Spheres wall effects

The data from the WS model in some cases deviated slightly from the full-bed models. This could be explained by the slightly different layout of the WS model. Some spheres had to be relocated in the WS model to create a two-layer periodicity from the six-layer periodicity in the full-bed models. The differences in velocity magnitudes were mainly found in the transition area between the wall layers and the center layers. The effect of slightly larger gaps between spheres from the nine-sphere wall layers and the three-sphere central layers, due to the sphere relocations, had a noticeable effect on the velocity profile. Differences were also found in the central layer area where the sphere positions were not identical. 

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 ... 

Fidleris, V. and Whitmore, R. L. Brit. J. App. Phys. 12 (1961) 490. Experimental determination of the wall effect for spheres falling axially in cylindrical vessels. 

Table 5.2 gives a new correlation, based on a critical examination of available data for spheres (N6). Results in which wall effects, compressibility effects, noncontinuum effects, support interference, etc. are significant have been excluded. The whole range of Re has been divided into 10 sub intervals, with a distinct correlation for each interval. Adjacent equations for match within 1% at the boundaries between sub intervals, but the piecewise fit shows slight gradient discontinuities there. The Re = 20 boundary corresponds to onset of wake formation as discussed above, the remaining boundaries being chosen for convenience. 

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. 

Achenbach based Re on flow conditions in the smallest cross section between sphere and tube. With this definition, wall effects increase Re, . 

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) ... 

Sutterby (S7) gave a useful tabulation of the viseosity ratio, defined in Eq. (9-8), for relatively low Re and a. These values, intended primarily to correct for departures from Stokes law in falling sphere viscometry, are shown in Fig. 9.6. Reynolds number is defined using the measured Uj and defined in Eq. (9-9). The curve for a = 0 accounts for departures from the creeping flow approximations in an unbounded fluid, and the relative displacement of the other curves indicates the wall effect. 

Ladenburg s. Correction for Falling Spheres—When a sphere falls in a cylinder whose dimensions are of the same order of magnitude as the sphere, Stokes equation must be modified to take account of the wall and bottom effects. This is necessary since the displacement action of the sphere interferes with its motion. Ladenburg (1907) supplied the necessary corrections. For the wall-effect, Ladenburg gave the correction as... 

This chapter will focus on infinitely-extended suspensions in which potential complications introduced by the presence of walls are avoided. The only wall-effect case that can be treated with relative ease is the interaction of a sphere with a plane wall (Goldman et ai, 1967a,b). The presence of walls can lead to relevant suspension rheological effects (Tozeren and Skalak, 1977 Brunn, 1981), which result from the existence of particle depeletion boundary layers (Cox and Brenner, 1971) in the proximity of the walls arising from the finite size of the suspended spheres. Going beyond the dilute and semidilute regions considered by the authors just mentioned is the ad hoc percolation approach, in which an infinite cluster—assumed to occur above some threshold particle concentration—necessarily interacts with the walls (cf. Section VI). 

Patzold (1980) compared the viscosities of suspensions of spheres in simple shear and extensional flows and obtained significant differences, which were qualitatively explained by invoking various flow-dependent sphere arrangements. Goto and Kuno (1982) measured the apparent relative viscosities of carefully controlled bidisperse particle mixtures. The larger particles, however, possessed a diameter nearly one-fourth that of the tube through which they flowed, suggesting the inadvertant intrusion of unwanted wall effects. 

Wall Effects. In the above discnssion, we have assnmed that the reaction is homogeneous (i.e., no catalytic reaction at the walls of the reaction bnlb). The fact that the data give first-order kinetics is not a proof that wall effects are absent. This point can be checked by packing a reaction bnlb with glass spheres or thin-walled tnbes and repeating the mea-snrements under conditions where the surface-to-volume ratio is increased by a factor of 10 to 100. This will not be done in this experiment, but the system chosen for study must be free from serious wall effects or it may not be possible to discnss the experimental results in terms of the theory of nnimolecular reactions. 

Thus, by measuring the value of rj may be found. The constant velocity u is often called the terminal velocity. The formula holds only if ur is small compared with rjy i.e. for very viscous liquids in the case of spheres of moderate size there are also corrections for the boundary conditions of the walls and base of the cylinder containing the liquid (the formula (1) being deduced for an infinite volume of liquid). If the liquid column is divided into three equal parts, and the centre one is used in timing the fall of the sphere, the correcting factor on the velocity for the wall effect is ... 

The packing structure near a wall differs from the bulk structure. This phenomenon has important implications with regard to fluid channeling and heat transfer near reactor walls. Fig. 1 is a graphic from a numerical simulation of flow in a chemical reactor. The velocity quivers indicate preferential fluid flow near the walls. This particular simulation contains only a few particles across the reactor diameter. For larger ratios (dbed/dparticies > 30), wall effects decay quite rapidly, penetrating approximately two sphere diameters in from the wall. ... 

Use the problem setup in Problem 10.12 and deduce the wall effect when measuring the terminal velocity by dropping a sphere into a fluid. Compare with Perry and Green (1997, p. 6-54). 

Chamber Dimensions and geometry A key consideration in chamber design is the minimisation of reactant / aerosol losses and hence die maximisation of experimental duration. A larger chamber provides smaller wall area per unit volume and therefore a reduction in die wall effects. The geometry minimising the surface area per volume is a sphere. For practical purposes and cost effectiveness a nearly cubic chamber of 18 has been chosen. The volume of air in the chamber will reduce through an experiment due to the sample volume requirement of the instruments. The chamber is mounted on three pairs of rectangular extruded... 

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