Solid Particle in a Fluid Flow

This chapter deals with the movement of a small solid particle in a fluid flow. We start by presenting the equations governing particle movements, which we refer to as the Basset, Boussinesq, Oseen, and Tchen (BBOT) equations, to name a few key contributors to this modeling. Rather than deriving the equations, we endeavor to identify and discuss the physical meaning of the different terms acceleration, added mass, Basset term, etc. 

This approach is embodied by applying the BBOT equations to describe the behavior of a particle in three configurations of particular significance by their applications  

the movement of a fluid particle under the effect of gravity in a fluid at rest, 

the movement of a particle in a unidirectional sheared fluid flow, and 

The BBOT equations allow the determination of the characteristic time with which the dynamics of a solid particle placed in a fluid flow adapts to its environment. This chapter is quite theoretical, although we have presented few derivations. This enables the reader to understand the hypotheses used in Chapters 15 (behavior of particles within gravity field) and 17 (centrifugation). 

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Chapters 16 and 18 discuss more academic matters. Chapter 16 deals with the movement of a solid particle in a fluid flow, while in Chapter 18 we introduce some essential notions on granular media. Such notions are useM to an engineer specializing in agri-food process engineering, to understand what happens inside a silo. 

Let us consider a three-phase system composed by an elastic body with large pores where a mixture of two immiscible fluids (or a suspension of solid particles in a fluid or a fluid component carrying an adsorbate) flows through. 

The centrifugal flows considered in this chapter are those dominated by rotatioa The azimuthal component of the velocity is preponderant, that is, Ur ue and Uz Ue in the cylindrical coordinate system. In such a configuration, the flow tends to become two-dimensional in a plane perpendicular to the Oz axis. The Ur and Ue components of the velocity are quasi-independent from coordinate z. The proof of this property goes beyond the scope of this chapter. Our goal is to describe the centrifugation of solid particles in a rotating flow. We simply choose to consider steady-state axisymmetric fluid flows, whose velocity and pressure fields possess the following kinematic characteristics ... 

Torobin, L. B. and Gauvin, W. H. Can. J. Chem. Eng. 38 (1959) 129, 167, 224. Fundamental aspects of solids-gas flow. Part I Introductory concepts and idealized sphere-motion in viscous regime. Part II The sphere wake in steady laminar fluids. Part III Accelerated motion of a particle in a fluid. 

In the present study, two-dimensional Two-Fluid Eulerian model was used to describe the steady state, dilute phase flow of a wet dispersed phase (wet solid particles) in a continuous gas phase through a pneumatic dryer. The predictions of the numerical solutions were compared successfully with the results of other one-dimensional numerical solutions and experimental data of Baeyens et al. [5] and Rocha [13], Axial and the radial distributions of the characteristic properties were examined. 

The dynamic response of a particle in gas-solid flows may be characterized by the settling or terminal velocity at which the drag force balances the gravitational force. The dynamic diameter is thus defined as the diameter of a sphere having the same density and the same terminal velocity as the particle in a fluid of the same density and viscosity. This definition leads to a mathematical expression of the dynamic diameter of a particle in a Newtonian fluid as... 

We first consider brielly why a polymer solution would be expected to have a higher viscosity than the liquid in which it is dissolved. We think initially of a suspension of solid particles in a liquid. The particles are wetted by the fluid, and the suspension is so dilute that the disturbance of the flow pattern of the suspending medium by one particle docs not overlap with that caused by another. Consider now the flow of the fluid alone through a tube which is very large compared to the dimensions of a suspended particle. If the fluid wets the tube wall its velocity profile will be that shown in Fig. 3-5a. Since the walls are wetted, liquid on the walls is stationary while the flow rate is greatest at the center of the tube. The flow velocity v increases from the wall to the center of the tube. The difference in velocities of adjacent layers of liquid (velocity gradient = civ/dr) is greatest at the wall and zero in the center of the tube. 

Fluidization refers to the state of solid particles in a suspended condition owing to the flow of fluid, gas, and/or liquid. Contact schemes of fluidized bed systems can be classified on the basis of the states of solid motion. For a batch-solids system, the fluid at a low velocity merely percolates through the voids between packed particles, while the particles remain motionless. The solids in this case are in the fixed bed state. With an increase in the fluid velocity, particles move apart and become suspended the bed then enters the fluidization state. The fluidization characteristics vary, depending on whether gas, liquid, or gas-liquid is the fluidizing medium. 

Langlois WE (1964) Slow Viscous Flow. Macmillan, New York Laux H (1998) Modeling of dilute and dense dispersed fluid-particle flow. Dr Ing Thesis, Norwegian University of Science and Technology, Trondheim, Norway Lawler MT, Lu P-C (1971) The role of lift in radial migration of particles in a pipe flow. In Zandi 1 (ed) Advances in Solid-Liquid Flow in Pipes and its Apphcations. Pergamon Press, Oxford, Chap 3, pp. 39-57 Lee SL (1987) Particle drag in a dilute turbulent two-phase suspension flow. Int J Multiphase Flow 13(2) 247-256... 

Proppants Solid particles in a suspension that is injected into a petroleum reservoir to maintain open fractures that have been artificially induced. The proppant-filled fractures then remain permeable to the flow of fluids. The suspension used for this purpose is sometimes termed proppant-laden fluid. See also Chapter 11. 

Mass transfer in a fluid flowing around solid particles, e.g., catalyst particles, ion exchange resins, particles of a dissolving solid. 

Consider a single-entry separator used either for separating solid particles from a fluid or separating solid particles having sizes above a particular value from those having sizes below the particular value. Let u/Jy. be the total mass flow rate of solids in the feed fluid whose total volumetric flow rate is Qf. In such a separator, there are only two product streams, the overflow (/ = 1) and the underflow (/ = 2). The total mass flow rate of solids in the overflow and the underflow are, respectively, and u/Jj- The overflow is identified with essentially the carrier fluid and the finer particles, whereas the underflow is assumed to have most of the coarser particles and small amounts of carrier fluid. Figure 2.4.1 illustrates this for a hydrocyclone (Talbot, 1980). Sizes of various natural and industrial particles are shown in Figure 2.4.2. 

When Pi p2, the location of the interface becomes unstable. The above relation also shows that if rzo is increased, rj t is also increased, pushing the liquid-liquid interface closer to the wall of the bowl. Meanwhile, if there are any solid particles in the fluids, they are thrown to the bowl wall and can be collected from the bottom. Such liquid-liquid centrifoges have a narrower and taller bowl than those used for particle separations (Figure 7.3.12). Note that the bulk flow of each liquid phase is perpendicular to the direction of the external force causing the separation. 

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