Fluid Spheres

The straight line for Ap = 0 represents diffusion in a stagnant medium [Eq. (3-44)]. In air spheres with diameters less than about 30 pm have transfer rates essentially equal to those in a stagnant medium, while in water the diameter for this to occur must be less than about 3 pm. In water the mass transfer coefficient is only weakly dependent on diameter, a prediction which has been verified experimentally (C2). For free fall in air, the transfer coefficient exhibits a larger decrease with diameter. The following expressions fit the predictions of Figs. 5.22 and 5.23 over the ranges indicated  

As noted in Chapter 2, bubbles and drops remain nearly spherical at moderate Reynolds numbers (e.g., at Re = 500) if surface tension forces are sufficiently strong. For drops and bubbles rising or falling freely in systems of practical importance, significant deformations from the spherical occur for all Re 600 (see Fig. 2.5). Hence the range of Re covered in this section, roughly 1 Re 600, is more restricted than that considered in Section II for solid spheres. Steady motion of deformed drops and bubbles at all Re is treated in Chapters 7 and 8. 

When a fluid sphere exhibits little internal circulation, either because of high K = Pp/p or because of surface contaminants, the external flow is indistinguishable from that around a solid sphere at the same Re. For example, for water drops in air, a plot of versus Re follows closely the curve for rigid spheres up to a Reynolds number of 200, corresponding to a particle diameter of approximately 0.85 mm (B5). In fact, many of the experimental points used in Section II to determine the standard drag curve refer to spherical drops in gas streams, where high values of k ensure negligible internal circulation. 

Here we consider three theoretical approaches. As for rigid spheres, numerical solutions of the complete Navier-Stokes and transfer equations provide useful quantitative and qualitative information at intermediate Reynolds numbers (typically Re 300). More limited success has been achieved with approximate techniques based on Galerkin s method. Boundary layer solutions have also been devised for Re 50. Numerical solutions give the most complete and 

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). 

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HABERMAN, W. L. and SAYRE, R. M. David Taylor Model Basin Report No 1143 (Oct 1958) Motion of rigid and fluid spheres in stationary and moving liquids inside cylindrical tubes. 

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

W. L. Haberman R. M. Sayre (1958) Motion of Rigid and Fluid Spheres in Stationary and Moving Liquids Inside Cylindical Tubes, David Taylor Model Basin Report No. 1143, U. S. Navy, Washington, D. C. 

Using a similar attack for a fully circulating fluid sphere in a stationary field but using only n = 2 due to inconsistency of the equations for higher order functions, their wall correction factor was... 

Haberman and Sayre also derived correction factors for a fluid sphere moving along the axis of a cylinder in which the field fluid was also in axial motion ... 

Fig. 8. Equatorial plane velocities and Hadamard streamlines for fluid sphere (H2).

One of the most important analytic solutions in the study of bubbles, drops, and particles was derived independently by Hadamard (HI) and Rybczynski (R5). A fluid sphere is considered, with its interface assumed to be completely free from surface-active contaminants, so that the interfacial tension is constant. It is assumed that both Re and Rep are small so that Eq. (1-36) can be applied to both fluids, i.e.,... 

Stokes s solution (S9) for steady creeping flow past a rigid sphere may be obtained directly from the results of the previous section with co. The same results are obtained by solving Eq. (3-1) with Eqs. (3-4) to (3-6) replaced by the single condition that Uq O a.tr = a. The corresponding streamlines are shown in Figs. 3.3a and 3.4a. As for fluid spheres, the particle causes significant... 

The Hadamard-Rybczynski theory predicts that the terminal velocity of a fluid sphere should be up to 50% higher than that of a rigid sphere of the same size and density. However, it is commonly observed that small bubbles and drops tend to obey Stokes s law, Eq. (3-18), rather than the corresponding Hadamard-Rybczynski result, Eq. (3-15). Moreover, internal circulation is essentially absent. Three different mechanisms have been proposed for this phenomenon, all implying that Eq. (3-5) is incomplete. 

By assuming that the surface tension on the surface of a fluid sphere varied from the surfactant-free value, at the nose to zero at the rear, Savic also deduced a relationship between velocity and Eotvos number, shown in Fig. 3.7, which agrees qualitatively with the experimental results of Bond and Newton. Modifications of this approach for cases where the maximum change in local interfacial tension is less than have been devised for bubbles (D5, G7) and... 

Equation (3-33) shows how the inertia term changes the pressure distribution at the surface of a rigid particle. The same general conclusion applies for fluid spheres, so that the normal stress boundary condition, Eq. (3-6), is no longer satisfied. As a result, increasing Re causes a fluid particle to distort towards an oblate ellipsoidal shape (Tl). The onset of deformation of fluid particles is discussed in Chapter 7. 

Fig. 3.10 External Sh for spheres in Stokes flow (1) Exact numerical solution for rigid and circulating spheres (2) Brenner (B6) rigid sphere, Pe 0, Eq. (3-45) (3) Levich (L3) rigid sphere, Pe 00, Eq. (3-47) (4) Acrivos and Goddard (A 1) rigid sphere, Pe oo, Eq. (3-48) (5) Approximate values fluid spheres.

Equation (3-45) is analogous to the Oseen correction to the Stokes drag, and is accurate to 0[Pe]." It applies for any rigid or fluid sphere at any Re, provided that Pe - 0 and the velocity remote from the particle is uniform. Figure 3.10 shows that Eq. (3-45) is accurate for Pe < 0.5. Acrivos and Taylor (A2) extended the solution to higher terms, but, as for drag, the additional terms only yield slight improvement at Pe < 1. 

For fluid spheres with k = 0, Eq. (3-39) has been solved numerically with the Hadamard-Rybczynski velocity field (Ol), and the resulting variation of Sh with Pe is shown in Fig. 3.10. The values are approximated within 6% for all Pe by... 

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

Complete solutions are available for Pe oo for rigid spheres (K4) and for fluid spheres (C5, R4) subject to the limitation of Eq. (3-54). Approximations good within 3% are, for rigid spheres ... 

The duration of the unsteady period, denoted by C, the time required for Sh to come within 100x% of the steady value, is different for rigid and fluid spheres. For a rigid sphere at high Pe, T ocPe "/". From Stokes s law, Eq. (3-18), Uj oc a" hence C is independent of particle size for a given fluid. However, for a fluid sphere, oc (1 + K )/Pe thus Ujt /a is a constant, and a given fractional approach to steady state is achieved when the particle has moved a fixed number of radii, e.g.. 

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. 

The external resistance has been evaluated under steady-state conditions using the assumption of a thin concentration boundary layer on the outer surface of a fluid sphere. Surface velocities calculated by each of the three methods described in Section B above have been used in conjunction with Eq. (3-51). 

Experimental data for mass transfer from freely circulating fluid spheres are difficult to obtain because of deformation and because of the presence of surface-active agents which reduce circulation. Shown in Fig. 5.30 are data from three studies on water droplets in isobutanol where the droplets were nearly spherical and were observed to be circulating. The data are in fair agreement with each other and with Eq. (5-39). The effects of shape changes and surface-active agents are discussed in Chapter 7. 

The case of a fluid sphere moving at constant velocity and suddenly exposed to a step change in the composition of the continuous phase has been treated by solving Eq. (3-56), with Eqs. (3-40), (3-41), (3-42), and (3-57) as boundary conditions for potential flow (R14). The transient external resistance is given within 3% by... 

Comparison of these equations shows that the area-free Sherwood number is only slightly affected by eccentricity e.g. Sh/Pe for a spheroid with E = 0.4 is only 8.5% larger than that for the equivalent sphere while the area ratio A/A is 17% larger. Therefore, we expect little effect of deformation on the area-free Sherwood number for bubbles and drops at high Re. This is borne out by the agreement of the data in Fig. 7.14 with Eq. (5-39), derived for fluid spheres. 

For circulating fluid particles without shape oscillations the internal resistance varies with time in a way similar to that discussed in Chapter 5 for fluid spheres. The occurrence of oscillation, with associated internal circulation, always has a strong effect on the internal resistance. If the oscillations are sufficiently strong to promote vigorous internal mixing, the resistance within the particle becomes constant. 

The surface velocities of Haberman and Sayre (HI), when used in the thin concentration boundary layer equation for circulating spheres, Eq. (3-51), yield the mass transfer factors and X d shown in Fig. 9.7 for k <2. For a fluid sphere in creeping flow the relationship between the mass transfer factors is... 

Similar analyses have been developed for fluid spheres (S8, S9) and for rigid spheroids (L3) moving parallel to their axes. The main conclusions are discussed below. 

Sy et al (S8, S9) and Morrison and Stewart (M12) analyzed the initial motion of fluid spheres with creeping flow in both phases. For bubbles (y = 0, k = 0), the condition that internal and external Reynolds numbers remain small is sufficient to ensure a spherical shape. However, for other k and y, the Weber number must also be small to prevent significant distortion (S9). For k = 0, the equation governing the particle velocity may be transformed to an ordinary differential equation (Kl), to give a result corresponding to Eq. (11-16), i.e.,... 

As for steady motion, shape changes and oscillations may complicate the accelerated motion of bubbles and drops. Here we consider only acceleration of drops and bubbles which have already been formed formation processes are considered in Chapter 12. As for solid spheres, initial motion of fluid spheres is controlled by added mass, and the initial acceleration under gravity is g y - l)/ y + ) (El, H15, W2). Quantitative measurements beyond the initial stages are scant, and limited to falling drops with intermediate Re, and rising... 

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