Drag coefficient wall effects

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) ... 

The conditions of flow around particles situated in the stream itself are not the same as the conditions of flow around particles attached to the walls. The air velocity in the main part of the stream is distributed more or less uniformly. In flow around adherent particles lying in the boundary layer, the flow velocity varies from zero up to a certain value. This variation has a substantial effect on the effective air-flow velocity, which determines the drag. Moreover, the particle drag coefficient Cx that appears in Eq. (X.3) depends on the Reynolds number, which in turn is a function of flow velocity, i.e., =/i(Re) and Re =f2(v). 

Prior to 1993 the results of theoretical and experimental smdies on the flow of non-Newtonian fluids past a sphere have been reviewed by Chabra". Since then a number of research smdies have been published, most notably by Machac and co-workers at the University of Pardubice. They investigated experimentally the drag coefficients and settling velocities of spherical particles in power law and Herschel-BuUdey model fluids, in Carreau model fluids (spherical in ref. 13 and non-spherical in ref. 14) and also the effect of the wall in a rectangular ceU, for power law fluids. ... 

Unnikrishnan and Chhabra (1990) Cylinders (axial) lA L/d) 2 Power law Experimental results on wall effects and drag coefficient in creeping flow region... 

Chhabra (1992) Cylinders (axial) 1.4 (L/d)<2 Power law Results on wall effects and drag coefficients up to Re 20... 

The value of We ranges from 0.03 to 0.4, suggesting that the value of drag coefficient in viscoelastic fluids can be up to four times larger than that in purely viscous fluids. However, more work is needed to generalize these observations over wider ranges of conditions, especially the non-Newtonian fluid parameters. No information is available concerning the severity of wall effects on nonspherical particles in viscoelastic media. 

Sharma, M. K. and R. P. Chhabra, A experimental study of free fall of cones in Newtonian and non-Newtonian media drag coefficient and wall effects, Chem. Eng. Proce.ss. 50 61-67 (1991). 

Unnikrishnan, A. and R. P. Chhabra, Slow parallel motion of cylinders in non-Newtonian media wall effects and drag coefficient, Chem. Eng. Process. 25 121-126 (1990). 

To date, no analogous results on wall effects are available in non-Newtonian liquids. Some workers (De Kee et al., 1986 Miyahara and Yamanaka, 1993) have minimized wall effects [on the basis of Eq. (16)], whereas others (Calderbank et al., 1970) have corrected their data using Eq. (16). Yet others (Acharya et al., 1977 Haque et al., 1988) have altogether ignored wall effects. Clearly, neither of these procedures is generally justifiable. Based on the behavior of rigid particles (Chhabra, 1993), one can perhaps conjecture that the wall correction is likely to be smaller in non-Newtonian media than in Newtonian liquids. Finally, it should be borne in mind that a 10% error in velocity will lead to a 20% error in drag coefficient ... 

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

The presence of container walls has a much smaller effect on Sherwood number than on drag since the mass transfer coefficient is only proportional to the one-third power of the surface vorticity. For a sphere with given settling on the axis of a cylindrical container, the Sherwood number decreases with 2, but it is still within 8% of the Sherwood number in an infinite fluid for 2 — 0.5. No data are available to test these predictions. 

When the solute dimension is no longer orders of magnitude smaller than the pore dimensions, the solute molecules experience an additional transport resistance due to the proximity of the pore wall. The effective diffusion coefficient is reduced further by a hindrance factor/ drag factor Gm i i,rp). If the solute molecules are assumed spherical of radius r, and diffusing through the centerline of the pore of radius, tp, Faxen s expression may be used to estimate Gor for (r,rp) < 0.5 (Lane, 1950 Renkin, 1954) ... 

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