Reynolds number sphere

The particle size deterrnined by sedimentation techniques is an equivalent spherical diameter, also known as the equivalent settling diameter, defined as the diameter of a sphere of the same density as the irregularly shaped particle that exhibits an identical free-fall velocity. Thus it is an appropriate diameter upon which to base particle behavior in other fluid-flow situations. Variations in the particle size distribution can occur for nonspherical particles (43,44). The upper size limit for sedimentation methods is estabHshed by the value of the particle Reynolds number, given by equation 11 ... 

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... 

UfQ = terminal velocity of a single sphere (infinite dilution) c = volume fraction sohd in the suspension n = function of Reynolds number Re = dpUto /[L as given Fig. 6-58... 

David W. Taylor Model Basin, Washington, September 1953 Jackson, loc. cit. Valentin, op. cit.. Chap. 2 Soo, op. cit.. Chap. 3 Calderbank, loc. cit., p. CE220 and Levich, op. cit.. Chap. 8). A comprehensive and apparently accurate predictive method has been publisned [Jami-alahamadi et al., Trans ICE, 72, part A, 119-122 (1994)]. Small bubbles (below 0.2 mm in diameter) are essentially rigid spheres and rise at terminal velocities that place them clearly in the laminar-flow region hence their rising velocity may be calculated from Stokes law. As bubble size increases to about 2 mm, the spherical shape is retained, and the Reynolds number is still sufficiently small (<10) that Stokes law should be nearly obeyed. 

The value of the Reynolds number which approximately separates laminar from turbulent flow depends, as previously mentioned, on the particular conhg-uration of the system. Thus the critical value is around 50 for a him of liquid or gas howing down a hat plate, around 500 for how around a sphere, and around 2500 for how tlrrough a pipe. The characterishc length in the dehnition of the Reynolds number is, for example, tire diameter of the sphere or of the pipe in two of these examples. 

Note - In designing a system based on the settling velocity of nonspherical particles, the linear size in the Reynolds number definition is taken to be the equivalent diameter of a sphere, d, which is equal to a sphere diameter having the same volume as the particle. 

For a eritieal value of the Reynolds number Rep <0.2 Stokes showed analytieally that the net foree aeting on a sphere is given by... 

Figure 2.2 Drag coejficients for the sphere as a function of particle Reynolds number...

Many materials of practical interest (such as polymer solutions and melts, foodstuffs, and biological fluids) exhibit viscoelastic characteristics they have some ability to store and recover shear energy and therefore show some of the properties of both a solid and a liquid. Thus a solid may be subject to creep and a fluid may exhibit elastic properties. Several phenomena ascribed to fluid elasticity including die swell, rod climbing (Weissenberg effect), the tubeless siphon, bouncing of a sphere, and the development of secondary flow patterns at low Reynolds numbers, have recently been illustrated in an excellent photographic study(18). Two common and easily observable examples of viscoelastic behaviour in a liquid are ... 

I I. Tayi.or, T.D. Physics of Fluids 6 (1963) 987. Heat transfer from single spheres in a low Reynolds number slip flow. 

Garner and Keey(52 53) dissolved pelleted spheres of organic acids in water in a low-speed water tunnel at particle Reynolds numbers between 2.3 and 255 and compared their results with other data available at Reynolds numbers up to 900. Natural convection was found to exert some influence at Reynolds numbers up to 750. At Reynolds numbers greater than 250, the results are correlated by equation 10.230 ... 

Figure 6. Flow around a sphere. The system size is 50 x 25 x 25 with y = 8 particles per cell. The gravitational field strength was g = 0.005 and the rotation angle for MPC dynamics was a — ti/2. Panel (a) is for a Reynolds number of Re — 24 corresponding to X = 1.8 while panel (b) is the flow for Re = 76 and X = 0.35. (From Ref. 30.)...

A. Najafi and R. Golestanian, Simple swimmer at low Reynolds number three linked spheres, Phys. Rev. E 69, 062901 (2004). 

However, more general correlations can be found for Kr in Equation 8.3. If in addition to assuming the particles to be rigid spheres, it is also assumed that the flow is in the laminar region, known as the Stoke s Law region, for Reynolds number less than 1 (but can be applied up to a Reynolds number of 2 without much error) ... 

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 ... 

SK Friedlander. Mass and heat transfer to single spheres and cylinders at low Reynolds numbers. AIChE J 3 43-48, 1957. 

Overall mass-transfer rates at a sphere in forced flow, and mass-transfer rate distribution over a sphere as a function of the polar angle have been measured by Gibert, Angelino, and co-workers (G2, G4a) for a wide range of Reynolds numbers. The overall rate dependence on Re exhibited two distinct regimes with a sharp transition at Re = 1250. Local mass-transfer rates were deduced from measurements in which the sphere was progressively coated by an insulator, starting from the rear. 

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. 

The viscosity of a Newtonian fluid can be determined by measuring the terminal velocity of a sphere of known diameter and density if the fluid density is known. If the Reynolds number is low enough for Stokes flow to apply (fVRe < 0.1), then the viscosity can be determined directly by rearrangement of Eq. (11-10) ... 

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... 

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... 

Beetstra, R., van der Hoef, M. A., and Kuipers, J. A. M. Drag force from lattice Boltzmann simulations of intermediate Reynolds number flow past mono- and bidisperse arrays of spheres, Manuscript submitted to AIChE J. (2006). 

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