Drag coefficient spherical bubble

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 ... 

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... 

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. 

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... 

In the case of larger non-spherical bubbles the drag coefficient no longer depends on the level of contamination and on Rep, but only on the Eotvos number. One commonly adopted dependence is... 

Moore32 generalized the boundary-layer analysis for a spherical bubble to include the deformation to an oblate ellipsoidal shape. His analysis is not reproduced here, but it is worth recording the final result for the drag coefficient, which takes the form... 

The last expression for Cf is known as the Stokes law for the drag coefficients of solid spherical particles. This law is confirmed by experiments for Re < 0.1. The drag law (2.2.18) for spherical bubbles holds only for extremely pure liquids... 

In the case of a spherical bubble in a translational flow at small Reynolds numbers, the solution of Oseen s equation (2.3.1) results is a two-term asymptotic expansion for the drag coefficient [476] ... 

Here Cf(0, Re) is the drag coefficient of the spherical bubble, which can be calculated by the formula (2.4.3), and Cf(oo, Re) is the drag coefficient of a solid spherical particle, which can be calculated by (2.3.8). The approximate expression (2.4.6) gives three correct terms of the expansion for small Reynolds numbers for 0 < Re < 50, the maximum error is less than 5%. 

The drop may conserve its spherical form until Re = 300 [94]. Since usually the boundary layer on a drop or a bubble is considerably thinner than on a solid sphere, one can use methods based on the boundary layer theory even for 50 < Re < 300. By using these methods, the following formula was obtained in [94] for the drag coefficient for Re 1 ... 

The drag coefficient for freely falling spherical droplets (or rising gas bubbles) of Newtonian fluids in power-law liquids at low Reynolds munber has been approximately evaluated and, in the absence of smface tension effects, it is given by equation (5.4), i.e. 

For higher Reynolds numbers, analytical solutions do not exist, so the numerical solutions must be considered. When k->-oo, this problem corresponds to the viscous flow around a rigid particle and was studied by several authors [10—15]. When k = 0, this problem corresponds to the viscous flow around a spherical bubble and was also studied by several authors [15-18]. The significant phenomena are very well explained in the books of Clift et al. [1] and Sadhal et al. [2]. Values of drag coefficients from numerical solutions for bubbles and rigid spheres are presented in Table 5.2, which shows a good agreement between the different studies. 

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