Fluid-particle drag

Glicksman (1984) showed that the list of controlling dimensionless parameters could be reduced if the fluid-particle drag is primarily viscous or primarily inertial. The standard viscous and inertial limits for the drag coefficient apply. This gives approximately... 

Furthermore, the closures for the fluid—particle drag and the particle-phase stresses that we discussed were all derived from data or analysis of nearly homogeneous systems. In what follows, we refer to the TFM equations with closures deduced from nearly homogeneous systems as the microscopic TFM equations. The kinetic theory based model equations fall in this category. 

Holloway, W., Yin, X. Sundaresan, S. 2010 Fluid-particle drag in inertial polydisperse gas-solid suspensions. AIChE Journal 56 (8), 1995-2004. 

Lateral Concentration Profiles in Horizontal Pipes. Previous theoretical studies for horizontal pipe and channel flow have concentrated on the variation of solids concentration in the vertical direction. In this case, gravity (including buoyancy), fluid-particle drag, turbulent diffusion, and particle-particle interaction effects must occur. In this section, the variation of solids concentration in a horizontal plane through the pipe axis is examined. 

The fluid-phase forces are readily obtained from the particle-phase relations for fluid-particle interaction (drag and the pressure gradient force), which acts in the opposite direction on the fluid, together with gravity and the effect of the fluid pressure gradient across the control volume fluid viscosity effects are considered only in so far as they as they contribute to fluid-particle drag. Thus we have for the axial component Pfi-... 

Holloway W, Yin XL, Sundaresan S Fluid-particle drag in inertial polydisperse gas-sohd suspensions, AIChEJ 56 1995-2004, 2010. http //dx.doi.org/10.1002/aic.12127. 

FIG. 17-2 Schematic phase diagram in the region of upward gas flow. W = mass flow solids, lh/(h fr) E = fraction voids Pp = particle density, Ih/ft Py= fluid density, Ih/ft Cd = drag coefficient Re = modified Reynolds uum-her. (Zenz and Othmei Fluidization and Fluid Particle Systems, Reinhold, New York, 1960. )... 

A relationship between these four variables is required in order to prediet partiele veloeity in a variety of eireumstanees. In doing so, it is noted that as a partiele moves through a fluid it experienees drag and viee versa as the fluid moleeules move aeross and around the surfaee of the partiele. There is thus a fluid-particle interaction due to interfaeial surfaee drag. 

This result can also be applied directly to coarse particle swarms. For fine particle systems, the suspending fluid properties are assumed to be modified by the fines in suspension, which necessitates modifying the fluid properties in the definitions of the Reynolds and Archimedes numbers accordingly. Furthermore, because the particle drag is a direct function of the local relative velocity between the fluid and the solid (the interstitial relative velocity, Fr), it is this velocity that must be used in the drag equations (e.g., the modified Dallavalle equation). Since Vr = Vs/(1 — Reynolds number and drag coefficient for the suspension (e.g., the particle swarm ) are (after Barnea and Mizrahi, 1973) ... 

The procedure for determining APs that will be presented here is that of Molerus (1993). The basis of the method is a consideration of the extra energy dissipated in the flow as a result of the fluid-particle interaction. This is characterized by the particle terminal settling velocity in an infinite fluid in terms of the drag coefficient, Cd ... 

The scope of coverage includes internal flows of Newtonian and non-Newtonian incompressible fluids, adiabatic and isothermal compressible flows (up to sonic or choking conditions), two-phase (gas-liquid, solid-liquid, and gas-solid) flows, external flows (e.g., drag), and flow in porous media. Applications include dimensional analysis and scale-up, piping systems with fittings for Newtonian and non-Newtonian fluids (for unknown driving force, unknown flow rate, unknown diameter, or most economical diameter), compressible pipe flows up to choked flow, flow measurement and control, pumps, compressors, fluid-particle separation methods (e.g.,... 

Note that depending on the manner in which the drag force and the buoyancy force are accounted for in the decomposition of the total fluid particle interactive force, different forms of the particle motion equation may result (Jackson, 2000). In Eq. (36), the total fluid-particle interaction force is considered to be decomposed into two parts a drag force (fd) and a fluid stress gradient force (see Eq. (2.29) in Jackson, 2000)). The drag force can be related to that expressed by the Wen-Yu equation, /wen Yu> by... 

In addition, it is dubious whether this new correlation due to Brucato et al. (1998) should be used in any Euler-Lagrangian approach and in LES which take at least part of the effect of the turbulence on the particle motion into account in a different way. So far, the LES due to Derksen (2003, 2006a) did not need a modified particle drag coefficient to attain agreement with experimental data. Anyhow, the need of modifying particle drag coefficient in some way illustrates the shortcomings of the current RANS-based two-fluid approach of two-phase flow in stirred vessels. 

A particle drag coefficient Cd can now be defined as the drag force divided by the product of the dynamic pressure acting on the particle (i.e. the velocity head expressed as an absolute pressure) and the cross-sectional area of the particle. This definition is analogous to that of a friction factor in conventional fluid flow. Hence... 

The terminal velocity of a fluid particle in creeping flow is obtained by equating the total drag to the net gravity force, 47ia Ap /3, giving ... 

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 for other types of fluid particle, the internal circulation of water drops in air depends on the accumulation of surface-active impurities at the interface (H9). Observed internal velocities are of order 1% of the terminal velocity (G4, P5), too small to affect drag detectably. Ryan (R6) examined the effect of surface tension reduction by surface-active agents on falling water drops. 

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