Fire Technology

, Volume 51, Issue 3, pp 513–521 | Cite as

Superposition with Non-linear Boundary Conditions in Fire Sciences

  • Johan Sjöström
  • Ulf Wickström
Short Communication


Linear response theory is widely used in science and engineering but its use in fire sciences is rare. This communication reviews shortly the Duhamel superposition technique for solving problems in fire sciences where the response of a material maybe assumed linear but the boundary conditions (BC) are non-linear. The method can be used as an alternative to e.g. finite element methods for problems where analytical solutions are not available. Examples include temperature distribution in solids with time-varying and non-linear heat flux boundaries using a simple spreadsheet solution technique. Supplementary material contains an Excel spreadsheet solving problems with non-linear BC.


Linear response Heat transfer Duhamel superposition principle 



Financially supported by the Swedish Fire Research Board—Brandforsk.

Supplementary material (118 kb)
Supplementary material 1 (ZIP 118 kb)


  1. 1.
    Kubo R (1957) Statistical mechanical theory of irreversible processes I. J Phys Soc Jpn 12:570–586CrossRefMathSciNetGoogle Scholar
  2. 2.
    Beck JV (1968) Surface heat flux determination using an integral method. Nucl Eng Des 7:170–178CrossRefGoogle Scholar
  3. 3.
    Zubair SM, Chaudhry MA (1993) Heat conduction in a semi-infinite solid subject to time-dependent surface heat fluxes: an analytical study. Heat Mass Transf 28:357–364Google Scholar
  4. 4.
    Wang ZH, Tan KH (2006) Green’s function solution for transient heat conduction in concrete-filled CHS subjected to fire. Eng Struct 28:1574–1585CrossRefGoogle Scholar
  5. 5.
    Wang ZH, Tan KH (2007) Temperature prediction for multi-dimensional domains in standard fire. Commun Numer Methods Eng 23:1035–1055CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Rocket JA, Milke JA (2008) Conduction of heat in solids. In: SFPE handbook on fire protection engineering, 4th edn, NFPA, QuincyGoogle Scholar
  7. 7.
    Jiji LM (2009) Heat conduction, 3rd edn. Springer, BerlinCrossRefMATHGoogle Scholar
  8. 8.
    Carslaw HS, Jaeger JC (1959) Conduction of heat in solids, 2nd edn. Oxford Press, LondonGoogle Scholar
  9. 9.
    VanSant JH (1980) Conduction heat transfer solutions. Lawrence Livermore National Laboratory, LivermoreGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.SP Technical Research Institute of SwedenBoråsSweden
  2. 2.Luleå Technical UniversityLuleåSweden

Personalised recommendations