Velocity mean settling

Tube Reynolds number, Re 2RpF p M Particle Reynolds number", Re 2a < p p P Particle Reynolds number in an unbounded stationary fluid, p Settling velocity-mean fluid velocity ratio. Eccentric equilibrium position of sphere center. 

We have an emulsion of oil in water that we need to separate. The oil droplets have a mean diameter of lO " m, and the specific gravity Of the oil is 0.91. Applying a sedimentation centrifuge to effect the separation at a spedd of 5,000 rpm, and assuming that the distance of a droplet to the axis of rotation is 0.1 m, determine the droplet s radial settling velocity. 

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... 

When the size of a particle approaches the same order of magnitude as the mean free path of the gas molecules, the settling velocity is greater than predicted by Stokes law because of molecular slip. The slip-flow correction is appreciable for particles smaller than 1 pm and is allowed for by the Cunningham correction for Stokes law (Lapple, op. cit. Licht, op. cit.). The Cunningham correction is applied in... 

Perforated plate distributors are widely used in industry because they are cheap and relatively easy to manufacture. Simple perforated plate-type distributors suffer from particles passing back through to the plenum despite mean gas velocities well above the settling velocity for the particles. This is because of imbalances in gas flow between the orifices, which is difficult to eliminate. Hence, such plates take the form of either a layer of mesh sandwiched between two perforated plates or two staggered perforated plates without a mesh screen (Kunii and Levenspiel, 1991). However, these structures often lack rigidity and need to be reinforced or sometimes curved (concave to the bed) to withstand heavy loads. The diameter of the orifices in a perforated plate distributor varies from 1 or 2mm in small beds used for research or very small-scale production to 50 mm in very large chemical reactors. Most food applications are likely to use apertures of intermediate size. 

We will apply equation (5.20) to solve for the concentration profile of suspended sediment in a river, with some simplifying assumptions. Suspended sediment is generally considered similar to a solute, in that it is a scalar quantity in equation (5.20), except that it has a settling velocity. We will also change our notation, in that the bars over the temporal mean values will be dropped. This is a common protocol in turbulent transport and will be followed here for conformity. Thus, if an eddy diffusion coefficient, e, is in the transport equation,... 

The mean suspended sediment concentration of a 3-m-deep river is lOg/m. The mean sediment settling velocity is 0.01 m/s, and the river slope is 2 x 10 . Assuming that is constant with z, what is the concentration profile for sediment in the river Does the solution make physical sense at all boundaries Explain. 

Yet, in view of the often strong currents in rivers, it would be difficult to interpret ks as a vertical settling velocity divided by the mean depth of the river bed as done in Eq. 23-16. In fact, ks should be understood as an empirical coefficient describing the specific first-order removal rate of suspended particles from the river. 

At the disengagement droplet size of Dpg, the terminal settling velocity, U,g, is calculated by means of Eq. (2b), and the corresponding disengagement velocity constant, Kg, from Eq, (5c). Equating this subscript-altered fonm of Eq. (2b) to Eq. (1) gives ... 

The vertical motion of solids in a horizontal suspension flow is strongly influenced by the ratio of the terminal settling velocity to the friction velocity [Blatch, 1906 Chien and Asce, 1956]. In a circular pipe, the mean stream velocity can be related to the friction velocity by [Taylor, 1954]... 

For condition 1 the settling velocity (dx/dt) is -0.015 cm/s (negative means rising) and the rise time -x /(dx/dt) is 89 min, which is much greater than 45 min. In this case the droplets need much more time to rise than is allowed in the vessel so recovery efficiency will be poor. 

It appears to us that this apparently random motion, superposed on the steady mean-settling velocity, is analogous to that of Brownian motion in the sense that it arises from the fact that the continuum is not structureless. 

Aerodynamic diameter Diameter of a unit density sphere (density = 1 g/cms) having the same aerodynamic properties as the particle in question. This means that particles of any shape or density will have the same aerodynamic diameter if their settling velocity is the same. 

In the design of upflow, three phase bubble column reactors, it is important that the catalyst remains well distributed throughout the bed, or reactor space time yields will suffer. The solid concentration profiles of 2.5, 50 and 100 ym silica and iron oxide particles in water and organic solutions were measured in a 12.7 cm ID bubble column to determine what conditions gave satisfactory solids suspension. These results were compared against the theoretical mean solid settling velocity and the sedimentation diffusion models. Discrepancies between the data and models are discussed. The implications for the design of the reactors for the slurry phase Fischer-Tropsch synthesis are reviewed. 

In the above equations, AG, AL, and As are the gas-phase, liquid-phase and calalyst-surface concentrations of the reacting species, ACi is the average gas-phase concentration at the reactor inlet, Z is the axial distance from the reactor inlet, L is the total length of the reactor, m = H/RgT, where H is the Henry s law constant (cm3 atm g-mol" ), Rg is the universal gas constant, and T is the temperature of the reactor. UG is the mean gas velocity, Us is the mean settling velocity of the particles, t is the time, k is the first-order rate constant, W is the catalyst loading, zc and ZP are the axial dispersion coefficients for the gas and solid phases, respectively. Following the studies of Imafuku et al.19 and Kato et al.,21 the axial dispersion coefficient for the liquid phase was assumed to be the same as that for the solid phase, w is the concentration of the particles and hG the fractional gas holdup. Other parameters have the same meaning as described earlier. 

Due to density differences the particles have the tendency to settle. Thus, solid concentration profiles result which can be described on the basis of the sedimentation-dispersion model (78,79,80). This model involves two parameters, namely, the solids dispersion coefficient, E3, and the mean settling velocity, U5, of the particles in the swarm. Among others Kato et al. (81) determined 3 and U3 in bubble columns for glass beads 75 and 163 yum in diameter. The authors propose correlations for both parameters, E3 and U3. The equation for E3 almost completely agrees with the correlation of Kato and Nishiwaki (51) for the liquid phase dispersion coefficient. 

---