Diffusion, definition spherical

Neal and Nader [260] considered diffusion in homogeneous isotropic medium composed of randomly placed impermeable spherical particles. They solved steady-state diffusion problems in a unit cell consisting of a spherical particle placed in a concentric shell and the exterior of the unit cell modeled as a homogeneous media characterized by one parameter, the porosity. By equating the fluxes in the unit cell and at the exterior and applying the definition of porosity, they obtained... 

Compared with one-dimensional diffusion in a plane sheet, for the same a value (such as a = 0.5), the fraction of mass loss is much larger for the solid sphere (T=0.9484) than for the planar slab (7=0.5622). To reach the same degree of mass loss (such as 50%), a smaller value is needed for the sphere (a = 0.0305) than for the thin slab = 0.1963). The difference in terms of is a factor of 6.4. Because the definition of is jD(f)df/a, by comparing with Equation 5-67b, we see that is proportional to 1/G. The difference in the value of for the same degree of mass loss for spherical and plane sheet shapes roughly explains the difference between G values (55/8.65 = 6.4) for spherical and plane sheet shapes. 

The second term in Eq. 80 that is proportional to r describes the contribution from spherical diffusion when it is 10 times larger than the planar diffusion term [i.e., (ti/JoO 1 s, this would lead to a definition of UMEs as stated in... 

In most cases, Do is about 10 cm s, and it then follows that the characteristic radius of an UME is smaller than 6 [im. However, it is important to notice that Do is sometimes appreciably smaller than 10 cm s if O is a very large molecule, or, more likely, if a viscous or glass-like medium is used planar diffusion would then prevail even at a radius of 6 pm. Note also that a microelectrode with a radius of 100 pm experiences a substantial contribution from spherical diffusion. If the geometry of the electrode differs from the simple disk or microsphere shape, the definition is less clear. In general, the characteristic behavior of a UME is observed if at least one of the dimensions is in the micrometer range. For instance, a band electrode with a width of less than 10 pm behaves as an UME even if it is several mm long. 

Equations 8.2.14-8.2.20, derived for the fluxes across a planar film, apply essentially unchanged for diffusion in cylindrical and spherical films. All that needs to be done is to use the appropriate definition of the characteristic length from Figure 8.3. The flux then calculated from Eqs. 8.2.14 or 8.2.15 would be that at the plane 77 = 0. 

What is the critical value of the intrapeUet Damkohler number for onedimensional diffusion and zeroth-order irreversible chemical reaction in catalytic pellets with spherical symmetry The radius of the sphere is used as the characteristic length in flie definition of the Damkohler number. 

Figure 20-2 Dimensionless correlations between the effectiveness factor and the intrapellet Damkohler number for radial diffusion and nth-order irreversible chemical kinetics in porous catalysts with spherical symmetry (i.e., n = 0, 1). The quantity on the horizontal axis is A, not A. The spherical radius R is the characteristic length in the definition of A.

Obtain an analytical expression for the effectiveness factor (i.e., E vs. tjcriticai) in Spherical catalysts when the chemical kinetics are zeroth-order and the intrapeUet Damkohler number is greater than its critical value. Use the definition of the effectiveness factor that is based on mass transfer via diffusion across the external surface of the catalyst. 

Consider a problem on definition of collision frequency of small spherical particles executing Brownian motion in a quiescent liquid. In Section 8.2, Brownian motion was considered as diffusion with a effective diffusion factor. It was supposed that suspension is sufficiently diluted, so it is possible to consider only the pair interactions of particles. To simplify the problem, consider a bi-disperse system of particles, that is, a suspension consisting of particles of two types particles of radius ai and particles of radius a2. In this formulation, the problem was first considered by Smolukhowski [59]. 

If the suspension is entirely homogeneous and contains only particles of one definite size, the boundary between the solution and the pure solvent is sharp (apart from diffusion in the case of very small particles, which will be referred to later) and we may conclude that a monodisperse system exists. It should, of course, be emphasized that this holds only for spherical or at least for nearly spherical particles, for with elliptical particles, there may be a different adjustment of the particle axes to the direction of the field, resulting in a different rate of sedimentation of particles of the same size but of different orientation. ... 

Since the only possible way in which n(0) can change is due to the current 7Af(0), we have a situation analogous to the diffusion in Fig 10.10. As noted earlier, in spherical coordinates we have the following definitions ... 

---