Theoretical analysis of steady nonadiabatic premixed laminar flames

Author:
Stephen B. Margolis

Journal:
Quart. Appl. Math. **38** (1980), 61-89

MSC:
Primary 80A30; Secondary 34E05

DOI:
https://doi.org/10.1090/qam/575833

MathSciNet review:
575833

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Abstract: A one-dimensional, steady, non-adiabatic, premixed laminar flame is assumed semi-infinite with the burner at the origin, and the investigation centers on the asymptotic behavior of the temperature and species mass fractions in the burned region near infinity. After a consideration of the general -species problem, specific results are obtained for the global two-step reaction , , where , , denote reactant, intermediate, and product species, respectively, and , , , are stoichiometric coefficients. Assuming Arrhenius kinetics, it is shown that the classical linearized asymptotic theory is not applicable unless , in which case the approach to burned equilibrium is an exponential decay. Consequently, a nonlinear theory applicable to arbitrary and is presented which shows that in general the asymptotic decay is algebraic. It is further shown that boundedness of the solution at infinity permits the arbitrary specification of only three boundary conditions on the original sixth-order differential system. This result is illustrated by a comprehensive analytical example and the computational implications for the general -species problem are discussed.

**[1]**F. A. Williams,*Combustion theory*(Addison-Wesley, Palo Alto, 1965), Chapter 5 and Appendix B**[2]**J. O. Hirschfelder, C. F. Curtiss, and D. E. Campbell, in*Fourth symposium (International) on combustion*(Williams and Wilkins, Baltimore, 1953), p. 190**[3]**W. B. Bush and F. E. Fendell, Combust. Sci. and Tech.**1**, 421 (1970)**[4]**J. F. Clarke, Combust. Sci. and Tech.**10**, 189 (1975)**[5]**G. F. Carrier, F. E. Fendell, and W. B. Bush, Combust. Sci. and Tech.**18**, 33 (1978)**[6]**D. B. Spalding, Phil. Trans. Roy. Soc. London**A249**1 (1956)**[7]**D. B. Spalding and P. L. Stephenson, Proc. Roy. Soc. London**A324**, 315 (1971)**[8]**J. O. Hirschfelder, C. F. Curtiss, and D. E. Campbell, J. Phys. Chem.**57**, 403 (1953)**[9]**K. A. Wilde, Combust. Flame**18**, 43 (1972)**[10]**G. Dixon-Lewis, Proc. Roy. Soc. London**A317**, 235 (1970)**[11]**J. K. Hale,*Ordinary differential equations*(Wiley-Interscience, New York, 1969), p. 106 MR**0419901****[12]**S. B. Margolis, J. Computational Physics**27**, 410 (1978)**[13]**D. B. Spalding, P. L. Stephenson, and R. G. Taylor, Combust. Flame**17**, 55 (1971)**[14]**L. Biedjian, Combust. Flame**20**, 5 (1973)**[15]**V. S. Berman and Iu. S. Riazantsev, J. Appl. Math. and Mech. (PMM)**37**, 995 (1973)**[16]**A. M. Kanury,*Introduction to combustion phenomena*(Gordon and Breach, New York, 1975), p. 11**[17]**G. F. Carrier, F. E. Fendell, and F. E. Marble, SIAM J. Appl. Math.**28**, 463 (1975) MR**0418766****[18]**F. A. Williams, in*Annual review of fluid mechanics*, Vol.**3**(Annual Reviews, Inc., Palo Alto, 1971), p. 171**[19]**L. Sirovich,*Techniques of asymptotic analysis*(Springer-Verlag, New York, 1971), p. 285 MR**0275034****[20]**H. B. Keller,*Numerical solution of two-point boundary-value problems*(SIAM, Philadelphia, 1976) MR**0433897****[21]**A. C. Hindmarsh, Lawrence Livermore Laboratory Report UCID-30130 (1976)**[22]**M. R. Scott and H. A. Watts, in*Fifth symposium on computers in chemical engineering*(Vysoke Tatry, Czechoslovakia, 1977); also Sandia Laboratories Report SAND77-0091**[23]**R. M. Kendall and J. T. Kelly, Aerotherm TR-75-158 (1975)**[24]**N. K. Madsen and R. F. Sincovec, ACM Trans. on Math. Software, to appear (1980)**[25]**G. H. Markstein, ed.,*Non-steady flame propagation*(MacMillan, New York, 1964)**[26]**S. B. Margolis, Combust. Sci. Tech., to appear (1980)

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DOI:
https://doi.org/10.1090/qam/575833

Article copyright:
© Copyright 1980
American Mathematical Society