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 reaction, 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 proposed 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). Moreover, it is admitted the numerical method. that chemical reactions yield a boundary layer whose thick- The article is organized as follows. First, the problem ness is scaled by the characteristic scale of the chemical statement is presented in the second section in which the kinetics. Advection leads also to the formation of boundary general and the boundary layer formulations are given fol- layer when the diffusion coefficient is sufficiently small. The lowed by the numerical results and discussion. After the con- interaction between these two boundary layers plays an clusion, two appendices are provided to detail the numerical important role on the determination of mass-transfer coeffi- methods and exact solutions for instantaneous chemical cient in chemical processes. It is inherent to all situations in reaction. which chemical reactions are involved. The main purpose of our work is to provide a comprehensive understanding of the Problem Statement interplay between diffusion, advection, and chemical reaction Transport equations for the species during mass transfer across the bubble interface. More pre- cisely, we focus on the determination of the enhancement We consider a spherical bubble, rising in a liquid at rest. factor due to the chemical reaction on the rate of mass Its radius, a, is assumed constant even in the presence of transfer. mass transfer. This assumption has been validated by For glass melting applications, the hydrodynamics is gen- Pigeonneau et al.22 where it is shown that the time scale of erally related to creeping flows, that is, small Reynolds num- the bubble radius variation is typically three orders of mag- ber. Indeed, the smallest dynamic viscosity in glass furnace nitude lower than the characteristic time of mass transfer. is around 10 Pa s, while the typical bubble size is smaller The interface between the bubble and the liquid is assumed than 1 cm giving a Reynolds number much smaller than 1. completely mobile (shear free boundary condition). In the The diffusion coefficients of gas species in molten glass are industrial context of glass melting, the last assumption has very small (around 10210 m2 s21 at 1400C) leading to a been verified experimentally by Jucha et al.,23 Hornyak and Schmidt number equal to 107, see for more details on the Weinberg,24 and Li and Scheider.25 We assume the Reynolds number to be small which corresponds to small bubbles ris- Dimensionless formulation ing in a viscous liquid (see Clift et al.26 for practical condi- The numerical solution is achieved for a set of dimension- tions of spherical bubble and creeping flow to be matched less equations. Only four dimensionless numbers are simultaneously). Consequently, the general solution provided involved in the normalized mass balance equations. The by Hadamard27 or Rybczynski28 is used to describe the flow characteristic length scale is the bubble diameter, 2a. The motion around the bubble. The balance between the drag velocity field is scaled by V . The dimensionless molar con- and the buoyancy forces gives the terminal velocity t centrations of A and B are written as follows 2 ga C 2C1 C 2K C1 Vt5 ; (1) 5 A A ; 5 Beq A 3m CA S 1 CB S 1 (8) CA2CA Keq CA2CA where g is the gravity acceleration, m 5 l=q is the kinematic Beside the equilibrium constant, already defined in Eq. (3), viscosity of the liquid and q the liquid density. the three other dimensionless numbers are We consider two chemical species, A, the reactant, and B, 2 1 the product, consumed and produced by a homogeneous 2aVt 4a k DB reversible chemical reaction in the liquid PeA5 ; Da5 ; D5 (9) DA DA DA ! A B (2) The first is the Peclet number based on the diffusion coef- with the equilibrium constant equal to ficient of A. The second dimensionless number is the Dam- kohler€ number written as the ratio of diffusion time of CBeq Species A to chemical reaction time of Eq. (2). Finally, the Keq 5 (3) CAeq last dimensionless number is the ratio of the two diffusion coefficients. The quantities CA and CB are the molar concentrations of Without the bar over the dimensionless variables, the the Solutes A and B in the liquid, respectively. dimensionless mass balance equations become Due to symmetry, cylindrical polar coordinates, ðr; u; zÞ, originating from the bubble center are used for which the u- @CA @CA @CA 1 1ur 1uz 5 component of the velocity and all derivatives with respect to @t @r@z PeA 2 (10) this coordinate are equal to zero. The general equations for 1 @ @CA @ CA Da 3 r 1 2 2 ðÞCA2CB the transport of CA and CB in the liquid experiencing advec- r @r @r @z PeA tion, diffusion, and chemical reaction are @C @C @C D B 1u B 1u B 5 @CA @CA @CA @t r @r z @z Pe 1ur 1uz 5DA A (11) @t @r@z 1 @ @C @2C Da 2 (4) 3 r B 1 B 1 C 2C 1 @ @CA @ CA 1 CB 2 ðÞA B r @r @r @z Keq PeA 3 r 1 2 2k CA2 r @r @r @z Keq The boundary conditions write @CB @CB @CB 1ur 1uz 5DB 5 ; C ; 5 ; x @t @r @z CA 1 on b CA 0 when !1 (12) 2 (5) 1 @ @CB @ CB 1 CB @CB 3 r 1 2 1k CA2 50; on Cb; CB50; when x !1 (13) r @r @r @z Keq @n where t is the time, DA and DB are the diffusion coefficients The dimensionless expressions for ur and uz, that is, veloc- 1 of A and B, respectively. The quantity k is the kinetic con- ity components divided by Vt,are stant of reaction (2) toward production of B. The two veloc- zr r212z2 ity components of the steady flow, u and u , are determined ur5 ; uz5211 (14) r z 2 2 3=2 2 2 3=2 according to Refs. 27 and 28 and will be given latter. 4ðÞz 1r 4ðÞz 1r Remark that the problem is written under unsteady formula- When steady state is reached, the solution will be used to tion although we will analyze only steady state while tran- determine the Sherwood number over the bubble interface sient effects have a negligible role on processes running 24 for Solute A by integration of the normal gradient of CA h a day. over the bubble interface area as follows To solve the coupled equations, boundary conditions are ð 1 @C required. The bubble-liquid interface, Cb (see Figure A1), is A ShA52 dS (15) p @n assumed permeable to A and impermeable to B (zero flux Cb Cb boundary condition). In the bulk, at the infinity, the chemical equilibrium between A and B is achieved. To summarize, This is a dimensionless measure of mass transfer averaged the boundary conditions are the following over the bubble surface (total mass transfer scaled by the pure diffusion solution). S 1 Equations (10) and (11) with a velocity given by (14) and CA5C ; on Cb; CA5C ; when jjxjj ! 1 (6) A A boundary conditions (12) and (13) have been solved by a @CB Finite element technique for which all details have been 50; on C ; C 5K C1; when jjxjj ! 1 (7) @n b B eq A reported in Appendix A. in which x represents the position of a point in the domain Boundary layer approximation given by the coordinates (r, z) in the cylindrical polar coor- When the Peclet number becomes large (typically larger 3 dinate system. than 10 ), concentrations CA and CB present strong gradients close to the bubble interface. The Sherwood number scales Da52a PeA (22) with the square root of the Peclet number when PeA is larger than 103 for pure transfer.29 In this section, a boundary layer In the last relation, a is a new dimensionless number approximation of the coupled equations is proposed to estab- which can be written using the definition of Da and PeA lish the asymptotic behaviour when the Peclet number is suf- given by (9) by ficiently large. ak1 To write the boundary layer equations, spherical coordi- a5 (23) Vt nates ðR; h; uÞ, with origin at the center of thep bubble,ffiffiffiffiffiffiffiffiffiffiffiffi are used. The radial coordinate, R, is equal to r21z2 and Consequently, a is the product of the kinetic constant of h5arctan ðr=zÞ, the polar angle. According to Hadamard– the reaction by the time scale of advection meaning that this Rybczynski’s solution, the radial uR and polar uh velocity quantity will be large when the chemical kinetics is very components are given in dimensionless form (divided by the fast. To keep the reaction source terms in the boundary layer terminal velocity) by formulation, we assume that a is of order one. According to the principle of least degeneracy of bound- 1 1 30,32 u 52 12 cos h; u 5 12 sin h (16) ary conditions, d has to be equal to R 2R h 4R rffiffiffiffiffiffiffiffi 2 In spherical coordinates, conservation equations on C d5 (24) A PeA and CB written under steady-state regime become Therefore, the final expressions of the boundary layer @C u @C 1 A h A equations at the zeroth order using the new variable uR 1 5 2 @R R @h R PeA l52cos h (25) @ @C 1 @ @C 3 R2 A 1 sin h A (17) @R @R sin h @h @h are 2 2 Da @CA 12l @CA @ CA 2 ðÞCA2CB fl 1 5 2 2aðÞCA2CB (26) PeA @f 2 @l @f 2 2 @CB uh @CB D @CB 12l @CB @ CB a uR 1 5 2 fl 1 5D 1 ðÞCA2CB (27) @R R @h R PeA @f 2 @l 2 K @f eq @ @C 1 @ @C 3 R2 B 1 sin h B (18) with the boundary conditions @R @R sin h @h @h CA51; for f50; lim CA50 (28) Da f!1 1 ðÞCA2CB Keq PeA @CB 50; for f50; lim CB50 (29) @f f!1 To find the boundary layer equations, the radial coordinate is written as follows The Sherwood number in the boundary layer formulation 11df is given by the following relation R5 (19) rffiffiffiffiffiffiffiffið 1 2 PeA @CA ShA52 dl (30) where d is a constant proportional to the boundary layer 2 21 @f f50 thickness and f is the inner coordinate.30 Using (19), Eqs. (17) and (18) become With the reversible reaction, we did not find any self- similar solution of the system of Eqs. (26) and (27). Conse- quently, a numerical method will be used to solve the uR @CA uh @CA 2 1 5 2 2 boundary layer equations. The details about the numerical d @f 11df @h d PeAðÞ11df method are presented in Appendix A. @ @C 1 @ @C 3 ðÞ11df 2 A 1 sin h A (20) @f @f sin h @h @h Results and Discussion Da 2 ðÞCA2CB Numerical results are presented successively with the two 2PeA methods used in this work starting with the boundary layer

uR @CB uh @CB 2D formulation and then with the ﬁnite element method for 1 5 2 2 direct numerical simulations. To verify the numerical accu- d @f 11df @h d PeAðÞ11df racy, asymptotic solutions will be established and also used @ @C 1 @ @C to understand the physics of the problem. 3 ðÞ11df 2 B 1 sin h B (21) @f @f sin h @h @h Da Boundary layer approximation 1 ðÞCA2CB 2Keq PeA In this section, the results provided by the boundary layer equations (see Section Boundary layer approximation under The velocity components, uR and uh have been expanded Problem Statement) are presented. When the chemical reac- 20,31 as a function of d. In this approach PeA is no longer an tion consumes the gas transferring from the bubble, the mass independent parameter as it is assumed large, we write the transfer is expected to increase. The enhancement factor is Damkohler€ number proportional to the Peclet number, that is deﬁned by 21 Figure 1. EA as a function of a for (a) D 5 1=2 and (b) D 5 2 and for Keq 5 10 ð᭺Þ; Keq 5 1 ðwÞ, and Keq 5 10 ð᭛Þ. Solid lines correspond to the asymptotic solution given by Eq. (33) for large a.

ShAðPeA; aÞ region the gradient of CA1Keq DCB is constant (see details in EA5 (31) ShAðPeA; a50Þ Appendix B) and that the gradient of CB is equal to zero at the bubble interface, the interfacial mass flux of A is directly where ShAðPeA; a50Þ corresponds to the Sherwood number given by Eq. (B5). This provides an analytic prediction of the without chemical reaction obtained in the boundary layer regime enhancement factor assuming chemical equilibrium by29 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi EAðPeA; a !1Þ5 ð11Keq Þð11DKeq Þ (33) ShAðPeA; a50Þ5 pffiffiffiffiffiffi PeA 0:651 PeA (32) 3p This is exactly the same relation established by Olander9 in The boundary layer formulation depends on three dimen- the Cartesian geometry for unsteady transfer. This corresponds to the surface-renewal mechanism which is actually the situa- sionless parameters: a, D, and Keq while PeA is assumed to be large. To study the influence of the reaction kinetics, we tion experienced by a bubble rising in a liquid. Remark that 33 performed numerical simulations over a large range of a. Crank proposed a model to describe diffusion process Figure 1 presents the enhancement factor as a function of a coupled with an instantaneous irreversible reaction for immo- for two values of D equal to 1/2 and 2. Three values of the bilized reactant A. Here, we extend this model for a nondiffu- equilibrium constant are tested: 10, 1, and 0.1. As expected, sive product, Species B. Indeed, if D tends top zeroffiffiffiffiffiffiffiffiffiffiffiffiffiffi in Eq. (33), when a is very small the enhancement factor tends to one the enhancement factor is simply equal to 11Keq . Similar meaning that the chemical reaction does not play any role on approach has been followed to describe the mass transfer of oxygen in molten glass in Refs. 20 and 34 where the enhance- mass transfer. When a becomes larger than one, EA increases significantly to reach an asymptotic value at very high a. ment factors are quite similar to our results. The results show that the equilibrium constant influences To verify those theoretical predictions, the solution of both the magnitude of the enhancement factor and the range boundary layer equations has been obtained to determine the of a over which E is well approximated by the asymptotic enhancement factor assuming chemical equilibrium. Figure 2 A 8 regime at large a. The asymptotic behavior is reached for shows EA vs. Keq for a 5 10 and for four diffusion ratios. smaller value of a when the equilibrium constant is small. When the chemical reaction is in favor of production of B Mass transfer of Species A is, therefore, enhanced when (large value of Keq) mass transfer is significantly enhanced. 24 the equilibrium constant is larger than one corresponding to For a diffusion ratio equal to 10 , the enhancement factor 2 consumption of A and production of B. An increase of the is approximatelypffiffiffiffiffiffiffiffiffiffiffiffiffiffi equal to 10 when Keq 510 which agrees diffusion coefficient of Species B is also an important factor with 11Keq . Moreover, the chemical reaction is more and to promote the role of the chemical reaction. more efficient when D > 1 (higher diffusion of B than A). To determine the enhancement factor when the chemical The solid line in Figure 2 represents the solution (33) reaction becomes instantaneous, that is, infinite value of a,an which is in perfect agreement with the numerical results exact solution of the boundary layer equations according to obtained from the boundary layer formulation. When Olander9 has been provided in Appendix B. It is shown that Keq 1, two scalingpffiffiffiffi laws can be derived for thepffiffiffiffiffiffiffiffi enhance- the two transport equations of A and B concentrations can be ment factor EA DKeq for D1 and EA Keq for D reduced to only one equation without chemical source term for 1 (see Figure 2 where the two asymptotic behaviors have been plotted with dashed lines when K is larger than 10). the quantity equal to CA1Keq CB. The solution is obtained by eq decomposing the domain outside the bubble in two regions: the former corresponds to the inner region close to the bubble Numerical solutions for arbitrary PeA interface where the chemical reaction does not reach equilib- The mass transfer involving a reversible chemical reaction rium and the latter where the chemical equilibrium is reached between two species is investigated over a large range of meaning that CA5CB. Using simultaneously that in the inner Peclet number by direct simulations of transport equations. Figure 3 shows the evolution of the Sherwood number for Species A computed with relation (15) as a function of PeA when D51=2, and (a) Keq 510, (b) Keq 51, and (c) 21 Keq 510 . Seven values of Da have been investigated ranging from 1 to 106. Results with the chemical reaction are compared to the correlation given by Clift et al.26 without reaction (Figure 3). As expected, the chemical reaction enhances the mass transfer for any value of the Peclet number. At a fixed value of the Peclet number, the Sherwood number increases with the Damkohler€ number to reach an asymptotic value at very large Da. For a fixed Damkohler€ number, we observe an unexpected behavior. Indeed, when the Peclet number increases, the effect of chemical reaction on mass transfer becomes less important. As shown in Figure 3a, the Sher- Figure 2. E as a function of K obtained with a 5 108 A eq wood number obtained in the presence of chemical reaction when D 5 1024 ðÞD᭺ 5 1=2 ðÞDw 5 2 ð᭛Þ, and becomes closer to the solution without reaction when the D 5 4 ð~Þ. Peclet number increases. The solid line is the solution given by Eq. (33). As observed in the previous results obtained from the boundary layer approximation, the enhancement due to The numerical results have been obtained for Peclet number chemical reaction depends strongly on the equilibrium con- 23 6 ranging from 10 to 10 . As in the previous subsection, the stant. When Keq 510, the increase of the Sherwood number equilibrium constant is equal to 10, 1, or 1021. is significant (Figure 3a). The enhancement remains

2 3 4 5 6 Figure 3. ShA as a function of PeA for D 5 1=2, for Da 5 0 ð᭺Þ; 1 ðwÞ; 10 ð᭛Þ; 10 ðDÞ,10 (1), 10 (3), 10 (*), and 10 21 (3) and for (a) Keq 5 10, (b) Keq 5 1, and (c) Keq 5 10 . The solid line is the correlation given by Clift et al.26 The dashed line is the curve obtaining from the boundary layer solution for an instantaneous reaction following the asymptotic behavior, Eq. (33). 2 3 4 5 6 Figure 4. EA as a function of PeA for D 5 1=2, for Da 5 1 ðwÞ; 10 ð᭛Þ,10 (D), 10 (1), 10 (3), 10 (*), and 10 (3) and 21 for (a) Keq 5 10, (b) Keq 5 1, and (c) Keq 5 10 .

21 moderate for Keq 51 and finally vanishes for Keq 510 (Fig- larger than 1.07 meaning that mass-transfer rate increases only ure 3c). When the kinetics of the chemical reaction becomes a few percent in presence of chemical reaction. Note that very important (large value of Damkohler€ number), the Sher- whatever the value of equilibrium constant, EA behaves non- wood number is expected to reach an asymptotic value. From monotonically as a function of the Peclet number. A particular the previous results provided by the boundary layer solution, optimum of the reaction-induced enhancement is observed for the behavior of the Sherwood number for an instantaneous each Damkohler€ number. reaction can be determined. The dashed lines in Figure 3 rep- For low Peclet numbers, the enhancement factor becomes resent the solution given by Eq. (33). Results obtained from constant corresponding to the diffusive regime. Moreover, the finite element method for the largest Damkohler€ number EA increases with the Damkohler€ number to reach an asymp- tend to the asymptotic behavior when the Peclet number is suf- totic value at very high Da. To predict the evolution of the ficiently large to stand in the boundary layer regime. enhancement factor when advective terms vanish, an exact To highlight the respective importance of the advection and solution of the transport equations of A and B is derived in chemical reaction, the enhancement factor is determined as the Appendix B (solution of the Laplacian operator in spherical ratio of the Sherwood number at a specific value of Da to the polar coordinates). Thanks to the exact solution (B15), the Sherwood number obtained without chemical reaction. This is Sherwood number can be estimated as follows similar to the one given by Eq. (31) for which the Sherwood rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi number becomes a function of Pe , and Da. Figure 4 presents A 112 Keq D EA as a function of PeA when D51=2 and for seven values of ðÞ11Keq D Da qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the Damkohler€ number and with (a) Keq 510, (b) Keq 51, and lim ShA5211Keq D (34) 21 Pe!0 ðÞ11Keq D Keq D (c) Keq 510 . For the equilibrium constant Keq 510, the 112 Da enhancement factor reaches a value close to 7 while the maxi- mum of EA is only equal to 1.7 when Keq 51. For the smallest For a fast chemical reaction corresponding to very large 21 equilibrium constant, Keq 510 , the enhancement factor is no Da, the Sherwood number reaches the following limit lim ShA5211Keq D (35) Pe!0; Da!1 which shows that the mass-transfer coefficient increases with the equilibrium constant and the ratio of diffusion coefficients D. Remark that the enhancement factor EA is equal to ShA given by (34) and (35) divided by two (which corresponds to pure diffusion) when the Peclet number tends to zero. It is noteworthy that when the diffusion coefficient of B tends to zero, the enhancement factor becomes equal to one even in the limit of high Damkohler€ number in the diffusive regime. This behavior is not observed for large Peclet number for which the enhancement factor is always larger than one when D tends to zero. This is due to the strong reduction of the convective terms in the limit of small Peclet number. At large Peclet number, the evolution of the Sherwood number becomes close to the solution without chemical reac- Figure 5. ShA as a function of PeA obtained from the finite tion. The precise value of PeA for which it happens depends element method for Keq 510 and for D 5 1=2and on both the Damkohler€ and Peclet numbers due to the com- Da 5 105 ð᭺Þ; D 5 1=2andDa 5 106 ðwÞ; D 5 2 5 6 petition between the advection and chemical reaction. Figure and Da 5 10 ð᭛Þ; D 5 2andDa 5 10 ðDÞ. 4 shows that EA reaches a value close to one for a Peclet The boundary layer solution is plotted for D 5 1=2andDa 5 number depending on and The larger the Damkohler€ 105 (solid line), D 5 1=2andDa 5 106 (dotted line), D 5 2 Da Keq. 5 6 number, the larger has to be the Peclet number to match this and Da 5 10 (dashed line) and D 5 2andDa 5 10 (dotted- dashed line). particular regime. Moreover, the value of PeA for which EA is close to one decreases with the equilibrium constant. The reduction of the chemical reaction effect when the the mass transfer is expected to increase with the Damkohler€ Peclet number increases can be explained by the interaction number, the Sherwood number obtained at Da5106 must be between the two boundary layers (due to the advection or larger than one obtained at Da5105 which is verified in Fig- the chemical reaction). From Eqs. (20) and (21), two bound- ure 5. Furthermore, the threshold for which the advection ary layer thicknesses can be estimated. The first corresponds dominates the chemical process is effectively observed for to the boundary layer for the Species A and the second for the same value of the Peclet number whatever the diffusion the Species B. The typical thicknesses of the boundary layers ratio. This conclusion can be drawn from Figure 5 where the are curves obtained for two values of diffusion ratio merge at the same Peclet number. Consequently, the competition ffiffiffiffiffiffiffiffi1 1ffiffiffiffiffiffi dA5min p ; p (36) between advection and the reaction kinetics is indeed driven PeA Da rffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi! by the boundary layer of Species B. D DKeq dB5min ; (37) PeA Da Conclusions Mass transfer around a spherical bubble rising in a viscous From Eq. (37), advection becomes dominant when the fluid has been studied when a homogeneous reversible chem- Peclet number exceeds the ratio Da=Keq . This criterion is ical reaction between two species takes place in the liquid. very well verified in Figure 4 where we can see that for a The hydrodynamics is assumed to be in the Stokes regime 3 Damkohler€ number equal to 10 , advection controls mass corresponding to low Reynolds number based on the slip 4 transfer when PeA is approximately equal to 10 for an equi- velocity and the bubble size. We addressed in particular the librium constant, Keq 510, whereas the Peclet number must effect of the chemical reaction on the mass-transfer coeffi- 6 21 be equal to 10 for Keq 510 which is two orders of magni- cient for a permeable reactant. The product is assumed tude larger. Such results show that the competition between impermeable at the bubble interface. The problem is written advection and chemical reaction is driven by the boundary under dimensionless form leading to four parameters control- layer observed on Species B and this is not depending on ling the transfer. The first two parameters are related to the the diffusion ratio D. chemical reaction: the equilibrium constant controlling the The results obtained from the two methods proposed in direction of the chemical reaction and the Damkohler€ num- this article can be compared more evidently. In the previous ber measuring the importance of the chemical kinetics com- section devoted to the boundary layer analysis, the effect of pared to diffusion. The two last are related to the physical the reaction kinetics has been studied thanks to the parame- process: the diffusion coefficient ratio comparing the relative ter a. Because this last quantity is proportional to the ratio importance of product and reactant diffusion and the Peclet of Da to PeA, those results can be determined in the diagram number which is the ratio of the characteristic time of advec- ShA vs. PeA. In Figure 5, this has been done for Keq 510 tion to the characteristic time of diffusion. The two mass and for two diffusion ratios D equal to 1/2 and 2. The results balance equations are solved using two methods: direct for two Damkohler€ numbers 105 and 106 have been numerical simulation based on a finite element method lead- presented. ing to results over a large range of Peclet number and The behavior of the Sherwood number obtained for a boundary layer approximation to emphasize the coupling given Damkohler€ number when the Peclet number increases between the chemical kinetics and the advection process is in agreement for the two numerical methods. Indeed, as when the Peclet number is sufficiently large. In general, an enhancement of the mass transfer is 8. Danckwerts PV. Significance of liquid-film coefficients in gas expected when a chemical reaction is involved. This is absorption. Ind Eng Chem. 1951;43(6):1460–1467. exactly what we found but it is noteworthy that the enhance- 9. Olander DR. Simultaneous mass transfer and equilibrium chemical reaction. AIChE J. 1960;6(2):233–239. ment can be limited due to the importance of the advection 10. Ruckenstein E, Dang VD, Gill WN. Mass transfer with chemical process. Indeed, the chemical reaction leads to the develop- reaction from spherical one or two components bubbles or drops. ment of a boundary layer which depends on the kinetic con- Chem Eng Sci. 1971;26:647–668. stant. Consequently, the mass transfer is enhanced due to the 11. Soung WY, Sears JT. Effects of reaction order and convection formation of this boundary layer. Nevertheless, when advec- around gas-bubbles in a gas-liquid reacting system. Chem Eng Sci. 1975;30(6):1353–1356. tion becomes more important (large value of the Peclet num- 12. Kleinman LS, Reed XBJ. Interphase mass transfer from bubbles, ber) the boundary layer created by the flow transport drops, and solid spheres: diffusional transport enhanced by external contributes to mass transfer, as well. If the typical size of chemical reaction. Ind Eng Chem Res. 1995;34:3621–3631. this advective boundary layer, scaling as the square root of 13. Kleinman LS, Reed XBJ. Unsteady conjugate mass transfer between the Peclet number, becomes smaller than the one due to the a single droplet and an ambient flow with external chemical reaction. Ind Eng Chem Res. 1996;35(9):2875–2888. chemical reaction, the effect of the chemical reaction 14. Juncu G. Unsteady heat and/or mass transfer from a fluid sphere in vanishes. creeping flow. Int J Heat Mass Transfer. 2001;44:2239–2246. We pointed out the influence of the equilibrium constant 15. Juncu G. The influence of the Henry number on the conjugate mass which favors the production or consumption of species. transfer from a sphere: II - mass transfer accompanied by a first- When the equilibrium constant is larger than one meaning order chemical reaction. Heat Mass Transfer. 2002;38:523–534. 16. Koynov A, Khinast JG, Tryggvason G. Mass transfer and chemical that the reaction is directed toward the production of B, the reactions in bubble swarms with dynamic interfaces. AIChE J. 2005; chemical reaction plays an important role on the interfacial 51(10):2786–2800. mass transfer of A: the rate of mass transfer can be multi- 17. Wylock CE, Cartage T, Colinet P, Haut B. Coupling between mass plied by a factor larger than 10. The enhancement of the transfer and chemical reactions during the absorption of CO2 in a mass-transfer coefficient is also more important when the NaHCO3-Na2CO3 brine: experimental and theoretical study. Int J Chem React Eng. 2008;6:1–22. diffusion coefficient of the product is larger than the diffu- 18. Wylock CE, Larcy A, Colinet P, Cartage T, Haut B. Direct numeri- sion coefficient of the reactant. Inversely, when the equilib- cal simulation of bubble-liquid mass transfer coupled with chemical rium constant is smaller than one, the chemical reaction does reactions: influence of bubble shape and interface contamination. not play a significant role on mass transfer (the results we Colloids Surf A. 2011;381(1–3):130–138. obtained are close to the situation of pure transfer without 19. Bothe D, Kroger€ M, Warnecke HJ. A VOF-based conservative method for the simulation of reactive mass transfer from rising bub- chemical reaction). bles. Fluid Dyn Mater Process. 2011;7(3):303–316. Finally, we showed that the Peclet number for which the 20. Pigeonneau F. Mass transfer of rising bubble in molten glass with advection process dominates does not depend on the diffu- instantaneous oxidation-reduction reaction. Chem Eng Sci. 2009;64: sion ratio but is related to the equilibrium constant. This 3120–3129. means that the competition between the chemical process 21. Scholze H. Glass: Nature, Structures and Properties. Berlin: Springer-Verlag, 1990. and the advection is controlled by the physical transport of 22. Pigeonneau F, Martin D, Mario O. Shrinkage of oxygen bubble ris- the product. ing in a molten glass. Chem Eng Sci. 2010;65:3158–3168. Apart from the numerical investigation, approximated sol- 23. Jucha RB, Powers D, McNeill T, Subramanian RS, Cole R. Bubble utions have been also established in this work in the limit of rise in glassmelts. J Am Ceram Soc. 1982;65:289–292. no advection and in the boundary layer when the chemical 24. Hornyak EJ, Weinberg MC. Velocity of a freely rising gas bubble in a soda-lime silicate glass melt. J Am Ceram Soc. 1984;67:C244– reaction becomes instantaneous. The enhancement due to C246. chemical reaction can be easily estimated from these simpli- 25. Li KWK, Schneider A. Rise velocities of large bubbles in viscous fied relations. These results justify the model developed in Newtonian liquids. J Am Ceram Soc. 1993;76:241–244. Refs. 20 and 34 to describe the important role of the 26. Clift R, Grace JR, Weber ME. Bubbles, Drops, and Particles. New oxidation-reduction reaction of iron in molten glass. York: Academic Press, 1978. This work has been done in Stokes regime. Nevertheless, 27. Hadamard J. Mouvement permanent lent d’une sphe`re liquide et vis- queuse dans un liquide visqueux. C R Acad Sci Paris. 1911;152: many applications in chemical engineering are facing non- 1735–1738. zero Reynolds numbers and nonspherical bubbles. So, this 28. Rybczynski W. Uber die fortschreitende bewegun einer flussingen work has to be extended to other hydrodynamic regimes. kugel in einem zaben medium. Bull de l’Acad des Sci de Cracovie. Moreover, different reaction models and simultaneous multi- 1911;1:40–46. ple reactions must be investigated. 29. Levich VG. Physicochemical Hydrodynamics. Englewood Cliffs, NJ: Prentice Hall, 1962. 30. Dyke MV. Perturbations Methods in Fluid Mechanics. Stanford, CA: Literature Cited The Parabolic Press, 1975. 1. Shelby JE. Introduction to Glass Science and Technology. Cam- 31. Ruckenstein E. Mass transfer between a single drop and a continu- bridge: The Royal Society of Chemistry, 1997. ous phase. Int J Heat Mass Transfer. 1967;10:1785–1792. 2. Mazumdar D, Guthrie RIL. The physical and mathematical model- 32. Zeytounian RK. Modelisation asymptotique en mecanique des fluides ling of gas stirred ladle systems. ISIJ Int. 1995;35(1):1–20. newtoniens. Paris: Springer-Verlag, 1994. 3. Sherwood TK, Wei JC. Interfacial phenomena in liquid extraction. 33. Crank J. The Mathematics of Diffusion. Oxford: Clarendon Press, Ind Eng Chem. 1957;49(6):1030–1034. 1956. 4. Lewis WK, Whitman WG. Principles of gas absorption. Ind Eng 34. Pigeonneau F, Muller S. The impact of iron content in oxidation Chem. 1924;16(12):1215–1220. front in soda-lime silicate glasses: an experimental and comparative 5. Sherwood TK, Pigford RL. Absorption and Extraction. New York: study. J Non-Cryst Solids. 2013;380:86–94. McGraw-Hill Book Company, Inc., 1952. 35. Ern A, Guermond JL. Elements finis: Theorie, applications, mise en 6. Levenspiel O. Chemical Reaction Engineering. New York: Wiley, œuvre. Paris: Springer, 2002. 1972. 36. Hughes TJR, Franca LP, Hulbert GM. A new finite element formula- 7. Higbie R. The rate of absorption of a pure gas into a still liquid dur- tion for computational fluid dynamics: VIII. The Galerkin/least- ing short periods of exposure. Tran Am Inst Chem Eng. 1935;31: squares method for advective-diffusive equations. Comput Methods 365–389. Appl Mech Eng. 1989;73:173–189. 37. Hauke G, Hughes TJR. A comparative study of different sets of variables for solving compressible and incompressible flows. Comput Methods Appl Mech Eng. 1998;153:1–44. 38. Stoer J, Bulirsch R. Introduction to Numerical Analysis. New York: Springer-Verlag, 1993. 39. Guo W, Labrosse G, Narayanan R. The Application of the Chebyshev-Spectral Method in Transport Phenomena, Vol. 68 of Lecture Note in Applied and Computational Mechanics. Berlin: Springer-Verlag, 2012. 40. Collatz L. The Numerical Treatment of Differential Equations. Ber- lin: Springer-Verlag, 1960.

Appendix A: Numerical Methods

Direct numerical solution Equations (10) and (11), describing the transport phenomena under dimensionless formulation, are solved with a finite element technique. First, the problem must be computed in a finite domain whose geometry is given in Figure A1. The bubble contour Cb is centered at the origin of the coordinate system. The position of the outer boundary is adapted to the strength of the Figure A1. Axisymetric computation domain around a convective terms to reduce domain size effect on the solution. bubble. When the Peclet number is small, Eqs. (10) and (11) present an The domain on the left is chosen for computations cor- elliptic character. In such situation, it is better to take a domain responding to Peclet number lower than 1. When the centered on the bubble as illustrated on the left of Figure A1. Peclet number is larger than 1, the domain on the Conversely, when the Peclet number is larger than 1, the con- right is used to accommodate advection mechanism. The real scale is not respected in this figure. The vective terms dominate. In consequence, the outer boundary, CD radius of the outer boundary is 20 times larger than [CN is not centered on the bubble in order to capture the wake the bubble radius. formation near the bubble20 as it is shown on the right of Figure A1. For the numerical simulations, the radius of the outer boundary is 20 times larger than the bubble radius. bubble. With P1 interpolation, the number of degrees of free- On the bubble surface, the boundary condition is fixed by the dom is equal to 682,650 to solve both equations. physics while the outer domain is divided in two equal sections Based on the weak formulation, ordinary differential equa- CD and CN. Furthermore, the boundary conditions on the outer tions form a linear implicit system depending on time solved frontier have been adapted to the finite size domain. Indeed, a using backward differentiation formula. This method of second- problem dominated by advection in a finite domain is not well order accuracy is used with an adaptive time step.38 posed if Dirichlet conditions are imposed everywhere.35 On the In the following, the problem is solved for Peclet number 23 6 contour CD, a Dirichlet condition is imposed both for CA and ranging from 10 to 10 , the Damkohler€ number varies 6 CB, whereas on CN, the cancelation of the diffusive fluxes of CA between 1 and 10 . Three values of the equilibrium constant 21 and CB is used. To summarize, the boundary conditions are the will be investigated: 10 , 1, and 10 and two values of the following reduced diffusion coefficient of CB equal to 1/2 and 2. @C C 51; B 50; on C (A1) A @n b A coupled finite difference-Chebyshev spectral method to solve the boundary layer equations CA5CB50; on CD (A2) A numerical tool of high accuracy level has been developed to @CA @CB 5 50; on C (A3) solve the boundary layer equations, (26) and (27) with the @n @n N boundary conditions (28) and (29). Remark that there is no The finite element method needs a weak formulation of the sys- boundary condition on m which appears only at the first order. tem of Eqs. (10) and (11). A Galerkin formulation is used which Moreover, when l 561, that is, at each poles of the bubble leads to unstable solution when the advection term is dominant. interface, the terms proportional to the first derivative in m dis- This is due to the lost of elliptic character of the equations. To appear. The system of equations is parabolic and can be seen as stabilize the numerical method, the weak formulation is written a Cauchy problem for the m coordinate. Consequently, the val- 35,36 using a Galerkin/least square method which is recommended ues of CA and CB for l 521 are used as initial conditions. even to solve partial differential equations yielding shocks.37 We develop a specific numerical method to solve (26) and The computational domain is discretized using triangular ele- (27) for which the discretization on f is achieved with a 39 ments with a linear interpolation, P1. When the Peclet and Chebyshev-spectral method and a finite difference method on Damkohler€ numbers become large, a thin boundary layer m with second-order accuracy. For the Chebyshev-spectral appears close to the bubble surface, Cb. To have high resolution method, a collocation method is used for which collocation accuracy of the boundary layer, a fine grid is used close to the points are taken following the Gauss-Lobatto distribution.39 bubble surface. The minimum size used for this current work is Moreover, the domain has to be truncated on the f axis. In 431024 which is enough as the largest values of Peclet and numerical simulations, the finite extent is taken between 10 and Damkohler€ numbers are fixed to 106 corresponding to a thick- 100. In the polar direction, a uniform discretization is used. The ness of the boundary layer approximately equal to 1023 (see jus- first derivative in m is determined following an upward off- tification below). Figure A2 presents the mesh around the centered scheme with three points.40 The numerical procedure is based on a fully implicit scheme coupling the two equations on CA and CB. When the chemical reaction is not accounted for, results of this numerical method can be compared to the self-similar solution provided by Levich29 in order to validate the numerical accuracy of the method. For instance, with a number of collocation points for the spectral method equal to 513 and a resolution on the polar angle of 0.1, the Sherwood number obtained from the boundary layer numerical solutionpffiffiffiffiffiffi gives a prefactor equal to 0.65147 to be compared to 2= 3p of the Levich’s solution29 (see Eq. 32). The relative error is then equal to 4:2231026%.

Appendix B: Exact Solutions for Instantaneous Chemical Reaction Boundary layer regime To predict the enhancement factor when the chemical reaction kinetics is very fast, the method proposed by Olander9 can be extended to the boundary layer formulation. When a is larger than 1, chemical equilibrium can be assumed outside the boundary f f Figure A2. Mesh grid around the bubble (the finest size layer for larger than eq . Using a linear combination of Eqs. (26) and (27), the chemical sink and source terms cancel by studying element is 431024 close to the bubble sur- 1 face when the Peclet number is larger the transport of CA Keq CB. Moreover, assuming chemical equi- 5 than 1). librium, that is, CA CB, we obtain the following equation

Figure B1. Concentrations obtained from the numerical solution of the boundary layer equations for CA (᭺), and ^ 6 CB (w) vs. f when a 5 10 and D51=2.

The solid line is the asymptotic solution of CA given by (B9) and the dashed line is the asymptotic solution of CB given by (B10). 23 Figure B2. Concentration profiles of CAðRÞð᭺Þ and CBðRÞðwÞ as a function of R when Pe510 ; Keq 510; D51=2 and for (a) Da51, (b) Da5103, and (c) Da5105.

The solid line is the exact solution for CA given by Eq. (B5) and the dashed line, the exact solution for CB given by Eq. (B6).

@C 12l2 @C 11K D @2C outer solution can be used to compute the total mass flux of C fl A 1 A 5 eq A (B1) A 2 1K C given by @f 2 @l 11Keq @f eq B qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 2ffiffiffiffiffiffi similar to the situation without chemical reaction for which the ShA1Keq B5 p ð11Keq Þð11Keq DÞ PeA (B5) solution is well known29 and given by the self-similar relation 3p

CA5erfc ðgÞ (B2) To establish the inner solution, the inner variable due to the chemical reaction is introduced according to the classical way of where the self-similar variable is given by the perturbation technique30 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi f 11K ^f5f a (B6) g5 eq (B3) 1 gðlÞ 1 Keq D In this case, the inner equations become @2C 29 A The function g(m) is given by 2 2ðÞCA2CB 50 (B7) rffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @^f 3 2 213l2l 2 @ CB 1 gðlÞ52 2 (B4) 3 12l D 2 1 ðÞCA2CB 50 (B8) @^f Keq The solution (B2) is also valid for CB. Nevertheless, this solution is not fulfilled in the entire domain as the boundary condi- which are solved exactly with the boundary conditions at the tion at the bubble interface is not verified for C following Eq. bubble interface and matched to the outer solution for a distance B ^ (42). So, this solution is only valid outside of feq corresponding equal to feq . The molar concentrations, CA and CB, are then to the outer solution. According to the approach of Olander,9 the given by c^f 2c^f ^ B1. Consequently, the gradient of the C at the bubble surface CA5A0e 1A1e 1A2f1A3 (B9) A decreases when the equilibrium constant decreases. ^ 2 2c^f 2 c^f CB5A01A1f1A2ð12c Þe 1A3ð12c Þe (B10) where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Exact Solution of the Transport Equations in a 11K D Quiescent Fluid c5 eq (B11) Keq D When advective terms are canceled in Eqs. (17) and (18), the problem is reduced to a coupling between diffusion and reaction and where the exact solution can be easily established. The concen- ^ ^ tration profiles of A and B become independent of h and take 2cfeq 2 2cfeq A152e A0; A252c 12c 11e A0; the following forms (B12) ^ 2cfeq A3512 12e A0 A1Be2bR sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C 5 (B15) A R 2^f 11K A 5 hieq eq 0 ^ ^ 2 2bR 2cfeq 2 2cfeq pa 11K D ADa1BðDa2b Þe gðlÞ 12e 1ðÞ12c c 11e eq C 5 (B16) B Da R (B13) where A and B are determined from the boundary conditions Finally, in the inner region the gradient of CA1Keq DCB is ^ ð21bÞðDa2b2Þ Da eb=2 constant and equal to A2 which permits to determine feq by A5 ; B52 (B17) matching the outer solution 2b½Da2b222b b½Da2b222b ^ ^ and b is 2cfeq 2 2cfeq ^ 12e 1 12c c 11e ð12feq Þ50 (B14) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 11K D Da b5 eq (B18) This theoretical solution can be used to evaluate the accuracy Keq D of the numerical solution of the boundary layer equations. So, Figure B2 presents the concentration profiles C ðRÞ and C ðRÞ three computations have been carried out for a 5 106 when D is A B obtained with the finite element method when the Peclet number equal to 1/2 and for the equilibrium constant equal to 10, 1, and is equal to 1023. The equilibrium constant is equal to 10 and 1021. Figure B1 shows the concentration profiles of A and B at D51=2. Three values of the Damkohler€ number are investigated: the north pole of the bubble, l 521, over the range of ^f eq 1, 103, and 105. The numerical results obtained for this small obtained from the numerical solution of the boundary layer for- value of Peclet number are very well predicted by the exact solu- mulation (513 collocation points for the spectral method have tion without advection. The mass-transfer enhancement is clearly been used for spatial resolution over f coordinate). Solid lines in seen in Figure B2. The region for which the two concentrations Figure B1 represent the exact solution given earlier. The numer- are distinct becomes thinner when the Damkohler€ number ical solution reproduces very well the asymptotic solution increases (remind that CA5CB corresponds to chemical reaction although the thickness of the inner region is very small (1023 at equilibrium). The concentration of CB at the bubble interface is when a 5 106). Remark that when the equilibrium constant closer and closer to 1 when Da increases. The slope of CA rises decreases the thickness over which the inner solution is valid up strongly as a function of Da. becomes smaller and smaller. The difference between CB and CA becomes very small as seen in the scale of y axis in Figure