Heat Transfer Equations#

In Lethe, it is possible to solve the heat transfer in various loading conditions. The main equation is derived from the energy equation in incompressible flows. Assuming a constant heat capacity \(C_p\) and dynamic viscosity \(\mu\), the equation takes the following form:

\[\rho C_p \frac{\partial T}{\partial t} + \rho C_p (\mathbf{u} \cdot \nabla)T - \nabla \cdot (k \nabla T) = - \phi + Q\]

where:

  • \(T\) is the temperature;

  • \(\mathbf{u}\) is the velocity of the fluid;

  • \(\nabla\) is the del operator;

  • \(\rho\) is the density;

  • \(C_p\) is the isobaric heat capacity;

  • \(k\) is the thermal conductivity;

  • \(Q\) is the energy source term or heat generation;

  • \(\phi\) is the viscous dissipation term. For an incompressible fluid it takes the following form: \(\phi = \mu (\nabla \mathbf{u} + \nabla \mathbf{u}^T):\nabla \mathbf{u}\), where \(\mu\) is the dynamic viscosity;

Depending on the physics involved, the terms \(\phi\) and \(Q\) can be included or excluded and can take various definitions.

Finite Element Formulation#

For the finite element formulation, we start from the strong form of the equation as shown above. We consider a domain \(\Omega\) with boundary \(\Gamma\). The applied boundary condition can vary depending on the problem being solved, but the main ones are the following:

  • Dirichlet boundary condition: \(T = T_0\) on \(\Gamma_D\);

  • Neumann boundary condition: \(\mathbf{n} \cdot \nabla T = q\) on \(\Gamma_N\);

  • Robin boundary condition: \(\mathbf{n} \cdot \nabla T + h(T - T_{\infty}) = 0\) on \(\Gamma_R\);

where \(\mathbf{n}\) is the normal vector to the boundary, \(q\) is the heat flux, \(h\) is the heat transfer coefficient, and \(T_{\infty}\) is the ambient temperature.

The weak form of the heat equation is obtained by multiplying the strong form by a test function \(v\) and integrating over the domain \(\Omega\). After applying the integration by part and the Gauss-Ostrogradsky theorem, the weak form of the heat equation is given by the following equation:

\[\int_{\Omega} \left[ v \rho C_p \frac{\partial T}{\partial t} + v \rho C_p (\mathbf{u} \cdot \nabla)T + k (\nabla v \cdot \nabla T) \right] \;d\Omega - \int_{\Gamma} v k (\nabla T \cdot \mathbf{n}) \;d\Gamma = - \int_{\Omega} v \left[ \phi - Q \right] \;d\Omega\]

Note that this formulation treats the thermal conductivity \(k\) as a constant.

Stabilization Techniques#

Under construction