Mass or heat transfer from spheroidal gas bubbles rising through a stationary liquid
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 aswhere is the aspect ratio ( in our study) and Peeq is the Peclet number based on the equivalent diameter. 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):
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 , 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:
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).Since Moore's derivation provides a very good prediction for the drag for , 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 , .
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)
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 (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 and kinematic viscosity is therefore moving at velocity at a given position . The dimensionless variables are chosen in terms of the reference values:
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 . According to Magnaudet and Mougin (2007), the onset of the recirculation zone corresponds to the critical values 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 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
Ab bubble surface area a minor semi-axis of the ellipsoid b major semi-axis of the ellipsoid c dimensionless concentration (or temperature) of a particular chemical species in the mixture CD drag coefficient CL lift coefficient chemical species concentration far from the interface or bulk concentration chemical species concentration at the interface D diffusion coefficient deq=2req equivalent diameter unit vector along the x-direction unit vector along the y-direction correction factor
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.
References (37)
Forced convection heat and mass transfer at small Peclet numbers from a particle of arbitrary shape
Chem. Eng. Sci.
(1963)Mass transfer around oblate spheroidal drops at low Reynolds numbers
Chem. Eng. Sci.
(2010)- et al.
Unsteady mass transfer around spheroidal drops in potential flow
Chem. Eng. Sci.
(2005) - et al.
Oxygen transfer prediction in aeration tank using CFD
Chem. Eng. Sci.
(2007) - et al.
Effet de l’accélération d’un écoulement sur le transfert thermique ou massique à la surface d’une bulle sphérique
C. R. ACad. Sci. Paris Srie IIb
(1999) - et al.
Mass transfer in the continuous phase around axisymmetric bodies of revolution
Chem. Eng. Sci.
(1964) - et al.
Mass transfer from very small bubbles—the optimum bubble size for aeration
Chem. Eng. Sci.
(1978) - et al.
Correction of the penetration theory based on mass-transfer data from bubble columns operated in the homogeneous regime under high pressure
Chem. Eng. Sci.
(2007) - et al.
Theoretische berechnung des stofftransports in der umgebung einer einzelblase
Chem. Eng. Sci.
(1973) - et al.
DNS-based prediction of the selectivity of fast multiphase reactions: hydrogenation of nitroarenes
Chem. Eng. Sci.
(2008)
On heat transfer between vapour bubbles in motion and the boiling liquid from which they are generated
Chem. Eng. Sci.
Bubble wake visualization by using photochromic dye
Chem. Eng. Sci.
Gas dissolution process of spherical rising bubbles
Chem. Eng. Sci.
Letters to the editor
Chem. Eng. Sci.
Reversal of the lift force on an oblate bubble in a weakly viscous linear shear flow
J. Fluid Mech.
The structure of the axisymetric high-Reynolds number flow around an ellipsoidal bubble of fixed shape
Phys. Fluids
Calcul du pouvoir refroidissant des courants fluides
J. Math. Pure Appl.
Large-eddy simulation of high-Schmidt number mass transfer in a turbulent channel flow
Phys. Fluids
Cited by (57)
Sparging-based fission gas separation technology for molten salt reactor
2023, Annals of Nuclear EnergyA mathematical model for single CO<inf>2</inf> bubble motion with mass transfer and surfactant adsorption/desorption in stagnant surfactant solutions
2023, Separation and Purification TechnologyCitation Excerpt :There is a significant body of studies on the dynamics and mass transfer of single bubbles in viscous liquids. These studies mainly focus on the bubble rising velocity [12–25], bubble shape and trajectory [12,15,16,18,21,24,26–31], wake structure and dynamics [21,31–35], and mass transfer [36–55]. It has shown that the mass transfer coefficient is mainly affected by the bubble size, bubble surface properties (surface mobility), hydrodynamic conditions at the bubble surface, and physical properties of the liquid [56].
Direct numerical simulation of gas-liquid mass transfer around a spherical contaminated bubble in the stagnant-cap regime
2022, International Journal of Heat and Mass TransferOxygen transfer of microbubble clouds in aqueous solutions – Application to wastewater
2022, Chemical Engineering Science