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Heat solid

Strong formulation

The transfer of heat in solids is a thermal conduction process and is governed by the equation:

βˆ‚Tβˆ‚t=βˆ’βˆ‡β‹…q+Q\begin{equation} \frac{\partial T}{\partial t} = -\nabla \cdot \boldsymbol{q} + Q \end{equation}

where,

  • q\boldsymbol{q} is the heat flux density (W/m2W/m^2)
  • QQ is the volumetric heat source (W/m3W/m^3)
  • TT is the Temperature field (KK)

Constitutive equation

Heat flux density due to conduction is given by Fick’s law of diffusion:

q=βˆ’ΞΊβˆ‡T\begin{equation} \boldsymbol{q} = -\kappa \nabla T \end{equation}

where ΞΊ\kappa is the thermal conductivity of the material (W/m2KW/m^2 K).

The governing equation becomes:

βˆ’βˆ‚Tβˆ‚t+βˆ‡β‹…(ΞΊβˆ‡T)+Q=0\begin{equation} -\frac{\partial T}{\partial t} + \nabla \cdot (\kappa \nabla T) + Q = 0 \end{equation}

Weak formulation

The weak form of solid mechanics is obtained by multiplying the partial differential equation in (3) with the test function of Temperature field Tβ€²T^\prime and integrating over the whole domain Ξ©\Omega.

βˆ«Ξ©β€…β€Šβˆ’βˆ‚Tβˆ‚tβ€…β€ŠTβ€²β€…β€ŠdΞ©+βˆ«Ξ©β€…β€Šβˆ‡β‹…(ΞΊβˆ‡T)β€…β€ŠTβ€²β€…β€ŠdΞ©+βˆ«Ξ©β€…β€ŠQβ€…β€ŠTβ€²β€…β€ŠdΞ©=0 \int_{\Omega}\; -\frac{\partial T}{\partial t} \; T^{\prime} \; d\Omega + \int_{\Omega}\; \nabla \cdot (\kappa \nabla T) \; T^{\prime} \; d\Omega + \int_{\Omega}\; Q \; T^{\prime} \; d\Omega = 0

Applying Leibniz rule on the divergence term, we get

βˆ«Ξ©β€…β€Šβˆ’βˆ‚Tβˆ‚tβ€…β€ŠTβ€²β€…β€ŠdΞ©+βˆ«Ξ©β€…β€Šβˆ’ΞΊβˆ‡Tβ‹…βˆ‡Tβ€²β€…β€ŠdΞ©+βˆ«Ξ©β€…β€Šβˆ‡β‹…(ΞΊβˆ‡Tβ€…β€ŠTβ€²)β€…β€ŠdΞ©+βˆ«Ξ©β€…β€ŠQβ€…β€ŠTβ€²β€…β€ŠdΞ©=0 \int_{\Omega}\; -\frac{\partial T}{\partial t} \; T^{\prime} \; d\Omega + \int_{\Omega}\; -\kappa \nabla T \cdot \nabla T^{\prime} \; d\Omega + \int_{\Omega}\; \nabla \cdot (\kappa \nabla T \; T^{\prime}) \; d\Omega + \int_{\Omega}\; Q \; T^{\prime} \; d\Omega = 0

Applying Divergence theorem on the divergence term, we get

βˆ«Ξ©β€…β€Šβˆ’βˆ‚Tβˆ‚tβ€…β€ŠTβ€²β€…β€ŠdΞ©+βˆ«Ξ©β€…β€Šβˆ’ΞΊβˆ‡Tβ‹…βˆ‡Tβ€²β€…β€ŠdΞ©+βˆ«Ξ“β€…β€Š(ΞΊβˆ‡Tβ‹…n)β€…β€ŠTβ€²β€…β€ŠdΞ“+βˆ«Ξ©β€…β€ŠQβ€…β€ŠTβ€²β€…β€ŠdΞ©=0 \int_{\Omega}\; -\frac{\partial T}{\partial t} \; T^{\prime} \; d\Omega + \int_{\Omega}\; -\kappa \nabla T \cdot \nabla T^{\prime} \; d\Omega + \int_{\Gamma}\; (\kappa \nabla T \cdot \boldsymbol{n}) \; T^{\prime} \; d\Gamma + \int_{\Omega}\; Q \; T^{\prime} \; d\Omega = 0

Substituting (2) in the boundary term, we get the final form weak formulation:

βˆ«Ξ©β€…β€Šβˆ’βˆ‚Tβˆ‚tβ€…β€ŠTβ€²β€…β€ŠdΞ©+βˆ«Ξ©β€…β€Šβˆ’ΞΊβˆ‡Tβ‹…βˆ‡Tβ€²β€…β€ŠdΞ©+βˆ«Ξ“β€…β€Š(qβ‹…n)β€…β€ŠTβ€²β€…β€ŠdΞ“+βˆ«Ξ©β€…β€ŠQβ€…β€ŠTβ€²β€…β€ŠdΞ©=0 \int_{\Omega}\; -\frac{\partial T}{\partial t} \; T^{\prime} \; d\Omega + \int_{\Omega}\; -\kappa \nabla T \cdot \nabla T^{\prime} \; d\Omega + \int_{\Gamma}\; (\boldsymbol{q} \cdot \boldsymbol{n}) \; T^{\prime} \; d\Gamma + \int_{\Omega}\; Q \; T^{\prime} \; d\Omega = 0