation. It can also be ignored in the root boundary condition unless C > X/Pc or A, < Pe. Inspection of the table of standard parameter values (Table 2) shows that this is never the case for realistic soil and root conditions. Inspection of Table 2 also reveals that the term relating to nutrient efflux, e, can also be ignored because e < Pe [Pg.343]

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. 

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