Transport in Porous Media

, Volume 89, Issue 3, pp 399–419 | Cite as

A Continuous–Discontinuous Approach to Simulate Heat Transfer in Fractured Media

  • P. MoonenEmail author
  • L. J. Sluys
  • J. Carmeliet


A macroscopic framework to model heat transfer in materials and composites, subjected to physical degradation, is proposed. The framework employs the partition of unity concept and captures the change from conduction-dominated transfer in the initial continuum state to convection and radiation-dominated transfer in the damaged state. The underlying model can be directly linked to a mechanical cohesive zone model, governing the initiation and subsequent growth and coalescence of micro-cracks. The methodology proved to be applicable for quasi-static, periodic, and transient problems.


Failure Damage Cohesive zone model Partition of unity (PU) Heat transfer Continuous–discontinuous framework 

List of Symbols

Latin Symbols


Area (m2)


Amplitude of the periodic temperature fluctuation on the inner wall surface (K)


Matrix containing the derivatives of the finite element shape functions (m−1)


Specific heat capacity (J/(kg K))


Black body constant (W/(m2 K4))


Emission factor


Temperature factor

hi, he

Heat transfer coefficient (indoor, outdoor) (W/(m2 K))

\({{\mathcal{H}}_{\Gamma_{\rm d}}}\)

Heaviside function


Effective thermal conductivity (W/(m K))

\({K_{\Gamma_{\rm d}}}\)

Thermal conductivity of the material bond (W/(m K))


Thermal conductivity matrix of the continuum material (W/(m K))


Normal vector


Vector containing finite element shape functions


Dimensionless Nusselt number


Energy flow rate (J/s)


Total heat flux vector (J/(m2 s))


Energy exchange via the undamaged material bond (J/(m2 s))


Energy exchange via an internal or external boundary (J/(m2 s))

\({R_{\Gamma_{\rm d}}}\)

Effective thermal resistance of the material bond ((m K)/W)


Time [s]


Absolute temperature (K)


Regular component of the absolute temperature (K)

\({\tilde {T}}\)

Enhanced component of the absolute temperature (K)


Variational temperature field (K)

Greek Symbols


Relative interface position


1 Unit meter (m)


Thermal conductivity of the medium inside the discontinuity (W/(m K))

\({\phi_{\rm i}}\)

Phase angle of the periodic temperature fluctuation on the inner wall surface (°)


Average mass density (kg/m3)


Damage variable









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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Katholieke Universiteit Leuven (KUL)LeuvenBelgium
  2. 2.Delft University of Technology (TUD)DelftThe Netherlands
  3. 3.Swiss Federal Institute of Technology Zürich (ETH)ZürichSwitzerland
  4. 4.Swiss Federal Laboratories for Materials Science and Technology (EMPA)DübendorfSwitzerland

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