Drag coefficient spherical

Assuming spherical particles, the drag coefficient, in the laminar, the Stokes flow regime is... 

In addition, dimensional analysis can be used in the design of scale experiments. For example, if a spherical storage tank of diameter dis to be constmcted, the problem is to determine windload at a velocity p. Equations 34 and 36 indicate that, once the drag coefficient Cg is known, the drag can be calculated from Cg immediately. But Cg is uniquely determined by the value of the Reynolds number Ke. Thus, a scale model can be set up to simulate the Reynolds number of the spherical tank. To this end, let a sphere of diameter tC be immersed in a fluid of density p and viscosity ]1 and towed at the speed of p o. Requiting that this model experiment have the same Reynolds number as the spherical storage tank gives... 

The drag force is exerted in a direction parallel to the fluid velocity. Equation (6-227) defines the drag coefficient. For some sohd bodies, such as aerofoils, a hft force component perpendicular to the liquid velocity is also exerted. For free-falling particles, hft forces are generally not important. However, even spherical particles experience lift forces in shear flows near solid surfaces. 

The drag coefficient for rigid spherical particles is a function of particle Reynolds number, Re = d pii/ where [L = fluid viscosity, as shown in Fig. 6-57. At low Reynolds number, Stokes Law gives 24... 

Equations (6-236) to (6-239) are based on experiments on cube-oc tahedrons, octahedrons, cubes, and tetrahedrons for which the sphericity f ranges from 0.906 to 0.670, respectively. See also Chft, Grace, and Weber. A graph of drag coefficient vs. Reynolds number with y as a parameter may be found in Brown, et al. (Unit Operations, Whey, New York, 1950) and in Govier and Aziz. 

C = Overall Drag Coefficient, Dimensionless Dp = Diameter of Spherical Particle,ft. 

Figure 2.10 Relationship between drag coefficient lfd) and Reynolds number (Re) for a spherical particle settling in a liquid.

If the relative velocity is sufficiently low, the fluid streamlines can follow the contour of the body almost completely all the way around (this is called creeping flow). For this case, the microscopic momentum balance equations in spherical coordinates for the two-dimensional flow [vr(r, 0), v0(r, 0)] of a Newtonian fluid were solved by Stokes for the distribution of pressure and the local stress components. These equations can then be integrated over the surface of the sphere to determine the total drag acting on the sphere, two-thirds of which results from viscous drag and one-third from the non-uniform pressure distribution (refered to as form drag). The result can be expressed in dimensionless form as a theoretical expression for the 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 the limit of very high voidage, the drag coefficient can be related to the single particle drag coefficient. For the case of spherical particles,... 

In Eq. (39), Cdo is a function of the particle Reynolds number, Rep — pedp V — Vp /p. For rigid spherical particles, the drag coefficient CD0 can be estimated by the following equations (Rowe and Henwood, 1961) ... 

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... 

There are two difficulties which soon become apparent when attempting to assess the very large amount of experimental data which are available on drag coefficients and terminal falling velocities for non-spherical particles. The first is that an infinite number of non-spherical shapes exists, and the second is that each of these shapes is associated with an infinite number of orientations which the particle is free to take up in the fluid, and the orientation may oscillate during the course of settling. 

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... 

Fig. 10.15 Drag coefficient for rigid spherical particles in air as a function of Mach number with Reynolds number as parameter, for the case where the absolute temperatures of the particle and fluid are essentially the same.

Fig. 1. Drag coefficient vs particle Reynolds number for spherical particles where (-) corresponds to the theoretical value of CD = 24/Re (eq. 4).

A. Haider and O. Levenspiel, Drag coefficient and terminal velocity of spherical and nonspherical particles, Powder Technol., 1989, 58, 63-70. 

In this expression yd is a characteristic area of a spherical particle, 2Pfluid is a characteristic kinetic energy for the flow, and Cd is the drag coefficient [27]. 

The origin of the spherical polar coordinate system (r, 9, cp) is held fixed at the center of one particle and the polar axis (9 = 0) is set parallel to E. Let the electrolyte be composed of M ionic mobile species of valence zt and drag coefficient A,-(/ = 1, 2,. . . , M), and let nf be the concentration (number density) of the ith ionic species in the electroneutral solution. We also assume that fixed charges are distributed with a density of pflx. We adopt the model of Debye-Bueche where the polymer segments are regarded as resistance centers distributed in the polyelectrolyte... 

The terminal settling velocity of a spherical particle having a diameter of 0.6 mm is 0.11 m/s. What is the mass density of the particle Assume the settling is type 1 and the temperature of the water is 22°C. What is the drag coefficient ... 

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