Drag coefficient sphere

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... 

Energy dissipation rate per unit mass of fluid (ranges 570 < Ns < 1420) fluid and sphere, m/s. Cq,. = drag coefficient for single particle fixed in fluid at velocity i>,.. See 5-27-G for calculation details and other applica- ... 

FIG. 6-57 Drag coefficients for spheres, disks, and cylinders =area of particle projected on a plane normal to direction of motion C = over-... 

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. )... 

Therefore, the inertia forces have an insignificant influence on the sedimentation process in this regime. Theoretically, their influence is equal to zero. In contrast, the forces of viscous friction are at a maximum. Evaluating the coefficient B in equation 55 for a = 1 results in a value of 24. Hence, we have derived the expression for the drag coefficient of a sphere, = 24/Re. 

AA-gpjj. Conditionally, the ionic atmosphere is regarded as a sphere with radius r. The valnes of approach the size of colloidal particles, for which Stokes s law applies (i.e., the drag coefficient 9 = where r is the liquid s viscosity) when they... 

FIGURE 6.9 Dependence of viscoelastic parameters on solvent quality. The (A) static force, (B) drag coefficient at 10 kHz, (C) dynamic spring constant, and (D) dispersion parameter are shown as a function of the surface-sphere distance. The results for water, propanol, and a 50/50 water/propanol mixture are given. Reprinted with permission from Benmouna and Johannsmann (2004). 

Here, p is the density of the fluid, V is the relative velocity between the fluid and the solid body, and A is the cross sectional area of the body normal to the velocity vector V, e.g., nd1/4 for a sphere. Note that the definition of the drag coefficient from Eq. (11-1) is analogous to that of the friction factor for flow in a conduit, i.e.,... 

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 ... 

Figure 11-2 Drag coefficient for spheres, cylinders, and disks. (From Perry, 1984.) o Eq. (11-5), spheres. Eq. (11-7), cylinders.

Although Eq. (11-6) is more accurate than Eq. (11-5) at intermediate values of /VRe, Eq. (11-5) provides a sufficiently accurate prediction for most applications. Also it is simpler to manipulate, so we will prefer it as an analytical expression for the sphere drag coefficient. 

For flow past a circular cylinder with L/d > normal to the cylinder axis, the flow is similar to over for a sphere. An equation that adequately represents the cylinder drag coefficient over the entire range of NRc (up to... 

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. 

With regard to the drag on a sphere moving in a Bingham plastic medium, the drag coefficient (CD) must be a function of the Reynolds number as well as of either the Hedstrom number or the Bingham number (7V Si = /Vne//VRe = t0d/fi V). One approach is to reconsider the Reynolds number from the perspective of the ratio of inertial to viscous momentum flux. For a Newtonian fluid in a tube, this is equivalent to... 

Figure 26. Error in cluster drag coefficient for fixed u0/u using Cd for a solid sphere. (From Glicksman et ah, 1993b.)...

Beetstra, R., van der Hoef, M. A., and Kuipers, J. A. M., A lattice-Boltzmann simulation study of the drag coefficient of clusters of spheres, Comput. Fluids 35, 966-970 (2006). 

A glass sphere, of diameter 6 mm and density 2600 kg/m3, falls through a layer of oil of density 900 kg/m3 into water. If the oil layer is sufficiently deep for the particle to have reached its free falling velocity in the oil, how far will it have penetrated into the water before its velocity is only 1 per cent above its free falling velocity in water It may be assumed that the force on the particle is given by Newton s law and that the particle drag coefficient R /pu2 = 0.22. 

When the Reynolds number Rep reaches a value of about 300000, transition from a laminar to a turbulent boundary layer occurs and the point of separation moves towards the rear of the sphere as discussed above. As a result, the drag coefficient suddenly falls to a value of 0.10 and remains constant at this value at higher values of Rep. 

Henderson 575 presented a set of new correlations for drag coefficient of a single sphere in continuum and rarefied flows (Table 5.1). These correlations simplify in the limit to certain equations derived from theory and offer significantly improved agreement with experimental data. The flow regimes covered include continuum, slip, transition, and molecular flows at Mach numbers up to 6 and at Reynolds numbers up to the laminar-turbulent transition. The effect on drag of temperature difference between a sphere and gas is also incorporated. 

Table 5.1. Correlations for Drag Coefficient of a Single Sphere in Continuum and Rarefied Flows1575 ...

Drag coefficient Q/l. .Re) is the drag coefficient calculated using the correlation for subsonic flow with Ma=l. f V/1.75, Re ) is the drag coefficient calculated using the correlation for supersonic flow with Ma00=1.75 Mach number based on relative velocity between gas and sphere ... 

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. ... 

From dimensional considerations, the drag coefficient is a function of the Reynolds number for the flow relative to the particle, the exponent, nm, and the so-called Bingham number Bi which is proportional to the ratio of the yield stress to the viscous stress attributable to the settling of the sphere. Thus ... 

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