Transport phenomena of bubbles in a high viscous ﬂuid · Transport phenomena of bubbles in a high viscous ﬂuid F. Pigeonneau Surface du Verre et Interfaces, UMR 125 CNRS/Saint-Gobain,

Transport phenomena of bubbles ina high viscous fluid

F. PigeonneauSurface du Verre et Interfaces, UMR 125

CNRS/Saint-Gobain, Aubervilliers (France)

Presented at the COMSOL Conference 2010 Paris

http://www.comsol.com/conf_cd_2011_eu

Industrial context

u Glass melting is a chemical process :

SiO2 + CaCO3 + Na2CO3 ⇒ SiO2 · CaO · Na2O+CO2.

u Large production of CO2 : 0.2 kg of CO2 per kg of glass.

u Low solubility of CO2 ⇒ large quantity of bubbles in moltenglass.

u Introduction of “fining” species leads to the bubble removalat high temperature (1450 ◦C).

Industrial context

u Glass melting is a chemical process :

SiO2 + CaCO3 + Na2CO3 ⇒ SiO2 · CaO · Na2O+CO2.

u Large production of CO2 : 0.2 kg of CO2 per kg of glass.

u Low solubility of CO2 ⇒ large quantity of bubbles in moltenglass.

u Introduction of “fining” species leads to the bubble removalat high temperature (1450 ◦C).

The mass transfer between bubbles and moltenglass is essential in glass melting

Industrial context

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crown

burners

raw mat.foam hot spot

cold spot

electrodethroat

bathFig. 1 : Sketch of furnace.

Outline

1. Main features of mass transfer between a bubble andmolten glass

2. Experimental study of the shrinkage of O2 bubble

3. Determination of the Sherwood number for O2

4. Modeling of the shrinkage of O2 bubble

1. Main features of mass transfer between a bubbleand molten glass

molten glass

CO2 : raw mat.

02, SO2 : fining agents

H2O, N2 : atmosphere


molten glass

CO2 : raw mat.

02, SO2 : fining agents

H2O, N2 : atmosphere

Mass transfer with a multicomponent bubble.


u At 1500 ◦C, for a bubble radius of 1 mm, the Reynoldsnumber is 10−3.

u Due to low diffusion coefficient, the Péclet number isgreater than 103.

Fig. 2 : O2 concentration around a rising bubble at Pe = 103, Comsol,Multiphysicsr 3.5.


u At 1500 ◦C, for a bubble radius of 1 mm, the Reynoldsnumber is 10−3.

u Due to low diffusion coefficient, the Péclet number isgreater than 103.

Fig. 2 : O2 concentration around a rising bubble at Pe = 103, Comsol,Multiphysicsr 3.5.

Mass transfer is mainly driven by advection.


u Iron is the major transition metal present in glass :uNaturally present in mineral elements used as rawmaterials ;

uAdded to change the optical properties of glass.uIron content : [0.01; 5] wt %.

u Iron undergoes the following oxidation-reduction reaction :

Fe3+ +12

O2− Fe2+ +

14

O2. (1)


u Iron is the major transition metal present in glass :uNaturally present in mineral elements used as rawmaterials ;

uAdded to change the optical properties of glass.uIron content : [0.01; 5] wt %.

u Iron undergoes the following oxidation-reduction reaction :

Fe3+ +12

O2− Fe2+ +

14

O2. (1)

The transport of O2 is also controlled by iron.

2. Experimental study of the shrinkage of O2 bubbleExperimental set-up

top view

crucible

crucible

furnacefront view

silica tube

glass

silica window

vide

oca

m.

Fig. 3 : Sketch of the laboratory furnace.

(a) the bubble formation (b) a rising bubble

Fig. 4 : Photographs of the bubble in theexperiment.

uTo increase the residence time of a bubble, it is trapped and transferedwith a silica tube (“Shuttle method”)1.

uThe bubble size is determined by a counting of pixels.

1J. Kloužek, and L. Nemec (2003) Ceramics 47, 155-161.

2. Experimental study of the shrinkage of O2 bubbleStudied glasses

u Two glasses are studied based on the same composition(flat glass).

u Only, the iron content changes :uGlass 1 : 0.03 wt % of iron ;uGlass 2 : 0.11 wt % of iron.

u No sulfate ⇒ very low concentration of SO2.

2. Experimental study of the shrinkage of O2 bubble

0 100 200 300 400 500 600 700 800 900 1000

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

a/a0

t (s)

Glass 1, a0 = 1 mmGlass 2, a0 = 1.3 mm

Fig. 5 : Bubble size vs. t with glasses at low and high iron content obtained atT = 1400 ◦C.

3. Determination of the Sherwood number for O2Present model

u Sherwood number for 02 ⇒ solve theadvection/diffusion/reaction equation :

DCO2

Dt= DO2∇

2CO2 + rO2 . (2)

u Assumptions :uThe flow around the bubble is in the Stokes regime.uInterface between the bubble and glass is fully mobile2.uOxidation-reduction reaction of iron oxide is in chemicalequilibrium.

uDiffusion of iron is assumed very low.

rO2 = − CFeKFe

16C3/4O2

(KFe + C1/4O2

)2

DCO2

Dt. (3)

2 E. J. Hornyak & M. C. Weinberg (1984) J. Am. Ceram. Soc. 67, C244-C246.

3. Determination of the Sherwood number for O2Present model

u Four dimensionless numbers are involved :

Pe =2aVT

DO2

, (4)

Sa =C∞

O2

CSO2

. (5)

NFe =C∞

Fe2+(1 −R∞)Sa1/4

16CSO2

, (6)

R∞ =C∞

Fe2+

C∞

Fe2++ C∞

Fe3+

. (7)

u The problem is solved by two methods :uBoundary layer theory for large Péclet number.uA full numerical method developed in Comsol,Multiphysicsr 3.5.

3F. Pigeonneau (2009) Chem. Eng. Sci., 64(13), 3120-3129.

3. Determination of the Sherwood number for O2Effect of iron content

100

101

102

103

10410

0

101

102

Pe

Sh

without reactionCFe=0.01 weight %CFe=0.05 weight %CFe=0.1 weight %CFe=0.2 weight %CFe=0.3 weight %

Boundary layer sol.

Fig. 6 : Sherwood number versus Péclet number at T = 1500 ◦C, R∞

= 0.2.

3. Determination of the Sherwood number for O2Effect of iron content

100

101

102

103

10410

0

101

102

Pe

Sh

without reactionCFe=0.01 weight %CFe=0.05 weight %CFe=0.1 weight %CFe=0.2 weight %CFe=0.3 weight %

Boundary layer sol.

Fig. 6 : Sherwood number versus Péclet number at T = 1500 ◦C, R∞

= 0.2.

Enhancement of mass transfer coefficient dueto the chemical reaction for all values of Pe.

4. Modeling of the shrinkage of O2 bubble

u The bubble size changes due to the mass transfer ofvarious species :

dni

dt= 2πaShiDi(C

∞

i − CSi ), (8)

CSi = LiP

βii , (9)

3RTNg∑

i=1

ni

4πa3 = P0 + ρg(H − z) +2σ

a, (10)

dzdt

= VT . (11)

u Equation system solved by a fourth-order Runge-Kuttamethod.

4. Modeling of the shrinkage of O2 bubbleComparison of various Sherwood numbers

0 250 500 750 1000 1250 1500 1750 2000 2250 2500t (s)

0.4

0.6

0.8

1

1.2

1.4

a (mm)

Exp. dataNum. sol., Sh ∝ 3

√Pe

Fig. 7 : a versus time obtained for the glass 1 (low iron content) at T = 1400◦C.


0 250 500 750 1000 1250 1500 1750 2000 2250 2500t (s)

0.4

0.6

0.8

1

1.2

1.4

a (mm)


√Pe

Num. sol., Sh ∝√

Pe



0 250 500 750 1000 1250 1500 1750 2000 2250 2500t (s)

0.4

0.6

0.8

1

1.2

1.4

a (mm)


√Pe


PeNum. sol., present model



0 250 500 750 1000 1250 1500 1750 2000 2250 2500t (s)

0.4

0.6

0.8

1

1.2

1.4

a (mm)


√Pe


PeNum. sol., present model


The relevance of oxidation-reduction reaction ofiron is clearly established.

4. Modeling of the shrinkage of O2 bubbleEffect of iron content

0 100 200 300 400 500 600 700 800 900 1000

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

a/a0

t (s)

Glass 1Glass 2

Fig. 8 : Bubble size vs. time for the two glasses, at T = 1400 ◦C.

4. Modeling of the shrinkage of O2 bubbleEffect of iron content4

0 100 200 300 400 500 600 700 800 900 1000

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

a/a0

t (s)

Glass 1, exp. res.Glass 2, exp. res.Glass 1, num. res.Glass 2, num. res.


4. Modeling of the shrinkage of O2 bubbleEffect of iron content4

0 100 200 300 400 500 600 700 800 900 1000

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

a/a0

t (s)

Glass 1, exp. res.Glass 2, exp. res.Glass 1, num. res.Glass 2, num. res.


Results obtained without data fitting.4F. Pigeonneau, D. Martin & O. Mario (2010) Chem. Eng. Sci., 65(10),

3158-3168.

Conclusion & perspectives

u Determination of a Sherwood number taking into accountthe redox reaction of iron oxides.

u The experimental results obtained with two iron contentsare well reproduced with the proposed model :

uThe larger the reduced iron content, the larger theshrinkage of the O2 bubble.

u Perspectives :u Development of numerical model to study mass transfer

interaction between molten glass and a large bubblepopulation (bubbly flow model).

u Bubble motion in a crucible (Level-set model).

Documents

Transport phenomena of bubbles in a high viscous ﬂuid · Transport phenomena of bubbles in a high viscous ﬂuid F. Pigeonneau Surface du Verre et Interfaces, UMR 125 CNRS/Saint-Gobain,