MELTING AND FREEZING IN A FINITE SLAB DUE TO A LINEARLY DECREASING FREE-STREAM TEMPERATURE OF A CONVECTIVE BOUNDARY CONDITION

Abstract

One-dimensional melting and freezing problem in a finite slab with time-dependent convective boundary condition is solved using the heat-balance integral method. The temperature, Tinf 1(t), is applied at the left face and decreases linearly with time while the other face of the slab is imposed with a constant convective boundary condition where Tinf 2 is held at a fixed temperature. In this study, the initial condition of the solid is subcooled (initial temperature is below the melting point). The temperature, Tinf 1(t) at time t = 0 is so chosen such that convective heating takes place and eventually the slab begins to melt (i. e., Tinf 1(0) > Tf > Tinf 2). The transient heat conduction problem, until the phase-change starts, is also solved using the heat-balance integral method. Once phase-change process starts, the solid-liquid interface is found to proceed to the right. As time continues, and Tinf,1(t) decreases with time, the phase-change front slows, stops, and may even reverse direction. Hence this problem features sequential melting and freezing of the slab with partial penetration of the solid-liquid front before reversal of the phase-change process. The effect of varying the Biot number at the right face of the slab is investigated to determine its impact on the growth/recession of the solid-liquid interface. Temperature profiles in solid and liquid regions for the different cases are reported in detail. One of the results for Biot number, Bi2 = 1.5 are also compared with those obtained by having a constant value of Tinf 1(t).

Keywords

Dates

  • Submission Date2008-09-24
  • Revision Date2009-03-10
  • Acceptance Date2009-03-20

DOI Reference

10.2298/TSCI0902141R

References

  1. Alexiades, V., Solomon, A. D., Mathematical Modeling of Melting and Freezing Processes, Hemisphere Publishing Corporation, Washington, DC, USA, 1993, Chapter 2
  2. Fukusako, S., Seki, N., Fundamental Aspects of Analytical and Numerical Methods on Freezing and Melting Heat Transfer-Problems, Annual Review of Numerical Fluid Mechanics and Heat Transfer, 1 (1987), 1, pp. 351-402
  3. Kar, A., Mazumder J., Analytic Solution of the Stefan Problem in Finite Mediums, Quarterly of Applied Mathematics, 52 (1994), 1, pp. 49-58
  4. Boley, B. A., A General Starting Solution for Melting and Solidifying Slabs, Int. J. Engng. Sci., 6 (1968), 89, pp. 89-111
  5. Boley, B. A., Yagoda, H. P., The Starting Solution for Two-Dimensional Heat Conduction Problems with Change of Phase, Quarterly of Applied Mathematics, 27 (1969), 2, pp. 223-246
  6. Goodman, T. R., Shea, J. J., The Melting of Finite Slabs, J. Appl. Mechanics, 27 (1960), 1, pp. 16-24
  7. Zhang, Y. W., et al., An Analytical Solution to Melting in a Finite Slab with a Boundary Condition of Second Kind, Trans. ASME J. Heat Transfer, 115 (1993), 2, pp. 463-467
  8. Lock, G. S. H, Latent Heat Transfer - An Introduction to Fundamentals, Oxford University Press, Oxford, UK, 1994, Chapter 4
  9. Yuen, W. W., Application of Heat Balance Integral to Melting Problems with Initial Sub Cooling, Int. J. Heat Mass Transfer, 23 (1980), 8, pp. 1157-1160.
  10. Chan, A. M. C., Smereka, P., Shoukri, M., An Approximate Analytical Solution to the Freezing Problem Subject to Convective Cooling and with Arbitrary Initial Liquid Temperatures, Int. J. Heat Mass Transfer, 26 (1983), 11, pp. 1712- 1715
  11. Gutman, L. N., On the Problem of Heat Transfer in Phase-Change Materials for Small Stefan Numbers, Int. J. Heat Mass Transfer, 29 (1986), 6, pp. 921-926
  12. Roday, A. P, Kazmierczak, M. J., Analysis of Phase-Change in Finite Slabs Subjected to Convective Boundary Conditions: Part I - Melting, International Review in Chemical Engineering (Rapid Communications), 1 (2009), 1, pp. 87-99
  13. Roday, A. P, Kazmierczak, M. J., Analysis of Phase-Change in Finite Slabs Subjected to Convective Boundary Conditions: Part II - Freezing, International Review in Chemical Engineering (Rapid Communications), 1 (2009), 1, pp. 100-108
Volume 13, Issue 2, Pages141 -153