Bubble drag coefficient

For larger Reynolds numbers (1 < NRe < 500), Rivkind and Ryskind (see Grace, 1983) proposed the following equation for the drag coefficient for spherical drops and bubbles ... 

In some way, introducing an increased particle drag by means of Eq. (17) resembles the earlier proposal raised by Bakker and Van den Akker (1994b) to increase viscosity in the particle Reynolds number due to turbulence (in agreement with the very old conclusion due to Boussinesq, see Frisch, 1995) with the view of increasing the particle drag coefficient and eventually the bubble holdup in the vessel. Lane et al. (2000) compared the two approaches for an aerated stirred vessel and found neither proposal to yield a correct spatial gas distribution. 

The drag of the bubble is expressed in terms of the drag coefficient as... 

Assuming the bubbles to be spherical, the frontal area Af of the bubble becomes nd2/4. Further, by making use of the expression v = Qjnd2, the drag force can be written in terms of Froude number, the drag coefficient, and the volume of the bubble. Thus... 

Fd was evaluated from the drag coefficient at bubble Reynolds number and the projected area of the bubble. As the Reynolds number varied from 2 to 700, the drag coefficient CD was evaluated by the Schiller and Naumann (S4) equation ... 

Ci Substitution given in the text C2 Substitution given in the text CD Drag coefficient dimensionless d Diameter of the bubble, drop, droplet, cm... 

Treatment of liquid drops is considerably more complex than bubbles, since the internal motion must be considered and internal boundary layers are difficult to handle. Early attempts to deal with boundary layers on liquid drops were made by Conkie and Savic (C8), McDonald (M9), and Chao (C4, W7). More useful results have been obtained by Harper and Moore (HIO) and Parlange (PI). The unperturbed internal flow field is given by Hill s spherical vortex (HI3) which, coupled with irrotational flow of the external fluid, leads to a first estimate of drag for a spherical droplet for Re 1 and Rep 1. The internal flow field is then modified to account for convection of vorticity by the internal fluid to the front of the drop from the rear. The drag coefficient. 

Fig. 5.29 Drag coefficients for bubbles in pure systems predictions of numerical, Galerkin, and boundary layer theories compared with selected experimental results.

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.

Equations (8-10) to (8-12) have been confirmed many times [e.g. (D4, W7)]. For M > 10, bubbles and drops change directly from spherical to spherical-cap, as noted in Chapter 2. The drag coefficient is then closely approximated by... 

There has been relatively little work on the motion of bubbles and drops in well-characterized turbulent flow fields. There is some evidence (B3, K7) that mean drag coefficients are not greatly altered by turbulence, although marked fluctuations in velocity (B3) and shape (K7) can occur relative to flows which are free of turbulence. The effect of turbulence on splitting of bubbles and drops is discussed in Chapter 12. 

Q) = drag coefficient for rigid sphere Cj = inlet oil drop population, drops/cm3 db - bubble diameter, cm dp = drop diameter, cm = single-bubble collision efficiency i = average collision efficiency... 

Karamanev [op. cit.] provided equations for bubble rise velocity based on the Archimedes number and on use of the bubble projected diameter d, in the drag coefficient and the bubble equivalent diameter in Ar. The Archimedes number is as defined in Eq. (6-236) except that the density difference is liquid density minus gas density, and dv is replaced by d . 

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]). 

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