The integral method of boundary characteristics is considered as applied to the solution of the Stefan problem with a Dirichlet condition. On the basis of the multiple integration of the heat-conduction equation, a sequence of identical equalities with boundary characteristics in the form of n-fold integrals of the surface temperature has been obtained. It is shown that, in the case where the temperature profile is defined by an exponential polynomial and the Stefan condition is not fulfilled at a moving interphase boundary, the accuracy of solving the Stefan problem with a Dirichlet condition by the integral method of boundary characteristics is higher by several orders of magnitude than the accuracy of solving this problem by other known approximate methods and that the solutions of the indicated problem with the use of the fourth–sixth degree polynomials on the basis of the integral method of boundary characteristics are exact in essence. This method surpasses the known numerical methods by many orders of magnitude in the accuracy of calculating the position of the interphase boundary and is approximately equal to them in the accuracy of calculating the temperature profile.
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References
J. Crank, The Mathematics of Diffusion, Clarendon Press, Oxford (1975), pp.129–135.
J. M. Hill, One-Dimensional Stefan Problems: An Introduction, Chapman & Hall, London (1989).
V. Alexiades and A. D. Solomon, Mathematical Modeling of Melting and Freezing Processes, Hemisphere–Taylor & Francis, Washington (1983).
J. M. Chadam and H. Rasmussen, Free Boundary Problems Involving Solids, Pitman Research Notes in Mathematics Series 281, Longman, Essex (1993).
S. C. Gupta, The Classical Stefan Problem. Basic Concepts, Modelling and Analysis, Elsevier, Amsterdam (2003).
V. J. Lunardini, Heat Transfer with Freezing and Thawing, Elsevier, London (1991).
D. A. Tarzia, Explicit and approximated solutions for heat and mass transfer problems with a moving interface, Adv. Topics Mass Transf., Chapter 20, InTech Open Access Publisher, Rijeka (2011), pp. 439–484. Available from: http://www.intechopen.com/articles/show/title/explicit-and-approximated-solutions-for-heat-and-mass-transfer.
Cannon J. R. The One-Dimensional Heat Equation, Addison–Wesley, Menlo Park, CA (1984).
T. R. Goodman, The heat-balance integral and its application to problems involving a change of phase, Trans. ASME, 80, 335–342 (1958).
T. R. Goodman, Application of integral methods to transient nonlinear heat transfer, in: T. F. Irvine and J. P. Hartnett (Eds.), Advances in Heat Transfer, Vol. 1, Academic Press, New York (1964), pp. 51–122.
B. Noble, Heat balance methods in melting problems, in: J. R. Ockendon and W. R. Hodgkins (Eds.), Moving Boundary Problems in Heat Flow and Diffusion, Clarendon, Oxford (1975).
M. C. Olguin, M. A. Medina, M. C. Sanziel, and D. A. Tarzia, Behavior of the solution of a Stefan problem by changing thermal coefficients of the substance, Appl. Math. Comput., 190, 765–780 (2007).
G. Poots, On the application of integral methods to the solution of problems involving the solidification of liquids initially at fusion temperature, Int. J. Heat Mass Transf., 5, No. 6, 525–531 (1962).
S. L. Mitchell and T. G. Myers, Heat balance integral method for one-dimensional finite ablation, J. Thermophys. Heat Transf., 22, No. 3, 508–514 (2008). DOI: 10.2514/1.31755.
S. L. Mitchell and T. G. Myers, Application of heat balance integral methods to one-dimensional phase change problems, Int. J. Differ. Equ., 2012, Article ID 187902 (2012). DOI:10.1155/2012/187902.
E. Case and J. Taysch, An integral equation method for spherical Stefan problems, Appl. Math. Comput., 218, No. 23, 11451–11460 (2012).
S. L. Mitchell and T. G. Myers, Application of standard and refined heat balance integral methods to one-dimensional Stefan problems, SIAM Rev., 52, No. 1, 57–86 (2010).
S. L. Mitchell and T. G. Myers, Improving the accuracy of heat balance integral methods applied to thermal problems with time dependent boundary conditions, Int. J. Heat Mass Transf., 53, No. 17, 3540–3551 (2010).
T. G. Myers, Optimal exponent heat balance and refined integral methods applied to Stefan problems, Int. J. Heat Mass Transf., 53, Nos. 5–6, 1119–1127 (2010).
T. G. Myers, Optimizing the exponent in the heat balance and refined integral methods, Int. Commun. Heat Mass Transf., 36, No. 2, 143–147 (2009).
N. Sadoun, E. K. Si-Ahmed, P. Colinet, and J. Legrand, On the Goodman heat-balance integral method for Stefan-like problems: further considerations and refinements, Therm. Sci., 13, No. 2, 81–96 (2009).
J. Caldwell and Y. Y. Kwan, Numerical methods for one-dimensional Stefan problems, Commun. Numer. Meth. Eng., 20, No. 7, 535–545 (2004).
J. Caldwell and Y. Y. Kwan, Numerical solution of the Stefan problems by the heat balance integral method, Part I — Cylindrical and spherical geometries, Commun. Numer. Meth. Eng., 16, 535–545 (2000).
J. Caldwell and Y. Y. Kwan, Starting solutions for the boundary immobilization method, Commun. Numer. Meth. Eng., 21, No. 6, 289–295 (2005).
N. Sadoun and E. K. Si-Ahmed, A new analytical expression of the freezing constant in the Stefan problem with initial superheat, Proc. 9th Int. Conf. “Numerical Methods in Thermal Problems,” USA, Atlanta, GA (1995), Vol. 9, pp. 843–854.
N. Sadoun and E. K. Si-Ahmed, On the double integral method for solving Stefan-like problems, Proc. 1st Int. Thermal and Energy Congress, Marrakesh, Morocco (1993), Vol. 1, pp. 87–91.
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, Appl. Math. Model., 30, No. 6, 531–544 (2006).
N. Sadoun, E. K. Si-Ahmed, and J. Legrand, On heat conduction with phase change: accurate explicit numerical method, J. Appl. Fluid Mech., 5, No. 1, 105–112 (2012).
V. N. Volkov and V. K. Li-Orlov, A refinement of the integral method in solving the heat conduction equation, Heat Transf. Sov. Res., 2, No. 2, 41–47 (1970).
R. S. Gupta and N. C. Banik, Approximate method for the oxygen diffusion problem, Int. J. Heat Mass Transf., 32, No. 4, 781–783 (1989).
R. S. Gupta and N. C. Banik, Diffusion of oxygen in a sphere with simultaneous absorption, Appl. Math. Model., 14, No. 3, 114–121 (1990).
R. S. Gupta and N. C. Banik, Constrained integral method for solving moving boundary problems, Comput. Meth. Appl. Mech. Eng., 67, No. 2, 211–221 (1988).
S. L. Mitchell, Applying the combined integral method to one-dimensional ablation, Appl. Math. Model., 36, No. 1, 127–138 (2012).
T. G. Myers and S. L. Mitchell, Application of the combined integral method to Stefan problems, Appl. Math. Model., 35, No. 9, 4281–4294 (2011).
S. L. Mitchell, Applying the combined integral method to two-phase Stefan problems with delayed onset of phase change, J. Comp. Appl. Math., 281, 58–73 (2015).
S. L. Mitchell and M. Vynnycky, Finite-difference methods with increased accuracy and correct initialization for one-dimensional Stefan problems, Appl. Math. Comput., 215, 1609–1621 (2009).
J. Caldwell, S. Savovic, and Y. Y. Kwan, Nodal integral end finite difference solution of one-dimensional Stefan problem, J. Heat Transf. (ASME), 125, 523–527 (2003).
S. Kutluay, A. R. Bahadir, and A. Ozdes, The numerical solution of one-phase classical Stefan problem, J. Comput. Appl. Math., 81, No. 1, 134–144 (1997).
Rizwan-Uddin, A nodal method for phase change moving boundary problems, Int. J. Comput. Fluid Dynam., No. 11, 211–221 (1999).
V. Voller and M. Cross, An explicit numerical method to track a moving phase change front, Int. J. Heat Mass Transf., 26, 147–150 (1983).
S. L. Mitchell and M. Vynnycky, The oxygen diffusion problem: Analysis and numerical solution, Appl. Math. Model., 39, No. 9, 2763–2776 (2015).
Whye-Teong Ang, A numerical method on integro-differential formulation for solving a one-dimensional Stefan problem, Numer. Meth. Partial Differ. Equ., 24, No. 3, 939–949 (2008).
V. A. Kot, Method of boundary characteristics, J. Eng. Phys. Thermophys., 88, No. 6, 1390–1408 (2015).
V. A. Kot, Boundary characteristics for the generalized heat-conduction equation and their equivalent representations, J. Eng. Phys. Thermophys., 89, No. 4, 985–1007 (2016).
D. Langford, The heat balance integral method, Int. J. Heat Mass Transf., 16, No. 12, 2424–2428 (1973).
T. R. Goodman, The heat-balance integral –– further considerations and refinements, Trans. ASME, Ser. C, No. 1, 83–93 (1961).
A. S. Wood, A new look at the heat balance integral method, Appl. Math. Model., 25, No. 10, 815–824 (2001).
M. S. El-Genk and A. W. Cronenberg, Some improvements to the solution of Stefan-like problems, Int. J. Heat Mass Transf., 22, 167–177 (1979).
V. Voller and M. Cross, Accurate solutions of moving boundary problems using the enthalpy method, Int. J. Heat Mass Transf., 24, 545–556 (1981).
J. Caldwell and C. C. Chan, Numerical solution of Stefan problems in annuli, in: C. A. Brebbia (Ed.), Advanced Computational Methods in Heat Transfer VI, B, WIT Press, Southampton and Boston (2000), pp. 215–225.
J. Caldwell and Y. Y. Kwan, Spherical solidification by the enthalpy method and heat balance integral method, in: C. A. Brebbia (Ed.), Advanced Computational Methods in Heat Transfer VII, B, WIT Press, Southampton and Boston (2002), pp. 165–174.
S. Kutluay, A. R. Bahadir, and A. Ozdes, The numerical solution of one-phase classical Stefan problem, J. Comput. Appl. Math., 81, 135–144 (1997).
N. Sadoun, P. Colinet, and E. K. Si-Ahmed, New analytical expression for the freezing constant using the refined integral method with cubic approximant, Multiscale Complex Fluid Flows and Interfacial Phenomena Conf. (MULTIFLOW 2010), Brussels, Belgium (2010); https://www.researchgate.net/profile/Nacer_Sadoun/publication/
F. Mosally, A. S. Wood, and A. Al-Fhaid, An exponential heat balance integral method, Appl. Math. Comput., 130, No. 1, 87–100 (2002).
R. S. Gupta and D. Kumar, A modified variable time step method for the one-dimensional Stefan problem, Comput. Math. Appl. Mech. Eng., 23, 101–109 (1980).
R. S. Gupta and D. Kumar, Variable time step methods for one-dimensional Stefan problem with mixed boundary condition, Int. J. Mass Heat Transf., 24, 251–259 (1981).
S. Kutluay, Numerical schemes for one-dimensional Stefan-like problems with a forcing term, Appl. Math. Comput., 168, No. 2, 1159–1168 (2005).
S. Kutluay and A. Elsen, An isotherm migration formulation for one-phase Stefan problem with time-dependent Neumann condition, Appl. Math. Comput., 150, No. 1, 59–67 (2004).
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Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 89, No. 5, pp. 1301–1327, September–October, 2016.
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Kot, V.A. Integral Method of Boundary Characteristics in Solving the Stefan Problem: Dirichlet Condition. J Eng Phys Thermophy 89, 1289–1314 (2016). https://doi.org/10.1007/s10891-016-1499-0
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DOI: https://doi.org/10.1007/s10891-016-1499-0