Particle wall effect

Another relationship was derived by considering the general motion of particles and taking into consideration acceleration of fluid displaced by the particles, wall effects, increase in area available for upward flow, increase in apparent viscosity (which is related to momentum transfer in the suspension), and decrease in gravitational force due to increase in buoyancy as the suspension becomes denser (17). The resulting relationship is... 

Fig. 5 Predictions of various models for drag coefficient for a spherical particle Wall Effects on Drag Coefficient...

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-... 

Fig. 10(a) presents a comparison of computer simulation data with the predictions of both density functional theories presented above [144]. The computations have been carried out for e /k T = 7 and for a bulk fluid density equal to pi, = 0.2098. One can see that the contact profiles, p(z = 0), obtained by different methods are quite similar and approximately equal to 0.5. We realize that the surface effects extend over a wide region, despite the very simple and purely repulsive character of the particle-wall potential. However, the theory of Segura et al. [38,39] underestimates slightly the range of the surface zone. On the other hand, the modified Meister-Kroll-Groot theory [145] leads to a more correct picture. 

Achieving steady-state operation in a continuous tank reactor system can be difficult. Particle nucleation phenomena and the decrease in termination rate caused by high viscosity within the particles (gel effect) can contribute to significant reactor instabilities. Variation in the level of inhibitors in the feed streams can also cause reactor control problems. Conversion oscillations have been observed with many different monomers. These oscillations often result from a limit cycle behavior of the particle nucleation mechanism. Such oscillations are difficult to tolerate in commercial systems. They can cause uneven heat loads and significant transients in free emulsifier concentration thus potentially causing flocculation and the formation of wall polymer. This problem may be one of the most difficult to handle in the development of commercial continuous processes. 

Values for the various parameters in these equations can be estimated from published correlations. See Suggestions for Further Reading. It turns out, however, that bubbling fluidized beds do not perform particularly well as chemical reactors. At or near incipient fluidization, the reactor approximates piston flow. The small catalyst particles give effectiveness factors near 1, and the pressure drop—equal to the weight of the catalyst—is moderate. However, the catalyst particles are essentially quiescent so that heat transfer to the vessel walls is poor. At higher flow rates, the bubbles promote mixing in the emulsion phase and enhance heat transfer, but at the cost of increased axial dispersion. 

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. 

Great care must be exercised when using graduated cylinders because decreases in the diameter of small containers can produce a wall effect, which often affects the settling rate or ultimate sedimentation volume of flocculated suspensions. Such small containers have a tendency to hold up the suspensions due to adhesive forces acting between the container s inner surface and the suspended particles. 

The wall effect for particles settling in non-Newtonian fluids appears to be significantly smaller than for Newtonian fluids. For power law fluids, the wall correction factor in creeping flow, as well as for very high Reynolds... 

Either a liquid or a gas can be used as the carrier fluid, depending on the size and properties of the particles, but there are important differences between hydraulic (liquid) and pneumatic (gas) transport. For example, in liquid (hydraulic) transport the fluid-particle and particle-particle interactions dominate over the particle-wall interactions, whereas in gas (pneumatic) transport the particle-particle and particle-wall interactions tend to dominate over the fluid-particle interactions. A typical practical approach, which gives reasonable results for a wide variety of flow conditions in both cases, is to determine the fluid only pressure drop and then apply a correction to account for the effect of the particles from the fluid-particle, particle-particle, and/or particle-wall interactions. A great number of publications have been devoted to this subject, and summaries of much of this work are given by Darby (1986), Govier and Aziz (1972), Klinzing et al. (1997), Molerus (1993), and Wasp et al. (1977). This approach will be addressed shortly. 

Glicksman and Farrell (1995) constructed a scale model of the Tidd 70 MWe pressurized fluidized bed combustor. The scale model was fluidized with air at atmospheric pressure and temperature. They used the simplified set of scaling relationships to construct a one-quarter length scale model of a section of the Tidd combustor shown in Fig. 34. Based on the results of Glicksman and McAndrews (1985), the bubble characteristics within a bank of horizontal tubes should be independent of wall effects at locations at least three to five bubble diameters away from the wall. Low density polyurethane beads were used to obtain a close fit with the solid-to-gas density ratio for the combustor as well as the particle sphericity and particle size distribution (Table 6). 

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. 

The validation of CFD codes using pressure drop is most reliable when actual experimental data are taken in equipment identical to the situation that is being simulated. Existing literature correlations such as the Ergun equation are known to have shortcomings with respect to wall effects, particle shape effects, application to ordered beds and validity at high Re. The applicability of literature correlations to typical CFD simulation geometries needs to be examined critically before fruitful comparisons can be made. 

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. 

The influence of K on g is caused by the so-called wall effect and an increase of void volume that is accessible for small particles with K. Both of these effects correspond to the appearance of the excluded volume in the places of contact of rigid particles between themselves (Figure 9.18) or a rigid wall (Figure 9.20). 

Figure 19.20 A scheme of a possible appearing of an additional void volume in a region of contact of particles with the wall of a container (wall effect).

The major directions of changing porosity in DRP are schematically shown in Figure 9.22 [3,61], As a starting point, one can use the porosity of DRP of monospheres s0 = 0.36-0.42. Values of >s0 increase with particles anisotropy, roughness, and internal porosity, and also with the influence of wall effects at K> 0.1, and DIH > 0.05. Values of < (J are characteristic for polydis-perse particles when denser zones with ordered or unidirectional packings are formed, and also under forced densification and deformation of particles, correspondingly. 

Equation 3.43 was obtained for the Stokes law regime. It overestimates the wall effect, however, at higher particle Reynolds number (Re > 0.2). 

Several expressions of varying forms and complexity have been proposed(35,36) for the prediction of the drag on a sphere moving through a power-law fluid. These are based on a combination of numerical solutions of the equations of motion and extensive experimental results. In the absence of wall effects, dimensional analysis yields the following functional relationship between the variables for the interaction between a single isolated particle and a fluid ... 

Wall effect. In a packed bed, the particles will not pack as closely in the region near the wall as in the centre of the bed, so that the actual resistance to flow in a bed of small diameter is less than it would be in an infinite container for the same flowrate per unit area of bed cross-section. A correction factor fw for this effect has been determined experimentally by Coulson(15). This takes the form ... 

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