Diffusion equation direct solution

The convective-diffusion equation for solute (e.g., tracer) transport in both the axial and radial direction is... 

The Case of Diffuse Illumination Direct Solution of the Diffusion Equation... 

The first and second integrals have their coordinate systems centered on the catalytic C and noncatalytic N spheres, respectively. The local nonequilibrium average microscopic density field for species a is pa(r) = [Y = 5(r - ( )) The solution of the diffusion equation can be used to estimate this nonequilibrium density, and thus the velocity of the nanodimer can be computed. The simple model yields results in qualitative accord with the MPC dynamics simulations and shows how the nonequilibrium density field produced by reaction, in combination with the different interactions of the B particles with the noncatalytic sphere, leads to directed motion [117],... 

As a last example in this section, let us consider a sphere situated in a solution extending to infinity in all directions. If the concentration at the surface of the sphere is maintained constant (for example c — 0) while the initial concentration of the solution is different (for example c = c°), then this represents a model of spherical diffusion. It is preferable to express the Laplace operator in the diffusion equation (2.5.1) in spherical coordinates for the centro-symmetrical case.t The resulting partial differential equation... 

Constraints may be introduced either into the classical mechanical equations of motion (i.e., Newton s or Hamilton s equations, or the corresponding inertial Langevin equations), which attempt to resolve the ballistic motion observed over short time scales, or into a theory of Brownian motion, which describes only the diffusive motion observed over longer time scales. We focus here on the latter case, in which constraints are introduced directly into the theory of Brownian motion, as described by either a diffusion equation or an inertialess stochastic differential equation. Although the analysis given here is phrased in quite general terms, it is motivated primarily by the use of constrained mechanical models to describe the dynamics of polymers in solution, for which the slowest internal motions are accurately described by a purely diffusive dynamical model. 

If the diffusion medium is isotropic in terms of diffusion, meaning that diffusion coefficient does not depend on direction in the medium, it is called diffusion in an isotropic medium. Otherwise, it is referred to as diffusion in an anisotropic medium. Isotropic diffusion medium includes gas, liquid (such as aqueous solution and silicate melts), glass, and crystalline phases with isometric symmetry (such as spinel and garnet). Anisotropic diffusion medium includes crystalline phases with lower than isometric symmetry. That is, most minerals are diffu-sionally anisotropic. An isotropic medium in terms of diffusion may not be an isotropic medium in terms of other properties. For example, cubic crystals are not isotropic in terms of elastic properties. The diffusion equations that have been presented so far (Equations 3-7 to 3-10) are all for isotropic diffusion medium. 

It Is seen from equation (7) that the optimum velocity is, directly proportional to the diffusivity of the solute in the mobile phase. To a lesser extent it also appears to be inversely dependant on the particle diameter of the packing (the particle size is an optional choice) and the film thickness of the stationary phase. The film thickness of the stationary phase is determined by the physical form of the packing, that is, in the case of silica gel, the nature of the surface and in the case of a reverse phase, on the bonding chemistry. 

It is interesting to note from equation (14) that when a column is run at its optimum velocity, the maximum efficiency attainable from a capillary column is directly proportional to the inlet pressure and the square of the radius and inversely proportional to the solvent viscosity and the diffusivity of the solute in the mobile phase. This means that the maximum efficiency attainable from a capillary column increases with the column radius. Consequently, very high efficiencies will be obtained from relatively large diameter columns. 

Equation (13) gives the minimum analysis time that can be obtained from an open tubular column, when separating a mixture of defined difficulty, under given chromatographic conditions. It is seen that, in a similar manner to the packed column, the analysis time is inversely proportional to the fourth power of the function (a-1) and inversely proportional to the inlet pressure. The contribution of the function of (k1), to the analysis time is not clear and can be best seen by calculation. It is also seen (perhaps a little surprisingly) that the analysis time is completely independent of the diffusivity of the solute in the mobile phase but is directly proportional to the viscosity of the mobile phase. 

It is seen from equation (18) that the solvent consumption is directly proportional to the charge placed on the column and the capacity ratios of the first peak of the critical pair and the last eluted peak respectively. It is also seen that, as with the optimized analytical column, the diffusivity of the solute and the viscosity of the mobile phase play no part in controlling the solvent economy, it should be pointed out, however, that this is only true for a completely optimized column... 

Usually, experiments are performed with steady-state photolysis or radiolysis of the solution and the yield of scavenger products determined optically or by ESR methods. There is no direct interest in the actual time evolution of the density or recombination (survival) probability. Consequently, the creation of ion-pairs may be pictured as occurring at a constant rate, say 1 s 1, from time t0 = 0 to infinity. The steady-state ion-pair density distribution, which arises when dp/dt = 0, is the balance between continuous creation of ion pairs at a rate Is-1, recombination and scavenging. Removing the instantaneous creation of an ion-pair at time t = t0 (i.e. removing the 6(f — f0) in the source term), means that ion-pairs were continuously formed from time t = — 00 to t. At long times, f > — oo the density distribution is independent of t and, of course, t0. Let pss(r cs r0) = /i p(r, t cs t0, 0)d 0 be the steady-state ion-pair density distribution for ion pairs continuously formed at r0, and note d/dt J" f pd 0 = 0. The diffusion equation (169) becomes... 

In the formulation and solution of conservation equations, we tend to prefer the direct evaluation of the diffusion velocities as discussed in the previous section. However, it is worthwhile to note that the Stefan-Maxwell equations provide a viable alternative. At each point in a flow field one could solve the system of equations (Eq. 3.105) to determine the diffusion-velocity vector. Solution of this linear system is equivalent to determining the ordinary multicomponent diffusion coefficients, which, in this formulation, do not need to be evaluated. 

The diffusion equation is the partial-differential equation that governs the evolution of the concentration field produced by a given flux. With appropriate boundary and initial conditions, the solution to this equation gives the time- and spatial-dependence of the concentration. In this chapter we examine various forms assumed by the diffusion equation when Fick s law is obeyed for the flux. Cases where the diffusivity is constant, a function of concentration, a function of time, or a function of direction are included. In Chapter 5 we discuss mathematical methods of obtaining solutions to the diffusion equation for various boundary-value problems. 

Fisher has produced a relatively simple solution for a specimen geometry that is convenient for experimentalists and which has been widely used in the study of boundary self-diffusion by making several approximations which are justified over a range of conditions [9, 10]. The geometry is shown in Fig. 9.8 it is assumed that the specimen is semi-infinite in the y direction and that the boundary is stationary. The boundary condition at the surface corresponds to constant unit tracer concentration, and the initial condition specifies zero tracer concentration within the specimen. Rapid diffusion then occurs down the boundary slab along y, while tracer atoms simultaneously leak into the grains transversely along x by means of crystal diffusion. The diffusion equation in the boundary slab then has the form... 

Here, the response functions of the diffusion equation for a number of discrete input signals were calculated based on the solution for a Dirac impulse input signal. A set of transformation relations for the injected mass M and the axial coordinate was used to obtain the solutions for a reacting gas directly from the solutions of a non-reacting gas ... 

In this equation A is the molecular diffusion coefficient expressed as m2/sec, and is the number of liquid moles of component i. This equation can be simplified using approximate analytical solutions for the transient diffusion equation in the vertical direction to ... 

The selection of diffusion equation solutions included here are diffusion from films or sheets (hollow bodies) into liquids and solids as well as diffusion in the reverse direction, diffusion controlled evaporation from a surface, influence of barrier layers and diffusion through laminates, influence of swelling and heterogeneity of packaging materials, coupling of diffusion and chemical reactions in filled products as well as permeation through packaging. 

The first complication for the application of the diffusion equation Eq. (7-12) comes when the complete left half of the plastic bar, x < 0, is uniformly and completely colored with coloring agent which can diffuse in the direction of x > 0 (Fig. 7-5b). The concentration of the color is expressed by the finite concentration of Cq. In order to find a solution to the problem the colored region x < 0 is thought of as being divided into an infinite number of layers perpendicular to the x-axis. In doing this, the problem can be related to an infinite number of diffusion sources and the mathematical solution can be arrived at by overlapping many solutions of the form of Eq. (7-18). 

In the same manner, one obtains a solution to the diffusion equation starting with a colored layer having a finite thickness 2d and an initial concentration Co in both directions of the unbounded x-axis (Fig. 7-7) ... 

Eq. (7-38) is a solution of the diffusion equation (7-12) for the models shown in Fig. 7-10. Where mt is the mass of i diffusing up to time t from L through the boundary surface A into the package or opposite direction and m is the amount which has migrated at equilibrium. 

Boundary layer formulation. Many membrane processes are operated in cross-flow mode, in which the pressurised process feed is circulated at high velocity parallel to the surface of the membrane, thus limiting the accumulation of solutes (or particles) on the membrane surface to a layer which is thin compared to the height of the filtration module [2]. The decline in permeate flux due to the hydraulic resistance of this concentrated layer can thus be limited. A boundary layer formulation of the convective diffusion equation can give predictions for concentration polarisation in cross-flow filtration and, therefore, predict the flux for different operating conditions. Interparticle force calculations are used in two ways in this approach. Firstly, they allow the direct calculation of the osmotic pressure at the membrane. This removes the need for difficult and extensive experi-... 

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