Drops drag coefficient

Czerwonatis N, Eggers R. Disintegration of liquid jets and drop drag coefficient in pressurized nitrogen and carbon dioxide. Chem Eng Technol 2001 24 619-624. 

The term essentially a drag coefficient for the dust cake particles, should be a function of the median particle size and particle size distribution, the particle shape, and the packing density. Experimental data are the only reflable source for predicting cake resistance to flow. Bag filters are often selected for some desired maximum pressure drop (500—1750 Pa = 3.75-13 mm Hg) and the cleaning interval is then set to limit pressure drop to a chosen maximum value. 

Between about Rop = 350,000 and 1 X 10 , the drag coefficient drops dramatically in a drag crisis owing to the transition to turbulent flow in the boundary layer around the particle, which delays aft separation, resulting in a smaller wake and less drag. Beyond Re = 1 X 10 , the drag coefficient may be estimated from (Clift, Grace, and Weber) ... 

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. )... 

AP is the pressure drop, cm of water Pg is the gas density, g/cm Ap is the total projected area of an entire row of baffles in the direction of inlet gas flow, cm" and At is the duct cross-sectional area, cm". The value jd is a drag coefficient for gas flow past inclined flat plates taken from Fig. 14-113, while L/ is the actual gas velocity, cm/s, which is related to the superficial gas velocity by U = L/g/cos 0. It must be noted that the angle of incidence 0 for the second and successive rows of baffles is twice the angle of incidence for the first row. Most of Calverts work was with 30° baffles, but the method correlates well with other data on 45° bafiles. 

AP is the pressure drop, cm of water p and Pg are the density of the scrubbing liquid and gas respectively, g/cm L/g is the velocity of the gas at the throat inlet, cm/s QtIQg is the volumetric ratio of liquid to gas at the throat inlet, dimensionless It is the length of the throat, cm Coi is the drag coefficient, dimensionless, for the mean liquid diameter, evaluated at the throat inlet and d[ is the Sauter mean diameter, cm, for the atomized liquid. The atomized-liquid mean diameter must be evaluated by the Nuldyama and Tanasawa [Trans. Soc Mech Eng (Japan), 4, 5, 6 (1937-1940)] equation ... 

As seen in Fig. 11-2, the drag coefficient for the sphere exhibits a sudden drop from 0.45 to about 0.15 (almost 70%) at a Reynolds number of about 2.5 x 105. For the cylinder, the drop is from about 1.1 to about 0.35. This drop is a consequence of the transition of the boundary layer from laminar to turbulent flow and can be explained as follows. 

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 ... 

The solids contribution to the pressure drop, APls, is a consequence of both the particle-wall and particle-particle interactions. The latter is reflected in the dependence of the friction factor fs on the particle diameter, drag coefficient, density, and relative (slip) velocity by (Hinkel, 1953) ... 

For viscous dominated flows, it can be assumed that the gas inertia and the gas gravitational forces are negligible. By dropping the gas inertia and gravity time from the gas momentum equation and simplifying the dimensionless drag coefficient to the linear viscous term, the set of dimensionless equations does not include gas-to-solid density ratio as a parameter. 

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. 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... 

The shape of a drop moving under the influence of gravity may be affected by interfacial motions the drop may also wobble and move sideways (S27, W3). In one system (S22) the terminal velocity was reduced yielding a drag coefficient nearly equal to that of a solid particle. Interfacial convection tends to increase the rate of mass transfer above that which would occur in the absence of interfacial motion. The interaction between mass transfer and interfacial convection has been treated by Sawistowski (S7) and Davies (D4, D5). 

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. 

Drops accelerated by an air stream may split, as described in Chapter 12. For drops which do not split, measured drag coefficients are larger than for rigid spheres under steady-state conditions (R2). The difference is probably associated more with shape deformations than with the history and added mass effects discussed above. For micron-size drops where there is no significant deformation, trajectories may be calculated using steady-state drag coefficients (SI). 

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... 

Spilhaus 8P) treats the shape of falling drops and the variation of the drag coefficient. Langmuir and Blodgett 6P) present a mathematical investigation of water droplet trajectories. 

The vaporization rates and drag coefficients for 2,2,4-trimethylpentane (iso-octane) sprays in turbulent air streams were determined experimentally by Ingebo (40), who reported that the effect of relative velocity on the evaporation rate was represented by the 0.6 power of the Reynolds number and that the drag coefficient varied inversely with the relative velocity of the drops in the spray. By assuming that the evaporation rate was independent of velocity and the drag coefficient for droplets obeyed Stokes s law, the present author derived a mathematical theory for the ballistics of droplets injected into an air stream for which the velocity varied linearly with distance (57) and... 

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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