Journal of Materials Science

, Volume 30, Issue 20, pp 5043–5048 | Cite as

A mathematical model of the reduction of carbon/alumina powder mixture in a flowing nitrogen stream

  • H. K. Chen
  • C. I. Lin


A physico-chemical model has been formulated to provide a description of the reduction of carbon/alumina powder mixture in a flowing nitrogen stream. Simultaneous differential equations were derived on the basis of this model. These equations were solved by numerical methods. The chemical reaction rate expression, which was determined in the chemical reaction control region, was used after the model had been employed for interpreting the experimental data. The expressions for effective gas diffusivities, which have been left as a fitting parameter for calculation of theoretical predictions, were determined as
$$\begin{gathered} D_{e_{Al_2 OCO} } = 4.29 x 10^{ - 5} exp ( - 25971 J mol^{ - 1} /RT) m^2 s^{ - 1} \hfill \\ D_{e_{Al_2 OCO} } = 2.49 x 10^{ - 5} exp ( - 31512 J mol^{ - 1} /RT) m^2 s^{ - 1} \hfill \\ D_{e_{Al_2 ON_2 } } = 3.13 x 10^{ - 5} exp ( - 29718 J mol^{ - 1} /RT) m^2 s^{ - 1} \hfill \\ \end{gathered} $$

The correlation between the geometrical factor g and half thickness of the sample was determined as g=1/(1+64.7L), and the correlation between the Sherwood number and Reynolds number was found to be Nsh=0.46N Re 0.42 . The reaction occurring between nitrogen, aluminium oxide and carbon was predicted fairly well by this model.


Polymer Aluminium Reynolds Number Theoretical Prediction Control Region 
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\(C_{Al_2 O_3 } ,C_C ,C_{A1N} \)

Concentrations of solids Al2O3, carbon and AlN, respectively, (kgmolm−3)

\(\begin{gathered} D_{e_{Al_2 O_3 CO} } , D_{e_{Al_2 O_3 CO_2 } } , \hfill \\ D_{e_{Al_2 O_3 N_2 } } ,D_{e_{COCO_2 } } , \hfill \\ D_{e_{CO_2 N_2 } } \hfill \\ \end{gathered} \)

Effective normal diffusivities (m2s−1)


Geometric factor (m−1)

ka, kb, kc

Reaction rate constants (m3s−1kg−1mol−1)


Equilibrium constant


Half thickness of solid matrix (m)

\(\begin{gathered} N_{Al_2 O} ,N_{CO} ,N_{CO_2 } , \hfill \\ N_{N_2 } , \hfill \\ \end{gathered} \)

Fluxes of gases Al2O, CO, CO2 and N2, respectively (kgmols−1 m−2)


Reynolds number


Sherwood number


Schmidt number

\(\begin{gathered} P_{Al_2 O} ,P_{CO} ,P_{CO_2 } , \hfill \\ P_{N_2 } , \hfill \\ \end{gathered} \)

Partial pressures of Al2O3, CO, CO2 and N2, respectively (atm)


Total pressure (atm)


Gas constant (kJ kg−1mol−1K−1)


Reaction time (s)


Absolute temperature (K)

\(X_{Al_2 O_3 } ,X_C ,Y_{AlN} \)

Conversions of solids Al2O3, carbon and yield of solid AlN, respectively


Coordinate (m)




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  1. 1.
    N. Kuramoto and H. Taniguchi, US Pat. 4618592 (1986).Google Scholar
  2. 2.
    H. L. Wang, MS thesis, National Cheng Kung University, Tainan, Taiwan (1988).Google Scholar
  3. 3.
    S. Hirai, T. Miwa, M. Ozawa and H. G. Katayama, J. Jpn. Inst. Metals 53 (1989) 1035.CrossRefGoogle Scholar
  4. 4.
    H. Inoue, A. Tsuge and M. Kasori, J. Mater. Sci. 25 (1990) 2359.CrossRefGoogle Scholar
  5. 5.
    H. K. Chen and C. I. Lin, ibid. 29 (1994) 1352.CrossRefGoogle Scholar
  6. 6.
    Y. K. Rao and B. P. Jalan, Metall. Trans. 3 (1972) 2465.CrossRefGoogle Scholar
  7. 7.
    E. A. Mason, R. B. Evans and A. P. Malinauskas, J. Chem. Phys. 46 (1967) 3199.CrossRefGoogle Scholar
  8. 8.
    J. M. Smith, “Chemical Engineering Kinetics”, 3rd Edn (McGraw-Hill, New York, 1981).Google Scholar
  9. 9.
    IMSL “Users Manual, IMSL Math/Library, Fortran Subroutines for Mathematical Applications,” 11 Edn. (IMSL, Houston, Texas, 1989).Google Scholar
  10. 10.
    G. E. Forsythe, M. A. Malcolm and C. B. Molar, “Computer Methods for Mathematical Computations” (Prentice-Hall, Engelwood Cliffs, New Jersey, 1977).Google Scholar
  11. 11.
    H. K. Chen, C. I. Lin and C. Lee, J. Am. Ceram. Soc. 77 (1994) 1753.CrossRefGoogle Scholar
  12. 12.
    G. F. Froment and K. B. Bischoff, “Chemical Reactor Analysis and Design”, 2nd Edn. (Wiley, New York, 1976).Google Scholar
  13. 13.
    R. B. Bird, W. E. Stewart and E. N. Lightfoot, “Transport Phenomena” (Wiley, New York, 1970).Google Scholar
  14. 14.
    H. K. Chen, PhD thesis, National Taiwan Institute of Technology, Taipei, Taiwan (1994).Google Scholar

Copyright information

© Chapman & Hall 1995

Authors and Affiliations

  • H. K. Chen
    • 1
  • C. I. Lin
    • 1
  1. 1.Department of Chemical EngineeringNational Taiwan Institute of TechnologyTaipeiTaiwan

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