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

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Abstract

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.

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Abbreviations

A :

Area (m2)

A i :

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

B :

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

c :

Specific heat capacity (J/(kg K))

C b :

Black body constant (W/(m2 K4))

e :

Emission factor

F T :

Temperature factor

hi, he:

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

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

Heaviside function

k :

Effective thermal conductivity (W/(m K))

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

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

K Ω :

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

n :

Normal vector

N :

Vector containing finite element shape functions

Nu :

Dimensionless Nusselt number

Q :

Energy flow rate (J/s)

q :

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

q eff :

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

\({\bar{{q}}}\) :

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)

t :

Time [s]

T :

Absolute temperature (K)

\({\hat{{T}}}\) :

Regular component of the absolute temperature (K)

\({\tilde {T}}\) :

Enhanced component of the absolute temperature (K)

w :

Variational temperature field (K)

α :

Relative interface position

γ :

1 Unit meter (m)

λ :

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 (°)

ρ :

Average mass density (kg/m3)

ω :

Damage variable

Γ:

Boundary

Γd :

Discontinuity

Ω:

Body

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Authors and Affiliations

  1. Katholieke Universiteit Leuven (KUL), Leuven, Belgium

    P. Moonen

  2. Delft University of Technology (TUD), Delft, The Netherlands

    P. Moonen & L. J. Sluys

  3. Swiss Federal Institute of Technology Zürich (ETH), Zürich, Switzerland

    P. Moonen & J. Carmeliet

  4. Swiss Federal Laboratories for Materials Science and Technology (EMPA), Dübendorf, Switzerland

    P. Moonen & J. Carmeliet

Authors
  1. P. Moonen
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  2. L. J. Sluys
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  3. J. Carmeliet
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Correspondence to P. Moonen.

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Moonen, P., Sluys, L.J. & Carmeliet, J. A Continuous–Discontinuous Approach to Simulate Heat Transfer in Fractured Media. Transp Porous Med 89, 399–419 (2011). https://doi.org/10.1007/s11242-011-9777-y

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