Spherical diffusion, finite

The problem of finite-length diffusion in spherical and cylindrical symmetry was solved by Jacobsen and West. ... 

Anderson, J. and C. Reed, Diffusion of spherical macromolecules at finite concentration. Journal of Chemical Physics, 1976, 64, 3240-3250. 

This equation gives the change of concentration in a finite volume element with time. In the approach of Barrer and Jost, the diffusivity is assumed to be isotropic throughout the crystal, as Dt is independent of the direction in which the particles diffuse. Assuming spherical particles. Pick s second law can be readily solved in radial coordinates. As a result, all information about the exact shape and connectivity of the pore structure is lost, and only reflected by the value of the diffusion constant. 

Transient Finite (Symmetric) Spherical Diffusion So far, we have only examined ID (Cartesian) examples of Fick s second law. Solving Fick s second law in alternative coordinate systems (e.g., for radial, spherical, 2D, or 3D problems) is not really any different. As an example, we examine here the case of transient finite spherical diffusion, which is essentially analogous to the transient finite planar diffusion problem that we just finished discussing. 

Consider the transient finite spherical diffusion problem illustrated in Figure 4.15, which describes the diffusion of H2 into a spherical particle. 

The derivation of a steady-state solution requires boundary conditions, but no initial condition. Steady-state can be seen as the asymptotic solution (so never mathematically reached at any finite time [43]) of the transient, which -for practical purposes - can be approached in a reasonably short time. For instance, limiting-flux diffusion of a species with diffusion coefficient Di = 10-9 m2 s 1 towards a spherical organism of radius rQ = 1 jxm is practically attained at t r jDi = 1 ms. 

Oxygen, substrate and biomass are all transported by diffusion within the liquid phase contained in the aggregate. The modelling of this process is achieved via the use of a finite differencing technique. In this, the spherical aggregate is divided into a number of shells, as seen in Fig. 1. 

For three-dimensional diffusion, if there is spherical symmetry (i.e., concentration depends only on radius), the diffusion equation can be transformed to a one-dimensional type by redefining the concentration variable w = rC. This transformation would work for a solid finite sphere, a spherical shell, an infinite sphere with a spherical hole in the center, or an infinite sphere. 

At finite polymer concentrations, the intermolecular hydrodynamic interaction may also alter polymer dynamics. Except for spherical particles, the hydrodynamic calculations of the effective diffusion coefficients including this... 

Figure 19.18 Dynamics of diffusive uptake of a chemical by a spherical particle suspended in a fluid of finite volume. The numbers on the curves correspond to y defined by Eq. 19-86, that is, to the fraction of the chemical taken up by the sphere when equilibrium is reached. From Wu and Gschwend (1988).

Figure 12.1 Finite-element diagram for diffusion to electrodes. The coordinate, x, represents the direction normal to the surface of the electrode. The origin of x is at the central axis in the cylindrical or spherical case.

In the grain model, it is assumed that the CaO consists of spherical grains of uniform size distributed in a porous matrix. The rate of reaction is controlled by the diffusion of SO2 through the porous matrix and the product CaSO layer formed on each grain (11). Allowance can be made for a finite rate of the CaO/SC reaction (12). The models have been found to describe experimental data for many limestones (13) by adjusting the constants in the model, most notably the diffusivity through the product layer. 

Fig. 3- Cose 1. Sherwood numbers computed for the convective-diffusion of particles of finite sine to the surface of a spherical collector by neglecting interaction forces. The dashed line is the Levich-LighthilJ equation (19) which is valid when a diffusion boundary-layer exists and the particles are infinitesimal.

Fig. 5. Case 3. Sherwood numbers far the transpart of finite size particles through a stagnant fluid to n spherical cellectnr under the action of diffusion and London forces...

Fig. 6. Sherwood numbers computed for the transport of finite particles to a spherical collector under the combined action of convective-diffusion and London forces. Values of the aspect ratio are (a) ft - 104, (b) ft — 105, (c) ft =- 10s, and (d) ft = 10. For each aspect ratio, the value of A/kT was taken (upper curves to lower curves) as ID2, 1, 10 2 and 10 4. Dashed lines represent the Levich-Lighthill equation (19), while the dotted curves represent Sherwood numbers deduced from Figure 4 which ignores the transport from diffusion.

While 8 is somewhat greater than 1, both p and / are somewhat smaller than 1. Therefore, for approximate calculations, epf can be set 1. The fission factor rj varies appreciably with the energy of the neutrons, as shown in Fig. 11.4 for The neutron losses in a reactor of finite dimensions can be taken into account approximately by the sum L +1 , where Ls is the mean slowing-down length of the fission neutrons in the moderator and L the mean diffusion length in the fuel-moderator mixture. For a spherical reactor of radius R the approximate relation is... 

Pairwise Brownian dynamics has been primarily used for the analysis of diffusion controlled reactions involving the reaction between isotropic molecules with complex reactive sites. Since its introduction by Northrup et al. [58], the pairwise Brownian dynamics method has been considerably refined and modified. Some of the developments include the use of variable time steps to reduce computational times [61], efficient calculation methods for charge effects [63], and incorporation of finite rates of reaction [58,61,62]. We review in the following sections, application of the method to two example problems involving isotropic translational diffusion reaction of isotropic molecules with a spherical reaction surface containing reactive patches and the reaction between rodlike molecules in dilute solution. 

Spherical reaction site with reactive patches We apply both the methods for finite rates of reaction discussed above to the case of a reactive molecule with reactive patches as shown in Figure 12, for comparison. The diffusing molecules are isotropic, and the site molecule is large enough to be stationary. In this case, the probabilities depend only on theangle 6 because of symmetry. Thus, for the survival probability method we have... 

As mentioned in Section 5.8.4, the electrical field, in general, polarizes the EDL around a charged particle. This means that the spherical symmetry of the ion cloud breaks down, and the additional force appearing between the charged particle and the distorted ion atmosphere must be taken into account for proper description of the particle dynamics. If the external field is suddenly switched off, some finite period of time is needed for restoration of the spherically symmetric configuration. This time can be estimated from the ion diffusivity and from the characteristic path length, / the ions should travel ... 

At low ionic strength (kR 1), other effects connected with the finite diffusivity of the small ions in the EDL surrounding the particle are present. The noninstantaneous diffusion of the small ions (with respect to the Brownian motion of the colloid particle) could lead to detectable reduction of the single particle diffusion coefficient, Dq, from the value predicted by the Stokes-Ein-stein relation. Equation 5.447. For spherical particles, the relative decrease in the value of Dq is largest at k/ 1 and could be around 10 to 15%. As shown in the normal-mode theory, the finite diffusivity of the small ions also affects the concentration dependence of the collective diffusion coefficient of the particles. Belloni et al. obtained an explicit expression for the contribution of the small ions in Ac)... 

Total electrode impedance consists of the contributions of the electrolyte, the electrode solution interface, and the electrochemical reactions taking place on the electrode. First, we consider the case of an ideally polarizable electrode, followed by semi-infinite diffusion in linear, spherical, and cylindrical geometry and, finally a finite-length diffusion. 

Ideally, first the measurement modeling should be carried out. The number and the nature of the circuit elements should be identified and then the process modeling should be carried out. Such a procedure is relatively elementary for a circuit containing simple elements R, C, and L. It may also be carried out for circuits containing distributed elements that can be described by a closed-form equation CPE, semi-infinite, finite length, or spherical diffusion, etc. However, many different conditions arise from the numerical calculations (e.g., for correct solution for porous electrodes, for... 

---