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/ 73 Phénomènes de transfert dans les liquides réactifs à haute viscosité. Application au procédé d élaboration du verre F. Pigeonneau Habilitation à Diriger des Recherches Université Pierre et Marie Curie

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/ 73 Phénomènes de transfert dans les liquides réactifs à haute viscosité. Application au procédé d élaboration du verre F. Pigeonneau Habilitation à Diriger des Recherches Université Pierre et Marie Curie 25 mai 22 2 / 73 I started at Saint-Gobain Recherche (SGR) in 2 to study the physics of glass melting. I am currently working in the Joined lab Surface du Verre et Interface. Treat topics at the crossroad between industry and research: Take an applied subject to transform an academic subject. In the team Heteregeneous and Reactive Materials The main purposes: Improve the basic knowledge in Glass melting Building materials I will speak only about of soda-lime silica glasses. 3 / 73 I started at Saint-Gobain Recherche (SGR) in 2 to study the physics of glass melting. I am currently working in the Joined lab Surface du Verre et Interface. Treat topics at the crossroad between industry and research: Take an applied subject to transform an academic subject. In the team Heteregeneous and Reactive Materials The main purposes: Improve the basic knowledge in Glass melting Building materials I will speak only about of soda-lime silica glasses. 4 / 73 Glass melting Glass melting is a chemical process: sand limestone soda ash (7%) (5%) (5%) silicate SiO 2 + CaCO 3 + Na 2 CO 3 SiO 2 CaO Na 2 O+CO 2 (g). () Glass melting (SGR) Glass melting 5 / 73 Glass melting 6 / 73 7 / 73 Outline natural convection in glass furnaces mass transfer between bubbles and molten glass transport phenomena glass foam stability 8 / 73 Natural convection in glass furnaces. Origins 2. Horizontal convection 3. Synthesis 4. Perspectives 9 / 73 Natural convection in glass furnaces Origins hot spot 55-6 C cold spot 5-2 C Figure : Sketch of the glass furnace. 5 C / 73 Natural convection in glass furnaces Origins hot spot 55-6 C cold spot 5-2 C Figure : Sketch of the glass furnace. 5 C / 73 Natural convection in glass furnaces Origins Natural convection drives heat transfer; glass homogenization; spreading of the residence time distribution. 2 / 73 Natural convection in glass furnaces Origins,2 mean residence time 2.7 days,,8 E (h ),6,4,2 one week to renew 95 % the liquid minimum of the time residence 7 hours t (h) Figure 2: Residence time distribution for a float furnace. 3 / 73 Natural convection in glass furnaces 2 Horizontal convection Heat and mass transfer are important to know for the glass furnace designers. Find the control parameters of the heat and mass transfer in glass furnaces Work achieved in collaboration with J.-M. Flesselles (Glass melting Dept., SGR). 4 / 73 Natural convection in glass furnaces 2 Horizontal convection Larger than height. Wider than height. Heating from above. 5 / 73 Natural convection in glass furnaces 2 Horizontal convection Temperature applied on one horizontal boundary. Closed system. Figure 3: Rectangular cavity with kinematic and heat boundary conditions. 6 / 73 Natural convection in glass furnaces 2 Horizontal convection Configuration studied in geophysics Overturning of oceans due to the heating of the Sun. Convection appears without threshold. Three dimensionless numbers have to be taken into account Pr = ν κ A = H L Ra = gβ TH3 νκ (2a) (2b) (2c) 7 / 73 Natural convection in glass furnaces 2 Horizontal convection Configuration studied in geophysics Overturning of oceans due to the heating of the Sun. Convection appears without threshold. Three dimensionless numbers have to be taken into account Pr = ν κ 3 (2a) A = H..5 (2b) L Ra = gβ TH3 4 7 νκ The Prandtl number is irrelevant when Pr : (2c) Only two parameters : A and Ra. 8 / 73 Natural convection in glass furnaces 2 Horizontal convection Figure 4: Isotherms and streamlines in the cavity of A = /5. Two regimes are observed: Conductive regime; Convective regime. 9 / 73 Natural convection in glass furnaces 2 Horizontal convection What is the control parameter when A and Ra change? What are the laws of heat and mass transfer in the two regimes? 2 / 73 Natural convection in glass furnaces 2 Horizontal convection What is the control parameter when A and Ra change? What are the laws of heat and mass transfer in the two regimes? From the scale analysis, we show : Ra A 2 is the only parameter. Ra A 2 = H2 /κ L/U, (3) U = Ra κ L. (4) Flesselles & Pigeonneau, C. R. Mécanique, 332: , 24. Natural convection in glass furnaces 2 Horizontal convection 2 2 Pe A 2 = 4.6 (Ra A 2 ) 2/5 5 2 Pe A Pe A 2 = π Ra A Ra A 2 A = /5 A = /4 A = /3 A = /2 A = / A = /7 A = /5 Harm. aver Ra A 2 Figure 5: Pe A 2 vs. Ra A 2 where Pe = u maxl κ. 2 F. Pigeonneau & J.-M. Flesselles. Int. J. Heat Mass Transfer, 55: , / 73 / 73 Natural convection in glass furnaces 2 Horizontal convection 2 Nu =.245(Ra A 2 ) /5 5 Nu =.8 4 (Ra A 2 ).4 Nu - -2 A = /5 A = /4 A = /3 A = /2 A = / A = /7 A = /5 Gen. average Ra A 2 Figure 6: Nu vs. Ra A 2 where Nu = A /A ( θ y ) 2 (x)dx. 2 F. Pigeonneau & J.-M. Flesselles. Int. J. Heat Mass Transfer, 55: , 23 / 73 Natural convection in glass furnaces 3 Synthesis Only, one parameter: Ra A 2. Two regimes with very well established scaling laws. The glass furnaces are in the convective regime. The typical velocity is given by ( ) β Tg 2/5 u κ 3/5 L /5. (5) ν Since κ β R : Strong influence of the infrared absorption of glass. H is irrelevant to describe the heat and mass transfer. 24 / 73 Natural convection in glass furnaces 4 Perspectives, applied works Find a link between the residence time distribution and the control parameter of the flow, Ra A 2 : Important to predict the period of transition between two glasses. Develop the same study for electric furnaces: Energy source in the bulk (Coll. with S. Adjoua, SGR). 25 / 73 Natural convection in glass furnaces 4 Perspectives, fundamental works Stability of horizontal convection: The flow becomes unsteady when Ra increases. E. Uguz (Ph. D. student of Univ. of Florida) has developed a spectral solver: Collaboration with G. Labrosse (Univ. d Orsay) and R. Narayanan (Univ. of Florida). Determination of the stability diagram Ra vs. A. 26 / 73 Natural convection in glass furnaces 4 Perspectives, fundamental works Unsteady flow in the cavity 27 / 73 Natural convection in glass furnaces 4 Perspectives, fundamental works 7 Pr = 3 Pr = 7 Ra A Figure 7: Critical Rayleigh number versus A for Pr = 3 and Pr = 7. 28 / 73 Natural convection in glass furnaces 4 Perspectives, fundamental works Siggers et al. 3 established using a variational method that the heat flux (Nusselt number) behaves as Ra /3. However, the numerical simulations give always a trend as Ra /5? Outline 3 J. H. Siggers et al. J. Fluid Mech., 57:55-7, 24. 29 / 73 Mass transfer around a bubble. Main features 2. Experiment with O 2 bubbles 3. Sherwood number determination 4. Experiment vs. numerical 5. Synthesis 6. Perspectives 3 / 73 Mass transfer around a bubble Main features molten glass H 2 O, N 2 : atmosphere CO 2 : raw mat. 2, SO 2 : fining agents 3 / 73 Mass transfer around a bubble Main features molten glass H 2 O, N 2 : atmosphere CO 2 : raw mat. 2, SO 2 : fining agents Mass transfer with a multicomponent bubble. 32 / 73 Mass transfer around a bubble Main features At 5 C, for a bubble radius of mm, Re = V T 2a ν 3. Due to low diffusion coefficient, Pe = V T 2a D 3. Figure 8: O 2 concentration around a rising bubble at Pe = 3. 33 / 73 Mass transfer around a bubble Main features At 5 C, for a bubble radius of mm, Re = V T 2a ν 3. Due to low diffusion coefficient, Pe = V T 2a D 3. Figure 8: O 2 concentration around a rising bubble at Pe = 3. Mass transfer is mainly driven by advection. 34 / 73 Mass transfer around a bubble 2 Experiment with O 2 bubbles top view furnace front view (a) the bubble inflation (b) a rising bubble crucible silica tube crucible video cam. silica window glass Figure : Snapshots of the bubble in the experiment (O. Mario & E. Grignon, SGR). Figure 9: Sketch of the laboratory furnace. To increase the residence time of a bubble, it is trapped and transfered with a silica tube ( Shuttle method ) 4. The bubble size is determined by a counting of pixels. 4 J. Kloužek, and L. Němec (23) Ceramics 47, 55-6. 35 / 73 Mass transfer around a bubble 2 Experiment with O 2 bubbles Two glasses were studied based on the same composition (flat glass). Only, the iron content changes: Glass :.3 wt % of iron; Glass 2:. wt % of iron. No sulfate very low concentration of SO2. 36 / 73 Mass transfer around a bubble 2 Experiment with O 2 bubbles a/a.6.5 Glass, a = mm Glass 2, a =.3 mm t (s) Figure : Bubble size vs. t with glasses at low and high iron content obtained at T = 4 C. Mass transfer around a bubble 2 Experiment with O 2 bubbles a/a.6.5 Glass, a = mm Glass 2, a =.3 mm t (s) Figure : Bubble size vs. t with glasses at low and high iron content obtained at T = 4 C. Need to explain the effect of the iron content. 37 / 73 Mass transfer around a bubble 3 Sherwood number determination Sherwood number for O2 solve the equation: DC O2 Dt = D O2 2 C O2 + ṙ O2. (6) Assumptions: The flow around the bubble is in the Stokes regime. Interface between the bubble and glass is fully mobile 5. Oxidation-reduction reaction of iron oxide is in chemical equilibrium. Diffusion of iron is assumed very low. ṙ O2 = 6C 3/4 C Fe K Fe DC O2. (7) O 2 (K Fe + C /4 ) 2 Dt O 2 5 E. J. Hornyak & M. C. Weinberg (984) J. Am. Ceram. Soc. 67, C244-C / 73 39 / 73 Mass transfer around a bubble 3 Sherwood number determination 6 2 without reaction C Fe =. weight % C Fe =.5 weight % C Fe =. weight % C Fe =.2 weight % C Fe =.3 weight % Sh Boundary layer sol Pe Figure 2: Sherwood number versus Péclet number at T = 5 C. 6 F. Pigeonneau. Chem. Eng. Sci., 64(3):32-329, 29. Mass transfer around a bubble 4 Experiment vs. numerical a/a Glass, exp. res. Glass 2, exp. res. Glass, num. res. Glass 2, num. res t (s) Figure 3: Bubble size vs. time for the two glasses, at T = 4 C. Comparison between experimental and numerical results. 7 F. Pigeonneau, D. Martin & O. Mario. Chem. Eng. Sci., 65(): , 2. 4 / 73 4 / 73 Mass transfer around a bubble 5 Synthesis Effect of the iron content established. Determination of the modified Péclet number taking into account the iron content. The scaling law of O2 bubble resorption has been done 8. 8 F. Pigeonneau. Int. J. Heat Mass Transfer, 54:448:455, 2. 42 / 73 Mass transfer around a bubble 6 Perspectives, applied works Describe a bubble population: Bubble consumption of gas chemical equilibrium of the glass. Two ways: Lagrangian method (Ph. D. thesis of M. Perrodin, coll. E. Climent, IMFT). Development of a population balance equation and solve it. Determine the bubble flux toward the free surface of the glass bath: Important to know to determine the foam layer. Outline 43 / 73 Glass foam stability. Motivations 2. Bubble drainage on the free surface 3. Synthesis 4. Perspectives Glass foam stability Motivations 44 / 73 45 / 73 Glass foam stability Motivations Primary foam Secondary foam 46 / 73 Glass foam stability Motivations Foam is a thermal screen: Reduction of 6 % of radiative fluxes: Increase of fuel consumption. Reduction of temperature in the bath. Increase of temperature in combustion space: Wear of the crown; Rising of pollutant emissions. What are the main phenomena leading the glass foam stability? Chemistry? Heat transfer? 47 / 73 Glass foam stability Motivations Mesoscopic studies: Bubble drainage on a free surface: Experimental work of Ph. D. thesis of H. Kočárková (coll. with F. Rouyer, Lab. Navier, Ecoles des Ponts et Chausées). Numerical method (coll. with A. Sellier, LadHyX); Stability of vertical film: Numerical method; Experimental work of Ph. D. thesis of H. Kočárková (coll. with F. Rouyer). 48 / 73 Glass foam stability Motivations Mesoscopic studies: Bubble drainage on a free surface: Experimental work of Ph. D. thesis of H. Kočárková (coll. with F. Rouyer, Lab. Navier, Ecoles des Ponts et Chausées). Numerical method (coll. with A. Sellier, LadHyX); Stability of vertical film: Numerical method; Experimental work of Ph. D. thesis of H. Kočárková (coll. with F. Rouyer). 49 / 73 Glass foam stability 2 Bubble drainage on the free surface Rising of a bubble 5 / 73 Glass foam stability 2 Bubble drainage on the free surface a γ2πa V g Figure 4: Force balance for a bubble close to a free surface. a 3 ρg aγ = Bo = ρg(2a)2, Nombre de BOND. (8) γ 5 / 73 Glass foam stability 2 Bubble drainage on the free surface z U g Figure 5: Film drainage with free shear interfaces. r µ U 2a ρg(2a)2 ρg2a = U =, τ = µ µ ρg2a. (9) Glass foam stability 2 Bubble drainage on the free surface 3 mm CCD camera laser mirror silica window cooling alumina tube heating elements Pt crucible refractory 325 mm 65 mm mm mm 7 mm 475 mm T T F S systems Pt tube with gas inlet Figure 6: Experimental set-up to study the bubble drainage on molten glass (H. Kočárková). 52 / 73 53 / 73 Glass foam stability 2 Bubble drainage on the free surface Experimental set-up has been duplicated at room temperature. Four liquids have been studied: Two glasses (high temperature): classical flat glass; glass rich in Al 2 O 3. Two oils (room temperature, exp. achieved by S. Metallaoui, Master ): Castor oil; UCON TM oil. 54 / 73 Glass foam stability 2 Bubble drainage on the free surface h h (a) Castor oil Bo =, 58 Bo = 4, 4 Bo = 26, 9 (b) UCON TM oil Bo = 3, 75 Bo = 2, 5 Bo = 7, t (c) Classical flat glass Bo = 5, 22 Bo =, 54 Bo = 2, 34 (d) Glass with Al 2O 3 Bo = 5, 65 Bo = 24, 5 Bo = 38, t Figure 7: Film thickness versus time for the four liquids. 55 / 73 Glass foam stability 2 Bubble drainage on the free surface Since the molten glass is a high viscous fluid: Reynolds number. Stokes equations can be used to describe the motion of fluid. Stokes equations have two important properties: Linear equations fundamental solutions are known. Existence of a reciprocity relationship (Principle of Lorentz reciprocity). Integral formulation of Stokes equations solved by a Boundary Element Method 9 9 F. Pigeonneau & A. Sellier. Phys. Fluids, 23:922, 22. 56 / 73 Glass foam stability 2 Bubble drainage on the free surface Bo = Figure 8: Shapes of a bubble near the free surface. Comparison with the previous work of Princen. H. M. Princen. J. Colloid Interface Sci., 8:78-95, 963. 57 / 73 Glass foam stability 2 Bubble drainage on the free surface Bo = Figure 8: Shapes of a bubble near the free surface. Comparison with the previous work of Princen. H. M. Princen. J. Colloid Interface Sci., 8:78-95, 963. 58 / 73 Glass foam stability 2 Bubble drainage on the free surface Bo = Figure 8: Shapes of a bubble near the free surface. Comparison with the previous work of Princen. H. M. Princen. J. Colloid Interface Sci., 8:78-95, 963. 59 / 73 Glass foam stability 2 Bubble drainage on the free surface Bo =, exact sol., Bart (968) Bo =.2 Bo = 3.6 Bo = 6 Bo = 2 Bo = 6 Bo = 2 h t Figure 9: Film thickness at the top of the bubble. For Bo =, The Bart s solution is used. E. Bart. Chem. Eng. Sci., 23:93:2, 968. 6 / 73 Glass foam stability 2 Bubble drainage on the free surface Strong influence of the bubble deformation: drainage controlled by the pressure due to the buoyancy force on the cap. Assuming a pure extensional flow: dh = 4π. () h dt 9S cap liquid film (spherical cap) hcap D c static meniscus θ R cap bubble interior interface Figure 2: Bubble shape obtained by the static equilibrium, Bo =. Glass foam stability 2 Bubble drainage on the free surface (a) Castor oil (c) Classical flat glass τ d τ d Sim. num. 7 τ d determined in the ref. (b) UCON TM oil (d) Glass with Al2O Bo Bo Figure 2: τ d = h dh dt versus Bo for the four liquids. 2 H. Kočárková, F. Rouyer, F. Pigeonneau. Phys. Fluids, under consideration, / 73 Glass foam stability 3 Synthesis Effect of the bubble size established. UCON TM oil and Castor oil are liquid models to represent molten glass. Observation of the daughter bubbles due to the rupture of the large bubbles in molten glass: daughter When a dynamical viscosity lesser than Pa s. On the vertical film: Stability has been observed on molten glass experiment due to Na 2 O evaporation 3. 3 F. Pigeonneau, H. Kočárková & F. Rouyer. Colloids Surf.,A. in press, / 73 63 / 73 Glass foam stability 4 Perspectives, applied works A foam is formed by a large quantity of bubbles: Can foam stability be explained with the stability on one bubble? Need to develop new experiments and new theoretical models: Glass foam aging should be investigated; Heat transfer has to take into account. Development of new products based on glass foam. 64 / 73 Glass foam stability 4 Perspectives, fundamental works Fundamental aspect of hydrodynamic interaction of a bubble with a free surface: Behavior of hydrodynamic force when Bo is small but not equal to zero? Extension of the work of Berdan and Leal 4. Work in progress in the Ph. D. thesis of M. Guémas (coll. A. Sellier). Development of 3D solver based on the B.E.M. to describe the bubble dynamics (M. Guémas): Study the mixing induced by the rising of bubbles; Rheology of bubbly flows. Experimental work on vertical film of molten glass: Extension for various glass natures (work in progress, J.-C. Guillard, Master 2). Frankel experiment: Film thickness vs. pulling-out velocity. 4 C. Berdan & L. G. Leal J. Colloid Interface Sci., 87:62-8, 982. 65 / 73 Conclusion Glass is a reactive media where the measures are difficult to do: Difficult to develop theoretical models. A lot of things stay to do. The general method is to try a simpler model to obtain a maximum of results to comeback after an original media. Connections with other science domains: Geophysics (oceanographies, vulcanology, mantle convection) close collaboration with M. Toplis (Obs. Midi-Pyrénées). Metallurgy. 66 / 73???? 67 / 73 Figure 22: Daughter bubbles observed for molten glass with a dynamical viscosity lesser than Pa s. Synthesis 68 / 73 69 / Figure 23: Bubble shape for ˆµ = 5 and Bo =. 7 / 73 Fining process Iron content Thanks to thermodynamics, it is possible to find the ion and gas concentrations: Constant equilibrium + equilibrium between a liquid phase and gas phase. Iron content play a role at low temperature: A larger consumption of sulfate; Glass is more reduced. 7 / 73 Fining process Iron content.25. wt % Fe.5 wt % Fe wt % Fe.2 SO 2 4 (wt % of SO3) T ( C) Figure 24: Concentration of SO 2 4 vs T with three iron contents. 72 / 73 Fining process Iron content wt % Fe.5 wt % Fe wt % Fe a (mm) t (s) Figure 25: a vs t with three iron contents. 73 / 73 Fining process Iron content.2 R =. R =.2.8 a (mm) t (min) Figure 26: a vs t with two redox, R = C Fe2+ /C tot.

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