THERMAL SCIENCE

International Scientific Journal

Authors of this Paper

External Links

Jordan Y. Hristov

AN ALTERNATIVE INTEGRAL-BALANCE SOLUTIONS TO TRANSIENT DIFFUSION OF HEAT (MASS) BY TIME-FRACTIONAL SEMI-DERIVATIVES AND SEMI-INTEGRALS: FIXED BOUNDARY CONDITIONS

ABSTRACT
A new approach to integral-balance solutions of the diffusion equation of heat (mass) with constant transport properties by applying time-fractional semi-derivatives and semi-integrals of Riemann-Liouville sense has been developed. The time-fractional semiderivatives and semiintegrals replace the surface gradient (temperature) which in the classical Heat-balance integral method (HBIM) of Goodman and the Double-integration method (DIM) should be expressed through the assumed profile. The application of semiderivatives and semiintegrals reduces the approximation errors to levels less than the ones exhibited by the classical HBIM and DIM. The method is exemplified by solutions of Dirichlet and Neumann boundary condition problems.
KEYWORDS
transient conduction, approximate solutions, integral-balance method, time-fractional derivatives, semiderivative, semiintegral
PAPER SUBMITTED: 2015-09-17
PAPER REVISED: 2016-01-06
PAPER ACCEPTED: 2016-01-06
PUBLISHED ONLINE: 2016-01-30
DOI REFERENCE: https://doi.org/10.2298/TSCI150917010H
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2016, VOLUME 20, ISSUE Issue 6, PAGES [1867 - 1878]
REFERENCES
  1. Goodman, T.R, The heat balance integral and its application to problems involving a change of phase, Transactions of ASME, 80 (1958), 1-2, pp.335-342.
  2. Langford , D., The heat balance integral method, Int. J. Heat Mass Transfer, 16 (1973), 12, pp.2424-2428.
  3. Thulin, D., Solutions apporimatives de l'equation de diffusion thermique, Int. J. Heat Mass Transfer, 17 (1974), 4, pp. 497-503.
  4. Wood, A.S., Kutluay, S., A heat balance integral model of the thermistor, Int. J. Heat Mass Transfer, 38 (1995), 10, pp. 1831-1840.
  5. Moghtaderi, B., Novozhilov, V., Fletcher, D., Kent, J.H.,An integral model for transient pyrolysis of solid materials, Fire and Materials, 21, (1997), 1, pp. 7-16.
  6. Staggs, J.K.J., Approximate solutions for the pyrolysis of char forming and filled polymers under thermally thick conditions, Fire and Materials, 24 (2000), 6, pp.305-308.
  7. Kutluay, S., Wood, A.S., Esen, A., A heat balance integral solution of the thermistor problem with a modified electrical conductivity, Applied Mathematical Modelling, 30 (2006), 4, pp. 386-394
  8. Hristov, J.Y., An Inverse Stefan Problem Relevant to Boilover: Heat Balance Integral Solutions and Analysis, Thermal Science, 11 (2007), 2, pp. 141-160.
  9. Wood, A.S., A new look at the heat balance integral method, Appl. Math. Model.. 25, (2001), 10, pp. 815-824.
  10. Myers, J.G. Optimizing the exponent in the heat balance and refined integral methods. Int Comm Heat Mass Transfer 36 (2009),2, pp. 143-147
  11. Mitchell, S. L., Myers, T. G. Application of standard and refined heat balance integral methods to onedimensional Stefan problems. SIAM Review 52 (2010) ,1, pp. 57-86
  12. Mosally, F., Wood, A.S., Al-Fhaid, A., On the convergence of the heat balance integral method, Appl. Math. Model., 29, (2005), 10, pp. 903-912.
  13. Braga, W., Mantelli, M., A new approach for the heat balance integral method applied to heat conduction problems, 38th AIAA Thermophysics Conference, Toronto, Ontario, June 6-9, 2005, paper AIAA-2005- 4686.
  14. Theuns, E., Merci, B., Vierendeeels, Vandevelde, P., Critical evaluation of an integral model for the pyrolysis of charring materials, Fire Safety Journal, 40 (2005), 2, pp.121-140.
  15. Sahu, S.K., Das, P.K., Bhattacharyya, S., A comprehensive analysis of conduction-controlled rewetting by the Heat balance Integral Method, Int. J. Heat Mass Transfer, 49 , (2006), 25-26, pp.4978-4986.
  16. Hristov J (2009) The heat-balance integral method by a parabolic profile with unspecified exponent: Analysis and Benchmark Exercises. Thermal Science 13(2009) , 2, pp.27-48
  17. Roday, A.P., Kazmierczak MJ (2009) Analysis of phase-change in finite slabs subjected to convective boundary conditions: Part I - Melting. Int Rev Chem Eng 1 (2009), 1, pp. 87-99
  18. Roday, A.P., Kazmierczak MJ (2009) Analysis of phase-change in finite slabs subjected to convective boundary conditions: Part II - Freezing. Int Rev Chem Eng 1 (2009), 1, pp. 100-108
  19. Hristov, J. The Heat-Balance Integral: 1. How to Calibrate the Parabolic Profile? Comptes Rendues Mechanique, 340 (2012),7, pp. 485-492
  20. Goodman T.R., Application of Integral Methods to Transient Nonlinear Heat Transfer, Advances in Heat Transfer, T. F. Irvine and J. P. Hartnett, eds., 1 (1964), Academic Press, San Diego, CA, pp. 51-122.
  21. Volkov, V. N., Li-Orlov, V. K., A Refinement of the Integral Method in Solving the Heat Conduction Equation, Heat Transfer Sov. Res., 2 (1970), 2, pp. 41-47.
  22. Sadoun N, Si-Ahmed EK, ColinetP (2006) On the refined integral method for the one-phase Stefan problem with time-dependent boundary conditions. Appl Math Model 30: 531-544
  23. Oldham , K.B., Spanier, J., The Fractional calculus , Academic Press, New York, 1974.
  24. Kulish ,V.V., Lage,J.L. Fractional-diffusion solutions for transient local temperature and heat flux, J. Heat Transfer, 122 (2000) 372-376.
  25. Agrawal ,O.P., Application of Fractional Derivatives in Thermal Analysis of Disk Brakes , Nonlinear Dynamics , 38( 2004) 191-206.
  26. Carslaw,H.S., Jaeger,J.C., Conduction of Heat in Solids. Oxford University Press, London, 1959.
  27. Zener C. , Theory of growth of spherical precipitates from solid solutions. J Appl Phys 20 (1949), 10, pp. 950-953
  28. Hristov J., Integral solutions to transient nonlinear heat (mass) diffusion with a power-law diffusivity: a semi-infinite medium with fixed boundary conditions, Heat Mass Transfer, in press, DOI: 10.1007/s00231-015-1579-2
  29. Mitchell, S.L., Myers, T.G., Application of heat balance integral methods to one-dimensional phase change problems, Int. J. Diff. Eqs., volume 2012, Article ID 187902, doi: 10.1155/2012/187902.
  30. Zien,T.-F., Approximate calculation of transient heat conduction, AIAA J. , 14 (1976), 3, pp. 404-406.
  31. Zien,T.-F., Integral solutions of ablation problems with time-dependent heat flux, AIAA J. , 14 (1978), 12, pp. 1287-1295.
PDF VERSION [DOWNLOAD]
An alternative integral-balance solutions to transient diffusion of heat (mass) by time-fractional semi-derivatives and semi-integrals: Fixed boundary conditions

© 2024 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence