THERMAL SCIENCE

International Scientific Journal

ON THE GOODMAN HEAT-BALANCE INTEGRAL METHOD FOR STEFAN LIKE-PROBLEMS: FURTHER CONSIDERATIONS AND REFINEMENTS

ABSTRACT
Since the pioneering studies of Goodman on the application of the integral method to transient non-linear heat diffusion, much attention has been devoted nowadays to what is called heat balance integral method. The present paper considers this technique fifty years later. The genesis and earlier developments, when applied to Stefan like-problems, are reported hereafter. Its simplicity and efficiency are demonstrated. Some numerical results obtained using methods developed on the basis of the heat balance integral are compared. Furthermore, for problems including temperature profile behavior, such as Stefan problem with forcing term (source or sink) this technique gives highly precise results and may, in some cases, lead to exact solutions.
KEYWORDS
PAPER SUBMITTED: 2008-04-04
PAPER REVISED: 2008-05-15
PAPER ACCEPTED: 2008-06-15
DOI REFERENCE: https://doi.org/10.2298/TSCI0902081S
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2009, VOLUME 13, ISSUE 2, PAGES [81 - 96]
REFERENCES
  1. Goodman, T. R., The Heat-Balance Integral and It's Application to Problems Involving a Change of Phase, Trans. ASME, 80 (1958), 2, pp. 335-345
  2. Goodman, T. R., This Week's Citation Classic, Current Contents, 23 (1983), 1, p.18
  3. Sucec, J., Extension of Modified Integral Method to Boundary Conditions of Prescribed Surface Heat Flux, Int. J. Heat Mass Transfer, 22 (1979), 5, pp. 771-774
  4. Zien, T. F., Approximate Analysis of Heat Transfer in Transpired Boundary Layers with Effects of Prandtl Number, Int. J. Heat Mass Transfer, 19 (1976), 5, pp. 513-521
  5. Kutluay, S., Wood, A. S., Esen, A., A Heat Balance Integral Solution the Thermistor Problem with a Modified Electrical Conductivity, Appl. Math. Modelling, 30 (2006), 4, pp. 386-394
  6. Wood, A. S., Kutluay, S., A Heat Balance Integral Model for Thermistor, Int. J. Heat Mass Transfer, 38 (1995), 10, pp. 1831-1840
  7. 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
  8. Hristov, J., An Inverse Stefan Problem Relevant to Boilover: Heat Balance Integral Solutions and Analysis, Thermal Science, 11 (2007), 2, pp. 141-160
  9. El-Genk, M. S., Improvements to the Solution of Stefan-Like Freezing and Melting Problems, with Application to LMFBR Safety Analysis, Approximate Analysis of Heat Transfer in Transpired Boundary Layers with Effects of Prandtl Number, Ph. D. dissertation, University of New Mexico, Albuquerque, N. Mex., USA, 1978
  10. Myers, T. G., et al., A Cubic Heat Balance Integral Method for One-Dimensional Melting of a Finite Thickness Layer, Int. J. Heat Mass Transfer., 50 (2007), 25-26, pp. 5305-5317
  11. Mitchell, S. L., Myers, T. G., Approximate Methods for One-Dimensional Solidification from an Incoming Fluid, Appl. Math. Comput., (2008), doi:10.1016/j.amc.2008.02.031
  12. Poots, G., On the Application of Integral-Methods to the Solution of Problems Involving the Solidification of Liquids Initially at Fusion Temperature, Int. J. Heat Mass Transfer, 5 (1962), 6, pp. 525-531
  13. Caldwell, J., Kwan, Y. Y., Starting Solutions for the Boundary Immobilisation Method, Commun. Numer. Meth. Engng, 21 (2005), 6, pp. 289-295
  14. Caldwell, J., Chan, C. C., Spherical Solidification by Enthalpy Method and the Heat Balance Integral Method, Appl. Math. Modelling, 24 (2000), 1, pp. 45-53
  15. Bell, G. E., Solidification of a Liquid about Cylindrical Pipe, Int. J. Heat Mass Transfer, 22 (1979), 12, pp. 1681-1686
  16. Ren, H. S., Application of the Heat Balance Integral to an Inverse Stefan Problem, Int. J. Thermal Sciences, 46 (2007), 2, pp. 118-127
  17. Poots, G., An Approximate Treatment of a Heat Conduction Problem Involving a Two-Dimensional Front, Int. J. Heat Mass Transfer, 5 (1962), 5, pp. 339-348
  18. Riley, D. S., Duck, P. W., Application of the Heat-Balance Integral Method to the Freezing of a Cuboid, Int. J. Heat Mass Transfer, 19 (1976), 3, pp. 294-296
  19. Goodman, T. R., The Heat Balance Integral - Further Considerations and Refinements, Trans. ASME J. of Heat Transfer, 83 (1961), 1, pp. 83-88
  20. 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
  21. Olguin, M. C., et al., Behaviour of the Solution of a Stefan Problem with Changing Thermal Coefficients of the Substance, Appl. Math. Comput., 190 (2007), 1, pp. 765-780
  22. Goodman, T. R., Application of Integral Methods to Transient Nonlinear Heat Transfer (Eds. T. F. Irvine Jr. & J. P. Hartnett ), in: Advances in Heat Transfer, Academic Press, New York, USA, 1964, Vol. I, pp. 51-122
  23. Mosally, F., Wood, A. S., Al-Fhaid, A., An Exponential Heat Balance Integral Method, Appl. Math. Comput.,130 (2002), 1, pp. 87-100
  24. Vujanovic, B., Djukic, Dj., On One Variational Principle of Hamilton's Type for Nonlinear Heat Transfer Problem, Int. J. Heat Mass Transfer, 15 (1972), 5, pp. 1111-1123
  25. Langford, D., The Heat Balance Integral Method, Int. J. Heat Mass Transfer, 16 (1973), 12, pp. 2424-2428
  26. Sadoun, N., Si-Ahmed, E. K., On the Double Integral Method for Solving Stefan Like-Problems, Proceedings, 1st International Thermal and Energy Congress, Marrakesh, Morocco, 1993, Vol. 1, pp. 87-91
  27. Hamill, T. D., Bankoff, S. G., Maximum and Minimum Bounds of Freezing-Melting Rates with Time Dependent Boundary Conditions, A. I. Ch. E. Journal, 9 (1963), 6, pp. 741-744
  28. Elmas, M., On the Solidification of the Warm Liquid Flowing over a Cold Wall, Int. J. Heat Mass Transfer, 13 (1970), 6, pp. 1060-1062
  29. Mennig, J., Özisik, M. N., Coupled Integral Approach for Solving Melting and Solidification, Int. J. Heat Mass Transfer, 28 (1985), 8, pp. 1481-1485
  30. Bell, G. E., A Refinement of the Heat Balance Integral Method Applied to Melting Problem, Int. J. Heat Mass Transfer, 21 (1978), 11, pp. 1357-1362
  31. Noble, B., Heat Balance Methods in Melting Problems, in: Moving Boundary Problems in Heat Flow and Diffusion (Eds. J. R. Ockendon, W. R. Hodgkins), Clarendon Press, Oxford, UK, 1975, pp. 208-209
  32. Stefan, J., On the Theory of the Ice, in Particular about the Ice in Polar Seas (in German), Annalen der Physik und Chemie, 42 (1891), 2, pp. 269-286
  33. Carslaw, H. S., Jaeger, J. C., Conduction of Heat in Solids, Oxford University Press, Oxford, UK, 1956
  34. Crank, J., Free and Moving Boundary Problems, Clarendon Press, Oxford, UK, 1984
  35. von Kármán Th., On the Laminar and Turbulent Friction (in German), Zs. F. Angew. Math. U. Mech. Bd., 1 (1921), 4, pp. 233-253
  36. Pohlhausen, K., On the Approximate Integration of the Laminar Boundary Layer Differential Equation (in German), Zs. F. Angew. Math. U. Mech. Bd., 1 (1921), 4, pp. 252-268
  37. Fox, L., What Are the Best Numerical Methods?, in: Moving Boundary Problems in Heat Flow and Diffusion (Eds. J. R. Ockendon, W. R. Hodgkins), Clarendon Press, Oxford, UK, 1975, pp. 210-241
  38. Wood, A. S., A New Look at the Heat Balance Integral Method, Appl. Math. Modelling, 25 (2001), 10, pp. 815-824
  39. Tani, I., On the Solution of the Laminar Boundary Layer Equations, Jour. Camb. Phil. Soc., 50 (1954), 3, pp. 454-465
  40. Sadoun, N., Si-Ahmed, E. K., A New Analytical Expression for the Freezing Constant in the Stefan Problem with Initial Superheat, in: Numerical Methods in Thermal Problems (Eds. R. W. Lewis, P. Durbetaki), Pineridge Press, Swansea, UK, 1995, Vol. 2, pp. 843-854
  41. Sadoun, N., Si-Ahmed, E. K., Colinet, P., On the Refined Heat Balance Integral Method for the One-Phase Stefan Problem with Time-Dependent Boundary Conditions, Appl. Math. Modelling, 30 (2006), 6, pp. 531-544
  42. El-Genk, M. S., Cronenberg, A. W., Some Improvements to the Solution of Stefan Like-Problems, Int. J. Heat Mass Transfer, 22 (1979), 1, pp. 167-170
  43. Bell, G. E., Accurate Solution of One-Dimensional Melting Problems by the Heat Balance Integral Method, in: Numerical Methods in Thermal Problems (Eds. R. W. Lewis, K. Morgan), Pineridge Press, Swansea, UK, 1979, pp. 196-203
  44. Mosally, F., Wood, A. S., Al-Fhaid, A., On the Convergence of the Heat Balance Integral Method, Appl. Math. Modelling, 29 (2005), 10, pp. 903-9012
  45. Bell, G. E., Abbas, S. K., A Convergence Properties of Heat Balance Integral Method, Num. Heat Transfer, Part A: Application, 8 (1985), 3, pp. 373-382
  46. Sadoun, N., Si-Ahmed, E. K., Legrand, J., A Refined Exponential Heat Balance Integral Method for One-Phase Stefan Problem, Proceedings, 2nd International Conference on Thermal Engineering: Theory and Applications (Eds. S. Chacha et al.), Al-Ain, United Arab Emirates, 2006, pp. 528-532
  47. Fasano, A., Primicerio, M., Free Boundary Problems for Nonlinear Parabolic Equations with Nonlinear Free Boundary Conditions, J. Math. Anal. Appl., 72 (1979), 1, pp. 247-273

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