International Scientific Journal

Thermal Science - Online First

online first only

Integral balance methods applied to non-classical Stefan problems

We consider two different Stefan problems for a semi-infinite material for the nonclassical heat equation with a source that depends on the heat flux at the fixed face. One of them, with constant temperature at the fixed face, was already studied in literature and the other, with a convective boundary condition at the fixed face, is presented in this work. Due to the complexity of the exact solution it is of interest to compare with approximate solutions obtained by applying heat balance integral methods, is carried out by using the parameters: Stefan number and the generalized assuming a quadratic temperature profile in space. A dimensionless analysis Biot number. In addition it is studied the case when Biot number goes to infinity, recovering the approximate solutions when a Dirichlet condition is imposed at the fixed face. Some numerical simulations are provided in order to verify the accuracy of the approximate methods.
PAPER REVISED: 2018-10-28
PAPER ACCEPTED: 2018-10-29
  1. Gupta, S., The classical Stefan problem. Basic concepts, modelling and analysis, Elsevier, Amsterdam, Netherlands, 2003
  2. Tarzia, D., Explicit and approximated solutions for heat and mass transfer problem with a moving interface, in Advanced Topics in Mass Transfer, (Ed. M. El-Amin), InTech Open Access Publisher, Rijeka, 2011, pp. 439-484
  3. Cannon, J.R., Yin H. M., A class of nonlinear nonclassical parabolic equations, Journal of differential equations, 79 (1989), 2, pp. 266-288
  4. Briozzo, A., Tarzia, D., Exact solutions for Nonclassical Stefan problems, International Journal of Differential Equations, Article ID 868059 (2010), pp. 1-19,
  5. Tarzia, D., Relationship between Neumann solutions for two phase Lamé-Clapeyron-Stefan problems with convective and temperature boundary conditions, Thermal Science, 21 No 1A (2017), pp. 187-197
  6. Zubair, S., Chaudhry, M., Exact solutions of solid-liquid phase-change heat transfer when subjected to convective boundary conditions, Heat and Mass Transfer, 30 (1994), 2, pp. 77-81
  7. Goodman, T., The heat balance integral methods and its application to problems involving a change of phase, Transactions of the ASME, 80 (1958), pp. 335-342
  8. Bollati, J., et al., Heat balance integral methods applied to the one-phase Stefan problem with a convective boundary condition at the fixed face, Applied Mathematics and Computation, 331 (2018), pp. 1-19
  9. Hristov, J., Research note on a parabolic heat-balance integral method with unspecified exponent: An Entropy Generation Approach in Optimal Profile Determination, Thermal Science, 13 (2009), pp. 49-59
  10. Mitchell, S., Applying the combined integral method to one-dimensional ablation, Applied Mathematical Modelling, 36 (2012), pp. 127-138
  11. Mitchell, S., Myers, T., Application of Heat Balance Integral Methods to One-Dimensional Phase Change Problems, International Journal of Differential Equations, 2012 (2012), pp. 1-22
  12. Hristov, J., Multiple integral-balance method: Basic Idea and Example with Mullin's Model of Thermal Grooving, Thermal Science, 21 (2017), pp. 1555-1560
  13. MacDevette, M., Myers, T., Nanofluids: An innovative phase change material for cold storage systems?, International Journal of Heat and Mass Transfer, 92 (2016), pp. 550-557
  14. Hristov, J., An approximate analytical (integral-balance) solution to a non-linear heat diffusion equation, Thermal Science, 19 (2015), pp. 723-733
  15. Fabre, A., Hristov, J., On the integral-balance approach to the transient heat conduction with linearly temperature dependent thermal diffusivity, Heat Mass Transfer, 53 (2017), pp. 177-204
  16. Hristov, J., Integral solutions to transient nonlinear heat (mass) diffusion with a power-law diffusivity: a semiinfinite medium with fixed boundary conditions, Heat Mass Transfer, 52 (2016), pp. 635-655
  17. Hristov, J., Integral-balance solution to a nonlinear subdiffusion equation, in Frontiers in Fractional Calculus, (Ed. S. Bhalekar), Bentham Publishing, 2017, pp. 71-106
  18. Wood, A., A new look at the heat balance integral method, Applied Mathematical Modelling, 25 (2001), pp. 815-824
  19. Sadoun, N., et al., On the refined integral method for the one-phase Stefan problem with time-dependent boundary conditions, Applied Mathematical Modelling, 30 (2006), pp. 531-544
  20. Solomon, A., An easily computable solution to a two-phase Stefan problem, Solar energy, 33 (1979), pp. 525-528