Mean velocity particle

MUNICIPAL WATER treatment). Scale-up of orthokinetic flocculators, generally in the form of paddle devices, is based on the product of mean velocity gradient and time, for a constant volume concentration of the flocculating particles. 

FIGURE 14.6 Log-normal probability size distributions, illustrating geometrical transposition between number, mass, and linear momentum curves and the mean size particle d,, which can be used in estimating the free-falling velocity of the particle group. 

Elutriation differs from sedimentation in that fluid moves vertically upwards and thereby carries with it all particles whose settling velocity by gravity is less than the fluid velocity. In practice, complications are introduced by such factors as the non-uniformity of the fluid velocity across a section of an elutriating tube, the influence of the walls of the tube, and the effect of eddies in the flow. In consequence, any assumption that the separated particle size corresponds to the mean velocity of fluid flow is only approximately true it also requires an infinite time to effect complete separation. This method is predicated on the assumption that Stokes law relating the free-falling velocity of a spherical particle to its density and diameter, and to the density and viscosity of the medium is valid... 

Hao et al. (2007) investigated the water flow in a glass tube with diameter of 230 Lim using micro particle velocimetry. The streamwise and mean velocity profile and turbulence intensities were measured at Reynolds number ranging from 1,540 to 2,960. Experimental results indicate that the transition from laminar to turbulent flow occurs at Re = 1,700—1,900 and the turbulence becomes fully developed at Re > 2,500. 

The stress acting on particles is due to a relative velocity between the particles and the fluid. If their mean velocities also differ, contact between the particles or between a particle and the tank wall or the impeller elements leads to impact stress. However, this impact stress is negligible if the density differences and the particle concentrations are low. 

Fig. 1. Relationship between reactor mean velocity gradient and particle size for isoelectric soya protein precipitate. Open symbols represent precipitate diameter for 50% oversize Closed symbols represent precipitate diameter for 90% oversize [51]...

Given that a collision takes place, the nature of the momentum transfer between the cells must be specified. This should be done in such a way that the total momentum and kinetic energy on the double cell are conserved. There are many ways to do this. A multiparticle collision event may be carried out on all particles in the pair of cells. Alternatively, a hard sphere collision can be mimicked by exchanging the component of the mean velocities of the two cells along da,... 

Here V = (N Vj + N2S2) / N + N2) is the mean velocity of the pair of cells. By summing over the particles in cell 1 it is easy to verify that velocities normal to collision rule treats the particles in the two cells as groups that undergo elasticlike collisions. 

The attachment coefficient is a function of the aerosol particle diameter, d, and mean velocity, v, as well as the unattached progeny diameter, d, and its mean velocity v. Since in most situations d d and v v, equation (2) reduces to... 

For each of the three particles sizes studied, the deposition efficiency in the trachea was much greater than predicted if uniform deposition is assumed. In each data set the ratio at the lower flow rate exceeded that at a higher flow. If the increase results from turbulence introduced by the larynx it might be expected to be more effective at the higher flow rate. The discrepency may result from the jet formed downstream of the larynx. The center line velocity substantially exceeds the mean velocity of the air stream and secondary circulation patterns are set up near the wall which can act as dead zones (Ultman, 1 985) ... 

Note that when solving the CFD transport equations, the mean velocity and turbulence state variables can be found independently from the mixture-fraction state variables. Likewise, when validating the CFD model predictions, the velocity and turbulence predictions can be measured in separate experiments (e.g., using particle-image velocimetry [PIV]) from the scalar field (e.g., using planar laser-induced fluorescence [PLIF]). 

To define the mean velocity of particles in a relativistic ideal gas, as always, we use that... 

Solid particles hydraulically conveyed in a vertical pipe have a mean velocity which is less than the mean velocity of the liquid. This is because of the tendency of the particles to settle. The volume fraction a in equation 9.42 is the delivered concentration. This is less than the volume fraction in the vertical pipe. 

Since the mean velocity and Reynolds-stress fields are known given the joint velocity PDF /u(V x, t), the right-hand side of this expression is closed. Thus, in theory, a standard Poisson solver could be employed to find (p)(x, t). However, in practice, (U)(x, t) and (u,Uj)(x, t) must be estimated from a finite-sample Lagrangian particle simulation (Pope 2000), and therefore are subject to considerable statistical noise. The spatial derivatives on the right-hand side of (6.61) are consequently even noisier, and therefore are of no practical use when solving for the mean pressure field. The development of numerical methods to overcome this difficulty has been one of the key areas of research in the development of stand-alone transported PDF codes.38... 

Inter-cell spatial transport by the mean velocity and turbulent diffusivity change the memberships of the sets 0 /, but not the particle composition vectors 4> n). 

The random selection in step (iii) is carried out by generating uniform random numbers U e [0, 1], For example, the index of a random particle selected from a set of N particles will be n = intup(//N) where intuP() rounds the argument up to the nearest integer. Note that for constant-density, statistically stationary flow, the effective flow rates will be constant. In this case, steps (i) and (ii) must be completed only once, and the MC simulation is advanced in time by repeating step (iii) and intra-cell processes. For variable-density flow, the mean density field ((p)) must be estimated from the notional particles and passed back to the FV code. In the FV code, the non-uniform density field is held constant when solving for the mean velocity field.15... 

As described above, spatial transport in an Eulerian PDF code is simulated by random jumps of notional particles between grid cells. Even in the simplest case of one-dimensional purely convective flow with equal-sized grids, so-called numerical diffusion will be present. In order to show that this is the case, we can use the analysis presented in Mobus et al. (2001), simplified to one-dimensional flow in the domain [0, L (Mobus et al. 1999). Let X(rnAt) denote the random location of a notional particle at time step m. Since the location of the particle is discrete, we can denote it by a random integer i X(mAt) = iAx, where the grid spacing is related to the number of grid cells (M) by Ax = L/M. For purely convective flow, the time step is related to the mean velocity (U) by16... 

During the MC simulation, boundary conditions must be applied at the edges of the flow domain. The four most common types are outflow, inflow, symmetry, and a zero-flux wall. At an outflow boundary, the mean velocity vector will point out of the flow domain. Thus, there will be a net motion of particles in adjacent grid cells across the outflow boundary. In the MC simulation, these particles are simply eliminated. By keeping track of the weights... 

The mean velocity and turbulent diffusivity should approach zero at solid walls. In theory, this should be enough to keep particles from crossing wall boundaries. In practice, due to the finite time step, some particles will eventually cross wall boundaries and must be accounted for. 

In these equations, the mean velocity and turbulence frequency are available from the FV code at (for example) grid-cell centers, and must be interpolated to the particle positions. 

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... 

Carlos and Latif both fluidised glass particles in dimethyl phthalate. Data on the movement of the tracer particle, in the form of spatial co-ordinates as a function of time, were used as direct input to a computer programmed to calculate vertical, radial, tangential and radial velocities of the particle as a function of location. When plotted as a histogram, the total velocity distribution was found to be of the same form as that predicted by the kinetic theory for the molecules in a gas. A typical result is shown in Figure 6.11(41 Effective diffusion or mixing coefficients for the particles were then calculated from the product of the mean velocity and mean free path of the particles, using the simple kinetic theory. 

It has been pointed out over the years that the simple exponential function of the form where / is travel time from the source, appears to approximate the Lagrangian velocity autocorrelation function R t) rather well (Neumann, 1978 Tennekes, 1979). If R(t) = exp(-l/r), then the mean square particle displacement is given by (Taylor, 1921)... 

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