Particle 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) ... 

In the limit of very high voidage, the drag coefficient can be related to the single particle drag coefficient. For the case of spherical particles,... 

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. 

In some way, introducing an increased particle drag by means of Eq. (17) resembles the earlier proposal raised by Bakker and Van den Akker (1994b) to increase viscosity in the particle Reynolds number due to turbulence (in agreement with the very old conclusion due to Boussinesq, see Frisch, 1995) with the view of increasing the particle drag coefficient and eventually the bubble holdup in the vessel. Lane et al. (2000) compared the two approaches for an aerated stirred vessel and found neither proposal to yield a correct spatial gas distribution. 

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 glass sphere, of diameter 6 mm and density 2600 kg/m3, falls through a layer of oil of density 900 kg/m3 into water. If the oil layer is sufficiently deep for the particle to have reached its free falling velocity in the oil, how far will it have penetrated into the water before its velocity is only 1 per cent above its free falling velocity in water It may be assumed that the force on the particle is given by Newton s law and that the particle drag coefficient R /pu2 = 0.22. 

Terminal falling velocity and particle drag coefficient... 

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 question these correlations ask is why does the entrainment rate decrease for smaller particles for some systems whereas in other systems, the entrainment rate correlates with the particle terminal velocity or particle drag. Baeyens infers that particles may be clnstering due to an interparticle adhesion force that becomes dominant at some critical particle diameter. However, no evidence of particle clnsters was reported. Baeyens assnmption was based on fitting their data. Therefore, the role of particle clnstering on entrainment rates was difficult to establish from first principles. 

The pressure drop reflects the sum of single-particle drag-force contributions, which is (Foscolo and Gibilaro, 1984)... 

With a venturi having an area ratio of 3.3 1, the velocity ratio and thus, particle size range trapped in the venturi is in theory, 11 1. This means if the velocity at the throat is set to suspend an 11 mm particle then particles just less than 1 mm would be lost to the filters. In reality this bandwidth was diminished by the layout of the test facility and variable particle Cd. In the layout shown as Fig. 2 the material entered from a drop tube at the top of the venturi, this allowed the particles to accelerate for nearly 1.2 m prior to the venturi constriction. If the throat velocity were set to suspend an 11 mm particle then the gravity force is balanced by the particle drag at the throat, however, there will be no net acceleration. This means that if an 11 mm particle entered the throat with any vertical velocity there would be no deceleration of that particle and it would pass through. Thus, the bandwidth of the unit is limited by the method in which the particles are introduced into the separator. In the testing work undertaken, a nominal bandwidth of 6 1 was observed. This could have been increased by altering the feed from a vertical drop to a lower horizontal entry. 

By equating the pressure gradient and the gas-particle drag force, the general momentum balance for the gas phase can be expressed as... 

The movement of each particle drags the surrounding fluid that convects and rotates the other particle. 

Mueller, P and Reh, L. Particle Drag and Pressure Drop in Accelerated Gas-Solid Flow," in Circulating Fluidized Bed Technology IV (Amos A. Avidan, eds.), pp. 193-198. Somerset, Pennsylvania (1993). 

Cdo = single particle drag coefficient at culated from Cdo = Ns. A Usuper cal- m... 

In regions with lower particle volume fractions a modified form of the single particle drag correlation is used instead ... 

Morsi and Alexander s (MA) correlation represents the single-particle drag curve accurately. Ma and Ahmadi s correlation predicts values comparable with the MA correlation. Molerus correlation deviates from the MA correlation at higher Reynolds numbers. Patel s correlation is found to give a better fit with the MA correlation than Richardson s correlation. Dalla Ville s correlation overpredicts values of drag coefficient compared to the MA correlation. 

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