Particle Velocity Distributions

The complete velocity distribution function f(v, s) is usually defined in six spaces (v v v x z). However, for convenience in evaluating the particle speed distribution in a specific direction in the velocity space, the vector velocity y is represented by its magnitude, the speed v, and an angle coin the vector direction of v. The velocity distribution function is, therefore, denoted by J (v, co, x), such that f (v, co, z)d odv represents the fraction of particles located at x with velocity vectors in an element of solid angle dw centered about the vector v and with speed between v-dv/2 and v+dv/2, and 

The function/ (v, co, x) is the normalized function and is assumed to be time independent. The normalized speed distribution function f(v, x) is then introduced such that f(v, x)dv is the fraction of particles in the speed range dv, centered at speed v, moving in all possible directions at location . Clearly, 

FIGURE 9.17 Solids mean speed distributions in a cylindrical fluidized bed for 500- im glass particles at ujumf = 2 

FIGURE 9.18 Directional distribution functions of particles that hit two surfaces of a 12.7-cm-diameter cylinder 

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As it stands, the time-independent transport equation (105) still applies to the complete iV-particle velocity distribution... 

What we really need is the electrical current (49), and for this it is sufficient to know the one-particle velocity distribution function <5 i(P t) indeed, we may write Eq. (49) as ... 

Methods for analysis of the particle size distribution in the aerosol cloud include techniques such as time of flight measurement (TOE), inertial impaction and laser diffraction. Dynamic light scattering (photon correlation spectroscopy) is confined to particles (in suspension) in the submicron range. In addition to the size distribution, the particle velocity distribution can be measured with the Phase Doppler technique. 

S. Reinhart, "Application of the Engrave-ment Method to the Study of Particle Velocity Distribution in Explosively Loaded Cylinders , JApplPhys 26, 1431-35(1955) NOTS 1385, NavQrd 5040, (2 Mar 1956) 9) C.H. Bagley... 

With the dipole force neglected, the direct integration of Eq. (E3.14) yields the particle velocity distribution as... 

In order to evaluate the collisional integrals Pc, qc, and y, explicitly, it is important to know the specific form of the pair distribution function /(2)(vi, ri V2, ry. t). The pair distribution function /(2) may be related to the single-particle velocity distribution function f by introducing a configurational pair-correlation function g(ri, r2). In the following, we first introduce the distribution functions and then derive the expression of /(2) in terms of f by assuming /(1) is Maxwellian and particles are nearly elastic (i.e., 1 — e 1). 

Assume that a complete pair distribution function can be expressed as the product of the spatial pair distribution function and the two single-particle velocity distribution functions. Thus, we have... 

Assuming that the single-particle velocity distribution is Maxwellian and takes the form... 

Single-particle velocity distribution function /<2) Complete pair distribution function... 

Figure 6.13. Effect of Cp and Nup on the dynamic properties of phases over a plane shock in a gas-solid suspension with a Mach number of 1.5, 10 /tm glass beads, and mp = 0.2 (from Rudinger, 1969) (a) Temperature distributions (b) Pressure and particle velocity distributions.

Particles migrate to the wall region by means of particle-particle collisions and diffusion, and through particle-wall collision effects which tend to widen the particle velocity distribution in the radial direction. 

If the NDF is a function of the particle velocity then the solution of the GPBE provides the modeler with the essential information for calculating the real-space advection term. This approach is used whenever the particle Stokes number is not small, and will result in the development of a particle-velocity distribution. More details on this topic can be found in Chapter 8. An alternative approach consists of integrating the NDF with respect to the particle velocity. Let us consider, for example, a generic NDF n(t,x, p, p), which is a function of the time t, space x, particle velocity Vp, and internal coordinates p. By integrating out the particle velocity the following NDF is obtained ... 

In summary, the Eulerian two-fluid model is represented by Eqs. (5.112) and (5.113) in addition to a constitutive model for the fluid stress tensor Tf. As already mentioned, Eq. (5.112) was derived under the assumption that the particle-velocity distribution is very narrow (i.e. small particle Stokes number), and the particles must have the same internal coordinates. If these simplifications do not hold, for example under dense conditions when particle-particle collisions become important, then particle-velocity fluctuations must be taken into account, as discussed at the end of Chapter 4. 

The next step is to provide a closure for the pair correlation function appearing in the collision source and collisional-flux terms. For moderately dense flows, the collision frequency for finite-size particles is known to be larger than that found using the Boltzmann Stofizahlansatz (Carnahan Starling, 1969 Enksog, 1921). In order to account for this effect, the pair correlation function can be modeled as the product of two single-particle velocity distribution functions and a radial distribution function ... 

As explained throughout the book, disperse multiphase systems are characterized by multiple phases, with one phase continuous and the others dispersed (i.e. in the form of distinct particles, droplets, or bubbles). The term polydisperse is used in this context to specify that the relevant properties characterizing the elements of the disperse phases, such as mass, momentum, or energy, change from element to element, generating what are commonly called distributions. Typical distributions, which are often used as characteristic signatures of multiphase systems, are, for example, a crystal-size distribution (CSD), a particle-size distribution (PSD), and a particle-velocity distribution. 

In order to obtain the instantaneous particle velocity directly from the signals of the reflected light of the probe for particle velocity measurement, Qin and Liu (1982) proposed a method to acquire the instantaneous particle velocity distribution with a logic discrimination method. 

FIGURE 4-37 Typical particle velocity distribution as measured by a five-fiber optic probe (Sand, dp of 150 im, by free fall system). 

Pneumatic transportation of solids is important to many industrial processes, for example transporting coal and powder particles. To an operator of such a pneumatic conveyor, the mass flow rate of the solids is the primary process parameter to be measured accurately. A solid/gas flow is very difficult to control because it behaves quite differently from solid/liquid flows. A recent review (Yan, 1996) discussed several variables that may affect the performance of a flow instrument. The distribution of solids in a pneumatic pipeline can be highly inhomogeneous consequently, the particle velocity distribution over the pipe cross section can be widespread. Figure 6.28 shows examples in which the roping type flow is particularly difficult to understand and monitor. 

In order to derive the particle velocity distribution of a gas we need a trick. We imagine all the particles with the same velocity v to be molecules of a substance B(o) and the entire gas as a mixture of a large number of such substances. However, we run into a difficulty at this point. The number of particles having... 

The state of a monatomic gas is described by the one-particle velocity distribution function fit, r, v), where t is the time, r is the position vector, and v is a velocity of molecules. The function fit, r, v) is defined so that the quantity fit, r, v) dr dv is the number of particles in the phase volume dr dv near the point (r, v) at the time t. AU macro-characteristics of gas flow can be calculated via the velocity distribution function ... 

Zeta Potential Measurement, Fig. 3 Stationary level, fluid velocity, and particle velocity distribution in an electrophoresis cell [1]... 

Carlos, N. F. and Richardson, J. F. Solids movements in liquid fluidized beds. 1. Particle velocity distribution. Chem. Engng Sci. 23, 813-824 (1968). 

Fig. 18.13. Water particle velocity distribution around the breakwater head.

Carlos CR, Richardson IF. Solids movement in Uquid fluidised beds—I. Particle velocity distribution. Chem Eng Sci 23 813-824, 1968a. 

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