Advertisement

Metallurgical and Materials Transactions B

, Volume 48, Issue 3, pp 1850–1867 | Cite as

A Mathematical Model for Reactions During Top-Blowing in the AOD Process: Derivation of the Model

  • Ville-Valtteri VisuriEmail author
  • Mika Järvinen
  • Aki Kärnä
  • Petri Sulasalmi
  • Eetu-Pekka Heikkinen
  • Pentti Kupari
  • Timo Fabritius
Article

Abstract

In an earlier work, a fundamental mathematical model was proposed for side-blowing operation in the argon–oxygen decarburization (AOD) process. The purpose of this work is to present a new model, which focuses on the reactions during top-blowing in the AOD process. The model considers chemical reaction rate phenomena between the gas jet and the metal bath as well as between the gas jet and metal droplets. The rate expressions were formulated according to a law of mass action-based method, which accounts for the mass-transfer resistances in the liquid metal, gas, and slag phases. The generation rate of the metal droplets was related to the blowing number theory. This paper presents the description of the model, while validation and preliminary results are presented in the second part of this work.

Keywords

Reaction Interface Sherwood Number Slag Phase Metal Droplet Average Residence Time 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Nomenclature

Symbols

a

Activity

A

Surface area (m2)

Ai

Parameter of the Kronig–Brink solution

Cp

Molar heat capacity at constant pressure (J/(mol K))

cp

Specific heat capacity at constant pressure (J/(kg K))

dt

Nozzle throat diameter (m)

dlimit

Fineness parameter of the RRS distribution (m)

d32,md

Sauter mean diameter of the metal droplets (m)

D

Diffusion coefficient (m2/s)

fi

Mass fraction of size class i at place of birth

f

Residual vector

g

Standard gravity (m/s2)

ΔG°

Change in standard Gibbs free energy of reaction (J/mol)

ΔGtot

Change in total Gibbs free energy (J/mol)

hcav

Depth of the cavity (m)

hlance

Distance of the top lance from the surface of the metal bath (m)

H°

Standard enthalpy (J/mol)

ΔH°

Change in standard reaction enthalpy (J/mol)

J

Jacobian matrix

Jeff

Droplet generation rate multiplication factor

kf

Forward reaction rate coefficient

K

Equilibrium constant

L

Characteristic length (m)

m

Mass (kg)

\( \dot{m}_{\text{md}} \)

Metal droplet generation rate (kg/s)

\( \dot{m}_{{{\text{md}},{\text{eff}}}} \)

Effective metal droplet generation rate (kg/s)

M

Molar mass (kg/mol)

nlance

Number of exit ports in a nozzle

n

Distribution exponent of the RRS distribution

p

Partial pressure

pcav

Arc length of the cavity (m)

pamb

Ambient pressure (Pa)

p0

Stagnation pressure at upstream part of the top lance (Pa)

rcav

Top radius of the cavity (m)

R

Gas constant (J/(mol K))

R

Reaction rate (kg/(m2 s))

R2

Correlation coefficient

RF

Cumulative weight fraction

S°

Standard entropy (J/(mol K))

ΔS°

Change in standard reaction entropy (J/(mol K))

tmd,i

Residence time of metal droplet size class i (s)

\( \bar{t}_{\text{md}} \)

Average residence time of the metal droplets (s)

T

Temperature (K)

T*

Interfacial temperature (K)

uG

Critical gas velocity (m/s)

uj

Axial velocity of the gas jet (m/s)

umd,i

Terminal velocity of metal droplet size class i (m/s)

\( \bar{u}_{\text{md}} \)

Average terminal velocity of the metal droplets (m/s)

uτ

Turbulent shear stress velocity (m/s)

\( \dot{V}_{\text{G}} \)

Volumetric gas flow rate (Nm3/s)

\( \dot{V}_{\text{G}}^{\prime } \)

Modified volumetric gas flow rate (Nm3/s)

x

Molar fraction

X

Cation fraction

Δx

Correction vector

y

Mass fraction

y*

Interfacial mass fraction

x|2

l 2-norm

Greek Symbols

α

Heat-transfer coefficient (W/(m2 K))

α

Interaction energy between cations (J)

β

Mass-transfer coefficient (m/s)

γ

Activity coefficient

γ°

Activity coefficient at infinite dilution

δN

Thickness of the diffusion boundary layer (m)

δPr

Thickness of the thermal boundary layer (m)

ɛ

First-order molar interaction parameter

η

Constant

\( \bar{\eta }_{\text{H}} \)

Average microkinetic efficiency of heat transfer

\( \bar{\eta }_{\text{M}} \)

Average microkinetic efficiency of mass transfer

θ

Inclination angle of each nozzle relative to lance axis (deg)

κ

Constant

λ

Heat conductivity (W/(m·K))

λi

Parameter of the Kronig–Brink solution

μ

Dynamic viscosity (Pa·s)

ν

Stoichiometric coefficient

\( \bar{\nu } \)

Mass-based stoichiometric coefficient

π

Mathematical constant

ρ

Density (kg/m3)

σ

Surface tension (N/m)

ϕ

Volume fraction

Dimensionless Numbers

E

Equilibrium number

FoH

Fourier number for heat transfer

FoM

Fourier number for mass transfer

\( \overline{\text{Gr}} \)

Mean Grashof number

GrH

Grashof number for heat transfer

GrM

Grashof number for mass transfer

NB

Blowing number

NB

Modified blowing number

Nu

Nusselt number

Sc

Schmidt number

Sh

Sherwood number

Pr

Prandtl number

Re

Reynolds number

Subscripts and Superscripts

cav

Cavity

bath

Metal bath

em

Gas–metal–slag emulsion

G

Gas phase

H

Henrian standard state

in

Gas flow into the system

jet

Gas jet

L

Liquid metal phase

md

Metal droplet

out

Gas flow out of the system

plume

Gas plume

R

Raoultian standard state

rel

Relative

S

Slag phase

STP

Standard temperature and pressure according to the DIN 1343 standard[49]: 273.15 K (0 °C) and 101325 Pa

slag

Top slag

(l)

Liquid state

(s)

Solid state

Indices

i

Size class

i

Species

n

Number of species

r

Number of reactions

ψ

Phase

ω

Reaction interface

Notes

Acknowledgments

This research has been conducted within the framework of the DIMECC SIMP research program. The funding for this work from Outokumpu Stainless Oy, the Finnish Funding Agency for Technology and Innovation (TEKES), the Graduate School in Chemical Engineering (GSCE), the Academy of Finland (Projects 258319 and 26495), the Finnish Foundation for Technology Promotion, the Finnish Science Foundation for Economics and Technology, and the Tauno Tönning Foundation is gratefully acknowledged. The first author thanks Professor Herbert Pfeifer for the possibility to conduct part of the research work at RWTH Aachen University. In addition, the authors are grateful to Professor Rauf Hürman Eriç, Kevin Christmann, and Tim Haas for their valuable comments on this manuscript.

References

  1. 1.
    B.V. Patil, A.H. Chan, and R.J. Choulet: in The Making, Shaping and Treating of Steel. Steel Making and Refining, 11th ed., R.J. Fruehan, ed., The AISE Steel Foundation, Pittsburgh, 1998, pp. 715–41.Google Scholar
  2. 2.
    J.-H. Wei, J.-C. Ma, Y.-Y. Fan, N.-W. Yu, S.-L. Yang, S.-H. Xiang and D.-P. Zhu: Ironmaking Steelmaking, 1999, vol. 26, pp. 363-371.CrossRefGoogle Scholar
  3. 3.
    P. Ternstedt, A. Tilliander, P. G. Jönsson and M. Iguchi: ISIJ Int., 2010, vol. 50, pp. 663-667.CrossRefGoogle Scholar
  4. 4.
    J.-H. Wei, H.-L. Zhu, H.-B. Chi and H.-J. Wang: ISIJ Int., 2010, vol. 50, pp. 26-34.CrossRefGoogle Scholar
  5. 5.
    J.-H. Wei, Y. He and G.-M. Shi: Steel Res. Int., 2011, vol. 82, pp. 693-702.CrossRefGoogle Scholar
  6. 6.
    H. Gorges, W. Pulvermacher, W. Rubens and H.-A. Dierstein: Stahl Eisen, 1979, vol. 99, pp. 1310-1312.Google Scholar
  7. 7.
    P. R. Scheller and F.-J. Wahlers: ISIJ Int., 1996, vol. 36, pp. S69-S72.CrossRefGoogle Scholar
  8. 8.
    H.-L. Zhu, J.-H. Wei, G.-M. Shi, J.-H. Shu, Q.-Y. Jiang and H.-B. Chi: Steel Res. Int., 2007, vol. 78, pp. 305-310.CrossRefGoogle Scholar
  9. 9.
    T. Watanabe and T. Tohge: Tetsu-to-Hagané, 1973, vol. 59, pp. 1224-1236.Google Scholar
  10. 10.
    S. Asai and J. Szekely: Metall. Trans., 1974, vol. 5, pp. 651-657.CrossRefGoogle Scholar
  11. 11.
    J. Szekely and S. Asai: Metall. Trans., 1974, vol. 5, pp. 1573-1580.CrossRefGoogle Scholar
  12. 12.
    R. J. Fruehan: Ironmaking Steelmaking, 1976, vol. 3, pp. 153-158.Google Scholar
  13. 13.
    T. Ohno and T. Nishida: Tetsu-to-Hagané, 1977, vol. 63, pp. 2094-2099.CrossRefGoogle Scholar
  14. 14.
    T. Deb Roy and D. G. C. Robertson: Ironmaking Steelmaking, 1978, vol. 5, pp. 198-206.Google Scholar
  15. 15.
    T. Deb Roy, D. G. C. Robertson and J. C. C. Leach: Ironmaking Steelmaking, 1978, vol. 5, pp. 207-210.Google Scholar
  16. 16.
    A. E. Semin, A. P. Pavlenko, T. Andzhum and E. A. Shuklina: Steel USSR, 1983, vol. 13, pp. 95-97.Google Scholar
  17. 17.
    T. Tohge, Y. Fujita, and T. Watanabe: Proceedings of the 4th Process Technology Conference, Chicago, IL, 1984, pp. 129–36.Google Scholar
  18. 18.
    P. Sjöberg: Doctoral thesis, Royal Institute of Technology, Stockholm, Sweden, 1994.Google Scholar
  19. 19.
    J. Reichel and J. Szekely: Iron Steelmaker, 1995, vol. 22, pp. 41-48.Google Scholar
  20. 20.
    M. Görnerup and P. Sjöberg: Ironmaking Steelmaking, 1999, vol. 26, pp. 58-63.Google Scholar
  21. 21.
    J.-H. Wei and D.-P. Zhu: Metall. Mater. Trans. B, 2002, vol. 33, pp. 111-119.CrossRefGoogle Scholar
  22. 22.
    J.-H. Wei and D.-P. Zhu: Metall. Mater. Trans. B, 2002, vol. 33, pp. 121-127.CrossRefGoogle Scholar
  23. 23.
    N. Kikuchi, K. Yamaguchi, Y. Kishimoto, S. Takeuchi and H. Nishikawa: Tetsu-to-Hagané, 2002, vol. 88, pp. 32-39.Google Scholar
  24. 24.
    B. Deo and V. Srivastava: Mater. Manuf. Process., 2003, vol. 18, pp. 401-08.CrossRefGoogle Scholar
  25. 25.
    B. Kleimt, R. Lichterbeck and C. Burkat: Stahl Eisen, 2007, vol. 127, pp. 35-41.Google Scholar
  26. 26.
    G.-M. Shi, J.-H. Wei, H.-L. Zhu, J.-H. Shu, Q.-Y. Jiang and H.-B. Chi: Steel Res. Int., 2007, vol. 78, pp. 311-317.CrossRefGoogle Scholar
  27. 27.
    M. Järvinen, A. Kärnä and T. Fabritius: Steel Res. Int., 2009, vol. 80, pp. 429-436.Google Scholar
  28. 28.
    J.-H. Wei, Y. Cao, H.-L. Zhu and H.-B. Chi: ISIJ Int., 2011, vol. 51, pp. 365-374.CrossRefGoogle Scholar
  29. 29.
    M. P. Järvinen, S. Pisilä, A. Kärnä, T. Ikäheimonen, P. Kupari and T. Fabritius: Steel Res. Int., 2011, vol. 82, pp. 638-649.CrossRefGoogle Scholar
  30. 30.
    S. E. Pisilä, M. P. Järvinen, A. Kärnä, T. Ikäheimonen, T. Fabritius and P. Kupari: Steel Res. Int., 2011, vol. 82, pp. 650-657.CrossRefGoogle Scholar
  31. 31.
    D. R. Swinbourne, T. S. Kho, B. Blanpain, S. Arnout and D. E. Langberg: Miner. Process. Extr. Metall., 2012, vol. 121, pp. 23–31.CrossRefGoogle Scholar
  32. 32.
    N. Å. I. Andersson, A. Tilliander, L. T. I. Jonsson and P. G. Jönsson: Steel Res. Int., 2012, vol. 83, pp. 1039-1052.CrossRefGoogle Scholar
  33. 33.
    N. Å. I. Andersson, A. Tilliander, L. T. I. Jonsson and P. G. Jönsson: Steel Res. Int., 2013, vol. 84, pp. 169-177.CrossRefGoogle Scholar
  34. 34.
    N. Å. I. Andersson, A. Tilliander, L. T. I. Jonsson and P. G. Jönsson: Ironmaking Steelmaking, 2013, vol. 40, pp. 390-397.CrossRefGoogle Scholar
  35. 35.
    N. Å. I. Andersson, A. Tilliander, L. T. I. Jonsson and P. G. Jönsson: Ironmaking Steelmaking, 2013, vol. 40, pp. 551-558.CrossRefGoogle Scholar
  36. 36.
    V.-V. Visuri, M. Järvinen, P. Sulasalmi, E.-P. Heikkinen, J. Savolainen and T. Fabritius: ISIJ Int., 2013, vol. 53, pp. 603-612.CrossRefGoogle Scholar
  37. 37.
    V.-V. Visuri, M. Järvinen, J. Savolainen, P. Sulasalmi, E.-P. Heikkinen and T. Fabritius: ISIJ Int., 2013, vol. 53, pp. 613-621.CrossRefGoogle Scholar
  38. 38.
    R. J. Fruehan: Metall. Trans. B, 1975, vol. 6, pp. 573-578.CrossRefGoogle Scholar
  39. 39.
    Y. Tang, T. Fabritius and J. Härkki: Steel Res. Int., 2004, vol. 75, pp. 373-381.CrossRefGoogle Scholar
  40. 40.
    J. Riipi, T. Fabritius, E.-P. Heikkinen, P. Kupari and A. Kärnä: ISIJ Int., 2009, vol. 49, pp. 1468-1473.CrossRefGoogle Scholar
  41. 41.
    R. Ding, B. Blanpain, P. T. Jones and P. Wollants: Metall. Mater. Trans. B, 2000, vol. 31, pp. 197-206.CrossRefGoogle Scholar
  42. 42.
    Y. Uchida, N. Kikuchi, K. Yamaguchi, Y. Kishimoto, S. Takeuchi, and H. Nishikawa: Proceedings of the 2nd International Conference on Process Development in Iron and Steelmaking, Luleå, Sweden, 2004, pp. 69–78.Google Scholar
  43. 43.
    J.-H. Wei and Y. Li: Steel Res. Int., 2015, vol. 86, pp. 189-211.CrossRefGoogle Scholar
  44. 44.
    V.-V. Visuri, M. Järvinen, A. Kärnä, P. Sulasalmi, E.-P. Heikkinen, P. Kupari, and T. Fabritius: Metall. Mater. Trans. B., DOI: 10.1007/s11663-017-0960-5.
  45. 45.
    M. Järvinen, V.-V. Visuri, S. Pisilä, A. Kärnä, P. Sulasalmi, E.-P. Heikkinen and T. Fabritius: Mater. Sci. Forum, 2013, vol. 762, pp. 236-241.CrossRefGoogle Scholar
  46. 46.
    M. Järvinen, V.-V. Visuri, E.-P. Heikkinen, A. Kärnä, P. Sulasalmi, C. De Blasio and T. Fabritius: ISIJ Int., 2016, vol. 56, pp. 1543-1552.CrossRefGoogle Scholar
  47. 47.
    Z. Song: Doctoral thesis, Royal Institute of Technology, Stockholm, Sweden, 2013.Google Scholar
  48. 48.
    Y. Tang, T. Fabritius and J. Härkki: Appl. Math. Model., 2005, vol. 29, pp. 497-514.CrossRefGoogle Scholar
  49. 49.
    Deutsches Institut für Normung e.V.: DIN 1343, Referenzzustand, Normzustand, Normvolumen; Begriffe und Werte, DIN1343, Referenzzustand, Normzustand, Normvolumen; Begriffe und Werte, 1990.Google Scholar
  50. 50.
    R. Taylor and R. Krishna: Multicomponent Mass Transfer, p. 126, John Wiley & Sons, Inc., New York, NY, USA, 1993.Google Scholar
  51. 51.
    R. I. L. Guthrie: Engineering in Process Metallurgy, p. 282, Clarendon Press, Oxford, United Kingdom, 1989.Google Scholar
  52. 52.
    B. Deo and R. Boom: Fundamentals of Steelmaking Metallurgy, Prentice Hall International, Hertfordshire, 1993, p. 170/176.Google Scholar
  53. 53.
    H.-J. Odenthal, U. Falkenreck, and J. Schlüter: Proceedings of the European Conference on Computational Fluid Dynamics, Egmond aan Zee, The Netherlands, 2006, p. 21.Google Scholar
  54. 54.
    N. Molloy: J. Iron Steel Inst., 1970, vol. 208, pp. 943-950.Google Scholar
  55. 55.
    S. Sabah and G. Brooks: ISIJ Int., 2014, vol. 54, pp. 836-844.CrossRefGoogle Scholar
  56. 56.
    X. Zhou, M. Ersson, L. Zhong, J. Yu and P. Jönsson: Steel Res. Int., 2014, vol. 85, pp. 273-281.CrossRefGoogle Scholar
  57. 57.
    S. K. Sharma, J. W. Hlinka and D. W. Kern: Iron Steelmaker, 1977, vol. 24, pp. 7-18.Google Scholar
  58. 58.
    D. Nakazono, K.-I. Abe, M. Nishida and K. Kurita: ISIJ Int., 2004, vol. 44, pp. 91-99.CrossRefGoogle Scholar
  59. 59.
    F. R. Cheslak, J. A. Nicholls and M. Sichel: J. Fluid Mech., 1969, vol. 36, pp. 55-63.CrossRefGoogle Scholar
  60. 60.
    S. N. Krivoshapko and V. N. Ivanov: Encyclopedia of Analytical Surfaces, p. 110, Springer International Publishing, Cham, Switzerland, 2015.Google Scholar
  61. 61.
    C. K. Lee, J. H. Neilson and A. Gilchrist: Iron Steel Int., 1977, vol. 50, pp. 175-184.Google Scholar
  62. 62.
    C. K. Lee, J. H. Neilson and A. Gilchrist: Ironmaking Steelmaking, 1977, vol. 4, pp. 329-337.Google Scholar
  63. 63.
    S. C. Koria and K. W. Lange: Steel Res., 1987, vol. 58, pp. 421-426.CrossRefGoogle Scholar
  64. 64.
    K.W. Lange and S.C. Koria: Wechselwirkung zwischen Sauerstoffstrahl und Roheisenschmelze beim Sauerstoffaufblasverfahren,, Publ. Wiss. Film. Techn. Wiss./Naturw., Ser. 8, No. 9, Institut für den Wissenschaftlichen Film, Göttingen, Germany, 1983, Film D 1386.Google Scholar
  65. 65.
    J.-H. Wei and L. Zeng: Steel Res. Int., 2012, vol. 83, pp. 1053-1070.CrossRefGoogle Scholar
  66. 66.
    S.C. Koria: Doctoral thesis, RWTH Aachen University, Aachen, Germany, 1981.Google Scholar
  67. 67.
    Subagyo, G.A. Brooks, K.S. Coley, and G.A. Irons: ISIJ Int., 2003, vol. 43, pp. 983–89.Google Scholar
  68. 68.
    E. Schürmann and K. Rosenbach: Arch. Eisenhüttenwes., 1973, vol. 44, pp. 761-768.CrossRefGoogle Scholar
  69. 69.
    W. Rubens: Doctoral thesis, Clausthal University of Technology, Clausthal-Zellerfeld, Germany, 1988.Google Scholar
  70. 70.
    K. Koch, W. Münchberg, H. Zörcher and W. Rubens: Stahl Eisen, 1992, vol. 112, pp. 91-99.Google Scholar
  71. 71.
    T. X. Zhu, K. S. Coley and G. A. Irons: Metall. Mater. Trans. B, 2012, vol. 43, pp. 751-757.CrossRefGoogle Scholar
  72. 72.
    E. T. Turkdogan: Chem. Eng. Sci., 1966, vol. 21, pp. 1133-1144.CrossRefGoogle Scholar
  73. 73.
    N. Standish and Q. L. He: ISIJ Int., 1989, vol. 29, pp. 455-461.CrossRefGoogle Scholar
  74. 74.
    I. Hahn: Doctoral thesis, RWTH Aachen University, Aachen, Germany, 1999.Google Scholar
  75. 75.
    A. Feiterna, D. Huin, F. Oeters, P.-V. Riboud, and J.-L. Roth: Steel Res., 2000, vol. 71, pp. 61-69.CrossRefGoogle Scholar
  76. 76.
    Z. Han and L. Holappa: ISIJ Int., 2003, vol. 43, pp. 292-297.CrossRefGoogle Scholar
  77. 77.
    Z. Han and L. Holappa: ISIJ Int., 2003, vol. 43, pp. 1698-1704.CrossRefGoogle Scholar
  78. 78.
    S. C. Koria and K. W. Lange: Ironmaking Steelmaking, 1983, vol. 10, pp. 160-168.Google Scholar
  79. 79.
    K.-Y. Lee, H.-G. Lee and P. C. Hayes: ISIJ Int., 1998, vol. 38, pp. 1242-1247.CrossRefGoogle Scholar
  80. 80.
    H. W. Meyer, W. F. Porter, G. C. Smith and J. Szekely: J. Met., 1968, vol. 20, pp. 35-42.Google Scholar
  81. 81.
    G. Lindstrand, P.G. Jönsson, and A. Tilliander: Proceedings of the ISIJ-VDEh-Jernkontoret Joint Symposium, Osaka, Japan, 2013, pp. 106–13.Google Scholar
  82. 82.
    A. Nordquist, A. Tilliander, K. Grönlund, G. Runnsjö and P. Jönsson: Ironmaking Steelmaking, 2009, vol. 36, pp. 421-431.CrossRefGoogle Scholar
  83. 83.
    W. Kleppe and F. Oeters: Arch. Eisenhüttenwes., 1977, vol. 48, pp. 139-143.CrossRefGoogle Scholar
  84. 84.
    S. Sabah, M. Alam, G. Brooks, and J. Naser: 4th International Conference on Process Development in Iron and Steelmaking, Luleå, Sweden, 2012, pp. 125–34.Google Scholar
  85. 85.
    S. Sabah and G. Brooks: Metall. Mater. Trans. B, 2015, vol. 46, pp. 863-872.CrossRefGoogle Scholar
  86. 86.
    M. Alam, G. Irons, G. Brooks, A. Fontana and J. Naser: ISIJ Int., 2011, vol. 51, pp. 1439-1447.CrossRefGoogle Scholar
  87. 87.
    S. Sarkar, P. Gupta, S. Basu and N. B. Ballal: Metall. Mater. Trans. B, 2015, vol. 46, pp. 961-976.CrossRefGoogle Scholar
  88. 88.
    B. K. Rout, G. Brooks, Subagyo, M. A. Rhamdhani and Z. Li: Metall. Mater. Trans. B, 2016, vol. 47, pp. 3350-3361.Google Scholar
  89. 89.
    S. C. Koria and K. W. Lange: Metall. Trans. B, 1984, vol. 15, pp. 109-116.CrossRefGoogle Scholar
  90. 90.
    C. Cicutti, M. Valdez, T. Pérez, J. Petroni, A. Gómez, R. Donayo, and L. Ferro: Proceedings of the 6th International Conference on Motel Slags, Fluxes and Salts, Stockholm, Sweden—Helsinki, Finland, 2000, pp. 1–17.Google Scholar
  91. 91.
    S. C. Koria and K. W. Lange: Ironmaking Steelmaking, 1986, vol. 13, pp. 236-240.Google Scholar
  92. 92.
    S.-Y. Kitamura and K. Okohira: Tetsu-to-Hagané, 1990, vol. 76, pp. 199-206.Google Scholar
  93. 93.
    B. Deo, A. Karamcheti, A. Paul, P. Singh and R. P. Chhabra: ISIJ Int., 1996, vol. 36, pp. 658-666.CrossRefGoogle Scholar
  94. 94.
    Q. L. He and N. Standish: ISIJ Int., 1990, vol. 30, pp. 356-361.CrossRefGoogle Scholar
  95. 95.
    G. Brooks, Y. Pan, Subagyo, and K. Coley: Metall. Mater. Trans. B, 2005, vol. 36B, pp. 525–35.Google Scholar
  96. 96.
    R. C. Urquhart and W. G. Davenport: Can. Metall. Q., 1973, vol. 12, pp. 507-516.CrossRefGoogle Scholar
  97. 97.
    Subagyo and G.A. Brooks: ISIJ Int., 2002, vol. 42, pp. 1182–84.Google Scholar
  98. 98.
    H. Gou, G. A. Irons and W.-K. Lu: Metall. Mater. Trans. B, 1996, vol. 27, pp. 195-201.CrossRefGoogle Scholar
  99. 99.
    F. Oeters: Metallurgie der Stahlherstellung, p. 162/174/337, Verlag Stahleisen mbH, Düsseldorf, Germany, 1989.Google Scholar
  100. 100.
    H. Lohe: Fortschr. -Ber. VDI-Z., 1967, Ser. 3, No. 15, pp. 1–59.Google Scholar
  101. 101.
    N. Dogan, G. A. Brooks and M. A. Rhamdhani: ISIJ Int., 2011, vol. 51, pp. 1102-1109.CrossRefGoogle Scholar
  102. 102.
    F. Memoli, C. Mapelli, P. Ravanelli and M. Corbella: ISIJ Int., 2004, vol. 44, pp. 1342-1349.CrossRefGoogle Scholar
  103. 103.
    R. L. Steinberger and R. E. Treybal: AIChE Journal, 1960, vol. 6, pp. 227-232.CrossRefGoogle Scholar
  104. 104.
    K. W. Lange: Arch. Eisenhüttenwes., 1971, vol. 42, pp. 233-241.CrossRefGoogle Scholar
  105. 105.
    R. Kronig and J. C. Brink: Appl. Sci. Res., 1951, vol. 2, pp. 142-154.CrossRefGoogle Scholar
  106. 106.
    D. Colombet, D. Legendre, A. Cockx and P. Guiraud: Int. J. Heat Mass Tran., 2013, vol. 67, pp. 1096-1105.CrossRefGoogle Scholar
  107. 107.
    P. M. Heertjes, W. A. Holve and H. Talsma: Chem. Eng. Sci., 1954, vol. 3, pp. 122-142.CrossRefGoogle Scholar
  108. 108.
    R. Clift, J. R. Grace and M. E. Weber: Bubbles, drops and particles, Academic Press, New York, USA, 1978.Google Scholar
  109. 109.
    P. H. Calderbank: Chem. Engr., 1967, vol. 45, pp. 209-233.Google Scholar
  110. 110.
    G. K. Sigworth and J. F. Elliott: Met. Sci., 1974, vol. 8, pp. 298-310.CrossRefGoogle Scholar
  111. 111.
    Outotec Oyj: HSC Chemistry 8, 2015.Google Scholar
  112. 112.
    A. D. Pelton and C. W. Bale: Metall. Trans. A, 1986, vol. 17, pp. 1211-1215.CrossRefGoogle Scholar
  113. 113.
    A. V. Alpatov and S. N. Paderin: Russ. Metall., 2010, vol. 2010, pp. 557-564.CrossRefGoogle Scholar
  114. 114.
    W.E. Slye and R.J. Fruehan: Proceedings of the 57th Electric Furnace Conference, Pittsburgh, PA, 1999, pp. 401–12.Google Scholar
  115. 115.
    S. Ueno, Y. Waseda, K.T. Jacob and S. Tamaki: Steel Res., 1988, vol. 59, pp. 474-483.CrossRefGoogle Scholar
  116. 116.
    K. V. Malyutin and S. N. Paderin: Russ. Metall., 2007, vol. 2007, pp. 545-551.CrossRefGoogle Scholar
  117. 117.
    J. Szekely and N. J. Themelis: Rate Phenomena in Process Metallurgy, p. 459, John Wiley & Sons, Inc., New York, NY, USA, 1971.Google Scholar
  118. 118.
    K. Nagata, Y. Ono, T. Ejima, and T. Yamamura: in Handbook of Physico-chemical Properties at High Temperatures, Y. Kawai and Y. Shiraishi, eds., The Iron and Steel Institute of Japan, Tokyo, Japan, 1988, pp. 181–204.Google Scholar
  119. 119.
    IAEA: Thermophysical Properties of Materials for Nuclear Engineering: A Tutorial and Collection of Data, International Atomic Energy Agency, Vienna, Austria, 2008, p. 169.Google Scholar
  120. 120.
    B.J. Keene and K.C. Mills: in Verein Deutscher Eisenhüttenleute: Slag Atlas, 2nd ed., Verlag Stahleisen GmbH, Düsseldorf, Germany, 1995, pp. 313–48.Google Scholar
  121. 121.
    C. R. Wilke: J. Chem. Phys., 1950, vol. 18, pp. 517-519.CrossRefGoogle Scholar
  122. 122.
    R. B. Bird, W. E. Stewart and E. N. Lightfoot: Transport Phenomena, p. 23, John Wiley & Sons, Inc., Singapore, 1960.Google Scholar
  123. 123.
    G. H. Geiger and D. R. Poirier: Transport phenomena in metallurgy, p. 11, Addison-Wesley Publishing Company, Reading, MA, USA, 1973.Google Scholar
  124. 124.
    L. D. Cloutman: A Database of Selected Transport Coefficients for Combustion Studies, p. 5, Lawrence Livermore National Laboratory, Livermore, CA, USA, 1993.CrossRefGoogle Scholar
  125. 125.
    L. Forsbacka, L. Holappa, A. Kondratiev and E. Jak: Steel Res. Int., 2007, vol. 78, pp. 676-684.CrossRefGoogle Scholar
  126. 126.
    D. G. Thomas: J. Colloid Sci., 1965, vol. 20, pp. 267-277.CrossRefGoogle Scholar
  127. 127.
    C. R. Wilke and C. Y. Lee: Ind. Eng. Chem., 1955, vol. 47, pp. 1253-1257.CrossRefGoogle Scholar
  128. 128.
    K.C. Mills: in Verein Deutscher Eisenhüttenleute: Slag Atlas, 2nd ed., Verlag Stahleisen GmbH, Düsseldorf, Germany, 1995, pp. 541–56.Google Scholar
  129. 129.
    C.F. Wuppermann: Doctoral thesis, RWTH Aachen University, Aachen, Germany, 2013.Google Scholar
  130. 130.
    O. Wijk: in Principles of Metal Refining, T.A. Engh, ed., Oxford University Press, Oxford, United Kingdom, 1992, pp. 280–301.Google Scholar
  131. 131.
    G. Urbain: Steel Res., 1987, vol. 58, pp. 111-116.CrossRefGoogle Scholar
  132. 132.
    A. Einstein: Ann. Phys., 1906, vol. 19, pp. 289-306.CrossRefGoogle Scholar
  133. 133.
    E. Guth and R. Simha: Kolloid Z., 1936, vol. 74, pp. 266-275.CrossRefGoogle Scholar
  134. 134.
    E. Guth: J. Appl. Phys., 1945, vol. 16, pp. 20-25.CrossRefGoogle Scholar
  135. 135.
    H. M. Smallwood: J. Appl. Phys., 1944, vol. 15, pp. 758-766.CrossRefGoogle Scholar
  136. 136.
    H. C. Brinkman: J. Chem. Phys., 1952, vol. 20, pp. 571.CrossRefGoogle Scholar
  137. 137.
    J. Happel: J. Appl. Phys., 1957, vol. 28, pp. 1288-1292.CrossRefGoogle Scholar
  138. 138.
    T. Kitano, T. Kataoka and T. Shirota: Rheol. Acta, 1981, vol. 20, pp. 207-209.CrossRefGoogle Scholar
  139. 139.
    E. Kreyszig, H. Kreyszig and E. J. Norminton: Advanced Engineering Mathematics Tenth Edition, p. 909, John Wiley & Sons, Inc., Hoboken, NJ, USA, 2011.Google Scholar

Copyright information

© The Minerals, Metals & Materials Society and ASM International 2017

Authors and Affiliations

  • Ville-Valtteri Visuri
    • 1
    Email author
  • Mika Järvinen
    • 2
  • Aki Kärnä
    • 1
  • Petri Sulasalmi
    • 1
  • Eetu-Pekka Heikkinen
    • 1
  • Pentti Kupari
    • 3
  • Timo Fabritius
    • 1
  1. 1.Process Metallurgy Research UnitUniversity of OuluUniversity of OuluFinland
  2. 2.Department of Mechanical EngineeringAalto UniversityAaltoFinland
  3. 3.Outokumpu Stainless OyTorneFinland

Personalised recommendations