Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 15.6.1 - Mass transfer of a rising bubble in a molten glass
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 Material Information
Title: 15.6.1 - Mass transfer of a rising bubble in a molten glass Reactive Multiphase Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Pigeonneau, F.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: bubble
mass transfer
chemical reaction
molten glass
 Notes
Abstract: The mass transfer around a rising bubble has been studied within the field of glass melting processes. Due to the large value of liquid viscosity, creeping flow is used. The rising bubble is assumed to have a clean interface with a total mobility and the exact solution of Hadamard or Rybczynski is used to define the velocity field around the bubble. The mass transfer of oxygen in the soda-lime-silica glass melt where oxidation-reduction reactions of iron oxides occur is described. The dimensionlessmass transfer coefficient, Sherwood number, is determined as a function of the Péclet number based on the terminal rise velocity of the bubble. Two different techniques have been used: the first based on the boundary layer theory and the second using a finite element method. In order to take into account the oxidation-reduction reaction in a unified framework, a modified Péclet number has been defined as a function of two dimensionless numbers. The first is strongly linked to the equilibrium constant of the chemical reaction and the second is the glass saturation, defined as the ratio of oxygen concentration in the bulk to that at the bubble surface. The Sherwood number, taking into account the chemical reactions, increases with iron content as well as with glass reduction (i.e. small saturation level). The numerical model used to describe bubble shrinkage is based on the preceding results. A comparison between the experimental and numerical results shows the importance of the oxidation-reduction reaction of iron in the mass transfer of oxygen. The shrinkage rate of a pure O2 bubble is enhanced with reduced molten glass iron content.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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Volume ID: VID00384
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Holding Location: University of Florida
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Resource Identifier: 1561-Pigeonneau-ICMF2010.pdf

Full Text
ICMF 2010, Tampa, FL, May 30 June 4, 2010


Introduction

Mass transfer between dispersed bubbles and a contin-
uous liquid phase occurs in many industrial chemical
engineering applications such as liquid-liquid extrac-
tion, boiling, fermentation and other examples, some
of which are listed by Clift et al. (1978) and Leven-
spiel (1972). This is especially true for the glass indus-
try where bubbles are formed by the physical trapping,
within a highly viscous fluid, of atmospheric gases and
the decomposition of raw materials (Shelby 1997). The
elimination of gaseous inclusions occurs by a combina-
tion of mass transfer processes and bubble migration to
the free surface. Therefore, improved understanding of
the shrinkage or growth of a bubble in a complex chem-
ical environment is important in order to optimize the
melting process.
Research on bubble features in molten glass has been
ongoing for many years, both experimentally and the-
oretically. Among the first contributions, Greene and
Gaffuey (1959) developed a rotating experimental de-
vice to stabilize and observe a bubble. Greene and Ki-


tano (1959), Greene and Lee (1965) and Greene and
Platts (1969) used the same experimental set-up to study
pure O, and SO, bubbles in soda-lime glasses. They
demonstrated, for example, that a bubble shrinks faster
in glasses containing As20s. N~mec (1980a,b) devel-
oped an experimental method where a single bubble
is released in a transparent crucible containing molten
glass and its size recorded via a camera. More recently,
Klouiek and N~mec (2003) proposed a method by trans-
ferring the former from the top to the bottom of the cru-
cible when it approaches the free surface to increase in-
teraction time between the bubble and the molten glass.
Doremus (1960) provides one of the first theoreti-
cal contributions studying the shrinkage of a bubble in
molten glass. He determined the diffusion coefficient
of oxygen for different glass compositions based on a
diffusion-controlled process. Readey and Cooper (1966)
studied, via a finite difference method, the mass trans-
fer following a pure diffusion process where the tem-
poral evolution of the bubble's radius was introduced
into the numerical model. In order to take into account
the effect of chemical reactions on the oxygen resorp-


Mass transfer of a rising bubble in a molten glass


F. Pigeonneau*

Surface du Verre et Interfaces, UMR 125 CNRS/Saint-Gobain, Aubervilliers, France

franck. pigeonneau~saint-gobain. com

Keywords: bubble, mass transfer, chemical reaction, molten glass




Abstract

The mass transfer around a rising bubble has been studied within the field of glass melting processes. Due to the large
value of liquid viscosity, creeping flow is used. The rising bubble is assumed to have a clean interface with a total
mobility and the exact solution of Hadamard or Rvbczvnski is used to define the velocity field around the bubble. The
mass transfer of oxygen in the soda-lime-silica glass melt where oxidation-reduction reactions of iron oxides occur is
described.
The dimensionless mass transfer coefficient, Sherwood number, is determined as a function of the P~clet number based
on the terminal rise velocity of the bubble. Two different techniques have been used: the first based on the boundary
layer theory and the second using a finite element method.
In order to take into account the oxidation-reduction reaction in a unified framework, a modified P~clet number has
been defined as a function of two dimensionless numbers. The first is strongly linked to the equilibrium constant of
the chemical reaction and the second is the glass saturation, defined as the ratio of oxygen concentration in the bulk
to that at the bubble surface. The Sherwood number, taking into account the chemical reactions, increases with iron
content as well as with glass reduction (i. e. small saturation level).
The numerical model used to describe bubble shrinkage is based on the preceding results. A comparison between
the experimental and numerical results shows the importance of the oxidation-reduction reaction of iron in the mass
transfer of oxygen. The shrinkage rate of a pure O, bubble is enhanced with reduced molten glass iron content.




ICMF 2010, Tampa, FL, May 30 June 4, 2010


tion pointed out by Greene and Lee (1965), Subrama-
nian and Chi (1980) developed a model where a first-
order irreversible reaction was introduced. The role of
oxidation-reduction reactions of polyvalent cations has
been studied by Yoshikawa and Kawase (1997) for an
immobile bubble. The oxidation-reduction reaction was
assumed to be very fast but limited by the diffusion of
oxygen following a first development by Beerkens and
de Waal (1990). It was pointed out by Yoshikawa and
Kawase (1997) that bubble growth is enhanced by the
oxidation-reduction reaction.
As indicated above, various species interact with a
bubble in molten glass. A multicomponent bubble has
been studied by Ramos (1986) where bubble motion was
introduced using mass transfer coefficients taken from
classical textbooks as (Levich 1962; Clift et al. 1978:
Sadhal et al. 1997). N~mec and Mithlbauer (1980) pro-
posed an equivalent model, valid for bubbles considered
as solid particles with the mass transfer dominated by
advection. Moreover, the mass transfer coefficients are
identical for all gases. Itoh et al. (1997a) also studied
bubble modeling and tested various mass transfer co-
efficients. In order to take into account the oxidation-
reduction reaction, Itoh et al. (1997b) changed the mass
transfer coefficient by introducing an effective diffusion
coefficient. An equivalent method has been recently pro-
posed by Beerkens (2002, 2003).
Despite these contributions, the influence of
oxidation-reduction reactions has not been studied
carefully, especially in the case of a rising bubble. The
mass transfer coefficient for On must be established
and, moreover, precise comparisons with experimental
results are lacking. The influence of oxygen on mass
transfer is studied by taking two glass compositions
with different iron contents and oxidation states. In
order to focus on oxygen transfer, sulfate was not
included in the raw materials apart from as an impurity.
The theoretical model describing the behavior of a
bubble is presented in section 1. The determination of
the mass transfer coefficient for oxygen is given in sec-
tion 2. The section 3 is devoted to the shrinkage of a pure
O, bubble where experimental and numerical results are
compared.



1 Mass transfer of a bubble in molten glass

As mentioned above, a model for a multicomponent
bubble with N, gaseous species is presented here and
has already been described in detail in Pigeonneau
(2007); Pigeonneau et al. (2010) and will be briefly re-
called here.
The bubble radius,

plicit equation


3RT n;
i=1= Po + py(H z) + 2a
4xca <


where R is the universal gas constant, T the temperature,
n; the number of moles of gaseous species i, Po the at-
mospheric pressure, p the glass density, y the gravity, H
the liquid height, z the local position of the bubble and
a the surface tension. In this last equation, each gaseous
species is designed by the index / varying from 1 to N,.
Eq. (1) represents the equilibrium between the pres-
sure obtained from the ideal gas law and the pressure
found by the summation of atmospheric, hydrostatic and
Laplace pressures. In order to solve eq. (1), the number
of moles of each species and the local position, z must
be determined.
The quantity ni is evaluated by the following ordinary
differential equation


Hi= 2i~aShi;D;(C,o OfS),
di


corresponding to the mass balance between the bubble
and its surrounding. In eq. (2), D; is the diffusion coef-
ficient, Sh; the Sherwood number described below, C,
the bulk molar concentration of gaseous species i. The
molar concentration on the bubble surface, Of is given
by Henry's law:
Cs L;P i, (3)

where L; is the solubility coefficient, P, the partial pres-
sure in the bubble of species i. The exponent, P;, is
equal to unity for most species but 1/2 for water since it
is chemically dissolved in molten glass (Paul 1990).
The local position of the bubble is determined by


where here the molten glass is assumed to be on rest.
Experiments done on borate glass by Jucha et al.
(1982), on soda-lime-silica glass by Hornvak and Wein-
berg (1984), and on nuclear waste glass by Li and
Schneider (1993) showed that the rise velocity is given
by the Hadamard-Rybczvnski formula (Hadamard 1911;
Rybczvnski 1911). These results cover a large range
of bubble sizes (micrometers to centimeters). Conse-
quently, the interface between the bubble and the liquid
is considered as totally mobile. So, the terminal rising
velocity of the bubble, VT, is given for a "clean bubble"
by the relationship (Clift et al. 1978: Sadhal et al. 1997)


T3v'




ICMF 2010, Tampa, FL, May 30 June 4, 2010


the diffusion of iron. Nevertheless, as it has been shown
by Cochain et al. (2009), the oxidation of a glass for
temperature greater than the glass transition temperature
is dominated by the diffusion of Oa. So, the iron self-
diffusion is assumed lesser than the oxygen diffusion:


in which, v is the kinematic viscosity defined as the ratio
p/p where p is the dynamical viscosity of the liquid.
The Shenvood number of gaseous species i represents
the ratio of the mass transfer with diffusion and advec-
tion to the mass transfer without relative motion between
the bubble and the molten glass. This quantity is crucial
to accurately describe the physics occurring in the prob-
lem.
In glass melting process, it is needed to separate be-
tween gases not taking into account in chemical re-
actions and gases taking into account in chemical re-
actions. In this work, the only gas occurring in an
oxidation-reduction reaction is O,. Before to present
how the Shenvood number for oxygen is determined, the
Shenvood number for the other species is taken follow-
ing (Clift et al. 1978):


;Do, > ;D,. 3


(11)


Due to the conservation of the iron, only one oxide of
iron could be considered. If the iron self-diffusion is as-
sumed low, the material derivative of molar concentra-
tion of oxidised iron can be related to its reaction rate:


Dt


(12)


From this last equation, the reaction rate of O, can be
expressed as follows


Sh; 1 + 1+ 01.564Pc: 13/4

where the P~clet number, Pei, is defined as


Pe

2 Shenrwood number of oxygen


1 c',
rF~3 =
-24 Dt


(13)


(7) The coefficient -1/4 comes directly from the stoi-
chiometry of reaction (9).
If the chemical reaction is assumed instantaneous, the
equilibrium constant


2.1 Theoretical model of oxygen transport

Considering the case of a rising bubble, the molar
concentration of oxygen, ('o,, is described by the
advection-diffusion-reaction equation

DC'o
=div (;Do2 gradl'o, ) + f o,, (8)

where ;Do, is the diffusion coefficient of oxygen. The
left hand side of this equation represents the material
derivative of molar concentration of oxygen. The reac-
tion rate, ros, is due to the oxidation-reduction of iron
given by

Fe3+ + 1o O - F +1 Oo (9)
'> 4 '

where, Fe3+ is the oxidised state and Fe2+ the reduced
state. The overall oxidation state of the glass is generally
measured as the ratio of the molar concentration of iron
in its reduced state to the total concentration of iron:

R = (10)

where (',,2+ and L've3 are respectively the molar con-
centrations of Fe2+ and Fe +. In the following, C'p =
(',,2+ + ('ve3+ (corresponding to the total quantity of
iron) remains constant due to the conservation of mass.
Since, the transport of O, is coupled with iron due to
the reaction (9), it should be needed to take into account


, via


(14)


of reaction (9) is always verified. By taking the deriva-
tive of the last equation with respect to time, the reaction
rate of oxygen is thus defined as


l'FeKp, DC'o,

lc:16' K + C,1/4 DL
oa Fe o


(15)


This expression depends exclusively on the molar
concentration of oxygen. The reaction rate is, thus, di-
rectly proportional to the material derivative of Co,. At
the stage, we have everything necessary to solve the
transport of O, and determine the Shenvood number de-
fined by


Sho, =D,


(16)


where ko, is the average molar flux at the surface of a
bubble defined by


4i7rct"Co,


(17)


The bubble surface is S and, ac'o, = C',S C0002




ICMF 2010, Tampa, FL, May 30 June 4, 2010


2.2 Dimensionless formulation


~f (C) = N~e/ [So~ + (1I Sau)C:]" (R Sal/q

(1 -R") [So (1 -Si)C] ) ( 23)

The dimensionless group NFe is given by


Before to compute the oxygen concentration, the flow
field around the bubble must be specified. Since the
Reynolds number of the rising bubble in a molten glass
is small, the flow is assumed in the Stokes's regime, see
for more detail (Pigeonneau 2009). The velocity field
is taken from the general solution given by Hadamard
(1911) or Rybczynski (1911). The problem is also as-
sumed to occur under a steady-state regime. Any un-
steadiness may come from the variation of the bubble
size due to shrinkage or growth. However, the ratio be-
tween the time scales for shrinkage or growth and for
mass transfer is sufficiently large to assume the mass
transfer problem as quasi-steady state. This ratio can
be easily determined from results given in (Pigeonneau
et al. 2010) where the typical value is 103.
The problem is solved under the assumption of revo-
lution symmetry. The velocity field is normalized by the
terminal velocity, Vr, given by equation (5) and the spa-
tial variables are written under dimensionless form with
the characteristic length scale equal to 2a. For a bubble
in a liquid, the two dimensionless velocity components
written in spherical coordinates are given by


CFe2+(1 R 0)Sa1/4
160 S


(24)


where So represents the saturation of oxygen in molten
glass given by


&
Co,
and C'" is the molar concentration of Fe
Fe2
the bubble written as follows

C ,2 = ROOCFe.

The boundary conditions for C(r, 9) are


(25)

far away


(26)


C(1/2, 9) =1,
linq C(r, 9)= 0.


(27a)
(27b)


1
cos 9
2r '
1
- sin 9.
4r


2.3 Shenrwood number for a large Pbclet
number

Even if the Reynolds number is small, mass transfer is
mainly driven by the relative motion between bubble and
liquid due to the small diffusion coefficient of oxygen.
This point can be characterized by the determination of
the Schmidt number


i (


where u, and us are the velocity components along the
radial and polar directions respectively and r is the ra-
dius with 9 the polar angle.
The oxygen concentration is normalized by the rela-
tionslup


Sc Y= D s


(28)


Co,
oa


Of
oa


which varies between 106 at 1500 C and 5 10s at 1200
C for a classical soda-lime-silica glass. Therefore, the
P~clet number's order of magnitude, Re Sco,, is 103 for
a bubble radius equal to 10-3 m. So, the boundary layer
theory can be used to solve equation (22). This method
has already been proposed by Levich (1962); however,
the self-similar solution of Ruckenstein (1967) is used
here. Detail are not reproduced in the paper and can be
found in (Pigeonneau 2009). Only the results are sum-
marized in the paper.
In order to identify the importance of the chemical
reaction, we compare the solution with one without re-
action where the Sherwood number is well known and is
given by (Levich 1962):


where CS is the molar concentration of oxygen at the
bubble surface given by equation (3). The concentration
in the liquid far away the bubble, C@, can be deter-
mined with the equilibrium constant as follows

[~V2=KFe(1 R")14" (1

where R" is the oxidation state far away the bubble.
Finally, C is a solution of the partial differential equa-
tion given by

[1+ f (C)] u,~d + -s -
Br r 89 Pe of '

~ r ar si 98 sin 9 (22)

where f(C) comes from the reaction rate and is given
by


She = 0.6516.


(29)


The subscript 0 is added in order to specify that this ex-
pression is obtained without reaction.




ICMF 2010, Tampa, FL, May 30 June 4, 2010


co" (c) so()R"=04,









Figure l: Sh ,2/Shi as a function of NFe for T
1300, 1400, and 1500 oC and (a) R" 0.1,
(b) R" = 0.2, (c) R" = 0.3 and (d) R" =
0.4.


the Sherwood number is determined for three temper-
atures, four oxidation states and a range of iron contents
equal to [0.01; 0.5] weight %. The mass transfer coeffi-
cient is given in Figure 1 by plotting the ratio Sh2/Shi
as a function of N~e. As expected, the Sherwood num-
ber increases with iron content due to the fact that the di-
mensionless group NFe is directly proportional to CGe.
When the iron content approaches zero, the Sherwood
number reaches the value without reaction, Eq. (29).
The representation in terms of Sh2/Shi shows that Sh
can be written as follows

Sho2 = Sho 1 + aNFe, (30)

where a is a numerical coefficient depending on temper-
ature and oxidation state.
The influence of the reaction is important at low tem-
peratures and also for reduced glass (high CFe2+). This
effect can easily be understood due to the feature of
f (C) as a function of saturation state, Sa. Indeed, when
both the temperature is low and the oxidation state is
small (high value of R), Sa can take small values. In
this case, f(C) becomes greater and greater. Conse-
quently, mass transfer is greatly increased as it is pos-
sible to see in Figure 1 where, for a glass at R"O 0.4
and T 1300 oC, the Sherwood number can be ten
times greater than without reaction when the iron con-
tent reaches 0.5 weight %.
The behavior of the Sherwood number as a function of
NFe shown in Figure 1 suggests that when the oxidation-
reduction reaction is taken into account, boundary layer
thickness is proportional to 1/ JPe~o2 Fe
In order to find the behavior of a, a large number
of numerical simulations of the boundary layer solu-
tion were performed as a function of oxidation state and
temperature. As already shown, the relevant parameter
seems to be glass saturation. Thanks to this extensive


Figure 2: a as a function of So for various oxidation
states.


analysis, a can be plotted as a function of So as is done
here in Figure 2.
As expected, the values of a obtained for various tem-
peratures and oxidation states match very well when
these data are plotted as a function of saturation state.
By using the numerical data, it is possible to find an ap-
proximate expression of a given by the relationship


(31)


a 3.05Sa-o.sys 1.28Sa_ .


This equation is represented by a solid line in Figure
2 where good agreement with the numerical data is ob-
tained. This relationship is determined by using a har-
monic average of the two asymptotic trends observed
both at small and large values of Sa. Indeed, when the
glass is undersaturated and So is small, a is a function
of Sa-o.srs and for larger values of Sa, a is a function
of Sa 1
The effect of the oxidation-reduction reaction of iron
oxide can be seen as an enhancement of the advection
term. Inversely, the reaction's effect may be seen as a
reduction of the diffusion coefficient as already pointed
outby Beerkens and de Waal (1990). Whatever the point
of view, the effective P~clet number increases. From
these developments, a modified P~clet number can be
written as follows


Pe'o, = Peo2 [1+ a(Sa)NFe] ,


(32)


where a(Sa) is given by equation (31).
Finally, the Sherwood number can be written using
the same expression independent of chemical reaction,
Eq. (29), where Peo2 iS replaced by Pe'o2

2.4 Shenrwood number for a moderate Pbclet
number

In order to establish a solution for a large range of PC-
clet numbers, a numerical method has been used to solve




ICMF 2010, Tampa, FL, May 30 June 4, 2010


(a) T = 1400 oC, R" = 0.2


(a) T = 1400 oC, R" = 0.2
-withoutreaction g
C Oe=0 01 weight %
Ove= 005weight %
- CFe=0 1 weight %
- CFe=0 2 weight %
SCFe=0 3 weight %


- withoutreaction
CFe=0 01 weight %
CjFe=0 05 weight %
CFe=0 3 weight %


IBoundary layer sol


Peo2
(b) T = 1400 oC, R" = 0.3


(b) T =1400 OL, To


without reaction
CFe=0 01 weight %
CFe=O 05 weighl 3

CFe=0 3 weight %


--
~.,
,.,,
,,,
~-p I
a
~-~- .p _- a
B
88

i


Figure 4: Sherwood number as a function of a modified
P~clet number at T 1400 oC and R"
0.2 and 0.3 where the total iron content varies
in the range 0.01-0.3 weight %.



iron leads to an enhancement of the mass transfer co-
efficient. The comparison with boundary layer theory
shows that the agreement between the two methods is
relatively good.

A modified P~clet number has been proposed above,
leading to a unified Sherwood number. This modified
P~clet number is a function of the dimensionless group
Nee and the glass saturation, Sa. In order to validate
that this modification can be extended to a large range of
P~clet numbers, the Sherwood numbers obtained using
finite element method are plotted in Figure 4 as function
of Pe'o2 giVen by equation (32). All numerical values
of the Sherwood number correspond well to the master
curve equivalent to that obtained without considering the
chemical reaction.

Consequently, the Sherwood number can be written


Figure 3: Sherwood number as a function of P~clet
number at T 1400 oC and R" 0.2 and
0.3 where the total iron content varies in the
range 0.01-0.3 weight %.



eq. (22) by finite element method. The convection dom-
inated transport problem requires a stabilization tech-
nique. Streamline diffusion is employed where only an
artificial diffusion along the streamlines is added and, in
order to minimize numerical error, the mesh is adapted
with an estimation of a posteriori error. For more detail,
see (Pigeonneau 2009).
The Sherwood number obtained from the finite ele-
ment analysis is given in Figure 3 where CFe varies over
the range of [0.01; 0.3] weight % for T 1400 0 C. The
oxidation state in the glass bulk is taken to be equal to
0.2 and 0.3. The numerical solutions are compared with
that obtained without chemical reaction represented as
a solid line in Figure 3. The solution obtained from
boundary layer theory presented in the preceding sub-
section is also given where the Sherwood numbers are
plotted using dashed lines.
The oxidation-reduction reaction enhances the mass
transfer coefficient over the whole range of P~clet num-
bers. For a glass with R"O 0.3 at T 1400 0 C,
the Sherwood number is multiplied by 2 when the PC-
clet number is equal to 1. The increase of the Sherwood
number is more important when the P~clet number is
high since, for a glass with R"O = 0.3 at T = 1400
oC, Sho2 iS 3.8 times greater than the value without re-
action for Peo2 104. The concentration of reduced


Sho2 = + (1 + 0.564Pe@ ~~)/


(33)


which is the same correlation proposed by Clift et al.
(1978) where the classical P~clet number is replaced by
the modified P~clet number introduced in this work.


Boundar layr s




ICMF 2010, Tampa, FL, May 30 June 4, 2010


3 Shrinkage of a pure oxygen bubble


(a) T =1400 'C


Exp data
Nom sol


The behavior of a pure oxygen bubble is now inves-
tigated where the numerical simulations of the mass
transfer process following equations (1), (2) and (4)
are obtained using a fourth-order Runge-Kutta method.
The Sherwood numbers are taken from equation (6) for
gaseous species not taken into account in chemical re-
actions. For oxygen, the development presented above
is used. The numerical computations are carried out us-
ing solubilities, and diffusion coefficients of all species
taken from the literature (Beerkens 2003).
The effect of iron content on the shrinkage rate had
been studied experimentally on two glasses where the
iron content is equal to 0.03 weight % for the glass 1 and
0.1 weight % for the glass 2. The glass compositions are
given in (Pigeonneau et al. 2010). The experimental set-
up is not presented in this article where all details can
be found in (Pigeonneau et al. 2010). Since the bubble
is composed initially with O,, the bubble size decreases
with the time. In the following, the results are focused
on the size behavior.
The bubble size as a function of time at T = 1400 and
T = 1450 oC for glass 1 (low iron content) is given in
Figure 5 and it can be seen that there is an adequate fit
between experimental and numerical results. Two dis-
tinct stages are observed: bubble shrinkage followed by
a plateau where the bubble size does not change signifi-
cantly. Remark however, the bubble size is numerically
overestimated.
Bubble radius versus time for glass 2 is given (for
two temperatures) in Figure 6. The numerical results
shows good agreement with the experimental data and
the bubble sizes obtained numerically and experimen-
tally at T 1450 oC are very close over the time period
studied.
These results have been obtained without data fitting.
The only change is the introduction of the new Sher-
wood number proposed in the preceding section. In
order to show the influence of the iron content more
clearly, The experimentally observed and numerically
calculated temporal evolution of bubble size (normal-
ized by its initial value) is plotted as a function of time
for the two glass compositions at T 1400 oC in Fig-
ure 7. Even though the temperature is identical, mean-
ing that the diffusion coefficients and dynamic viscosity
are equal, the shrinkage rate changes from one glass to
another. This trend can be described by the theoretical
model used in the present work: the shrinkage rate of an
oxygen bubble increases with the ferrous iron content.
The importance of oxidation-reduction on the shrink-
age of oxygen bubbles in molten glass has previ-
ously been underlined by Greene and Platts (1969) who
showed an enhanced shrinkage rate for a glass contain-


a (mm)












a (mm)


Figure 5: Bubble size as a function of time for glass 1
at (a) T =1400 oC, (b) T =1450 oC. Com-
parison between numerical model with a new
mass transfer for O, and experimental results.


(a) T =1300 'C

40 Expndat




.-~
-- 9.


almm)












a (mme


Figure 6: Bubble size as a function of time for glass 2
at (a) T 1300 0C, (b) T 1400 0C. Com-
parison between numerical model with a new
mass transfer for O, and experimental results.







.

-- -a -mm


--Glass1, num res
-Glass 2, num res


al ao


tas)

Figure 7: Bubble size divided by its initial value as
a function of time for the two glasses at
T 1400 oC. Comparison between numer-
ical model with mass transfer and experimen-
tal results.


ing arsenic oxides. The present work has focused on
glass containing iron oxides and not arsenic oxides, ney-
ertheless, oxygen dissolution is of the same nature in-
volving diffusion, advection, and reaction. The ability of
molten glass to absorb oxygen is related to the quantity
of reduced iron. The increase in Sherwood number cou-
pled with the reduced state of iron and total iron content
obtained in the preceding section expresses this ability.


Conclusion

Mass transfer around a rising bubble in molten glass
has been investigated. A model to describe a multi-
component bubble is first presented. The work is then
focused on coupling advection-diffusion to chemical re-
action processes for oxygen in molten glass. The bub-
ble/1iquid interface was assumed to be completely mo-
bile which is the case for a clean bubble, and chemical
equilibrium of the iron oxide oxidation-reduction reac-
tion has also been assumed. A model to describe the
mass transfer around a bubble under a quasi-steady state
regime was presented. The dimensionless mass transfer
coefficient, the Sherwood number, was determined via
two methods: the first founded on boundary layer the-
ory and the second on a numerical resolution of the full
problem. The influence of increasing iron content was
investigated and it was found that it increases the Sher-
wood number. A definition of a modified P~clet number
taking into account the oxidation-reduction reaction has
also been proposed, which leads to a unified relationship
between the Sherwood number and the modified P~clet
number.
The last development is applied to describe the mass
transfer of a pure O, bubble. By comparing the two glass
compositions studied in the current work, it was seen
that the shrinkage rate of a pure oxygen bubble increases
with reduced iron content in the molten glass. One crit-


ICMF 2010, Tampa, FL, May 30 June 4, 2010


ical conclusion for glass melting is that the oxidation-
reduction of iron at high temperature is mainly driven
by oxygen transfer.




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