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| [[Category:benchmark]] | | [[Category:benchmark]] |
| [[Category:parametric system]]
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| [[Category:linear system]] | | [[Category:linear system]] |
| [[Category:first order system]] | | [[Category:first order system]] |
| [[Category:physical parameters]]
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| [[Category:four parameters]]
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| [[Category:time invariant]] | | [[Category:time invariant]] |
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| 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. |
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| ==Description of the device==
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| There is a large demand for gas sensing devices in various domains. They are
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| desired in e. g. safety applications where combustible or toxic gases are present or
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| in comfort applications, such as climate controls of buildings and vehicles where
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| good air quality is required. Additionally, gas monitoring is needed in process
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| control and laboratory analysis. All of these applications demand cheap, small
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| and user-friendly gas sensing devices which show high sensitivity, selectivity and
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| stability with respect to a given application.
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| A micromachined gas sensor is not only a challenge with respect to thermal
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| design but also with respect to mechanical design. Only by choosing the right
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| mechanical design a large intrinsic or thermal-induced membrane stress leading
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| to membrane deformation/ breaking of the membrane can be avoided. It is further
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| necessary to build a chemometrics calibration model which correlates the set of
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| sensor resistance measurements to the sensed gas concentration. Prior to fabrication,
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| a thermal simulation is performed to determine the heating efficiency and
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| temperature homogeneity of the gas sensitive regions. As the device is connected to circuitry for heating power
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| control and sensing resistor readout, a system-level simulation is also needed.
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| Hence, a compact thermal model must be generated. (The text above is taken from [1].)
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| ==Description of the model==
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| The heat transfer within a hotplate is described through the governing heat transfer equation [2]
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| <math>
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| \nabla \cdot (\kappa \nabla T) + Q- \rho c_p \frac{\partial T}{\partial t}=0, \quad
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| Q=j^2R(T), \quad (1)
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| </math>
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| 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
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| specific heat capacity in <math>J /(kg*K), \, \rho(r)</math> is the mass density in
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| <math>kg /m^3,</math> and <math>T(r,t)</math> is the temperature distribution. We assume a
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| homogeneous heat generation rate over a lumped resistor:
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| <math>
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| Q = \frac{u^2(t)}{R(T)}
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| </math>
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| with unit <math>W/m^3</math>.
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| We use the initial condition <math> T_0 = 273K </math>, and the
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| Dirichlet boundary condition <math> T = 273 K </math> at the bottom of
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| the computational domain. The convection boundary condition at the top of the membrane is
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| <math>
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| q=h(T-T_{air}),
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| </math>
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| where <math>h</math> is the heat transfer coefficient between the membrane and the ambient air in <math>W/(m^2 K)</math>.
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| Assuming <math>T_{air}=0</math>, spatial discretization of the heat transfer model in (1) leads to the parametrized system as below,
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| <math>
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| (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
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| y=C^T \cdot T.
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| </math>
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| 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>.
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| 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...
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| 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.)
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| ==Data information==
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| 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>.
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| ==References== | | ==References== |
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| University of Freiburg, 2005. | | University of Freiburg, 2005. |
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| [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).
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| Contact information: | | Contact information: |
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| '' [[User:Feng|Lihong Feng]] ''
| | Tamara Bechtold |
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| | tamara.bechtold@imtek.uni-freiburg.de |
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