Drag on a particle

Drag on a particle at nonnegligible Kn, but low Ma and Re, is conveniently expressed by the slip correction factor ... 

From the examples of experimental studies discussed, it is clear that it is impossible to predict with any confidence either the magnitude or the direction of the drag on a particle when the relative velocity and acceleration are not... 

Lee [86] examined earlier studies on cylinders in turbulent flow fields, and found in turbulent flow that the wake would decrease with increasing turbulence intensity, and then disappear, and the mean flow would behave in the same manner as for Stokes flow. He therefore proposed, and showed, that the drag on a particle in turbulent flow could be determined from a Stokes law with the molecular viscosity replaced by an effective viscosity calculated from the volumetric concentration of particles, the flow Reynolds number, the density ratio, and the Proude number. The Proude number, determining the ratio between the initial- and gravity forces, is defined as ... 

At the level of the Stokes-Einstein equation, infinite dilntion is assumed. For calculation of the hydrodynamic drag, analysis of a single sphere is sufficient and particle-particle forces such as the charge effect need not be considered. When small concentration effects are included, up to the level indicated in Equation 8.74, the drag on a particle needs to be considered in the presence of a second... 

Figure 3.3 Voidage functions for drag on a particle in a particle bed Continuous curve, the common adopted form, open squares, the viscous regime form, 3.33(1 e)/e + 1 solid squares, the inertial regime form, 3.55...

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. 

Particle Velocity on a. Surfa.ce. Smaller particles, those that are more irregular in shape and/or those that have a higher surface roughness, typically have a higher frictional drag on a hopper or chute surface. 

Nonsplierical Rigid Particles The drag on a nonspherical particle depends upon its shape and orientation with respect to the direction of motion. The orientation in free fall as a function of Reynolds number is given in Table 6-8. 

Finer particles ( < 3 pm), termed respirable particles, pass beyond the ex-trathoracic airways and enter the tracheobronchial tree. Impaction plays a significant role near the tracheal jet, but sedimentation predominates as the effects of rapid conduit expansion dampen in the distal trachea and beyond. Sedimentation occurs when gravitational forces exerted on a particle equal drag forces, i.e., when particle velocity falls to u . As mean inspiratory air-stream velocity gradually declines along the tracheobronchial tree, particle momentum diminishes and 0.5-3 pm MMAD particles settle out of the airflow and onto mucosal surfaces. 

There are essentially three forces that act on a particle moving through a fluid. They are (i) the external force, gravitational or centrifugal (ii) the buoyant force, which acts parallel with the external force, but in the opposite direction and (iii) the drag force which appears... 

In a series of papers, Chhabra (1995), Tripathi et al. (1994), and Tripathi and Chhabra (1995) presented the results of numerical calculations for the drag on spheroidal particles in a power law fluid in terms of CD = fn(tVRe, ). Darby (1996) analyzed these results and showed that this function can be expressed in a form equivalent to the Dallavalle equation, which applies over the entire range of n and tVRe as given by Chhabra. This equation is... 

Tripathi A, RP Chhabra. Drag on spheroidal particles in dilatant fluids. AIChE J 41 728, 1995. 

The solids flux depends on the local concentration of solids, the settling velocity of the solids at this concentration relative to the liquid, and the net velocity of the liquid. Thus the local solids flux will vary within the thickener because the concentration of solids increases with depth and the amount of liquid that is displaced (upward) by the solids decreases as the solids concentration increases, thus affecting the upward drag on the particles. As these two effects act in opposite directions, there will be some point in the thickener at which the actual solids flux is a minimum. This point determines the conditions for stable steady-state operation, as explained below. 

The drag force that the gas phase exerts on a particle a, consistent with the source term Sp in expression Eq. (41), reads... 

As the fluid velocity is increased the drag on the particles increases and a point is reached where the pressure drop balances the effective weight of bed per unit cross-sectional area. At this point the fluid drag just supports the solid particles. A small increase in the flow rate causes a slight expansion of the bed from its static, packed state. Further increase in the flow rate allows the bed to expand more and the particles become free to move around and the bed is said to be fluidized. The state when the bed just becomes fluidized is known as incipient, or minimum, fluidization. The fluid velocity required to cause incipient fluidization is called the minimum fluidization velocity and is denoted by umf. 

If the fluid is moving relative to some surface other than that of the particle, there will be a superimposed velocity distribution and the drag on the particle may be altered. Thus, if the particle is situated at the axis of a vertical tube up which fluid is flowing in streamline motion, the velocity near the particle will be twice the mean velocity because of the... 

Rowe and Henwood(26) made similar studies by supporting a spherical particle 12.7 mm diameter, in water, at the end of a 100 mm length of fine nichrome wire. The force exerted by the water when flowing in a 150 mm square duct was calculated from the measured deflection of the wire. The experiments were carried out at low Reynolds numbers with respect to the duct (< 1200), corresponding to between 32 and 96 relative to the particle. The experimental values of the drag force were about 10 per cent higher than those calculated from the Schiller and Naumann equation. The work was then extended to cover the measurement of the force on a particle surrounded by an assemblage of particles, as described in Chapter 5. 

The curve for R /pu2 against Re may be divided into four regions, (a), (b), (c) and (d), as before. In region (a) the flow is entirely streamline and, although no theoretical expressions have been developed for the drag on the particle, the practical data suggest that a law of the form ... 

Several expressions of varying forms and complexity have been proposed(35,36) for the prediction of the drag on a sphere moving through a power-law fluid. These are based on a combination of numerical solutions of the equations of motion and extensive experimental results. In the absence of wall effects, dimensional analysis yields the following functional relationship between the variables for the interaction between a single isolated particle and a fluid ... 

From Table 3.9 it is seen that, depending on the value of n, the drag on a sphere in a power-law fluid may be up to 46 per cent higher than that in a Newtonian fluid at the same particle Reynolds number. Practical measurements lie in the range 1 < Y < 1.8, with considerable divergences between the results of the various workers. 

It is seen therefore that, in the absence of any entirely satisfactory theoretical approach or reliable experimental data, it is necessary to adopt a highly pragmatic approach to the estimation of the drag force on a particle in a non-Newtonian fluid. 

In a concentrated suspension, the drag force on a particle will be a function of its velocity up relative to the liquid and will be influenced by the concentration of particles that is, it will be a function of the voidage e of the suspension. 

Thus, for any voidage the drag force on a sedimenting particle can be calculated, and the corresponding velocity required to produce this force on a particle at the same voidage in the model is obtained from the experimental results. All the experiments were carried out at particle Reynolds number greater than 500, and under these conditions the observed sedimentation velocity is given by equations 5.76 and 5.84 as ... 

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