Average particle velocity

Uhlemann and Mori [93] described the calculation of the average particle velocity wp within the fluidized bed. The impact time of a particle can be calculated with the help of particle distance ap and average particle velocity. Analogous to the previous derivations, the probability of destruction Pq> of the wetted particles surface - and thus of the liquid film - can be calculated by the ratio of the wetted surface to the total surface of the particle. 

The function y(co) characterizes the relaxation of the average particle velocity. For instance, with y(co) as given by Eq. (140), one can show that the average velocity at time t corresponding to an initial velocity v(f,) = v,- is given by [52]... 

In each volume element the average particle velocities of each substance can be different, so that the convection of the volume elements overlaps the relative movement of the particles of different substances. This macroscopic relative movement is known as diffusion. The average velocity of the particles of substance A is... 

A number average particle velocity may be introduced to simplify the expression for the convective terms ... 

Figure 5. Three-dimensionsal simulation for 800 /jm particles with a viscosity input model in the IIT slurry bubble column with a 5 cm width (a) instantaneous particle velocity vector plots and solid volume fractions (color bar) at 25 s, and (b) time-averaged particle velocity at the center (c) near the wall from 15 to 36 5. (Fi=2.02cm/s, VG=3,37cm/s)...

Because it will be useful when developing numerical methods in Chapter 5, we will consider next the case in which pi = Vp and p2 = Mp are constant. This case corresponds to all particles having the same volume and mass, so that the only phase-space variables are the velocities Vp and Vf. Starting from Eq. (4.63), we find that for this case p = MpN = PpOTp. Likewise, the average particle velocities are all equal, so we can set them equal to the mean particle velocity Up = Um = Uy = Un. Under these conditions, Eq. (4.64) becomes... 

Time-averaged particle mass flux in general can be expressed in terms of the time-averaged particle velocity Up, time-averaged mass concentration CJ (phase density), and particle diffusive mass flux m (covariance of particle mass concentration and velocity) as... 

The average transit time, rav, can be obtained from equation 22. The average particle velocity, vp, can be calculated from... 

Eqs. (4.7.6), (4.7.13), (6.3.13) Cross-sectional average particle velocity Radial drift velocity of particles in centrifugation... 

From the Taylor-Aris formulation for times t> a lD, where a is the capillary radius and D the Stokes-Einstein diffusion coefficient of the particle, the particle of radius will have had sufficient time to sample the full velocity profile. With the local particle velocity taken to be equal to that of the fluid (Eq. 4.2.14), the average particle velocity over the tube cross-section IJ is given by... 

Thus for a giv temperature diere is a corresponding average particle velocity. However, the individual particles are found to move around with slightly different velocities. J. C. Maxwell has calculated the velocity distribution of the particles in a gas. For the simplest possible system it is a Boltzmann distribution. In a system of n particles the number of particles n that have kinetic ergy > is given by... 

The average continuum velocity / in each cell / is obtained by averaging the x- and y-velocities on the bounding edges. Continuity of the mean velocity requires that the averaged particle velocity in this cell is equal to uy. 

Verver and van Swaaij [8] found a good agreement between calculated and experimentally determined values of average particle velocity (us) with optimal values of empirical parameters. However, both empirical parameters (the effective bed void and initial velocity) were found to have different values for each of the flowing solids materials tested. Kiel and van Swaaij [23] obtained a good agreement between computed and experimental results, for best values of empirical parameters found to be —0.05 m/sec for initial velocity and 1.35 for effectiveness factor. 

The length of fall in a void of equivalent diameter will vary depending on position, from L = 0 to L = dy. For a rough estimation, an average particle velocity in a void can be taken to be ... 

Velocity based on Kolmogorov s power dissipation approach (m/s) Time-averaged particle velocity (m/s)... 

The chemical diffusion coefficient D is necessary and sufficient for a phenomenological description of binary diffusion. For one-dimensional diffusion, in which a constant diffusional cross-section is assumed, the average particle velocities Vi (z = 1, 2) in an arbitrary coordinate system are given as ... 

For further investigation, sensitivity of the avalanching angle and average particle velocity to the material properties such as size, cohesion, and Young s modulus, as well as interaction properties such as sliding friction between the particles and... 

The advantage of PIV is that the entire, instantaneous dow deld can be obtained from two high-speed images of the bed. It is a well-known technique and several software packages exist that allow to postprocess the images. PIV, however, does not take the particle fraction in each interrogation zone into account, and therefore requires a correction step if one is interested in the particle dux rather than the average particle velocity. This correction step is discussed in Section 2.1.5. 

Although PIV measures the instantaneous, average particle velocity in every interrogation zone, the measurement technique does not account for the different number of particles in different interrogation zones. A combination of the two techniques discussed above is essential for a good... 

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