Buoyancy force fluid-solid system

Summarizing the forces introduced above, tests carried out in different multiphase systems have shown that the order of importance of the different forces involved typically ranks buoyancy and drag in the first positions and then lift and virtual-mass forces for fluid-solid systems and virtual-mass and lift forces for fluid-fluid systems (see, for example, the studies on non-drag forces by Diaz et al (2008) and Barton (1995)), whereas the most common values for the corresponding constants are Cl = 0.25 and Cv = 0.5 both for fluid-fluid and for fluid-solid systems. Naturally, since it is straightforward to implement all the forces in a computational code (Vikas et al, 201 lb), it is best to include them all for the sake of generality. 

The particle and bulk densities are commonly used in mass balance equations, since the mass and the external volume of the particles are involved. On the other hand, the hydraulic density should be preferably used in hydrodynamic calculations, because buoyancy forces are involved, and so the total mass of the particle should be taken into account, including the fluid in the open pores. It is obvious that the particle density is equal to the skeletal and hydrodynamic density in the case of nonporous particles. Moreover, in the case of a porous solid in a gas-solid system, the gas density is much lower than the particle density, and tlius... 

Catalysts for fluidized-bed reactors have to be spherical as well. The appropriate particle size fraction for gas-solid systems can be estimated after Geldart [1] from the density difference between soKd and gas. Most widely used catalysts for fluidized beds and risers are Geldart-type B powders with particle diameters ranging from 40 to 500 pm or solid densities between 1.4 X 10 and 4 x 10 kg/m, respectively. When fluidization is provided by a Kquid as in ebullated-bed reactors, the particle sizes may be substantially larger because of the higher buoyancy in these systems. However, all types of fluidized-bed catalysts must exhibit high mechanical stability because they are exposed to abrasion on reactor walls and internals, collisions between particles and shear forces exerted by the surrounding fluid. 

Convection in Melt Growth. Convection in the melt is pervasive in all terrestrial melt growth systems. Sources for flows include buoyancy-driven convection caused by the solute and temperature dependence of the density surface tension gradients along melt-fluid menisci forced convection introduced by the motion of solid surfaces, such as crucible and crystal rotation in the CZ and FZ systems and the motion of the melt induced by the solidification of material. These flows are important causes of the convection of heat and species and can have a dominant influence on the temperature field in the system and on solute incorporation into the crystal. Moreover, flow transitions from steady laminar, to time-periodic, chaotic, and turbulent motions cause temporal nonuniformities at the growth interface. These fluctuations in temperature and concentration can cause the melt-crystal interface to melt and resolidify and can lead to solute striations (25) and to the formation of microdefects, which will be described later. 

---