Velocity wall effect

Wall Effects When the diameter of a setthng particle is significant compared to the diameter of the container, the settling velocity is reduced. For rigid spherical particles settling with Re < 1, the correction given in Table 6-9 may be used. The factor k is multiplied by the settling velocity obtained from Stokes law to obtain the corrected set-... 

Jet interaction should not be taken into account when the jets are closely adjacent to each other, are propagated in confined conditions, and entrainment of the ambient air is restricted. This may be the case for concentrated air supply when air diffusers are uniformly positioned across the wall and the jets are replenished by the reverse flow, which decreases the jet velocity. This effect should be taken into consideration using the confinement coefficient discussed in Section 7.4.5. For the same reason, jet interaction should not be taken into consideration when air is supplied through the ceiling-mounted air diffusers and they are uniformly distributed across the ceiling. 

In equation (2) Rq is the equivalent capillary radius calculated from the bed hydraulic radius (l7), Rp is the particle radius, and the exponential, fxinction contains, in addition the Boltzman constant and temperature, the total energy of interaction between the particle and capillary wall force fields. The particle streamline velocity Vp(r) contains a correction for the wall effect (l8). A similar expression for results with the exception that for the marker the van der Waals attraction and Born repulsion terms as well as the wall effect are considered to be negligible (3 ). 

Most, if not all, velocity measurements in the bulk, other than NMR/MRI, make measurements through dear end-walls of either long or short cylinders. End-walls, because of friction with the particles, change the dynamic angle of repose near the end-wall and can also cause convections in long cylinders with components of velocity in the axial direction [34, 35], NMR/MRI experiments can avoid the end-wall effects by making measurements far from the end-walls in a long cylinder. 

When all three draft tubes were operated at similar velocities, the pressure drops across all draft tubes and downcomers were comparable. However, solid particle velocities in outside downcomers close to the walls were substantially less due to wall effect and redistribution of downcomer aeration flow. Smooth operations under these conditions were possible. The solid particle velocities in outside downcomers can be increased by enlarging the downcomer cross-section or by increasing downcomer aeration through separate plenums to minimize wall effects. 

An advantage of this approach to model large-scale fluidized bed reactors is that the behavior of bubbles in fluidized beds can be readily incorporated in the force balance of the bubbles. In this respect, one can think of the rise velocity, and the tendency of rising bubbles to be drawn towards the center of the bed, from the mutual interaction of bubbles and from wall effects (Kobayashi et al., 2000). In Fig. 34, two preliminary calculations are shown for an industrial-scale gas-phase polymerization reactor, using the discrete bubble model. The geometry of the fluidized bed was 1.0 x 3.0 x 1.0 m (w x h x d). The emulsion phase has a density of 400kg/m3, and the apparent viscosity was set to 1.0 Pa s. The density of the bubble phase was 25 g/m3. The bubbles were injected via 49 nozzles positioned equally distributed in a square in the middle of the column. 

Dispersion in packed tubes with wall effects was part of the CFD study by Magnico (2003), for N — 5.96 and N — 7.8, so the author was able to focus on mass transfer mechanisms near the tube wall. After establishing a steady-state flow, a Lagrangian approach was used in which particles were followed along the trajectories, with molecular diffusion suppressed, to single out the connection between flow and radial mass transport. The results showed the ratio of longitudinal to transverse dispersion coefficients to be smaller than in the literature, which may have been connected to the wall effects. The flow structure near the wall was probed by the tracer technique, and it was observed that there was a boundary layer near the wall of width about Jp/4 (at Ret — 7) in which there was no radial velocity component, so that mass transfer across the layer... 

Both phases are substantially in plug flow. Dispersion measurements of the liquid phase usually report Peclet numbers, uLdp/D, less than 0.2. With the usual small particles, the wall effect is negligible in commercial vessels of a meter or so in diameter, but may be appreciable in lab units of 50 mm dia. Laboratory and commercial units usually are operated at the same space velocity, LHSV, but for practical reasons the lengths of lab units may be only 0.1 those of commercial units. 

Velocity Profile Effects Many variables can influence the accuracy of specific flow measurement methods. For example, the velocity profile in a closed conduit affects many types of flow-measuring devices. The velocity of a fluid varies from zero at the wall and at other stationary solid objects in the flow channel to a maximum at a distance from the wall. In the entry region of a conduit, the velocity field may approach plug flow and a constant velocity across the conduit, dropping to zero only at the wall. As a newtonian fluid progresses down a... 

At the other extreme of Re, Achenbach (Al) investigated flow around a sphere fixed on the axis of a cylindrical wind tunnel in the critical range. Wall effects can increase the supercritical drag coefficient well above the value of 0.3 arbitrarily used to define Re in an unbounded fluid (see Chapter 5). If Re is based on the mean approach velocity and corresponds to midway between the sub- and super-critical values, the critical Reynolds number decreases from 3.65 x 10 in an unbounded fluid to 1.05 x 10 for k = 0.916. 

Figure 9.4 shows curves for the drag coefficient (based on the velocity for a freely settling sphere and the mean approach velocity for a fixed or suspended sphere) and for the fractional increase in drag caused by wall effects, Kp — 1). Up to Re of order 50, the results are approximated closely by an equation proposed by Fay on and Happel (F2) ... 

All studies of drops and bubbles have been carried out in containers of finite dimensions hence wall effects have always been present to a greater or lesser extent. However, few workers have set out to determine wall effects directly using a series of different columns of varying diameter. Where studies have been carried out, the sole aim has usually been to determine the influenee of X on the terminal velocity. While it is known that the eontaining walls tend to... 

General Generally, from a macroscopic point of view, maldistribution can be divided into two different phenomena (Stanek, 1994). The first one is small-scale maldistibution, which is connected mainly to the so-called preferred paths. It is the case where the liquid follows specific paths through bed and travels with velocities considerably higher than the mean. The same phenomenon is characterized as chaneling. The second case is large-scale maldistribution, which is connected to the nonhomogeneous (nonunifonn) initial distribution of the liquid and is referred to as wall effects. The concepts of distributor quality and liquid maldistribution in fixed beds are frequently found in the related technical literature, and these concepts are connected to each other—the better die distributor quality, the better the liquid distribution and flow into bed (Klemas and Bonilla, 1995). 

Daiton el al. (1977) and Werther (1983) presented different relationships for bubble diameter and bubble velocity for Group A and Group B particles (for bubbling fluidization). The mean rise velocity of a bubble in the bed ( bljb) can also be evaluated using the following equations, which include a wall effect collection (Darton et al., 1977 Werher, 1983 Wen, 1984). 

EXAMPLE 12.4 Electrophoretic Mobility of Bacteria. It is proposed to evaluate the electrophoretic mobility of the bacteria cells shown in Figure 12.10a by multiplying the appropriate value of time-1 by the distance of particle displacement and then dividing by E. Criticize or defend the following proposition It is appropriate to use the maximum apparent velocity since this is measured at the center of the cell and is therefore subject to the least interference by wall effects. 

Above a Reynolds number of the order of magnitude of 1000, bubbles assume a helmet shape, with a flat bottom (Eckenfelder and Barnhart, loc. cit. and Leibson et al., loc. cit.). After bubbles become large enough to depart from Stokes law at their terminal velocity, behavior is generally complicated and erratic, and the reported data scatter considerably. The rise can be slowed, furthermore, by a wall effect if the diameter of the container is not greater than 10 times the diameter of the bubbles, as shown by Uno and Kintner [AlChE 2, 420 (1956) and Collins, J. Fluid Meek, 28(1), 97 (1967)]. Work has... 

Fig. 12 shows the radial profile of the liquid velocity at different axial positions. Due of the baffles, the liquid is redistributed in the radial direction and the turbulent intensity is increased. The radial profile of the liquid velocity is almost uniform after passing the internal. Liquid velocity is lower at the center and higher near the wall as compared with that below the internal. With increasing distance from the internal, the turbulence intensity diminishes and the wall effect becomes more apparent, that is, the liquid velocity increases at the center and decreases near the wall. The radial profile obtained at the position of 114 cm from the internal is similar to that obtained below the internal and is the same as that at the position of 144 cm. 

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