Sphere flow around

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. 

Certain hydrodynamical problems, as well as mass-transfer problems in the presence of surface-active agents, have been investigated theoretically under steady-state conditions (L3, L4, L10, R9). However, if we take into account the fact that in gas-liquid dispersions, the nonstationary term must appear in the equation of mass- or heat-transfer, it becomes apparent that an exact analysis is possible if a mixing-contacting mechanism is adopted instead of a theoretical streamline flow around a single bubble sphere. 

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.)...

Free convection flow around horizontal cylinders and spheres is laminar for moderate values of GrSc (see Table VII, Part C) mass-transfer rates obey correlations of the same type as that for a vertical plate electrode, Eq. (29a) ... 

As the fluid flows over the forward part of the sphere, the velocity increases because the available flow area decreases, and the pressure decreases as a result of the conservation of energy. Conversely, as the fluid flows around the back side of the body, the velocity decreases and the pressure increases. This is not unlike the flow in a diffuser or a converging-diverging duct. The flow behind the sphere into an adverse pressure gradient is inherently unstable, so as the velocity (and lVRe) increase it becomes more difficult for the streamlines to follow the contour of the body, and they eventually break away from the surface. This condition is called separation, although it is the smooth streamline that is separating from the surface, not the fluid itself. When separation occurs eddies or vortices form behind the body as illustrated in Fig. 11-1 and form a wake behind the sphere. 

The flow toward the surface is caused by the pressure under the indenter. It is analogous to the upward flow around a sphere dropped into a liquid. It is also analogous to inverse extrusion. A model of the flow has been proposed by Brown (2007) in terms of rotational slip. This model reproduces some of the observed behavior, but it is a continuum model and does not define the mechanism of rotational slip. 

Figure 10. Velocity field of creeping flow around a sphere [24], Each arrow represents the velocity at the origin of the depicted vector. The length of the arrow corresponding with the free velocity v would equal the radius of the sphere...

In contrast to single-phase flow in a pipe of constant cross section, flow around a sphere or other bluff object exhibits several different flow regimes at different values of the Reynolds number. 

This equation too is solved with the same boundary conditions as Eq. (148). A series of equations results when different combinations of fluids are used. There is no change for the first stage. All the terms of equation of motion remain the same except the force terms arising out of dispersed-phase and continuous-phase viscosities. The main information required for formulating the equations is the drag during the non-Newtonian flow around a sphere, which is available for a number of non-Newtonian models (A3, C6, FI, SI 3, SI 4, T2, W2). Drop formation in fluids of most of the non-Newtonian models still remains to be studied, so that whether the types of equations mentioned above can be applied to all the situations cannot now be determined. 

HAMIELEC, A. E. and Johnson, A. I. Can. J. Chem. Eng. 40 (1962) 41. Viscous flow around fluid spheres at intermediate Reynolds numbers. 

Many authors (B4, G3, H2, L2, Sll) have considered the flow pattern and wakes involved in the flow of a fluid past a rigid sphere. Nearly every book on fluid mechanics contains a chapter on flow around submerged shapes. Flow around fluid shapes is only touched upon by a few advanced treatises such as that by Lamb (L2). 

Elzinga and Banchero (El) use Meksyn s boundary layer equation (M2) for flow around a rigid sphere, with the boundary condition that the interfacial velocity is not zero, to calculate a shift in the boundary-separation ring from an equivalent rigid-sphere location. Their calculated positions are slightly less than their observed shifts but confirm the thesis that these shifts are due to internal circulation. Similar quantitative results are reported by Garner and Tayeban (G7). 

In the limit of slow flow over a sphere Sh/j = 2.0, and this corresponds to diffusion to or from a sphere surrounded by a stagnant fluid. When the sphere diameter is sufifrciently small. Re becomes sufficiently small that Sh/j = 2.0 in many common situations of flow around spheres. 

Now we need and h, the mass and heat transfer coefficients around a sphere. These come from Sherwood and Nusselt numbers, respectively, for flow around a sphere,... 

Analytic solutions for flow around and transfer from rigid and fluid spheres are effectively limited to Re < 1 as discussed in Chapter 3. Phenomena occurring at Reynolds numbers beyond this range are discussed in the present chapter. In the absence of analytic results, sources of information include experimental observations, numerical solutions, and boundary-layer approximations. At intermediate Reynolds numbers when flow is steady and axisym-metric, numerical solutions give more information than can be obtained experimentally. Once flow becomes unsteady, complete calculation of the flow field and of the resistance to heat and mass transfer is no longer feasible. Description is then based primarily on experimental results, with additional information from boundary layer theory. 

Numerical solutions of the flow around and inside fluid spheres are again based on the finite difference forms of Eqs. (5-1) and (5-2) (BIO, H6, L5, L9). The necessity of solving for both internal and external flows introduces complications not present for rigid spheres. The boundary conditions are those described in Chapter 3 for the Hadamard-Rybczynski solution i.e., the internal and external tangential fluid velocities and shear stresses are matched at R = 1 (r = a), while Eq. (5-6) applies as R- co. Most reported results refer to the limits in which k is either very small (BIO, H5, H7, L7) or large (L9). For intermediate /c, solution is more difficult because of the coupling between internal and external flows required by the surface boundary conditions, and only limited results have been published (Al, R7). Details of the numerical techniques themselves are available (L5, R7). 

Johansson (Jl) reported numerical calculations of the flow around a sphere fixed on the axis of a Poiseuille flow (Fig. 9.1 with b = 0,U = 0). Only solutions for 2 = 0.1 were considered, and wake formation was predicted for Re = 20.4 based on the centerline velocity Uq. ... 

At the other extreme of Re, Achenbach (Al) investigated flow around a sphere fixed on the axis of a cylindrical wind tunnel in the critical range. Wall effects can increase the supercritical drag coefficient well above the value of 0.3 arbitrarily used to define Re in an unbounded fluid (see Chapter 5). If Re is based on the mean approach velocity and corresponds to midway between the sub- and super-critical values, the critical Reynolds number decreases from 3.65 x 10 in an unbounded fluid to 1.05 x 10 for k = 0.916. 

The values of n and m vary depending on the system and the geometry, but typical values are n = m = 0.5 for creeping flow around a sphere in a gas-liquid system, and n = m = 0.33 for creeping flow around a sphere in liquid-solid systems. 

A number of authors [46 to 48] employ the single sphere model in which the packed bed is considered as a set of equal spheres that are under the same state of extraction, and the fluid flowing around them is solute-free. That is, equation (3.4-90) would be valid, but without the generation term [46], The transport at the solid-fluid interface obeys the boundary condition (Eqn. 3.4-94) with C = 0 (fluid-flows at a large velocity). Under these assumptions, there is an analytical solution to the above problem (without axial dispersion) in terms of the Biot number (Bi = k, R/De), included in the following equation ... 

In the laminar-film contactor shown in Fig. 4.116, the supporting surface has the form of a sphere held in place by a wire<3>. This arrangement has the advantage that, in the regions where the liquid runs on to the sphere, and where it leaves the sphere, the surface areas exposed to the gas are relatively small so that, even if the hydrodynamics of the liquid flow in these regions is not ideal, the effect on the rate of absorption of the gas will be small. As in the case of the cylindrical tube, the contact time for each element of liquid as it flows around the sphere can be calculated from the liquid flowrate, although the mathematics of the analytical treatment is somewhat more complicated than in the case of the tube. 

Sample isotherm simulation results for a glass sphere of 200 /tm in contact with a heat transfer surface surrounded by static air are shown in Fig. 12.5 for two contact times, i.e., 1.2 ms and 52.4 ms [Botterill and Williams, 1963]. The initial temperature difference between the sphere and the surface is 10°C. It is seen that at the instant of contact heat begins to flow around the upper surface of the sphere and significant heat transfer takes place at... 

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