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Mass Or Heat Transfer Inside a Spherical Gas Bubble at Low to Moderate Reynolds Number Damien Colombet, Dominique Legendre, Arnaud Cockx, Pascal Guiraud

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Mass Or Heat Transfer Inside a Spherical Gas Bubble at Low to Moderate Reynolds Number Damien Colombet, Dominique Legendre, Arnaud Cockx, Pascal Guiraud Mass or heat transfer inside a spherical gas bubble at low to moderate Reynolds number Damien Colombet, Dominique Legendre, Arnaud Cockx, Pascal Guiraud To cite this version: Damien Colombet, Dominique Legendre, Arnaud Cockx, Pascal Guiraud. Mass or heat transfer inside a spherical gas bubble at low to moderate Reynolds number. International Journal of Heat and Mass Transfer, Elsevier, 2013, vol. 67, pp.1096-1105. 10.1016/j.ijheatmasstransfer.2013.08.069. hal-00919752 HAL Id: hal-00919752 https://hal.archives-ouvertes.fr/hal-00919752 Submitted on 17 Dec 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Open Archive TOULOUSE Archive Ouverte (OATAO) OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited version published in : http://oatao.univ-toulouse.fr/ Eprints ID : 10540 To link to this article : DOI:10.1016/j.ijheatmasstransfer.2013.08.069 URL : http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.08.069 To cite this version : Colombet, Damien and Legendre, Dominique and Cockx, Arnaud and Guiraud, Pascal Mass or heat transfer inside a spherical gas bubble at low to moderate Reynolds number. (2013) International Journal of Heat and Mass Transfer, vol. 67 . pp. 1096-1105. ISSN 0017-9310 Any correspondance concerning this service should be sent to the repository administrator: [email protected] Mass or heat transfer inside a spherical gas bubble at low to moderate Reynolds number ⇑ D. Colombet a,b,c,d,e,f, D. Legendre d,e, , A. Cockx a,b,c, P. Guiraud a,b,c a Université de Toulouse, INSA, UPS, INP, LISBP, 135 Avenue de Rangueil, F-31077 Toulouse, France b INRA, UMRA792 Ingénierie des Systémes Biologiques et des Procédés, F-31400 Toulouse, France c CNRS, UMR5504, F-31400 Toulouse, France d Université de Toulouse, INPT, UPS, IMFT (Institut de Mécanique des Fluides de Toulouse), Allée Camille Soula, F-31400 Toulouse, France e CNRS, IMFT, F-31400 Toulouse, France f Rhodia – Solvay, 85, Avenue des Fréres Perret, BP 62, 69192 Saint Fons, France abstract Mass (or heat) transfer inside a spherical gas bubble rising through a stationary liquid is investigated by direct numerical simulation. Simulations were carried out for bubble Reynolds number ranging from 0.1 to 100 and for Péclet numbers ranging from 1 to 2000. The study focuses on the effect of the bubble Rey- nolds number on both the interfacial transfer and the saturation time of the concentration inside the bub- ble. We show that the maximum velocity Umax at the bubble interface is the pertinent velocity to describe both internal and external transfers. The corresponding Sherwood (or Nusselt) numbers and the satura- Keywords: tion time can be described by a sigmoid function depending on the Péclet number Pe = U d /D (d Bubble max max b b and D being the bubble diameter and the corresponding diffusion coefficient). Fluid mechanics Heat/mass transfer Numerical simulation 1. Introduction side Sherwood number Sh has been obtained for a fixed concentra- tion cs at the surface and an initial uniform concentration c0 inside In a great number of processes such as chemical engineering or the sphere. The corresponding Sherwood number tends to an water treatment, bubbly flows are used for mass transfer. In some asymptotic constant value Sh = Sh(t ? ): 1 1 situations the resistance of the transfer resides in the gas phase. 2p2 For physical absorption or desorption of very soluble gases, as pre- Sh Re 0; Pe 0 6:58 1 dicted by the Lewis–Whitman two-film model [1], the mass trans- 1ð ¼ ¼ Þ¼ 3 ð Þ fer liquid-phase resistance can be negligible and mass transfer is This solution can be used for a bubble fixed relative to the sur- then controlled by the gas phase resistance. A typical example is rounding liquid i.e. for a bubble Reynolds number Re = qLUbdb/ ammonia removal from water where mass transfer is limited by lL = 0 and a Péclet number Pe = Ubdb/D = 0 where db is the bubble the solute concentration transport inside bubbles. For gas absorp- diameter, Ub is the bubble relative velocity, qL is the liquid density, tion followed by an extremely fast chemical reaction in the liquid lL its dynamic viscosity and D is the diffusion coefficient. In prac- phase, mass transfer can also be controlled by gas-side transfer tice, this solution is expected to be valid in the limits Re ? 0 and resistance [2,3]. A typical example is sulfur dioxide absorption into Pe ? 0. alkali solutions. In such cases, the estimation of the mass transfer Using the Stokes (or creeping flow) solution for the description requires the knowledge of the gas-side transfer. A large amount of the flow inside the bubble [9,10], Kronig and Brink [11] obtained of studies have considered the external mass transfer [4–7] but less numerically the instantaneous Sherwood number in the limit of attention has been paid to the internal mass transfer. high Péclet number. The corresponding asymptotic value of the Newman [8] has derived the analytical solution of the mass Sherwood number Sh , valid in the limits Re ? 0 and Pe ? ,is transfer controlled by pure diffusion inside a sphere. From the 1 1 also found to be constant: instantaneous concentration profile, the time evolution of the in- Sh Re 0; Pe 17:90 2 1ð ! !1Þ ð Þ ⇑ Corresponding author at: Université de Toulouse, INPT, UPS, IMFT (Institut de As pointed out by Clift et al. [4], the solution (2) is very close to the Mécanique des Fluides de Toulouse), Allée Camille Soula, F-31400 Toulouse, France. solution (1) when considering an effective diffusion coefficient E-mail address: [email protected] (D. Legendre). Deff = 2.5D. The experiments of Calderbank and Korchinski [12] on List of symbols 3 c mass concentration, kg mÀ tsat saturation time, s 3 2 cs saturation mass concentration, kg mÀ t0sat dimensionless saturation time, tsat0 tsatD=rb 3 ¼ c0 initial mass concentration, kg mÀ tsatà normalized saturation time 1 c0 dimensionless mass concentration, c0 =(c c0)/(cs c0) uk velocity of phase k,msÀ À À 3 1 c volume average concentration in the gas bubble, kg mÀ Umax maximal velocity at the bubble interface, m sÀ h i 1 CD bubble drag coefficient Ub bubble rising velocity, m sÀ 1 db bubble diameter, m uh local liquid velocity at the bubble surface, m sÀ 2 1 D molecular diffusion coefficient, m sÀ F dimensionless mass concentration, F = c /(cs c0) Greek symbols 2 1 h i À J surface average mass flux, kg mÀ sÀ dD concentration boundary layer thickness, m 2 1 J local mass flux, kg mÀ sÀ loc Din internal grid size at the interface, m n unit vector normal to the surface under consideration Dout external grid size at the interface, m p pressure, Pa lk dynamic viscosity of phase k, Pas Pe Péclet number, Pe = U d /D ⁄ ⁄ b b l dynamic viscosity ratio, l = lG/lL Pe effective Péclet number 2 1 eff mk kinematic viscosity of phase k,m sÀ Pe Péclet number based on U , Pe = U d /D 3 max max max max b qk density of phase k,kgmÀ r radial coordinate ⁄ ⁄ q density ratio, q = qG/qL r0 dimensionless radial coordinate, r0 = r/rb s viscous shear stress, Pa rb bubble radius rb = db/2, m R computational domain radius, m 1 Superscripts Re bubble Reynolds number, Re = qLUbdb/lL ext external transfer Sc Schmidt number, Sc = mG/D Sh instantaneous Sherwood number Sh asymptotic Sherwood number, Sh = Sh(t ? ) Subscripts 1 1 1 G gas phase Shà normalized asymptotic Sherwood number t 1 time, s k phase k 2 L liquid phase t0 dimensionless time (Fourier number), t0 tD=r ¼ b t unit vector tangential to the bubble surface heat transfer inside bromobenzene circulating drops falling in Pe 6 104. The simulations confirm the increase of the Sherwood water–glycerol solutions also show an agreement with relation number with the increase of the bubble Reynolds number. The (2) with a measured effective diffusion coefficient Deff = 2.25D. study also shows that the scaling proposed by Oliver and De Witt Using the Hadamard–Rybczynski solution for the internal flow [13] (Eq. (3)) is only adapted for the description of the transfer and numerical simulations for the transport of the mass concentra- for small to moderate effective Péclet number, i.e. Peeff 6 200. tion, Clift et al. [4] have shown that the increase of the Péclet num- In order to determine the transfer in the limit of high Reynolds ber produces a gradual increase of the asymptotic Sherwood numbers, Zaritzky and Calvelo [16] have used the internal poten- number Sh , from the value Sh 6.58 given by (1) up to a value tial flow solution [17] for the flow inside the bubble in order to 1 1 close to Sh 17.90 given by (2). solve by numerical simulation the concentration field. Surprisingly, 1 Few studies have considered the internal transfer at moderate Sh is close to the value 17.90 given by (2) in the limit of small 1 Reynolds number [13,14].
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