Reynolds number for sphere

FIGURE 9.25 Dependence of drag coefficient on Reynolds number for spheres, cylinders, and discs. 

The zero vorticity cell model has been extended numerically to intermediate Reynolds numbers for spheres (LeClair and Hamielec, 1968), and both the zero vorticity and the free-surface cell models (unmodified) to spheroids at low Reynolds numbers (Epstein and Masliyah, 1972). However, these extensions are more applicable to immobilized packed beds than to fluidized beds, and only in the absence of turbulence, which for unexpanded, fixed packed beds of spheres develops at Re > 110—150 (foils and Hanratty, 1966). 

The behavior of Co with particle Reynolds number for spheres, disks, and cylinders is shown in Figure 4-1. Note that the curves appear to pass through regions of behavior. Also, note the obvious bend in the curves for spheres and cylinders in the vicinity of Rcp = 10. These kinks are due to a phenomenon called boundary layer separation, which takes place when the fluid s velocity change is so large that the fluid no longer adheres to the solid surface. 

The behavior of a rotating sphere or hemisphere in an otherwise undisturbed fluid is like a centrifugal fan. It causes an inflow of the fluid along the axis of rotation toward the spherical surface as shown in Fig. 1(a). Near the surface, the fluid flows in a spirallike motion towards the equator as shown in Fig. 1(b) and (c). On a rotating sphere, two identical flow streams develop on the opposite hemispheres. The two streams interact with each other at the equator, where they form a thin swirling jet toward the bulk fluid. The Reynolds number for the rotating sphere or hemisphere is defined as ... 

The usual approach for non-Newtonian fluids is to start with known results for Newtonian fluids and modify them to account for the non-Newtonian properties. For example, the definition of the Reynolds number for a power law fluid can be obtained by replacing the viscosity in the Newtonian definition by an appropriate shear rate dependent viscosity function. If the characteristic shear rate for flow over a sphere is taken to be V/d, for example, then the power law viscosity function becomes... 

Based on these data, particle-liquid Reynolds numbers were calculated to range from Re = 25 (50 rpm) to Re = 90 (150 rpm) for coarse grade particles with a median diameter of 236 pm. In contrast, Reynolds numbers for a batch of micronized powder of the same chemical entity with a median diameter of 3 pm were calculated to be significantly lower (Re < 1), indicating less sensitivity towards convective hydrodynamics [(10), Chapter 12.3.8]. Based on the aforementioned considerations for spheres, bulk Reynolds numbers of about Re > 50 appear to be sufficient to produce the laminar-turbulent transition around a rough drug particle of coarse grade dimensions. 

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 ... 

In Chapter 3, relations are given that permit the calculation of Re 0(uodp/p), the particle Reynolds number for a sphere at its terminal falling velocity n0, also as a function of Galileo number. Thus, it is possible to express Re mp in terms of Re 0 and u ,f in terms Of Uq. 

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.

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]). 

FIGURE 11.2 Friction factor, f, versus Reynolds number for a sphere. Adapted from Eisner [3]. 

In case 3 the relative size of the particles (with respect to the computational cells) is large enough that they contain many hundreds or even thousands of computational cells. It should be noted that the geometry of the particles is not exactly represented by the computational mesh and special, approximate techniques (i.e., body force methods) have to be used to satisfy the appropriate boundary conditions for the continuous phase at the particle surface (see Pan and Banerjee, 1996b). Despite this approximate method, the empirically known dependence of the drag coefficient versus Reynolds number for an isolated sphere could be correctly reproduced using the body force method. Although these computations are at present limited to a relatively low number of particles they clearly have their utility because they can provide detailed information on fluid-particle interaction phenomena (i.e., wake interactions) in turbulent flows. 

Table 6.1 The experimental relationship between drag coefficient and Reynolds number for a sphere settling in a liquid [1]...

Fig. 6.5 Drag coefficient versus Reynolds number for particles of different volume shape coefficients by microscopy compared with data for a sphere...

The characteristic length for a circular cylinder or sphere is taken to be the external diameter D. Thus, the Reynolds number is defined as Re = VD/v where V is Ihe uniform velocity of Ihe fluid as it approaches the cylinder or splicre. The critical Reynolds number for flow across a circular cylinder or sphere is about Re s 2 X 10. That is, the boundar) layer remains laminar for about Re < 2 X K) and becomes turbulent forRc 2 X l(y. ... 

For Re 1, we have creeping flow, and the drag coefficient decreases with increasing Reynolds number. For a sphere, it is Cp 24/Re. There is no flbwrseparation in this regime. 

Figure 4 shows the experimentally determined friction factor as a function of the Reynolds number for a laboratory PPR module with six 4-mm-thick catalyst slabs of 68-mm width and 500-mm height, spaced apart with a pitch of 11 mm, and made up from 2.2-mm-diameter glass spheres enclosed in 0.5-mm gauze mesh [6]. It can be seen that the transition of laminar to turbulent flow occurs already at a low Reynolds number (approximately 1(XX)), which is attributable to the roughness of the channel walls caused by the wire gauze. 

The heat itransfer correlation relating the Nusselt nmtiber to the Prandtl and Reynolds numbers for flow around a sphere is ... 

Proudman J, Pearson JRA (1957) Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder. Journal of Fluid Mechanics 2 237-262... 

Problem 7-6. Linearity and Superposition. Determine the flow between two concentric spheres of radii a and b that are rotating about different axes with angular velocities 2 and 2f, respectively, assuming that the Reynolds number for the motion is small. Also determine the torque exerted by the fluid on each sphere. 

Problem 7-22. The Viscosity of a Multicomponent Membrane. An interesting generalization of the Einstein calculation of the effective viscosity of a dilute suspension of spheres is to consider the same problem in two dimensions. This is relevant to the effective viscosities of some types of multicomponent membranes. Obtain the Einstein viscosity correction at small Reynolds number for a dilute suspension of cylinders of radii a whose axes are all aligned. Although there is no solution to Stokes equations for a translating cylinder, there is a solution for a force- and torque-free cylinder in a 2D straining flow. The result is... 

Problem 7-24. Sedimentation of a Colloidal Aggregate. Colloidal particles often aggregate because of London-van der Waals or other attractive interparticle forces unless measures are taken to stabilize them. The aggregation kinetics are such that the aggregate formed has a fractal dimension Df, which is often less than the spatial dimension. The fractal dimension measures the amount of mass in a sphere of radius R, i.e., mass R D<. For a fractal aggregate composed of Aprimary particles of radius Op with mass mp, estimate the sedimentation velocity of the aggregate when the Reynolds number for the motion is small. What is the appropriate Reynolds number ... 

S. Kaplan and P. A. Lagerstrom, Asymptotic expansions of Navier-Stokes solutions for small Reynolds numbers, J. Math. Mech. 6, 585-593 (1957) S. Kaplan, Low Reynolds number flow past a circular cylinder, J. Math. Mech. 6, 595-603 (1957) I. Proudman and 1. R. A. Pearson, Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder, J. Fluid Mech. 2, 237-262 (1957). 

Numerous available numerical solutions the Navier-Stokes equations, as well as experimental data (see a review in [94]), provide a detailed analysis of the flow pattern for increasing Reynolds numbers. For 0.5 < Re < 10, there is no flow separation, although the fore-and-aft symmetry typical of inertia-free Stokes flow past a sphere is more and more distorted. Finally, at Re = 10, flow separation occurs at the rear of the particle. 

The Reynolds number at which the attached boundary layer becomes turbulent is called the critical Reynolds number for drag. The curve for spheres shown in Fig. 7.3 applies only when the fluid approaching the sphere is non-turbulent or when the sphere is moving through a stationary static fluid. If the approaching fluid is turbulent, the critical Reynolds number is sensitive to the scale of turbulence and becomes smaller as the scale increases. For example, if the scale of turbulence, defined as is 2 percent, the critical Reynolds... 

The foregoing pertains entirely to spheres. We can use Eq. 6.53 for other shapes, if we agree on what area A represents. Generally, in drag measurements it refers to the frontal area perpendicular to the flow that is the definition on which the coefficients in Fig. 6,22 are based. Moreover, we must decide on which dimensions to base the Reynolds number in our correlation of Cj versus in Fig. 6.22 the Reynolds number for cylinders takes the cylinder diameter as /), and that for disks takes the disk diameter. 

---