Heat Conduction in a Solid

Heat Conduction in a Solid#

Mathematical Model#

Consider the heat conduction equation in a solid domain (\Omega):

\[ \begin{align} \rho C_p \frac{\partial T}{\partial t} - \nabla \cdot (k \nabla T) = Q \end{align} \]

Boundary Conditions#

  • Dirichlet: \(T = T_{\text{inlet}}\) on the inlet face.

  • Neumann: \(-k \frac{\partial T}{\partial n} = 0\) on the outlet face.

  • Insulated: \(-k \frac{\partial T}{\partial n} = 0\) on the lateral faces.

Initial Conditions#

Assume an initial temperature distribution \(T_0(x, y, z)\) at \(t = 0\).

Weak Formulation#

Find \(T \in L^2(\Omega)\) such that:

\[ \begin{align} \int_{\Omega} \rho C_p \frac{\partial T}{\partial t} \phi \, d\Omega + \int_{\Omega} k \nabla T \cdot \nabla \phi \, d\Omega = \int_{\Omega} Q \phi \, d\Omega \end{align} \]

for all test functions \(\phi \in H_0^1(\Omega)\).

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