Difference between revisions of "Silicon Nitride Membrane"

Revision as of 10:49, 21 November 2012

Description of the model

A silicon nitride membrane (SiN membrane) [1] can be a part of a gas sensor, but also a part of an infra-red sensor, a michrothruster, an optimical filter etc. This structure resembles a microhotplate similar to other micro-fabricated devices such as gas sensors [2] and infrared sources [3]. 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 $W m^{-1} K^{-1}$ , cp 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 $T_{air}=0$ , 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 respectively $\kappa \in [2, 5]$ , $c_p \in [400, 750]$ , $\rho \in [3000,3200]$ , $h \in [10, 12]$ . The frequency range is $f \in [0,50]Hz$ .

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 information

The system matrices are in MatrixMarket format (http://math.nist.gov/MatrixMarket/) and can be downloaded here File:Matrices gassensor.tgz. The files named by *. $A_i, \, i=0,1,2$ correspond to the system matrices $A_i, \, i=0,1,2$ , respectively. The files named by $*.E_i, \, i=0,1$ correspond to $E_i, \, i=0,1$ . The file named by $*.B$ corresponds to the load vector $B$ and the file named by $*.C$ corresponds to the output matrix $C$ .

References

[1] T. Bechtold, D. Hohfeld, E. B. Rudnyi and M. Guenther, "Efficient extraction of thin-film thermal parameters from numerical models via parametric model order reduction," J. Micromech. Microeng. 20(2010) 045030 (13pp).

[2] J. Spannhake, O. Schulz, A. Helwig, G. Müller and T. Doll,"Design, development and operational concept of an advanced MEMS IR source for miniaturized gas sensor systems ," Proc. Sensors, 762-765, 2005.

[3] M. Graf, D. Barrettino, S. Taschini, C. Hagleitner, A. Hierlemann and H. Baltes, "Metal oxide-based monolithic complementary metal oxide semiconductor gas sensor microsystem," Anal. Chem., 76:4437-4445, 2004.

Fig.1

Contact information:

Lihong Feng

Tamara Bechtold (tamara.bechtold@imtek.uni-freiburg.de)

@@ Line 45: / Line 45: @@
 where the volumetric heat capacity <math>\rho c_p</math>, thermal conductivity
 <math>\kappa</math> and the heat transfer coefficient <math>h</math> between the membrane
-are kept as parameters. The volumetric hear capacity <math>\rho c_p</math> is the product of two independent variables, i.e. the specific hear capacity <math>c_p</math> and the density <math>\rho</math>. The range of interest for the four independent variables are respectively <math>\kappa \in [2, 5]</math>, <math>c_p \in [400, 750]</math>, <math>\rho \in [3000,3200]</math>, <math> h \in [10, 12]</math>. The frequency range is <math>f \in [0,10]Hz</math>.
+are kept as parameters. The volumetric hear capacity <math>\rho c_p</math> is the product of two independent variables, i.e. the specific hear capacity <math>c_p</math> and the density <math>\rho</math>. The range of interest for the four independent variables are respectively <math>\kappa \in [2, 5]</math>, <math>c_p \in [400, 750]</math>, <math>\rho \in [3000,3200]</math>, <math> h \in [10, 12]</math>. The frequency range is <math>f \in [0,50]Hz</math>.
 Here <math>R(T)</math> is either a constant heat resistivity <math>R(T)=R_0</math>, or <math>R(T)=R_0(1+\alpha T)</math>, which depends linearly on the temperature. Here we use <math>R_0=274.94 \Omega</math> and temperature coefficient <math>\alpha=2.293 \pm 0.006 \times 10^{-4}</math>. The model was created and meshed in ANSYS. It contains a constant load vector <math>B</math> corresponding to the constant input power of <math>2.49mW</math>. The number of degrees of freedom is <math>n=60,020</math>.