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Mass-Transfer Enhancement by a Reversible Chemical Reaction Across the Interface of a Bubble Rising Under Stokes Flow

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Mass-Transfer Enhancement by a Reversible Chemical Reaction Across the Interface of a Bubble Rising Under Stokes Flow CORE Metadata, citation and similar papers at core.ac.uk Provided by Open Archive Toulouse Archive Ouverte 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 : 15691 To link to this article : DOI:10.1002/aic.14520 URL : http://dx.doi.org/10.1002/aic.14520 To cite this version : Pigeonneau, Franck and Perrodin, Marion and Climent, Eric Mass-Transfer Enhancement by a Reversible Chemical Reaction Across the Interface of a Bubble Rising Under Stokes Flow. (2014) AIChE Journal, vol. 60 (n° 9). pp. 3376-3388. ISSN 0001-1541 Any correspondence concerning this service should be sent to the repository administrator: [email protected] Mass-Transfer Enhancement by a Reversible Chemical Reaction Across the Interface of a Bubble Rising Under Stokes Flow Franck Pigeonneau Surface du Verre et Interfaces, UMR 125 CNRS/Saint-Gobain, 39 Quai Lucien Lefranc, BP 135, 93303 Aubervilliers Cedex, France Marion Perrodin Surface du Verre et Interfaces, UMR 125 CNRS/Saint-Gobain, 39 Quai Lucien Lefranc, BP 135, 93303 Aubervilliers Cedex, France Institut de Mecanique des Fluides de Toulouse, Universite de Toulouse, CNRS UMR 5502, 2 Allee du Professeur Camille Soula, 31400 Toulouse, France Eric Climent Institut de Mecanique des Fluides de Toulouse, Universite de Toulouse, CNRS UMR 5502, 2 Allee du Professeur Camille Soula, 31400 Toulouse, France Mass transfer around a bubble rising in a liquid under Stokes regime is investigated when a reversible chemical reac- tion, A !B, is taken into account. Four dimensionless parameters control the interfacial transfer rate: the Peclet and Damkohler€ numbers, the ratio of the diffusion coefficient of both species, and the reaction equilibrium constant. The mass-transfer equations are solved numerically with a finite element technique. A boundary layer approach is also pro- posed and solved with a coupled technique of finite difference and Chebyshev-spectral method. The equilibrium constant and the ratio of diffusion coefficients have a strong influence on the coupling between the chemical reaction and mass transfer leading to an increase of the Sherwood number. The interaction between the chemical reaction and advection is clearly established by the simulations. Conditions corresponding to Peclet number larger than the Damkohler€ number reduces the effect of the chemical reaction. Keywords: Sherwood number, chemical reaction, Stokes flow, boundary layer theory, bubble Introduction a comprehensive understanding of the transport phenomena Many chemical engineering processes are based on the between the two phases close to interfaces before investigat- absorption or desorption of gaseous material into a liquid ing the industrial process at large scale. phase. During glass melting, bubbles are created due to Mass transfer in two-phase flows occurs often coupled with chemical reactions between raw materials and undergo, fur- chemical reactions3 in many industrial applications. The predic- ther, mass transfer due to fining process enhancing the rate tion of interfacial transfer coefficient due to simultaneous diffu- of bubbles removal.1 Among many other industrial manufac- sion and chemical reaction is usually based on the assumption turing processes, the treatment of molten steel by blowing that the resistance to diffusion is localized in a thin film adjacent gas in ladles2 is concerned with mass transfer between dis- to the gas–liquid interface.4,5 The thickness of this thin film is persed bubbles and molten steel. For all processes, the per- then related to the effect of the kinetics of chemical reactions. formance of conversion depends on the mass-transfer In such case, it is generally observed that the chemical reactions coefficient between the dispersed and continuous phases and enhance mass transfer6 when the reactant transfers through the the interfacial specific area. This multiscale problem requires interface. Higbie7 and later Danckwerts8 developed alternative models under the assumption of continual renewal of fluid at the interface. Olander9 studied the effect on mass transfer of Current address of Marion Perrodin: Saint-Gobain Recherche, 39 Quai Lucien various first- and second-order reversible chemical reactions in Lefranc, BP 135, 93303 Aubervilliers Cedex, France one-dimensional problems. He compared carefully the two the- Correspondence concerning this article should be addressed to F. Pigeonneau at oriesproposedinRefs.7and8. [email protected]. From the first contributions devoted to mass transfer with chemical reactions around a bubble or a drop moving in a liquid, Ruckenstein et al.10 determined the mass-transfer physical data in molten glass, the book of Scholze.21 Conse- coefficient for a first-order irreversible reaction both for quently, the Peclet number, product of the Reynolds number creeping and potential flows. Soung and Sears11 investigated by the Schmidt number is quite large (106) meaning that the effect of the irreversible chemical reactions in unsteady even if the flow motion in molten glass is in creeping regime. Kleinman and Reed12 determined the mass transfer regime, mass transfer is mainly driven by advection. In this for a quiescent spherical inclusion with a first-order irreversi- article, the following issues are addressed: what is the effect ble reaction. They focused their investigation on the effect of coupling phenomena such as advection, diffusion, and of reaction kinetics on the enhancement of mass transfer. chemical reaction around a single rising bubble? What is the These authors studied in Ref. 13 the situation of a moving evolution of the enhancement factor when advection is more drop, bubble or particle experiencing irreversible first-order and more important? More specifically, we want to evaluate chemical reaction. More recently, Juncu14,15 determined the the interplay between those phenomena on the resulting mass transfer around drop or bubble over a limited range of boundary layer thickness (theoretical analysis leading to scal- Peclet and Damkohler€ numbers for first- or second-order ing laws). By varying the equilibrium constant different irreversible reactions. When the Reynolds number based on chemical regimes can be investigated where species are pref- the relative bubble velocity is larger than 10, an accurate erentially consumed or produced. description of the two-phase flow is required. In the article,16 As it is impossible to embrace the study of mass transfer Koynov et al. solved the two-phase flow problem with a coupled with chemical reactions completely, this work is front tracking interface method to follow the bubble/liquid limited to the situation of one reversible chemical reaction interface coupled with mass transfer coupled to two irrevers- between two species, that is, A !B. The effect of the relative ible reactions. The authors focused their work on the effect diffusion of both species is also addressed. The main purpose of bubble swarms and interaction between few bubbles. is to raise general results as a first step toward more complex Wylock et al.17 gave an important contribution in the field of situations. mass transfer coupled with chemical reactions by determin- To investigate the coupling between transport phenomena ing the mass-transfer coefficients where four species and two and chemical reaction over large range of Peclet and Dam- reversible reactions are involved. They achieved a comple- kohler€ numbers, two numerical approaches are used. Over a mentary work taking into account the interface contamina- large range of Peclet number, a finite element technique is tion and the bubble shape in Ref. 18. Using volume of fluid used to solve directly the general problem for low to moder- method, Bothe et al.19 determined the mass transfer for an ate reaction kinetics. Nevertheless, when the chemical irreversible chemical reaction of sulfite oxidation. The kinetics becomes very fast meaning that the Damkohler€ numerical simulations have been compared to experimental number is large or when the Peclet number is larger than 6 results using laser-induced fluorescence measurement tech- 10 , it is difficult to enforce accuracy on spatial resolution nique. The determination of the mass-transfer coefficient for without a dramatic increase of computing time. Moreover, in instantaneous reaction has been done by Pigeonneau20 for a these two limits, difficulties appear in term of numerical sta- particular application devoted to the oxidation-reduction of bility. To investigate properly the cases of large Damkohler€ iron in molten glass. and Peclet numbers, a coupled spectral-Chebyshev/finite dif- Despite these contributions, a large overview of the evolu- ference method to solve the boundary layer formulation in tion of mass-transfer coefficients for large variation of Peclet the limit of high Peclet number has been developed for and Damkohler€ numbers is still lacking. Numerical modeling which the chemical kinetics is studied over a large range. It of bubbly reactors needs closure laws for mass-transfer coef- is then possible to obtain mass-transfer coefficients when the ficient (corresponding in dimensionless for to Sherwood chemical reactions become instantaneous. We can compare number as a function of the flow properties, i.e., Peclet, the simulation results with asymptotic solutions validating Damkohler,€ and Schmidt numbers).
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