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. ... 

The surface viscosity effect on terminal velocity results in a calculated drag curve that is closer to the one for rigid spheres (K5). The deep dip exhibited by the drag curve for drops in pure liquid fields is replaced by a smooth transition without a deep valley. The damping of internal circulation reduces the rate of mass transfer. Even a few parts per million of the surfactant are sometimes sufficient to cause a very radical change. 

Not until the above effects can be mathematically related can we expect to progress beyond the experimental stage. To predict such items as size of drop formed at a nozzle, terminal velocity, drag curves, changes of oscillations, and speed of internal circulation, one must possess experimental data on the specific agent in the specific system under consideration. Davies (Dl, D2) proposes the use of the equation... 

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. 

Heywood gave drag curves for various values of k (H3), and tabulated the velocity correction factor (H2). Figure 6.15 shows plotted from Heywood s table. There is empirical evidence for the validity of this approach (Dl). As with sphericity, comparison for specific shapes is informative. For oblate spheroids (for which Ja is the equatorial diameter) and Re < 100,... 

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 ... 

Morsi and Alexander s (MA) correlation represents the single-particle drag curve accurately. Ma and Ahmadi s correlation predicts values comparable with the MA correlation. Molerus correlation deviates from the MA correlation at higher Reynolds numbers. Patel s correlation is found to give a better fit with the MA correlation than Richardson s correlation. Dalla Ville s correlation overpredicts values of drag coefficient compared to the MA correlation. 

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. 

DRAG COEFFICIENT. The quantitative use of Eqs. (7.29) to (7.34) requires that numerical values be available for the drag coefficient C ). Figure 7.3, which shows the drag coefficient as a function of Reynolds number, indicates such a relationship. A portion of the curve of versus for spheres is reproduced in Fig. 7.6. The drag curve shown in Fig. 7.6 applies, however, only under restricted conditions. The particle must be a solid sphere, it must be far from other particles and from the vessel walls so that the flow pattern around the particle is not distorted, and it must be moving at its terminal velocity with respect to the fluid. The drag... 

These relations all form tangents to the curve in Fig. 4.3 in their effective range. Kelbaliyev and Ceylan [6] came up with a single correlation, which would fit the drag curve very nicely for Reynolds numbers smaller than that of the drag crisis. Their relation is in the form of... 

Fig. 4.12 (a) Droplet distortion, y, and (b) Various experimental drag curves (from Rudinger [47]. Reprinted by permission. Copyright (1980), Elsevier Scientific Publication Company.)... 

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. 

Unlike the flows considered in Chapter 3 which were essentially imidirectional, the fluid flows in particulate systems are either two- or three-dimensional and hence are inherently more difficult to analyse theoretically, even in the creeping (small Reynolds number) flow regime. Secondly, the results are often dependent on the rheological model appropriate to the fluid and a more generalised treatment is not possible. For instance, there is no standard non-Newtonian drag curve for spheres, and the relevant dimensionless groups depend on the fluid model which is used. Most of the information in this chapter relates to time-independent fluids, with occasional reference to visco-elastic fluids. 

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]. 

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