Metallurgical and Materials Transactions B

, Volume 49, Issue 4, pp 1985–1994 | Cite as

Mathematical Modeling of Decarburization in Levitated Fe-Cr-C Droplets

  • Lei Gao
  • Zhe Shi
  • Yindong Yang
  • Donghui Li
  • Guifang Zhang
  • Alexander McLean
  • Kinnor Chattopadhyay


Using carbon dioxide to replace oxygen as an alternative oxidant gas has proven to be a viable solution in the decarburization process, with potential for industrial applications. In a recent study, the transport phenomena governing the carbon dioxide decarburization process through the use of electromagnetic levitation (EML) was examined. CO2/CO mass transfer was found to be the principal reaction rate control step, as a result gas diffusion has gained significant attention. In the present study, gas diffusion during decarburization process was investigated using computational fluid dynamics (CFD) modeling coupled with chemical reactions. The resulting model was verified through experimental data in a published paper, and employed to provide insights on phenomena typically unobservable through experiments. Based on the results, a new correction of the Frössling equation was presented which better represents the mass transfer phenomena at the metal-gas interface within the range of this research.



Sherwood number


Reynolds number, \( \left( { = \frac{{d_{\text{p}} v\rho_{\text{g}} }}{{\mu_{\text{g}} }}} \right) \)


Schmidt number, \( \left( { = \frac{{\mu_{\text{g}} }}{{\rho_{\text{g}} D_{\text{AB}} }}} \right) \)


Prandtl number, \( \left( { = \frac{{Cp\mu_{\text{g}} }}{k}} \right) \)


Rayleigh number, \( \left( { = Gr^{\prime}{\text{Sc}}} \right) \)

\( Gr^{\prime} \)

Mean Grashof number, (= Grm + GrH(Sc/Pr)0.5)


Grashof number for mass transfer, \( \left( { = \frac{{\rho_{\text{g}} gd_{\text{p}}^{3} (C_{\text{i}} - C_{\text{b}} )}}{{v_{\text{g}}^{2} }}} \right) \)


Grashof number for heat transfer, \( \left( { = \frac{{gd_{\text{p}}^{3} (T_{\text{i}} - T_{\text{b}} )}}{{T_{\text{f}} v_{\text{g}}^{2} }}} \right) \)


Velocity, m s−1


Pressure, Pa


Specific heat, J g−1 K−1


Thermal conductivity, W m−1 K−1


The mass source due to chemical reaction

\( \overline{\overline{\tau }} \)

Stress tensor, Nm−2


Density, kg m−3

\( {\vec{\text{F}}} \)

External body forces, N


Effective conductivity of the gas mixture


Temperature, K


Enthalpy, \( \left( {\int_{298}^{T} {c_{p,j} dT} } \right),\,J \)

\( \vec{J}_{j} \)

Diffusion flux of species j


Source term, includes the heat of chemical reaction and other volumetric heat source


Sensible enthalpy of gas flow, (= ∑jYjhj)


Mass fraction of species j

Di, m

Mass diffusion coefficient for species i in the mixture

DT, i

Thermal diffusion coefficient


Forward rate constant for reaction r


Backward rate constant for reaction r


Molar concentrations of gaseous species


Molar concentrations of site species

\( \eta^{\prime}_{i,g,r} {\text{ and }}\eta^{\prime\prime}_{\"i ,g,r} \)

The rate exponents for the ith gaseous species as reactant and product, respectively

\( \eta^{\prime}_{j,s,r} {\text{ and }}\eta^{\prime\prime}_{j,s,r} \)

The rate exponents for the jth site species as reactant and product, respectively


Pre-exponential factor (consistent units)


Temperature exponent (dimensionless)


Activation energy for the reaction, J kmol−1


Universal gas constant, J kmol−1 K−1


Equilibrium constant for rth reaction

\( \Delta S_{r} \)

Entropy change, J mol−1 K−1

\( \Delta H_{r} \)

Enthalpy change, J


Atmospheric pressure, 101325 Pa


The number of different types of sites


The number of different types of gases


The number of site species of type k


Site density of site type k

\( \nu^{\prime\prime}_{\"i ,r} {\text{ and }}\nu^{\prime}_{i,r} \)

Stoichiometric coefficients of product i and reactant i in reaction r, separately

\( \nu^{\prime\prime}_{j,k,r} {\text{ and }}\nu^{\prime}_{j,k,r} \)

Stoichiometric coefficients of the jth site species of type k in reaction r


Mass fraction of carbon in the droplet


Time, s

\( J_{{{\text{CO}}_{ 2} }} \)

Flux of CO2, mol m−2 s−1


Superficial area of the droplet, m2


Weight of the droplet, kg


Mutual diffusion coefficient in gas phase, m2 s−1

\( X_{{{\text{CO}}_{ 2} }}^{\text{b}} \)

Mole fraction of CO2 in the bulk gas

\( X_{{{\text{CO}}_{ 2} }}^{i} \)

Mole fraction of CO2 at the gas-metal interface


Diameter of the droplet, m


Film temperature, \( \left( { = \frac{{T_{i} + T_{b} }}{2}} \right) \)


Gas–metal interface temperature, K


Bulk gas temperature, K


Mass transfer coefficient, mol m−2 s−1 atm−1


Characteristic length, m


Gas density, kg m−3


Dynamic viscosity, Pa s


Kinematic viscosity, m2 s−1


Hydraulic diameter, m


Radius of the pipe, m


Radius of the droplet, m


Reynolds number, \( \left( { = \frac{{d_{h} v_{h} \rho_{g} }}{{\mu_{g} }}} \right) \)


Velocity near droplet surface, m s−1


Sherwood number due only to natural convection and not forced convection


Coefficient in Eq. [20]


Coefficient in Eq. [21]



Acknowledgments are expressed to the Natural Sciences and Engineering Research Council of Canada for the support of steel-related research at the University of Toronto, the National Natural Science Foundation of China under Grant No. 51664036 and Science Foundation of Yunnan Provincial Department of Education under Grant No. 2016CYH07. The authors also would like to thank the Chinese Scholarship Council for funding, and ANSYS Inc., and SimuTech Group for their support towards the mathematical modeling performed in this study. The comments from Alvin Ma and Dr. Paul Wu are gratefully acknowledged.


  1. 1.
    S. I. Bakhtiyarov and D. A. Siginer: FDMP, 2008, vol. 4, pp. 99-112.Google Scholar
  2. 2.
    A. Seidel, W. Soellner, and C. Stenzel: J. Phys. Conf. Ser., 2011, vol. 327, pp. 1-11.Google Scholar
  3. 3.
    D. Herlach: Appl. Mech. Mater., 2014, vol. 526, pp. 21-27.CrossRefGoogle Scholar
  4. 4.
    C. Yang and J. Gao: J. Cryst. Growth., 2014, vol. 394, pp. 24-27.CrossRefGoogle Scholar
  5. 5.
    A. McLean: Metall. Mater. Trans. B, 2006, vol. 37B, pp. 319-32.CrossRefGoogle Scholar
  6. 6.
    J. Little, M. Van Oosten, and A. McLean: Can. Metall. Q., 1968, vol. 7, pp. 235-46.CrossRefGoogle Scholar
  7. 7.
    P. Kershaw, A. McLean, and R.G. Ward: Can. Metall. Q., 1972, vol. 11, pp. 327-36.CrossRefGoogle Scholar
  8. 8.
    J. G. Dondelinger, D. A. R. Kay, and A. McLean: Metall. Mater. Trans. B, 1971, vol. 2, pp. 3203-08.CrossRefGoogle Scholar
  9. 9.
    J. Siwka: ISIJ Int., 2008, vol. 48, pp. 385-94.CrossRefGoogle Scholar
  10. 10.
    J. Siwka and A. Hutny: Metalurgija., 2009, vol. 48, pp. 23-27.Google Scholar
  11. 11.
    M. Beaudhuin, K. Zaidat, T. Duffar, and M. Lemiti: J. Mater. Sci., 2010, vol. 45, pp. 2218-22.CrossRefGoogle Scholar
  12. 12.
    P. Wu, Y. Yang, M. Barati and A. McLean: High Temperature Material and Process, 2014, Vol. 33, pp. 477-83.Google Scholar
  13. 13.
    P. Wu, Y. Yang, M. Barati, and A. McLean: Metall. Mater. Trans. B, 2014, vol. 45, pp. 2211-21.CrossRefGoogle Scholar
  14. 14.
    W.E. Ranz and W.R. Marshall: Chem. Eng. Prog., 1952, vol. 48, pp. 141-46.Google Scholar
  15. 15.
    R.L. Steinberger and R. E. Treybal: AIChE J., 1960, vol. 6, pp. 227-32.CrossRefGoogle Scholar
  16. 16.
    L. Gao, Z. Shi, D. Li, A. McLean, and K. Chattopadhyay: Metall. Mater. Trans. B, 2016, vol. 47, pp. 1905-15.CrossRefGoogle Scholar
  17. 17.
    L. Gao, Z. Shi, D. Li, Y. Yang, G. Zhang, A. McLean, and K. Chattopadhyay: Metall. Mater. Trans. B, 2016, vol. 47, pp. 67-75.CrossRefGoogle Scholar
  18. 18.
    J. Lee, D.M. Matson, S. Binder, M. Kolbe, D. Herlach, and R.W. Hyers: Metall. Mater. Trans. B, 2013, vol. 45, pp. 1018-23.Google Scholar
  19. 19.
    J. Lee, X. Xiao, D.M. Matson, and R.W. Hyers: Metall. Mater. Trans. B, 2014, vol. 46, pp. 199-207.Google Scholar
  20. 20.
    L. Gao, Z. Shi, D. Li, G. Zhang, Y. Yang, A. McLean, and K. Chattopadhyay: Metall. Mater. Trans. B. 2016, vol. 47, pp. 537-47.CrossRefGoogle Scholar
  21. 21.
    ANSYS Inc. ANSYS Fluent Theory Guide,, 2013.
  22. 22.
    G. R. Belton and D.R. Sain: Metall. Mater. Trans. B, 1976, vol. 7, pp. 235-44.Google Scholar
  23. 23.
    F. P. Incropera, D. P. Devitt, T. L. Bergman, and A. S. Lavine: Fundamentals of Heat and Mass Transfer, 6nd ed., John Wiley & sons Inc., Hoboken, NJ,2006, pp. 381-989.Google Scholar
  24. 24.
    D. J. Zuliani and A. McLean: Can. Metall. Q., 1979, vol. 18, pp. 323-31.CrossRefGoogle Scholar
  25. 25.
    W.G. Mathers, A.J. Madden and E.L. Piret: Fluid Mechanics in Chemical Engineering., 1957, vol. 49, pp. 961-68.Google Scholar

Copyright information

© The Minerals, Metals & Materials Society and ASM International 2018

Authors and Affiliations

  • Lei Gao
    • 1
    • 2
  • Zhe Shi
    • 1
  • Yindong Yang
    • 2
  • Donghui Li
    • 2
  • Guifang Zhang
    • 1
  • Alexander McLean
    • 2
  • Kinnor Chattopadhyay
    • 2
  1. 1.Faculty of Metallurgical and Energy Engineering, State Key Laboratory of Complex Non-Ferrous Metal Resources Clean UtilizationKunming University of Science and TechnologyKunmingChina
  2. 2.Department of Materials Science and Engineering, Process Metallurgy and Research Labs (PMRL)University of TorontoTorontoCanada

Personalised recommendations