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

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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.

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Abbreviations

A, B:

Coefficients, defined by Eqs. 23 and 24

a, b, c:

Coefficients, defined by Eq. 29

c p :

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

b :

At the boundary

f :

Freezing

0, 1:

Coefficients associated with power 0 and 1

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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).

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Authors and Affiliations

  1. Institute of Thermal Science and Power Systems, Zhejiang University, Hangzhou, 310027, People’s Republic of China

    Zi-Tao Yu & Ya-Cai Hu

  2. Mechanical Engineering Department, Auburn University, Auburn, AL, 36849-5341, USA

    Li-Wu Fan

  3. State Key Laboratory of Clean Energy Utilization, Zhejiang University, Hangzhou, 310027, People’s Republic of China

    Ke-Fa Cen

Authors
  1. Zi-Tao Yu
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  2. Li-Wu Fan
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  3. Ya-Cai Hu
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  4. Ke-Fa Cen
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Correspondence to Zi-Tao Yu.

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Yu, ZT., Fan, LW., Hu, YC. et al. Perturbation solution to heat conduction in melting or solidification with heat generation. Heat Mass Transfer 46, 479–483 (2010). https://doi.org/10.1007/s00231-010-0596-4

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  • DOI: https://doi.org/10.1007/s00231-010-0596-4

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