International Journal of Thermophysics

, Volume 34, Issue 2, pp 284–305 | Cite as

Inverse Identification of Temperature-Dependent Volumetric Heat Capacity by Neural Networks



An artificial neural network (NN)-based solution of the inverse heat conduction problem of identifying the temperature-dependent volumetric heat capacity function of a solid material is presented in this paper. The inverse problem was defined according to the evaluation of the BICOND thermophysical property measurement method. The volumetric heat capacity versus temperature function is to be determined using the measured transient temperature history of a single sensor. In this study, noiseless and noisy artificial measurements were generated by the numerical solution of the corresponding direct heat conduction problem. The inverse problem was solved by back-propagation and radial basis function type neural networks applying the whole history mapping approach. The numerical tests included the comparison of two different data representations of the network inputs (i.e., temperature vs. time and time vs. temperature) and accuracy analysis of the two network types with noiseless and noisy inputs. Based on the results presented, it can be stated that feed-forward NNs are powerful tools in a non-iterative solution of function estimation inverse heat conduction problems and they are likely to be very effective in evaluation of real measured temperature histories to determine the volumetric heat capacity as an arbitrary function of temperature.


Finite difference method Inverse heat conduction problem  Neural network Volumetric heat capacity 



The work was supported by the Hungarian Scholarship Board. This work was connected to the scientific program of the “Development of quality-oriented and harmonized R+D+I strategy and functional model at BME” project. This project was supported by the New Széchenyi Plan (Project ID: TÁMOP-4.2.1/B-09/1/KMR-2010-0002). The work was supported by the Grant OTKA 82024.


  1. 1.
    C. Yang, Int. J. Heat Mass Transf. 43, 1261 (2000)MATHCrossRefGoogle Scholar
  2. 2.
    J. Zmywaczyk, Arch. Thermodyn. 27/2, 37 (2006)Google Scholar
  3. 3.
    S. Zhao, B. Zhang, S. Du, X. He, Int. J. Thermophys. 30, 2021 (2009)ADSCrossRefGoogle Scholar
  4. 4.
    J. Zmywaczyk, Arch. Thermodyn. 27/3, 39 (2006)Google Scholar
  5. 5.
    V.T. Borukhov, V.I. Timoshpol’skii, J. Eng. Phys. Thermophys. 78, 695 (2005)CrossRefGoogle Scholar
  6. 6.
    C. Huang, J. Yan, Int. J. Heat Mass Transf. 38, 3433 (1995)CrossRefGoogle Scholar
  7. 7.
    J.V. Beck, B. Blackwell, C.R. St Clair Jr., Inverse Heat Conduction (Wiley, New York, 1985)Google Scholar
  8. 8.
    M.N. Özisik, H.R.B. Orlande, Inverse Heat Transfer: Fundamentals and Applications (Taylor & Francis, New York, 2000)Google Scholar
  9. 9.
    S. Vakili, M.S. Gadala, Numer. Heat Transf. B 56, 119 (2009)ADSCrossRefGoogle Scholar
  10. 10.
    N. Tian, Numer. Heat Transf. B 60, 73 (2011)ADSCrossRefGoogle Scholar
  11. 11.
    M. Raudensky, J. Horsky, J. Krejsa, L. Slama, Int. J. Numer. Methods Heat Fluid Flow 6, 19 (1996)MATHCrossRefGoogle Scholar
  12. 12.
    S. Garcia, J. Guynn, E.P. Scott, Numer. Heat Transf. A 33, 149 (1998)ADSCrossRefGoogle Scholar
  13. 13.
    A. Imani, A.A. Ranjbar, M. Esmkhani, Inverse Prob. Sci. Eng. 14, 767 (2006)Google Scholar
  14. 14.
    A. Ranjbar, M. Famouri, A. Imani, Int. J. Numer. Methods Heat Fluid Flow 20, 201 (2010)CrossRefGoogle Scholar
  15. 15.
    Q. Guo, D. Shen, Y. Guo, C.H. Lai, Int. J. Comput. Math. 84, 241 (2007)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    S.W. Phillips, W. Aquino, W.M. Chirdon, J. Eng. Mech. 133, 1341 (2007)CrossRefGoogle Scholar
  17. 17.
    A. Adili, N. Hasni, C. Kerkeni, S. Ben Nashrallah, Int. J. Therm. Sci. 49, 889 (2010)CrossRefGoogle Scholar
  18. 18.
    V. Gorbatov, Yu. Okulovskii, P. Skripov, A. Smotritskiy, A. Starostin, J. Eng. Thermophys. 19, 144 (2010)CrossRefGoogle Scholar
  19. 19.
    B. Czél, Gy. Gróf, Int. J. Thermophys. 30, 1975 (2009)Google Scholar
  20. 20.
    B. Czél, Gy. Gróf, Int. J. Heat Mass Transf. 55, 4254 (2012)Google Scholar
  21. 21.
    B. Czél, Gy. Gróf, Int. J. Thermophys. 33, 1023 (2012)Google Scholar
  22. 22.
    B. Czél, Gy. Gróf, Period. Polytech. Mech. Eng. (2008). doi: 10.3311/
  23. 23.
    B. Czél, Gy. Gróf, L. Kiss. Jpn. J. Appl. Phys. (2011). doi: 10.1143/JJAP.50.11RE05
  24. 24.
    M.T. Hagan, H.B. Demuth, M.H. Beale, Neural Network Design (PWS Publishing, Boston, 1996)Google Scholar
  25. 25.
    S. Deng, Y. Hwang, Int. J. Heat Mass Transf. 49, 4732 (2006)MATHCrossRefGoogle Scholar
  26. 26.
    S. Deng, Y. Hwang, Int. J. Heat Mass Transf. 50, 2089 (2007)MATHCrossRefGoogle Scholar
  27. 27.
    F.T. Mikki, E. Issamoto, J.I. da Luz, P.P.B. de Oliveira, H.F. Campos-Velho, J.D.S. da Silva, A neural network approach in a backward heat conduction problem, in Proceedings of the Brazilian Conference on Neural Networks (Sao Jose dos Campos, Brazil, 1999), pp. 19–24Google Scholar
  28. 28.
    S.S. Sablani, Chem. Eng. Process. 40, 363 (2001)CrossRefGoogle Scholar
  29. 29.
    S.S. Sablani, A. Kacimov, J. Perret, A.S. Mujumdar, A. Campo, Int. J. Heat Mass Transf. 48, 665 (2005)MATHCrossRefGoogle Scholar
  30. 30.
    S. Lecoeuche, G. Mercere, S. Lalot, Inverse Prob. Sci. Eng. 14, 97 (2006)Google Scholar
  31. 31.
    L. Zhang, L. Li, H. Ju, B. Zhu, Energy Convers. Manag. 51, 1898 (2010)CrossRefGoogle Scholar
  32. 32.
    K.A. Woodbury, Application of genetic algorithms and neural networks to the solution of inverse heat conduction problems: a tutorial, inverse problems in engineering: theory and practice, in Proceedings of the 4th International Conference on Inverse Problems in Engineering (Angra dos Reis, Brazil, 2002)Google Scholar
  33. 33.
    L. Boillereoux, C. Cadet, A. Le Bail, J. Food Eng. 57, 17 (2003)CrossRefGoogle Scholar
  34. 34.
    M.T. Sun, C.H. Chang, B.F. Lin, Appl. Therm. Eng. 29, 1818 (2009)CrossRefGoogle Scholar
  35. 35.
    S. Chudzik, Meas. Sci. Technol. 22/7, 1 (2011)Google Scholar
  36. 36.
    C. Balaji, T. Padhi, Int. J. Heat Mass Transf. 53, 5440 (2010)MATHCrossRefGoogle Scholar
  37. 37.
    S. Ghosh, D.K. Pratihar, B. Miati, P.K. Das, Inverse Prob. Sci. Eng. 19, 337 (2011)Google Scholar
  38. 38.
    M. Raudensky, J. Horsky, J. Krejsa, Int. Commun. Heat Mass Transf. 22, 661 (1995)CrossRefGoogle Scholar
  39. 39.
    J. Krejsa, K.A. Woodbury, J.D. Ratliff, M. Raudensky, Inverse Prob. Eng. 7, 197 (1999)Google Scholar
  40. 40.
    E.H. Shiguemori, F.P. Harter, H.F. Campos Velho, J.D.S. da Silva, Tendencias em Mathematica Aplicada e Computacional 3, 189 (2002)MATHGoogle Scholar
  41. 41.
    E.H. Shiguemori, J.D.S. da Silva, H.F.D. Velho, Inverse Prob. Sci. Eng. 12, 317 (2004)Google Scholar
  42. 42.
    L. Kiss, Determination of Thermal Properties. C.Sc. Thesis, Hungarian Academy of Sciences, Budapest, 1983 [in Hungarian]Google Scholar
  43. 43.
    B. Czél, Gy. Gróf, ICHMT Digit. Libr. (2008). doi: 10.1615/ICHMT.2008.CHT.1700
  44. 44.
    B. Czél, Determination of the Thermal Conductivity and the Volumetric Heat Capacity by Genetic Algorithm. Ph.D. Thesis, Budapest University of Technology and Economics, Budapest, 2011 [in Hungarian] (

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Balázs Czél
    • 1
    • 2
  • Keith A. Woodbury
    • 1
  • Gyula Gróf
    • 2
  1. 1.Department of Mechanical EngineeringThe University of AlabamaTuscaloosaUSA
  2. 2.Department of Energy EngineeringBudapest University of Technology and EconomicsBudapestHungary

Personalised recommendations