Heat equation

1. Transient Case

1.1. Differential Formulation

Set a domain \(\Omega_c \in \mathbb{R}^3\) with bound \(\Gamma_c = \partial \Omega_c\), bound of Neumann condition \(\Gamma_{Nc}\) and bound of Robin condition \(\Gamma_{Rc}\), such as \(\Gamma_c = \Gamma_{Nc} \cup \Gamma_{Rc}\).

The heat equation is written :

Heat equation
\[\text{(Heat)} \left\{ \begin{matrix} \rho C_p \, \frac{\partial T}{\partial t} - k \Delta T = Q \text{ on } \Omega_c \\ \frac{\partial T}{\partial \mathbf{n}} = 0 \text{ on } \Gamma_{Nc} \\ k \, \frac{\partial T}{\partial \mathbf{n}} = h \, \left( T - T_c \right) \text{ on } \Gamma_{Rc} \\ \end{matrix} \right.\]

With :

  • \(T\) : temperature \((K)\)

  • \(Q\) : source term \((K)\)

  • \(\rho\) : density \((kg/m^3)\)

  • \(h\) : convection coefficient \((W \, m^{-2} / K)\)

  • \(C_p\) : thermal capacity \((J/K/kg)\)

  • \(T_c\) : cooling temperature \((K)\)

  • \(k\) : thermal conductivity \((W/m/K)\)

1.2. Weak Formulation

By making scalar product with \(\mathbf{\phi} \in H^1(\Omega_c)\) and integration of Heat equation :

\[ \int_{\Omega_c}{ \rho C_p \, \frac{\partial T}{\partial t} \, \phi } - \int_{\Omega_c}{ k \Delta T \, \phi } = \int_{\Omega_c}{ Q \, \phi }\]

By Second Formula of Green :

\[ \int_{\Omega_c}{ \rho C_p \, \frac{\partial T}{\partial t} \, \phi } + \int_{\Omega_c}{ k \, \nabla T \cdot \nabla \phi } + \int_{\Gamma_c}{ \left( - k \frac{\partial T}{\partial \mathbf{n}} \right) \, \phi } = \int_{\Omega_c}{ Q \, \phi }\]
\[ \int_{\Omega_c}{ \rho C_p \, \frac{\partial T}{\partial t} \, \phi } + \int_{\Omega_c}{ k \, \nabla T \cdot \nabla \phi } + \int_{\Gamma_{Nc}}{ \left( - k \frac{\partial T}{\partial \mathbf{n}} \right) \, \phi } + \int_{\Gamma_{Rc}}{ \left( - k \frac{\partial T}{\partial \mathbf{n}} \right) \, \phi } = \int_{\Omega_c}{ Q \, \phi }\]

We put the boundary conditions :

  • Neumann : \(\frac{\partial T}{\partial \mathbf{n}} = 0\) on \(\Gamma_{Nc}\)

  • Robin : \(-k \, \frac{\partial T}{\partial \mathbf{n}} = h \, \left( T - T_c \right) \) on \(\Gamma_{Rc}\)

Finally :

Weak Formulation of Heat Equation
\[ \text{(Weak Heat)} \int_{\Omega_c}{ \rho C_p \, \frac{\partial T}{\partial t} \, \phi } + \int_{\Omega_c}{ k \, \nabla T \cdot \nabla \phi } = \int_{\Omega_c}{ Q \, \phi } - \int_{\Gamma_{Rc}}{ h \, (T-T_c) \, \phi }\]

1.3. Time Discretization

We discretize in time with time step \(\Delta t\). We note \(f^n(\mathbf{x}) = f(n\Delta t, \mathbf{x})\).

If, we know \(T^n\), the equations become :

Time Discretization of Heat Equation
\[\small{ \int_{\Omega_c}{ \rho C_p \, \frac{T^{n+1}}{\Delta t} \, \phi } + \int_{\Omega_c}{ k \, \nabla T^{n+1} \cdot \nabla \phi } = \int_{\Omega_c}{ Q \, \phi } - \int_{\Gamma_{Rc}}{ h \, (T-T_c) \, \phi } + \int_{\Omega_c}{ \rho C_p \, \frac{T^n}{\Delta t} \, \phi } }\]

2. Stationary Case

This section presents the AV-Formulation for the stationary regime as a special case of MQS. We assume that all parameters are time independent.

2.1. Differential Formulation

The Differential Formulation becomes :

Differential Formulation of Static Heat Equation
\[\text{(Heat Static)} \left\{ \begin{matrix} - k \Delta T = Q \text{ on } \Omega_c \\ \frac{\partial T}{\partial \mathbf{n}} = 0 \text{ on } \Gamma_{Nc} \\ - k \, \frac{\partial T}{\partial \mathbf{n}} = h \, \left( T - T_c \right) \text{ on } \Gamma_{Rc} \\ \end{matrix} \right.\]

2.2. Weak Formulation

The Weak Formulation becomes :

Weak Formulation of Static Heat Equation
\[ \text{(Weak Heat Static)} \int_{\Omega_c}{ k \, \nabla T \cdot \nabla \phi } = \int_{\Omega_c}{ Q \, \phi } - \int_{\Gamma_{Rc}}{ h \, (T-T_c) \, \phi }\]