7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Coupling chemical reactions with mass transfer around a bubble rising in molten
glass
Marion PERRODIN *t~ Eric CLIENT *t Edouard BRUNET and Frank PIGEONNEAU 9
University de Toulouse; INPT, UPS ; IMFT; Allee du Professeur Camille Soula, F31400 Toulouse, France
iCNRS ;IMFT, Toulouse, France
7 Elaboration des Verres, SAINTGOBAIN RECHERCHE, 39 quai Lucien Lefrane 93303 Aubervilliers Cedex
SUMR 125 CNRS/SAINTGOBAIN, 39 quai Lucien Lefrane 93303 Aubervilliers Cedex
perrodin ~imft.fr climent~imft.fr Edouard.Brunet~ saintgobain.com and Franck.Pigeonneau ~saintgobain.com
Keywords : bubble, molten glass, chemical reaction, simulation, interfacial transfer
Abstract
During glass production, interfacial mass transfer supplemented by chemical reactions can be used to remove tiny gas
bubbles present in the molten glass (this is a keystep of the quality control of the endproduct). In this communication,
we present experiments on bubbly glass melt highlighting the influence of the coupling between hydrodynamics, mass
transfer and chemical reactions. Varying the iron content and temperature shows the influence of physicalchemical
properties of the glass on interfacial transfer. Results of the modeling and simulations of these experiments are
discussed. Some preliminary results of the simulation of a single bubble rising in molten glass show that convection
largely dominated diffusion. Chemical reactions enhanced the gaz transfer.
Nomenclatu re
Roman symbols
g gravity acceleration (m1 s )
C; concentration of i species (m1ol 111)
D; diffusivity of the i species (m12 s1
clb radius of the oxygen bubble (11)
cliy reaction order of the lcs species considered
as product in the reaction r; ()
EA acceleration factor ()
I. reaction order of the lcs species considered
as reactive in the reaction r; ()
Ha Hatta number ()
k+ kinetic constant for the direct way of
the reaction
k kinetic constant for the reverse way of
the reaction
Equilibrium constant of the reaction ()
L characteristic length of the domain (m1)
N,
Ns
Pe
r;
Re
Sh
c;
Greek
Xi
6
Ti
~il'
total number of reactions ()
total number of chemical species ()
Peclet number ()
reaction i ()
Reynolds number ()
Sherwood number ()
degree of conversion of the reaction i (11ol)
symbols
characteristic parameter for iron reaction
characteristic parameter for sulphate reaction
concentration of j species (molm )
thickness of the mass boundary layer (11)
stoichiometric coefficient in the right hand
side of the reaction r; for the Xj species ()
scaled distance (y/L) ()
scaled distance (y/ )~ ()
stoichiometric coefficient in the left hand
side of the reaction r; for the Xj species ()
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
E multiplying factor ()
Dynamic viscosity (Pa s)
Skinematic viscosity (111 s )
< source term of chemical reactions (mol e1 )
p density (kg 1113)
Pb density of the bubble (kg 1113)
Pf density of the fluid (kg 1113)
9; scaled concentration of i species ()
Subscripts
0 at the interface of the bubble
L distance to the boundary of the domain
located far from the interface
00 far from the bubble, in the glass melt
1/2 O, + SO, + O2 SO'
As the O' species is largely in excess in the glass melt,
the previous equations yield :
1/4 O, + Fe2 Fe and 1/2 O, + SO, = SO .
Our first experimental studies is intended to avoid
convection for highlighting the coupling of diffusion and
reaction phenomena. Thus the impact of reactions on the
species diffusion in the glass melt could be observed. We
have chosen to consider a single species (the oxygen)
and to focus on the oxidationreduction of iron (1). The
equilibrium constant of this reaction in the glass melt is :
K, Fe 7.8395 at T 1450"C according to Beer
kens (2003). The reaction is then significantly leading
to the consumption of oxygen gas. The term redoxox"'
characterises the reduction state of the glass. It corres
ponds to the ratio of Fe2 concentration on the total
iron concentration :
CFe2+
c,,,o,,l
redorr:
Introduction
Glass production is an ancient process, as glass is pro
duced since more than 3,000 years. The raw materials
are sand, in particular silica, limestone and soda. No
wadays it represents a production of millions of tonnes
per year, and glass is used for several applications, as
automobile industry, solar panels or house equipment.
When these three components are molten at high tem
perature, an important gas release (mainly CO,) is ob
served. The refining step of glass production consists in
removing from the glass melt the microbubbles which
were trapped. Due to the high viscosity of liquid glass,
it can be very long to remove tiny bubbles thanks so
lely to gravity. Using chemical reactions which consume
or produce dissolved gases improves the refining pro
cess. On one hand, the interfacial transfer increases the
bubbles size decreasing the rising time. On the other
hand, it consumes the bubbles gaseous content leading
to complete resorption. The refining step is subjected to
physical (hydrodynamics, interfacial mass transfer) and
chemical phenomena, which will control bubbles dyna
mics. We have to deal with the interaction of convec
tion/diffusion phenomena and chemical reactions.
Experimental measurements
The main reactions occurring in the glass melt are the
oxidationreduction reactions of iron (equation 1) and
sulphate (equation 2).
The experiments presented in this paper were carried out
at T 1450"C. High temperature and possible coupling
with other reactions complicate the results analysis.
Oxidation front
The first step of the experiments consists in produ
cing a reduced glass, it means a glass with a high re
dox. The reduced glass is placed in a platinum crucible.
The crucible is then inserted in the furnace at a fixed
temperature, upon contact with a gas. This gas contains
a controlled proportion of oxygen. The diffusion of the
oxygen from the gas to the glass is observed. We inves
tigate the impact of several parameters : the tempera
ture, the glass composition, the composition of the gas
contacting the glass interface and the duration of the ex
periment. The glass we used contains negligible concen
tration of sulphate, as we try to characterize the impact
of iron content solely. The technique we used measures
the oxygen diffusion dynamics by spectrometry : an op
tical fiber scans the whole depth of the sample (glass
spectra are obtained every millimeters) yielding a verti
cal profile of glass redox.
The figure 1 shows some spectra obtained thanks to
the fiber optic spectrometer. The transmission around
the wavelength A 1000 nm is related to the Fe2+
concentration. Because the total iron content in the glass
is known, the corresponding redox can easily be deter
mined.
Interpretation
1/4 O, + Fe2 Fe3+ + 1/2 O
The figure 2 shows a redox profile along the depth of
(1) the sample. A clear trend of redox reduction is observed
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
sence of iron reaction consuming the oxygen gas leads
to a lower transport of oxygen in the melt.
Modelling
The second pant of the article is devoted to the mo
delling. The validation of the simulation code on cases
j~p~~Sr ~ with diffusion and reaction is presented. Finally the si
1000 110 mulations of cases corresponding to our experiments are
discussed.
400 500 600 700 800
Wavelength (nm)
FIGURE 1: Profile of light transmission across the glass
sample.
0 o.7 x 9 +
~gXx ++
Gouverning equations
Our numerical simulations have been carried out with
the JADIM code, which is widely described in Magnau
det (1995) and Legendre (1998). This code is particu
larly welladapted for the detailed study of hydrodyna
mics around bubbles. It allows to solve directly the full
NavierStokes equations coupled to the transport of spe
cies in the liquid. The equations governing the hydrody
namics are solved with a velocitypressure formulation
in a system of orthogonal curvilinear coordinates. The
spatial discretization is based on a staggered mesh and
the equations are spatially integrated with a method ba
Sed on finite volumes. The time advancement is carried
out by a RungeKutta/CrankNicholson algorithm, accu
rate at the second order, whereas the continuity equation
providing the incompressibility is satisfied at the end of
each time step by the solving of a Poisson equation for
an auxiliary potential.
These equations are coupled to the transport equations
of the Ns chemical species Xj including source/sink
terms 04 which takes into account the presence of che
mical reactions in the liquid. The system of equations
solved is :
0.005 0.01 0.015 0.02 0.025 0.03
Depth (m)
0.035
FIGURE 2: Profile of redox across the glass sample.
in the first centimeter and in the last millimeters. This
corresponds to the oxidation of the glass. The oxidation
of the upper pant of the sample is due to the oxygen trans
fer through the gas/1iquid interface. We suspect that the
oxidation of the lower part is due to a reaction between
the platinum of the crucible and the iron content of the
glass melt.
The results obtained for various values of the tempera
ture, glass composition, proportion of oxygen in the gas
and duration of experiments allow us to draw the follo
wing conclusions on the effective diffusivity of oxygen
in the melt :
No effect of the presence of sulphate was observed
on the diffusivity coefficients.
A lower temperature of the glass melt leads to lower
diffusivity of the oxygen.
A higher temperature of the glass melt leads to fas
ter diffusivity of the oxygen.
Increasing the iron content in the glass melt leads
to lower diffusivity of the oxygen.
The diffusion coefficient determined in this experiment
is an effective coefficient as it takes into account the
reaction. If we compared the diffusivities we determined
to diffusivity cited in the literature. The comparison, as
Dolmesurea << Doltizeuretical, confirmed that the pre
v.U
PS (go v v UU))
atnl [v]v. ([v,]u
vp v Ovo vuT ,~ U) ~
v. (DV[xj])+ 4,
The source/sink terms 04 take into account all the si
multaneous N, chemical reactions. In order to have a
unique framework whatever the number of species and
reactions, we write all the Xj species on each side of the
N, equations. The source/sink terms 04 c'l' gydi are
given by the equation 4, with the degree of conversion
vi of each reaction (equation 5) :
cl' [Xjl] (6;y y; ) v; <1 (4)
=1i
myi and 6;y are respectively the stoichiometric coeffi
cients on the left hand side and the right hand side of the
^
v; = k+ Adl"'"
d=1
^ s
k, Ad 1ldij
3=
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
reaction ri for the species Xj. The stoichiometric coeffi
cients *744 (respectively 6ij) take then a null value when
the species Xj is not actually a reagent (respectively a
product) in the reaction ri. The coefficients ., (respec
tively : .) are the orders of reaction of the species Xj
considered as reagents (respectively as products) in the
reaction ri. These coefficients ., and dij can then take
null values when the species Xj is not actually involved
in the kinetics of the reaction ri. This framework allows
to consider as many species and simultaneous chemical
reactions as necessary.
oL
CC, =co = n
(a)
CAC E B BL
^~ B
CAC AO B 8
Validation Case
A tB
(c)
FIGURE 3: Definition of boundary conditions for vali
dation cases
In this section, we focus on diffusion and a single
reversible chemical reaction A B. The kinetic
constants for production of B species and production
of A species are respectively kt and k. The reaction
reaches the equilibrium state when the molar concentra
tions of A and B species correspond to Keq= k /k
On/CA We validate the code on the particular case
of a reversible reaction, as many oxidationreduction re
versible reactions occur in the melting glass. The study
of interactions between mass diffusion and a chemical
reactions led to several publications (Olander (1960))
and the textbook of Danckwerts (1970). Some other pu
blications are related to our research topic : simulta
neous absorption of two gases in the presence of an ins
tantaneous reversible reaction (Ramachandran (1971))'
reactive transfer in unsteady state (Ruckenstein (1971)),
mass transfer with chemical reaction in the presence of
an interface (Subramanian (1980)). In spherical coordi
nates, the diffusion equations of A and B species are
coupled by the source/sink term related to the reaction.
The transport equations of A and B species around a
spherical bubble, with a constant radius, can be simpli
fied using the symmetry and applying the following sub
stitution of variables : CA C}/r and On CL/r.
We obtain equations similar to the Cartesian geometry :
S= DA + kC) k 01 (6)
t B T2 B A ()
The problem of diffusion/reaction around a spherical
bubble can therefore be solved by an analysis in 1D Car
tesian coordinate (the radial variable r is now substitute
to the variable y in Cartesian coordinates).
Steady state solution
Diffusion and chemical reaction with finite kinetics Far
from the bubble, A and B are considered to be in che
mical equilibrium at y L. We want to determine the
mass flux of A at the interface (y 0 ). The figures 3(a),
3(b) and 3(c) represent three distinct combinations of
boundary conditions. The physical problem corresponds
to the diffusion of species A which reacts with a non
volatile species present in the solution (ionic species for
the glass).
Thus, the boundary condition CAo is known thanks to
the thermodynamic equilibrium at the interface, which
depends on the gas solubility in the glass melt. This cor
responds to the boundary conditions sketched on the fi
gure 3(b). The equations are solved in nondimensional
form defining OA CA/!CAL, OB = c", and
r = y/L. Therefore, when the chemical equilibrium
is reached BA = 8B. In order to obtain an exact solu
tion for this case in steady state, the simplest situation,
with four Dirichlet boundary conditions (figure 3(a)),
has been solved analytically. The analytical solution of
this configuration is summarized by the equations 8 and
The nondimensional number w2 is called the Dam
khoiler number and represents the ratio between the cha
racteristic times of chemical kinetics and pure diffusion
of species. Finally, a nonlinear equation has to be solved
numerically by fixing Cno to a value which verifies the
no flux condition at the interface y = 0 for B. The flux
of A at the interface is finally deduced by calculating the
value of ~ at rl 0 The constants E, F, G, H, I, J
and the coefficients of the function f(r) are determined
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
thanks to boundary conditions.
eA = G cosh(w r) + H sinh(w r) +I
Os = I cosh(W r) + J sinh(w r) +
k L E
DA DB w2 CAL
L2 I (E L r F)
with : f(rl)
DA Dg CAL
GCA0 L2 I F
CAL DA Dg CAL 2
0 (10)
H = 1 G oshw)L2 I (E L F)1 1
DA Dg CAL ,2 sinh(w)
DA Dg
E (CAL AO) (BL CBO)
L L
F = DA CAo DeCBo
CBo L2 p p
Keg CAL DA Dg CAL W2
J= 1 cosh~w) L2 I (E L F)1 1
DA Dg CAL ,2 sinh(w)
kc+ L2 c L2
DA Dg
The figure 4 shows that the reaction is out of equi
librium in a boundary layer very close to the interface
whereas a large part of the liquid stays at the equilibrium
eA = eB. A very good agreement between the simula
tions and the exact solution is observed. We can notice
also that the thickness of the mass boundary layer (6 on
the figure 4) decreases when the Damkhoiler parameter
increases corresponding to a very fast reaction. We ob
served that this decrease of the thickness of the boundary
layer varies as w In the case of infinite chemical ki
netics, this thickness tends to zero.
Diffusion and chemical reaction with infinite kinetics The
sketch 3(c) corresponds to an infinite w (ie. instanta
neous reaction or infinite rate of reaction). The boun
dary layer collapses towards the interface while the
concentrations CA and On verifies the equality Keq
k~k /k Os/CA in the bulk and also on the
boundaries. If we consider an infinite rate of reaction
(Vy, Keq k /k O n/CA) in steady state, we ob
tain the following analytical solution for BA et On :
CA0 CA0
CAL CAL
The interfacial flux of A in r 0 is obtained by the
sum of the flux by diffusion of A and the flux of B. As
suming that the boundary layer is collapsed on the in
terface, the moles of A are transferred either in the form
of A or in the form of B (ie. transformed by the chemi
cal reaction). It is the asymptotic value of the flux at the
interface when the kinetics is infinite.
I1 y/
0.4 0 6
n =v/L
(b)
0 0.2 0.4 0.6
n = V/l,
(c)
0.8 1
FIGURE 4: Spatial profiles of dimensionless concentra
tions in steady state for various values of w :
(a) w 8.66 ,(b) w 15 ,(c) w 42.43
,6 being the thickness of the mass boun
dary layer and : + : BA numerical simu
lations ; x : On numerical simulations ;
:analytical BA for finite chemical ki
netics ; : analytical On for finite chemical
kinetics ;  : eA = eB for infinite kine
tics.
with ex +~ and D B n the limit of npur
diffusion (no reaction), we recover the solution of pure
diffusion in a semiinfinite domain. The general case of
a finite kinetics was studied by Sherwood (1952). We
compared the spatiotemporal evolutions of the concen
tration of A with the two asymptotic cases correspon
ding to pure diffusion and reactive diffusion with an in
finite rate of reaction (figure 6). For all cases the kinetic
constants kt and k were varied keeping Ke, constant.
The results are presented in the figure 6, where the evolu
tion of the nondimensional concentration yA versus il
is shown. It is interesting to note that the solution for
pure diffusion (w = 0) corresponds to the limit of very
slow kinetics, whereas for fast kinetics (w > 50), re
sults obtained with Jadim are very close to the analytic
prediction for infinite kinetics. For intermediate rate of
reaction, the simulation results stand between these two
limits. In these cases, the solution is not autosimilar at
all times : at short times, the solution is close to the pure
diffusion state and later evolves to a profile which de
pends on physical/chemical parameters.
0.8
S+
0.6
0.4 il+
o +
0.2 O +
o + s
+ 's
oaz * h
f 8
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
il' = J, the analytic solution is written :
7
YA = 1 erf(r" ~) = YB
10. to
FIGURE 5: Evolution of the acceleration factor EA with
w : 0 : pure diffusion (w > 0) ; x : JADIM ;
 : exact solution for a finite kinetics..
Enhancement factor Enhancement of mass transfer
through the interface by the chemical reaction can be
characterized by an acceleration factor E 4, which com
pares the interfacial flux in presence of reaction to
the flux due to pure diffusion (equation 12, see for
example Olander(1960) and Danckwerts(1970)). Our re
sults were obtained for : K, 10 and D DB/DA 4
0.2. If we assume an infinite kinetics, we obtain an ana
lytic relation for the acceleration factor. The two asymp
totic limits of the acceleration factor for w tending to
zero : E 4 = 1 (pure diffusion) and for w tending to infi
nity: EA = 1+ D Key.
1 89 4
E 4 (12)
The figure 5 shows clearly the monotonic evolution
of the acceleration factor from 1 towards 1 + D Key.
Moreover, numerical simulations with the Jadim code
match very well with the theoretical predictions.
Validation in unsteady state
Finally, we propose a last configuration of validation :
reactive transfer in unsteady state from an initial condi
tion such as the chemical imbalance is only imposed at
the interface. The concentrations ('4 and C's at che
mical equilibrium are fixed in the bulk at t = 0. We
scale the results following the analogy with pure diffu
sive transport in an infinite media :
C(14Y, t) C'4' and a = (Y, t) CL'
C An C6o CBx O En k
For an infinite domain (as on figure 3(c) with L >
Oc) an autosimilar analytic solution can be found if the
reaction kinetics is infinite. Using the reduced variable
0 0.5
1.5 2
FIGURE 6: Nondimensional profiles of concentration :
x : w 670.82; 0 : w 474.34; a. : w
54.77 ; +: w 17.32 ; 0: w= 0;
O : w 670.82 with no mass flux at y 0
pour B ;  : analytic solution for pure
diffusion ; : analytic solution for an
infinitely rapid reaction (13).
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
na^
OO DI[IllD o o 0o 0 0 DEIIIIIEE~ilmEIII
Variation of the acceleration factor with the
equilibrium state
We are now studying the evolution of the accelera
tion factor for various values of Keq. When Keq iS Very
low (Keq << 1), the reaction A W B is strongly fa
voured in the way A 4 B, and when Keq iS Very large
(Ke >> 1), the reaction A W B is strongly favoured in
the way A + B. The evolution of EA is spread between
these two behaviours. The expressions of the accelera
tion factor for the two limiting cases corresponding to
irreversible reactions are :
(cAL CAo cosh(al)) aI
AMB : EA = (14)
(cAL CA0) sinh(a~)
A + B : EA 1 BL B ch() (15)
DA CAL CA0
with: ,et7 l L
When the chemical kinetics tends to the infinity, the two
expressions (equations 14 and 15) can be simplified :
*for the reaction A 4 B :
cAL Ao cosh(a) aI
EA
cAL Ao sinh(~)
FIGURE 7: Evolutions of the acceleration factor EA
functions of the Damkhoiler parameter w, for
various values of the equilibrium constant
Ke of the reaction A W B : 0 : Ke,
10Lq; x :, Kq 1; 0 :, Kq 5;
r>:, Ke 10 ; a :, Ke 20 ; + : Ke
100 ; < : Keq = 1000 ;  : exact solution
A 4 B ; : exact solution A + B.
?On Oxidationreduction
For the real case of a glass melt, we consider diffusion
and one single reaction : 1/4 A + B W C. This reaction
corresponds to the oxidationreduction reaction of iron :
1/4 Og + Fe2 Fe3 + 1/2 O2
Because the species O is in excess in the
glass melt, the reaction is then reduced to
1/4 Og + Fe2 Fe +. Its equilibrium constant
is : Keq Fe 7.8395 at T 1450oC according to
Beerkens (2003). The objective is to obtain the evolu
tion of the acceleration factor with the chemical kinetics.
Scaling the transport/reaction equation of A leads to
the following equation :
8t* 842 4 A B
I I+ L2 CBL
with : a(16)
DA o "L4
This parameter takes into account the chemical kinetics,
the characteristic length of the domain, the diffusivity
of A and the equilibrium concentrations of A and B
species.
According to the figure 7, and as Keq Fe 7.8395 ~
10, we expected to obtain a curve presenting two pla
teaus : pure diffusion for low ai and another plateau with
a larger value for high a The figure 8 corresponds to
coshi~b~~ CA0) o
aj co a(o) CAL CA0
*for the reaction A B :
Dg 1 cosh(y)
EA 1 BLDAAL A0
CA0
CA0 ( AL
DA CAL A0
DA CAL AO
In the figure 7, these results are confirmed : at low
Ke,, ie. Keq < 10 the acceleration factor tends to
a constant value when the parameter a is high enough
while the curves with a high Ke,, ie Keq > 10 tends
to a quadratic increase. The Jadim code allows therefore
to reproduce the evolutions of the acceleration factor for
any value of Keq. This analysis, in the simplified case of
one reaction like A W B, help us to anticipate the trends
for the cases of iron and sulphate oxidationreductions.
We expect for the acceleration factor an increasing evo
lution from the diffusion state to the solution of infinite
rate kinetics for the iron, as its equilibrium constant iS
about 10, and a very steep negative evolution for the
sulphate, as its equilibrium constant is very low (about
10s
Uncoupled oxidationreduction reactions in melting
glass
The code has been validated for academic configura
tions. We are now using it for actual reactions occurring
in the glass melt.
xoxot6
102
I I
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
7 shows an evolution of the acceleration factor towards
negative values for the sulphate case. This is confirmed
on the figure 9. The curve has a plateau at low a then
it decreases towards negative values. This negative value
of the acceleration factor corresponds to a flux of oxy
gen penetrating in the bubble due to its production by
the reaction.
this prediction. We have a system where the flux of A
species gets out of the bubble due to the consumption
of oxygen by the reaction. As only the chemical kinetics
was varied, it is clear that the acceleration factor is an
increasing function of a up to a certain value beyond
which the flux is constant. The transfer is then limited
by the diffusion of the oxygen.
2.5 r .
x 10
P ~1
10 10
FIGURE8: Evolution of the acceleration factor as
a function of the parameter a~ for the
oxidationreduction reaction of iron : + : E 4
JADIM.
FIGURE 9: Evolution of the acceleration factor func
tion of the parameter a' for the oxidation
reduction reaction of sulphate : + : E 4 JA
DIM.
The effect of convection
Several theoretical, experimental and numerical stu
dies have been carried out on mass transfer around ri
sing bubbles (McLaughlin (1996)). The mass transfer
is characterized by the Sherwood number, which com
pares the interfacial flux when the bubble is rising to
the pure diffusion flux for a bubble at rest. In the text
book Clift (1978), an expression valid for a spherical
bubble under Stokes flow is proposed : Sh 1+
(1 + 0.564 Pe 2/3) ,4 which is valid for any value of
the Peclet number Pe (Pe = Vbubble dbubble/D). Lo
chiel (1964) proposed Sh = 0.651 Pe 1/2, which is
the asymptotic limit for high Peclet numbers. This ex
pression is valid for low Reynolds numbers and high
Schmidt numbers. This corresponds to bubbles in the
glass : bubbles are very small and the glass melt is very
viscous (Re b < < 1). Moreover diffusivities of dissolved
gaSCS arT Very low, so we have Pe > > 1. The simulation
code have been validated for transfer without reaction
OVeT a ar~ge THnge Of Peclet numbers (figure 10). The first
results of this study are concerned with reactive transfer
between a pure oxygen bubble and molten glass at T=
1450 "C. The dissolved gas coming from the bubble is
consumed by the oxidationreduction reaction of iron
Sulphate oxidationreduction
We follow the same analysis for the oxidation
reduction of sulphate, which is :
1/2 O, + SO, + O' = SO
Because 02 is in excess in the glass melt,
we obtain : 1/2 O, + SO, SO or :
1/2 24 + B = C. The equilibrium constant is in
this case : Ke, Surpheate 1.01363 x 10" at
T 1450 "C according to Beerkens (2003). It corres
ponds to the reaction SO ~ 1/2 O, + SO,
A value of the acceleration factor characterizing the
reaction 1/2 24 + B C is now expected. Scaling the
transport/reaction equation of A leads to the following
equation :
c ac
idt* ili
a (C 41/2 CD
with : a (17
Because the equilibrium constant has a very low va
lue (Ke, Sulpi,,te 1.01363 10" ~ 10"), the figure
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
(Fe /lFe3 ).The Reynolds number has been fixed to :
Re 102. Kinetics constants of oxidationreduction
reaction of iron in molten glass are not well known. Va
rious simulations with increasing Hatta numbers (Ha
Da([O~o[O~oo) ) have been carried out. The simu
lations were carried out around a single bubble of pure
oxygen in a domain corresponding to 120 radii of the
bubble. All the physical quantities will be scaled by
the radius db and by the rising velocity of the bubble
Vt. A polar 2Daxisymmetric mesh has been used. The
cells were refined near the interface in order to resolve
the mass boundary layer for all species. The concen
tration field of O, is initialized to the spatial distribu
tion obtained thanks to a diffusionconvection (without
reaction) calculation converged with Pe[o2] 5.10 4
The concentrations [Fe +] and [Fe3 ] are initialized to
the equilibrium concentrations far from the bubble. At
the initial time t* 0, the system is therefore chemi
cally unbalanced in the bubble wake. In the figure 11
the concentration fields of the species Og, Fe2 and
Fe3+ are represented at different times of the simula
tion. We can observe that the oxygen excess is consum
med, and that the thickness of the mass boundary layer
decreseases as time goes on. Therefore the reduction of
the characteristic size of the boundary layer increases
the [O,] gradient around the bubble and the interfacial
transfer is increased (increase of Sherwood number for
increasing values of Hatta number). When the chemical
reaction is activated, an enhancement of the mass trans
fer is observed (see figure 10) in agreement with recent
results of Pigeonneau (2009).
Experimental results obtained on various devices can
be compared using the mass flux area density. The dif
ferent configurations are : a single bubble rising freely
in the glass melt, a single static bubble in the glass
melt and the oxidation front experiments. The values
of the mass flux area density are respectively : ~1 ~
6.8 10 kg/m /s for the single bubble rising freely,
2~ ~ 5.4 101 kg/m2/s for the single static bubble,
and a3 ~ 8.8 1010kg/m2/s for the oxidation front.
It is clear that convection increases mass transfer at least
by two orders of magnitude. The difference between #2
and a3 can be due to the experimental conditions.
Conclusion
Several results concerning mass transfer in the pre
sence of a reversible chemical reaction have been pre
sented in this paper. First of all the simulation code
was validated on reference configurations, by compa
ring exact solutions or asymptotic solutions (infinite rate
chemical kinetics) to numerical results. Then, the nu
merical tool was used to characterize the effect of the
0,1 1 10 100 1000 10000 le+05
Pe
FIGURE 10l: Sherwood number functions of Peclet
number for various Hatta number,
[O2] [Fe2+] [Fe3+]
t*= 0 t*= 0,5 t*= 0 t*= 0,5 t*= 0 t*= 0,5
FIGURE 11: Nondimensional concentration fields for
Og Fe2 and Fe3+ species for the
2D axisymmetric simulation. Initial state
(t* = 0) on lefthand side, and steady state
(t* 0.5) on righthand side of each fi
gure.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Pigeonneau F., Mass Transfer of a Rising Bubble in Mol
ten Glass with Instantaneous Oxidationreduction Reac
tion, Chemical Engineering Science, vol. 64, issue 13, p
31203129, 2009
Ramachandran PA. and Sharma MM., Simultaneous ab
sorption of two gases accompanied by reversible instan
taneous reactions, Chemical Engineering Science, vol.
26, p 349360, 1971
Ruckenstein E. and Dang V.D. and Gill, WN, Mass
transfer with chemical reaction from spherical one or
two component bubbles or drops, Chemical Engineering
Science, vol. 26, issue 5, 1971
Sherwood T. and Pigford R., Absorption and Extraction,
Chemical Engineering Series, 2nd edition, 1952
Subramanian R.S. and Chi B.,Bubble dissolution with
chemical reaction, Chemical Engineering Science, vol.
35, issue 10, p 21852194, 2009
convection on the mass transfer around a single rising
bubble. We showed the effect on the interfacial mass
flux due to the iron oxidationreduction in the glass
melt. There is still many uncertainties concerning the
physicalchemical properties of the glass melt but the
trends are clear. The experiments allow a better unders
tanding of the interaction between diffusion and reaction
phenomena, but a quantitative comparison between ex
perimental results and numerical simulations is not yet
achieved.
Acknowledgements
We thank the Glass Melting Department of Saint
Gobain Recherche, and more specifically the Quality
Group, for supporting this study. Mrs Perrodin thanks
also Mrs Pedrono for her great help concerning the code
development and Mr Sarrot for his contribution during
the first months of her thesis.
Bibliographie
Beerkens R. G. C., Analysis of advanced and fast fi
ning processes for glass melts,American Ceramic So
ciety, vol.219,p 324, 2003
Clift R. and Grace J. R. and Weber M. E., Bubbles, drops
and particles, Academic Press, New York, 1978
Danckwerts PV. and Lannus A., GasLiquid Reactions,
J. of The Electrochemical Soc., vol. 117, p 369C, 1970
Legendre D. and Magnaudet J., The lift force on a sphe
rical bubble in a viscous linear shear flow, Joumnal of
Fluid Mechanics, vol. 368, p 81126, 1998
Lochiel A.C. and Calderbank P.H., Mass transferin the
continuous phase around axisymetric bodies of revolu
tion,Chem. Eng. Sci., vol. 19, p. 471484, 1964
McLaughlin J.B., Numerical simulation of bubble mo
tion in water, J. Coll. Int. Sci., vol. 184 (2), p. 614, 1996
Magnaudet J. and Rivero M. and Fabre J.,Accelerated
flows past a rigid sphere or a spherical bubble. Partl.
Steady straining flow,Joumal of Fluid Mechanics, vol.
284, p 97135, 1995
Nemec L. and Klouzek J., Modelling of Glass Refining
Kinetics Part 1. Single Bubbles, Ceramics Silikaty 47,
p 8187, 2003
Olander D., Simultaneous Mass Transfer and Equili
brium Chemical Reaction, AIChE Joumnal, 6(2), 233,
1960
