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


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.


Heat Generation Phase Change Material Perturbation Parameter Stefan Problem Perturbation Solution 
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List of symbols

A, B

Coefficients, defined by Eqs. 23 and 24

a, b, c

Coefficients, defined by Eq. 29


Specific heat (J/kg K)


Thermal conductivity (W/m K)


Latent heat of fusion (J/kg)


Characteristic length (m)


Volumetric heat generation (W/m3)


Stefan number, defined in Eq. 5


Interface location (m)


Temperature (K)


Time (s)


Coordinate (m)


Thermal diffusivity (W/m2)


Dimensionless heat generation


Perturbation parameter


Dimensionless temperature


Dimensionless interface location


Dimensionless coordinate


Density (kg/m3)


Dimensionless time



At the boundary



0, 1

Coefficients associated with power 0 and 1



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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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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