- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
International Journal of Differential Equations
Volume 2012 (2012), Article ID 187902, 22 pages
Application of Heat Balance Integral Methods to One-Dimensional Phase Change Problems
1MACSI, Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland
2Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, Bellaterra, 08193 Barcelona, Spain
Received 15 December 2011; Accepted 12 February 2012
Academic Editor: Ebrahim Momoniat
Copyright © 2012 S. L. Mitchell and T. G. Myers. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- J. M. Hill, One-Dimensional Stefan Problems: An Introduction, Longman Scientific & Technical, New York, NY, USA, 1987.
- S. L. Mitchell and T. G. Myers, “Heat balance integral method for one-dimensional finite ablation,” Journal of Thermophysics and Heat Transfer, vol. 22, no. 3, pp. 508–514, 2008.
- T. G. Myers, J. P. F. Charpin, and S. J. Chapman, “The flow and solidification of a thin fluid film on an arbitrary three-dimensional surface,” Physics of Fluids, vol. 14, no. 8, pp. 2788–2803, 2002.
- T.G. Myers, S.L. Mitchell, and G. Muchatibaya, “Unsteady contact melting of a rectangular cross-section phase change material on a flat plate,” Physics of Fluids, vol. 20, no. 10, 2008.
- T. G. Myers, S. L. Mitchell, G. Muchatibaya, and M. Y. Myers, “A cubic heat balance integral method for one-dimensional melting of a finite thickness layer,” International Journal of Heat and Mass Transfer, vol. 50, no. 25-26, pp. 5305–5317, 2007.
- T.R. Goodman, “The heat-balance integral and its application to problems involving a change of phase,” Transactions of the ASME, vol. 80, pp. 335–342, 1958.
- T. R. Goodman, “Application of integral methods to transient nonlinear heat transfer,” Advances in Heat Transfer, vol. 1, pp. 51–122, 1964.
- T. R. Goodman and J. J. Shea, “The melting of finite slabs,” vol. 27, pp. 16–24, 1960.
- K. Pohlhausen, “Zur naherungsweisen Integration der Differentialgleichunger der laminaren Grenzschicht,” Zeitschrift für Angewandte Mathematik und Mechanik, vol. 1, pp. 258–252, 1921.
- H. Schlichting and K. Gersten, Boundary-Layer Theory, Springer, Berlin, Germany, 18th edition, 2000.
- J. Hristov, “The heat-balance integral method by a parabolic profile with unspecified exponent: analysis and benchmark exercises,” Thermal Science, vol. 13, no. 2, pp. 27–48, 2009.
- S. L. Mitchell and T. G. Myers, “Application of standard and refined heat balance integral methods to one-dimensional Stefan problems,” SIAM Review, vol. 52, no. 1, pp. 57–86, 2010.
- A. Antic and J. M. Hill, “The double-diffusivity heat transfer model for grain stores incorporating microwave heating,” Applied Mathematical Modelling, vol. 27, no. 8, pp. 629–647, 2003.
- T. G. Myers, “An approximate solution method for boundary layer flow of a power law fluid over a flat plate,” International Journal of Heat and Mass Transfer, vol. 53, no. 11-12, pp. 2337–2346, 2010.
- T. G. Myers and S. L. Mitchell, “Application of the heat-balance and refined integral methods to the Korteweg-de Vries equation,” Thermal Science, vol. 13, no. 2, pp. 113–119, 2009.
- V. Novozhilov, “Application of heat-balance integral method to conjugate thermal explosion,” Thermal Science, vol. 13, no. 2, pp. 73–80, 2009.
- S. K. Sahu, P. K. Das, and S. Bhattacharyya, “How good is goodman's heat-balance integral method for analyzing the rewetting of hot surfaces?” Thermal Science, vol. 13, no. 2, pp. 97–112, 2009.
- N. Sadoun, E. K. Si-Ahmed, and P. Colinet, “On the refined integral method for the one-phase Stefan problem with time-dependent boundary conditions,” Applied Mathematical Modelling, vol. 30, no. 6, pp. 531–544, 2006.
- S. L. Mitchell and T. G. Myers, “Improving the accuracy of heat balance integral methods applied to thermal problems with time dependent boundary conditions,” International Journal of Heat and Mass Transfer, vol. 53, pp. 3540–3551, 2010.
- T. G. Myers and S. L. Mitchell, “Application of the combined integral method to Stefan problems,” Applied Mathematical Modelling, vol. 35, no. 9, pp. 4281–4294, 2011.
- D. Langford, “The heat balance integral method,” International Journal of Heat and Mass Transfer, vol. 16, no. 12, pp. 2424–2428, 1973.
- T. G. Myers, “Optimizing the exponent in the heat balance and refined integral methods,” International Communications in Heat and Mass Transfer, vol. 36, no. 2, pp. 143–147, 2009.
- T. G. Myers, “Optimal exponent heat balance and refined integral methods applied to Stefan problems,” International Journal of Heat and Mass Transfer, vol. 53, no. 5-6, pp. 1119–1127, 2010.
- E. Guseva-Lozinski, “The mathematical modelling of salinity changes in ice and frozen soils as a result of thermal variations,” Annals of Glaciology, vol. 31, pp. 295–299, 2000.
- Q. T. Pham, “Modelling heat and mass transfer in frozen foods: a review,” International Journal of Refrigeration, vol. 29, no. 6, pp. 876–888, 2006.
- G. I. Poots, Ice and Snow Accretion on Structures, Springer, London, UK, 1996.
- M. Vynnycky, “An asymptotic model for the formation and evolution of air gaps in vertical continuous casting,” Proceedings of the Royal Society A, vol. 465, no. 2105, pp. 1617–1644, 2009.
- W. F. Braga, M. B. H. Mantelli, and J. L. F. Azevedo, “Analytical solution for one-dimensional semi-infinite heat transfer problem with convection boundary condition,” in Proceedings of the 38th AIAA Thermophys Conference (AIAA '05), June 2005.
- S. K. Thomas, R. P. Cassoni, and C. D. MacArthur, “Aircraft anti-icing and de-icing techniques and modeling,” Journal of Aircraft, vol. 33, no. 5, pp. 841–854, 1996.
- T. G. Myers and J. P. F. Charpin, “A mathematical model for atmospheric ice accretion and water flow on a cold surface,” International Journal of Heat and Mass Transfer, vol. 47, no. 25, pp. 5483–5500, 2004.
- T. G. Myers and D. W. Hammond, “Ice and water film growth from incoming supercooled droplets,” International Journal of Heat and Mass Transfer, vol. 42, no. 12, pp. 2233–2242, 1999.
- H. E. Huppert, “Phase changes following the initiation of a hot turbulent flow over a cold solid surface,” Journal of Fluid Mechanics, vol. 198, pp. 293–319, 1989.
- J. R. King and D. S. Riley, “Asymptotic solutions to the Stefan problem with a constant heat source at the moving boundary,” The Royal Society of London, vol. 456, no. 1997, pp. 1163–1174, 2000.
- S. L. Mitchell, “Applying the combined integral method to one-dimensional ablation,” Applied Mathematical Modelling, vol. 36, no. 1, pp. 127–138, 2012.
- S. L. Mitchell and M. Vynnycky, “Finite-difference methods with increased accuracy and correct initialization for one-dimensional Stefan problems,” Applied Mathematics and Computation, vol. 215, no. 4, pp. 1609–1621, 2009.