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Difference between revisions of "Gas Sensor"

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[[Category:benchmark]]
 
[[Category:benchmark]]
[[Category:parametric system]]
 
 
[[Category:linear system]]
 
[[Category:linear system]]
 
[[Category:first order system]]
 
[[Category:first order system]]
[[Category:physical parameters]]
 
[[Category:four parameters]]
 
 
[[Category:time invariant]]
 
[[Category:time invariant]]
   
This is an extension of the non-parametrized model of Gas sensor in the Oberwolfach Model Reduction Benchmark Collection (http://simulation.uni-freiburg.de/downloads/benchmark Gas sensor(38880)) to a parametrized model.
+
This is a non-parametrized first order linear model. The description of the model can be found in the Oberwolfach Model Reduction Benchmark Collection (http://simulation.uni-freiburg.de/downloads/benchmark Gas sensor(38880)). The data of the system can also be downloaded there.
   
==Description of the device==
 
   
There is a large demand for gas sensing devices in various domains. They are
 
desired in e. g. safety applications where combustible or toxic gases are present or
 
in comfort applications, such as climate controls of buildings and vehicles where
 
good air quality is required. Additionally, gas monitoring is needed in process
 
control and laboratory analysis. All of these applications demand cheap, small
 
and user-friendly gas sensing devices which show high sensitivity, selectivity and
 
stability with respect to a given application.
 
 
A micromachined gas sensor is not only a challenge with respect to thermal
 
design but also with respect to mechanical design. Only by choosing the right
 
mechanical design a large intrinsic or thermal-induced membrane stress leading
 
to membrane deformation/ breaking of the membrane can be avoided. It is further
 
necessary to build a chemometrics calibration model which correlates the set of
 
sensor resistance measurements to the sensed gas concentration. Prior to fabrication,
 
a thermal simulation is performed to determine the heating efficiency and
 
temperature homogeneity of the gas sensitive regions. As the device is connected to circuitry for heating power
 
control and sensing resistor readout, a system-level simulation is also needed.
 
Hence, a compact thermal model must be generated. (The text above is taken from [1].)
 
 
==Description of the model==
 
 
The heat transfer within a hotplate is described through the governing heat transfer equation [2]
 
 
<math>
 
\nabla \cdot (\kappa \nabla T) + Q- \rho c_p \frac{\partial T}{\partial t}=0, \quad
 
Q=j^2R(T), \quad (1)
 
</math>
 
 
where <math>\kappa(r)</math> is the thermal conductivity in <math>W/(m*K)</math> at the position <math>r, \, c_p</math> is the
 
specific heat capacity in <math>J /(kg*K), \, \rho(r)</math> is the mass density in
 
<math>kg /m^3,</math> and <math>T(r,t)</math> is the temperature distribution. We assume a
 
homogeneous heat generation rate over a lumped resistor:
 
 
<math>
 
Q = \frac{u^2(t)}{R(T)}
 
</math>
 
 
with unit <math>W/m^3</math>.
 
We use the initial condition <math> T_0 = 273K </math>, and the
 
Dirichlet boundary condition <math> T = 273 K </math> at the bottom of
 
the computational domain. The convection boundary condition at the top of the membrane is
 
 
<math>
 
q=h(T-T_{air}),
 
</math>
 
 
where <math>h</math> is the heat transfer coefficient between the membrane and the ambient air in <math>W/(m^2 K)</math>.
 
 
Assuming <math>T_{air}=0</math>, spatial discretization of the heat transfer model in (1) leads to the parametrized system as below,
 
 
<math>
 
(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 \cdot T.
 
</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=1.469 \times 10^{-3}</math>. The model was created and meshed in ANSYS. It contains a constant load vector corresponding to the constant input power of <math>2.49mW</math>. The number of degrees of freedom is <math>n=60,020</math>.
 
 
The input function <math>u(t)</math> is a step function with the value <math>1</math>, which disappears at the time <math>0.02s</math>. This means between <math>0s</math> and <math>0.02s</math> input is one and after that it is zero. However, be aware that <math>u(t)</math> is just a factor with which the load vector B is multiplied and which corresponds to the heating power of <math>2.49mW</math>. This means if one keeps <math>u(t)</math> as suggested above, the device is heated with <math>2.49mW</math> 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 <math>5mW</math>, then <math>u(t)</math> has to be set equal to two, etc...
 
When <math>R(T)=R_0(1+\alpha T)</math>, it is a function of the state vector <math>T</math> 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 *.<math>A_i, \, i=0,1,2</math> correspond to the system matrices <math>A_i, \, i=0,1,2</math>, respectively. The files named by <math>*.E_i, \, i=0,1,2</math> correspond to <math>E_i, \, i=0,1</math>. The file named by <math>*.B</math> corresponds to the load vector <math>B</math> and the file named by <math>*.C</math> corresponds to the output matrix <math>C</math>.
 
   
 
==References==
 
==References==
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University of Freiburg, 2005.
 
University of Freiburg, 2005.
   
[2] T. Bechtold, D. Hohfel, 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).
 
   
   
 
Contact information:
 
Contact information:
   
  +
Tamara Bechtold
'' [[User:Feng|Lihong Feng]] ''
 
  +
  +
tamara.bechtold@imtek.uni-freiburg.de

Revision as of 16:32, 16 November 2012


This is a non-parametrized first order linear model. The description of the model can be found in the Oberwolfach Model Reduction Benchmark Collection (http://simulation.uni-freiburg.de/downloads/benchmark Gas sensor(38880)). The data of the system can also be downloaded there.


References

[1] T. Bechtold, "Model Order Reduction of Electro-Thermal MEMS", PhD thesis, Department of Microsystems Engineering, University of Freiburg, 2005.


Contact information:

Tamara Bechtold

tamara.bechtold@imtek.uni-freiburg.de