International Scientific Journal

Authors of this Paper

External Links


The heat-balance integral method of Goodman has been thoroughly analyzed in the case of a parabolic profile with unspecified exponent depending on the boundary condition imposed. That the classical Goodman's boundary conditions defining the time-dependent coefficients of the prescribed temperature profile do not work efficiently at the front of the thermal layers if the specific parabolic profile at issue is employed. Additional constraints based on physical assumption enhance the heat-balance integral method and form a robust algorithm defining the parabola exponent. The method has been compared by results provided by the Veinik's method that is by far different from the Goodman's idea but also assume formation of thermal layer penetrating the heat body. The method has been demonstrated through detailed solutions of 4 1-D heat-conduction problems in Cartesian co-ordinates including a spherical problem (through change of variables) and over-specified boundary condition at the face of the thermal layer.
PAPER REVISED: 2008-08-20
PAPER ACCEPTED: 2008-09-10
CITATION EXPORT: view in browser or download as text file
  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., Approximative Solutions of the Equation for Thermal Diffusion (in French), 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. Leong, K. C., Lu, G. Q., Rudolph, V. , An Upper Bound Solution for Coating Thickness of Cylinders in a Fluidized Bed, Chemical Engineering Science, 54 (1999), 8, pp. 1145-1149
  7. Staggs, J. K. J., A Theory for Quasi-Steady Single-Step Thermal Degradation of Polymers, Fire and Materials, 22 (1998), 3, pp. 109-118
  8. 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
  9. 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
  10. Moghtaderi, B., The State-of-the-Art in Pyrolysis Modelling of Lignocellulosic Solid Fuels, Fire and Materials, 30 (2006),1, pp. 1-34
  11. Ren, H. S., Application of the Heat-Balance Integral to an Inverse Stefan Problem, Int. J. Thermal Sciences, 46 (2007), 2, pp. 118-127
  12. Hristov, J. Y., An Inverse Stefan Problem Relevant to Boilover: Heat Balance Integral Solutions and Analysis, Thermal Science, 11 (2007), 2, pp. 141-160
  13. Wood, A. S., A New Look at the Heat Balance Integral Method, Applied Mathematical Modelling, 25 (2001), 10, pp. 815-824
  14. Mosally, F., Wood, A. S., Al-Fhaid, A., An Exponential Heat Balance Integral Method, Applied Mathematics and Computation, 130 (2002), 1, pp. 87-100
  15. Mosally, F., Wood, A. S., Al-Fhaid, A., On the Convergence of the Heat Balance Integral Method, Applied Mathematical Modelling, 29 (2005), 10, pp. 903-912
  16. Gomes, F. A. A., Silva, J. B. C., Diniz, A. J., Radiation Heat Transfer with Ablation in a Finite Plate, Engenharia Termica (Thermal Engineering), 4 (2005), 2, pp. 190-196
  17. Braga, W., Mantelli, M., A New Approach for the Heat Balance Integral Method Applied to Heat Conduction Problems, 38th AIAA Thermophysics Conference, Toronto, Ont., Canada, June 6-9, 2005, paper AIAA-2005-4686
  18. Theuns, E., Merci, B., Vierendeels, J., Vandevelde, P., Critical Evaluation of an Integral Model for the Pyrolysis of Charring Materials, Fire Safety Journal, 40 (2005), 2, pp. 121-140
  19. 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
  20. Sanzo, D., An Approximate Analytical Solution of the Problem of Melting and Evaporation During Disruption in Magnetic Fusion Reactors, Nuclear Engineering and Design/Fusion, 4 (1987), 1, pp. 191-210
  21. Goodman, T. R., Application of Integral Methods to Transient Nonlinear Heat Transfer, in: Advances in Heat Transfer (Eds.T. F. Irvine, Jr. J. P. Hartnett), Vol. 1, 1964, Academic Press, San Diego, Cal., USA, pp. 51-122
  22. Mokrushin, S. A., Integral Method to Solution of Heat Transfer Conduction Problem (in Russian), in: Hydrodynamics and Heat Transfer, (Eds. V. P. Skripov, A. G. Sheikman), Ural Center of RAS, Sverdlovsk (Yekaterinburg), Russia, 1972, pp. 3-6
  23. Mokrushin, S.A., Transient Temperature Fields in a Plate Heated by Radiation and Convection (in Russian), in: Hydrodynamics and Heat Transfer (Eds. V. P. Skripov, A. G. Sheikman), Ural Center of RAS, Sverdlovsk (Yekaterinburg), Russia, 1972, pp. 12-15
  24. Braga, W. F., Thermophysics and Characterization of Materials at High Temperatures (in Portuguese), M. Sc. thesis, Federal University of Santa Catarina, Florianopolis, Santa Catarina, Brazil, 2006
  25. Pasichnyi, V. V., Uryukov, B. A., Theoretical Models of Surface Heat Treatment of Products in Solar Furnaces, 1. Uniform Radiant-Energy Flux, Journal of Engineering Physics and Thermophysics, 75 (2002), 6, pp. 1429-1436
  26. Pasichnyi, V. V., Uryukov, B. A. Theoretical Models of Surface Heat Treatment of Products in Solar Furnaces, 2. Nonuniform Radiant-Energy Flux, Journal of Engineering Physics and Thermophysics, 75 (2002), 6, pp. 1437-1444
  27. Dombrovsky, L. A, Sazhin, S. S., A Parabolic Temperature Profile Model for Heating of Droplets. ASME J Heat Transfer, 125 (2003), 3, 125:535-7
  28. Dombrovsky, L. A, Sazhin, S. S., Absorption of Thermal Radiation in a Semi-Transparent Spherical Droplet: A Simplified Model, Int. J Heat Fluid Flow, 24 (2003), 6, pp. 919-27
  29. Antic, A., Hill, J. M., A Mathematical Model for Heat Transfer in Grain Store Microclimates, Australian & New Zealand Industrial and Applied Mathematics Journal (ANZIAM J), 42 (2000), 1, pp. C117-C133
  30. Tarzia, D. A., A Variant of the Heat Balance Integral Method and a New Proof of the Exponentially Fast Asymptotic Behaviour of the Solutions in the Heat Conduction Problems with Absorption, International Journal of Engineering Science, 28 (1990), 12, pp. 1253-1259
  31. Ozisik, M. N., Boundary Value Problems of Heat Conduction, Dover Publishing, Mineola, N. Y., USA, 1989, pp. 301-346
  32. Ozisik, M. N., Heat Conduction, Wiley, New York, USA,1993, pp. 325-371
  33. Spearpoint, M. J., Quintiere, J. G., Predicting the Burning of Wood Using an Integral Model, Combustion and Flame, 123 (2000), 3, pp. 308-324
  34. Spearpoint, M. J., Quintiere, J. G., Predicting Ignition of Wood in the Cone Calorimeter Using an Integral Model-Effect of Species, Grain Orientation and Heat Flux, Fire Safety Journal, 36 ( 2001), 4, pp. 391-415
  35. Ho, C. D., Yeh, H. M., Wang, W. P., Thermal Characteristic of Ice under Constant Heat Flux and Melt Removal, Heat Transfer Engineering, 23 (2001), 5, pp. 36-44
  36. Zhu, N., Vafai, K., Analytical Modelling of the Startup Characteristics of Asymmetrically Flat-Plate and Disk-Shaped Heat Pipes, Int. J. Heat Mass Transfer, 41 (1998), 17, pp. 2619-2637
  37. Jayaram, B., Strieder, W., An Analysis of Substrate Heat Loses in Stefan's Problems with a Constant Flux, Int. J. Heat Mass Transfer, 26 (1983), 5, pp. 786-790
  38. Lunardini, V. J. , Cylindrical Phase Change Approximation with Effective Thermal Diffusivity, Cold Regions Science and Technology, 4 (1981), 2, pp. 147-154
  39. Harrach, R. J., Analytical Solutions for Laser Heating and Burnthrough of Opaque Solid Slabs, Journal of Applied Physics, 48 (1977), 6, pp. 2370-2383
  40. Frangi, A., Fontana, M., Charring Rates and Temperature Profiles of Wood Sections, Fire and Materials, 27 (2003), 1, pp. 91-102
  41. Janssen, M., White, .H. Temperature Profiles in Wood Members Exposed to Fire, Fire and Materials, 18 (1994), 2, pp. 263-265
  42. Reszka, P., Torero, J. L., In-Depth Temperature Measurements of Timber in Fire, Proceedings, 4th International Workshop on Structures in Fire, Aveiro, Portugal, 2006, pp. 921-930
  43. Reszka, P., Torero, J. L., In-Depth Temperature Measurements in Wood Exposed to Intense Radiant Energy, Experimental Thermal and Fluid Science, 32 (2008), 7, pp. 1405-1411
  44. Carslaw, H. S., Jaeger, J. C., Conduction of Heat in Solids, 2nd ed., Oxford Science Publications, Oxford University Press, Oxford, UK, 1992 (a photocopy edition of 2nd ed., 1959)
  45. Veinik, A. I., Approximate Methods in Heat Conduction (in Russian), Gosenergoizdat, Moscow, 1959
  46. Rosenband, V. I., Barzikin, V. V., Application of an Integral Method to Calculation of Heterogeneous Ignition Characteristics (in Russian), Fizika Gorenia i Vzriva (now Combustion and Shock Waves), 10 (1974), 1, pp. 52-56
  47. Mioura, K., Method of Thermal Layer Approximation: an Approximate Method for Transient Thermal and Thermo-Elastic Problems (in Japanese), Reports of Japan Aerospace Exploration Agency, 3 (1962), 2, pp. 143-177
  48. Hristov, J. Y., Planas, E., Arnaldos, J, Casal, J., Accidental Burning of a Fuel Layer on a Waterbed: A Scale Analysis Study of the Heat Transfer Models Predicting the Pre-Boilover Time and Scaling to Published Data, Int. J. Thermal Sciences, 43 (2004), 3, pp. 221-239
  49. Hristov, J. Y., Research Note on a Parabolic Heat-Balance Integral Method with Unspecified Exponent: an Entropy Generation Approach in Optimal Profile Determination, Thermal Science, 13 (2009), 2, pp. 49-61 (present issue)

© 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