Silicon Nitride Membrane

Silicon Nitride Membrane

Background
Benchmark ID	siliconNitrideMembrane_n60020m1q2
Category	misc
System-Class	AP-LTI-FOS
Parameters
nstates	60020
ninputs	1
noutputs	2
nparameters	2
components	A, B, C, E
Copyright
License	NA
Creator	Lihong Feng
Editor	Lihong Feng Christian Himpe
Location
NA

Description

Figure 1: silicon nitride membrane temperature profile

A silicon nitride membrane (SiN membrane) can be a part of a gas sensor, but also a part of an infra-red sensor, a microthruster, an optical filter etc. This structure resembles a microhotplate similar to other micro-fabricated devices such as gas sensors and infrared sources (See also Gas Sensor Benchmark). See Fig. 1, the temperature profile for the SiN membrane.

The governing heat transfer equation in the membrane is:

$\nabla \cdot (\kappa \nabla T)+Q - \rho c_p \cdot \frac{\partial T}{\partial t}=0$

where $\kappa$ is the thermal conductivity in $W m^{-1} K^{-1}$, $c_p$ is the specific heat capacity in $J kg^{-1} K^{-1}$, $\rho$ is the mass density in $kg m^{-3}$ and $T$ is the temperature distribution. We assume a homogeneous heat generation rate over a lumped resistor:

$Q = \frac{u^2(t)}{R(T)}$

with $Q$ the heat generation rate per unit volume in $W m^{-3}$. We use the initial condition $T_0 = 273K$, and the Dirichlet boundary condition $T = 273 K$ at the bottom of the computational domain.

The convection boundary condition at the top of the membrane is

$q=h(T-T_{air}),$

where $h$ is the heat transfer coefficient between the membrane and the ambient air in $W m^{-2} K^{-1}$.

Discretization

Under the above convection boundary condition and assuming $T_{air}=0$, a finite element discretization of the heat transfer model leads to the parametrized system as below,

$(E_0+\rho c_p \cdot E_1) \dot{T} +(A_0 +\kappa \cdot A_1 +h \cdot A_2)T = B \frac{u^2(t)}{R(T)}, \quad y=C T,$

where the volumetric heat capacity $\rho c_p$, thermal conductivity $\kappa$ and the heat transfer coefficient $h$ between the membrane are kept as parameters. The volumetric hear capacity $\rho c_p$ is the product of two independent variables, i.e. the specific hear capacity $c_p$ and the density $\rho$. The range of interest for the four independent variables are $\kappa \in [2, 5]$, $c_p \in [400, 750]$, $\rho \in [3000,3200]$ and $h \in [10, 12]$ respectively. The frequency range is $f \in [0,25]Hz$. What is of interest is the output in time domain. The interesting time interval is $t \in [0,0.04]s$.

Here $R(T)$ is either a constant heat resistivity $R(T)=R_0$, or $R(T)=R_0(1+\alpha T)$, which depends linearly on the temperature. Here we use $R_0=274.94 \Omega$ and temperature coefficient $\alpha=2.293 \pm 0.006 \times 10^{-4}$. The model was created and meshed in ANSYS. It contains a constant load vector $B$ corresponding to the constant input power of $2.49mW$. The number of degrees of freedom is $n=60,020$.

The input function $u(t)$ is a step function with the value $1$, which disappears at the time $0.02s$. This means between $0s$ and $0.02s$ input is one and after that it is zero. However, be aware that $u(t)$ is just a factor with which the load vector B is multiplied and which corresponds to the heating power of $2.49mW$. This means if one keeps $u(t)$ as suggested above, the device is heated with $2.49mW$ for the time length of 0.02s and after that the heating is turned off. If for whatever reason, one wants the heating power to be $5mW$, then $u(t)$ has to be set equal to two, etc. When $R(T)=R_0(1+\alpha T)$, it is a function of the state vector $T$ and hence, the system has non-linear input (It is also called a weakly nonlinear system.).

Data

The model is generated in ANSYS. The system matrices are in MatrixMarket format and can be downloaded here: SiN_membrane.tgz.

Dimensions

System structure:

$\begin{array}{rcl} (E_1 + \rho c_p E_2)\dot{x}(t) &=& -(A_1 + \kappa A_2 + h A_3)x(t) + Bu(t) \\ y(t) &=& Cx(t) \end{array}$

System dimensions:

$E_{1,2} \in \mathbb{R}^{60020 \times 60020}$, $A_{1,2,3} \in \mathbb{R}^{60020 \times 60020}$, $B \in \mathbb{R}^{60020 \times 1}$, $C \in \mathbb{R}^{2 \times 60020}$.

Contact

Lihong Feng

Tamara Bechtold