Hadamard-Rybczynski solution

There are no solutions for transfer with the generality of the Hadamard-Rybczynski solution for fluid motion. If resistance within the particle is important, solute accumulation makes mass transfer a transient process. Only approximate solutions are available for this situation with internal and external mass transfer resistances included. The following sections consider the resistance in each phase separately, beginning with steady-state transfer in the continuous phase. Section B contains a brief discussion of unsteady mass transfer in the continuous phase under conditions of steady fluid motion. The resistance within the particle is then considered and methods for approximating the overall resistance are presented. Finally, the effect of surface-active agents on external and internal resistance is discussed. 

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). 

Figure 7-15. Streamlines for the Hadamard-Rybczynski solution for translation of a spherical drop through a quiescent fluid (plotted in a frame of reference that is fixed to the drop).

This result is known as the Hadamard-Rybczynski solution. It describes the velocity field for a spherical drop that is translating through an otherwise motionless fluid. A plot of the streamlines is shown in Fig. 7-15 for several different ratios of the internal to external viscosity. Obviously, for A. = 10, the interior fluid is moving slowly, and this is reflected in the small number of streamlines inside the drop. In addition, the exterior fluid is required to come almost to a stop on the drop surface, and this results in rather large velocity gradients (the streamlines are close together). As k is decreased, on the other hand, the interior fluid moves more freely, and the velocity gradients in the exterior fluid are reduced. 

Flow separation in the case of a drop is delayed compared with the case of a solid particle, and the vorticity region (wake) is considerably narrower. While in the case of a solid sphere, the flow separates and the rear wake occurs at Re 10 (the number Re is determined by the sphere radius), in the case of a drop there may be no separation until Re = 50. For 1 < Re < 50, numerical methods are widely used. The results of numerical calculations are discussed in [94], For these Reynolds numbers, the internal circulation is more intensive than is predicted by the Hadamard-Rybczynski solution. The velocity at the drop boundary increases rapidly with the Reynolds number even for highly viscous drops, In the limit case of small viscosity of the disperse phase, /3 —> 0 (this corresponds to the case of a gas bubble), one can use the approximation of ideal fluid for the outer flow at Re > 1. 

The mathematical statement of the problem on the concentration distribution outside a drop is described by Eq. (4.4.3) and the boundary conditions (4.4.4) and (4.4.5), in which the dimensionless stream function satisfies the Hadamard-Rybczynski solution (see Section 2.2)... 

Let us consider a transient solute concentration field in a liquid outside and inside a spherical drop of radius a moving at a constant velocity U in an infinite fluid medium. We assume that the fluid velocity fields for the continuous and disperse phases are determined by the Hadamard-Rybczynski solution [177, 420], obtained for low Reynolds numbers. The concentration far from the drop is maintained constant and equal to C,. At the initial time f = 0, the concentration outside the drop is everywhere uniform and is equal to C inside the drop, it is also uniform, but is equal to Co-... 

Internal circulation patterns have been observed experimentally for drops by observing striae caused by the shearing of viscous solutions (S7) or by photographing non-surface-active aluminum particles or dyes dispersed in the drop fluid [e.g. (G2, G3, J2, L5, Ml, SI)]. A photograph of a fully circulating falling drop is shown in Fig. 3.5a. Since the internal flow pattern for the Hadamard-Rybczynski analysis satisfies the complete Navier-Stokes equation... 

Section II shows that the dimensionless external velocity field uJU, UqIU) is a function of dimensionless position r/a, 0) and k for creeping flow. The dimensionless concentration defined in Eq. (1-45) is a function of these quantities and of the Peclet number, Pe = 2aU/. Hence the Sherwood number, Sh = is a function of k and Pe (with additional dependence on Re unless the creeping flow approximation is valid). The exact solution of Eqs. (3-39) to (3-42) with the Hadamard-Rybczynski velocity field is not available for all values of Pe and K, but several special cases have been treated. 

For a circulating sphere with Pep/(1 + /c) oc, the time required for diffusion is much greater than that for fluid circulation, so that surfaces of uniform concentration coincide with the Hadamard-Rybczynski streamlines. Kronig and Brink (K6) showed that the solution is then... 

Solutions have been obtained for a rigid sphere with Pep = 0 (G8), and the results are shown in Fig. 3.21. We have complemented these with solutions for a sphere circulating with the Hadamard-Rybczynski velocities at PCp/(l + k) 00, assuming Shi proportional to sin 0 and with the overall mean Sh used to define Bi. These results are shown in Fig. 3.22. For Bi oo (i.e., negligible external resistance), the limiting curves are the Newman solution in Fig. 3.21 and the Kronig-Brink solution in Fig. 3.22. For Bi < 0.2 the internal resistance... 

An approximate solution to this problem can also be obtained with the particularization of the Hadamard-Rybczynski problem [6.30, 6.31] from which it is found that C = 1/18 so that ... 

A theoretical analysis of the Stokes flow problem for a noimeutraUy buoyant droplet is clearly called for. Germane to this problem is the theoretical analysis of Haberman (H3), dealing with axially symmetric Stokes flow relative to a liquid droplet at the axis of a circular tube, and Taylor and Acrivos (T2c) extension of the classical Hadamard-Rybczynski liquid droplet problem to the case of nonzero Reynolds numbers. In particular, Haberman shows that the assumption of a spherical shape for the droplet in a tube is incompatible with the differential equations and boundary conditions. Taylor and Acrivos (T2c) point out that, though Hadamard (H3a) and Rybczynski (RIO) were able to solve the Stokes flow problem by assuming a spherical shape for a liquid droplet, irrespective of the magnitude of the interfacial tension, the correctness of their a priori assumption was, to some extent, fortuitous. These remarks are undoubtedly pertinent to the resolution of Haberman s paradox and, ultimately, to the solution of the nonaxially symmetric droplet problem. 

For creeping flow (0 < Re < 1), the solutions of the conduction-convection equation with flow held of the Hadamard-Rybczynski or Stokes are given by numerical integration [1]. The numerical results show that the concentration contours are not symmetrical (Figure 5.1 and Figure 5.2) and that the how inside and outside the sphere largely inhuences heat or mass transfer. In the case of a sphere with weak viscosity ratio, the heat or mass transfer is facilitated. 

As constant interfacial tension is no guarantee of absence of contamination, measurement of the upward velocity of a bubble in solution seems a much more sensitive criterion. Thus, the verification of the Hadamard-Rybczynski equation could be considered as a criterion, indicating that the solution has reached a purity beyond which there is no longer any contamination [22]. 

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.,... 

A similar solution for creeping flow past a spherical droplet of fluid were derived independently by Hadamard [55] and Rybczynski [124], In this case the fluid stream has a velocity V at infinity and viscosity p/, while the droplet has a viscosity /j,p and a fixed interface. The boundary conditions at the droplet interface are (1) zero radial velocities and (2) equality of surface shear and... 

In a more recent study, which is an extension of the previous works, Saboni et al. [15] proposed a predictive equation for drag coefficients covering Reynolds numbers in the range 0.01 < Re < 400 and viscosity ratio from 0 to 1000. This correlation, which is reduced to the solution of Hadamard [4] and Rybczynski [5] for Re 0, is as follows ... 

---