High Temperature

, Volume 56, Issue 2, pp 223–228 | Cite as

The Variational Form of the Mathematical Model of a Thermal Explosion in a Solid Body with Temperature-Dependent Thermal Conductivity

  • V. S. Zarubin
  • G. N. Kuvyrkin
  • I. Yu. Savel’eva
Heat and Mass Transfer and Physical Gasdynamics


The variational form of a mathematical model of a thermal explosion has been developed based on a variational formulation of a nonlinear problem of stationary thermal conductivity in a homogeneous solid body. The model takes the temperature dependence of the thermal conductivity of a solid body into account. The presented example of quantitative analysis of the model demonstrates a method for finding the combination of parameters for determining a thermal explosion in a plate with an exponential temperature dependence of the thermal conductivity. At the same time, the analysis allows one to identify the number of steadystate temperature distributions inside a body whose energy release intensifies with a temperature increase.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Kirillov, P.L. and Bogoslovskaya, G.P., Teplomassoobmen v yadernykh energeticheskikh ustanovkakh (Heat and Mass Transfer in Nuclear Power Plants), Moscow: Energoatomizdat, 2000.Google Scholar
  2. 2.
    Frank-Kamenetskii, D.A., Diffuziya i teploperedacha v khimicheskoi kinetike (Diffusion and Heat Transfer in Chemical Kinetics), Moscow: Nauka, 1987.Google Scholar
  3. 3.
    Eliseev, V.N. and Tovstonog, V.A., Teploobmen i teplovye ispytaniya materialov i konstruktsii aerokosmicheskoi tekhniki pri radiatsionnom nagreve (Heat Exchange and Thermal Testing of Materials and Structures of Aerospace Engineering under Radiation Heating), Moscow: Mosk. Gos. Tekh. Univ. im. N.E. Baumana, 2014.Google Scholar
  4. 4.
    Bityukov, V.K., Petrov, V.A., and Smirnov, I.V., High Temp. 2015, vol. 53, no. 1, p. 27.CrossRefGoogle Scholar
  5. 5.
    Formalev, V.F., Kuznetsova, E.L., and Rabinskiy, L.N., High Temp. 2015, vol. 53, no. 4, p. 548.CrossRefGoogle Scholar
  6. 6.
    Izmailova, G.R., Kovaleva, L.A., and Nasyrov, N.M., High Temp. 2016, vol. 54, no. 1, p. 56.CrossRefGoogle Scholar
  7. 7.
    Rubtsov, N.A., Sleptsov, S.D., and Grishin, M.A., High Temp. 2016, vol. 54, no. 2, p. 252.CrossRefGoogle Scholar
  8. 8.
    Zarubin, V.S., Kuvyrkin, G.N., and Savel’eva, I.Yu., J. Eng. Phys. Thermophys., 2014, vol. 87, no. 4, p. 820.CrossRefGoogle Scholar
  9. 9.
    Zarubin, V.S., Kotovich, A.V., and Kuvyrkin, G.N., Izv. Ross. Akad. Nauk, Energ. 2016, no. 1, p. 127.Google Scholar
  10. 10.
    Fizika vzryva (Physics of Explosion), 2vols., Orlenko, L.P., Ed., Moscow: Fizmatlit 2002, vol. 1.Google Scholar
  11. 11.
    Zarubin, V.S., Modelirovanie (Simulation), Moscow: Akademiya, 2013.Google Scholar
  12. 12.
    Zarubin, V.S. and Kuvyrkin, G.N., Mat. Model. Chislennye Metody 2014, no. 1, p. 5.Google Scholar
  13. 13.
    Samarskii, A.A., Galaktionov, V.A., Kurdyumov, S.P., and Mikhailov, A.P., Rezhimy s obostreniem v zadachakh dlya kvazilineinykh parabolicheskikh uravnenii (Regimes with Peaking in Problems for Quasilinear Parabolic Equations), Moscow: Nauka 1987.Google Scholar
  14. 14.
    Zarubin, V.S. and Kuvyrkin, G.N., High Temp. 2003, vol. 41, no. 2, p. 257.CrossRefGoogle Scholar
  15. 15.
    Zarubin, V.S. and Rodikov, A.V., High Temp. 2007, vol. 45, no. 2, p. 243.CrossRefGoogle Scholar
  16. 16.
    Zarubin, V.S. and Selivanov, V.V., Variatsionnye i chislennye metody mekhaniki sploshnoi sredy (Variation and Numerical Methods of Continuum Mechanics), Moscow: Mosk. Gos. Tekh. Univ. im. N.E. Baumana, 1993.Google Scholar
  17. 17.
    Van’ko, V.I., Ermoshina, O.V., and Kuvyrkin, G.N., Variatsionnoe ischislenie i optimal’noe upravlenie (Variational Calculus and Optimal Control), Moscow: Mosk. Gos. Tekh. Univ. im. N.E. Baumana 2001.Google Scholar
  18. 18.
    Vlasova, E.A., Zarubin, V.S., and Kuvyrkin, G.N., Priblizhennye metody matematicheskoi fiziki (Approximate Methods of Mathematical Physics), Moscow: Mosk. Gos. Tekh. Univ. im. N.E. Baumana 2004.Google Scholar
  19. 19.
    Glagolev, K.V. and Morozov, A.N., Fizicheskaya termodinamika (Physical Thermodynamics), Moscow: Mosk. Gos. Tekh. Univ. im. N.E. Baumana, 2007.Google Scholar
  20. 20.
    Zarubin, V.S., Inzhenernye metody resheniya zadach teploprovodnosti (Engineering Methods for Solving Heat Conduction Problems), Moscow: Energoatomizdat, 1983.Google Scholar
  21. 21.
    Zarubin, V.S., Ivanova, E.E., and Kuvyrkin, G.N., Integral’noe ischislenie funktsii odnogo peremennogo (Integral Calculus of One Variable Function), Moscow: Mosk. Gos. Tekh. Univ. im. N.E. Baumana 2006.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • V. S. Zarubin
    • 1
  • G. N. Kuvyrkin
    • 1
  • I. Yu. Savel’eva
    • 1
  1. 1.Bauman Moscow State Technical UniversityMoscowRussia

Personalised recommendations