Cylindrical Particle

The rate expression for reaction of a cylindrical particle (radius r) is [469]... 

Gorin has extended this analysis to include (1) the effects of the finite size of the counterions in the double layer of spherical particles [137], and (2) the effects of geometry, i.e. for cylindrical particles [2]. The former is known as the Debye-Huckel-Henry-Gorin (DHHG) model. Stigter and coworkers [348,369-374] considered the electrophoretic mobility of polyelectrolytes with applications to the determination of the mobility of nucleic acids. 

Electrophoresis The physical situation of relative motions of a solution and another (insulating) phase during electrophoresis is exactly the same as in electroosmosis. Hence, the linear velocity of a cylindrical particle (which is the equivalent of a cylindrical pore) is given by the value following from Eq. (31.4). With particles of dilferent shape, this velocity can be written as... 

The above presentation due to Hubb (Z) is a good initial approach which gives one physical insight into the problem, but a more detailed presentation for the release of elastic particles form rigid surfaces (specifically, cylindrical particles) may be found in Chamoun (11). In addition, extensive experimental data on the release of particles from various surfaces is reported by this author. An analysis of this experimental data is presented in Shirzadi et al. (12). 

A concise approach for the analysis of isotropic scattering curves of spherical and cylindrical particles with a radial density profile has been developed by Burger [207], In practice it is useful for the study of latices and vesicles in solution. 

The analytical structural model for the topology of the nanostructure is defined in Isr (5). For many imaginable topologies such models can be derived by application of scattering theory. Several publications consider layer topologies [9,84,231] and structural entities built from cylindrical particles [240,241], In the following sections let us demonstrate the principle procedure by means of a typical study [84],... 

More recently, we have created full-bed periodic two-layer models for packings of cylindrical particles (Taskin et al., 2006). The geometry shown in Fig. 3b was created specifically for comparison of the WS model with cylindrical particles described in the following section, with the structures identical within a 120° segment of the bed. 

Fig. 32. (a) Sequence of transformations l->2->3->4to place bottom front cylindrical particle (b) Midplane cross-section of the WS packed with cylinders, showing control volumes found by selection algorithm, marked as darkest cells. 

Repeat Example 9-1 and problem 9-1 for an isothermal cylindrical particle of radius R and... 

If the particles all have the same residence time, the contribution to 1 - /B depends on particle size. For spherical or cylindrical particles of radius R, the contribution is the product of 1 - fB(t, Rj for that size and the fraction of particles of that size. By integrating over all sizes, we obtain a form analogous to that in equation 22.2-1 ... 

This value of /B may also be obtained, by means of the E-Z Solve software, by simultaneous solution of equation 22.2-17 and numerical integration of 22.2-13 (with user-defined function fbcr(t, f,), for cylindrical particles with reaction control see file ex22-3.msp). This avoids the need for analytical integration leading to equation 22.2-18. 

As shown in Example 22-3, for solid particles of the same size in BMF, the form of the reactor model resulting from equation 22.2-13 depends on the kinetics model used for a single particle. For the SCM, this, in turn, depends on particle shape and the relative magnitudes of gas-film mass transfer resistance, ash-layer diffusion resistance and surface reaction rate. In some cases, as illustrated for cylindrical particles in Example 22-3(a) and (b), the reactor model can be expressed in explicit analytical form additional results are given for spherical particles by Levenspiel(1972, pp. 384-5). In other f l cases, it is convenient or even necessary, as in Example 22-3(c), to use a numerical pro-... 

Figure 22.4 Example 22-4 Dependence of mean fractional conversion (/B) on mean residence time (t )-effect of rate-limiting process data of Example 22-3 (cylindrical particles of all same size in BMF SCM n = 1)...

Although it is possible to obtain analytical expressions for the reactor model for spherical particles for parts (a), (b), and (c) using the approach in Example 22-3(a) and (b) for cylindrical particles (expressions for fB for spherical particles are given by Levenspiel, 1972, pp. 384-5), we use the numerical procedure described in Example 22-3 (c) for all four parts of this example, (a) to (d), since this procedure must be used for (d). 

If cylindrical particles of two different sizes (R2 = 2R ) of species B undergoing reaction... 

Figure 5.11 Calculated degree of polycondensation related to the axis of PET chips, shown as axial profiles for a cylindrical particle T, 220 °C hydraulic diameter (dh). 2.9 mm [12b]. From Weger, F., Solid-state postcondensation of polyesters and polyamides, presentation given at the Frankl and Thomas Polymer Seminar, 16 June, 1994, Greenville, SC, USA, and reproduced with permission of EMS Inventa-Fischer, GmbH Co. KG...

Several workers have used numerical methods for solving the equations of motion for flow at higher Reynolds numbers relative to spherical and cylindrical particles. These include, Jenson(4), and le Clair, Hamielec and Pruppacher . ... 

Correlations for Terminal Reynolds Number of Cylindrical Particles... 

Interconnected cylinders (L2 ) are formed in two water content range (5.5 < vv < 11 and 30 < vv < 35). Syntheses in these two regions of the phase diagram show very strong correlation and similar data. Spherical and cylindrical particles are formed in both cases. No other particle shapes... 

In this chapter we consider theories of scattering by particles that are either inhomogeneous, anisotropic, or nonspherical. No attempt will be made to be comprehensive our choice of examples is guided solely by personal taste. First we consider a special example of inhomogeneity, a layered sphere. Then we briefly discuss anisotropic spheres, including an exactly soluble problem. Isotropic optically active particles, ones with mirror asymmetry, are then considered. Cylindrical particles are not uncommon in nature—spider webs, viruses, various fibers—and we therefore devote considerable space to scattering by a right circular cylinder. 

The quantity /is a numerical factor that depends on the ratio a/K, where a is the radius of the spherical or cylindrical particle. In other words, / depends on the ratio of the radius of the particle to the effective thickness K 1 of the diffuse layer. When alKX is large, (the particle large in comparison with the diffuse-charge thickness), the numerical factor is always equal to irrespective of the shape of the particle. When the particle is small compared with the thickness of the double layer,/is i for cylindrical particles parallel to the field and for spherical particles (Fig. 6.140). 

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