Slip velocity between particles

Solution Assuming that the slip velocity between particles and air in the core region is equal to the particle terminal velocity and Dc can be approximated by D, the particle velocity in the core region can be determined from Eq. (10.25), i.e.,... 

For particles ranging in size from 30 nm to 2 pm, the particle dissolution is influenced mainly by Brownian motion and experimentally Dr values were found to vary from 1.0 to 2.0. For bigger particles (non-Brownian) between 10 pm and 5 mm, dissolution is mainly controlled by the relative slip velocities between particles and surrounding fluid. The bigger the particles, the higher is the relative slip velocity resulting in faster dissolution rate. Therefore values of Dr for these particles (2.0 < Dr < 3.0) are higher than those affected by Brownian motion. For both the Brownian and non-Brownian particles, the dissolution is by erosion of the external surface. 

Concerning the liguid/solid mass transfer coefficient ks, two different theories have been used to correlate data from so-lid/liquid suspensions one uses the terminal slip velocity between particles and the liquid. The other approach is based on Kolmogoroff s theory of isotropic turbulence and uses energy dissipation rate C= Go8 correlation. Sanger [30] recommend the equation... 

By substituting the relative velocity, or slip velocity, between particles and fluid for the velocity of classical fluidization, Kwauk (1963,1992) extended... 

What this shows is that, from the definition of off-bottom motion to complete uniformity, the effect of mixer power is much less than from going to on-bottom motion to off-bottom suspension. The initial increase in power causes more and more solids to be in active communication with the liquid and has a much greater mass-transfer rate than that occurring above the power level for off-bottom suspension, in which slip velocity between the particles of fluid is the major contributor (Fig. 18-23). 

Finally, consider the case when the magnitude of the slip velocity between the particles and the gas is close to umf/e everywhere in the fluidized bed. With the vertical pressure drop equal to the particle weight, the following holds for any value of the particle Reynolds number,... 

The results of an example calculation for a recirculating fluidized bed coal devolatilizer of 0.51 m in diameter handling coal of average size 1200 pm at 870°C and 1550 kPa are presented in Fig. 11. The calculation is based on operating the fluidized bed above the draft tube at 4 times the minimum fluidization velocity. It is also based on the selection of a distributor plate to maintain the downcomer at the minimum fluidization condition. If the two-phase theory applies, this means that the slip velocity between the gas and the particles in the downcomer must equal to the interstitial minimum fluidizing velocity as shown below. 

The solids particle velocity in the gas-solid two-phase jet can be calculated as shown in Eq. (27), assuming that the slip velocity between the gas and the solid particles equals the terminal velocity of a single particle. It should be noted that calculation of jet momentum flux by Eq. (26) for concentric jets and for gas-solid two-phase jets is only an approximation. It involves an implicit assumption that the momentum transfer between the concentric jets is very fast, essentially complete at the jet nozzle. This assumption seems to work out fine. No further refinement is necessary at this time. For a high velocity ratio between the concentric jets, some modification may be necessary. 

It is readily apparent that finer and finer structures get resolved as the number of spatial grids is increased. Statistical quantities, such as average slip velocity between the gas and particle phases, obtained by averaging over the whole domain, were found to depend on the grid resolution employed in the simulations and they became nearly grid-size independent only when grid sizes of the order of a few ( 10) particle diameters were used. Thus, if one sets out to solve microscopic TFM equations, grid sizes of the order of few particle... 

A final piece of the proof-of-concept calculations is to compare the predictions obtained by solving the filtered TFM equations with highly resolved simulations of the microscopic TFM equations. For this purpose, Andrews and Sundaresan (2005) performed simulations of the microscopic TFM equations in a 16 x 32 cm periodic domain at various resolutions (e.g., see Fig. 29). From these simulations, they extracted domain-average quantities in the statistical steady state (see Agrawal et al., 2001 for a discussion of how these data are gathered). Fig. 33 shows the domain-average slip velocity between the gas and particle phases at various grid resolutions (shown by the squares connected by... 

Analogous to the slip velocity between gas and particle at Kn above the continuum flow range discussed in Section A above, a temperature discontinuity exists close to the surface at high Kn. Such a discontinuity represents an additional resistance to transfer. Hence, transfer rates are generally lowered by compressibility and noncontinuum effects. The temperature jump occurs over a distance 1.996kl 2 — a )/Fva k + 1) (K2, Sll) where is the thermal accommodation coefficient, interpreted as the extent to which the thermal energy of reflected molecules has adjusted to the surface temperature. 

While the particle is undergoing the accelerative motion as described above, heat is being transfered between it and the surrounding gas stream, also in an unsteady state. By using the Nusselt number Nu for evaluating the heat-transfer coefficient h from the slip velocity between the particles and the gas (Kramers, 1964),... 

The high slip velocity between the gas and the particles favors both G/S mass and heat transfer and treatment of cohesive solids. 

Imafuku et al.46 measured the gas holdup in a batch (i.e., no liquid flow) three-phase fluidized-bed column. They found that the presence of solids caused significant coalescence of bubbles. They correlated the gas holdup with the slip velocity between the gas and liquid. They found that the gas holdup does not depend upon the type of gas distributor or the shape of the bottom of the column when solid particles are completely suspended. Kato et al.53 found that the gas holdup in an air-water-glass sphere system was somewhat less than that of the air-water system and that the larger solid particles showed a somewhat smaller... 

Terminal velocity-slip velocity theory In this theory, the steady slip velocity between solid and liquid is used in the correlation for the Sherwood number. For low particle Reynolds number, Friedlander33 gave... 

Up — Uc represents the resultant slip velocity between the particulate and continuous phase. Some other commonly used drag coefficient correlations are listed in Appendix 4.2. For fluid particles such as gas bubbles or liquid drops, the drag coefficient may be different than that predicted by the standard drag curve, due to internal circulation and deformation. For example, Johansen and Boysen (1988) proposed the following equation to calculate Cd, which is valid for ellipsoidal bubbles in the range 500 < Re < 5000 ... 

The first condition means that in principal, the constraint imposed for equidistant sampled data by the sampling theorem can be circumvented. Thus spectral content of velocity fluctuations can be estimated beyond half of the mean data rate, however this is generally achieved at the expense of estimator variability. There exists a large body of literature on the estimation of spectra from randomly sampled velocity data, such as with LDA, and this remains an active area of research (Adrian and Yao 1987, Gaster and Roberts 1977, Roberts and Ajmani 1986, Nobach et al. 1996). One aspect of these developments which is of particular interest when measuring in two-phase flows, is that of signal reconstruction, i.e. the estimation of fluid velocity between particles (Muller et al. 1994 a) b), Veynante and Candel 1988). In this way the velocity of the continuous phase can be approximated at the instance when the dispersed phase is measured and can thus lead to improved estimators of the slip velocity (Prevost et al. 1996). 

The appropriate velocity term for the particle Reynolds number in equations 6.64 and 6.65 is the slip velocity, i.e. the relative velocity between particle and fluid. The slip velocity is usually assumed to be the free fall velocity of the particle, but this quantity is not easy to predict. 

Slip velocity between fluid and particle phase defined by... 

Effect of Drum Speed on Solids and Fluid Mean Residence Times. Due to the difference in the axial velocity of the two phases (slip velocity) the mean residence time of the solid particles in the drum is in general higher than that of the fluid. The ratio of the mean residence time (t/Xj) is a measure of the slip velocity between... 

Effect of Drum Speed on C/C. As a result of the slip velocity between the particles and the fluid, the mean solids concentration in the drum is higher than that in the feed. Figure 27 shows the mean slurry concentration in the drum... 

The Maxey-Riley equation does not include inertial effects such as the lift force. Saffman [48] derived an expression for the lift force on a small rigid sphere in a linear shear flow. The leading order lift force on the sphere is caused by an interaction between the slip velocity between the particle and the flow and the shear. Fig. 4 shows a schematic illustration of a particle in a... 

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