Standard drag curve

FIG. 6-60 Drag coefficient for water drops in air and air hiihhles in water. Standard drag curve is for rigid spheres. (From Clift, Grace, and Weher, Biih-hles. Drops and Particles, Academic, New York, 1978. )... 

Despite the long history of determinations of the standard drag curve, Eq. (14.21), values for the drag coefficient under idealized conditions are still a matter of dispute. A comprehensive review has been represented by R. Clift and W. H. Gauvin. ... 

Extensive comparisons of predictions and experimental results for drag on spheres suggest that the influence of non-Newtonian characteristics progressively diminishes as the value of the Reynolds number increases, with inertial effects then becoming dominant, and the standard curve for Newtonian fluids may be used with little error. Experimentally determined values of the drag coefficient for power-law fluids (1 < Re n < 1000 0.4 < n < 1) are within 30 per cent of those given by the standard drag curve 37 38. ... 

Fig. 5.12 Drag coefficient of a sphere as a function of Reynolds number (standard drag curve).

The conventional correlation for the drag on a sphere in steady motion is presented as a graph, see Fig. 5.12, called the standard drag curve , where is plotted as a function of Re. Many empirical or semiempirical equations have been proposed to approximate this curve. Some of the more popular are listed in Table 5.1. None of these correlations appears to consider all available data. 

Recommended Drag Correlations Standard Drag Curve, w = logjo Re... 

When a fluid sphere exhibits little internal circulation, either because of high K = Pp/p or because of surface contaminants, the external flow is indistinguishable from that around a solid sphere at the same Re. For example, for water drops in air, a plot of versus Re follows closely the curve for rigid spheres up to a Reynolds number of 200, corresponding to a particle diameter of approximately 0.85 mm (B5). In fact, many of the experimental points used in Section II to determine the standard drag curve refer to spherical drops in gas streams, where high values of k ensure negligible internal circulation. 

Equations (5-32) and (5-33) are only expected to be valid at relatively low k (S14), typically k <2, and for Re > 50 (H8). They should not be used when T predicted by Eq. (5-33) is less than 0.5, or when from Eq. (5-32) exceeds the value from the standard drag curve for rigid spheres at the same Re. In these cases, the true drag will be close to the rigid sphere value, provided that the drop is nearly spherical. 

Results for Re < 300 were included in the data used to derive the standard drag curve in Chapter 5. Numerical results for spherical raindrops (valid for Re < 200) are also discussed in Chapter 5. 

Fig. 7.2 Drag coefficient as function of Reynolds number for water drops in air and air bubbles in water, compared with standard drag curve for rigid spheres.

Figure 6-60 gives the drag coefficient as a function of bubble or drop Reynolds number for air bubbles in water and water drops in air, compared with the standard drag curve for rigid spheres. Information on bubble motion in non-Newtonian liquids may be found in Astarita and Apuzzo (AIChE J., 11, 815-820 [1965]) Calderbank, Johnson, and Loudon (Chem. Eng. Sci., 25, 235-256 [1970]) and Acharya, Mashelkar, and Ulbrecht (Chem. Enz. Sci., 32, 863-872 [1977]). 

Up — Uc represents the resultant slip velocity between the particulate and continuous phase. Some other commonly used drag coefficient correlations are listed in Appendix 4.2. For fluid particles such as gas bubbles or liquid drops, the drag coefficient may be different than that predicted by the standard drag curve, due to internal circulation and deformation. For example, Johansen and Boysen (1988) proposed the following equation to calculate Cd, which is valid for ellipsoidal bubbles in the range 500 < Re < 5000 ... 

The standard drag curve refers to a plot ot Cp as a function of Re for a smooth rigid sphere in a steady uniform flow field. The best fit of the cumulative data that have been obtained for this drag coefficient is shown in Fig 5.2. Numerous parameterizations have been proposed to approximate this curve (e.g., many of them are listed by [22]). 

Fig. 5.2. The standard drag curve shows the drag coefficient of a rigid sphere as a function of particle Reynolds number. Reprinted from Clift et al [22] with permission from Elsevier.

However, there is generally considerable discrepancy in the data for the drag coefficient dependence on turbulence. The spread in the data obtained for the drag coefficient of a sphere in turbulent flows is indicated in Fig 5.6. For this reason the standard drag curve parameterizations are normally used and the effects of turbulence is disregarded. 

Here Rcp = dp ug — Up //ig is the droplet Reynolds number. The above correlation is valid for Re < 800. The constants a = 0.15 and b = 0.687 yield the drag within 5% from the standard drag curve. Modifications to the solid particle drag are applied to compute the drag on a liquid drop and are given below. 

In the intermediate Reynolds munber region, though some predictive expressions have been developed, e.g. see Chhabra [1993a] but most of these data are equally well in line with the standard drag curve for Newtonian liquids [Machac et al., 1995]. 

The relation between Co and Rep,ihe standard drag curve, is shown on a log-4og plot in Figure 3.2. 

This is the relationship between drag coefficent Cp and single particle Reynolds number RCp for particles of density 2500 kg/m having a terminal velocity of 0.15 m/s in a fluid of density 700kg/m3 and viscosity 0.5 x IQ- Pas. Since Cp/Rcp is a constant, this relationship will give a straight line of slope +1 when plotted on the log-log coordinates of the standard drag curve. 

These values are plotted on the standard drag curves for particles of different sphericity (Figure 2.3). The result is shown in Figure 2W5.1. 

Where the plotted line intersects the standard drag curve for a sphere ( / = 1), Rep = 130. 

For a cube having the same terminal velocity under the same conditions, the same Cd versus Rep relationship applies, only the standard drag curve is that for a cube (v / = 0.806). 

At the intersection of this standard drag curve with the plotted hne. Rep = 310. 

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