Sherwood number fluid spheres

Here is the maximum value of X and A is the surface area of the volume equivalent sphere. For a fluid particle the average Sherwood number is... 

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 a rigid sphere k = oo) on the axis of a cylindrical tube, the Sherwood number is larger than in an unbounded fluid with the same particle/fluid velocity. The ratio of Sherwood numbers is approximated within 3% for 2 < 0.6 by... 

The presence of container walls has a much smaller effect on Sherwood number than on drag since the mass transfer coefficient is only proportional to the one-third power of the surface vorticity. For a sphere with given settling on the axis of a cylindrical container, the Sherwood number decreases with 2, but it is still within 8% of the Sherwood number in an infinite fluid for 2 — 0.5. No data are available to test these predictions. 

In almost all the above models it is supposed that the solid surfaces instantaneously adsorb the mass, which is diluted within the fluid that flows through the medium. This approach has been initially proposed by Levich [8] who obtained analytical expression for the overall Sherwood number for the case of a single isolated sphere in an unbounded fluid. Significant improvements to this model, including the sphere-in-cell model and an intelligent representation of the mass transport conditions, have been presented later [9-11]. 

Once the upward flowing fluid has reached the minimum fluidisation velocity wml and with that the Reynolds number the value Remi = w dp/i/, point a in Fig. 3.40, a fluidised bed is formed. The heat and mass transfer coefficients hardly change with increasing fluid velocity The Nusselt and Sherwood numbers are only weakly dependent on the Reynolds number, corresponding to the slightly upwardly arched line a b in Fig. 3.40. After a certain fluid velocity has been reached, indicated here by point b in Fig. 3.40, the particles are carried upwards. At point b the heat and mass transfer coefficients are about the same as those for flow around a single sphere of diameter dP. 

The next three terms of the expansion of the Sherwood number corresponding to the asymptotic solution of problem (4.4.2)-(4.4.5) as Pe -> 0 were obtained in [7], These results were generalized in [401], where the solution found in [382] was used to describe the velocity field of the fluid around the sphere. Here we write out the final expression for the Sherwood number [401] ... 

Here Iq - II/a is the total flux to the particle in a stagnant fluid, f is the dimensionless vector equal to the ratio of the drag of the particle to the Stokes drag of a solid sphere of radius a (a is the characteristic length using which the dimensionless variables Pe, I, and Iq were introduced), and e is the unit vector codirected with the fluid velocity at infinity. The Sherwood number is given by Sh = I/S, where S is the dimensionless surface area of the particle. 

Mass transfer between spheres and surrounding fluids has been a topic of extensive study through the years. The most comprehensive situation is when the relative velocity between the solid and liquid phases is small, which is approximately the case in slurries of small particles. The general results are represented by the curve in Figure 8.12, showing the relationship between the Sherwood number, (k dp/D), and a Peclet number gdp/S.p/ % jD). The solid curve is given by [P.L.T. Brian and H.B. Hales, Amer. Inst. Chem. Eng. JL, 15, 419 (1969)]... 

As an incompressible fluid of infinite extent approaches and flows past either a spherical solid pellet or a gas bubble, a mobile component undergoes inteiphase mass transfer via convection and diffusion from the sphere to the fluid phase. The overall objective is to calculate the mass transfer coefficient and the Sherwood number at any point along the interface (i.e., the local transfer coefficients), as well as surface-averaged transfer coefficients. The results are applicable in the laminar flow regime (1) when the sphere is stationary and the fluid moves,... 

The primary focus of this chapter is to analyze the dimensionless equation of motion in the laminar flow regime and predict the Reynolds number dependence of the tangential velocity gradient at a spherical fluid-solid interface. This information is required to obtain the complete dependence of the dimensionless mass transfer coefficient (i.e., Sherwood number) on the Reynolds and Schmidt numbers. For easy reference, the appropriate correlation for mass transfer around a solid sphere in the laminar flow regime, given by equation (11-120), is included here ... 

A knowledge of mass transfer is essential for the understanding of the mechanism of combustion of coal in a turbulent fluidized bed. If the kinetic rate of combustion of the fuel is known, one can estimate the burning rate using the information on the mass transfer rate. The rate of transfer of oxygen from the bulk of the bed to the particle surface, k, is often expressed as the dimensionless Sherwood number, Sh = kgdp/Dg- For diffusion to a fixed single sphere in an extensive fluid, Sherwood number may be expressed as [28, 29]... 

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