Gas Sensor

Gas Sensor

Background
Benchmark ID	gasSensor_n66917m1q28
Category	oberwolfach
System-Class	LTI-FOS
Parameters
nstates	66917
ninputs	1
noutputs	28
nparameters	0
components	A, B, C, E
Copyright
License	NA
Creator	Lihong Feng
Editor	Lihong Feng Christian Himpe Judith Schneider
Location
NA

Description: Microhotplate Gas Sensor

Figure 1

Figure 2: Masks disposition (left) and the schematical position of the chosen output nodes (right).

The goal of European project Glassgas (IST-99-19003) was to develop a novel metal oxide low power microhotplate gas sensor . In order to assure a robust design and good thermal isolation of the membrane from the surrounding wafer, the silicon microhotplate is supported by glass pillars emanating from a glass cap above the silicon wafer, as shown in Fig. 1. In this design, four different sensitive layers can be deposited on the membrane. The thermal management of a microhotplate gas sensor is of crucial importance.

The benchmark contains a thermal model of a single gas sensor device with three main components: a silicon rim, a silicon hotplate and glass structure . It allows us to simulate important thermal issues, such as the homogeneous temperature distribution over gas sensitive regions or thermal decoupling between the hotplate and the silicon rim. The original model is the heat transfer partial differential equation.

The device solid model has been made and then meshed and discretized in ANSYS 6.1 by means of the finite element method (SOLID70 elements were used). It contains 68000 elements and 73955 nodes. Material properties were considered as temperature independent. Temperature is assumed to be in degree Celsius with the initial state of $0 C$. The Dirichlet boundary conditions of $T = 0 C$ is applied at the top and bottom of the chip (at 7038 nodes).

The output nodes are described in Table 1. In Fig. 2 the red marked nodes are positioned on the silicon rim. Their temperature should be close to the initial temperature in the case of good thermal decoupling between the membrane and the silicon rim. The black marked nodes are placed on the sensitive layers above the heater and are numbered from left to right row by row, as schematically shown in Fig 2. They allow us to prove whether the temperature distribution over the gas sensitive layers is homogeneous (maximum difference of $10C$ is allowed by design).

Table 1: Inputs and outputs for the gas sensor model.

Number	Code	Comment
1	aHeater	within a heater, to be used for nonlinear input
2-7	SiRim1 to SiRim7	silicon rim
8-28	Memb1 to Memb21	gas sensitive layer

The benchmark contains a constant load vector. The input function equal to $u(t) = 1$ corresponds to the constant input power of $340 mW$. One can insert a weak input nonlinearity related to the dependence of heater's resistivity on temperature given as:

$R(T) = R_{0}(1 + \alpha T)$

where $\alpha =1.469 \cdot 10^{-3} K^{-1}$. To this end, one has to multiply the load vector by a function:

$\frac{U^2 274.94 (1 + \alpha T)}{0.34 (274.94 (1 + \alpha T)+148.13)^2}$

where $U$ is a desired constant voltage. The temperature in the equation above should be replaced by the temperature at the input 1 (aHeater).

The linear ordinary differential equations of the first order are written as:

$\begin{array}{rcl} E \frac{\partial T}{\partial t} &=& A T(t) + B u(t) \\ y(t) &=& C T(t) \end{array}$

where $E$ and $A$ are the symmetric sparse system matrices (heat capacity and heat conductivity matrix), $B$ is the load vector, $C$ is the output matrix, and $T$ is the vector of unknown temperatures. The dimension of the system is $66917$, the number of nonzero elements in matrix $E$ is $66917$, in matrix $A$ is $885141$.

The outputs of the transient simulation at output 18 (Memb11) over the rise time of the device of $5 s$ for the original linear (with constant input power of $340 mW$) and nonlinear (with constant voltage of $14 V$) model are placed in files LinearResults and NonlinearResults respectively. The results can be used to compare the solution of a reduced model with the original one. The time integration has been performed in ANSYS with accuracy of about $0.1 \%$. The results are given as matrices where the first row is made of times, the second of the temperatures.

More information can also be found in .

Data

Download matrices in the Matrix Market format: GasSensor-dim1e5-GasSensor.tar.gz, (8 MB).

The matrix name is used as an extension of the matrix file. File *.C names contains a list of output names written consecutively. The system matrices have been extracted from ANSYS models by means of mor4fem.

The discussion of electro-thermal modeling related to the benchmark including the nonlinear input function can be found in .

Origin

This benchmark is part of the Oberwolfach Benchmark Collection; No. 38880, see .

Dimensions

System structure:

$\begin{align} E \dot{x}(t) &= Ax(t) + B \\ y(t) &= Cx(t) \end{align}$

System dimensions:

$E \in \mathbb{R}^{66917 \times 66917}$, $A \in \mathbb{R}^{66917 \times 66917}$, $B \in \mathbb{R}^{66917 \times 1}$, $C \in \mathbb{R}^{1 \times 66917}$.

Citation

To cite this benchmark, use the following references:

• For the benchmark itself and its data:

The MORwiki Community, Gas Sensor. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Gas_Sensor

@MISC{morwiki_gas,
  author =       {{The MORwiki Community}},
  title =        {Gas Sensor},
  howpublished = {{MORwiki} -- Model Order Reduction Wiki},
  url =          {https://modelreduction.org/morwiki/Gas_Sensor},
  year =         {20XX}
}

• For the background on the benchmark:

@INPROCEEDINGS{BecHWetal04,
  author =       {T. Bechtold, J. Hildenbrand, J. Wöllenstein, J. G. Korvink},
  title =        {Model Order Reduction of 3D Electro-Thermal Model for a Novel, Micromachined Hotplate Gas Sensor},
  booktitle =    {Proceedings of 5th International Conference on Thermal and Mechanical Simulation and Experiments in Microelectronics and Microsystems},
  pages =        {263--267},
  year =         {2004},
  doi =          {10.1109/ESIME.2004.1304049}
}

Contact

Lihong Feng

Tamara Bechtold