Sphere boundary layer

If the thickness of the diffusion boundary layer is smaller than b — a (and also smaller than a), one may consider that the diffusion takes place from the sphere to an infinite liquid. It should be emphasized here that the thickness of the diffusion boundary layer is usually about 10 % of the thickness of the hydrodynamic boundary layer (L3). Hence this condition imposes no contradiction to the requirements of the free surface model and Eq. (195). ... 

This formulation assumes that the continuum diffusion equation is valid up to a distance a > a, which accounts for the presence of a boundary layer in the vicinity of the catalytic particle where the continuum description no longer applies. The rate constant ky characterizes the reactive process in the boundary layer. If it approximated by binary reactive collisions of A with the catalytic sphere, it is given by kqf = pRGc(8nkBT/m)1 2, where pR is the probability of reaction on collision. 

The volume fraction dependence of /cq/(4>) is plotted in Fig. 21b and shows that it increases strongly with 4>- Recall that this rate coefficient is independent of 4> if simple binary collision dynamics is assumed to govern the boundary layer region. The observed increase arises from the obstacle distribution in the vicinity of the catalytic sphere surface. When obstacles are present, a reactive... 

For turbulent flow on a rotating sphere or hemisphere, Sawatzki [53] and Chin [22] have analyzed the governing equations using the Karman-Pohlhausen momentum integral method. The turbulent boundary layer was assumed to originate at the pole of rotation, and the meridional and azimuthal velocity profiles were approximated with the one-seventh power law. Their results can be summarized by the... 

ShaJSc113 as indicated by the thin solid line. This 0.67 power of Re agrees with the result of a turbulent heat transfer measurement on a rotating sphere [40], Since the flow induced by a rotating sphere is also characterized by an outflowing radial jet at the equator caused by the collosion of two opposing flow boundary layers on the sphere, the 0.67 power dependence on Re is clearly related to the radial flow stream away from the equator. 

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 flow over an immersed body (e.g., a sphere), the boundary layer grows from the impact (stagnation) point along the front of... 

You want to perform an experiment that illustrates the wake behind a sphere falling in water at the point where the boundary layer undergoes transition from laminar to turbulent. (See Fig. 11-4.) If the sphere is made of steel with a density of 500 lbm/ft3, what should the diameter be ... 

On increasing the Reynolds number further, a point is reached when the boundary layer becomes turbulent and the point of separation moves further back on the surface of the sphere. This is the case illustrated in the lower half of Figure 9.1 with separation occurring at point C. Although there is still a low pressure wake, it covers a smaller fraction of the sphere s surface and the drag force is lower than it would be if the boundary layer were laminar at the same value of Rep. 

Roughening the surface of a sphere causes the transition to a turbulent boundary layer to occur at a lower value of the Reynolds number. This explains the apparent anomaly that, at certain values of the Reynolds number, the drag will be lower for a sphere with a rough surface than for a similar sphere with a smooth surface. It is for the same reason that golf balls are made with a dimpled surface. 

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. 

Originally, the concept of the Prandtl boundary layer was developed for hydraulically even bodies. It is assumed that any characteristic length L on the particle surface is much greater than the thickness (<5hl) of the boundary layer itself (L > Ojil) Provided this assumption is fulfilled, the concept can be adapted to curved bodies and spheres, including real drug particles. Furthermore, the classical ( macroscopic ) concept of the hydrodynamic boundary layer is valid solely for high Reynolds numbers of Re>104 (14,15). This constraint was overcome for the microscopic hydrodynamics of dissolving particles by the convective diffusion theory (9). 

When Re exceeds about 2 x 105, the flow in the boundary layer changes from streamline to turbulent and the separation takes place nearer to the rear of the sphere. The drag force is decreased considerably and ... 

If Re is of the order of 105, the drag on the sphere may be reduced if the fluid stream is turbulent. The flow in the boundary layer changes from streamline to turbulent and the size of the eddies in the wake of the particle is reduced. The higher the turbulence of the fluid, the lower is the value of Re at which the transition from region (c) to region (d) occurs. The value of Re at which R /pu2 is 0.15 is known as the turbulence number and is taken as an indication of the degree of turbulence in the fluid. 

At still greater Reynolds numbers the boundary layer itself becomes turbulent and separation occurs at the rear of the sphere and closer to the particle. In this fully turbulent region, beyond Re = 2x 10 the drag coefficient falls further to a value of about 0.10. 

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). 

Model a campfire spark as a carbon sphere (pc = 3 g/cm ) surrounded by a boundary layer... 

Levich (L3) obtained an asymptotic solution to Eq. (3-39) for Pe oo, using the thin concentration boundary layer assumption discussed in Chapter 1. Curvature of the boundary layer and angular diffusion are neglected (i.e., the last term in Eq. (3-39) is deleted), so that the solution does not hold at the rear of the sphere where the boundary layer thickens and angular diffusion is significant. The asymptotic boundary layer formula, Eq. (1-59), reduces for a sphere to ... 

For a fluid sphere with Pe oo the thin concentration boundary layer approximation, Eq. (1-63), becomes... 

The dependence of Sh on Pe/(1 + k) at high Pe results because the Hadamard -Rybczynski analysis gives dimensionless velocities iiJU, iio/V) proportional to (1 + k) within and close to the particle (Eqs. (3-7) and (3-8)). Similar dependence is encountered for unsteady external transfer (Section B.2), and for internal transfer at all Pe (Section C.4). These results do not give the rigid sphere values as /c x, because of fundamental diflerences between the boundary layer approximations for the two cases (see Chapter 1), and arc only valid for /c < 2. 

Unsteady transfer with Pe oo has been treated using the thin concentration boundary layer assumptions. With this approximation, the last term in Eq. (3-56) is deleted. Hence, for small t where the convection term is negligible, the transfer rate for rigid or circulating spheres is identical to that for diffusion from a plane into a semi-infinite region ... 

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. 

As Re increases further and vortices are shed, the local rate of mass transfer aft of separation should oscillate. Although no measurements have been made for spheres, mass transfer oscillations at the shedding frequency have been observed for cylinders (B9, D6, SI2). At higher Re the forward portion of the sphere approaches boundary layer flow while aft of separation the flow is complex as discussed above. Figure 5.17 shows experimental values of the local Nusselt number Nuj c for heat transfer to air at high Re. The vertical lines on each curve indicate the values of the separation angle. It is clear that the transfer rate at the rear of the sphere increases more rapidly than that at the front and that even at very high Re the minimum Nuj. occurs aft of separation. Also shown in Fig. 5.17 is the thin concentration boundary layer... 

Fig. 5.17 Local Nusselt number for heat transfer from a sphere to air (Pr = 0.71). Experimental results of Galloway and Sage (Gl). Dashed lines are predictions of boundary layer theory by Lee and Barrow (LIO).

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