Buoyancy and drag forces

A constant particle mass does not imply zero net momentum change due to mass transfer. For example, the particle could be losing and gaining mass at the same rates so that dMp/dr = 0, but the momentum of the lost and gained masses need not be die same due to differences in dieir velocities. 

However, it would then be necessary to relate Vp and Ad to the internal coordinates used to describe the particle size (see Section 5.2.1). By dividing Eq. (5.53) by the particle mass (PpVp), the particle acceleration due to buoyancy and drag is readily calculated, and by assuming that all the particles are statistically identical the following expression for the pure advection velocity is obtained for an isolated sphere (ap = 1)  

The expression in Eq. (5.55) is of course valid only for a sphere characterized by very small slip velocity, and it is generally assumed to be valid for Rep 0.1. For spherical particles at higher particle Reynolds numbers, the following corrections can be used  

These equations are valid for isolated spherical particles when the surrounding continuous phase can be treated as a continuum (as opposed to a rarefied gas). In fact, it is important to quantify the ratio between the mean free path (i.e. average time interval between two subsequent collisions) of the molecules constituting the primary phase and the particle size  

A similar expression has been developed for spherical isolated bubbles in contaminated systems  

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Coverage has been limited to horizontal three-phase separators up to this point. Considering Fig. 4.9, oil and water must flow vertically downward and gas vertically upward. The same laws of buoyancy and drag force apply. Equation (4.3) may therefore be used in the oil phase for water separation. Equations (4.12), (4.13), and (4.7) (see Fig. 4.8) are applied to the gas phase and oil phase for oil-gas particle separations, as was equally done for horizontal separators. The equations for the horizontal separator from Fig. 4.8 may also be used for the water drop terminal velocity in the vertical separator. 

The change of momentum for a particle in the disperse phase is typically due to body forces and fluid-particle interaction forces. Among body forces, gravity is probably the most important. However, because body forces act on each phase individually, they do not result in momentum transfer between phases. In contrast, fluid-particle forces result in momentum transfer between the continuous phase and the disperse phase. The most important of these are the buoyancy and drag forces, which, for reasons that will become clearer below, must be defined in a consistent manner. However, as detailed in the work of Maxey Riley (1983), additional forces affect the motion of a particle in the disperse phase, such as the added-mass or virtual-mass force (Auton et al., 1988), the Saffman lift force (Saffman, 1965), the Basset history term, and the Brownian and thermophoretic forces. All these forces will be discussed in the following sections, and the equations for their quantification will be presented and discussed. 

Many researches adopted one of the aforanentioned approaches and modified it to include various aspects of the pneumatic drying process. Andrieu and Bressat [16] presented a simple model for pneumatic drying of polyvinyl chloride (PVC), particles. Their model was based on elementary momentum, heat and mass transfer between the fluid and the particles. In order to simplify their model, they assumed that the flow is unidirectional, the relative velocity is a function of the buoyancy and drag forces, solid temperature is uniform and equal to the evaporation temperature, and that evaporation of free water occurs in a constant rate period. Based on their simplifying assumptions, six balance equations were written for six unknowns, namely, relative velocity, air humidity, solid moisture content, equilibrium humidity, and both solid and fluid temperatures. The model was then solved numerically, and satisfactory agreanent with their experimental results was obtained. A similar model was presented by Tanthapanichakoon and Srivotanai [24]. Their model was solved numerically and compared with their experimental data. Their comparison between the experimental data and their model predictions showed large scattering for the gas temperature and absolute humidity. However, their comparisons for the solid temperature and the water content were failed. 

A theoretically better founded approach has been developed by Stichlmair (1978 Stichhnair and Fair 1998). A balance of weight, buoyancy, and drag forces on a single droplet whose size is determined via a critical Weber number yields... 

The gravitational settling of a particle is described by Stokes law, which is based on a force balance between gravitational, buoyancy, and drag forces acting on a particle. These relations can be manipulated to provide an estimate of the terminal settling... 

Equations 3 to 7 indicate the method by which terminal velocity may be calculated. Erom a hydrodynamic force balance that considers gravity, buoyancy, and drag, but neglects interparticle forces, the single particle terminal velocity is... 

Zeng et al. [3] used a force balance approach to predict the bubble diameter at departure. They included the surface tension, inertial force, buoyancy and the lift force created by the wake of the previously departed bubble. But there was empiricism involved in computing the inertial and drag forces. The study assumed a power law profile for growth rate with the proportionality constant exponent determined from the experiments. 

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. 

Answer Since there is no longer any acceleration when submerged objects achieve terminal velocity, the sum of all forces acting on the object must be zero. Hence, there is a balance between buoyancy, gravity, and hydrodynamic drag. The gravity force acts downward, and the buoyant and drag forces act in the opposite direction. Each force is calculated as follows ... 

Combining the expressions for the drag, buoyancy, and gravitational forces, the volumetric flow rate as a function of conditions gives ... 

When settling aggregates have reached their terminal velocity, the buoyancy force and drag force have reached an exact balance ... 

Consider a single particle falling under gravity in a static gas in the absence of any solids boundaries. We know that this particle will reach a terminal velocity when the forces of gravity, buoyancy and drag are balanced (see Chapter 2). If the gas of infinite extent is now considered to be moving upwards at a velocity equal to the terminal velocity of the particle, the particle will be stationary. If the gas is moving upwards in a pipe at a superficial velocity equal to the particle s terminal velocity, then ... 

The sphere will reach a terminal velocity and, assuming the cylinder radius is large compared to the sphere, equilibrium of the forces on the sphere (gravity, buoyancy and drag) will give. 

Transport Disengaging Height. When the drag and buoyancy forces exerted by the gas on a particle exceed the gravitational and interparticle forces at the surface of the bed, particles ate thrown into the freeboard. The ejected particles can be coarser and more numerous than the saturation carrying capacity of the gas, and some coarse particles and clusters of fines particles fall back into the bed. Some particles also coUect near the wall and fall back into the fluidized bed. 

A single particle settling in a gravity field is subjected primarily to drag force, gravity force, Ta-g-, and buoyancy, which have to be in... 

The sequence, flocculation — coalescence — separation, is compHcated by the fact that creaming or sedimentation occurs and that this process is determined by the droplet size. The sedimentation velocity is monitored by the oppositely directed forces which form the buoyancy and the viscous drag of the continuous phase on the droplet ... 

The hydrodynamic region has received considerable attention over the years. Equations (2-63) and (2-64) follow the buoyancy-drag force balance theory. If we... 

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... 

The prime difficulty of modeling two-phase gas-solid flow is the interphase coupling, which deals with the effects of gas flow on the motion of solids and vice versa. Elgobashi (1991) proposed a classification for gas-solid suspensions based on the solid volume fraction es, which is shown in Fig. 2. When the solid volume fraction is very low, say es< 10-6, the presence of particles has a negligible effect on the gas flow, but their motion is influenced by the gas flow for sufficiently small inertia. This is called one-way coupling. In this case, the gas flow is treated as a pure fluid and the motion of particle phase is mainly controlled by the hydrodynamical forces (e.g., drag force, buoyancy force, and so... 

These authors have assumed the bubble to be expanding at the orifice, and have used the force balance equation at the time of detachment. The various forces considered by these authors are buoyancy, force due to the addition of mass (P2), excess pressure force, surface tension force, drag force, and force due to the inertia of the liquid. 

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