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Heat and Mass Transfer

, Volume 46, Issue 4, pp 479–483 | Cite as

Perturbation solution to heat conduction in melting or solidification with heat generation

  • Zi-Tao YuEmail author
  • Li-Wu Fan
  • Ya-Cai Hu
  • Ke-Fa Cen
Technical Note

Abstract

The Stefan problem involving a source term is considered in this technical note. As an example, planar solidification with time-dependent heat generation in a semi-infinite plane is solved by use of a perturbation technique. The perturbation solution is validated by reducing the problem to the case without heat generation whose exact solution is available. An application to the case with constant heat generation is presented, for which a closed-form solution is obtained. The effects of heat generation and Stefan number on the evolution of solidification are examined using the perturbation solution.

Keywords

Heat Generation Phase Change Material Perturbation Parameter Stefan Problem Perturbation Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of symbols

A, B

Coefficients, defined by Eqs. 23 and 24

a, b, c

Coefficients, defined by Eq. 29

cp

Specific heat (J/kg K)

k

Thermal conductivity (W/m K)

L

Latent heat of fusion (J/kg)

l

Characteristic length (m)

q′″

Volumetric heat generation (W/m3)

Ste

Stefan number, defined in Eq. 5

s

Interface location (m)

T

Temperature (K)

t

Time (s)

x

Coordinate (m)

α

Thermal diffusivity (W/m2)

δ

Dimensionless heat generation

ε

Perturbation parameter

θ

Dimensionless temperature

λ

Dimensionless interface location

ξ

Dimensionless coordinate

ρ

Density (kg/m3)

τ

Dimensionless time

Subscripts

b

At the boundary

f

Freezing

0, 1

Coefficients associated with power 0 and 1

Notes

Acknowledgments

The financial support from the National Natural Science Foundation of China (NSFC) under Grant No. 50706044 is gratefully acknowledged. Z.-T. Yu wishes to appreciate a grant from the Program of Zhejiang University for Outstanding Young Faculty Members (Zi-Jin Program).

References

  1. 1.
    Özişik MN (1980) Heat conduction. Wiley, New YorkGoogle Scholar
  2. 2.
    Jiji LM (2009) Heat conduction, 3rd edn. Springer, BerlinzbMATHGoogle Scholar
  3. 3.
    Muehlbauer JC, Sunderland JE (1965) Heat conduction with freezing or melting. Appl Mech Rev 18:951–959Google Scholar
  4. 4.
    Gupta SC (2003) The classical Stefan problem. Elsevier, AmsterdamzbMATHGoogle Scholar
  5. 5.
    Lock GSH (1971) On the perturbation solutions of the ice-water layer problem. Int J Heat Mass Transf 14:642–644CrossRefGoogle Scholar
  6. 6.
    Pedroso RI, Domoto GA (1973) Exact solution by perturbation method for planar solidification of a saturated liquid with convection at the wall. Int J Heat Mass Transf 16:1816–1819CrossRefGoogle Scholar
  7. 7.
    Huang C-L, Shih Y-P (1975) Perturbation solutions of planar diffusion-controlled moving-boundary problems. Int J Heat Mass Transf 18:689–695zbMATHCrossRefGoogle Scholar
  8. 8.
    Huang C-L, Shih Y-P (1975) Perturbation solution for planar solidification of a saturated liquid with convection at the wall. Int J Heat Mass Transf 18:1481–1483CrossRefGoogle Scholar
  9. 9.
    Pedroso RI, Domoto GA (1973) Perturbation solutions for spherical solidification of saturated liquids. J Heat Transf 95:42–46Google Scholar
  10. 10.
    Huang C-L, Shih Y-P (1975) A perturbation method for spherical and cylindrical solidification. Chem Eng Sci 30:897–906CrossRefGoogle Scholar
  11. 11.
    Aziz A, Na TY (1984) Perturbation methods in heat transfer. Hemisphere Publishing Corporation, Washington, DCGoogle Scholar
  12. 12.
    Lunardini VJ, Aziz A (1993) Perturbation techniques in conduction-controlled freeze–thaw heat transfer. Monograph 93-1, Cold Regions Research & Engineering Laboratory, US Army Corps of EngineersGoogle Scholar
  13. 13.
    Caldwell J, Kwan YY (2003) On the perturbation method for the Stefan problem with time-dependent boundary conditions. Int J Heat Mass Transf 46:1497–1501zbMATHCrossRefGoogle Scholar
  14. 14.
    Basak T (2003) Analysis of resonances during microwave thawing of slabs. Int J Heat Mass Transf 46:4279–4301zbMATHCrossRefGoogle Scholar
  15. 15.
    Rattanadecho P (2004) Theoretical and experimental investigation of microwave thawing of frozen layer using a mircowave oven (effects of layered configurations and layer thickness). Int J Heat Mass Transf 47:937–945CrossRefGoogle Scholar
  16. 16.
    Li JF, Li L, Stott FH (2004) Comparison of volumetric and surface heating sources in the modeling of laser melting of ceramic materials. Int J Heat Mass Transf 47:1159–1174Google Scholar
  17. 17.
    Cheung FB, Chawla TC, Pedersen DR (1984) The effects of heat generation and wall interaction on freezing and melting in a finite slab. Int J Heat Mass Transf 27:29–37CrossRefGoogle Scholar
  18. 18.
    Jiji LM, Gaye S (2006) Analysis of solidification and melting of PCM with energy generation. Appl Therm Eng 26:568–575CrossRefGoogle Scholar
  19. 19.
    Crepeau J, Siahpush A (2008) Approximate solutions to the Stefan problem with internal heat generation. Heat Mass Transf 44:787–794CrossRefGoogle Scholar
  20. 20.
    Crepeau JC, Siahpush A, Spotten B (2009) On the Stefan problem with volumetric energy generation. Heat Mass Transf 46:119–128CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Institute of Thermal Science and Power SystemsZhejiang UniversityHangzhouPeople’s Republic of China
  2. 2.Mechanical Engineering DepartmentAuburn UniversityAuburnUSA
  3. 3.State Key Laboratory of Clean Energy UtilizationZhejiang UniversityHangzhouPeople’s Republic of China

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