Drag ratio

The next step is to estimate the lift/drag ratio... 

A typical graph of drag ratio as a function of superficial air velocity is shown in Figure 5.5 in which each curve refers to a constant superficial liquid velocity. The liquids in question exhibited power law rheology and the corresponding values of the Metzner and Reed Reynolds numbers ReMR based on the superficial liquid velocity uL (see Chapter 3) are given. The following characteristics of the curves may be noted ... 

For a given liquid, the value of the minimum drag ratio (superficial liquid velocity is decreased. 

The superficial air velocity required to give the minimum drag ratio increases as the liquid velocity decreases. 

For a more highly shear-thinning liquid, the minimum drag ratio becomes even smaller although more air must be added to achieve the condition. 

Figure 5.5. Drag ratio as function of superficial gas velocity (liquid velocity as parameter)...

It is convenient to define an equivalent sphere as in Chapter 2. Drag is then related to that on the equivalent sphere either by a drag ratio defined as... 

Ol). Results for thin disks are obtained in the limit as 0. Approximate relationships, obtained by treating the spheroid as a slightly deformed sphere (H3, SI), are also given. The drag ratio may conveniently be expressed as the ratio of the resistance of the spheroid to that of the sphere with the same equatorial radius a ... 

Figure 4.3 shows exact and approximate values for, the axial drag ratio. Corresponding curves for A 2 appear in Fig. 4.4 with the exact values... 

Fig. 4.3 Drag ratios for spheroids in axisymmetric flow. Drag ratio — Exact —- Approximate ------Drag Component (1) Friction (2) Form.

The drag ratio based on the sphere with equal volume is... 

Fig. 4.5 Drag ratios for spheroids compared to volume-equivalent spheres.

Analytic results for cylinders comparable to those discussed for spheroids are not available. However, Heiss and Coull (H4) reported accurate experimental determinations for cylinders, spheroids, and rectangular parallelepipeds, and developed a general correlation for settling factors. In terms of the volume drag ratio,, their results may be written ... 

The principal resistances may be obtained from the drag ratios as... 

Fig. 4.12 Drag ratios for rectangular parallelepipeds with square section.

For a cube of side /, Eq. (4-50) gives the resistance as 12.70/, compared with experimental values of 12.58/ (P6), 12.63/ (H4), and 12.71/ (Cl). To the accuracy of the determinations, the resistance can be taken as 4ti / (DI). It is noteworthy that Eqs. (4-26) and (4-27) predict that a spherically isotropic cylinder with aspect ratio 0.812 should have a drag ratio of 1.050, while Eq. (4-50) gives Ag = 1.054. Agreement is so favorable that Eq. (4-50) may be useful for spherically isotropic particles other than the simple shapes for which it was developed. 

The conductance of arbitrary axisymmetric particles may be approximated using the correlation given in Fig. 4.13. By analogy with the drag ratio, a conductance factor is defined as... 

Based on Bowen s (34) procedure, Lord and McGowen (35) developed a statistical based correlation of drag ratio versus fluid velocity. 

Figure 11 depicts a comparison of laboratory data to correlation predictions of drag ratio versus velocity. It can be seen that the correlation developed is diameter invariant. The laboratory correlation was... 

Figure 11. Comparison of laboratory data to correlation predictions of drag ratio vs. velocity. (Reproduced with permission from reference 35. Copyright 1986 Society of Petroleum Engineers.)...

---