Opus Heat

This test case has been proposed by Annabelle Le-Hyaric and Michel Fouquembergh formerly at AIRBUS.

We consider a 2D model representative of the neighboring of an electronic component submitted to a cooling air flow. It is described by four geometrical domains in \(\mathbb{R}^2\) named \(\Omega_i,i=1,2,3,4\), see figure. We suppose the velocity \(\vec{v}\) is known in each domain --- for instance in \(\Omega_4\) it is the solution of previous Navier-Stokes computations. --- The temperature \(T\) of the domain \(\Omega = \cup_{i=1}^4 \Omega_i\) is then solution of heat transfer equation :

\[\begin{equation} \label{eq:1} \rho C_i \Big( \frac{\partial T}{\partial t} + \vec{v} \cdot \nabla T \Big) - \nabla \cdot \left( k_i \nabla T \right) = Q_i,\quad i=1,2,3,4 \end{equation}\]

where \(t\) is the time and in each sub-domain \(\Omega_i\), \(\rho C_i\) is the volumic thermal capacity, \(k_i\) is the thermal conductivity. \(k_1\) and \(k_2\) are parameters of the model.

ICs dissipate heat, so the volumic heat dissipated \(Q_1\) and \(Q_2\) are also parameters of the model, while \(Q_3=Q_4=0\).

One should notice that the convection term in heat transfer equation may lead to spatial oscillations which can be overcome by Petrov-Galerkin type or continuous interior penalty stabilization techniques.

Integrated circuits (ICs) (domains \(\Omega_1\) and \(\Omega_2\) ) are respectively soldered on PCB at \(\mathbf{x1}=(e_{Pcb}, h_1)\) and \(\mathbf{x_2}=(e_{Pcb}, h_2)\). They are considered as rectangles with width \(e_{IC}\) and height \(h_{IC}\). The printed circuit board (PCB) is a rectangle \(\Omega_3\) of width \(e_{PCB}\) and height \(h_{PCB}\). The air(Air) is flowing along the PCB in domain \(\Omega_4\). Speed in the air channel \(\Omega_4\) is supposed to have a parabolic profile function of \(x\) coordinate. Its expression is simplified as follows (notice that \(\vec{v}\) is normally solution of Navier-Stokes equations; the resulting temperature and velocity fields should be quite different from that simplified model), we have for all \(0 \leq y \leq h_{PCB}\)

\[\begin{equation} \label{eq:2} \begin{array}[c]{rl} e_{Pcb} + e_{Ic} \leq x \leq e_{Pcb} + e_a, & \displaystyle \vec{v} = -V(x-(e_{Pcb}+e_{Ic}))(x-(e_{Pcb}+e_a))\vec{y}\\ e_{Pcb} \leq x \leq e_{Pcb} + e_{Ic}, & \vec{v} = 0 \end{array} \end{equation}\]

where V is a parameter of the model.

The medium velocity \(\vec{v}_i = \vec{0}, i=1,2,3\) in the solid domains \(\Omega_i, i=1,2,3\).

1. Boundary conditions

We set

on \(\Gamma_1\cup\Gamma_3\), a zero flux (Neumann-like) condition

\[\begin{equation} \label{eq:10} -k_i\ \nabla T \cdot \vec{n}_i\ =\ 0; \end{equation}\]

on \(\Gamma_2\), a heat transfer (Robin-like) condition

\begin{equation} \label{eq:7} -k_4\ \nabla T \cdot \vec{n}_4\ =\ h(T-T_0); \end{equation}

where \(h\) is a parameter of the model

on \(\Gamma_4\) the temperature is set (Dirichlet condition)

\[\begin{equation} \label{eq:11} T\ = T_0; \end{equation}\]

on other internal boundaries, the coontinuity of the heat flux and temperature, on \(\Gamma_{ij} = \Omega_i \cap \Omega_j \neq \emptyset\)

\[\begin{equation} \label{eq:6} \begin{array}{rl} T_i &= T_j \\ k_i\ \nabla T \cdot \vec{n}_i &= -k_j\ \nabla T \cdot \vec{n}_j. \end{array} \end{equation}\]

2. Initial condition

At \(t=0s\), we set \(T = T_0\).

3. Outputs

The output is the mean temperature \(s_1(\mu)\) of the hottest IC

\[\begin{equation} \label{eq:3} s_1(\mu) = \frac{1}{e_{IC} h_{IC}} \int_{\Omega_2} T \end{equation}\]

4. Parameters

The table displays the various fixed and variables parameters of this test-case.

Table 1. Table of model order reduction parameters
Name	Description	Range	Units
\(k_1\)	thermal conductivity	\([1,3\)]	\(W \cdot m^{-1} \cdot K^{-1}\)
\(k_2\)	thermal conductivity	\([1,3\)]	\(W \cdot m^{-1} \cdot K^{-1}\)
\(h\)	transfer coefficient	\([0.1,5\)]	\(W \cdot m^{-2} \cdot K^{-1}\)
\(Q_1\)	heat source	\([10^4, 10^{6}\)]	\(W \cdot m^{-3}\)
\(Q_1\)	heat source	\([10^4, 10^{6}\)]	\(W \cdot m^{-3}\)
\(V\)	Inflow rate	\([1,30\)]	\(m^2 \cdot s^{-1}\)

Table 2. Table of fixed parameters
Name	Description	Nominal Value	Units
\(t\)	time	\([0, 500\)]	\(s\)
\(T_0\)	initial temperature	\(300\)	\(K\)
IC Parameters
\(\rho C_{IC}\)	heat capacity	\(1.4 \cdot 10^{6}\)	\(J \cdot m^{-3} \cdot K^{-1}\)
\(e_{IC}\)	thickness	\(2\cdot 10^{-3}\)	\(m\)
\(h_{IC} = L_{IC}\)	height	\(2\cdot 10^{-2}\)	\(m\)
\(h_{1}\)	height	\(2\cdot 10^{-2}\)	\(m\)
\(h_{2}\)	height	\(7\cdot 10^{-2}\)	\(m\)

PCB Parameters
\(k_3 = k_{Pcb}\)	thermal conductivity	\(0. 2\)	\(W \cdot m^{-1} \cdot K^{-1}\)
\(\rho C_{3}\)	heat capacity	\(2 \cdot 10^{6}\)	\(J \cdot m^{-3} \cdot K^{-1}\)
\(e_{Pcb}\)	thickness	\(2\cdot 10^{-3}\)	\(m\)
\(h_{Pcb}\)	height	\(13\cdot 10^{-2}\)	\(m\)

Air Parameters
\(T_0\)	Inflow temperature	\(300\)	\(K\)
\(k_4 \)	thermal conductivity	\(3 \cdot 10^{-2}\)	\(W \cdot m^{-1} \cdot K^{-1}\)
\(\rho C_{4}\)	heat capacity	\(1100\)
\(J \cdot m^{-3} \cdot K^{-1}\)	\(e_{a}\)	thickness	\(4\cdot 10^{-3}\)