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Heat Conduction Equation

This chapter summarizes the governing equation of heat conduction addressed by FrontISTR at the continuum level and its boundary conditions. Time discretization and FEM discretization are left to the discretization section (L3d) and the solution algorithm section (L3e: solution_methods/05_transient_heat).

Governing Equation

The heat conduction equation in a continuum is given as follows.

\[\begin{equation} \rho c \frac{\partial T}{\partial t} = \frac{\partial}{\partial x}\left(k_x \frac{\partial T}{\partial x}\right) + \frac{\partial}{\partial y}\left(k_y \frac{\partial T}{\partial y}\right) + \frac{\partial}{\partial z}\left(k_z \frac{\partial T}{\partial z}\right) + Q \label{eq:gov_he_main} \end{equation}\]

Here, \(\rho=\rho(x)\) denotes mass density, \(c=c(x,T)\) specific heat, \(T=T(x,t)\) temperature, \(k=k(x,T)\) thermal conductivity, \(Q=Q(x,T,t)\) heat generation, \(x\) position, and \(t\) time. \(k_x, k_y, k_z\) are the thermal conductivities in the respective directions for anisotropic heat conduction.

For specific thermal properties, including temperature dependence and anisotropy, see Thermal Properties.

Boundary Conditions

Let the domain under consideration be \(S\), with its boundary denoted by \(\Gamma\). Assuming that either a Dirichlet-type or a Neumann-type boundary condition is prescribed everywhere on \(\Gamma\), the boundary conditions are given as follows.

\[\begin{equation} T = T_1(x,t), \; x \in \Gamma_1 \label{eq:gov_he_dirichlet} \end{equation}\]
\[\begin{equation} k \frac{\partial T}{\partial n} = q(x,T,t), \; x \in \Gamma_2 \label{eq:gov_he_neumann} \end{equation}\]

Here, the functional forms of \(T_1\) and \(q\) are assumed to be known. \(q\) is the heat flux leaving the boundary. The boundary heat flux \(q\) consists of the following three components.

\[\begin{equation} q=-q_s+q_c+q_r \label{eq:gov_he_flux} \end{equation}\]
\[\begin{equation} q_s=q_s(x,t) \label{eq:gov_he_qs} \end{equation}\]
\[\begin{equation} q_c=hc(T-Tc) \label{eq:gov_he_qc} \end{equation}\]
\[\begin{equation} q_r=hr(T^4-Tr^4) \label{eq:gov_he_qr} \end{equation}\]

Here, \(q_s\) is the distributed heat flux, \(q_c\) is the heat flux due to convective heat transfer, and \(q_r\) is the heat flux due to radiative heat transfer.

Here, \(Tc=Tc(x,t)\) is the ambient temperature for convective heat transfer, \(hc=hc(x,t)\) is the convective heat transfer coefficient, \(Tr=Tr(x,t)\) is the ambient temperature for radiative heat transfer, \(hr=\varepsilon \sigma F = hr(x,t)\) is the radiative heat transfer coefficient, \(\varepsilon\) is the emissivity, \(\sigma\) is the Stefan-Boltzmann constant, and \(F\) is the view factor.

For how to specify these conditions in an input file, see Boundary Conditions and Loads in the function reference.

Weak Form

Coupling with Structural Analysis