Post on 29-Jul-2018
Dissolution of Si in Molten Al with Gas Injection
by
Mehran Seyed Ahmadi
A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy
Graduate Department of Materials Science and Engineering University of Toronto
© Copyright by 2014 by Mehran Seyed Ahmadi
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ABSTRACT
Dissolution of Si in Molten Al with Gas Injection
Mehran Seyed Ahmadi Doctor of Philosophy
Graduate Department of Materials Science and Engineering University of Toronto
2014 Silicon is an essential component of many aluminum alloys, as it imparts a range of desirable
characteristics. However, there are considerable practical difficulties in dissolving solid Si in
molten Al, because the dissolution process is slow, resulting in material and energy losses. It
is thus essential to examine Si dissolution in molten Al, to identify means of accelerating the
process.
This thesis presents an experimental study of the effect of Si purity, bath temperature,
fluid flow conditions, and gas stirring on the dissolution of Si in molten Al, plus the results of
physical and numerical modeling of the flow to corroborate the experimental results. The
dissolution experiments were conducted in a revolving liquid metal tank to generate a bulk
velocity, and gas was introduced into the melt using top lance injection. Cylindrical Si
specimens were immersed into molten Al for fixed durations, and upon removal the
dissolved Si was measured. The shape and trajectory of injected bubbles were examined by
means of auxiliary water experiments and video recordings of the molten Al free surface.
The gas-agitated liquid was simulated using the commercial software FLOW-3D. The
simulation results provide insights into bubble dynamics and offer estimates of the
fluctuating velocities within the Al bath.
The experimental results indicate that the dissolution rate of Si increases in tandem with
the melt temperature and bulk velocity. A higher bath temperature increases the solubility of
Si at the solid/liquid interface, resulting in a greater driving force for mass transfer, and a
higher liquid velocity decreases the resistance to mass transfer via a thinner mass boundary
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layer. Impurities (with lower diffusion coefficients) in the form of inclusions obstruct the
dissolution of the Si main matrix. Finally, dissolution rate enhancement was observed by gas
agitation. It is postulated that the bubble-induced fluctuating velocities disturb the mass
boundary layer, which increases the mass transfer rate.
Correlations derived for mass transfer from solids in liquids under various operating
conditions were applied to the Al–Si system. A new correlation for combined natural and
forced convection mass transfer from vertical cylinders in cross flow is presented, and a
modification is proposed to take into account free stream turbulence in a correlation for
forced convection mass transfer from vertical cylinders in cross flow.
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ACKNOWLEDGEMENTS
I wish to express my sincere gratitude to my supervisors Professors S. A. Argyropoulos
and M. Bussmann. Thank you for giving your support and guidance during these few years. I
especially thank you for taking the time to discuss the many and varied aspects of this
project.
I would like to thank the members of my advisory committee members Professors N.
Ashgriz and M. Barati for taking time out of their busy schedule to meet with me and give
insightful suggestions. I would also like to thank Professors T. Coyle and M. Hasan for
attending my final exam and for their valuable comments. Special acknowledgements also go
to Dr. D. Doutre with whom I had many fruitful discussions.
A special thank you to those who helped in so many ways with love and words of
encouragement, especially: Dad, Mom, and my sister and brother.
Financial assistance is also acknowledged and appreciated from: the Natural Science and
Engineering Research Council of Canada (NSERC), the Government of Ontario, Novelis
Global Technology Centre, and Department of Materials Science and Engineering at the
University of Toronto.
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To my dear parents, Mojtaba and Maryam, whose love, support and encouragement have accompanied me throughout my life
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CONTENTS
ABSTRACT ________________________________________________________________________________ ii
ACKNOWLEDGEMENTS ___________________________________________________________________ iv
LIST OF TABLES ___________________________________________________________________________ ix
LIST OF FIGURES __________________________________________________________________________ x
NOMENCLATURE ________________________________________________________________________ xvi
INTRODUCTION ___________________________________________________________________________ 1
1.1 Background Theory _____________________________________________________________________ 4
1.2 Objectives _____________________________________________________________________________ 5
1.3 This Thesis ____________________________________________________________________________ 6
BACKGROUND ____________________________________________________________________________ 7
2.1 Dissolution of Solid Additions in Molten Al __________________________________________________ 9
2.1.1 Assimilation mechanism of solid additions ________________________________________________ 9
2.1.2 Diffusion coefficient of solid additions in liquids __________________________________________ 11
2.1.3 Natural and forced convection mass transfer _____________________________________________ 12
2.1.4 Effect of intermetallics on mass transfer _________________________________________________ 13
2.2 Dissolution of Solid Additions in Two-phase Flow ____________________________________________ 17
2.2.1 Fluid mechanics of bubbly systems ____________________________________________________ 17
2.2.1.1 Bubble formation and trajectory _____________________________________________________________ 19
2.2.1.2 Bubble-induced velocity field in the liquid ____________________________________________________ 20
2.2.2 Mass transfer in two-phase flow systems ________________________________________________ 23
2.2.2.1 Mass transfer from small particles ___________________________________________________________ 24
2.2.2.2 Mass transfer from large solid additions _______________________________________________________ 26
DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: SINGLE-PHASE FLOW _______________ 34
3.1 Pure Diffusion of Si in Al ________________________________________________________________ 34
3.2 Experimental Methodology ______________________________________________________________ 36
3.2.1 Materials _________________________________________________________________________ 36
3.2.2 Experimental set-up ________________________________________________________________ 37
3.2.3 Experimental procedure _____________________________________________________________ 40
3.3 Results and Discussion __________________________________________________________________ 40
3.3.1 Natural convection experiments _______________________________________________________ 40
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3.3.1.1 Heat transfer after immersion _______________________________________________________________ 42
3.3.1.2 Isothermal mass transfer ___________________________________________________________________ 46
3.3.1.3 Effect of impurities _______________________________________________________________________ 50
3.3.2 Forced convection experiments ________________________________________________________ 56
3.3.2.1 Local variation of dissolution rate ___________________________________________________________ 59
3.3.2.2 Comparison with a forced convection correlation _______________________________________________ 59
3.3.2.3 Comparison with a combined natural and forced convection correlation _____________________________ 61
3.4. Relationship Between Bath Temperature and Bulk Velocity ____________________________________ 64
3.5 Summary _____________________________________________________________________________ 65
FLUID DYNAMICS OF GAS-AGITATED LIQUID _____________________________________________ 67
4.1 Bubble Formation and Trajectory in an Air–Water System ______________________________________ 67
4.2 Bubble Formation and Trajectory in a N2–Al System __________________________________________ 76
4.2.1 Bubble size _______________________________________________________________________ 78
4.2.2 Bubble trajectory ___________________________________________________________________ 81
4.3. Numerical Modeling of Gas-Agitated Tank _________________________________________________ 82
4.3.1 Simulation set-up and parameters ______________________________________________________ 83
4.3.2 Temporal- and spatial-averaged velocities _______________________________________________ 89
4.3.3 Experimental validation with the air–water system ________________________________________ 90
4.3.4 Results and Discussion ______________________________________________________________ 96
4.3.4.1 Bubble distribution within the liquid Al _______________________________________________________ 96
4.3.4.2 Non-rotating tank ________________________________________________________________________ 98
4.3.4.3 Rotating tank ___________________________________________________________________________ 107
4.4 Summary ____________________________________________________________________________ 119
DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: TWO-PHASE FLOW ________________ 121
5.1 Experimental Procedure ________________________________________________________________ 121
5.2 Results and Discussion _________________________________________________________________ 123
5.2.1 Effect of bulk velocity at a given gas flow rate ___________________________________________ 124
5.2.2 Effect of gas flow rate ______________________________________________________________ 131
5.2.3 Effect of lance location _____________________________________________________________ 133
5.2.4 Correlating single- and two- phase flows _______________________________________________ 137
5.3 Comparison with a Correlation for Mass Transfer in a Gas‐stirred Tank ___________________________ 140
5.4 Comparison with a Correlation for Mass Transfer in Turbulent Cross Flow ________________________ 141
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5.5 Relationship Between Mass Transfer Coefficient and Energy Input into the Liquid in Single and
Two-phase Flows ________________________________________________________________________ 145
5.6 Summary ____________________________________________________________________________ 146
SUMMARY, CONCLUSIONS AND FUTURE WORK __________________________________________ 148
6.1 Summary ____________________________________________________________________________ 148
6.2 Conclusions _________________________________________________________________________ 149
6.3 Contributions ________________________________________________________________________ 154
6.4 Recommendations for Future Work _______________________________________________________ 155
LIST OF REFERENCES ___________________________________________________________________ 157
APPENDIX A: CORRELATIONS FOR MATERIAL PROPERTIES ______________________________ 168
A.1 Diffusion Coefficient of Si into liquid Al __________________________________________________ 168
A.2 Density and Viscosity of Liquid Al _______________________________________________________ 169
APPENDIX B: IMPROVING THE REPRODUCIBILITY OF Si DISSOLUTION ____________________ 170
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LIST OF TABLES
Table 2-1: Si content, attractive properties, and applications of some common cast Al–Si alloys [16]. 8
Table 2-2: Dissolution studies of stationary solid additions in liquid Al. 14
Table 2-3: Dissolution studies of rotating solid additions in liquid Al. 15
Table 2-4: Heat and mass transfer correlations based on cold model studies of gas-agitated metallurgical ladles. 32
Table 2-5: Heat and mass transfer correlations based on hot model studies of gas-agitated metallurgical ladles. 33
Table 3-1: Overall chemical analysis (in wt.%) of the various batches of Si used in this study. 37
Table 3-2: Results of Energy Dispersive X-ray analysis at points shown in Figure 3-11 (b) and Figure 3-12 (b). 53
Table 3-3. Dimensionless parameters involved in the combined convection from a Si vertical cylinder in liquid Al cross flow. 62
Table 4-1: List of FLOW‐3D simulations for the N2–Al system. 96
Table 4-2: Comparison of experimental measurements and FLOW-3D predictions for the equivalent bubble diameter at the lance exit in the N2–Al system (bulk velocity = 0). 97
Table 5-1: Mean mass transfer coefficient of MGSi–II specimens, mk , and estimated mass boundary layer thickness, δm, at various operating conditions. 131
Table 5-2: Predicted mass transfer coefficients using the general combining law for Equation (5-6) and natural convection, and a comparison with experimental values (bulk velocity = 0, θ = 30°). 144
Table 5-3: Predicted mass transfer coefficients using the general combining law for Equation (5-8) and natural convection, and a comparison with experimental values (θ = 30°). 144
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LIST OF FIGURES
Figure 2-1: Equilibrium Al–Si phase diagram [19]. 9
Figure 2-2: Schematics of (a) bottom stirring and (b) top lance injection. 18
Figure 2-3: (a) He bubble in water, and (b) He bubble in a liquid metal (mercury), both at the end of a stainless steel lance [60]. 20
Figure 3-1: Cylinder radius vs. time, considering only pure diffusion. 36
Figure 3-2: Schematic of a bilge-shaped crucible shows the location of thermocouples to examine the effect of preheating the specimen. All dimensions are in cm. 38
Figure 3-3: (a) The RLMT with solid Al charge, (b) top view shows the rotation direction and location of the immersed Si specimen and AA´ is perpendicular to the xy-plane and passes through the Si sample, and (c) side view with the Si sample at the AA´ plane. All dimensions are in cm. 39
Figure 3-4: MGSi–I specimens after natural convection dissolution for 3 min in an Al bath at various superheats. 41
Figure 3-5: The effect of bath superheat on the dissolved fraction, md/mi, of MGSi–I specimens after natural convection dissolution for 3 min. 42
Figure 3-6: Formation of an Al shell on MGSi–I cylinders immersed for short periods in an Al bath at SPH = 40 K. 43
Figure 3-7: Temperature history of a MGSi–I cylinder immersed into liquid Al (a) initially at room temperature, (b) preheated for 3 min prior to immersion. 45
Figure 3-8: Comparison of the dissolved fraction with and without preheating, for MGSi–I at SPH = 40 K. 46
Figure 3-9: Comparisons between measured and predicted natural convection mass transfer for liquid Al at (a) SPH = 40 K and (b) SPH = 80 K. 49
Figure 3-10: Effect of bath superheat on the dimensionless radius, r/r0, of MGSi–I, MGSi–II and EGSi specimens immersed for 3 min in liquid Al under natural convection. 50
Figure 3-11: (a) Secondary electron image of a MGSi–I specimen prior to immersion, (b) a higher magnification of the outlined area in (a). 51
Figure 3-12: (a) Secondary electron image of a MGSi–II specimen prior to immersion, (b) a higher magnification of the outlined area in (a). 52
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Figure 3-13: Backscattered electron image of a MGSi–I specimen with an attached Al after quenching in water from 973 K to 293 K. 53
Figure 3-14: Si specimens after natural convection dissolution for 3 min at SPH = 80 K; captions on top indicate the batch of Si. 55
Figure 3-15: Effect of bulk velocity on the dissolved fraction after 3 min immersion, (a) SPH = 40 K and (b) SPH = 80 K. 57
Figure 3-16: Dissolution of MGSi–I under natural and forced convection conditions; (a) SPH = 40 K and (b) SPH = 80 K. 58
Figure 3-17: Variation of local dissolution of MGSi–II cylindrical specimens in a cross flow of liquid Al at various bulk velocities (SPH = 40 K, immersion time = 3 min). 59
Figure 3-18: Comparison of experimental and estimated dimensionless radii for pure forced convection mass transfer from vertical Si cylinders immersed for 3 min in a liquid Al cross flow. 61
Figure 3-19: Comparison of experimental and estimated dimensionless radii for combined convection mass transfer from vertical Si cylinders immersed for 3 min in a liquid Al cross flow. 63
Figure 3-20: Comparison of experimental and estimated dimensionless radii for combined convection mass transfer from vertical Si cylinders immersed in a liquid Al cross flow at a bulk velocity of 7 cm s−1. 64
Figure 3- 21: Mass transfer coefficient vs. SPH (shown on the bottom abscissa) superimposed on mass transfer coefficient vs. bulk velocity (shown on the top abscissa) (MGSi–II, immersion time = 3 min). 65
Figure 4-1: (a) The transparent water tank on a rotating table, (b) top view shows the rotation direction and location of the gas injection lance, and AA´ represents a plane perpendicular to the xy-plane passing through the lance, (c) side view of the water tank and the lance at AA´. All dimensions are in cm. 68
Figure 4-2: Typical output voltage of the pressure transducer used to measure bubble formation frequency (air–water system, bulk velocity = 0). 69
Figure 4-3: Formation and rise of single bubbles in the air–water system (gas flow rate = 0.3 SLPM, bulk velocity = 0). 70
Figure 4-4: Formation of single bubbles and coalescence of two consecutive bubbles during rise in the air–water system (gas flow rate = 0.6 SLPM, bulk velocity = 0). 71
Figure 4-5: Formation of doublets in the air–water system (gas flow rate = 1.0 SLPM, bulk velocity = 0). 72
Figure 4-6: Equivalent bubble diameter vs. gas flow rate in the air–water system. 73
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Figure 4-7: Equivalent bubble diameter in the air–water system with and without cross flow. 74
Figure 4-8: Effect of bulk liquid velocity on the equivalent bubble diameter in the air–water system (db normalized by the bubble diameter when the bulk velocity = 0). 75
Figure 4- 9: Comparison of bubble trajectories at various bulk velocities (gas flow rate = 1 SLPM). 76
Figure 4-10: Schematic of the RLMT shows the rotation direction and location of the immersed Si specimen and the gas injection lance. (a) Top view where AA´ represents a plane perpendicular to the xy-plane passing through the lance, (b) side view of the RLMT. All dimensions are in cm. 78
Figure 4-11: Typical output voltage of the pressure transducer used to measure the frequency of bubble formation (N2–Al at SPH = 40 K, bulk velocity = 0). 79
Figure 4-12: Bubble size in liquid Al at bulk velocities of 0 and 7.0 cm s−1, and a comparison with the available correlation for a (Chlorine/N2)–Al system [57]. 80
Figure 4-13: Bubble arrival at liquid Al surface (bulk velocity = 0, gas flow rate = 0.50 SLPM, θ = 30º, the coated lance diameter is approximately 0.725 cm). 81
Figure 4-14: Bubble arrival at the liquid Al surface (bulk velocity = 3.5 cm s−1, gas flow rate = 0.50 SLPM, θ = 30º, the coated lance diameter is approximately 0.725 cm). 82
Figure 4-15. Cylindrical domain for the FLOW-3D simulations of the N2–Al system. All dimensions are in cm. 84
Figure 4-16: (a) Top view of the mesh planes, and (b) the mesh around the lance. 85
Figure 4-17. (a) Overall liquid kinetic energy of liquid per unit mass of liquid, Kl, and (b) ldK dt vs. time (bulk velocity = 0, gas flow rate = 0.50 SLPM, solid line is the sliding 1 s average). 88
Figure 4-18: Instantaneous liquid velocity magnitude (bulk velocity = 0, gas flow rate = 0.50 SLPM, r = 13.5 cm, θ = 30°, z = 11 cm: 4 cm below the free surface of liquid Al coincides with the center of an imaginary Si cylinder). 90
Figure 4-19: Qualitative comparison of air-water bubble formation (bulk velocity = 0, gas flow rate = 0.3 SLPM). 93
Figure 4-20: Qualitative comparison of air-water bubble formation (bulk velocity = 0, gas flow rate = 1.0 SLPM). 94
Figure 4-21: Quantitative comparison of the formation and rise time of many bubbles over a time period of 4 s as predicted by a FLOW‐3D simulation, and measured in the
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air–water experiments, at gas flow rates of (a) 0.3 SLPM and (b) 1.0 SLPM (bulk velocity = 0). 95
Figure 4-22: Predicted instantaneous bubble distributions at gas flow rates of (a) 0.50 SLPM and (b) 1.00 SLPM (3D view of the lance vicinity, bulk velocity = 0). 98
Figure 4-23: (a) Predicted instantaneous bubble-induced velocity magnitude in the θz-plane passing through the lance center, (b) the tangential velocity at various rz-planes, and (c) the vertical velocity at various rθ-planes (bulk velocity = 0, gas flow rate = 0.50 SLPM, t = 9.00 s). 101
Figure 4-24. Spatially averaged (a) absolute mean velocity, and (b) rms velocity fluctuations in the r, θ, and z directions (bulk velocity = 0, gas flow rate = 0.50 SLPM, θ = 30°). 103
Figure 4-25: Comparison of spatially-averaged (a) mean velocity and (b) rms velocity fluctuations at various angular positions relative to the lance (bulk velocity = 0, gas flow rate = 0.50 SLPM). 105
Figure 4-26: Comparison of spatially-averaged (a) mean velocity and (b) rms velocity fluctuations at various gas flow rates (bulk velocity = 0, θ = 30°). 106
Figure 4-27: Instantaneous (a) tangential velocity, uθ, and (b) vertical velocity, uz, on the θz-plane passing through the center of the lance (bulk velocity = 3.5 cm s−1, gas flow rate = 0.50 SLPM, t = 9.00 s). 108
Figure 4-28: Instantaneous tangential velocity on two rθ‐planes (bulk velocity = 3.5 cm s−1, gas flow rate = 0.50 SLPM, t = 9.00 s). 109
Figure 4-29: Instantaneous velocity magnitude, V, at various rz‐planes (bulk velocity = 3.5 cm s−1, gas flow rate = 0.50 SLPM, t = 9.00 s). 109
Figure 4-30: Spatially-averaged absolute mean velocity in the r, θ, and z directions (bulk velocity = 3.5 cm s-1, gas flow rate = 0 SLPM, θ = 30°). 110
Figure 4-31: Spatially-averaged rms velocity fluctuations and turbulence intensity (bulk velocity = 3.5 cm s−1, gas flow rate = 0 SLPM, θ = 30°). 111
Figure 4-32: Spatially-averaged (a) absolute mean velocity, and (b) rms velocity fluctuations in the r, θ, and z directions (bulk velocity = 3.5 cm s−1, gas flow rate = 0.50 SLPM, θ = 30°). 112
Figure 4-33: Comparison of spatially-averaged rms velocity fluctuations at various bulk velocities (gas flow rate = 0.50 SLPM, θ = 30°). 113
Figure 4-34: Comparison of spatially-averaged turbulence intensity in the rotating tank (gas flow rate = 0.50 SLPM, θ = 30°). 114
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Figure 4-35: Comparison of spatially-averaged absolute mean velocities in the (a) r, (b) θ, and (c) z directions at various gas flow rates (bulk velocity = 3.5 cm s−1, θ = 30°). 116
Figure 4-36: Comparison of spatially-averaged rms velocity fluctuations in the (a) r, (b) θ, and (c) z directions at various gas flow rates (bulk velocity = 3.5 cm s−1, θ = 30°). 117
Figure 4-37: Comparison of spatially-averaged (a) mean velocity, and (b) rms velocity fluctuations at various angular positions relative to the lance (bulk velocity = 3.5 cm s−1, gas flow rate = 0.50 SLPM). 118
Figure 5-1: The furnace lid includes various holes to immerse a thermocouple, Si specimens, and the lance into the liquid Al. 122
Figure 5-2: Dissolved MGSi–II cylindrical specimens at various gas flow rates (immersion time = 3 min, SPH = 40 K, bulk velocity = 3.5 cm s−1, θ = 30°). 123
Figure 5-3: Dissolved fraction of MGSi–II at SPH = 40 K vs. bulk velocity (immersion time = 3 min, gas flow rate = 0.50 SLPM, θ = 30°). 125
Figure 5-4: (a) ΔDF, and (b) EF for dissolution for MGSi–II vs. bulk velocity at a gas flow rate = 0.50 SLPM (immersion time = 3 min, θ = 30°, the lines indicate average values). 127
Figure 5-5: (a) ΔDF, and (b) EF for dissolution for MGSi–II vs. bulk velocity at a gas flow rate = 1.00 SLPM (immersion time = 3 min, θ = 30°, the lines indicate average values). 128
Figure 5-6: (a) ΔDF and (b) EF for MGSi–I vs. gas flow rate (immersion time = 3 min, bulk velocity = 3.5 cm s−1, θ = 30°, the lines indicate average values). 134
Figure 5-7: (a) ΔDF and (b) EF for MGSi–II vs. gas flow rate (immersion time = 3 min, bulk velocity = 3.5 cm s−1, θ = 30°, the lines indicate average values). 135
Figure 5-8: ΔDF vs. the relative positions of the gas injection lance and Si specimen for (a) MGSi–I, SPH = 40 K and (b) MGSi–II, SPH = 80 K (immersion time = 3 min, bulk velocity = 3.5 cm s−1, gas flow rate = 1.00 SLPM, the filled symbols indicate average values). 136
Figure 5-9: Mass transfer coefficient vs. gas flow rate (shown on the top abscissa) superimposed on mass transfer coefficient vs. bulk velocity (shown on the bottom abscissa) for (a) SPH = 40 K and (b) SPH = 80 K (MGSi–II, immersion time = 3 min, θ = 30°, error bars indicate the standard deviations). 138
Figure 5-10: Mass transfer coefficient vs. gas flow rate at a bulk velocity of 3.5 cm s−1 (shown on the top abscissa) superimposed on mass transfer coefficient vs. bulk velocity (shown on the bottom abscissa) for (a) SPH = 40 K and (b) SPH = 80 K
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(MGSi–II, immersion time = 3 min, θ = 30°, error bars indicate the standard deviations). 139
Figure 5-11: Normalized mass transfer coefficient for single-phase and two-phase flows vs. normalized liquid kinetic energy per unit mass of liquid. 146
Figure A-1: Diffusion coefficient of Si into Al as a function of temperature. 168
Figure A-2: Kinematic viscosity of liquid Al as a function of temperature. 169
Figure B-1: MGSi–I as‐drilled specimens after natural convection dissolution for 5 min in liquid Al at SPH = 40 K. 171
Figure B-2: Effect of sample surface preparation on the dissolved fraction of MGSi–I specimens immersed for 5 min in liquid Al under natural convection at SPH = 40 K. 171
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NOMENCLATURE
Acronyms AA Aluminum Association
ASTM American Society for Testing and Materials
BOF Basic Oxygen Furnace
CFD Computational Fluid Dynamics
DF Dissolved Fraction
DNS Direct Numerical Simulation
EDX Energy Dispersive X-ray spectroscopy
EF Enhancement Factor
EGSi Electronic Grade Silicon
ICP Inductively Coupled Plasma mass Spectrometry
LDV Laser Doppler Velocimetry
LES Large Eddy Simulation
LPD Liquid Phase Diffusion
MAC Marker-and-Cell
MGSi Metallurgical Grade Silicon
NLPM Normal Liter per Minute
RLMT Revolving Liquid Metal Tank
SLPM Standard Liter per Minute
SPH Superheat
SEM Scanning Electron Microscopy
VoF Volume of Fluid
English symbols Ac cross section area of lance or nozzle, m2
C concentration, kg m−3
C1 coefficient appears in the equation of Re based on energy dissipation rate, Equation (2-3)
D diffusion coefficient, m2 s−1
d diameter of a cylindrical or spherical specimen, m
d0 initial diameter of a cylindrical or spherical specimen, m
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db bubble diameter, m
db0 bubble diameter at zero bulk velocity, m
(db)max maximum stable bubble diameter, m
dlance,ID inner diameter of a lance or nozzle, m
dlance,OD outer diameter of a lance or nozzle, m
dp particle diameter, m
EK gas kinetic power, W
EB power due to buoyancy force of rising bubbles, W
Grm,l mass transfer Grashof number [ 3 Sat b 2Si T Si Sig( C ) l (C C )= ∂ρ ∂ − ρν ]
FB buoyancy force, N LDF lateral drag force, N
f frequency of bubble formation, s−1
g gravitational acceleration, m s−2
H liquid height inside a tank, m
I coefficient appears in the equation for mass flux from rotating disk, Equation (2-1)
j mass flux, kg m−2 s−1
k specific turbulent kinetic energy, m2 s−2
Kl liquid kinetic energy per unit mass of liquid, m2 s−2
km local mass transfer coefficient, m s−1
mk mean combined convection mass transfer coefficient, m s−1
Nmk mean natural convection mass transfer coefficient for vertical cylinder, m s−1
Fmk mean forced convection mass transfer coefficient for vertical cylinder in cross
flow, m s−1
L distance from the nozzle/ lance exit to the bath surface, m
Lv downward migration of gas upon injection from lance, m
l length of cylindrical specimen, m
lc characteristic length, m
M total number of data used in time-averaging
m exponent appears in the equation for mean Sh of natural convection, Equation (3-6)
md dissolved mass of cylindrical Si specimen, kg
mi immersed mass of cylindrical Si specimen, kg
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NV number of cells in a volume representing a Si cylinder
n coefficient appears in the equation for mean Sh of natural convection, Equation (3-6)
Nu mean Nusselt number
P static pressure of liquid at the lance/nozzle tip, Pa
P0 atmospheric pressure, Pa
Pr Prandtl number [= ν/α]
p exponent appears in the equation of the general combining law
Q gas flow rate at the lance exit, m3 s−1
QS gas flow rate at standard temperature and pressure
(273.15K and 100 kPa), m3 s−1
Ri inner diameter of a liquid tank, m
Ro outer diameter of a liquid tank, m
Re Reynolds number
ReT Reynolds number based on rms velocity fluctuations
r radius of a cylindrical specimen, or radial direction, m
r0 initial radius of a cylindrical specimen, m rot
S ir radial distance from the center of the tank to the point of immersion of a Si cylinder centerline, m
S location of solid/liquid interface, m
S0 initial location of solid/liquid interface, m
Sc Schmidt number [= ν/D]
Sh mean Sherwood number
T temperature, K
Tu turbulence intensity
t time, s
Ub bulk velocity of liquid normal to the centerline of a vertical cylinder or an injection lance, m s−1
ub bubble rise velocity, m s−1
up peripheral velocity of a rotating cylinder, m s−1
Up velocity of upward liquid flow in a ladle, m s−1
u rms velocity fluctuations, m s−1
ur, uθ, uz velocity components in cylindrical coordinates, m s−1
ru , uθ , zu time-averaged velocity components in cylindrical coordinates, m s−1
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ru′ , uθ′ , zu′ velocity fluctuation components in cylindrical coordinates, m s−1
V instantaneous velocity magnitude, m s−1
V mean velocity of liquid, m s−1
Vg gas entry velocity, m s−1
v volume, m−3
X, Y, Z general variables
x, y, z Cartesian coordinates
Greek symbols α thermal diffusivity, m2 s−1
δ momentum boundary layer thickness, m
δm mass boundary layer thickness, m
φ general quantity
ε energy dissipation rate per unit mass, m2 s−3
θ relative angular position with respect to a lance or a Si specimen, deg
θc contact angle, deg
λ ratio of interface velocity to mass transfer coefficient
μ dynamic viscosity, kg s−1 m−1
μ0 dynamic viscosity at the solid surface, kg s−1 m−1
ν kinematic viscosity, m2 s−1
ρ density, kg m−3
σ surface tension, N m−1
Ω rotational speed of a rotating disk, rad s−1
ω rotational speed of a liquid Al tank, rad s−1
Subscript 0 initial condition
Al liquid Al
c centerline of a cylindrical ladle
g gas
l liquid
Si solid Si
Superscript
b bulk liquid
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F forced convection
N natural convection
ref reference condition
rms root mean square
Sat saturation
Other | | absolute value
< > spatial averaging
1
CHAPTER 1
INTRODUCTION
Aluminum (Al) is ubiquitous: in the buildings we live in, the trains we take to work, the
networks that connect us, and the packaging that protects our food and drink. Al is in high
demand for several reasons: it is a light metal, has good corrosion resistance, high electric
and thermal conductivity, and Al is highly recyclable, which is crucial from an
environmental standpoint [1].
The direct use of pure Al is limited because of lack of strength and castability. To
improve on pure Al properties, the requirements for any particular application, including
formability, strength, and wear resistance must be taken into account. This necessitates the
addition of alloying elements. Al alloys with silicon (Si) as the major alloying element are of
great importance as they are characterized by excellent castability and good weldability. Si
levels in unalloyed Al are very low (typically 0.05 to 0.15 wt.%). When added in
combination with Mg, Si yields heat treatable casting alloys such as 356 (6 wt.% Si, 0.4 wt.%
Mg, widely used for wheels and suspension components), and wrought alloys of the 6000
series which can be extruded or rolled. Very high levels of Si are present in many die casting
alloys where the Si provides strengthening, resistance to hot tearing and wear resistance. Si is
also a major component of the 4000 series brazing alloys, where typical
CHAPTER 1. INTRODUCTION
2
Si levels range from 5 to 13%.
Al is produced by the electrolysis of alumina extracted from bauxite ore (primary Al), or
by recycling (secondary Al). Secondary Al accounted for 60% of the metal produced in
North America in 2011 [2]. Al properties are not compromised by recycling [3] and the
energy required to produce secondary Al is only 5% of that required to produce primary Al
[1]. Despite the energy savings associated with using recycled scrap, remelting is
nevertheless an energy intensive process, with a total energy consumption of about
10-20 GJ/tonne [4].
There is no "standard process" for the production of Al alloys. Every plant has certain
resources, and they exercise the most efficient ways to achieve the specific compositions of
their products. For example, to produce Al–Si alloys, one major producer of primary Al adds
lumps of Si to an empty furnace and then transfers hot Al from the reduction cells to the
furnace. Because of the high temperature of the melt (~900°C), the solubility of Si is very
high, and with a bottom-mounted electromagnetic stirrer, Si can be dissolved even before all
of the molten metal has been transferred [5]. At the end of the cycle, only melt refining
measures must be taken to reach an acceptable tolerance for composition and homogeneity.
Needless to say, such a process requires a separate source of hot metal.
In Al processing plants without a smelter (e.g. Novelis, the sponsor of this project at one
of its North American plants), metal in the form of unalloyed Al ingots from a smelter or
scrap Al is charged into melting or holding furnaces, to which alloying elements are added
[5]. Batch sizes are typically 20 to 50 tonnes, which represents an addition of several tonnes
of Si. At this scale, there are considerable practical difficulties in dissolving and mixing Si
into Al. The melting time for Al can vary significantly, but roughly speaking, it takes 2 to 4
hours to melt 20-50 tonnes of Al (700-740°C), and then an additional 20 to 30 minutes to
dissolve Si in the liquid Al. To make Al–Si alloys, Metallurgical Grade Si (MGSi) is
generally used in the form of crushed pieces (5 to 10 cm) which are dropped onto the surface
of the molten Al. The solid Si floats on the Al, but is mostly submerged (ρSi/ρAl = 0.98 at
CHAPTER 1. INTRODUCTION
3
700°C [6,7]). Some form of stirring is required to continuously expose the pieces to molten
Al, to speed up the dissolution process and obtain a uniform melt composition. This stirring
is most commonly carried out by a large steel rake attached to a fork truck, which requires
opening the furnace doors during the process. This results in heat loss, and the continuous
exposure of fresh liquid Al to the atmosphere forms dross.
The addition of Si can be a lengthy process, resulting in lost production time, energy loss
as heat when adding Si and stirring the melt, and material losses due to dross formation.
Significant savings could be achieved if the dissolution time could be reduced. This is the
motivation for this thesis: to examine Si assimilation in liquid Al, to identify means of
improving productivity.
The assimilation rate of a solid addition in melting/holding furnaces can be improved by
a number of means. One can raise the flame temperature of the furnace, as this increases the
solubility of Si into Al. However, this approach has many drawbacks: it can cause
overheating of the metal near the flame, leading to excessive oxidation and increasing the
metal loss. Also, increasing the flame temperature not only heats the metal, but also the roof
refractories and chimney system, which can significantly reduce thermal efficiency.
Another approach to accelerating the rate of Si dissolution is mechanical stirring using a
steel blade or an impeller to circulate the molten metal. This produces a bulk velocity in the
liquid; however, the use of either device entails considerable energy loss. In the case of
stirring with a blade, the estimated heat loss corresponds to approximately 10% of the total
energy used during the entire melt cycle [5], and as mentioned earlier, mechanical agitation
causes metal loss due to dross formation, which can be as important as the energy cost. A
more sophisticated method is electromagnetic stirring with the furnace doors closed, which
can be very efficient, but this is costly for new installations, and difficult to retrofit on old
furnaces [8].
Another method to increase the dissolution rate of Si is gas injection: light bubbles move
quickly through the liquid, agitating the flow, and resulting in bubble-induced stirring. Gas
CHAPTER 1. INTRODUCTION
4
stirring can be used by itself or with mechanical stirring. Gas injection is relatively
economical, and has been extensively used in the steelmaking industry for homogenization
and inclusion removal [9,10]. The implementation of gas injection is feasible in the Al
industry, as evidenced by the fact that, for example, hydrogen is removed from liquid Al by
purging with Ar and/or N2 through a simple tube [3], and using an inert gas, the pneumatic
injection of powdered alloying elements (e.g. Fe, Mn and Cr) into molten Al is practiced
[11].
1.1 Background Theory
Si has a higher melting temperature than Al, and so it dissolves rather than melts. For solid Si
dissolving in liquid Al, the dissolution rate is controlled by mass transfer through a mass
boundary layer between the solid addition and the liquid. The mass flux of the solute can be
expressed by Fick’s law of diffusion as j = −DSi/Al (dC/dx) at the interface. The rate of
dissolution of Si in Al is slow owing to low solute (Si) diffusivity (DSi/Al ≈ 10−8 m2 s−1). The
kinetics of dissolution can be improved by stirring the melt which will increase concentration
gradient, dC/dx, and so can be an effective method of enhancing the mass transfer rate.
Stirring drives convection, and then the mass flux is expressed as j = km (CSat − Cb) [12],
where km is the convective mass transfer coefficient. Assuming that the solute concentration
varies linearly over a thickness δm from the saturation value to the bulk value, then km is
inversely proportional to δm. The higher the liquid velocity, the thinner the mass boundary
layer, which results in a higher km.
The flow field and accordingly the liquid velocity depends on the stirring method. While
impellers generate a bulk velocity, injected bubbles induce a liquid bulk motion plus an
agitation that results in an effectively thinner mass boundary layer adjacent to any dissolving
Si. This study compares bubble-induced agitation with a uniform bulk velocity as a means of
accelerating Si dissolution.
CHAPTER 1. INTRODUCTION
5
In general, the assimilation kinetics of alloying additions in Al as a function of flow
conditions have not been sufficiently studied, although the assimilation process can be a rate
limiting step which affects the production rate of Al alloys. Specifically, no one has
examined Si dissolution in liquid Al with and without gas agitation. The only study of Si
alloying of Al showed that Si granules (0-1.2 cm) dissolved faster by a factor of two to three
than Si chunks (1-10 cm) [13]. However, the authors did not describe the Al flow and its
effect on the dissolution rate. Despite the more rapid dissolution of granules than chunks,
industry tends to dissolve larger pieces of Si so as not to lower recoveries [5], because small
pieces float on top of the molten metal and so tend to end up in dross.
The Al industry will benefit from this study, and there is the potential to apply the
methodology to the processing of other metals as well. The development of this technology
will yield economic and environmental benefits, by increasing production capacity, and
lowering the energy requirement.
1.2 Objectives
This thesis presents a quantitative comparison of the dissolution rate of Si in molten Al
without and with gas agitation, referred to as single-phase and two-phase flow conditions,
respectively. The specific objectives of the study were to:
• conduct a series of experiments to quantify the dissolution rate under natural and
forced convection without gas agitation;
• examine the effect of gas stirring on the enhancement of dissolution, and correlate the
effect of the gas flow rate to an equivalent bulk velocity associated with single-phase
flow;
• investigate the influence of impurity levels of Si specimens on the dissolution rate;
CHAPTER 1. INTRODUCTION
6
• study the shape and trajectory of injected bubbles, by means of auxiliary cold model
experiments;
• simulate gas agitation using commercial Computational Fluid Dynamics (CFD)
software, to visualize the bubble dynamics and to estimate the velocity field within
the opaque Al bath.
1.3 This Thesis
The rest of the thesis is organized as follows:
Chapter 2 begins with a review of the dissolution of high melting point additions in
single-phase flow, and is followed by a review of the fluid dynamics of bubbly systems, and
mass transfer from additions in a two-phase flow.
In chapter 3, the dissolution of solid Si in molten Al is presented as a function of molten
metal temperature and velocity, and the level of impurities present in the Si. The experiments
were conducted for a range of Al bath velocities, providing a quantitative comparison of the
dissolution rate under natural and forced convection conditions.
In chapter 4 results are presented of three studies of gas injection: (i) observations of the
molten Al free surface, as well as measurements of N2 bubble frequency in molten Al; (ii)
bubble characteristics upon injection from a downward lance in an analogous air–water
system; and (iii) simulation results of the gas-agitated liquid, to obtain estimates of the
bubble-induced velocities.
In chapter 5, the dissolution enhancement of cylindrical Si specimens in gas-agitated
molten Al is presented as a function of gas flow rate, injection lance position, uniform bulk
velocity, and bath temperature.
Chapter 6 concludes the thesis.
7
CHAPTER 2
BACKGROUND
Al alloys can be categorized as wrought or cast, and whether or not they are heat treatable.
Wrought alloys generally contain smaller amounts of alloying additions than cast alloys [14].
Wrought alloy products include plates, sheets, foils and wires used in beverage cans,
radiators, paneling and window frames. Even though most wrought Al products are alloyed,
there are applications such as electrical cables where near-to-pure Al is used [3].
Cast Al alloys are widely used in the transportation sector for engine blocks, cylinder
heads, pistons and wheels. Cast parts produced from Al alloys exhibit a unique high strength
to density ratio which make them desirable for the aerospace and automobile industries,
where structural weight is a critical property. Substituting steel and iron with light weight Al
is an accepted strategy to improve fuel economy and reduce CO2 emissions [3]. Pure Al has a
density of only 2700 kg m−3, approximately one-third that of steel (7830 kg m−3), while
alloying with precipitation-hardening elements generates mechanical properties equivalent to
those of steel.
In general, the physical and mechanical characteristics required of a part determine the
alloying elements that must be added to molten Al before casting. Si, Cu, Mg and Zn are the
main commercial alloying elements to Al. Among cast Al alloys, the Al–Si family makes up
85% or more of all Al cast parts [15]. The addition of Si promotes fluidity and low shrinkage
CHAPTER 2. BACKGROUND
8
in casting, brazing and welding applications, and so makes Al–Si alloys commercially
feasible.
Cast Al alloys are categorized based on their chemical compositions. They are numbered
according to a four digit system adopted by the Aluminum Association (AA) and approved
by the American Society for Testing and Materials (ASTM). The first digit indicates the
major alloying element in the group. For instance, 4xxx refers to Al–Si alloys and 3xxx
refers to Al–Si (+Cu or Mg) alloys [16]. Cast Al–Si alloys can be classified into three groups
[17]: (i) hypoeutectic (5-10 wt.% Si), (ii) eutectic and near eutectic (l0-13 wt.% Si), and (iii)
hypereutectic (13-25 wt.% Si). The composition of some common Si-containing alloys, along
with common applications, is given in Table 2-1 [16]. Note that there are applications of Al–
Si alloys with up to 18 wt.% Si.
Table 2-1: Si content, attractive properties, and applications of some common cast Al–Si alloys [16].
Alloy Si content (wt.% Si) Attractive properties Application
443.0 4.5-6 Good corrosion resistance Cooking utensils
332.0 8.5-10.5 Good wear resistance Pulleys and sheaves
413.0 11-13 Excellent castability Marine equipment
A390.0 16-18 Low thermal expansion, high hardness and good wear resistance
Engine pistons, compressor cylinders
Although wrought Al alloys generally contain smaller amount of additions, there are
wrought Al alloys with high Si concentration (e.g. AA4004 with 9.0-10.5 wt.% Si [18]),
because the Si lowers the melting point. These alloys commonly form part of a clad package
consisting of a high melting point core covered with a layer of the low melting point 4xxx
alloy. Such clad alloys are used for brazing applications, where the 4xxx alloy serves as the
brazing filler that seals and holds the finished assembly together.
CHAPTER 2. BACKGROUND
9
The thermodynamics of the Al–Si system have been well studied. The Al–Si binary
system is a simple eutectic, as shown in Figure 2-1 [19]. The eutectic temperature is
577 ± 1°C and the eutectic composition is 12.6 ± 1 wt.% Si. Si has a high solubility in molten
Al but its solubility in solid Al is low. The saturation concentration of Si in liquid Al can be
obtained from the liquidus curve.
Figure 2-1: Equilibrium Al–Si phase diagram [19].
2.1 Dissolution of Solid Additions in Molten Al
2.1.1 Assimilation mechanism of solid additions
This thesis focuses on the kinetics of Si assimilation in liquid Al. This section introduces
different dissolution routes for the melting/dissolution of additions in metallic systems, and
then identifies the route specific to the Al–Si system.
The assimilation of a solid addition into a liquid metal can occur as melting or
dissolution: the former takes place when the melting temperature of the addition is below that
of the liquid, whereas the latter occurs when the solid addition has a melting temperature
CHAPTER 2. BACKGROUND
10
higher than the liquid temperature. Solid additions to liquid metals can be categorized into
three groups [20]. Group I is characterized by melting; adding scrap Al to liquid Al is an
example. Group II is characterized by dissolution. Group III includes exothermic additions
with melting points higher than the liquid metal temperature. These exothermic additions
release substantial amounts of heat resulting in mass transfer that begins as dissolution and
progresses to melting. Mn in liquid Al is an example of a Group III addition [21].
Si has a melting point higher than that of Al, and so it dissolves in liquid Al, rather than
melts. When Si is added to liquid Al, there is negligible exothermicity, so it belongs to the
Group II of additions, for which assimilation takes place as two steps [22]:
Step I: Interface reaction at the solid/liquid interface: atoms migrate from the solid phase into
the melt. The absence of a relationship between dissolution rate and liquid agitation indicates
that the dissolution rate is controlled by this step.
Step II: Transport of dissolved species from the interface to the bulk liquid metal: the
dissolved species diffuses from the solid/liquid interface into the bulk liquid Al. The
liquid-side interface concentration of the dissolving species is defined by the liquidus curve
on the phase diagram at the melt temperature. When dissolution is controlled by liquid phase
diffusion (LPD), the flow in the vicinity of the addition will have a significant influence on
the dissolution rate. In this case, the mass transfer coefficient depends on the fluid flow
conditions.
The author is unaware of any systematic work on the dissolution of Si in Al under
different fluid flow conditions, but the dissolution of other high melting point additions in Al
has been investigated. These studies can be categorized based on whether the solid addition
was stationary or rotating, corresponding to natural and forced convection, respectively.
Studies of stationary solid additions, summarized in Table 2-2, were conducted to compare
the dissolution kinetics of different additions in liquid Al [23-25], investigate natural
convection mass transfer in liquid metal systems [24,26,27], and characterize the types of
intermetallic layers formed at interfaces [25]. Experiments using rotating disks [28-35] or
CHAPTER 2. BACKGROUND
11
cylinders [23,36] of the addition, summarized in Table 2-3, were conducted to measure solute
diffusion coefficients and identify the dependence of dissolution rate on fluid flow
conditions. In the following, we focus on the findings of: (i) the diffusion coefficient of solid
additions in liquid metals, as this value for the Al–Si system is required in order to compare
dissolution rates to dimensionless correlations for mass transfer; (ii) the dissolution behavior
under natural and forced convection, to study spatial and temporal dissolution patterns, the
mass transfer rate in different convection regimes, and discrepancies between experimental
results and available correlations; and (iii) intermetallics as a resistance to mass transfer.
2.1.2 Diffusion coefficient of solid additions in liquids
To obtain reliable measurements of solute diffusion coefficients, the thickness of the mass
boundary layer must be constant on the entire dissolving surface. Levich [37] showed that
this condition is achieved only by a rotating disk. Therefore, the rotating disk method has
been used to study the dissolution of high melting point alloying elements in many liquid
metals, including Al and steel.
In rotating disk experiments, by periodically sampling the melt, the mass transfer
coefficient and mass flux of the material can be calculated at various rotation speeds. The
rotation speed is chosen to ensure a laminar flow. To calculate the diffusion coefficient, the
equation formulated by Levich and modified by Kasnar [38] can be used, that is appropriate
for the range of Sc encountered in liquid metal systems,
1 2/3 1/6 1/2 Sat bj 0.544I D (C C )− −= ν Ω − (2-1)
I is a function of Sc. At a constant temperature, a linear dependence of dissolution rate on the
rotation speed of the disk implies that the dissolution mechanism is LPD-controlled, which is
the case for most additions. For these additions, the diffusion coefficient, solubility and
dissolution rate are described by Arrhenius-type equations [31,32].
CHAPTER 2. BACKGROUND
12
The diffusion coefficient for Si in liquid Al used in the current study is that of
Ershov et al. [39], albeit without the modification of Kasnar [38]. Even though the value is
different than earlier measurements [40], it is deemed more reliable as it was obtained by the
rotating disk method.
2.1.3 Natural and forced convection mass transfer
The solubility of elements in liquid Al determines relative dissolution rates. For example, Ti,
V, Cr, Fe, Co, and Ni dissolve in molten Al at rates that increase in the listed order [23]. The
radius of a cylindrical specimen varies linearly with time during the early stages of
dissolution [27], and the dissolution rate varies from the top to the bottom of an immersed
cylinder. For example, Ni [27] and Fe [26] cylinders in liquid Al dissolve more quickly at the
top than the bottom; Zn and Sn cylinders in liquid Hg dissolve more quickly at the bottom
[26]. This is because the mass boundary layer thickness across which diffusion takes place
varies along a specimen, which confirms that dissolution takes place under natural
convection [26,27]. Based on the available literature, the dissolution of stationary cylinders
was compared to dimensionless correlations developed for natural convection. In these
studies, the correlations assumed a fixed interface, by assuming a low mass transfer rate.
Pekguleryuz et al. [24] explained a discrepancy in the mass transfer coefficient on the lack of
reliable values for the diffusion coefficient. In another study, Kosaka et al. [26] modified the
general form of a Sherwood number correlation for turbulent natural convection to fit their
data, rather than ascribe the uncertainty to measured values of the diffusion coefficient.
To investigate the effect of forced convection on dissolution, rotating cylinder
experiments are usually conducted. Forced convection correlations for rotating cylinders
underpredict the dissolution of commercially pure iron, S45C, Fe–Cr, and Fe–Si in liquid Al
[36]. It was suggested that the higher dissolution rates were caused by either natural
convection effects at lower rotating speeds, or turbulent flow at the rough surfaces of the
CHAPTER 2. BACKGROUND
13
dissolving specimens at higher speeds. However, a velocity at which the transition from
natural convection to forced convection occurs was not identified.
2.1.4 Effect of intermetallics on mass transfer
In contrast to the Al–Si system which forms no intermetallic phase [19], the assimilation of
some additions in Al involves formation of intermetallics at the addition/liquid interface. It
has been reported that these intermetallic phases can retard the dissolution of a stationary
addition by resisting mass transfer [25].
Rotating specimens have been used to analyze intermetallic layer growth. In rotating
cylinder experiments, the intermetallic layers are thinner than those that form on stationary
samples [36]. Tunca et al. [31] showed that the thickness of the intermetallic phase depends
on the type of addition. Nevertheless, the dissolution of Mo, Nb and Cr rotating disks in
unsaturated liquid Al is LPD-controlled regardless of the thickness and number of
intermetallic layers at the solid/liquid interface [31].
CHAPTER 2. BACKGROUND
14
Table 2-2: Dissolution studies of stationary solid additions in liquid Al.
Sample type
Solid addition
Diameter (mm) Length (mm)
Al purity wt.%
Al bath size (mm)
Al bath mass or volume
Temperature (°C)
Sample preheating
Dissolution rate measurement Intermetallics Reference
Cylinder
Ti, 99.96 wt.% 31
12.7 immersed 99.996 Not reported
Not reported 705-818
1 hour, 2.5 cm above the
melt Mass loss Not reported [23]
V, 99.5 wt.% 31 Cr, High purity 25 Ni, 99.96 wt.% 31
Co, 99.816 wt.% 31 Fe, 99.885 wt.% 31
Al 90 wt.%–10wt.% Sr
15-18 20-30 99.99 Not reported 10 kg 675-775 Not reported Chemical analysis
of melt samples
Al4Sr
[24] Al 76 wt.%–Sr 10 wt.%–Si 14 wt.% SrAl2Si2
Al 45 wt.%–Sr 55 wt.% Al4Sr–Al2Sr Fe–Si
5 50 (30 immersed) 99.8 45.1 ID,
140.6 depth 350 gr 700-800 Not reported Mass loss FeAl3–Fe2Al5 [25] Fe–Ni Fe–Cu Fe–Mn Fe–C
Fe, 98.3 wt.% 0.6-1 4.1-8.35 immersed 99.3 53 ID,
100 depth 600 ± 10 gr 710-820 Above the melt Mass loss Not reported [26]
Ni, 99.7 wt.% 6.05 2.5 pure 3.5 ID, 6.35 depth 120 gr 675
Above the melt to reach
a required temperature
Diameter along the length Not reported [27]
CHAPTER 2. BACKGROUND
15
Table 2-3: Dissolution studies of rotating solid additions in liquid Al.
*mc = monocrystalline, **pc = polycrystalline
Sample type
Solid addition
Diameter (mm)
Length (mm)
Rotational speed (rpm)
Al purity wt.%
Al bath size (mm)
Al bath mass or volume
Temperature (°C)
Sample preheating
Dissolution rate measurement Intermetallics Reference
Disk
Cr, 99.94 wt.% 12 ± 0.5 5 63-351 99.995 Not reported Not reported 700-900 Not reported Chemical analysis of melt samples
(12-90 min) Al7Cr, Al11Cr2 [28]
Rh, mc*, 99.8 wt.% 12 ± 0.5 5 63-351 99.995 Not reported Not reported 700-900 Not reported Chemical analysis of melt samples
(18-90 min) Not reported [29]
Fe, 99.99 wt.%
12 ± 0.5 5
63-351
99.995 Not reported Not reported 700-900 Not reported
Chemical analysis of 6 melt samples regular intervals
(6-8 hours)
FeAl3(Fe2Al5)
[30] CO, 99.98 wt.% 63-351 Co2Al9
Ni, 99.97 wt.% 59-591 NiAl3, Ni2Al3
Mo (pc** 99.99 wt.%)
12.7 Not reported 95-306 99.999 45 ± 1 ID 150 g
735-915 Not reported
Chemical analysis of melt samples
(ICP)
Al8Mo3, Al17Mo4, Al22Mo5, Al4Mo
[31] Nb (99.997 wt.%) NbAl3
Cr (99.997 wt.%) Δ, γ, β phases
Y (99.9 wt.%) 700-800 Al3Y, Y-rich (68 wt.%)
Nb (pc, 99.5 wt.%)
(mc, 99.88 wt.%)
12 Not reported 38-382 99.995 36 ± 1 ID 100 g 700-850 Not reported
Chemical analysis of 6 melt samples regular intervals
(7.5 hours)
NbAl3
[32] Ta (mc, 99.99 wt.%) TaAl3
Mo (pc 99.95 wt.%)
(mc 99.994 wt.%) Al4Mo
W (mc, 99.999 wt.%) WAl5
Ti, 99.7 wt.% 10 10 0-960 99.6 Not reported Not reported 800-1300 Not reported Mass loss (30-60 min) Not reported [33]
Armco Fe, 99.0 wt.% 11 50 8.5-615.4 99.99 Not reported 100 ± 5 g 700-800 To reach
close to bath temperature
Chemical analysis of melt samples
FeAl3, Fe2Al5, FeAl2, Fe3Al,
Fe2Al5 [34]
Fe–Ni (5 to 90 wt.% Fe) 11.28 ± 0.01 6 62-787 99.995 Not reported 0.1 lit 700 To the
required temperature
Chemical analysis of melt samples
(150-2400 s) Not reported [35]
CHAPTER 2. BACKGROUND
16
Table 2-3: (Continued)
Sample type
Solid addition
Diameter (mm) Length (mm)
Rotational speed (rpm)
Al purity wt.%
Al bath size (mm)
Al bath mass or volume
Temperature (°C)
Sample preheating
Dissolution rate measurement Intermetallics Reference
Cylinder
Ti (99.96 wt.%) 31
12.7 immersed
21.6 (11.1 mm from the axis of
rotation) 99.996 Not reported Not reported 705-818
1 hour, 2.5 cm above the
melt Mass loss Not reported [23]
V(99.5 wt.%) 31 Cr (High purity) 25 Ni (99.96 wt.%) 31
Co (99.816 wt.%) 31
Fe (99.885 wt.%) 31
9-34 (11.1 mm from the axis of
rotation) Fe (99.6 wt.%)
5 50 (30 immersed)
7-130 (around the
addition axis)
99.8 45.1 ID, 140.6 depth 350 gr 700-800 Not reported Mass loss
(0-2500 s) FeAl3–Fe2Al5 [36] Fe–3C (2.94 wt.% C) Fe–Cr (2.94 wt.% Cr)
S45C Fe–Si (3 wt.% Si)
CHAPTER 2. BACKGROUND
17
2.2 Dissolution of Solid Additions in Two-phase Flow
Quantifying the effect of gas agitation for enhancing the dissolution rate of alloying additions
in liquid Al has been largely neglected in the literature. We refer to flow phenomena with
injected gas as "two-phase flow".
Dissolution is a localized phenomenon, and predicting the rate of the process requires an
estimate of the velocity field. Since we consider gas agitation as a means of dissolution
acceleration, the behavior of bubbles and characteristics of the bubble-induced velocity field
are of interest. Lance and Bataille [41] argue that rising bubbles induce a random velocity
field in the liquid that they refer to as "pseudo-turbulence", that is different than the
shear-induced turbulence or "true turbulence" of the liquid phase. Pseudo-turbulence can be
generated by liquid stirring due to the motion of bubbles, in addition to turbulence due to
vortex shedding in the wake of bubbles, and the deformation of gas/liquid interfaces.
2.2.1 Fluid mechanics of bubbly systems
The introduction of gas at moderate velocities into liquid metals is common in ladle refining
processes. In this case, bubbles move due to buoyancy (bubbling regime), as opposed to the
initial gas momentum (jetting regime). An example of the jetting regime is gas injection in
Basic Oxygen Furnaces (BOFs), where the gas flow rate can be on the order of
10,000 NLPM, an order of magnitude higher than in refining furnaces [42]. In this case, the
gas released from the lance does not form bubbles, but rather blows into the liquid metal as a
continuous jet.
Within the bubbling regime, bottom stirring and top lance injection, shown in Figure 2-2,
are widely practiced in furnaces, ladles, and other transfer vessels, to homogenize
temperature and composition. These two approaches can differ in the number and size of
bubbles [43]. In bottom stirring, a bubble plume forms that leads to rapid bubble break up
CHAPTER 2. BACKGROUND
18
and coalescence, resulting in a distribution of bubble size along and across the container. In
top lance injection, generally fewer large bubbles agitate the liquid, although generating a
bubble plume similar to bottom stirring is possible via higher gas flow rates and increased
lance immersion depth.
Bubble plumes, typically generated by bottom stirring, are so common in ladle
metallurgy that numerous investigations have characterized bubble formation [44-46], the
subsequent rise behavior [43-45,47,48], and the velocities within [47,49-52] and outside [49-
53] of the two-phase region. A good review of the fluid dynamics, heat and mass transfer in
these ladles is presented in [9,10].
(a) (b)
Figure 2-2: Schematics of (a) bottom stirring and (b) top lance injection.
By comparison, there exist comparatively few studies of two-phase flow by top lance
injection, where the lance intrudes on the rise trajectory of individual bubbles. This is the
injection type adopted in the current study, and so relevant knowledge of bubble formation
and rise is required. In addition, as top lance injection can result in a column of large bubbles,
previous experimental and numerical studies of the liquid velocity field induced by an
unhindered stream of large bubbles may be relevant, although the presence of the lance may
limit their direct applicability.
CHAPTER 2. BACKGROUND
19
2.2.1.1 Bubble formation and trajectory
Information on the size and distribution of bubbles is required to understand their interaction
with the surrounding liquid. Using top lance injection, bubble frequency and size have been
studied by visual observation in transparent liquids [54-56] and with indirect measurements
(e.g. pressure transducers, acoustic devices) in opaque liquid metals [46,57].
When gas is injected vertically downward into a liquid, the bubbles initially travel
downwards a distance, Lv, before beginning to rise. Using water experiments, Iguchi et al.
[58] proposed a correlation for Lv as a function of 2 5g l lance,IDρ Q (ρ gd ) and showed that for the
bubbling regime, this vertical migration is negligible.
The non-wetting characteristic of liquid metals must be considered in metallurgical
bubbling systems, and so water experiments must be interpreted carefully [59]. The contact
angle, θc, is used to quantify the wettability. For the case of a liquid drop on a solid surface,
liquid wets the solid when 0 < θc < 90º, and does not wet it when 90º < θc < 180º. As clearly
illustrated in Figure 2-3 [60], a He bubble forms in water at the end of a lance because the
water/stainless steel contact angle is small (θc < 90º). On the other hand, mercury does not
wet stainless steel (θc > 90º) and so the bubble wets the outer surface of the lance.
For a downward facing lance, non-wetting behavior combined with the upward buoyancy
force results in more rapid bubble detachment and rise, as the bubbles wick up onto the lance
and so experience a smaller drag force. As a result, Irons and Guthrie [46] reported 10 times
smaller bubbles (by volume) using a vertical downward facing nozzle than an upward facing
one, in pig iron. Using a poorly wetted lance in water, it was shown that Ar bubbles along the
lance rise faster than away from it [61]. Fu et al. [57] found that Cl2–Ar bubbles in liquid Al
wicked up onto a top submerged alumina lance; however, they were unable to measure the
bubble rise velocity using a two-microphone detection system, as the noise of bubbles
breaking close to lance was not distinct. Although the rise velocity of bubbles in liquid Al
was not reported in this case, its lower limit may be estimated using measurements from a
CHAPTER 2. BACKGROUND
20
J-shaped top submerged lance [57]. For flow rates and equivalent bubble diameters pertinent
to this study (1-2 cm), the rise velocity was measured to be 0.35-0.55 m s−1 [57,62].
(a) (b)
Figure 2-3: (a) He bubble in water, and (b) He bubble in a liquid metal (mercury), both at the end of a stainless steel lance [60].
The bubble arrival frequency at the melt free surface can be different than the bubble
detachment frequency, for two reasons: the formation of double bubbles, or "doublets", at the
lance exit [44,46], and bubble coalescence during rise. In liquid Al, using a radiographic
technique, the coalescence of Ar bubbles was observed at the exit of an upward facing nozzle
[62].
A liquid flow in the transverse horizontal direction relative to the nozzle results in an
early detachment of bubbles because of the shear force exerted at the lance exit [63]. An
investigation of a cross-flow on a top injection lance in water showed that by increasing the
transverse velocity at the lance exit, the bubble formation frequency begins to increase when
( )0.5blance,IDU gd > 0.204 , provided the generated bubble is not partially engulfed in the wake
of the lance, which usually occurs at higher flow rates [55].
2.2.1.2 Bubble-induced velocity field in the liquid
In a confined column of liquid, with bubbles rising individually as a chain along the center,
the liquid velocity field depends on the trajectory of the bubbles [64]: (a) rigid bubbles that
follow a straight line establish a bulk recirculating flow, (b) moderately wobbling bubbles
that follow a 2D zigzag path add uneven counter-rotating vortex pairs, and (c) violently
CHAPTER 2. BACKGROUND
21
wobbling bubbles that follow a spiral trajectory further generate cross flow and large
irregular circular flow. In a water tank including a free surface, large scale structures of a
bubble-induced flow were examined [65] using time-resolved image velocimetry. Aside from
mean recirculating flow and source-and/or sink-like flow related to bubble wakes, an
up-and-down motion of the free surface was identified as another dominant flow structure.
A superposition of the bubble-induced motions results in a random flow field with
velocity fluctuations that have been referred to as pseudo-turbulence [41]. It has been shown
that larger bubbles, associated with a higher gas flow rate, do not significantly alter the
recirculating mean velocity field; however, the turbulence is greatly enhanced [65].
Increasing the gas flow rate by a factor of 5 yielded only a 2.5 times increase in the overall
kinetic energy, while increasing the turbulent kinetic energy by a factor of 5.7. It was
postulated that the energy transfer from the bubbles to the liquid is non-linear due to the
oscillation of the free surface [65]. Laser Doppler Velocimetry (LDV) measurements of
bubble plumes in water confirmed a uniform turbulent energy in the bulk, but significantly
higher values in the two-phase region and near the free surface [49,52]. Because of gas/liquid
interaction the turbulence intensity in the plume region can be 0.5, much higher than the
turbulence intensity of typical single-phase flows (<0.2) [49]. A comparison of vertical and
horizontal velocity fluctuations showed anisotropy of turbulence skewed towards the vertical
direction due to flow entrainment in the wakes of bubbles [49,66].
Numerical modeling of gas–liquid two-phase flows: The above‐mentioned studies have
characterized bubble-induced flow; however, the quantitative results (e.g. bulk velocity and
velocity fluctuations) cannot be easily extrapolated to other studies, given that the flow is a
function of numerous parameters including the gas/liquid density ratio, the gas flow rate, the
size and distribution of bubbles, and the geometry of the tank or column. In particular, there
is a lack of data of bubble-induced velocity fields in high temperature liquid metals, in which
velocities cannot be easily measured. For such systems, CFD can be used to estimate local
velocities in a flow field, which is especially important when one seeks to quantify a
CHAPTER 2. BACKGROUND
22
localized rate phenomenon such as dissolution. Mathematical/numerical models used to
simulate two-phase flow are summarized below, including shortcomings and advantages as
they relate to the system of interest.
In early work, the liquid velocity field in a ladle with a central plume of many small
bubbles was approximated analytically by prescribing a Gaussian distribution for bubble
distribution and liquid velocity [67]. Then, simple quasi-single phase models were developed
that assumed a predetermined axisymmetric plume, with the buoyancy force acting on the
surrounding liquid. These models predicted the velocity field outside the plume satisfactorily
with little computation [68], assuming the free surface to be flat. For flows with larger
bubbles, such quasi-single phase models cannot be used, as the bubbles are larger than the
mesh size [43].
In turbulent multiphase flows, when none (k-ε model [69]) or only some (Large Eddy
Simulation or LES [70]) of the turbulent structures are resolved, turbulence closures must be
included to model mean turbulent quantities. It has been shown that k-ε model predictions of
fluctuating velocities agree with LES predictions and experimental measurements only when
extra source terms are added to the k and ε equations [70]. The source term for turbulent
kinetic energy is simply the work done by the bubbles (the interfacial force multiplied by the
slip velocity) [69].
In the last two decades, direct numerical simulations (DNS) of multiphase flows have
become the state-of-the-art. These simulations solve the governing equations on a sufficiently
fine grid so that all time and length scales are fully resolved (for laminar flow and more
recently for turbulent flows (e.g. [71,72])), using one of the following approaches: the
Eulerian-Lagrangian approach, or the Eulerian or "one-fluid" approach [73].
The Eulerian-Lagrangian method for computing multiphase flows refers to the use of
moving Lagrangian grids in conjunction with an Eulerian stationary grid [73]. Fluid
interfaces are represented by unstructured grids that move and deform over the Eulerian grid.
CHAPTER 2. BACKGROUND
23
This is a complex approach as the interaction of the Lagrangian grids with the stationary grid
must be managed, and the Lagrangian grids reconstructed dynamically (e.g [74]).
In the Eulerian models, a single set of Navier-Stokes equations is solved on a stationary
grid covering the entire domain, and different fluids are identified by a marker function that
is advected across the Eulerian grid, and that is in turn used to calculate fluid densities and
viscosities. Examples of Eulerian models are the Marker-and-Cell (MAC) method [75], the
Volume of Fluid (VoF) method developed by Hirt and Nichols [76], and the Level-Set
method introduced by Osher and Sethian [77]. Among these models, the VoF method is
relatively simple to implement and has demonstrated its usefulness for a large number of
problems. For example, using a 3-D VoF method, bubble-bubble interaction [78] and the
effect of cross flow on bubble formation [79] has been satisfactorily modeled. The VoF
method is widely used, and a number of commercial software packages employ it to
represent and track interfaces and free surfaces. In the VoF method, a volume fraction is
identified in each cell as the fraction of the volume of that cell filled with one of the two
fluids. As interfaces move with the flow, the volume fractions are advected, either by a
geometric reconstruction/advection scheme (e.g. [80]), or by any of a number of algebraic
schemes.
2.2.2 Mass transfer in two-phase flow systems
Dimensionless correlations for convective heat and mass transfer from solids to liquids are
based on boundary layer theory. When the fluid motion is generated by an external source (as
opposed to natural convection caused by thermal and density gradients), the classical
dimensionless mass transfer correlations (e.g. Ranz-Marshall correlation) are functions of Re
and Sc [81]:
( )Sh f Sc,Re= (2-2)
CHAPTER 2. BACKGROUND
24
This type of correlation is only applicable when the level of turbulence in the flow is very
low. However, ladle metallurgical processes involve the agitation of liquid in a large domain,
characterized by high Re that indicate very turbulent flow [10]. The effect of turbulence can
be taken into account by (i) modifying the definition of Re for a stirred flow, or by (ii)
introducing a new dimensionless number into a correlation, similar to what has been done for
heat and mass transfer phenomena in flows with grid generated turbulence [82-84]. In
metallurgical reactors, the choice of approach depends on the size of the alloying addition.
Mass transfer may occur from solid particles (e.g. Mn and Fe powder in liquid Al, with
particles smaller than 0.06 cm [11]) or from large additions (e.g. Si in liquid Al with lumps
larger than 1 cm). Approach (i) has been used for small particles, and approach (ii) for larger
additions. In the following, a brief review of the former is provided, for the sake of
completeness, and then the latter is discussed in more detail.
2.2.2.1 Mass transfer from small particles
The rate of dissolution of small particles is a function of the relative velocity between the
particles and the turbulent liquid flow. Unfortunately, that relative velocity cannot be simply
measured for erratically moving small particles due to the random velocity fluctuations
superimposed on the mean bulk flow velocities [11].
In a turbulent flow, larger eddies extract energy from the bulk flow and transfer it to
smaller eddies; smaller eddies in turn transfer their energy to even smaller eddies, and so on,
until the eddies are of the Kolmogroff length scale, and energy is dissipated by viscous
losses. During this cascade, the directional information of the large eddies (with a length
scale of the main flow) is gradually lost as the energy transfers from larger to smaller eddies
in multiple directions. Thus, at scales much smaller than the scale of the main flow, the
turbulence is isotropic, and this is referred to as Kolmogoroff’s theory of local isotropic
turbulence. For a small particle, mass transfer is a function of the flow in the small volume
CHAPTER 2. BACKGROUND
25
around the particle, and so average statistical properties of the turbulent flow can be
determined by proposing a Re based on the local energy dissipation rate [85]:
p
1/3 4 3p
d 1l
dRe C
ε=
ν (2-3)
where ε is the energy dissipation rate per unit mass of liquid. To estimate the energy
dissipation rate in turbulent stirred systems, it is assumed that the rate of energy supplied is
equal to the rate of energy dissipated [11]. The energy can be supplied either by impellers or
by injecting bubbles.
The energy input rate of impellers is typically estimated by measuring the torque on the
motor [11]. For instance, for two different impellers in a turbulent stirred Al bath, the
dissolution of suspended fine particles of Fe and Mn (equivalent spherical diameter less than
0.06 cm) was correlated to pdRe [11].
Injecting gas into a liquid adds energy in two ways [10]: as gas kinetic power,
3K g c gE = 0.5ρ A V , and due to the bubbles displacing the liquid. There are various expressions
to describe the latter [9]; a widely used one calculates the work done on the surrounding
liquid by the upward movement of a known gas volume [86,87]:
lB
0
gLE 854QT log(1 )Pρ
= + . (2-4)
In the bubbling regime in ladle metallurgy, the energy due to buoyancy is dominant
[9,88], and the gas kinetic power is generally neglected. Using the rate of energy input due to
buoyancy per unit mass of liquid, Kolmogoroff’s theory of local isotropic turbulence can be
employed to correlate the mass transfer rate of particles to the fluid properties and energy
dissipation rate. For instance, a cold model study (benzoic acid/water) was used to
investigate the assimilation of Direct Reduced Iron (DRI) particles charged into the slag layer
in electric furnaces, as a function of the rate of gas energy input [89].
CHAPTER 2. BACKGROUND
26
2.2.2.2 Mass transfer from large solid additions
When additions are in the form of larger pieces, as is of interest here, the associated length
scales are more similar to the dimensions of the main flow, and so the above approach cannot
be employed. This means the size of eddies which contribute to convective mass transfer lie
outside the range of isotropic turbulence, so that the convective mass transfer coefficients for
large solid additions will be affected by both the dimensions of the system and the means of
agitation. Therefore, both the bulk flow velocity and the intensity of turbulence must be
identified in the vicinity of the addition.
In a bubble-induced flow, the instantaneous flow variables are random and vary with time
either due to the pseudo-turbulence or shear-induced turbulence, even though in practice
these effects cannot be isolated [41]. In such a flow the velocity at any location can be
expressed in terms of a mean value and a fluctuating value. For instance, in cylindrical
coordinates:
r r ru u (r, , z, t) u (r, , z, t)′= θ + θ , (2-5)
u u (r, , z, t) u (r, , z, t)θ θ θ′= θ + θ , (2-6)
z z zu u (r, , z, t) u (r, , z, t)′= θ + θ , (2-7)
where ru , u θ , and zu are time-averaged velocities at the point (r, θ, z). The velocity
fluctuations are represented by ru′ , uθ′ , and zu′ . The kinetic energy of the turbulent
fluctuations per unit mass is then defined as [90]:
2 2 2r z
1k (u u u )2 θ′ ′ ′= + + (2-8)
which is half of the trace of the Reynolds stress tensor. Note that pseudo-turbulence
contributes to the kinetic energy and to the Reynolds stress tensor, and so the bubble-induced
flow can be categorized as a turbulent flow.
As mentioned earlier, in a turbulent flow, mass transfer processes are functions of the
detailed flow and turbulence characteristics near the solid addition. However, the dependence
CHAPTER 2. BACKGROUND
27
cannot be easily predicted from first principles due to the complex nature of turbulent
transport [91]. Various models have been developed based on many simplifying assumptions.
As discussed extensively by Mathpati and Joshi [91], Direct Numerical Simulation (DNS)
and Large Eddy Simulation (LES) represent the state of the art for critically evaluating the
assumptions behind heat and mass transfer models, and are capable of providing detailed
information on turbulent flow structures in the vicinity of an interface. However, to date such
methods are limited to low values of Re and Sc, as they are computationally intensive.
Simpler models fall into two categories: one category (referred to as analytical) is based on
turbulent viscosity, and the other on heuristic arguments.
The so-called analytical approach generally results in correlations for transport rates with
parameters obtained from experimental results; such models are semi-empirical in nature,
and do not necessarily help to understand the underlying physics of transport near solids. On
the other hand, heuristic models are useful to visualize the transport phenomena, but are
limited in applicability. Heuristic models suggest that fluid elements, referred to as eddies,
intermittently move from a turbulent core of the flow to a solid surface, periodically
renewing the interface by introducing fresh liquid. For example, the Random Eddy
Penetration (or Surface Rejuvenation) Model [92,93] may be used to understand the
enhancement of heat and mass transfer rates by gas agitation. According to this theory,
eddies from a bulk liquid intermittently travel sufficiently close to a solid to sweep away
some of the solute in a mass boundary layer [93]. The contact time, and the distance these
eddies travel with respect to the solid/liquid interface, are random. The eddies remove
accumulated solute in the concentration boundary layer from the approach distance out to
infinity and replace it with fresh fluid [93], causing a steep concentration gradient at the
approach point. Between the random encroachment of eddies, mass transfer in the boundary
layer is controlled by diffusion [93]. The net effect is an increase in the mass transfer rate.
Although heuristic models such as this describe a picture of mass transfer near the interface,
CHAPTER 2. BACKGROUND
28
they are not widely used because parameters such as the size, time and distance distributions
of the penetrating eddies are not well-developed.
Consequently, following an empirical approach that is appropriate for engineering
applications, mass transfer coefficients can be estimated by superimposing turbulent effects
on the bulk flow (Re) and fluid properties (Sc), by introducing a new dimensionless variable,
ReT [84,94]:
cT
ulRe =ν
, (2-9)
where u represents the root mean square (rms) velocity fluctuations [95,96]:
2 2 2
r zu u uu3θ′ ′ ′+ +
= (2-10)
(Note that for isotropic turbulence: 2 2 2r zu u u u 2 / 3kθ′ ′ ′= = = = ).
Similar to the pioneering work of Lavender and Pei [84], who studied heat transfer in
unidirectional grid-generated turbulent flow, mass transfer from solid additions immersed in
gas-agitated ladles is described as:
( )TSh f Sc,Re,Re= . (2-11)
In the above equation, ReT can be replaced by turbulence intensity, where the rms
velocity fluctuations are normalized by the local mean velocity [97,98]:
uTuV
= . (2-12)
A mean value of the local motion near a solid addition can be calculated by taking into
account the influence of multi-dimensional flow as in a gas-agitated system [96-98]:
2 2 2r zV(r, , z, t) u u uθθ = + + . (2-13)
Different transport correlations have been developed for gas-agitated systems; lists are
presented in Table 2-4 and Table 2-5. These correlations require flow velocities, which are
determined using one of three methodologies: analytical approaches based on an energy
CHAPTER 2. BACKGROUND
29
balance; solving the Navier-Stokes equations in conjunction with a turbulence model; and
LDV.
Several studies have utilized water modeling to correlate heat or mass transfer
coefficients from solid additions to either a bulk velocity or to the local mean and fluctuating
velocities of recirculating flow generated by gas injection (similar to Figure 2-2 (a)).
Tanguchi et al. [99] used an energy balance approach to estimate the upward bulk velocity. A
comparison of their results with the Whitaker [100] correlation (applicable to turbulent flows
with low turbulence intensities) was satisfactory.
In a study of the melting rate of horizontal ice rods [101], the heat transfer coefficients
were correlated to the measured (also computed elsewhere [52]) local mean velocity and
turbulence intensity based on the mean velocity at the centerline of the ladle. Iguchi et al.
[102] showed that for a bottom blowing water jet (single-phase flow) and bubbling jet
(two-phase flow), the measured turbulence intensity in the latter flow was much higher
around a melting ice sphere. The results for both single-phase and two-phase flow were
correlated by modifying the empirical correlation of Whitaker [100], by introducing Tu as a
new dimensionless number, to account for turbulence intensities higher than 5%. However,
the turbulence intensity was calculated based on the vertical velocity of the plume centerline.
In the spirit of the Lavender and Pei correlation [84], Mazumdar et al. [97] proposed a
dimensionless equation for the dissolution of solid benzoic acid cylinders in a gas-agitated
water ladle taking into account local mean and fluctuating velocities.
Although most studies have used cold models, there have been some attempts to
investigate gas agitation on heat and mass transfer phenomena in high temperature liquid
metals, presented in Table 2-5.
Argrypoulos et al. [103] examined the melting rate of Al cylinders in liquid Al without
(single-phase flow) and with gas agitation (two-phase flow) in a bottom stirred ladle. They
used an average plume velocity as the characteristic velocity of the two-phase flow. At a Re
CHAPTER 2. BACKGROUND
30
similar to that of a single-phase flow, higher heat transfer coefficients were obtained for the
two-phase flow owing to the higher velocity fluctuations.
Extending their ice/water study, Taniguchi et al. [104] compared the melting and
dissolution rates of Al spheres in Al and Al/Si baths with available correlations for heat [100]
and mass [105] transfer in room temperature fluids. The agreement was unsatisfactory
because they used the average plume velocity instead of local velocity values, neglected the
turbulence intensities, and as Argyropoulos et al. [106] argue, the classical heat transfer
correlations appropriate for high values of Pr are likely inadequate for metallic systems.
Szekely et al. [98] developed an axisymmetric numerical model for a cylindrical plume to
compute a velocity profile in an Ar-stirred steel ladle. Applying the bulk velocity values to
the modified Lavender and Pei correlation [84] for cylinders, and assuming a constant
turbulence intensity of 0.3, they demonstrated satisfactory agreement with the measured mass
transfer coefficient from graphite cylinders. Following a similar approach, but this time
prescribing a conical-shaped plume, Mazumdar et al. [107] predicted the local mean
velocities and rms velocity fluctuations in a 25 kg gas agitated melt [88], and showed that
using their previously-derived correlation for a cold system demonstrated good agreement
with the experimental data of Wright [88].
Wright [88] attempted to deduce the average plume velocities by measuring the mass
transfer rate from steel cylinders in a Fe–C melt. However, they used a correlation that had
been proposed for the dissolution of cylinders rotating at a certain peripheral velocity [108].
Thus, the analysis yielded equivalent peripheral velocities for steel cylinders based on their
dissolution rate. This, of course, did not provide information on the plume bulk velocity or
the level of overall turbulence in the gas-stirred ladle.
Finally, El-Kaddah et al. [109] dissolved graphite cylinders in a 4-tonne inductively
stirred melt of molten steel. The dissolution of the cylinders depended both on the mean
velocity and turbulence intensity. While the source of agitation was different than a
CHAPTER 2. BACKGROUND
31
gas-stirred melt, the study confirmed the significance of local mean velocities and velocity
fluctuations on the mass transfer process.
In general, the variety of dimensionless correlations that have been used, and the
sometimes poor agreement with measured data [97], suggest that they may be very specific
to the conditions used to derive them. In addition, none of the above studies directly
compared mass transfer in a single-phase forced convection system to mass transfer in a
gas-agitated one. The current study was undertaken for that reason: to investigate the effect
of gas injection on a constant bulk flow, and to describe the effect of gas agitation on the
mass transfer as an increment to the bulk velocity.
CHAPTER 2. BACKGROUND
32
Table 2-4: Heat and mass transfer correlations based on cold model studies of gas-agitated metallurgical ladles. Equation Description Method to estimate velocities Reference
( ) ( )1 41 2 2 3 0.4d d b 0
Nu 2
0.4 Re 0.06Re Pr
− =
+ μ μ
sphere
l pd
l
U d3.5 Re 76000
ρ≤ = <
μ0.71 Pr 380< <
( )b 01 3.2< μ μ <
Photography of tracers, macroscopic energy balance [99]
( )0.8 1 3d
Nu
0.388 Re Tu Pr
=
horizontal cylinder 2 2
l r zd
l
u u d100 Re 2000
ρ +≤ = ≤
μ
z,c
uTu 0.15u
= >
Pr 10=
Numerical solution of Navier-Stokes equation or
LDV measurements, both presented in [52]
[101]
( ) ( )1 41 2 2 3 0.4 1.36d d b 0
Nu 2
0.4Re 0.06Re Pr (1 Tu)
− =
+ μ μ +
sphere
l cd
l
u d1500 Re 16000ρ≤ = <
μ
2z,c
z,c
uTu 0.5
u
′= ≤
5.7 Pr 9.2< <
( )b 00.46 0.7< μ μ <
LDV measurements [102]
( ) ( ) ( )0.25 0.32 0.33d T
Sh
0.73 Re Re Sc
=
vertical cylinder 2 2
l r zd
l
u u d100 Re 3000
ρ +≤ = <
μ
lT
l
ud100 Re 800ρ≤ = <
μ
u 2 3k=
Sc 500=
Numerical solution of Navier-Stokes equation [97],
LDV measurements [110] [97,110]
CHAPTER 2. BACKGROUND
33
Table 2-5: Heat and mass transfer correlations based on hot model studies of gas-agitated metallurgical ladles. Equation Description Method to estimate velocities Reference
( ) ( )1 41 2 2 3 0.4d d b 0
Nu 2
0.4 Re 0.06Re Pr
− =
+ μ μ
sphere
l pd
l
U d3.5 Re 76000
ρ≤ = <
μ0.71 Pr 380< <
( )b 01 3.2< μ μ <
Macroscopic energy balance [104]
( )0.621 2NSh Sh 0.374 ReSc= + 1/ 2
d1 Re Sc 500000≤ <
1 2 1 4 1 3TSh 2 0.72Re Re Sc= +
vertical cylinder 2 2
l r zd
l
u u dRe
ρ +=
μ
lT
l
udRe ρ=
μ
Numerical solution of Navier-Stokes equation for ru and zu [98]
0.33 0.644md
p
k0.1121Re Scu
− =
vertical cylinders
l pd
l
u d100 Re 10000
ρ< = <
μ Estimated from dissolution data [88]
( )0.8 1 3dSh 0.388 Re Tu Sc=
vertical cylinders 2 2
l r zd
l
u u dRe
ρ +=
μ
c,max
uTuu
=
u 2 3k=
Numerical solution of Navier-Stokes equation [109]
34
CHAPTER 3
DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: SINGLE-PHASE FLOW
A study of Si dissolution in molten Al in single-phase flow is presented. We begin by
estimating the maximum time scale associated with the dissolution of Si in Al based on pure
diffusion. Then, we present experimental results of the dissolution rate of Si in Al as a
function of impurity levels of the solid Si, the Al bath temperature, and flow conditions. The
experimental results are compared with available natural and forced convection correlations.
3.1 Pure Diffusion of Si in Al
A one-dimensional model was developed to examine the pure diffusion of a Si cylinder into
liquid Al, to evaluate an upper bound on the total dissolution time. A one-dimensional, radial
mass diffusion equation was solved.
Consider a pure cylindrical Si addition (as used in the experimental study) immersed in
molten Al. The diffusion of the solid occurs due to the driving force of a concentration
gradient. The change in Si concentration at any point (r) in the infinite medium is then
determined by:
Chapter 3. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: SINGLE-PHASE FLOW
35
2
2Si/Al
1 C C 1 CD t r r r
∂ ∂ ∂= +
∂ ∂ ∂ (3-1)
The radius of the cylinder decreases as the Si diffuses, as described by the following
boundary condition,
SatSi Si Si/Al
r S(t )
dS C(C ) Ddt r =
∂−ρ = −
∂ (3-2)
Equation (3-1) was solved numerically for the Al–Si system, where S0 = 0.95 cm (the
initial radius of the specimens used in the experimental study), Sat 3SiC 504 kg m−= [19] (the
saturation concentration of Si in Al at T = 973 K), ρSi = 2331 kg m−3 [7], and DSi/Al = 1.25 ×
10−8 m2 s−1 [39]. The initial Si concentration in the liquid is assumed to be zero, and the
concentration of Si at the interface reaches the equilibrium value, SatSiC , instantaneously.
Figure 3-1 shows that the time required for the complete dissolution of the cylinder is
approximately 6 hours. This is the maximum possible total dissolution time; under natural or
forced convection conditions the overall mass transfer coefficient will be higher. The
variation of cylinder radius with time is not linear. Early on, dissolution proceeds at a fast
rate due to a higher concentration gradient (dC/dr), and then decelerates as the solute diffuses
into the infinite medium. The rate of dissolution accelerates again as the radius of the
specimen decreases. The dissolution rate is proportional to the concentration gradient at an
interface. For a given value of saturation concentration at an interface, the concentration
gradient as a function of distance from the interface decreases more sharply from a
cylindrical interface than a planar one. This is because in a cylindrical domain the area
through which the solute diffuses is proportional to r. When the specimen shrinks, this effect
leads to a higher dissolution rate as the interfacial area becomes very small.
Chapter 3. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: SINGLE-PHASE FLOW
36
Figure 3-1: Cylinder radius vs. time, considering only pure diffusion.
3.2 Experimental Methodology
An experimental setup was developed to study the effect of natural and forced convection on
the dissolution of cylindrical Si specimens.
3.2.1 Materials
Al ingots were of commercial purity with a typical composition of 99.87 wt.% Al,
0.04 wt.% Si and 0.09 wt.% Fe. Two different batches of metallurgical grade Si were used,
with higher and lower levels of impurities, referred to as the MGSi–I and MGSi–II
respectively; a few experiments were also conducted with monocrystalline electronic grade
Si, referred to as EGSi. Table 3-1 lists the main impurities in each batch, determined using
Inductively Coupled Plasma (ICP) mass spectrometry.
Chapter 3. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: SINGLE-PHASE FLOW
37
Table 3-1: Overall chemical analysis (in wt.%) of the various batches of Si used in this study.
The Si arrived in large pieces from which cylinders were core drilled to a diameter of
approximately 1.87 cm and a length of between 9 and 10 cm. The specimens were ground
with emery paper and cleaned in acetone before use1. Each Si specimen was weighed to
±0.1 gr.
In a few of the experiments, the specimen temperature was monitored by a K-type
thermocouple placed in a hole (0.6 cm diameter and 2.5 cm deep), drilled into the center top
of the cylinder. The hole was filled with tin (melting point of 231.9°C) which upon melting,
ensures proper thermal contact between the thermocouple and the specimen) drilled into the
center of the cylinder. A K-type thermocouple was also placed adjacent to the Si specimen
surface at the middle of the immersed length, recording the Al–Si interface temperature.
Thermocouple readings were unaffected by the heat of solution of Si in Al, which is low
(−1.6 kJ mol−1) [111].
3.2.2 Experimental set-up
Two different electric resistance furnaces were used.
Furnace A with stationary bath: A bilge-shaped graphite crucible (18 cm top ID, 13 cm
bottom ID, 25 cm deep) was filled with 11 kg of Al and placed in the furnace. A schematic of
1 Following this preparation procedure carefully played an important role in reducing the variability of the
experimental data. In an initial stage of this work, more than 300 Si dissolution experiments, with and without gas agitation, were carried out. To prepare the cylindrical Si specimens, the only measure taken was to use acetone to remove dirt prior to immersion; these were referred to as "as-drilled" specimens. Varying the operating parameters (i.e. temperature, velocity, gas flow rate, and immersion time), some trends could be deduced from the experimental results, but the scatter in the data was large. More accurate conclusions about the effect of the operating parameters were obtained by adding the grinding step (for more details see Appendix B). The results of the as-drilled specimens do not appear in this thesis.
Al Ca Fe Mn Ti MGSi–I 0.131 ± 0.017 0.083 ± 0.014 0.896 ± 0.136 0.012 ± 0.005 0.064 ± 0.009MGSi–II 0.081 ± 0.008 0.011 ± 0.001 0.233 ± 0.021 0.011 ± 0.002 0.026 ± 0.002
EGSi 0.001 <0.0001 0.001 <0.0001 <0.0001
Chapter 3. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: SINGLE-PHASE FLOW
38
this crucible is shown in Figure 3-2. The Si specimens were immersed in to the center of the
crucible for the natural convection experiments. The temperature of the furnace was
controlled with two K-type thermocouples connected to an electric control device, that
allowed the bath temperature to be controlled to ±2K. The bath temperature was monitored
by another K-type thermocouple immersed approximately 1.5 cm away from the crucible
wall and 3 cm beneath the Al free surface.
Figure 3-2: Schematic of a bilge-shaped crucible shows the location of thermocouples to examine the effect of preheating the specimen. All dimensions are in cm.
Furnace B with Revolving Liquid Metal Tank (RLMT): Figure 3-3 shows an image and
schematics of the RLMT (35 cm ID, 23.8 cm deep), with a usable capacity of 35 kg of Al. In
an effort to protect the crucible from the Al bath, the walls of the RLMT were painted with
boron nitride. The tank was placed inside a furnace and rotated by a variable speed motor.
The rotational speed of the tank could be controlled to ±0.5 rpm for the forced convection
experiments. A heavily insulated lid was used to minimize heat loss from the furnace top.
More detailed information on the furnace specifications is available from [112]. Through a
hole in the lid, the Si specimens were immersed at a radial distance of 13.5 cm from the
center of the tank. A single K-type thermocouple was placed at the center of the tank, 5 cm
Chapter 3. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: SINGLE-PHASE FLOW
39
beneath the molten metal free surface. The temperature of the liquid metal was controlled to
±2 K by manually adjusting the power to the furnace.
(a)
(b)
(c)
Figure 3-3: (a) The RLMT with solid Al charge, (b) top view shows the rotation direction and location of the immersed Si specimen and AA´ is perpendicular to the xy-plane and passes through the Si sample, and (c) side view with the Si sample at the AA´ plane. All dimensions are in cm.
Chapter 3. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: SINGLE-PHASE FLOW
40
3.2.3 Experimental procedure
A holding assembly was employed to immerse cylindrical specimens into the liquid Al in
both furnaces. For the RLMT, when the liquid Al reached the desired melt temperature, the
tank was rotated for a sufficient length of time prior to cylinder immersion to attain solid
body rotation. That time was estimated by ( )0.52AlH /ν ω [113], where H is the height of
liquid, νAl is the kinematic viscosity of the liquid Al, and ω is the RLMT rotational speed.
Approximately 11 min was required to spin up the tank to 1 rpm.
Once the Al reached the desired temperature and rotational speed, a Si cylinder was
immersed 8 cm into the liquid; after a certain duration, the cylinder was withdrawn and
allowed to cool. The solid Al that remained attached to the sample was removed by using a
38 vol.% HCl solution, which left the Si intact [114]. Then the partially dissolved Si
specimen was weighed to ±0.1 gr to determine the dissolved mass in the liquid Al. The data
presented in this thesis usually represents three experiments. The standard deviation of the
dissolved fraction measurements is represented by error bars.
With each Si cylinder immersion the Si level in the liquid Al increased. Whenever the Si
level in the Al bath exceeded 1.5 wt.% Si, the bath was discarded and replaced.
3.3 Results and Discussion
The results are presented in two parts: Si dissolution experiments under natural convection
conditions (without mechanical stirring) in Furnace A, and Si dissolution experiments under
forced convection conditions (by rotating the liquid metal) in Furnace B.
3.3.1 Natural convection experiments
The bath temperature was set to superheats (SPH) of 20, 40 60, 80 and 100 K for different
experimental runs. Figure 3-4 depicts typical MGSi–I samples after immersion for 3 min at
Chapter 3. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: SINGLE-PHASE FLOW
41
various bath superheats, and illustrates that after dissolution the sample lengths remained
unchanged.
Figure 3-4: MGSi–I specimens after natural convection dissolution for 3 min in an Al bath at various superheats.
Figure 3-5 depicts the dissolved fraction of MGSi–I specimens as a function of bath
superheat after 3 min immersion. The results show that the dissolved fraction increases
significantly with increasing bath superheat. The dissolution of the MGSi–I specimen at the
lowest superheat of 20 K is negligible, while approximately 50% of the sample dissolved at
the highest superheat of 100 K. In the industrial practice of Al–Si alloy making, the Al bath
temperature is typically in the range of 973-1013 K (700-740°C). Therefore, in the remainder
of this chapter, experimental data is only presented for superheats of 40 K and 80 K.
Chapter 3. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: SINGLE-PHASE FLOW
42
Figure 3-5: The effect of bath superheat on the dissolved fraction, md/mi, of MGSi–I specimens after natural convection dissolution for 3 min.
3.3.1.1 Heat transfer after immersion
When a Si addition, initially at room temperature, is first immersed into the liquid metal, an
Al shell solidifies around the cylinder. In the experiments, as shown in Figure 3-6, the
formation of an Al shell was confirmed by removing the solid Si from the Al bath after a
relatively short immersion time. As can be seen, the Al shell forms within a few seconds and
then melts back as the immersion time increases. The heat absorbed by the solid Si cylinder
upon immersion leads to a local temperature drop in the melt, that forms the shell. As the
temperature of the Si increases, the solidified layer melts back and eventually thermal
equilibrium is established between the Si cylinder and the liquid Al. Referring again to
Figure 3-6, as the shell melts back, it is likely that some of any remaining shell slips off of
the cylinder during withdrawal. As such, the portion of the Al shell that remains attached to
the cylinder upon withdrawal is likely less than the amount that was on the cylinder while
still submerged in the melt.
Chapter 3. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: SINGLE-PHASE FLOW
43
Figure 3-6: Formation of an Al shell on MGSi–I cylinders immersed for short periods in an Al bath at SPH = 40 K.
To investigate the influence of heat transfer after immersion on the onset and rate of
dissolution, experiments were run where the Si cylinder (and the specimen holding assembly)
was preheated above the Al free surface. The bath temperature, Al/Si interface temperature,
and the internal temperature of the cylinder were monitored, as shown in Figure 3-2.
Figure 3-7 (a) and (b) show typical temperature measurements without and with
preheating, respectively. The insets magnify the thermocouple readings near the initial bath
temperature. Without preheating, the specimen prior to immersion was at room temperature,
indicated by points A and F in Figure 3-7 (a), while the specimen centerline temperature
reached 793 K (520°C) after preheating for 3 min, shown by segment F´F´´ in Figure 3-7 (b).
Note that in Figure 3-7 (b), the kink at about 230°C on the Si centerline temperature is
associated with the melting point of the tin used to fill the specimen hole.
Figure 3-8 compares the dissolved fraction of Si with and without preheating at
SPH = 40 K. An extrapolation of the experimental data (linear regression) indicates that
preheating reduced the incubation time (the time to the onset of dissolution) from about 66 s
Chapter 3. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: SINGLE-PHASE FLOW
44
to about 10 s. The preheating not only shortens the incubation time, but also increases
slightly the subsequent dissolution rate by avoiding the initial large decrease in the liquid Al
temperature in the vicinity of the Si sample, as illustrated in the insets of Figure 3-7 (a) and
(b). Without preheating, the Al/Si interface temperature is 6 K lower than the liquid Al initial
temperature even after 3 min (point E in Figure 3-7 (a)); with preheating, the Al/Si interface
temperature is only 2 K lower than the initial bath temperature (point E´ in Figure 3-7 (b)).
Finally, Figure 3-7 (b) shows that the temperatures at the Al/Si interface and at the
specimen center (shown by segments A´B´B´´C´D´E´ and F´F´´G´H´I´, respectively) do not
exceed the initial temperature of the Al bath. This confirms that there is no appreciable
exothermicity involved in the dissolution process.
Chapter 3. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: SINGLE-PHASE FLOW
45
(a)
(b)
Figure 3-7: Temperature history of a MGSi–I cylinder immersed into liquid Al (a) initially at room temperature, (b) preheated for 3 min prior to immersion.
Chapter 3. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: SINGLE-PHASE FLOW
46
Figure 3-8: Comparison of the dissolved fraction with and without preheating, for MGSi–I at SPH = 40 K.
3.3.1.2 Isothermal mass transfer
The isothermal dissolution of Si makes the adjacent liquid denser, because Si T( ρ C ) =0.14∂ ∂
[116]. This means that upon dissolution, the fluid next to the Si cylinder flows downward,
which establishes a natural convection flow.
Since Si dissolution in liquid Al is LPD-controlled [39], the following equation describes
the steady-state rate of change of the cylindrical specimen mass at an instantaneous cylinder
radius:
( )2
Si N Sat bm Si Si
d r lk (2 r l) C C
dtρ π
⎡ ⎤= − π −⎣ ⎦ (3-3)
Further assuming that the Si bulk concentration bSiC =0 , and Siρ and l are constants,
Equation (3-3) reduces to:
Sat
N Sim
Si
Cdr kdt
= −ρ
(3-4)
Chapter 3. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: SINGLE-PHASE FLOW
47
Under the assumption that there is a negligible effect of curvature on the mass transfer
coefficient, Nmk , the radius of the Si specimen will vary according to the following equation:
Sat
N Si0 m
Si
Cr r k t= −ρ
(3-5)
To estimate the natural convection mass transfer coefficient, Nmk , a general form of a
dimensionless correlation is employed:
( )mm ,lSh n Gr Sc= , (3-6)
where Grm,l is defined as [117]:
3 Sat b
l Si T Si Sim,l 2
l l
g( C ) l (C C )Gr ∂ρ ∂ −=
ρ ν, (3-7)
and
l
Si/Al
ScDν
= . (3-8)
In the current study 8 8m,l5.5×10 < Gr < 7.4×10 , and 42 > Sc > 29. For this range of Sc,
the momentum boundary layer is thicker than the mass boundary layer and accordingly the
fluid properties are evaluated at the bulk concentration [108,118]. For 9 12m,l10 < Gr Sc <10 ,
two independent experimental studies [119,120] also involving mass transfer in liquid metals
suggest values of n that vary from 0.11 to 0.129, and m = 1/3, and so we adopt these values
as well.
As dissolution proceeds, the specimen radius decreases, and so the thickness of the mass
boundary layer changes with time and is truly transient [10]. In addition, when the densities
of solid and liquid phases are different, this induces a fluid velocity normal to the moving
interface (i.e. transpiration flow) [115]. In this thesis, these two effects are neglected when
predicting mass transfer rates:
(i) The effect of specimen shrinkage on the mass transfer rate is relatively independent of
bulk flow velocity, and has been characterized by the ratio of interface velocity to the mean
Chapter 3. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: SINGLE-PHASE FLOW
48
mass transfer coefficient with no transpiration and constant specimen radius [115]:
( ) mdr / dt kλ = − . The mass transfer coefficient for a shrinking specimen at large values of λ
will be lower than for a specimen of constant radius. When this ratio is large the solute
accumulates near the interface. In the current study 0.22 < λ < 0.25, for which the results of
[115] suggest that neglecting the effect of specimen shrinkage will overpredict the mass
transfer coefficient by less than 15%.
(ii) As mentioned in Chapter 1, solid Si in liquid Al is neutrally buoyant, and so the effect of
transpiration is negligible.
The experimental results of dissolved fraction were converted into changes in average
radius. A dimensionless cylinder radius after immersion (non-dimensionalized by the initial
radius of the cylinder) was calculated as:
d
0 i
mr 1r m= − (3-9)
Figure 3-9 shows comparisons of the predicted and measured cylinder radius versus
immersion time, at different bath superheats, where the incubation times determined from the
experimental results have been incorporated into the model predictions. The correlation
clearly overpredicts the rate of dissolution of the MGSi–I, while the rates of dissolution of
the MGSi–II and EGSi are much better predicted. A detailed discussion of the effect of
impurities is presented in the next section. Regardless of Si purity, increasing the temperature
of the liquid Al, which increases the saturation concentration at the solid/liquid
interface, SatSiC , increases the dissolution rate of the specimen, as shown in Figure 3-10.
Chapter 3. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: SINGLE-PHASE FLOW
49
(a)
(b)
Figure 3-9: Comparisons between measured and predicted natural convection mass transfer for liquid Al at (a) SPH = 40 K and (b) SPH = 80 K.
Chapter 3. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: SINGLE-PHASE FLOW
50
3.3.1.3 Effect of impurities
Figure 3-10 displays the effect of Al bath superheat on the dimensionless radius after 3 min,
for the three types of Si specimens: MGSi–I, MGSi–II, and EGSi. The specimens with fewer
and no impurities, MGSi–II and EGSi respectively, dissolved faster in liquid Al than the
specimens with more impurities. Similar trends were observed by Yeremenko et al. [32] who
showed that 99.994% pure polycrystalline Mo dissolved more quickly in molten Al than
99.95% pure Mo, and that there was no difference in the dissolution rates of mono and
polycrystalline Mo of the same purity.
Figure 3-10: Effect of bath superheat on the dimensionless radius, r/r0, of MGSi–I, MGSi–II and EGSi specimens immersed for 3 min in liquid Al under natural convection.
Further support for the finding that the level of impurities in an alloy may impact on
dissolution rate is provided by another study involving liquid steel as solvent. In that case,
two different grades of Ferroniobium (35 wt.% Fe, 65 wt.% Nb) with very different levels of
impurities dissolved at very different rates [121]. The FeNb with lower levels of impurities
dissolved more quickly than the FeNb with more impurities. The author postulated that the
Chapter 3. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: SINGLE-PHASE FLOW
51
impurities form various high melting point phases, that retard the FeNb dissolution rate
[121].
To characterize the distribution of impurities in the MGSi–I and MGSi–II samples, the
microstructure of a number of Si specimens was examined by Scanning Electron Microscopy
(SEM) equipped with Energy Dispersive X-ray (EDX) spectroscopy. (A similar analysis of
the EGSi microstructure found no distinct phases.) Figure 3-11 and Figure 3-12 illustrate
secondary electron images of MGSi–I and MGSi–II specimens, respectively. In both cases,
the microstructures shown are prevalent over the entire specimen sections. Figure 3-11 (b)
and Figure 3-12 (b) depict a higher magnification of the outlined areas in Figure 3-11 (a) and
Figure 3-12 (a), respectively.
(a) (b)
Figure 3-11: (a) Secondary electron image of a MGSi–I specimen prior to immersion, (b) a higher magnification of the outlined area in (a).
+3
+2
+1
Chapter 3. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: SINGLE-PHASE FLOW
52
(a) (b)
Figure 3-12: (a) Secondary electron image of a MGSi–II specimen prior to immersion, (b) a higher magnification of the outlined area in (a).
The corresponding results of the EDX analysis are listed in Table 3-2, and show
randomly distributed inclusions contain impurities that the ICP analysis also detected
(Table 3-1). The dominant elements are consistent with those found by other studies on
metallurgical grade Si [122,123]. However, the compositions vary from those in the
literature, as they do between MGSi–I and MGSi–II. The inclusions are primarily silicide
phases with formation temperatures above 1255 K (982°C), as reported by Meteleva-Fischer
et al. [123]; this temperature is higher than the liquid Al temperature in the current
experiments. Thus, these phases do not melt in liquid Al, but rather assimilation must take
place by the dissolution of the constituents into the molten metal. Figure 3-13 shows a
backscattered electron image of Al attached to a dissolving MGSi–I specimen, that was
obtained by quenching in water from 973 K to 293 K. This figure illustrates that the
inclusions do not melt (in accordance with the above discussion) or dissolve more quickly
than Si, but rather the inclusions migrate into the molten Al as the surrounding Si dissolves.
+1
+2
+3
Chapter 3. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: SINGLE-PHASE FLOW
53
Figure 3-13: Backscattered electron image of a MGSi–I specimen with an attached Al after quenching in water from 973 K to 293 K.
As presented in Table 3-2, different compositions of similar elements were identified in
the MGSi–I and MGSi–II batches. The area fraction of the inclusions is higher in MGSi–I
(implying a higher level of impurities) than in MGSi–II, as seen in Figure 3-11 (a) and
Figure 3-12 (a). Image analysis of the microstructures at a low magnification of ×30 showed
that the area fraction of inclusions was 4.6 ± 0.5 % in MGSi–I and only 0.8 ± 0.2 % in
MGSi–II.
Table 3-2: Results of Energy Dispersive X-ray analysis at points shown in Figure 3-11 (b) and Figure 3-12 (b).
Batch Element Point
1 2 3 Composition wt.%
MGSi–I
O 0.7 – – Al – 2.8 1.1 Si 99.3 50.0 32.2 Ca – 0.5 – Ti – – 29.0
Mn – – 0.9 Fe – 46.7 36.8
MGSi–II
O 0.4 – – Al – 23.9 4.5 Si 99.6 30.6 27.8 Ca – 7.7 0.6 Ti – – 29.1
Mn – 1.3 2.1 Fe – 36.5 35.9
Attached Al Solid Si
Chapter 3. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: SINGLE-PHASE FLOW
54
The major elemental impurities, including Fe, Mn, and Ti, have lower diffusion
coefficients in liquid Al than Si [23,39]. The uneven presence of these impurities as
inclusions obstructs the dissolution of the Si matrix. In other words, Si cannot dissolve
evenly from a specimen surface, resulting in the lower dissolution rate of the MGSi–I
cylinders, as well as rougher surfaces after dissolution, illustrated in Figure 3-14. Although
the impurity levels of the two batches are not dramatically different, the presence of the
impurities likely impedes the development of a two-dimensional convective flow along the
specimen length, and alters the dissolution pattern. The mass boundary layer is likely more
local and three-dimensional at a solid Al/Si interface with more inclusions, because when
liquid Al encounters an inclusion at the specimen surface, it will not dissolve as quickly as
the solid Si. Instead, liquid Al must dissolve the Si surrounding the inclusion, until the
inclusion transfers into the liquid Al, as shown in Figure 3-13.
The effect of impurities is more pronounced at a lower superheat due to the larger
difference between the diffusion coefficients of Si and the impurities in liquid Al. For
example, according to Ershov et al. [39], DSi/Al/DFe/Al decreases from 6 to 4.5 as the Al bath
temperature increases from 973 K (700°C) to 1013 K (740°C). This is qualitatively consistent
with the decrease in the variation of r/r0 with increasing bath superheat, as depicted in
Figure 3-10.
Chapter 3. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: SINGLE-PHASE FLOW
55
Figure 3-14: Si specimens after natural convection dissolution for 3 min at SPH = 80 K; captions on top indicate the batch of Si.
The dissolved surfaces become smoother as the impurities decrease, as shown in
Figure 3-14. This clearly indicates that reducing the level of impurities provides a more
accessible surface for the diffusion of Si. The necking at the top of the samples is most
evident for the EGSi specimen. This confirms that the liquid Al becomes denser upon
dissolution of Si, ( )l Si Tρ C > 0∂ ∂ [116], and that the leading edge of the natural convection
mass transfer boundary layer is located at the top of the specimen. Figure 3-14 also shows
that the local dissolution rate is nearly constant along the MGSi–II and EGSi specimen
surfaces. Natural convection mass transfer from a vertical cylinder in a turbulent regime
produces uniform dissolved surfaces, as was also observed during the dissolution of steel
specimens in a Fe–C melt [88]. The value of m,lGr Sc for the results presented here is in the
turbulent range, which is taken into account by the exponent m = 1/3 in Equation (3-6), and
indicates that the dissolution rate is independent of the immersed length.
Chapter 3. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: SINGLE-PHASE FLOW
56
3.3.2 Forced convection experiments
The RLMT was rotated at 1, 2.5, 5, 7, 10, and 15 rpm, and the bath superheat set to 40 K and
80 K. When rotating the liquid Al in the RLMT, the flow pattern at steady state is a solid
body rotation with zero velocity at the center of the tank and maximum velocity at the wall.
The tangential velocity of liquid metal at the position of the Si cylinder can be calculated as:
rot rotSi Siu (r r ) rθ = = ×ω (3-10)
where rotSir = 13.5 cm is the distance from the center of the tank to the point of immersion of
the cylinder center line. Therefore, the aforementioned rotational speeds of the RLMT
correspond to tangential velocities of 1.4, 3.5, 7.0, 9.9, 14.0, and 21.0 cm s−1, respectively.
These velocities change by only ±6% across the initial cylinder radius, and so from here on
the tangential velocity at the point of immersion is simply referred to as the bulk velocity, Ub.
Under forced convection conditions, two sets of tests were carried out, one varying the
bulk velocity while immersing specimens for 3 min, and the other varying the immersion
time at a constant bulk velocity. Figure 3-15 depicts the dissolved fractions of MGSi–I and
MGSi–II after 3 min at various bulk velocities, with liquid Al at 40 K and 80 K SPH. These
figures indicate that the dissolution rate increases in tandem with bulk velocity and bath SPH.
Figure 3-16 (a) and (b) show r/r0 versus time for the dissolution of MGSi–I at a bulk velocity
of 7.0 cm s−1 compared to natural convection dissolution, at bath superheats of 40 K and
80 K, respectively. The incubation time is not significantly affected, but the rate of
dissolution clearly increases by the forced convection.
Chapter 3. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: SINGLE-PHASE FLOW
57
(a)
(b)
Figure 3-15: Effect of bulk velocity on the dissolved fraction after 3 min immersion, (a) SPH = 40 K and (b) SPH = 80 K.
Chapter 3. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: SINGLE-PHASE FLOW
58
(a)
(b)
Figure 3-16: Dissolution of MGSi–I under natural and forced convection conditions; (a) SPH = 40 K and (b) SPH = 80 K.
Chapter 3. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: SINGLE-PHASE FLOW
59
3.3.2.1 Local variation of dissolution rate
Although the average mass transfer rate from the entire surface of a specimen is of primary
interest, the dissolution patterns of dissolved specimens reflect the local transfer rates.
Figure 3-17 demonstrates that the dissolution rate is highest at the stagnation point, and
lowest at the back of cylinder. It is also evident that the higher the Reynolds number
bd AlRe (= U d ν ) , the higher both the local and average transfer rates. The variation of the
local mass transfer around the periphery of a cylinder has been experimentally studied by
several investigators [124,125]. Among those, Bošković-Vragolović et al. [125] used the
adsorption method to measure mass transfer rates from a cylinder for 38 < Red < 3614. They
showed Sh to be a maximum at the stagnation point and decrease significantly around the
cylinder, in agreement with the current findings.
Figure 3-17: Variation of local dissolution of MGSi–II cylindrical specimens in a cross flow of liquid Al at various bulk velocities (SPH = 40 K, immersion time = 3 min).
3.3.2.2 Comparison with a forced convection correlation
An estimate of the dissolution of a Si cylinder in a cross flow of liquid Al can be obtained
from Equation (3-4). However, in this case a forced convection mass transfer coefficient
Chapter 3. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: SINGLE-PHASE FLOW
60
must be estimated from a reliable dimensionless mass transfer correlation. Such a correlation
has been proposed by Churchill and Bernstein using heat and mass transfer data [126]:
( )
4/55/81/2 1/3d d
1/42/3
0.62 Re Pr ReNu 0.3 12820001 0.4 Pr
⎡ ⎤⎛ ⎞= + +⎢ ⎥⎜ ⎟⎝ ⎠⎡ ⎤ ⎢ ⎥⎣ ⎦+⎣ ⎦
(3-11)
This equation is applicable for Re Pr > 0.2. The authors [126] presumed that
Equation (3-11) also holds for mass transfer, with Nu replaced by Sh and Pr by Sc. The
applicability of the mass transfer version of Equation (3-11) for the dissolution of horizontal
cylinders of benzoic acid in water has been established [127].
In the current study, the applicability of Equation (3-11) was examined. The
characteristic length for dRe is the diameter of the Si addition (for an initial diameter d0,
0d510 < Re < 3800 ). As the dissolution of the Si specimen progressed, the diameter
decreased, and so Red decreased accordingly.
Figure 3-18 displays a comparison of experimental data with estimates of the
dimensionless radii based on Equation (3-11), that only accounts for forced convection mass
transfer. The correlation predicts the MGSi–II data reasonably well, while the MGSi–I
cylinders dissolved much more slowly. The agreement between experimental and estimated
radii is especially good at higher liquid Al velocities. At lower velocities, the correlation
predicts a slower rate of dissolution than was measured experimentally, presumably due to
the fact that Equation (3-11) does not take into account natural convection effects that are
relatively more dominant at lower liquid Al velocities (i.e. < 3 cm s−1).
Chapter 3. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: SINGLE-PHASE FLOW
61
Figure 3-18: Comparison of experimental and estimated dimensionless radii for pure forced convection mass transfer from vertical Si cylinders immersed for 3 min in a liquid Al cross flow.
3.3.2.3 Comparison with a combined natural and forced convection correlation
For the vertical cylinders immersed in the RLMT, the buoyant force is normal to the cross
flow. Thus, the combined natural and forced convection is three-dimensional, for which a
mass transfer correlation is unavailable. A correlation for this type of combined convection
was developed as part of this work.
The characteristic length associated with the forced convection is the cylinder diameter,
and with the natural convection is the cylinder length. An asymptotic solution for buoyancy
effects on a boundary layer along an infinite cylinder in a cross flow [128] suggests that the
strength of the two convection regimes on heat or mass transfer is determined by the
magnitude of ( ) ( )4 2m,l dd l Gr Re⎡ ⎤
⎣ ⎦ . This criterion is different than the Richardson number
( )2m,d d= Gr Re , which is the expansion parameter for combined free and forced convection
Chapter 3. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: SINGLE-PHASE FLOW
62
from a horizontal cylinder in cross flow [129], and even ( )2m,l lGr Re , which quantifies the
effect of buoyancy on forced convection along a vertical cylinder [130].
The values of Grm,l and 0dR e for the experiments of this study are given in Table 3-3.
Notice that the values of ( ) ( )4 2m,l dd l Gr Re⎡ ⎤
⎣ ⎦ are equal to or larger than unity at bulk
velocities of 1.4 and 3.5 cm s−1, which implies that natural convection must be taken into
account to evaluate the mass transfer from the dissolving specimens at these velocities.
A general combining law for transfer processes which vary uniformly between two
known limiting solutions is given by [131]:
p p 1/pZ (X Y )= + , (3-12)
where X and Y are the limiting solutions for asymptotically small and large values of an
independent variable (i.e. ( ) ( )4 2m,l dd l Gr Re⎡ ⎤
⎣ ⎦ ). For the current problem the mass transfer
coefficient when the combined convection variable is zero is given by FmX = k . When the
independent variable goes to infinity, NmY = k represents the asymptotic solution. As for the
exponent, [132] recommends p = 4 for spheres and for cylinders in a flow normal to the
buoyant force. Hence, incorporating the mass transfer coefficient for combined free and
forced convection in Equation (3-4) yields:
Table 3-3. Dimensionless parameters involved in the combined convection from a Si vertical cylinder in liquid Al cross flow.
L (cm)
d0 (cm)
SPH (K) Sc Grm,l
Ub
(cm s−1) 0dRe ( ) ( )0
4 20 m,l dd l Gr Re
8 1.87
40 42 5.5 x 108 1.4 510 6.5 3.5 1260 1.0 7.0 2530 0.2 9.9 3540 0.13
80 29 7.4 x 108 1.4 540 7.4 3.5 1360 1.2 7.0 2710 0.3 9.9 3800 0.15
Chapter 3. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: SINGLE-PHASE FLOW
63
( ) ( )
Sat1/44 4N F Sim m
Si
Cdr k kdt
⎡ ⎤= − +⎢ ⎥⎣ ⎦ ρ (3-13)
Figure 3-19 shows the estimated change in radius based on the combined convection
correlation, compared to the experimental data, which now predicts the dissolution of the
MGSi–II specimens within the experimental error even at lower bulk velocities.
Figure 3-19: Comparison of experimental and estimated dimensionless radii for combined convection mass transfer from vertical Si cylinders immersed for 3 min in a liquid Al cross flow.
Figure 3-20 shows the dimensionless radius as a function of time at a bulk velocity of
7.0 cm s−1 using the combined convection correlation. (Recall that the correlation estimates
include the experimentally-measured incubation times.) The correlation predicts the
acceleration of dissolution with time that was observed in the experiments, by incorporating
the change in cylinder diameter when calculating the forced convection mass transfer
coefficient. The acceleration of the shrinkage rate with decreasing radius was also observed
for the melting of steel scrap specimens in liquid steel [133]. As expected, the correlation
underpredicts the MGSi–I experimental data for various superheats, due to the presence of
Chapter 3. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: SINGLE-PHASE FLOW
64
impurities, but predicts the dissolution of the higher purity MGSi–II with very good
agreement.
Figure 3-20: Comparison of experimental and estimated dimensionless radii for combined convection mass transfer from vertical Si cylinders immersed in a liquid Al cross flow at a bulk velocity of 7 cm s−1.
3.4. Relationship Between Bath Temperature and Bulk Velocity
For the ranges of bath SPH and bulk velocity examined, the two effects can be correlated via
mass transfer coefficients. The experimental mass transfer coefficients were deduced for
natural and forced convection from the dissolution of specimens immersed for 3 min,
incorporating the incubation times:
0 Sim Sat
Si
(r r)kt C− ρ
= . (3-14)
Figure 3- 21 superimposes plots of forced convection mass transfer coefficients vs. bulk
velocity (on the top abscissa) at two SPH, and natural convection mass transfer coefficients
vs. SPH (on the bottom abscissa). To demonstrate the use of this graph; consider the
Chapter 3. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: SINGLE-PHASE FLOW
65
following example. The mass transfer coefficient associated with natural convection at SPH
= 100 K (TT'), mk = 8.2 × 10−5 m s−1, can also be obtained at lower SPHs, of 40 K and 80 K,
by introducing bulk velocities of 6.1 (LL') and 3.9 cm s−1 (HH'), respectively. This figure is
of practical importance as it quantifies the extent to which mechanical stirring and increasing
bath temperature each contribute to a higher dissolution rate.
Figure 3- 21: Mass transfer coefficient vs. SPH (shown on the bottom abscissa) superimposed on mass transfer coefficient vs. bulk velocity (shown on the top abscissa) (MGSi–II, immersion time = 3 min).
3.5 Summary
The dissolution of solid Si in liquid Al was studied as a function of bath temperature, fluid
flow conditions, and Si purity. The delay in the onset of mass transfer (incubation time) was
reduced by preheating the addition prior to immersion. With no external stirring, the
dissolution process occurred under turbulent natural convection. The experimental data for
high purity Si showed a good agreement with a dimensionless correlation for vertical
cylinders; the dissolution was slower for Si with a higher level of impurities. For the forced
Chapter 3. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: SINGLE-PHASE FLOW
66
convection conditions, the mass transfer rate was fitted with a dimensionless correlation for a
cylinder in cross flow, and at higher bulk velocities, predicted the experimental results very
well. At lower bulk velocities (< 3 cm s−1) natural convection effects cannot be neglected. A
newly-developed correlation for combined forced and natural convection yields very good
predictions of the dissolution of high purity Si at all bulk velocities. This correlation offers a
method to estimate the mass transfer coefficient in similar liquid metal systems,
8 8m,l5.5×10 < Gr < 7.4×10 ,
0d510 < Re < 3800 , and 29 < Sc < 42 .
67
CHAPTER 4
FLUID DYNAMICS OF GAS-AGITATED LIQUID
This chapter considers the fluid dynamics of a gas-agitated tank. First, bubbling behavior
(size, frequency and trajectory) in non-rotating and rotating liquids is considered, both in an
air–water system, and based on observations of the molten Al free surface and measurements
of N2 bubble frequency. Then, CFD simulation results are presented to estimate
bubble-induced velocities in the opaque liquid Al.
4.1 Bubble Formation and Trajectory in an Air–Water System
Air–water experiments were conducted to characterize the two-phase flow (e.g. bubble
plume vs. individual bubbles) that results from top lance injection. The experiments were
carried out to investigate the formation and trajectory of air bubbles in water, as a function of
gas flow rate and the bulk velocity of the water. To perform the experiments, a transparent
acrylic tank filled with water was placed on a rotating table, shown in Figure 4-1. Top lance
injection (Ti tube with 4.37 mm ID and 6.25 mm OD) was used to blow air bubbles 10 cm
below the free surface of an 11 cm deep bath.
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
68
(a)
(b)
(c)
Figure 4-1: (a) The transparent water tank on a rotating table, (b) top view shows the rotation direction and location of the gas injection lance, and AA´ represents a plane perpendicular to the xy-plane passing through the lance, (c) side view of the water tank and the lance at AA´. All dimensions are in cm.
A pressure transducer was used to measure the frequency of bubble formation at the lance
exit. When no gas was blown the output voltage from the transducer was constant. Bubble
frequency was measured by processing a uniform pattern of pressure fluctuations in the gas
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
69
delivery line caused by the formation and release of bubbles. Figure 4-2 shows typical output
voltage variations vs. time for gas flow rates up to 1.0 SLPM. For a given gas flow rate, QS,
and bubble frequency, f, and assuming spherical bubbles, the equivalent diameter of each
bubble, db, is:
1/3
S 0 lb
6Q P Tdf P 273
⎛ ⎞= ⎜ ⎟π⎝ ⎠. (4-1)
This equation is more appropriate at lower gas flow rates [44], (e.g. 0.3 and 0.6 SLPM in
Figure 4-2), where each repeating pattern is clearly associated with a single bubble.
Figure 4-2: Typical output voltage of the pressure transducer used to measure bubble formation frequency (air–water system, bulk velocity = 0).
The frequency of bubble formation was also determined using a high speed camera (210
frames per second). The videos illustrate not only bubble formation but also bubble-bubble
interactions. Three different regimes were detected. In the first regime (< 0.5 SLPM,
Figure 4-3), individual bubbles form and rise to the free surface without coalescing or
breaking up.
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
70
Figure 4-3: Formation and rise of single bubbles in the air–water system (gas flow rate = 0.3 SLPM, bulk velocity = 0).
In the second regime (0.5 SLPM < QS < 0.7 SLPM, Figure 4-4), bubbles form
individually at the lance exit but coalesce during rise. The distance between two consecutive
bubbles decreases as the bubbles rise because the trailing bubble rises in the wake of the
leading one. This is consistent with [134], which shows that for a chain of bubbles rising
through a quiescent liquid, the velocity of a bubble is equal to the velocity of an isolated
bubble in a liquid of infinite extent, plus the wake velocity of bubbles ahead of it.
Lance
Ruler
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
71
(i) first bubble forms (ii) second bubble forms
(iii) two bubbles coalesce
Figure 4-4: Formation of single bubbles and coalescence of two consecutive bubbles during rise in the air–water system (gas flow rate = 0.6 SLPM, bulk velocity = 0).
In the third regime (0.7 SLPM < QS < 1.0 SLPM, Figure 4-5), two consecutive bubbles
combine into a single bubble near the exit of the lance. This affects the shape of the bubbles:
the first bubble is flatter while the second one elongates at the lance exit. This phenomena is
referred to as formation of "doublets" and is caused by the reduced pressure in the wake of
the leading bubble [44,46].
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
72
(i) leading bubble forms (ii) trailing bubble forms
(iii) two bubbles coalesce (iv) forming a single bubble
Figure 4-5: Formation of doublets in the air–water system (gas flow rate = 1.0 SLPM, bulk velocity = 0).
From the videos, bubble size was estimated with a standard deviation less than ±7% db,
and is plotted vs. gas flow rate at the lance exit in Figure 4-6. In the first regime, all the
bubbles are the same size at a given gas flow rate. In the second regime, the trailing bubbles
are slightly smaller than leading ones (less than 7% by diameter, and the size of the leading
bubble is plotted in Figure 4-6). In the third regime, the trailing bubble is 15‐20% smaller (by
diameter) than the leading bubble, but then the two bubbles coalesce at the lance exit to form
a larger bubble. Goda et al. [55] also measured bubble formation frequency in water using
top lance injection and suggested a correlation for a range of gas flow rates
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
73
0.3 LPM < Q < 3 LPM with −20% to +30% scatter, but did not comment on bubble
coalescence. As seen in Figure 4-6, in the absence of coalescence at the lance exit, the
correlation adequately predicts the current experimental results, where at higher flow rates
the correlation underpredicts the bubble size by neglecting the coalescence. Nevertheless, the
coalescence of air bubbles in water at the exit of a top injection lance has also been reported
by Tseuge et al. [54] at gas flow rates higher than a "critical value".
Figure 4-6: Equivalent bubble diameter vs. gas flow rate in the air–water system.
The transition from single bubbles to doublets (regime III) with increasing gas flow rate
helps interpret the pressure transducer data. At lower gas flow rates, Figure 4-2 shows peaks
of equal height, while at higher flow rates, the peaks are uneven which implies bubble size
variation. One would underestimate the size of bubbles in the liquid by processing only the
output voltage data, as the coalescence of bubbles at the lance exit or during rise would not
be taken into account.
The effect of a liquid cross flow at 20 cm s−1 on bubble size at various gas flow rates is
presented in Figure 4-7. Due to the lateral force, bubble size decreases (in agreement with the
experimental study of [55]) and the transitions between the bubble formation regimes shift to
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
74
lower flow rates. As seen in Figure 4-7, when the bulk velocity is 20 cm s−1, the transition
from single bubbles to doublet formation at the lance exit occurs at a lower gas flow rate (0.5
SLPM) than in the absence of a bulk flow (0.7 SLPM).
Figure 4-7: Equivalent bubble diameter in the air–water system with and without cross flow.
Figure 4-8 further illustrates the effect of cross flow, by plotting bubble size vs. Relance,OD,
the liquid Reynolds numbers at the lance exit. At 0.3 SLPM single bubbles formed, and at 0.8
SLPM doublets formed, independent of bulk velocity. For these cases, the bubble size
monotonically decreases with increasing bulk velocity.
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
75
Figure 4-8: Effect of bulk liquid velocity on the equivalent bubble diameter in the air–water system (db normalized by the bubble diameter when the bulk velocity = 0).
Bubble trajectory during rise depends on bubble size and bulk flow velocity; bubbles
nevertheless do not wet the lance. Without a bulk velocity, smaller single bubbles of
ellipsoidal shape wobble along a zig-zag path, in agreement with [135]. The path of larger
doublets is rectilinear with more dramatic changes in shape. In cross flow, the rising bubbles
move laterally and undergo large deformation, particularly at higher gas flow rates. As seen
in Figure 4- 9, bulk velocities of 10, 20, and 30 cm s−1 result in lateral displacements of
3.6 dlance,OD, 7.1 dlance,OD, and 9.1 dlance,OD respectively, by the time the bubbles reach the free
surface.
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
76
Bulk velocity = 0 Bulk velocity = 10 m s−1
Bulk velocity = 20 m s−1 Bulk velocity = 30 m s−1
Figure 4- 9: Comparison of bubble trajectories at various bulk velocities (gas flow rate = 1 SLPM).
4.2 Bubble Formation and Trajectory in a N2–Al System
The opacity of liquid Al precludes the direct observation of the formation and trajectory of
bubbles similar to the air–water system, and so we only measured the pressure fluctuations in
the gas delivery line, and video recorded the free surface of the liquid Al.
It should be noted first that suggesting relationships to relate bubble behavior in the air–
water and N2–Al systems is complex and beyond the scope of the current study, for two
reasons. First, as mentioned in Section 2.2.1.1, bubbles wet the injection lance in liquid Al,
but they do not in water; this significantly changes the formation and detachment
9.1 dlance,OD
3.6 dlance,OD
7.1 dlance,OD
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
77
characteristics of bubbles in the two liquids. Second, for water, the heat transfer between the
gas and the liquid is negligible; therefore, knowing the gas temperature at the lance entrance,
the actual gas flow rate can be accurately calculated from Equation (4-1). For liquid Al, the
temperature of the gas at the lance exit can only be approximated by assuming heat transfer
between the immersed cylindrical lance and the liquid Al as a function of gas flow rate. This
introduces uncertainty into the gas flow rates calculated at the lance exit (i.e. +5-8 % in the
reported values for every ΔT = +50 K error in the estimated exit temperature). Nevertheless,
the air–water experimental results are employed to interpret the pressure transducer data and
the observations of the liquid Al free surface.
To carry out two-phase flow experiments in molten Al, the gas flow rate of industrial
grade N2 was controlled to 0.01 SLPM using a mass flow controller. Tygon tubing delivered
N2 to the top of a straight Ti lance (the same used in the air–water experiments) immersed
10 cm below the molten Al surface (L/H = 2/3), as shown in Figure 4-10. The Ti lance was
coated with boron nitride of approximately 0.5 mm thickness. The lance was positioned in
the melt at the same radial distance as the Si specimen; however, its angular position relative
to the Si cylinder (shown by θ in Figure 4-10) was varied.
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
78
(a)
(b)
Figure 4-10: Schematic of the RLMT shows the rotation direction and location of the immersed Si specimen and the gas injection lance. (a) Top view where AA´ represents a plane perpendicular to the xy-plane passing through the lance, (b) side view of the RLMT. All dimensions are in cm.
4.2.1 Bubble size
Figure 4-11 shows a typical pressure transducer measurement. Similar to the air–water
system (Figure 4-2), at lower flow rates (e.g. 0.50 SLPM), the pattern is more uniform and
each peak can be associated with the formation of a uniform size bubble at the lance exit.
Although the pressure transducer data implies the formation of single bubbles at the lance
exit, bubbles may coalesce during the rise period, which cannot be confirmed in the liquid
Al. This means that at lower flow rates, the N2 bubbles are in Regime I or II as identified in
the air–water experiments.
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
79
At higher flow rates (e.g. 1.00 SLPM), consecutive bubbles appear to coalesce at the
lance exit, as implied by the uneven peaks and the irregular pattern. As such, the size of
bubbles at the lance exit may be larger than what is predicted from the pressure transducer
data, as was also the case in the air–water system in Regime III.
Figure 4-11: Typical output voltage of the pressure transducer used to measure the frequency of bubble formation (N2–Al at SPH = 40 K, bulk velocity = 0).
The pressure transducer data was used to obtain the bubble frequency and then the bubble
size was estimated (single bubbles) using Equation (4-1). Figure 4-12 depicts the equivalent
bubble diameter after detachment vs. volume flow rate at the lance exit, and a comparison
with a correlation of Fu et al. [57] based on measurements of bubble frequency of
Chlorine/N2 injection into molten Al. Based on the earlier discussion, the size of rising
bubbles can be underestimated by up to 26% (by diameter) if one neglects the coalescence of
two same size bubbles during rise. This may occur at gas flow rates higher than 0.50 SLPM
in Figure 4-12, where the pattern of pressure data is non-uniform.
Figure 4-12 also illustrates that the bubble size decreases slightly (i.e. 3-5%) by rotating
the tank (Ub = 7 cm s−1). Comparing this reduction to the results of the air–water system
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
80
(Figure 4-8), the effect of the lateral drag force due to the bulk flow in the molten Al is less
pronounced. This may be because the bubbles that form in liquid Al encompass the lance
(Figure 2-3 (b) and as will be shown in Section 4.2.2), while the bubbles in water form at the
lance tip (Figure 2-3 (a)), and therefore, can be more easily pinched off.
To decide whether N2 bubbles break into smaller ones before reaching the Al free
surface, we estimate the bubble size at the free surface with no coalescence. Assuming an
isothermal expansion, the diameter of an ascending bubble remains approximately
unchanged (+3%). A very conservative upper limit for the diameter of a stable N2 bubble in
liquid Al is [135],
( )b max Al Al g(d ) 4 g -= σ ρ ρ (4-2)
Using this equation, (db)max = 2.2 cm, much larger than any of the bubble size in
Figure 4-12. Therefore, the bubbles likely do not disintegrate. This is confirmed by images of
the liquid Al free surface (Figure 4-13), which vaguely show a bubble diameter on the order
of 1 cm.
Figure 4-12: Bubble size in liquid Al at bulk velocities of 0 and 7.0 cm s−1, and a comparison with the available correlation for a (Chlorine/N2)–Al system [57].
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
81
4.2.2 Bubble trajectory
Information on the trajectory of the bubbles was also obtained by visual examination of the
melt free surface. Figure 4-13 shows the lance and Si specimen 30º apart (this corresponds to
an arc distance of 7.1 cm). At zero bulk velocity (non-rotating tank), the bubbles rise along
the lance and break the free surface adjacent to the lance. In the rotating tank, Figure 4-14
also shows the bubbles at the free surface next to the lance, very different than what was
observed in the air–water system at a similar Relance,OD (Figure 4- 9). This is because of the
larger buoyancy force exerted on the rising bubbles in liquid Al (ρAl/ρwater ≈ 2.4), as well as
the higher wettability of the lance. A comparison of the buoyancy force, FB, for a bubble
diameter of 1-2 cm, and the lateral drag force LDF exerted on the bubbles by the bulk liquid
(at velocities of 1.4-7 cm s−1), reveals that the former is dominant ( LD BF F < 0.02 ). More
important, N2 bubbles wet the lance during rise, which strongly opposes their detachment and
transport downstream.
Figure 4-13: Bubble arrival at liquid Al surface (bulk velocity = 0, gas flow rate = 0.50 SLPM, θ = 30º, the coated lance diameter is approximately 0.725 cm).
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
82
Figure 4-14: Bubble arrival at the liquid Al surface (bulk velocity = 3.5 cm s−1, gas flow rate = 0.50 SLPM, θ = 30º, the coated lance diameter is approximately 0.725 cm).
Although bubbles do not appear to contact the dissolving Si specimen in the experiments,
the displacement of liquid by the rising bubbles induces velocities within the liquid Al. These
velocities will be estimated in the next section.
4.3. Numerical Modeling of Gas-Agitated Tank
Not only can one not directly observe bubbles in the melt, but it is hardly possible to measure
the bubble-induced liquid Al velocities. Hence, the commercial software FLOW-3D was
used to predict the bubble distribution beneath the free surface, and estimate the velocity
field. In the simulations, a pragmatic approach was adopted to estimate mean velocities and
turbulence intensities, rather than attempt to fully resolve the flow field. Considering the
large scale of the problem, this would require for more computational resources than were
available.
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
83
This section presents information on model set-up (domain and mesh), the initial and
boundary conditions, and the stopping criterion. The assumptions and simplifications are
clearly stated. Next, the model is validated by comparing with results of the air–water
experiment in the non-rotating tank. Then, the results of simulations of the N2–Al system, of
bubble frequency, bubble distribution and the flow field characteristics, are presented.
Finally, for both the non-rotating and rotating tanks, time and spatial averages of bubble-
induced velocities are computed at various gas flow rates and angular positions of the lance.
4.3.1 Simulation set-up and parameters
The simulations were time-varying, and started by blowing gas into a liquid at t = 0. The
flow was assumed to be isothermal, and as mass transfer was not considered, energy and
species balance equations were not solved. The liquid Al properties (density, viscosity, and
surface tension) were evaluated at 700°C, and the specified gas flow rate was adjusted for the
temperature and pressure at the lance exit.
The Navier‐Stokes equations were solved along with the standard two equation k-ε model
to account for the turbulent properties of the flow. Bubble/liquid interactions and the liquid
free surface motion were simulated with the volume of fluid (VoF) method [76]. The
interaction of bubble/liquid/lance was taken into account by a surface tension model and by
specifying an appropriate contact angle. Equations were solved iteratively with a minimum
time step of 10−6 s. The simulations were set up with the FLOW-3D Graphical User Interface
(GUI), version 10.0.3.1, and the flow variables were solved with solver version 10.0.3.5.
Domain: FLOW‐3D uses the VoF method to track interfaces, and the simulations were run
using the "one‐fluid" model that only solves for flow within the liquid phase, and not within
the air above the tank. The computational domain included the top injection lance and the
tank, with enough room above the liquid to allow for free surface fluctuations (Figure 4-15).
The Si cylinder was not modeled. Rather, the results of the simulations were used to assess
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
84
mean and fluctuating velocities at the various positions up and downstream of the lance
where Si cylinders were positioned in the experiments. A similar approach has been adopted
by other studies (e.g. [97,98,101,107,109,110]) in which flow was predicted in the absence of
an immersed melting or dissolving specimen.
Figure 4-15. Cylindrical domain for the FLOW-3D simulations of the N2–Al system. All dimensions are in cm.
Mesh: The computational domain was discretized with a 3D cylindrical mesh (the radial,
angular and vertical coordinates denoted by r, θ, and z, respectively), as shown in
Figure 4-15. The mesh was refined selectively within regions of interest (Figure 4-16 (a)).
Figure 4-16 (b) illustrates the extra mesh refinement around the lance to capture bubble
formation.
FLOW-3D uses an explicit approach to solve for fluid advection, and so the time step is
directly proportional to the control volume size; very small cells adversely affect the runtime.
To eliminate the very small cells at the center of the cylindrical domain (r = 0), a core of
4 cm radius (i.e. 5.2% of the liquid volume) was removed, and the mesh size in the radial
direction was gradually increased near the core to avoid very small cells.
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
85
(a)
(b)
Figure 4-16: (a) Top view of the mesh planes, and (b) the mesh around the lance.
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
86
Initial conditions: For a non-rotating tank, the entire velocity field began at rest. For the
rotating tank, the fluid velocity was initialized to the corresponding solid body rotation
velocity profile in the positive θ direction (counterclockwise).
Boundary conditions: The boundary conditions were specified based on the experimental
conditions. The gas volume flow rate was imposed as an inflow condition at the lance exit.
No-slip boundary conditions were imposed at the bottom (i.e., rθ-plane: z = 0), inside
(θz-plane: r = Ri) and outside tank walls (θz-plane: r = R0). When simulations were run of a
rotating-tank, appropriate velocities were specified at those walls. The pressure at the melt
surface (rθ-plane: z = H, H changes with time because of the free surface fluctuations) was
specified as the atmospheric pressure, P0.
Although the flow was modeled in the entire tank (0° ≤ θ ≤ 360°), FLOW-3D
nevertheless requires that one define a pair of boundaries in the angular direction (i.e. a pair
of rz‐planes). For the non-rotating tank, a periodic boundary condition was specified for the
pair of rz-plane, so that the liquid flow that leaves a periodic plane re-enters the other one.
These boundaries were located 180° from the lance to minimize the potential impact on
bubble generation and the flow in the vicinity of the lance.
For the rotating tank, the pair of rz-plane boundaries was set differently. A solid body
rotation velocity profile was imposed as the inflow condition at rz-plane 90° upstream of the
lance, and an outflow boundary condition was specified at the rz-plane 270° downstream.
This is a simplification of the experiments: it neglects the changes in free stream bulk
velocity, by assuming the solid body velocity profile is recovered far downstream of the
lance.
Stopping criterion: For a full domain simulation, the combination of the large number of
cells (~3.2 x 106) and small time steps (~10−6-10−5 s) resulted in very lengthy simulations.
The simulation runtimes were between 1 to 2 months on an Intel i7-4770 CPU @ 3.40 GHz
with 8 cores and 32 GB of RAM. The simulations were run until a quasi‐steady state was
reached, evaluated by monitoring a sliding 1 s average of the overall liquid kinetic energy per
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
87
unit mass of liquid, lK . The stopping criterion was when ldK dt was reduced by a factor of
5 from its initial value, as shown in Figure 4-17. This occurred at about 9 s of simulation time
for the non-rotating tank. For consistency, the simulations in the rotating tank were run for
the same time period.
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
88
(a)
(b)
Figure 4-17. (a) Overall liquid kinetic energy of liquid per unit mass of liquid, Kl, and (b) ldK dt vs. time (bulk velocity = 0, gas flow rate = 0.50 SLPM, solid line is the sliding 1 s average).
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
89
4.3.2 Temporal- and spatial-averaged velocities
The simulation results were used to obtain time- and spatial-averaged velocities. The results
were averaged over time to capture the effect of many bubbles without overlooking the
transient response of the system, and then they were averaged over a volume representing the
Si cylinder.
As mentioned in Section 2.2.2, instantaneous velocities at any location vary with time.
Figure 4-18 illustrates the velocity magnitude at the midpoint of an imaginary cylinder
immersed at θ = 30° relative to the lance. As the simulations are transient and start from the
onset of bubble injection, even the mean velocities are time-dependent. Time-averaged mean
velocities and the rms values of fluctuating components in each direction were calculated as
follows:
Mi
r ri 1
1u (r, , z, t) uM =
θ = ∑ , M
i
i 1
1u (r, , z, t) uMθ θ
=
θ = ∑ , M
iz z
i 1
1u (r, , z, t) uM =
θ = ∑ ; (4-3)
and,
( )2iMr r2
ri 1
u uu
M=
−′ = ∑ ,
( )2iM2
i 1
u uu
Mθ θ
θ=
−′ = ∑ ,
( )2iMz z2
zi 1
u uu
M=
−′ = ∑ . (4-4)
where M is the number of liquid velocity data. Substituting Equations (4-3) into Equation
(2-13) and Equations (4-4) into Equation (2-10) yields the mean velocity, V, and the rms
velocity fluctuations, u .
Time-averaging must be performed over a time period that is sufficiently longer than the
duration of any velocity fluctuation [12]. Therefore, a sliding temporal average was
calculated for a period of 1 s, during which about 30 bubbles were released, based on data at
every 0.005 s (i.e. M = 200), that is much larger than the typical time step (~ 10−5 s).
Sampling the data every 0.05 s (non-rotating Al tank at θ = 30° with a gas flow rate of 0.50
SLPM) yielded very similar time-averaged results.
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
90
In the experimental study, the focus was on the overall dissolution rate of Si cylinders,
and so spatial averages of mean and rms velocity fluctuations were calculated within the
volume that encompasses an imaginary Si specimen. Assuming an equal weight for each cell
in the volume:
v
vN
ϕ< ϕ >=
∑, (4-5)
where φ is a general quantity and vN is the number of cells (i.e. 15000-45000) in the
imaginary cylinder.
Figure 4-18: Instantaneous liquid velocity magnitude (bulk velocity = 0, gas flow rate = 0.50 SLPM, r = 13.5 cm, θ = 30°, z = 11 cm: 4 cm below the free surface of liquid Al coincides with the center of an imaginary Si cylinder).
4.3.3 Experimental validation with the air–water system
FLOW-3D was used to simulate the air–water visualization experiments in the non-rotating
tank. Water wets Ti, and so a contact angle of 58.5° was specified, as suggested by [136].
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
91
The simulation results predict a periodic bubble formation and a noticeable shift from
single bubbles (0.3 SLPM, Regime I) to doublets (1.00 SLPM, Regime I) by increasing the
gas flow rate, in agreement with the experiments (Section 4.1). Although the formation of
doublets was dominant at the highest flow rate, single bubble formation was also observed.
In addition, a simulation at a gas flow rate of 0.6 SLPM (Regime II) was run that predicted
bubble coalescence closer to the lance exit than is shown in Figure 4-4.
Figure 4-19 and Figure 4-20 compare the predicted evolution of bubbles at the lance exit
to images from the experiments, at gas flow rates of 0.30 and 1.00 SLPM, respectively. The
results were compared at various time intervals from the appearance of a single
bubble/doublet at the lance exit to its detachment. At each instant of time in Figure 4-19 and
Figure 4-20, the top and bottom images correspond to the experiment and simulation,
respectively. The lance is omitted from the simulation images to more clearly show the
bubbles; what appears as a section of the lance attached to the bubbles is in fact the inside of
the lance (4.37 mm ID).
Figure 4-19 shows the single bubble formation belonging to Regime I. The simulation
results, in accordance with the experiments, predict a bubble evolution consisting of three
distinct stages. Initially, the bubble emerges at the lance exit as the momentum of the gas
overcomes the liquid pressure, and the bubble grows in line with the lance centerline
(0-0.035 s). Then the bubble, while continuing to grow, moves preferentially towards one
side of the lance (0.035-0.070 s). Finally, the bubble detaches as the buoyancy force
increases with bubble size (0.07-0.105 s). The same evolution stages, with very similar
images, are reported in the experimental study of [137]. Note that liquid flow instabilities due
to the periodic formation and release of bubbles cause each bubble to move towards different
side of the lance.
Figure 4-20 depicts the doublet formation belonging to Regime III. The formation of the
leading bubble (0-0.08 s) is similar to the single bubble formation described above. However,
the bubble is larger because of the higher gas flow rate, and so the bubble forms more
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
92
quickly. At the lower flow rate, the leading bubble has a negligible effect on the trailing
bubble. In contrast, the effect of the leading bubble at the higher flow rate is clearly observed
in the experiments, and is well-predicted by the simulation. As shown in Figure 4-20, the
large leading bubble reduces the trailing bubble growth time, leading to a smaller bubble and
deforming it (0.08-0.12 s). This is because the leading bubble wake influences the formation
of the trailing bubble as the distance between the two bubbles is short [138]. FLOW-3D
satisfactorily predicts both the quicker detachment and the elongation of the trailing bubble.
In addition to a "picture norm" comparison, the FLOW-3D predictions were evaluated
quantitatively, by comparing the average formation time of the single bubbles (at 0.30
SLPM), and of the leading and trailing bubbles of the doublets (1.00 SLPM) over a time
period of 4 s, to the corresponding experimental results. The time for the bubble to reach the
free surface was also calculated and compared. This time was defined as follows: for a single
bubble, from detaching from the lance exit to exiting the liquid; for the doublets, from the
leading bubble detaching from the lance exit to the doublet exiting the liquid. Figure 4-21
shows the comparison at gas flow rates of 0.30 SLPM and 1.00 SLPM. The FLOW-3D
predictions and experimental results agree within the standard deviations. Overall, the
prediction of the numerical model is quite satisfactory, and so the validated model was used
to investigate the characteristics of the N2–Al system.
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
93
t = 0 s t = 0.035 s
t = 0.070 s t = 0.105 s
Figure 4-19: Qualitative comparison of air-water bubble formation (bulk velocity = 0, gas flow rate = 0.3 SLPM).
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
94
t = 0 t = 0.04 s
t = 0.08 s t = 0.10 s
t = 0.12 s
Figure 4-20: Qualitative comparison of air-water bubble formation (bulk velocity = 0, gas flow rate = 1.0 SLPM).
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
95
(a)
(b)
Figure 4-21: Quantitative comparison of the formation and rise time of many bubbles over a time period of 4 s as predicted by a FLOW‐3D simulation, and measured in the air–water experiments, at gas flow rates of (a) 0.3 SLPM and (b) 1.0 SLPM (bulk velocity = 0).
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
96
4.3.4 Results and Discussion
Table 4-1 lists the full tank FLOW-3D simulations of the N2–Al system. The results of these
cases are presented in the following section.
Table 4-1: List of FLOW‐3D simulations for the N2–Al system.
Bulk velocity of liquid at the lance exit (cm s−1)
Gas flow rate (SLPM)
Non-rotating tank 0 0.50 0.75 1.00
Rotating tank
1.4 0.50
3.5 0
0.50 1.00
7.0 0.50
4.3.4.1 Bubble distribution within the liquid Al
In the N2–Al experiments, the Ti lance was coated with boron nitride, which molten Al does
not wet. As a result, the N2 bubbles appear to wet the lance and rise along it, as confirmed
earlier in Figure 4-13.
FLOW-3D simulations in a non-rotating tank were run at both 0.50 and 1.00 SLPM, with
the Al/lance contact angle set to 160° [139,140]. Unlike both the air–water and N2–Al
experiments, the simulations predicted a less clear transition from single bubble (Regimes I
and II) to doublet (Regime III) formation, as both single bubbles and doublets formed at both
0.50 and 1.00 SLPM. Nevertheless, the average bubble size was reasonably well predicted by
FLOW-3D, as indicated in Table 4-2. At both flow rates the predicted bubble size was a few
percent (~8%) higher than the measured values, and the variations in the predicted bubble
size were higher than the experimental values obtained via pressure transducer data. The
simulations for a gas flow rate of 0.50 SLPM were also run with the contact angle set to 150°
and 170°. As expected, average bubble size increased and decreased, respectively. At 170°
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
97
and gas flow rate = 0.50 SLPM, the average size was 1.22 ± 0.18 cm, just slightly larger than
the measured size. Given the uncertainties: the contact angle, the temperature-adjusted gas
flow rate at the lance exit, and the transport properties (density, viscosity, and surface
tension) at high temperatures, the FLOW-3D results are surprisingly good.
Table 4-2: Comparison of experimental measurements and FLOW-3D predictions for the equivalent bubble diameter at the lance exit in the N2–Al system (bulk velocity = 0).
Gas flow rate (SLPM)
Bubble equivalent diameter (cm) Experimental
(Estimated from pressure transducer data)
FLOW-3D (Counting formed bubbles
at the lance exit) 0.50 1.16 ± 0.01 1.26 ± 0.15 1.00 1.47 ± 0.02 1.59 ± 0.20
Finally, Figure 4-22 shows sample results for gas flow rates of 0.50 and 1.00 SLPM, that
illustrate the bubbles rising along the lance. For a gas flow rate of 0.50 SLPM, bubbles rise
with a velocity of 0.63 ± 0.05 m s−1, higher than that of unhindered bubbles of a similar size
[57,62]. The attachment of the ascending bubbles to a poorly wetted lance has been
previously reported [60,61] in water model studies, and Watanabe and Iguchi [61] showed
that such bubbles rise more quickly than those away from a lance.
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
98
(a) (b)
Figure 4-22: Predicted instantaneous bubble distributions at gas flow rates of (a) 0.50 SLPM and (b) 1.00 SLPM (3D view of the lance vicinity, bulk velocity = 0).
4.3.4.2 Non-rotating tank
The simulation results of the RLMT provide insights into the behavior of the bubble-induced
bulk flow, and quantify the velocity field. This section presents the liquid flow characteristics
and estimated velocities in the non-rotating tank.
Fluid flow characteristics: Prior to gas injection the liquid is at rest; when the bubbles are
introduced, the gas/liquid interaction induces a flow in the melt. The general characteristics
of the gas-agitated flow are inferred from velocity profiles. Figure 4-23 shows contour plots
of instantaneous velocities at various planes in the cylindrical domain. The characteristics of
the flow field are similar on both sides of the lance. However, the flow field is not entirely
symmetric because of the non-linear instabilities due to the bubbly flow. The simulation
results indicate that the assumption of axisymmetric flow adopted in most previous studies of
gas injection in ladles (e.g. [43,52,97,98,107,141-143]) may not be appropriate, especially
where the lance is located off-center in a metallurgical reactor.
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
99
Figure 4-23 (a) is a contour plot of the instantaneous velocity magnitude in the θz-plane
that passes through the center of the lance (r = 13.5 cm). The velocity values are highest
adjacent to the lance, decreasing with a sharp gradient in the θ direction. The release of
bubbles at the free surface of the liquid leads to higher velocities in this region, while in the
zone beneath the lance, the flow is weak and nearly stagnant, as also reported by [58,141].
This is presumably due to the negligible downward movement of the bubbles.
Figure 4-23 (b) plots the tangential velocity, uθ, in a series of rz-planes. At smaller θ
values (e.g. θ = 30°), uθ is towards the lance as the liquid is drawn into the wake of the rising
bubbles. A strong outflow passes across the top surface and outside wall, which establishes a
recirculating bulk flow on each side of the lance, similar to what was observed
experimentally for top lance injection in a water ladle [144], albeit for a bubble plume instead
of individual bubbles. The entrainment of liquid becomes less significant as θ increases
(θ = 70° and 120°), evidenced by less flow towards the lance (negative tangential velocity),
and is negligible far from the lance, on the opposite side of the tank (θ = 180°).
As the bubbles rise, they displace the liquid above, inducing a vertical velocity.
Figure 4-23 (c) depicts the vertical velocity, uz, at two rθ-planes, at the exit of the lance (z =
5 cm), and 1 cm below the free surface (z = 14 cm). On the lower plane, there is an upward
bulk flow in the neighborhood of the lance due to momentum transfer from the rising bubbles
[141]; the flow then moves downwards near the wall. On the upper plane, no distinct bulk
flow can be detected due to the highly agitated free surface. An up-and-down motion at the
liquid Al free surface was also observed experimentally (i.e. the uneven surface in
Figure 4-13), that is an indication of higher turbulence intensities. The magnitude of velocity
fluctuations at the top surface may have been overpredicted, as the formation of dross
observed experimentally was not considered in the simulations. This oxide layer with a high
melting point likely damps the agitation of the free surface. Note that many previous studies
of gas-agitated ladles (e.g. [43,52,97,98,107,141-143]) assumed a flat free surface;
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
100
nevertheless, they observed higher values of turbulent kinetic energy (turbulence intensity) at
the free surface than in the bulk of the liquid [52].
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
101
(a)
(b)
(c)
Figure 4-23: (a) Predicted instantaneous bubble-induced velocity magnitude in the θz-plane passing through the lance center, (b) the tangential velocity at various rz-planes, and (c) the vertical velocity at various rθ-planes (bulk velocity = 0, gas flow rate = 0.50 SLPM, t = 9.00 s).
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
102
Velocity estimates: As mentioned in Section 2.2.2, both mean velocities and rms velocity
fluctuations affect localized mass transfer phenomena, and so are of interest. The simulation
results were used to estimate these velocities as functions of relative position to the lance,
and gas flow rate. In Chapter 5 these velocities will be used to develop correlations for mass
transfer in gas-agitated systems. 416 871 0222
In the dissolution experiments in the non-rotating tank, the Si cylinder was immersed at
θ = 30° with respect to the lance. Figure 4-24 shows a spatial average of the absolute mean
velocity and the rms velocity fluctuations in the r, θ, and z directions at this position, for a
gas flow rate of 0.50 SLPM. Figure 4-24 (a) illustrates the multidimensionality of the mean
flow: although the velocity components are of the same order of magnitude, the mean
tangential velocity is the largest, followed by the vertical and radial components. This result
shows that the approach [97,98,110] of assuming an axisymmetric flow relative to the lance,
and neglecting the velocities perpendicular to the plane that passes through the solid addition
and the lance, leads to an underestimation of the mean velocity. Figure 4-24 (b) shows that
the velocity fluctuations are appreciable in all directions, although the tangential and vertical
components are greater than the radial one, which implies a deviation from isotropic
turbulence. This is in accordance with the results of [145].
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
103
(a)
(b)
Figure 4-24. Spatially averaged (a) absolute mean velocity, and (b) rms velocity fluctuations in the r, θ, and z directions (bulk velocity = 0, gas flow rate = 0.50 SLPM, θ = 30°).
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
104
Figure 4-25 compares the mean velocity and rms velocity fluctuations as a function of θ,
and illustrates a stronger mean flow and turbulence closer to the gas/liquid region. As θ
increases from 5° to 10°, both < V > and < u> change abruptly (consistent with the sharp
velocity gradient near the lance shown in Figure 4-23), decreasing by factors of 2.6 and 4.3,
respectively. Further increasing θ from 10° to 30°, the velocities gradually decrease. At
θ = 5°, a portion of the volume over which the spatial averaging was performed encompasses
the gas and liquid mixture. The increase in velocity is in general agreement with the
prediction of Mazumdar et al. [97], who reported that the mean and fluctuating velocities
outside an axisymmetric plume (of a conical shape with specified gas volume fraction as a
function of bath height) are significantly lower than inside it.
Figure 4-26 shows the mean velocities and rms velocity fluctuations at various gas flow
rates in the non-rotaing tank. By increasing the gas flow rate from 0.50 to 1.00 SLPM, when
the startup period has passed, < V > is essentially the same, but < u> increases by 70%,
indicating that a higher gas flow rate yields a greater liquid agitation rather than a stronger
mean flow. This observation is in agreement with the experimental findings of [65].
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
105
(a)
(b)
Figure 4-25: Comparison of spatially-averaged (a) mean velocity and (b) rms velocity fluctuations at various angular positions relative to the lance (bulk velocity = 0, gas flow rate = 0.50 SLPM).
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
106
(a)
(b)
Figure 4-26: Comparison of spatially-averaged (a) mean velocity and (b) rms velocity fluctuations at various gas flow rates (bulk velocity = 0, θ = 30°).
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
107
4.3.4.3 Rotating tank
The tank rotation generates a liquid bulk velocity. As discussed in the N2–Al experiments,
this causes a small decrease in the bubble size and a shift of the rising bubbles to the
downstream side of the lance. FLOW-3D predicts these behaviors satisfactorily. By
increasing the bulk velocity from 0 to 3.5 cm s−1, at gas flow rates of 0.50 SLPM and 1.00
SLPM, the average bubble size decreases slightly from 1.26 cm to 1.24 cm, and from 1.49
cm to 1.48 cm, respectively. These reductions are negligible compared to the standard
deviations associated with the equivalent bubble diameter; a similar conclusion was drawn
from the pressure transducer measurements. Figure 4-27 illustrates that the bubbles rise
adjacent to the lance downstream of the bulk flow, in accordance with Figure 4-14.
Fluid flow characteristics: With the predicted bubble behavior in qualitative agreement with
experiments, the bubble-induced field can be examined. Figure 4-27 (a) shows a contour plot
of the tangential velocity. Downstream of the lance, the wakes behind rising bubbles cause
tangential velocities opposite to the bulk flow direction. Of course the magnitude of these
bubble-induced velocities decreases with distance from the lance. Upstream of the lance, the
approaching flow is significantly disturbed, as the bubbles generate both inward and outward
flows along the lance. Figure 4-27 (a) also shows that in the rotating tank the approaching
flow below the lance is accelerated in the presence of the two-phase region, in contrast to the
flow in the non-rotating tank, in which the liquid below the injection lance is nearly
unaffected (Figure 4-23 (a)).
Bubbles also induce velocities in the vertical direction. Figure 4-27 (b) is a contour plot
of the vertical velocity on the θz-plane passing through the lance. The rise of bubbles
downstream of the lance causes upward velocities skewed towards the direction of the bulk
flow. This yields a non-symmetric flow around the lance, where vertical velocities are
induced mostly in the downstream flow.
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
108
(a)
(b)
Figure 4-27: Instantaneous (a) tangential velocity, uθ, and (b) vertical velocity, uz, on the θz-plane passing through the center of the lance (bulk velocity = 3.5 cm s−1, gas flow rate = 0.50 SLPM, t = 9.00 s).
The bubble-induced velocities also vary vertically within the bath. Figure 4-28 is a
contour plot of tangential velocity at two heights. Both upstream and downstream flow
disturbances are magnified near to the free surface. At the lance exit, the tangential velocity
profile resembles the initial solid body rotation, while at 1 cm below the free surface the flow
pattern is more agitated.
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
109
Figure 4-28: Instantaneous tangential velocity on two rθ‐planes (bulk velocity = 3.5 cm s−1, gas flow rate = 0.50 SLPM, t = 9.00 s).
The variation of velocity in the flow direction is of interest because it illustrates how the
effect of gas injection is convected throughout the domain. Figure 4-29 is a contour plot of
velocity magnitude on various rz-planes (where the Si cylinders were immersed in the
experiments). Large velocity gradients are observed adjacent to the lance (Figure 4-27); the
velocity profiles are more similar on the planes further from the lance. Nevertheless, higher
velocities are predicted on the downstream planes near the free surface, due to the release of
bubbles.
Figure 4-29: Instantaneous velocity magnitude, V, at various rz‐planes (bulk velocity = 3.5 cm s−1, gas flow rate = 0.50 SLPM, t = 9.00 s).
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
110
Velocity estimates: Before quantifying mean and fluctuating velocity components due to gas
agitation in the rotating tank, it is instructive to consider the influence of the lance on the
flow field without injecting gas, since the simplifying assumption of solid body rotation is
not valid in the presence of the lance. Such a simulation offers an estimate of the velocity
fluctuations due to pure shear-induced turbulence. Figure 4-30 shows absolute mean
velocities at θ = 30° averaged over the imaginary cylinder volume. The tangential velocity
decreases from the initial 3.5 cm s−1 to a steady 2.5 cm s−1 due to the drag force exerted by
the lance. There are also small radial and vertical velocities (an order of magnitude less than
the tangential velocity) due to flow disturbances created by the lance. Figure 4-31 shows the
rms velocity fluctuations and turbulence intensity corresponding to the mean velocity. The
spatially-averaged turbulence intensity, <Tu>, is about 11% at steady state. This value is in
the expected range for single-phase flows [103,145].
Figure 4-30: Spatially-averaged absolute mean velocity in the r, θ, and z directions (bulk velocity = 3.5 cm s-1, gas flow rate = 0 SLPM, θ = 30°).
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
111
Figure 4-31: Spatially-averaged rms velocity fluctuations and turbulence intensity (bulk velocity = 3.5 cm s−1, gas flow rate = 0 SLPM, θ = 30°).
Figure 4-32 (a) shows the spatially-averaged mean velocity components at θ = 30° and a
gas flow rate of 0.50 SLPM. The results demonstrate radial and vertical velocities of the
same order of magnitude, albeit much smaller than the tangential velocity in the bulk flow
direction. This is a crucial distinction from previous studies on heat and mass transfer from
vertical cylinders in cross flow with grid-generated turbulence [82-84,146,147]. These
experiments were conducted in wind tunnels where the mean velocity was one-dimensional
and constant along the test section; correlations derived from such experiments would be
difficult to apply to a system with gas agitation as the source of velocity fluctuations
(pseudo-turbulence).
Figure 4-32 (b) indicates that the rms velocity fluctuations are significant in all
directions; fluctuations in the tangential direction are slightly higher than in the vertical and
radial directions because of the interaction of the bulk and bubble-induced flows. (This also
indicates a deviation from the isotropic turbulence present in grid-generated turbulence.)
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
112
(a)
(b)
Figure 4-32: Spatially-averaged (a) absolute mean velocity, and (b) rms velocity fluctuations in the r, θ, and z directions (bulk velocity = 3.5 cm s−1, gas flow rate = 0.50 SLPM, θ = 30°).
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
113
A comparison of Figure 4-24 (b) and Figure 4-32 (b) reveals that the rms velocity
fluctuations at θ = 30° in the rotating tank are generally higher than those in the non-rotating
one. This is further illustrated in Figure 4-33, which plots u< > at θ = 30° downstream of
the lance (where the dissolution of Si cylinders was examined in the experiments) for various
bulk velocities. Increasing the bulk velocity from 0 to 3.5 cm s−1, u< > increases by more
than 60% after the startup period (4.5 s < t < 9.0 s). The much higher rms velocity fluctuation
in the rotating tank indicates that the effect of gas agitation is further extended into the
domain when a bulk flow is present. Further doubling the bulk velocity; however, the rms
velocity fluctuation only increases by another 25%. As a result, the turbulence intensity,
<Tu>, decreases at the higher bulk velocity. Figure 4-34 shows the turbulence intensities at
two bulk velocities; the simulation results reveal a 40% decrease in <Tu> as the bulk velocity
increases from 3.5 cm s−1 to 7.0 cm s−1.
Figure 4-33: Comparison of spatially-averaged rms velocity fluctuations at various bulk velocities (gas flow rate = 0.50 SLPM, θ = 30°).
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
114
Figure 4-34: Comparison of spatially-averaged turbulence intensity in the rotating tank (gas flow rate = 0.50 SLPM, θ = 30°).
Figure 4-35 and Figure 4-36 show a comparison of mean velocities and rms velocity
fluctuations at various gas flow rates (0, 0.50, and 1.00 SLPM), respectively. Injecting
bubbles into the melt significantly enhances liquid bulk motion and agitation, as evidenced
by a comparison of single-phase and two-phase flow simulations.
As the gas flow rate increases from 0.50 SLPM to 1.00 SLPM, larger bubbles form at the
lance exit (Table 4-2), displacing a larger volume of liquid that leads to changes in the mean
flow and a higher degree of agitation (turbulence intensity). As demonstrated in
Figure 4-35 (b), in the downstream flow the mean tangential velocity decreases near the
lance due to the effect of larger bubble wakes, but the absolute mean velocities increase in
the radial (Figure 4-35 (a)) and in particular the vertical directions (Figure 4-35 (c)).
Increasing the gas flow rate from 0.50 to 1.00 SLPM, when the mean velocities become
nearly steady (4.5 s < t < 9.0 s), the mean tangential velocity decreases by about 10%, while
the absolute mean radial and vertical velocities increase 30% and 50%, respectively. This
implies that for transport phenomena taking place in bubble-induced flows, neglecting the
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
115
three-dimensionality of the velocity field may lead to inconsistencies in correlating the
operating parameters (e.g. gas flow rate) to the transfer rate (e.g. mass transfer).
Nevertheless, the net effect of the aforementioned changes in the components of mean
velocity is a negligible change in the mean velocity, < V >.
As shown in Figure 4-36, increasing the gas flow rate from 0.50 SLPM to 1.00 SLPM
increases the intensity of turbulence in all directions. The components of rms velocity
fluctuations increase in the radial, tangential and vertical directions by approximately 30%,
20%, and 20%, respectively, and so the overall rms velocity fluctuations, < u>, also increase
by 20%. This demonstrates that an increase in mass transfer rates at higher gas flow rates is
primarily associated with the enhanced velocity fluctuations.
Finally, Figure 4-37 (a) compares the mean velocity as a function of θ, and illustrates that
even in upstream locations, especially closer to the lance (θ = −30°) where liquid is drawn
into the wakes of rising bubbles, the mean velocity can be even higher than in the
downstream. Figure 4-37 (b), of the rms velocity fluctuations, shows that introducing bubbles
into a bulk flow can significantly disturb even the approaching liquid, yielding velocity
fluctuations in the upstream flow that are similar or higher than in the downstream.
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
116
(a)
(b)
(c)
Figure 4-35: Comparison of spatially-averaged absolute mean velocities in the (a) r, (b) θ, and (c) z directions at various gas flow rates (bulk velocity = 3.5 cm s−1, θ = 30°).
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
117
(a)
(b)
(c)
Figure 4-36: Comparison of spatially-averaged rms velocity fluctuations in the (a) r, (b) θ, and (c) z directions at various gas flow rates (bulk velocity = 3.5 cm s−1, θ = 30°).
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
118
(a)
(b)
Figure 4-37: Comparison of spatially-averaged (a) mean velocity, and (b) rms velocity fluctuations at various angular positions relative to the lance (bulk velocity = 3.5 cm s−1, gas flow rate = 0.50 SLPM).
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
119
4.4 Summary
The size and trajectory of bubbles were evaluated using air–water experiments and video
recordings of the N2–Al free surface. In liquid Al, large buoyancy‐driven bubbles (with an
equivalent diameter of 1‐2 cm) are formed at a frequency of approximately 33-38 s−1, and
rise adjacent to the lance in both rotating and non-rotating tanks. These bubbles induce
velocities in the neighboring liquid, rather than come into contact with the dissolving Si.
The commercial CFD software FLOW-3D was used to characterize the bubble-induced
flow and quantify the velocities. The simulation results were validated by comparison with
the air–water results (bubble formation and rise times). The simulation and experimental
results agreed within the respective standard deviations. In the N2–Al simulations, the
equivalent bubble diameters were overpredicted slightly (by about 8%), where the
experimental bubble size was evaluated via pressure transducer data.
In the non-rotating tank, simulations revealed bubble-induced flows that are similar but
not symmetric around the lance, and that the entrainment of the liquid due to the rising
bubble wakes establishes a recirculating flow around the lance. The results showed that mean
velocity and turbulence intensities increase sharply near the lance, and beyond the two-phase
region the velocities decrease gradually. At the position of the Si cylinder outside the
two-phase region, a two-fold increase in the gas flow rate yielded a 70% increase in the rms
velocity fluctuations, but only a very small increase (< 10%) in the mean velocity.
In the rotating tank, the bubble-induced flow patterns are more complicated because of
the interaction between the bulk flow and the wakes of the rising bubbles, especially close to
the proximity of the lance. The bubbles accelerate the flow in the upstream and create
velocities in the opposite direction to the bulk flow in the downstream. Since the bubbles rise
just downstream of the lance, the vertical velocities are highest in this region, especially near
the free surface. Although gas agitation increases both mean velocity and turbulence intensity
compared to a single-phase flow, increasing the gas flow rate by a factor of two mainly
CHAPTER 4. FLUID DYNAMICS OF GAS–AGITATED LIQUID
120
affects the rms velocity fluctuations (which increase by 20% at the location of the Si
cylinder), while the mean velocity remains unchanged. A higher bulk velocity further
convects the effect of gas agitation throughout the entire tank. Finally, the bubble-induced
motion and agitation are significant both upstream and downstream of the lance.
121
CHAPTER 5
DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: TWO-PHASE FLOW
The results of a study of the effect of gas agitation on the dissolution of Si in molten Al are
reported, examining the dissolution enhancement of cylindrical Si specimens in a two-phase
flow. Mass transfer coefficients with and without gas injection are compared, to correlate the
effect of gas injection as an equivalent bulk velocity. Finally, the mass transfer coefficients
with gas agitation are compared to available correlations for mass transfer from solids in
gas-agitated liquids (i.e. non-rotating tank), and in the cross flow of a turbulent liquid (i.e.
rotating tank), incorporating the predicted velocities of Chapter 4 into various dimensionless
groups.
5.1 Experimental Procedure
Similar to the forced convection single-phase flow experiments, Al was melted inside the
RLMT, as described in Chapter 3. As shown in Figure 5-1, the furnace lid has three
Chapter 5. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: TWO-PHASE FLOW
122
types of holes. The hole at the center was used to insert a bath thermocouple and the large
hole to immerse Si specimens. The remaining holes were used to insert a gas injection lance
at different angular positions with respect to the Si cylinder. The two-phase flow
experimental setup included a gas delivery system with a flow controller.
Figure 5-1: The furnace lid includes various holes to immerse a thermocouple, Si specimens, and the lance into the liquid Al.
Metallurgical grade Si specimens (both MGSi–I and MGSi–II) were used in the
two-phase flow experiments. A detailed discussion of the differences between the two
batches is reported in Chapter 3. The size of the specimens, the preparation method, and the
immersed length were the same as in the single-phase flow experiments.
Preparation for a typical two-phase flow experiment was similar to that described in
Section 3.2.3, up to the point that the melt reached a solid body rotation. Then the lance was
inserted into the melt, the gas was turned on, and N2 bubbles began to form. After adjusting
the gas flow rate to the desired value, a Si cylinder was immersed into the liquid. After
immersion, the specimens were treated in the same way as in the single-phase flow
experiments. As before, dissolution data presented in this chapter usually represents three
experiments, and the standard deviation of these measurements is represented by error bars.
Chapter 5. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: TWO-PHASE FLOW
123
5.2 Results and Discussion
The single-phase flow experiments (Chapter 3), and the studies of bubble behavior and liquid
flow (Chapter 4), allow us to understand and quantify the impact of gas injection on Si
dissolution.
Figure 5-2 illustrates the effect of gas agitation on the dissolution of the cylindrical Si
specimens. The overall dissolution rate increases with increasing gas flow rate, and the rate
of dissolution is highest near the top of each specimen, facing the bulk flow. This is
consistent with the simulation predictions which revealed higher bubble-induced velocities
and a more vigorous agitation near the Al bath surface.
Figure 5-2: Dissolved MGSi–II cylindrical specimens at various gas flow rates (immersion time = 3 min, SPH = 40 K, bulk velocity = 3.5 cm s−1, θ = 30°).
In the two-phase flow experiments, the effect of the following parameters was examined
by immersing cylinders for 3 min: (1) bath temperature (SPH = 40 and 80 K), (2) bulk
velocity (0, 1.4, 3.5, and 7.0 cm s−1), (3) gas flow rate (0 to 2.00 SLPM), and (4) the position
of the lance relative to the Si cylinder (θ = −50° to +70°, negative denotes a downstream
Chapter 5. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: TWO-PHASE FLOW
124
position, positive an upstream position with respect to the Si specimen). The effect of each
parameter is examined while keeping the others constant.
5.2.1 Effect of bulk velocity at a given gas flow rate
Figure 5-3 illustrates the effect of changing bulk velocity; at a given gas flow rate, superheat,
and lance position, and shows that for the entire range of bulk velocities tested, the
introduction of gas leads to higher dissolved fractions. The injected gas produces relatively
large bubbles that agitate the liquid, as confirmed by the observed rippling of the liquid Al
free surface (Figure 4-13). The bubble-induced bulk fluid motion and increase in turbulence
intensity (estimated in Chapter 4) result in a decrease in mass transfer resistance adjacent to
the dissolving cylinder. This finding may be explained heuristically by, for example, the
Random Eddy Penetration (or Surface Rejuvenation) Model of Harriott [93], introduced in
Section 2.2.2. According to this model, turbulent eddies from regions of uniform solute
concentration will regularly penetrate the mass boundary layer, sweeping away some of the
accumulated solute. In between arrival of these eddies, the mass transfer in the mass
boundary layer is by diffusion. Overall, this agitation of the boundary layer increases the
mass transfer rate.
Chapter 5. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: TWO-PHASE FLOW
125
Figure 5-3: Dissolved fraction of MGSi–II at SPH = 40 K vs. bulk velocity (immersion time = 3 min, gas flow rate = 0.50 SLPM, θ = 30°).
The increased rate of dissolution due to gas agitation can be expressed as an increase in
the dissolved fraction, ΔDF = DFwith gas − DFwithout gas. The relative enhancement of the
dissolution rate is then defined as:
without gas
DFEnhancement factor (EF) = 100DFΔ
× . (5-1)
EF expresses the percent increase in the dissolved fraction due to gas agitation.
Figures 5-4 (a) and 5-5 (a) illustrate ΔDF vs. bulk velocity for gas flow rates of 0.50 and
1.00 SLPM; the corresponding EF are shown in Figures 5-4 (b) and 5-5 (b). In general, two
regions can be identified in Figures 5-4 (a) and 5-5 (a), at lower and higher bulk velocities.
At lower velocities where natural convection is dominant, ΔDF is roughly constant. As the
bulk velocity increases there is a noticeable increase in ΔDF, corresponding to the forced
convection regime.
Whether there is no bulk velocity, or bulk velocity matters little (bulk velocity < 3
cm s−1), natural convection is the dominant mode of mass transfer, and the effect of gas
Chapter 5. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: TWO-PHASE FLOW
126
injection on the flow around the dissolving Si is similar. Therefore, the velocity fluctuations
induced by the same degree of gas agitation result in a comparable ΔDF. For instance, as
shown in Figure 5-5 (a), at SPH = 40 K and a gas flow rate of 1.00 SLPM, ΔDF is 0.065 and
0.06 at bulk velocities of 0 and 1.4 cm s−1, respectively.
When the bulk flow is sufficiently strong to shift the primary mode of mass transfer to
forced convection (bulk velocity > 3 cm s−1), agitating the flow by gas injection significantly
augments ΔDF and thus EF. For example, as shown in Figure 5-5 (a), at SPH = 40 K, a gas
flow rate of 1.00 SLPM, and a bulk velocity of 3.5 cm s−1, ΔDF = 0.13 is twice the value as
when natural convection is dominant. This indicates that gas agitation is more effective in the
presence of forced convection, which convects the agitation downstream. The FLOW-3D
results confirm this: Figure 4-23 (a) illustrates flow in a non-rotating tank and shows that the
agitation is limited to a small region near the lance. In a rotating tank, (Figure 4-27), the
agitated region is much larger. The simulation results also show this quantitatively: u< > at
the location of a Si cylinder 30° downstream of a lance is 0.46 in the non-rotating tank, and
0.76 in the rotating one (bulk velocity = 3.5 cm s−1) as shown in Figure 4-33.
Figures 5-4 (a) and 5-5 (a) also show that as the bulk velocity increases beyond
3.5 cm s−1, the enhancement of mass transfer rate does not increase. For instance, at SPH =
40 K and a gas flow rate = 1.00 SLPM, ΔDF is a relatively constant 0.12 at velocities of 3.5
and 7.0 cm s−1. As the bulk velocity increases, the turbulence intensity decreases,
(Figure 4-34) which leads to a smaller dissolution enhancement. This is in accordance with
describing the enhancement of heat or mass transfer rate as a function of Re1/2Tu [83] or
Tu [102].
Chapter 5. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: TWO-PHASE FLOW
127
(a)
(b)
Figure 5-4: (a) ΔDF, and (b) EF for dissolution for MGSi–II vs. bulk velocity at a gas flow rate = 0.50 SLPM (immersion time = 3 min, θ = 30°, the lines indicate average values).
Chapter 5. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: TWO-PHASE FLOW
128
(a)
(b)
Figure 5-5: (a) ΔDF, and (b) EF for dissolution for MGSi–II vs. bulk velocity at a gas flow rate = 1.00 SLPM (immersion time = 3 min, θ = 30°, the lines indicate average values).
Chapter 5. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: TWO-PHASE FLOW
129
Figures 5-4 and 5-5 also show that the effect of gas agitation on ΔDF and EF decreases
with an increase in the bath superheat, in particular in the forced convection regime (bulk
velocity > 3 cm s−1). Comings et al. [82] show that the effect of turbulence intensity on heat
transfer from cylindrical specimens decreases as the thickness of the boundary layer
increases. Since gas agitation disturbs the mass boundary layer, the observed trend may be
associated with how the thickness of the boundary layer varies with the liquid temperature.
The boundary layer thickness can be estimated from the mass transfer coefficients
deduced from the experimental results. Adjusting for incubation times associated with the
single-phase flow experiments (and acknowledging that these may be longer than the
incubation times in two-phase flow), a mean mass transfer coefficient for the entire
immersion period , mk , can be calculated from Equation (3-14), and then the mass boundary
layer thickness can be estimated as [23,108]:
Si/Alm
m
Dk
δ ≅ (5-2)
where DSi/Al increases with temperature [39]. The estimates of δm, from the MGSi–II
experimental data are presented in Table 5-1: they range from 0.11-0.24 mm. This range is of
the same order of magnitude as found by Kim and Pehlke [108], who estimated a boundary
layer thickness in the range of 0.48-0.86 mm for Sh pertinent to the current study.
Table 5-1 also shows that a higher temperature generally yields a slightly thicker mass
boundary layer. For instance, at a bulk velocity of 3.5 cm s−1 the mass boundary layer in
single-phase flow at SPH = 80 K is about 10% thicker than at SPH = 40 K. This is simply
because of a higher diffusion coefficient, DSi/Al, at a higher SPH.
When natural convection is dominant (bulk velocity < 3 cm s−1), the single-phase flow
experiments suggest that the mass boundary layer thickness at a higher temperature
(SPH = 80 K) remains relatively unchanged compared to a lower temperature (SPH = 40 K);
therefore, when gas is injected, the variation of ΔDF with temperature is less significant.
Chapter 5. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: TWO-PHASE FLOW
130
A dimensional analysis of the effect of transport properties on the mass boundary layer
supports the effect of liquid Al temperature. For simplicity assume a single-phase flow and a
laminar boundary layer around a cylinder (which is plausible, as Red based on the bulk
velocity is less than 2710). Then the momentum boundary layer thickness can be described
by [117] as:
1/2Re−δ ∼ (5-3)
and the mass boundary layer thickness can be estimated as [117]:
1/3m 1 Sc1.026
−δ=
δ. (5-4)
Combining Equations (5-3) and (5-4) at a given bulk velocity and a characteristic length
yields:
1/6 1/3m Si/AlDδ ν ×∼ (5-5)
Taking into account the variations of Al and Si transport properties as a function of
temperature (correlations in Appendix A were used to calculate the transport properties), the
mass boundary layer thickness increases by 10% when SPH increases from 40 K to 80 K
similar to the increase observed experimentally.
The above analyses suggest that when a thinner mass boundary layer (i.e. at lower SPH)
is disturbed by bubble-induced agitation, a relatively larger increase in the dissolved fraction
is obtained. Following the argument of Harriott [93]: δm~D1/3, and so at a lower temperature,
indicating a lower diffusion coefficient, eddies must come closer to the surface to remove
solute. However, following each solute removal, the time required to re-establish a steady
state gradient is proportional to D−1. This implies that the effect of relatively fewer eddies is
greater when the diffusion coefficient is lower.
Chapter 5. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: TWO-PHASE FLOW
131
Table 5-1: Mean mass transfer coefficient of MGSi–II specimens, mk , and estimated mass boundary layer thickness, δm, at various operating conditions.
SPH (K)
Bulk velocity, (cm s−1)
Gas flow rate, (SLPM)
Mass transfer coefficient, × 5
mk 10 (m s−1)
Estimated boundary layer thickness, δm
(mm)
40
0 0 5.17 ± 0.25 0.24
0.50 6.04 ± 0.25 0.20 1.00 6.48 ± 0.24 0.19
1.4 0 6.05 ± 0.01 0.20
0.50 6.77 ± 0.24 0.18 1.00 7.26 ± 0.07 0.17
3.5
0 6.42 ± 0.38 0.19 0.25 6.85 ± 0.44 0.18 0.50 8.36 ± 0.26 0.15 0.75 8.94 ± 0.45 0.14 1.00 9.08 ± 0.35 0.13
7.0 0 8.82 ± 0.35 0.14
0.50 11.00 ± 0.13 0.11 1.00 11.52 ± 0.19 0.11
80
0 0 6.93 ± 0.33 0.24
0.50 7.58 ± 0.33 0.22 1.00 7.87 ± 0.27 0.21
1.4 0 7.31 ± 0.25 0.22
0.50 7.82 ± 0.26 0.21 1.00 8.36 ± 0.13 0.20
3.5
0 7.92 ± 0.80 0.21 0.25 8.19 ± 0.26 0.20 0.50 8.85 ± 0.31 0.19 0.75 9.23 ± 0.51 0.18 1.00 9.84 ± 0.62 0.17
7.0 0 10.76 ± 0.56 0.15
0.50 11.64 ± 0.22 0.14 1.00 12.85 ± 0.40 0.13
5.2.2 Effect of gas flow rate
A higher gas flow rates imparts a higher energy to the liquid. While the addition of gas
kinetic energy is insignificant, the displacement of liquid due to the relative motion of the
Chapter 5. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: TWO-PHASE FLOW
132
rising bubbles plays an important role in dissolution enhancement. By changing the gas flow
rate from 0.25 to 1.00 SLPM, EB is in the range of 0.13-0.47 W (Equation (2-4)), while EK is
in the range of 10−6-10−4 W.
Figures 5-6 and 5-7 illustrate that ΔDF and EF increase with an increase in the gas flow
rate, regardless of the type of MGSi and bath temperature. Injecting gas at a higher flow rate
produces larger bubbles at roughly the same frequency, as shown in Figure 4-12 and
discussed in Section 4.3.5.1. The displacement of a larger volume of liquid significantly
increases turbulence intensity but not mean velocity, as evidenced by the FLOW-3D
simulations (Section 5.3.5.2) and by LDV measurements in water [65]. Nevertheless, as Sh is
a function of both Red based on the mean velocity component, and ReT based on the rms
velocity fluctuations at the position of the dissolving object [97], there is a noticeable
enhancement in mass transfer by increasing the gas flow rate.
A comparison of Figure 5-6 (b) and Figure 5-7 (b) also reveals that the EF for MGSi–I
are much higher than for MGSi–II at the same operating conditions, even though the
corresponding ΔDF values in Figure 5-6 (a) and Figure 5-7 (a) are similar. This is because
the MGSi–I specimens dissolved more slowly than the MGSi–II in a single-phase flow,
resulting in a much smaller denominator in Equation (5-1).
As mentioned, gas agitation increased the dissolved fraction of MGSi–I and MGSi–II to a
similar extent. By contrast, in the single-phase experiments, increasing the bulk velocity
increased the dissolution rate of Si specimens with fewer impurities (MGSi–II) more than the
rate of specimens with more impurities (MGSi–I). This may be because the MGSi–I
specimens dissolved more non-uniformly (e.g. Figure 3-14), and the solid Si/liquid Al mass
transfer interface was thus much more three-dimensional due to the presence of randomly
located inclusions. The bubble‐induced velocities associated with gas injection are
appreciably more three-dimensional (e.g. Figure 4-32 (a)) and it may be that this type of flow
was better suited to dissolving a more complex interface structure.
Chapter 5. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: TWO-PHASE FLOW
133
Finally, Figure 5-7 shows that at various gas flow rates, the relative effect of gas agitation
on dissolution enhancement decreases with increasing temperature, for the same reasons
mentioned previously.
5.2.3 Effect of lance location
The effect of the relative position of the lance with respect to the Si cylinder on the
dissolution rate was examined. As illustrated in Figure 5-8, although the mass transfer
enhancement decreases slightly with increasing distance between the lance and specimen, it
appears that the velocity fluctuations generated by the bubbles are convected throughout the
tank. This is in accordance with the simulation results (Figure 4-27 and Figure 4-28).
It is especially noteworthy that even when the lance is downstream of the Si specimen
(i.e. θ < 0), a higher mass transfer is achieved compared to the single-phase flow
experiments. This is also in agreement with the results of simulations, that predicted similar
mean velocity and rms velocity fluctuations both at upstream and downstream locations
(Figure 4-37). This is one of the characteristics of turbulent flow, that the motion at any
location can strongly affect the motion at other locations [148].
Chapter 5. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: TWO-PHASE FLOW
134
(a)
(b)
Figure 5-6: (a) ΔDF and (b) EF for MGSi–I vs. gas flow rate (immersion time = 3 min, bulk velocity = 3.5 cm s−1, θ = 30°, the lines indicate average values).
Chapter 5. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: TWO-PHASE FLOW
135
(a)
(b)
Figure 5-7: (a) ΔDF and (b) EF for MGSi–II vs. gas flow rate (immersion time = 3 min, bulk velocity = 3.5 cm s−1, θ = 30°, the lines indicate average values).
Chapter 5. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: TWO-PHASE FLOW
136
(a)
(b)
Figure 5-8: ΔDF vs. the relative positions of the gas injection lance and Si specimen for (a) MGSi–I, SPH = 40 K and (b) MGSi–II, SPH = 80 K (immersion time = 3 min, bulk velocity = 3.5 cm s−1, gas flow rate = 1.00 SLPM, the filled symbols indicate average values).
Chapter 5. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: TWO-PHASE FLOW
137
5.2.4 Correlating single- and two- phase flows
Chapter 4 presented CFD predictions of the effect of gas injection on both bulk velocity and
velocity fluctuations in liquid Al. Here from the experimental data, single-phase and
two-phase flow mass transfer coefficients are compared to correlate the effect of gas
injection (i.e. two-phase flow) to an equivalent bulk velocity (i.e. single-phase flow). Each of
Figure 5-9 (a) and (b) superimposes two plots: (i) two-phase mass transfer coefficients vs.
gas flow rate (on the top abscissa) at different bulk velocities, and (ii) single-phase mass
transfer coefficients vs. bulk velocity (on the bottom abscissa). Bubble-induced velocities can
then be interpreted as an equivalent increment to the bulk velocity of the liquid.
To illustrate this, Figure 5-10 is a simpler version of Figure 5-9, with the single-phase
flow results plotted against only the two-phase flow experiments at 3.5 cm s−1. At a SPH of
40 K and a two-phase bulk velocity of 3.5 cm s−1 (Figure 5-10 (a), dotted line with the open
circles), when gas is injected at a flow rate of 1.00 SLPM (TT'), the mass transfer coefficient
is about 9 × 10−5 m s−1; this same mass transfer coefficient is obtained in a single-phase flow
moving at about 7.3 cm s−1 (SS'). At the higher SPH of 80 K (Figure 5-10 (b)), for the same
two-phase flow bulk velocity and gas flow rate as above (TT"), the equivalent single-phase
flow bulk velocity (SS") is only 5.9 cm s−1. Figure 5-9 may be used in a similar manner at
other operating conditions to correlate the effect of gas injection on mass transfer rate to an
increase in the bulk velocity.
Chapter 5. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: TWO-PHASE FLOW
138
(a)
(b)
Figure 5-9: Mass transfer coefficient vs. gas flow rate (shown on the top abscissa) superimposed on mass transfer coefficient vs. bulk velocity (shown on the bottom abscissa) for (a) SPH = 40 K and (b) SPH = 80 K (MGSi–II, immersion time = 3 min, θ = 30°, error bars indicate the standard deviations).
Chapter 5. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: TWO-PHASE FLOW
139
(a)
(b)
Figure 5-10: Mass transfer coefficient vs. gas flow rate at a bulk velocity of 3.5 cm s−1 (shown on the top abscissa) superimposed on mass transfer coefficient vs. bulk velocity (shown on the bottom abscissa) for (a) SPH = 40 K and (b) SPH = 80 K (MGSi–II, immersion time = 3 min, θ = 30°, error bars indicate the standard deviations).
Chapter 5. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: TWO-PHASE FLOW
140
5.3 Comparison with a Correlation for Mass Transfer in a Gas‐stirred Tank
For the non-rotating tank, the results of the two-phase flow dissolution experiments are
compared to the correlation of Mazumdar et al. [97,107] for mass transfer from vertical
benzoic acid cylinders in a gas-stirred water tank [97], and vertical steel cylinders in
gas-stirred carbon saturated iron [107], both inside and outside the plume region:
( ) ( ) ( )0.25 0.32 0.33d TSh 0.73 Re Re Sc= . (5-6)
Red and ReT are based on local mean and rms velocity fluctuations, respectively.
Table 5-2 shows that Equation (5-6) underestimates the mass transfer coefficients, mk , by
27% and 17% at gas flow rates of 0.50 and 1.00 SLPM, respectively. The reason may be that
in [97,107], both V and u at the location of the dissolving specimens are much higher
(benzoic acid cylinders in water: 5 cm s−1 < V < 32 cm s−1, 1.1 cm s−1 < u < 7.5 cm s−1 [97];
steel cylinders in carbon saturated iron: 18 cm s−1 < V < 43 cm s−1, 1.3 cm s−1 < u < 5.0 cm
s−1 [107]) than the values predicted numerically in the current study (Figure 4-26 and
Table 5-2). As a result, natural convection effects were less important in those studies. To
take into account natural convection in the current study, the Mazumdar correlation values
were combined with the experimentally-deduced natural convection mass transfer
coefficients. Using the general combining law described in Section 3.3.2.3 yields a much
better agreement with the experimental data. The differences between the combined
correlation predictions and the experimental results are −5% and −3% at gas flow rates of
0.50 and 1.00 SLPM, respectively.
The small discrepancy may be associated with the prescribed incubation time. To
estimate the two-phase flow experimental mass transfer coefficients, the dissolution data was
used while taking into account the incubation times from the single-phase flow experiments.
It is very possible that the incubation times were shorter in the two-phase flow experiments.
Chapter 5. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: TWO-PHASE FLOW
141
By overestimating the incubation times, the experimental mass transfer coefficients would be
overpredicted.
5.4 Comparison with a Correlation for Mass Transfer in Turbulent Cross Flow
The Mazumdar correlation was also applied to the rotating tank (e.g. bulk velocity = 3.5
cm s−1, in which the gas-agitated velocity field was predicted in Chapter 4). However, even
when considering natural convection, the correlation underpredicts the experimental results
by 19% and 23% at gas flow rates of 0.50 SLPM and 1.00 SLPM, respectively. This is
because the Mazumdar fluid flow configuration with respect to the cylindrical specimen
[97,107] is different than in the rotating tank. For that reason, an effort was made to compare
the two-phase flow experimental results to a more appropriate turbulent flow configuration.
As mentioned in Section 4.3.4.3, there are a fair number of correlations that predict heat
and mass transfer from spheres and cylinders in cross flow, but usually in a gas phase
characterized by low turbulence intensities. In studies of liquid phase cross flow (i.e. water),
Iguchi et al. [102] and Szekely et al. [101] proposed a correlation for heat transfer based on
the melting rate of ice spheres and horizontal cylinders in a bottom stirred cylindrical water
tank, respectively. Iguchi et al. [102] defined Tu based on axial mean and rms velocity
fluctuations along the plume centerline; Szekely et al. [101] normalized the local velocity
fluctuations based on gas entry velocity. As a result, these correlations are not based on
appropriate local velocities, and so their general use is not possible. To the best of the
author’s knowledge, only Sandoval-Robles et al. [149] proposed a correlation for mass
transfer from solid spheres in a liquid phase cross flow with high turbulence intensities. In
that case, the turbulence was generated by a porous plate. They used the mean and rms
velocity fluctuations at the location of the solid to calculate Re and ReT (330 < Red < 1720
Chapter 5. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: TWO-PHASE FLOW
142
and 0.04 < Tu < 0.3, in the range of Red and Tu of the current study). They describe the mean
velocity by 1/2dRe , and account for the turbulence effect by 0.066
TRe .
This suggests that for a turbulent stream, ReT with the same exponent may be
incorporated into the Churchill and Bernstein correlation [126] for heat and mass transfer
from cylinders in cross flow (Equation (3-11)), rewritten here for mass transfer,
( )
4/55/81/2 1/3d d
1/42/3
0.62 Re Sc ReSh 0.3 12820001 0.4 / Sc
⎡ ⎤⎛ ⎞= + +⎢ ⎥⎜ ⎟⎝ ⎠⎡ ⎤ ⎢ ⎥⎣ ⎦+⎣ ⎦
(5-7)
Recall that the predictions of this correlation when combined with natural convection
were in a satisfactory agreement with the single-phase flow experiments (Section 3.3.2.3).
Therefore, Equation (5-7) may be adapted as follows:
( )
4/55/81/2 1/30.066d dT1/42/3
0.62 Re Sc ReSh 0.3 1 Re2820001 0.4 / Sc
⎡ ⎤⎛ ⎞= + +⎢ ⎥⎜ ⎟⎝ ⎠⎡ ⎤ ⎢ ⎥⎣ ⎦+⎣ ⎦
(5-8)
Table 5-3 presents a comparison of mass transfer coefficients obtained from
Equation (5-8) (based on the initial diameter of the cylinder, and combined with a natural
convection correlation) with experimental values. The agreement is much better than with the
Mazumdar correlation, yet the experimental data are under-predicted at lower velocities and
overpredicted at higher velocities.
The correlation may underpredict the experimental data for either of two reasons. The
first is associated with the incubation time, as described in Section 5.3. The second reason is
due to the use of the initial diameter of the cylindrical Si specimens for calculating the mass
transfer coefficients. The initial diameter underpredicts the mass transfer coefficient as the
dissolution progresses. As discussed in Section 3.3.2.3, as the radius of the specimen
decreases, the shrinkage rate increases, indicating a higher mass transfer coefficient.
On the other hand, the correlation may overpredict the experimental data because the
effect of gas agitation in Equation (5-8) is represented only by the magnitude of velocity
fluctuations, irrespective of the bulk velocity. As mentioned in Section 5.2.1, there are
Chapter 5. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: TWO-PHASE FLOW
143
studies that suggest that when predicting transport rate enhancement in a turbulent flow, that
the velocity fluctuations must be normalized by the liquid bulk velocity or its square root.
This reduction of dissolution enhancement at higher bulk velocities is observed
experimentally (Figure 5-4 and Figure 5-5), but is not incorporated in the correlation.
Chapter 5. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: TWO-PHASE FLOW
144
Table 5-2: Predicted mass transfer coefficients using the general combining law for Equation (5-6) and natural convection, and a comparison with experimental values (bulk velocity = 0, θ = 30°).
Bubble-induced velocities, cm s−1
(Average at steady state)
Re, based on initial specimen diameter
Mean transfer coefficient × 105, m s−1
Gas flow rate
(SLPM) V< > u< >
0dRe based
on V< > TRe based
on u< > Experimental
Predicted via
Equation (5-6)
Modified using combined convection correlation
Error (%) Combined convection correlation Experimental
100Experimental
−×=
0 – – – – 5.17 – – – 0.50 1.78 0.46 637 166 6.04 4.40 5.75 −5 1.00 1.90 0.81 680 290 6.48 5.35 6.26 −3
Table 5-3: Predicted mass transfer coefficients using the general combining law for Equation (5-8) and natural convection, and a comparison with experimental values (θ = 30°).
Bubble-induced velocities, cm s−1
(Average at steady state)
Re, based on initial specimen diameter Mean mass transfer coefficient × 105, m s−1
Gas flow rate
(SLPM)
Bulk velocity (cm s−1)
V< > u< > 0dRe based
on V< > TRe based
on u< > Experimental
Predicted via
Equation (5-8)
Modified using combined convection correlation
Error (%) Combined convection correlation Experimental
100Experimental
−×=
0.50 1.4 1.70 0.72 609 259 6.77 5.28 6.21 −8 0.50 3.5 3.43 0.76 1227 272 8.36 7.58 7.96 −5 1.00 3.5 3.46 0.91 1236 326 9.08 7.70 8.06 −11 0.50 7.0 6.91 0.96 2473 342 11.06 11.00 11.20 +2
Chapter 5. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: TWO-PHASE FLOW
145
5.5 Relationship Between Mass Transfer Coefficient and Energy Input into the Liquid in Single and Two-phase Flows
The effect of single-phase and two-phase flows on accelerating the dissolution rate of Si in
Al has been presented. We now briefly consider the implications for practical applications,
from an energy requirements point of view. The approach is to calculate the energy input to
the liquid for a certain degree of dissolution enhancement. Figure 5-11 shows normalized
mass transfer coefficient at the position of the Si specimen (θ = 30°, SPH = 40 K) vs. the
normalized time- and spatial-averaged liquid kinetic energy per unit mass of liquid
(calculated analytically for single-phase flow, and computed from the FLOW-3D results for
two-phase flow). The baseline is associated with the lowest single-phase flow velocity (bulk
velocity = 3.5 cm s−1).
To increase the dissolution rate, the required energy input into the liquid via gas injection
is less ( 1/3m lk K< > < >∼ ) than that required to increase the bulk flow velocity (
1/4m lk K< > < >∼ ). This may be surprising as in Figure 5-9 an increase in the bulk velocity
appears to be more effective at increasing the mass transfer coefficient. However, the results
suggest that the square root dependence (i.e. b 0.5l(U ) K< >∼ ) implies that increasing the
bulk velocity of a single-phase flow requires much more energy than is needed to induce
fluctuating velocities in a two-phase flow. (Deduction of a universal relationship between
energy input and the velocity fluctuations is not straightforward; however, the limited
simulation results suggest that at the position of the Si cylinder: 1.2lu K< > < >∼ .) This is a
significant practical implication, that dissolution can be enhanced by gas stirring with less
energy input to the liquid than is required by mechanical stirring.
Chapter 5. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: TWO-PHASE FLOW
146
Figure 5-11: Normalized mass transfer coefficient for single-phase and two-phase flows vs. normalized liquid kinetic energy per unit mass of liquid.
5.6 Summary
The dissolution of Si in Al in a gas-agitated liquid was examined. The effect of bath
temperature, bulk liquid velocity, gas flow rate and the lance position on dissolution was
quantified: gas agitation is relatively less effective at higher bath temperatures; gas injection
to agitate a bulk flow is more successful when forced convection is dominant; a higher gas
flow rate produces larger bubbles, which yield higher turbulence intensities resulting in a
higher dissolution rate; and gas agitation increased the dissolved fraction of Si specimens
both downstream and upstream of the lance.
At a given bulk velocity, the results offer quantitative guidance to enhance the dissolution
rate to a desired level, by increasing the bulk velocity or by blowing a prescribed amount of
gas. In this way, the effect of gas agitation on dissolution enhancement is expressed as an
equivalent increment to the bulk velocity.
Chapter 5. DISSOLUTION OF SOLID Si CYLINDERS IN MOLTEN Al: TWO-PHASE FLOW
147
The experimental results were compared to available mass transfer correlations
applicable to the rotating and non-rotating tank experiments. For the non-rotating tank, a
correlation developed for mass transfer from vertical cylinders in gas-agitated ladles
predicted the experimental results within 5%. For the rotating tank, the effect of free stream
turbulence was introduced into a correlation for heat and mass transfer from cylinders in
cross flow. The new correlation predicted the experimental results within 11%. The
discrepancies may be associated with the unknown incubation times in the two-phase flow,
an increase of the mass transfer coefficient with decreasing specimen size, and
overpredicting the effect of gas agitation at higher bulk velocities.
148
CHAPTER 6
SUMMARY, CONCLUSIONS AND FUTURE WORK
6.1 Summary
The dissolution of Si into Al is a bottleneck for the Al industries which produce Al–Si alloys
for cast parts. The slow dissolution results in material and energy losses, and reduces
productivity. Faster dissolution would yield economic savings and environmental benefits,
yet the effects of parameters that influence the dissolution rate have not been systematically
quantified.
By immersing more than 700 cylindrical Si specimens, this thesis represents a
comprehensive study of the dissolution rate of solid Si into molten Al under different fluid
flow conditions. The first series of experiments were conducted without gas agitation. The
dissolution of Si specimens with different impurity levels was examined both under natural
and forced convection conditions. In the next series of experiments, the Al bath was agitated
by introducing N2 bubbles into the liquid using top lance injection. The effects of bath
temperature, bulk liquid velocity, and gas flow rate on dissolution rate were related to each
other via deduced mass transfer coefficients.
CHAPTER 6. SUMMARY, CONCLUSIONS AND FUTURE WORK
149
To understand the localized mass transfer phenomena, the fluid dynamics of a
gas-agitated bath must be considered. A physical model (air–water system) was used to
provide insight into the bubble behavior (frequency, size, and trajectory), and the
gas-agitated free surface of the molten Al was video recorded. To estimate the velocity field,
the FLOW-3D software was used. The simulation results were validated by comparing with
the air–water results, and then simulations were run for some of the N2–Al experimental
cases.
The following are the conclusions drawn from the single-phase flow experiments, the
bubble characterization experiments, the simulations, and the two-phase flow experiments.
6.2 Conclusions
Single-phase flow experiments: The effects of solid Si impurities and Al bath temperature
and velocity on the dissolution rate of Si specimens were examined.
(1) Polycrystalline metallurgical grade Si (MGSi) with higher levels of impurities
dissolved more slowly than high purity polycrystalline MGSi, which showed a similar
dissolution rate to monocrystalline electronic grade Si (EGSi). The impurities were in the
form of inclusions composed of elements with lower diffusion coefficients in liquid Al
than that of Si in liquid Al. It is postulated that the inclusions obstruct the dissolution of
Si.
(2) For the high purity metallurgical grade Si, under natural convection, an increase in the
bath superheat from 40 to 80 K increases the mass transfer coefficient by 30%. Under
forced convection, increasing the bulk liquid velocity from 0 to 9.9 cm s−1 increases
the mass transfer coefficient by 100%.
CHAPTER 6. SUMMARY, CONCLUSIONS AND FUTURE WORK
150
(3) In the natural convection experiments, the value of GrmSc was in the turbulent
regime. For the high purity Si cylinders, the experimental data exhibited good
agreement with a dimensionless correlation for vertical cylinders:
1/3mSh (0.11 to 0.129)(Gr Sc)= , where 8 8
m,l5.5 10 Gr 7.4 10× < < × and 29 Sc 42< < .
(4) Under forced convection conditions (by rotating the liquid Al), the mass transfer rate
increased at higher liquid velocities, demonstrating that the dissolution process is
controlled by liquid phase diffusion (LPD). When the forced convection prevailed
over natural convection, the experimental data for high purity cylinders were fitted
with a dimensionless correlation for cylinders in cross flow [126]:
( )
4/55/81/2 1/3d d
1/42/3
0.62Re Sc ReSh 0.3 12820001 0.4 / Sc
⎡ ⎤⎛ ⎞= + +⎢ ⎥⎜ ⎟⎝ ⎠⎡ ⎤ ⎢ ⎥⎣ ⎦+⎣ ⎦
, where 0d510 < Re < 3800 and
29 Sc 42< < .
(5) At lower bulk velocities of the liquid (< 3 cm s−1), a combined effect of natural and
forced convection must be considered; this is important when the value of
( ) ( )4 2m,l dd l Gr Re⎡ ⎤
⎣ ⎦ is equal to or larger than unity. For this case, a new correlation
was presented based on the general combining law of Churchill and Usagi [131]:
( ) ( )1/44 4N F
m m mk k k⎡ ⎤= +⎢ ⎥⎣ ⎦.
Bubble characterization experiments: A series of bubble injection experiments in water and
in liquid Al were conducted to investigate the effect of gas flow rate and bulk velocity on
bubbling behavior.
(6) In water, using top lance injection, when air bubbles form they do not wet the lance.
Increasing the gas flow rate increases the bubble size and changes the formation
mechanism, from single bubbles, to the coalescence of bubble pairs during rise, to the
CHAPTER 6. SUMMARY, CONCLUSIONS AND FUTURE WORK
151
coalescence of bubbles at the lance exit. When bubble pairs form, the shape and size
of the trailing bubble is affected (i.e. the bubble is elongated and up to 45% smaller
by volume) by the leading one, especially in the third regime. Smaller wobbling
bubbles in water follow a zig-zag path, which changes to a rectilinear one with more
violent bubble interface deformation as the bubbles become larger.
(7) In liquid Al, using top lance injection, N2 bubbles form at a relatively constant
frequency (33-38 s−1),and so the bubble size increases with gas flow rate. The
estimated single bubble size was in the range of 1-2 cm, but it is postulated that the
bubbles may coalesce at the lance exit or during rise, especially at higher flow rates.
The bubbles wet the boron nitride coated lance, and so wick up the lance and reach
the free surface adjacent to it. The agitation of the liquid was noticeable due to the
rippling of the entire free surface.
(8) In liquid Al, increasing the liquid bulk velocity from 0 to 7 cm s−1, the estimated
bubble diameter decreases no more than 5%, and the bubbles still rise very near the
lance but on the downstream side. This is in contrast to air–water experiments where
the bubble diameter decreases more than 7% at a similar Re (based on lance
diameter), and the bubbles travel downstream of the lance by 4-7 lance diameters.
Fluid flow simulations: FLOW-3D software was used to examine top lance injection bubble
dynamics within water and molten Al, and to predict the 3D velocity field within a revolving
liquid metal tank.
(9) For the air–water system, the simulation and experimental results agree within the
respective standard deviations of bubble formation and rise time. For the N2–Al
system, FLOW-3D slightly overpredicts the bubble size (~8% by diameter); however,
the general bubble formation and rise behavior is predicted as expected.
CHAPTER 6. SUMMARY, CONCLUSIONS AND FUTURE WORK
152
(10) In the non-rotating tank, a recirculating flow is established around the lance due to
the rising bubbles. The region beneath the lance is nearly unaffected by the bubbles.
The largest induced velocities are observed near the lance and near the free surface.
(11) In the non-rotating tank, time and spatial averaging of velocities demonstrates that
liquid velocities decrease sharply near the lance, followed by a more gradual decrease
moving away from the two-phase region. Higher gas flow rates primarily increase the
turbulence intensities rather than the mean velocities. When the Si cylinder is outside
the two-phase region, FLOW-3D predicts that the rms velocity fluctuations increase
by 70%, but the mean velocity by less than 10%, when the gas flow rate is doubled.
(12) In the rotating tank, the bubble-induced flow causes an acceleration of the flow field
upstream of the lance, but induces velocities in the opposite direction of the bulk flow
downstream of the lane. As the bubbles rise just downstream of the lance, the vertical
velocities are highest in this region.
(13) In the rotating tank, time and spatial averaging of velocities reveals that gas injection
increases both mean velocity and turbulence intensity compared to the single-phase
flow. However, increasing the gas flow rate by a further factor of two, mainly affects
the rms velocity fluctuations (which increase by 20%), while the mean velocity
remains relatively unchanged. The bubble-induced motion and agitation are
significant both upstream and downstream of the lance, implying that the effect of gas
agitation is convected to the entire flow field.
Two-phase flow experiments: The effects of bath temperature, liquid bulk velocity, gas flow
rate, and the position of the lance on the dissolution rate of Si were examined:
(14) The effect of gas agitation on accelerating dissolution decreases with increasing bath
temperature, especially in the forced convection regime. This is because the mass
CHAPTER 6. SUMMARY, CONCLUSIONS AND FUTURE WORK
153
boundary layer thickness increases with increasing temperature: a thicker mass
boundary layer is less affected by gas agitation.
(15) Introducing bubbles to agitate the liquid yields a larger increase in the dissolution
rate under forced convection than under natural convection. This is because the bulk
velocity convects the effect of gas agitation throughout the domain. However, a
further increase in the bulk velocity yields a relatively smaller increase in the
dissolution rate.
(16) As the gas flow rate increases, the larger bubbles induce higher velocities, that
further decrease the mass transfer resistance adjacent to the solid Si, increasing the
dissolution rate. Increasing the gas flow rate from 0.25 SLPM to 1.00 SLPM
increases the mass transfer coefficient by 20-30%.
(17) When gas injection is combined with a bulk flow, gas agitation is effective both
upstream and downstream of the lance. This is in agreement with the simulation
results which reveal similar mean velocity and rms velocity fluctuations in upstream
and downstream positions.
(18) For high purity Si in the non-rotating tank, the experimental data is compared to a
dimensionless correlation for mass transfer from vertical solid cylinders in a gas
agitated liquid: ( ) ( ) ( )0.25 0.32 0.33d TSh 0.73 Re Re Sc= ; and the effect of natural
convection is included by applying a general combining law. At gas flow rates of 0.50
and 1.00 SLPM, the discrepancy between the predicted and experimental mass
transfer coefficients is −5% and −3%, respectively.
(19) For high purity Si in the rotating tank, a correlation for vertical cylinders in cross
flow is modified to take into account the turbulence in the bulk flow:
CHAPTER 6. SUMMARY, CONCLUSIONS AND FUTURE WORK
154
( )
4/55/81/2 1/30.066d dT1/42/3
0.62 Re Sc ReSh 0.3 1 Re2820001 0.4 / Sc
⎡ ⎤⎛ ⎞= + +⎢ ⎥⎜ ⎟⎝ ⎠⎡ ⎤ ⎢ ⎥⎣ ⎦+⎣ ⎦
. Combining this equation
with one for natural convection yields mass transfer coefficients within 11% of
experimental values.
6.3 Contributions
Owing to the difficulty of performing experiments at high temperatures, systematic
studies of transport rates in liquid metals, as a function of fluid flow conditions, are very
limited. By conducting a large number of dissolution experiments, corroborated with
physical and numerical modeling of the fluid flow, this thesis quantifies mass transfer rates
from a solid addition in a liquid metal under natural and forced convection conditions,
without and with high turbulence in the bulk flow. The turbulence was generated with top
lance injection method which, as suggested by the findings of this work, can be implemented
in alloying processes for enhancing the assimilation rates of metallic additions. Therefore, the
results of the current study have both theoretical and practical implications.
In the single-phase flow experiments the validity of an existing correlation for natural and
forced convection heat and mass transfer from vertical cylinders in a liquid metal system was
extended. Also, a correlation for mixed convection (at lower bulk velocities) was proposed,
for when the buoyancy force is normal to the direction of the bulk flow. In addition,
examining the effect of temperature and the bulk velocity of the liquid on the dissolution rate
yielded a quantitative guidance for correlating the effect of the two parameters on the
dissolution of Si in molten Al.
The numerical simulation of a gas-agitated ladle was validated with the results of
physical modeling in water, and measurements of bubble frequency in liquid Al. The
validated simulation tool, even though it did not fully resolve the flow, proved to be effective
CHAPTER 6. SUMMARY, CONCLUSIONS AND FUTURE WORK
155
for predicting bubble distribution and estimating the velocity field within the opaque liquid
metal, for the purpose of predicting mass transfer rates.
To the best of author’s knowledge, this is the first time the effect of gas injection
superimposed on a bulk liquid velocity was examined, in terms of the mass transfer rate from
a solid addition in a liquid metal. The results enabled us to present the effect of gas agitation
on the mass transfer rate as an equivalent increment to the bulk velocity. Finally, to extend
the application of the experimental results, an existing mass transfer correlation for a
gas-agitated liquid without an external bulk flow was used to fit the experimental data; with a
bulk velocity, a correlation for mass transfer from a vertical cylinder in cross flow was
modified to incorporate the effect of gas injection.
6.4 Recommendations for Future Work
The following are recommendations for improving and extending this work:
• The experimental study must be extended to analyze the recovery of Si and dross
formation under different operating conditions, to optimize the overall alloying
process.
• A major area of future research is to examine the correlations presented in this thesis
to other metallic systems. These correlations should also be tested for higher
velocities and degrees of agitation.
• In this thesis, the bubble-induced velocities were only approximated. The two-phase
flow could be resolved using LES or DNS, or reliable sensors for measuring
velocities in high temperature liquid metals should be developed.
CHAPTER 6. SUMMARY, CONCLUSIONS AND FUTURE WORK
156
• The analysis of the experimental results suggests that an increase in the liquid energy
by gas agitation yields a higher dissolution rate than by increasing the bulk velocity of
a single-phase flow. This finding must be developed further by identifying a
relationship between the energy input to a liquid and the induced velocity
fluctuations. More important, this laboratory observation must be confirmed at a plant
scale.
157
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168
APPENDIX A: CORRELATIONS FOR MATERIAL PROPERTIES
A.1 Diffusion Coefficient of Si into liquid Al
Using the rotating disk method, Ershov et al. [39] measured the diffusion coefficient of solid
Si into liquid Al in the temperature range 700-1000°C. The variation of diffusion coefficient
with temperature is described by an Arrhenius-type equation,
0QD D exp
RT⎛ ⎞= −⎜ ⎟⎝ ⎠
(A-1)
where D0 = 2.30 x 10−5 m2 s−1, Q = 61010 J mol−1, and R = 8.314 J mol−1 K−1. A linear
variation of −log D vs. 410 / RT is illustrated in Figure A-1.
Figure A-1: Diffusion coefficient of Si into Al as a function of temperature.
169
A.2 Density and Viscosity of Liquid Al
The density of liquid Al in the temperature range 660-917°C is given as [6],
Al 1 2 refc c (T T )ρ = − − (A-2)
where c1 = 2377.23 kg m−3, c2 = 0.311 kg m−3 K−1, and Tref = 933.47 K (melting point of Al).
The standard deviation of the correlation at the 95% confidence level is 0.0065.
The dynamic viscosity of liquid Al in the temperature range 660-997°C is given as [6],
0 210 1
alog ( ) aT
μ μ = − + (A-3)
where μ0 = 0.001 kg m−1 s−1, a1 = 0.7324, and a2 = 803.49 K. The standard deviation of the
correlation at the 95% confidence level is 0.137.
Using Equations (A-2) and (A-3), the kinematic viscosity of liquid Al ( / )ν = μ ρ is
plotted vs. temperature in Figure A-2.
Figure A-2: Kinematic viscosity of liquid Al as a function of temperature.
170
APPENDIX B: IMPROVING THE REPRODUCIBILITY OF Si DISSOLUTION
Significant anomalies in the data obtained from early dissolution experiments led to
inconclusive results. This required a thorough investigation, to improve the reproducibility of
the data. First, an X‐ray Photoelectron Spectroscopy (XPS) analysis was carried out to detect
surface impurities. Substantial amounts of impurities were identified on to the specimen
surfaces. The main elements were carbon and oxygen. The carbon was possibly deposited on
to the surface from the diamond drill used to make the Si cylinders. The oxygen was likely in
the form of SiO2 generated by Si oxidization in ambient air.
To further investigate the variability of the experimental results, 22 as‐drilled MGSi–I
specimens were immersed into 700°C liquid Al (SPH = 40 K) for 5 min. As seen in
Figure B-1, the dissolution of these samples varied significantly; both fully and partially
undissolved samples were retrieved after immersion. This suggested that there must be
dissolution barriers on the surface, possibly the impurities detected using the XPS analysis.
To confirm this, the impurities were mechanically removed by sand paper grinding
(removing 0.05-0.15 mm of the surface). This procedure was performed not more than 12
hours prior to the experiments, and then the ground specimens were kept in a desiccator to
minimize contact with air. Figure B-2 demonstrates that the scatter in the data is significantly
reduced following this preparation method. The ground specimens showed consistently
higher dissolved fractions compared to the as‐drilled ones. In addition, the grinding
eliminated the non‐uniform dissolution of specimens illustrated in Figure B-1.
171
Figure B-1: MGSi–I as‐drilled specimens after natural convection dissolution for 5 min in liquid Al at SPH = 40 K.
Figure B-2: Effect of sample surface preparation on the dissolved fraction of MGSi–I specimens immersed for 5 min in liquid Al under natural convection at SPH = 40 K.
Based on the above findings, all of the early data was discarded and the entire body of
experiments was repeated. Clearer conclusions on the effect of operating parameters were
obtained from the new results.