Sherwood number spherical bubble

Insofar as the mass-transfer coefficient for clean bubbles is concerned (see, for example, the review by Clift et al. (1978)), in the case of spherical bubbles moving under creeping (or Stokes) flow conditions, the following correlation has been proposed Sh = 1 + (1 + 0.564Pe ). For spherical particles with Rep > 70 the Sherwood number can be expressed by the following relationship (Lochiel Calderbank, 1964) ... 

Sherwood number. For solid particles, drops, and bubbles of spherical shape, the mean Sherwood number can be calculated according to the formula... 

In this section, some interpolation formulas are presented (see [367, 368]) for the calculation of the mean Sherwood number for spherical particles, drops, and bubbles of radius a in a translational flow with velocity U at various Peclet numbers Pe = aU /D and Reynolds numbers Re = aU-Jv. We denote the mean Sherwood number by Shb for a gas bubble and by Shp for a solid sphere. 

Spherical bubble as Re —> 0, 0 < Pe < oo. The problem of mass transfer to a spherical bubble in a translational flow as Re 0 was studied numerically in [321], The results for the mean Sherwood number can be approximated well by the expression... 

Spherical bubble at any Peclet numbers for Re > 35. For a spherical bubble in a translational flow at moderate and high Reynolds numbers and high Peclet numbers, the mean Sherwood number can be calculated by the formula [94]... 

Particles and bubbles. Using the method of asymptotic analogies, we shall derive formulas for the calculation of the Sherwood number in a laminar flow past spherical particles, drops, and bubbles for an arbitrary structure of the nonperturbed flow at infinity. We assume that closed streamlines are lacking in the flow. 

Drops in the entire range of phase viscosities. For low and moderate Peclet numbers in an arbitrary laminar flow past a spherical drop under limiting resistance of the continuous phase, it is expedient to calculate the mean Sherwood number by using formula (4.7.3), where Shp and Shb are the Sherwood numbers for the limit cases of a solid particle and a bubble. These quantities can be calculated by formulas (4.7.9) and (4.7.10). For high Peclet numbers, in the entire range of phase viscosities, the mean Sherwood number can be found by solving the cubic equation [359]... 

The mean Sherwood number for spherical solid particles, drops, and bubbles in a linear straining shear flow (Gkm = 0 for k m) at low Reynolds numbers and high Peclet numbers... 

We assume that the fluid velocity distribution remote from the interface is given by Eq. (4.5.1). The mean Sherwood number for a spherical particle, drop, or bubble does not change if we change all signs of the shear coefficients, that is, Sh(G m) = Sh(-G m). 

Table 4.8 presents a comparison of the mean Sherwood numbers calculated according to Eq. (4.12.3) with available data for various flows past spherical drops, bubbles, and solid particles at high Peclet numbers (in this table, we use the abbreviation DBLA for diffusion boundary layer approximation ). 

For the Stokes motion of spherical drops and bubbles, the cell Happel model (see Section 2.9) results in the following expression for the mean Sherwood number [503] ... 

Figure 5.1. The Sherwood number against the rate constant of second-order surface chemical reaction 1, by formula (5.1.5) 2, for a solid sphere 3, for a circular cylinder and 4, for a spherical drop or bubble...

In the case of a spherical bubble in a translational flow of a viscoplastic fluid with low Bingham numbers, one can use the following formula for the mean Sherwood number [37] ... 

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,... 

Both Hirose and Moo-Young (1969) and Bhavaraju et al. (1978) have obtained closed-form expressions for mass transfer from single bubbles rising slowly (Re 1) through power-law liquids. By analogy with the drag behavior, it is customary to express the results in terms of the deviation from the Newtonian result. It is, thus, instructive to recall that the Sherwood number for a spherical bubble with clean surface in a Newtonian liquid is given as (Levich, 1962)... 

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