Elsevier

Chemical Engineering Science

Volume 65, Issue 23, 1 December 2010, Pages 6296-6309
Chemical Engineering Science

Mass or heat transfer from spheroidal gas bubbles rising through a stationary liquid

https://doi.org/10.1016/j.ces.2010.09.018Get rights and content

Abstract

Mass transfer was studied for the case of a spheroidal bubble rising through a stationary liquid. A numerical code that solves the Navier–Stokes equations and the diffusion–advection equation for the concentration was used to characterize the transfer from the bubble to the surrounding liquid phase. Simulations were carried over systematically for Reynolds number ranging from 1 to 1000, Schmidt numbers from 1 to 500 and bubble aspect ratio from 1 to 3. It appears that the use of the equivalent diameter as the characteristic length is the more appropriate to describe the transfer. The effect of bubble aspect ratio on the Sherwood number has been analyzed. At first order the extension of Boussinesq expression using the equivalent diameter can be used for practical purposes. The evolution of the correction factor that compares the Sherwood number to the one of a sphere with same equivalent Peclet number is presented and described using simple correlations. The implementation of these results into Euler–Euler simulations of mass transfer is discussed. It appears that the modification of the interfacial area combined to the modification of the Sherwood number gives a significant contribution to the interfacial source term in the equation of the concentration. Note that the results can also be considered for heat transfer and used for inviscid drops.

Introduction

Mass and heat transfers are important from the chemical, environmental and industrial points of view. A great variety of processes such as fermentation or aerobic digestion imply the use of chemical and biological reactors, stirred vessels, bubble columns, etc. This work was motivated by a project related to mass transfer in waste water treatment plants, where biological reactors are aerated through the injection of small bubbles. Oxygen in gaseous phase coming from the bubbles is transferred through the gas–liquid interface, allowing for the aerobic respiration of micro-organisms present in the liquid phase. The modeling of such facilities involves biological, chemical and physical subsystems and their interaction. In general, local and large scale hydrodynamic phenomena have to be studied separately given the impossibility of describing the physics for each bubble in a reactor. Not even the most powerful and modern computer could deal with the direct numerical simulation of a few number of bubbles rising in a (very) small aeration vessel (Koynov et al., 2005). A more practical viewpoint for numerical simulations is to model large scale physics separately, based on ‘subgrid’ local models that feed the proper local behavior of the flow in terms of measurable quantities such as the gas/liquid volume fraction and mean bubble size. However, for these submodels to be precise and reliable, bubble dynamics, mass/heat transfer have to be studied at the scale of a single inclusion.

Transfer (of mass, heat, momentum) depends on both the inclusion geometry (area, shape) and the flow hydrodynamics. From the point of view of the latter, this interdependence between hydrodynamics and topology can be understood in terms of the forces acting on the bubble: inertia, surface tension, buoyancy and viscous stress. With these elements, three dimensionless groups can be defined, for example the Reynolds, the Eötvos and the Morton number. Useful correlations of the bubble shape and velocities in terms of these quantities can be found in the work of Clift et al. (1978) as a function of the aforementioned dimensionless groups (see also Magnaudet and Eames, 2000 for recent progress on spherical bubbles hydrodynamics). Bubble shape for moderate deformation can be approximated as oblate spheroids. For significant deformation (typically for an aspect ratio larger than 2.5 for air/water system) bubbles shape loses fore-and-aft-symmetry (Sanada et al., 2007, Zenit and Magnaudet, 2008).

The regime of flow that concerns us is characterized by oblate ellipsoidal bubbles that rise following vertical rectilinear paths. The rise velocity of such bubbles has been studied theoretically by Moore (1965), who calculated the bubble aspect ratio and drag force of slightly oblate ellipsoidal bubbles by solving the boundary layer between the bubble surface and the external (potential) flow. The effect of oblateness on the drag coefficient has also been recently revisited by Legendre (2007), who obtained the drag of a bubble from a simple correlation between the drag and the maximum vorticity produced on the bubble surface. The characteristic trajectories of these deformed rising bubbles was recently studied in detail by Ellingsen and Risso (2001), Sanada et al. (2007), Magnaudet and Mougin (2007), and Zenit and Magnaudet (2008).

Within the context of mass/heat transfer, the case of spheroidal bubbles has been considered theoretically by Lochiel and Calderbank (1964), who developed analytical expressions for the transfer coefficients (Sherwood number) of solid and fluid spheres and spheroids. These results are valid for the limiting cases of very small or very large Reynolds numbers.1 A general expression for the Sherwood number of an axisymmetric body (of arbitrary shape) was obtained. For the case of large Reynolds number, the transfer was derived using velocities coming from the potential flow. The resulting expression for the Sherwood number of an oblate spheroid (of major and minor semi-axis b and a) can be expressed asSheq(χ)=2πPeeq1/2f(χ)where χ=b/a is the aspect ratio (χ1 in our study) and Peeq is the Peclet number based on the equivalent diameter. f(χ) is the correction to the Sherwood number SheqB of a spherical bubble of same Peclet number Peeq given by the potential flow solution (Boussinesq, 1905):SheqB=2πPeeq1/2

The results were compared with some previous experimental investigations, showing differences that can reach as much as 20% (Lochiel and Calderbank, 1964). Due to the potential flow assumption, this correlation (Eq. (1)) is valid for the limit Reeq, corresponding to asymptotic vanishing of the vorticity boundary layer (Moore, 1963, Moore, 1965). The effect of the wake recirculation was not considered in this particular case. This aspect will be investigated in Section 5.1. More recently, the solution was extended to oblate and prolate spheroids (Favelukis and Ly, 2005) and oblate spheroids at low Reynolds number (Favelukis, 2010).

For moderate Reynolds number and high Schmidt numbers, Takemura and Yabe (1998) determined the oxygen concentration of millimetric (almost spherical) gas bubbles by means of precise partial pressure and (time-varying) diameter measurements. They compared previous expressions from the literature (Clift et al., 1978, Oellrich et al., 1973, Leclair and Hamielec, 1971, Winnikow, 1967) and proposed a semi-empirical relation that gives a better fit of their experiments and numerical simulations:SheqTY=2π1231(1+0.09Reeq2/3)3/41/2(2.5+Peeq1/2)

Among the expressions from the literature, we retain the analytical solution obtained by Winnikow (1967) which is based on the tangential velocity derived by Moore (1963).SheqW=2π12.89Reeq1/2Peeq1/2Since Moore's derivation provides a very good prediction for the drag for Reeq>50, its use for deriving the transfer is also expected to give a good correction to the Boussinesq solution. This will be discussed in the following. Note that Eq. (4) can be transformed in order to eliminate the divergence of the solution in the limit of small Reynolds numbers by considering that for Reeq1, (12.89Reeq1/2)1/2(1+2.89Reeq1/2)1/2.

In the opposite limit of small Reeq and Peeq numbers, the evolution of the Sherwood number of a spherical bubble is given by (Brenner, 1963)SheqBre=2+12Peeq

Theoretical and numerical studies have also been conducted to study transient transfer, transfer in unsteady and non uniform flows (Legendre and Magnaudet, 1999). A broad bibliographical review for momentum and mass/heat transfer with and without phase transition can be found in Michaelides (2003). As indicated before most of these studies have focused on spherical bubbles.

The objective of this work is to analyze the effect of bubble aspect ratio on the transfer, in order to compare it with previous investigations and to obtain useful correlations in terms of the pertinent parameters. For this purpose, direct numerical simulations (DNS) solving the Navier–Stokes equations and the diffusion–advection equation for the concentration are performed on orthogonal boundary-fitted grids.

Section snippets

Governing equations

Let us consider an oblate spheroidal bubble moving through a liquid at a constant velocity U (Fig. 1). It is assumed that the bubble has reached a stationary state (from a reference frame moving with the bubble), so its shape does not change with time. The liquid of constant density ρ and kinematic viscosity ν is therefore moving at velocity u at a given position x. The dimensionless variables are chosen in terms of the reference values:t=tUL,u=uU,x=xL,p=pρU2andc=(cc)(csc

Numerical procedure

The system of Eqs. (7), (8), (9) was solved via direct numerical simulations using the JADIM code. This code can solve the unsteady 3D Navier–Stokes equations in terms of velocity–pressure variables and the advection–diffusion for the concentration (or temperature), for any orthogonal curvilinear coordinate system. The discretization method is finite volumes, which is well adapted to properties conservation. Precision is second order in time and space (Runge–Kutta/Crank–Nicolson schemes) and

Bubble wake

There is an important specific aspect concerning oblate spheroids that may have an effect on the transfer. In contrast with spherical bubbles, the wake is characterized by the appearance of a recirculation zone behind the bubble. This transition takes place at large Reynolds numbers and aspect ratios χ. According to Magnaudet and Mougin (2007), the onset of the recirculation zone corresponds to the critical values χc=1.6 and Rec=130. The limits of appearance of this recirculation is, as a

Results

Simulations were carried over systematically for Reynolds number Re ranging from 1 to 1000, Schmidt numbers Sc from 1 to 500 and bubble shape χ from 1 to 3. Section 5.1 first presents the results concerning the local distribution of the transfer at the bubble surface. The mean Sherwood number is then given in Section 5.2 and a simple correlation that fits the numerical results is proposed. Finally, some unsteady effects are discussed in Section 5.3. Simulations reported in 5.1 Local transfer,

Discussion

The results discussed above are presented using the Sherwood number. It is the normalized total mass transferred at the bubble surface. Even if Sh could be seen as the mean transfer at the bubble surface, it does not really express the mass or heat exchange potential of a given bubble since surface of exchange increases as the bubble deforms.

A direct application of the results presented above is their use to describe mass transfer in bubbly situations. The effect of bubble deformation for model

Conclusion

The effect of bubble deformation on mass or heat transfer was estimated using numerical simulations of single axisymmetric (oblate) spheroidal bubbles. Our results are consistent with previous investigations from the hydrodynamical and mass/heat transfer viewpoints. Available theoretical and experimental results for the mean and local Sherwood (or Nusselt) numbers were compared to our simulations for spherical bubbles.

The local Sherwood numbers of deformed bubbles showed significant differences

Nomenclature

Abbubble surface area
aminor semi-axis of the ellipsoid
bmajor semi-axis of the ellipsoid
cdimensionless concentration (or temperature) of a particular chemical species in the mixture
CDdrag coefficient
CLlift coefficient
cchemical species concentration far from the interface or bulk concentration
cschemical species concentration at the interface
Ddiffusion coefficient
deq=2reqequivalent diameter
exunit vector along the x-direction
eyunit vector along the y-direction
f(χ)correction factor Sheq(χ)/Sheq(

Acknowledgements

This work was granted the support of the National Research Agency of France (Agence Nationale de la Recherche), Project O2STAR, Reference: ANR-07-ECOT-007-01. D.L. would like to thank A. Cockx and colleagues from the CNRS Federation FERMaT, the LISBP and the Cemagref d’Antony for making this collaboration possible. B.F. thanks Annaig Pedrono and the IMFT personnel for their help with the code and their suggestions during his stay in Toulouse.

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