Spherical fluid particles

Fig. 5.30 Mass transfer factor as a function of Reynolds number for spherical fluid particles ...

Surface-active contaminants play an important role in damping out internal circulation in deformed bubbles and drops, as in spherical fluid particles (see Chapters 3 and 5). No systematic visualization of internal motion in ellipsoidal bubbles and drops has been reported. However, there are indications that deformations tend to decrease internal circulation velocities significantly (MI2), while shape oscillations tend to disrupt the internal circulation pattern of droplets and promote rapid mixing (R3). No secondary vortex of opposite sense to the prime internal vortex has been observed, even when the external boundary layer was found to separate (Sll). 

This law is valid for Re < 1.4. With the assumption of spherical fluid particles and Tjy 7d -> 00 the relationship according to Haas (Haas et al. 1972)... 

Figure 15.4. Streamlines, in the relative frame attached to the center of the particle, of the flow at a low Reynolds number (Re = 0.01) inside and outside a spherical fluid particle. The particle is moving within the fluid at the gravitational settling velocity Iff (Figure by A. Saboni)...

For a Reynolds number exceeding 0.1, nonhnear terms have to be taken into account to determine the setthng velocity of a spherical fluid particle. From correlations given in literature, it is possible to derive the variation of the settling or rising velocity with the viscosity ratio ic and the Reynolds nnmber Re inside the interval 0.1 < Re < 400. For a value of the Reynolds nnmber greater than 400, the resolution of the flow can no longer be formnlated in the same context, as the deformation of the spherical particle has to be taken into acconnt. 

Several colloidal systems, that are of practical importance, contain spherically symmetric particles the size of which changes continuously. Polydisperse fluid mixtures can be described by a continuous probability density of one or more particle attributes, such as particle size. Thus, they may be viewed as containing an infinite number of components. It has been several decades since the introduction of polydispersity as a model for molecular mixtures [73], but only recently has it received widespread attention [74-82]. Initially, work was concentrated on nearly monodisperse mixtures and the polydispersity was accounted for by the construction of perturbation expansions with a pure, monodispersive, component as the reference fluid [77,80]. Subsequently, Kofke and Glandt [79] have obtained the equation of state using a theory based on the distinction of particular species in a polydispersive mixture, not by their intermolecular potentials but by a specific form of the distribution of their chemical potentials. Quite recently, Lado [81,82] has generalized the usual OZ equation to the case of a polydispersive mixture. Recently, the latter theory has been also extended to the case of polydisperse quenched-annealed mixtures [83,84]. As this approach has not been reviewed previously, we shall consider it in some detail. 

The singlet-level theories have also been applied to more sophisticated models of the fluid-solid interactions. In particular, the structure of associating fluids near partially permeable surfaces has been studied in Ref. 70. On the other hand, extensive studies of adsorption of associating fluids in a slit-like [71-74] and in spherical pores [75], as well as on the surface of spherical colloidal particles [29], have been undertaken. We proceed with the application of the theory to more sophisticated impermeable surfaces, such as those of crystalline solids. 

Motivated by a puzzling shape of the coexistence line, Kierlik et al. [27] have investigated the model with Lennard-Jones attractive forces between fluid particles as well as matrix particles and have shown that the mean spherical approximation (MSA) for the ROZ equations provides a qualitatively similar behavior to the MFA for adsorption isotherms. It has been shown, however, that the optimized random phase (ORPA) approximation (the MSA represents a particular case of this theory), if supplemented by the contribution of the second and third virial coefficients, yields a peculiar coexistence curve. It exhibits much more similarity to trends observed in... 

The simulations to investigate electro-osmosis were carried out using the molecular dynamics method of Murad and Powles [22] described earher. For nonionic polar fluids the solvent molecule was modeled as a rigid homo-nuclear diatomic with charges q and —q on the two active LJ sites. The solute molecules were modeled as spherical LJ particles [26], as were the molecules that constituted the single molecular layer membrane. The effect of uniform external fields with directions either perpendicular to the membrane or along the diagonal direction (i.e. Ex = Ey = E ) was monitored. The simulation system is shown in Fig. 2. The density profiles, mean squared displacement, and movement of the solvent molecules across the membrane were examined, with and without an external held, to establish whether electro-osmosis can take place in polar systems. The results clearly estab-hshed that electro-osmosis can indeed take place in such solutions. 

In quiescent liquids and in bubble columns, buoyancy-driven coalescence is more important. Large fluid particles with a freely moving surface will also have a low-pressure region at the edge of the particle where the velocity is maximum. This low-pressure region will not only allow the bubble to stretch out and form a spherical cap but also allow other bubbles to move into that area and coalesce. Figure 15.14 shows an example of this phenomenon. 

Many engineering operations involve the separation of solid particles from fluids, in which the motion of the particles is a result of a gravitational (or other potential) force. To illustrate this, consider a spherical solid particle with diameter d and density ps, surrounded by a fluid of density p and viscosity /z, which is released and begins to fall (in the x = — z direction) under the influence of gravity. A momentum balance on the particle is simply T,FX = max, where the forces include gravity acting on the solid (T g), the buoyant force due to the fluid (Fb), and the drag exerted by the fluid (FD). The inertial term involves the product of the acceleration (ax = dVx/dt) and the mass (m). The mass that is accelerated includes that of the solid (ms) as well as the virtual mass (m() of the fluid that is displaced by the body as it accelerates. It can be shown that the latter is equal to one-half of the total mass of the displaced fluid, i.e., mf = jms(p/ps). Thus the momentum balance becomes... 

In 1994, we reported the dispersion polymerization of MM A in supercritical C02 [103]. This work represents the first successful dispersion polymerization of a lipophilic monomer in a supercritical fluid continuous phase. In these experiments, we took advantage of the amphiphilic nature of the homopolymer PFOA to effect the polymerization of MMA to high conversions (>90%) and high degrees of polymerization (> 3000) in supercritical C02. These polymerizations were conducted in C02 at 65 °C and 207 bar, and AIBN or a fluorinated derivative of AIBN were employed as the initiators. The results from the AIBN initiated polymerizations are shown in Table 3. The spherical polymer particles which resulted from these dispersion polymerizations were isolated by simply venting the C02 from the reaction mixture. Scanning electron microscopy showed that the product consisted of spheres in the pm size range with a narrow particle size distribution (see Fig. 7). In contrast, reactions which were performed in the absence of PFOA resulted in relatively low conversion and molar masses. Moreover, the polymer which resulted from these precipitation... 

Fig. 31. Horizontal and vertical planes through the third-stacked WS, showing the temperature fields in the fluid and through the spherical catalyst particles, with 5% activity.

Khan, A. R. and Richardson, J. F. Chem. Eng. Comm. 78 (1989) 111. Fluid-particle interactions and flow characteristics of fluidized beds and settling suspensions of spherical particles. 

These results are useful reference conditions for real flows past spherical particles. For example, comparisons are made in Chapter 5 between potential flow and results for flow past a sphere at finite Re. Other potential flow solutions exist for closed bodies, but none has the same importance as that outlined here for the motion of solid and fluid particles. 

H3, Tl), it is unimportant that the Reynolds number of the internal motion was rather large for many flow visualization studies which set out to verify the Hadamard-Rybczynski predictions, so long as the Reynolds number based on the continuous fluid properties was small and the fluid particle spherical. The observed streamlines show excellent qualitative agreement with theory, although quantitative comparison is difficult in view of refractive mdex differences and the possibility of surface contamination. When a trace of surface-active contaminant is present, the motion tends to be damped out first at the rear of... 

For a particle which is spherically isotropic (see Chapter 2), the three principal resistances to translation are all equal. It may then be shown (H3) that the net drag is — judJ regardless of orientation. Hence a spherically isotropic particle settling through a fluid in creeping flow falls vertically with its velocity independent of orientation. 

All the work discussed in the preceding sections is subject to the assumptions that the fluid particles remain perfectly spherical and that surfactants play a negligible role. Deformation from a spherical shape tends to increase the drag on a bubble or drop (see Chapter 7). Likewise, any retardation at the interface leads to an increase in drag as discussed in Chapter 3. Hence the theories presented above provide lower limits for the drag and upper limits for the internal circulation of fluid particles at intermediate and high Re, just as the Hadamard-Rybzcynski solution does at low Re. 

As shown in Fig. 8.1, spherical-cap fluid particles are geometrically similar with a wake angle of approximately 50"" once Re is greater than about 150. The radius of curvature may then be related directly to either V or yielding... 

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