Subject spherical particles

Resistance functions have been evaluated in numerical compu-tations15831 for low Reynolds number flows past spherical particles, droplets and bubbles in cylindrical tubes. The undisturbed fluid may be at rest or subject to a pressure-driven flow. A spectral boundary element method was employed to calculate the resistance force for torque-free bodies in three cases (a) rigid solids, (b) fluid droplets with viscosity ratio of unity, and (c) bubbles with viscosity ratio of zero. A lubrication theory was developed to predict the limiting resistance of bodies near contact with the cylinder walls. Compact algebraic expressions were derived to accurately represent the numerical data over the entire range of particle positions in a tube for all particle diameters ranging from nearly zero up to almost the tube diameter. The resistance functions formulated are consistent with known analytical results and are presented in a form suitable for further studies of particle migration in cylindrical vessels. 

The microductile/compliant layer concept stems from the early work on composite models containing spherical particles and oriented fibers (Broutman and Agarwal, 1974) in that the stress around the inclusions are functions of the shear modulus and Poisson ratio of the interlayer. A photoelastic study (Marom and Arridge, 1976) has proven that the stress concentration in the radial and transverse directions when subjected to transverse loading was substantially reduced when there was a soft interlayer introduced at the fiber-matrix interface. The soft/ductile interlayer allowed the fiber to distribute the local stresses acting on the fibers more evenly, which, in turn, enhanced the energy absorption capability of the composite (Shelton and Marks, 1988). 

In the free troposphere, particles are subjected to gravitational forces. They also can be subjected to electrical forces in nature as well as in the course of detection and measurement. When such forces are applied, the particle moves relative to the gas and hence is subjected to a resistance force. Stokes law gives the force (FR) acting on smooth spherical particles due to the laminar flow of air over them ... 

According to the classical Stokes law, a spherical particle, i, of radius rif moving with velocity V through a static fluid of viscosity t] is subjected to a force f as shown by Eqs (7.11) or (7.12) ... 

Because the subject is vast, the presentation is limited to a discussion of the uptake of a tracer from the vapor phase by spherical particles. This is the viewpoint of one concerned with fallout formation. The reverse process—escape from spherical particles—is the viewpoint of one concerned with reactor fuels. For the idealized case the treatment is exactly the same for the two situations. The fact that we deal with trace quantities and concentration means that we can neglect changes in the particle properties as the reaction proceeds and that we need not be concerned with surface nucleation. 

Figure 13.7 Spherical particle pinning an interface between grains 1 and 2. The interface is subjected to a driving pressure that tends to move it in the y direction. From Nes et al. [11].

The 9-parameter is also subject to uncertainty. It is likely that Whitby s (1978) estimates of 9 (Table 10.1) do not reflect the true surface area distribution, since they were based on the average size spectrum of aerosols and assumed spherical particles. Adsorption/desorption kinetics (Kamens et al., 1995 Rounds and Pankow, 1990,1993) and relative humidity (Goss and Eisenreich, 1997 Lee and Tsai, 1994 Pankow et al., 1993 Storey et al., 1995 Thibodeaux et al., 1991) influence the adsorption of POPs onto aerosols, and the Junge-Pankow model does not take these factors into account. 

Each bead represents one catalyst as a member of a library of solid catalysts. It may consist of an unporous material like a-Al2C>3 or Steatit or of typical porous support materials - such as A1203, Si02, Ti02, or the like. These beads can be subjected to different synthesis procedures and sequences like impregnation, coating etc. In addition, full mixed metal oxide catalysts can also be formed as spherical particles. 

For isotropic spherical particles of given refractive index in a medium of known refractive index, the exact form of C can be found by matching the incident, internal and scattered electromagnetic waves at the particle surface, subject to certain boundary conditions7. The solution comes in the form of an infii te series ... 

The most widely used unsteady state method for determining diffusivities in porous solids involves measuring the rate of adsorption or desorption when the sample is subjected to a well defined change in the concentration or pressure of sorbate. The experimental methods differ mainly in the choice of the initial and boundary conditions and the means by which progress towards the new position of equilibrium is followed. The diffusivities are found by matching the experimental transient sorption curve to the solution of Fick s second law. Detailed presentations of the relevant formulae may be found in the literature [1, 2, 12, 15-17]. For spherical particles of radius R, for example, the fractional uptake after a pressure step obeys the relation... 

This general technique was employed to simulate the motion of a mono-layer of identical spherical particles subjected to a simple shear. Confinement to a monolayer represents tremendous economies of computer time compared with a three-dimensional simulation. Hopefully, these highly specialized monolayer results will provide comparable insights into the physics of three-dimensional suspensions. Periodic boundary conditions were used in the simulation, and the method of Evans (1979) was incorporated to reproduce the imposed shear. 

Suspended spherical particles, each containing a permanently embedded dipole (e.g., magnetic), are unable to freely rotate (Brenner, 1984 Sellers and Brenner, 1989) in response to the shear and/or vorticity field that they are subjected to whenever a complementary external (e.g., magnetic) field acts on them. This hindered rotation results from the tendency of the dipole to align itself parallel to the external field because of the creation of a couple arising from any orientational misalignment between the directions of the dipole and external field. In accordance with Cauchy s moment-of-momentum equation for continua, these couples in turn give rise to an antisymmetric state of stress in the dipolar suspension, representable as the pseudovector Tx = — e Ta of the antisymmetric portion Ja — (T — Tf) of the deviatoric stress T = P + Ip. 

Near rich limits of hydrocarbon flames, soot is sometimes produced in the flame. The carbonaceous particles—or any other solid particles— easily can be the most powerful radiators of energy from the flame. The function k(t) is difficult to compute for soot radiation for use in equation (21) because it depends on the histories of number densities and of size distributions of the particles produced for example, an approximate formula for Ip for spherical particles of radius with number density surface emissivity 6, and surface temperature is Ip = Tl nrle ns) [50]. These parameters depend on the chemical kinetics of soot production—a complicated subject. Currently it is uncertain whether any of the tabulated flammability limits are due mainly to radiant loss (since convective and diffusive phenomena will be seen below to represent more attractive alternatives), but if any of them are, then the rich limits of sooting hydrocarbon flames almost certainly can be attributed to radiant loss from soot. 

Sorption Kinetics. The adsorption and desorption data were analyzed in terms of a model based on the following main assumptions. Micropore diffusion within the sieve crystals is the rate-controlling process. Diffusion may be described by Fick s law for spherical particle geometry with a constant micropore diffusivity. The helium present in the system is inert and plays no direct role in the sorption or diffusion process. Sorption occurs under isothermal conditions. Sorption equilibrium is maintained at the crystal surface, which is subjected to a step change in gas composition. These assumptions lead to the following relation for the amount of ethane adsorbed or desorbed by a single particle as a function of time (Crank, 4). 

Maxwell (68) calculated the potential distribution for a single spherical particle immersed in a conducting medium and subjected to a uniform electrical field He solved Laplace s equation within the two regions subject to continuity of potential, and continuity of the normal component of the current density, at the surface of the particle. Maxwell then extended his single-sphere solution to dilute mixtures and obtained the following expression for... 

Intraparticle Diffusion and External Mass-Transfer Resistance For typical industrial conditions, external mass transfer is important only if there is substantial intraparticle diffusion resistance. This subject has been discussed by Luss, Diffusion-Reaction Interactions in Catalyst Pellets, in Carberry and Varma (eds.), Chemical Reaction and Reactor Engineering, Dekker, 1987. This, however, may not be the case for laboratory conditions, and care must be exerted in including the proper data interpretation. For instance, for a spherical particle with both external and internal mass-transfer limitations and first-order reaction, an overall effectiveness factor r, can be derived, indicating the series-of-resistances nature of external mass transfer followed by intraparticle diffusion-reaction ... 

Porous inorganic oxides are made through a sol-gel process. The sol is converted into a hydrogel that is subjected to dehydration to form a porous xerogel. Special techniques have been developed to combine the sol-gel transition with a shaping to spherical particles (Fig. 3.11). 

If such a medium is subjected to an external uniform electric held Eq [Eq. (258)], then the electric held E inside a spherical particle is by quasi electrostatics... 

FIGURE 5.35 Types of hydrodynamic interactions between two spherical particles (a) motion along and rotation around the line of centers (b) motion along and rotation around an axis perpendicular to the line of centers (c) the first particle moves under the action of an applied external force, F, whereas the second particle is subjected to the hydrodynamic disturbance created by the motion of the first particle. 

Proteins for fats In order for a protein to mimic a fat, it has to be cut into tiny particles that measure only from 0.1 to 3.0 pm in size. Simplesse is the only protein-based fat replacer available in the United States. In one version, egg white and milk protein are subjected to high heat and stress to form small, spherical particles. The small size of these protein particles causes them to be perceived by the mouth as smooth and creamy. Like carbohydrates, protein replacers provide only four Calories per gram. However, they are not suitable for frying, and they lack the fat flavor. 

The amount of charge at the interface depends on the field strength and the dielectric properties (conductivity and permittivity) of the particle and the electrolyte. However, there is a slight asymmetry in the charge density on the particle which gives rise to an effective or induced dipole across the particle. Note that if the field is removed the dipole disappears, it is induced . The magnitude of the dipole moment depends on the amount of charge and the size of the particle. For a spherical particle in an electrolyte subject to a uniform applied electric field, three cases can be considered ... 

---