Micropyros Thruster

Description

Figure 1

The goal of the European project Micropyros was to develop a microthruster array shown in xx--CrossReference--dft--fig1--xx. It is based on the co-integration of solid fuel with a silicon micromachined system. In addition to space applications, the device can be also used for gas generation or as a highly-energetic actuator. When the production of a bit-impulse is required, the fuel is ignited by heating a resistor at the top of a particular microthruster. Design requirements and modeling alternatives are described in . The discussion of electro-thermal modeling related to the benchmark can be found in .

The benchmark contains a simplified thermal model of a single microthruster to help with a design problem to reach the ignition temperature within the fuel and at the same time not to reach the critical temperature at neighboring microthrusters, that is, at the border of the computational domain. At the same time, the resistor temperature during the heating pulse should not become too high as this leads to the destruction of the membrane.

The benchmark suite has been made with the Micropyros software developed by IMTEK. There are four different test cases described in Table 1 with the goal to cover different cases of different computational complexity. Note that the results from different models cannot be compared directly with each other as the output nodes are located in slightly different geometrical positions and there is some difference in modeling for the 3D and 2D-axisymmetric cases.

Table 1: Microthruster benchmarks.

Code	Comment	Dimension	nnz(A)	nnz(E)
T2DAL	2D-axisymmetric, linear elements	$4\,257$	$20\,861$	$4\,257$
T2DAH	2D-axisymmetric, quadratic elements	$11\,445$	$93\,781$	$93\,781$
T3DL	3D, linear elements	$20\,360$	$265\,113$	$20\,360$
T3DH	3D, quadratic elements	$79\,171$	$2\,215\,638$	$2\,215\,638$

The device solid model has been made and meshed in ANSYS. The material properties assumed to be constant. Temperature is assumed to be in Celsius with the initial state of $0$Celsius.

The output nodes are described in Table 2. Nodes 2 to 5 show the fuel temperature distribution and nodes 6 and 7 characterize temperature in the wafer, nodes 5 and 7 being the most far away from the resistor.

Table 2: Outputs for the microthruster models.

#	Code	Comment
1	aHeater	within the heater
2	FuelTop	fuel just below the heater
3	FT-100	fuel 0.1 mm below the heater
4	FT-200	fuel 0.2 mm below the heater
5	FuelBot	fuel bottom
6	WafTop1	wafer top (touching fuel)
7	WafTop2	wafer top (end of computational domain)

The benchmark files contain a constant load vector, corresponding to the constant power input of $150$mW. In order to insert a weak nonlinearity related to the dependence of the resistivity on temperature, one has to multiply the load vector by a function

$f(T) = 1 + 9 \cdot 10^{-4} \cdot T_1 + 3 \cdot 10^{-7} \cdot T_1^2$

assuming the constant current. The Temperature $T_1$ in the equation above, is the temperature at the node 1.

The first order ordinary differential equations are written as

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

where $E$ and $A$ are the system matrices (both are symmetric), $B$ is the load vector, $C$ is the output matrix, and $T$ is the vector of unknown temperatures. System dimensions:

$E \in \mathbb{R}^{N \times N}$, $A \in \mathbb{R}^{N \times N}$, $B \in \mathbb{R}^{N \times 1}$, $C \in \mathbb{R}^{7 \times N}$

The ANSYS results for the original models as well as the reduced models obtained by mor4fem can be found at the micropyros page: choose EleThermo for T2DAL and T2DAH or EleThermo3D for T3DL and T3DH. The system matrices have been converted to the Matrix Market format by means of mor4fem.

The model reduction of the microthruster model by means of mor4fem is described in .

Origin

This benchmark is part of the Oberwolfach Benchmark Collection, No. 38847.

Data

Download matrices in the Matrix Market format:

• MicropyrosThruster-dim1e3-T2DAL.tar.gz (215.7 kB)
• MicropyrosThruster-dim1e4-T2DAH.tar.gz (1.6 MB)
• MicropyrosThruster-dim1e4-T3DL.tar.gz (2.1 MB)
• MicropyrosThruster-dim1e5-T3DH.tar.gz (36.7 MB)

The matrix name is used as an extension of the matrix file. File *.C.names contains a list of output names written consecutively.

Dimensions

System structure:

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

System dimensions:

$E \in \mathbb{R}^{N \times N}$, $A \in \mathbb{R}^{N \times N}$, $B \in \mathbb{R}^{N \times 1}$, $C \in \mathbb{R}^{7 \times N}$.

System variants:

T2DAL: $N = 4\,257$, T2DAH: $N = 11\,445$, T3DL: $N = 20\,360$, T3DH: $N = 79\,171$.

Citation

To cite this benchmark, use the following references:

• For the benchmark itself and its data:

Oberwolfach Benchmark Collection, Micropyros Thruster. hosted at MORwiki - Model Order Reduction Wiki, 2005. http://modelreduction.org/index.php/Micropyros_Thruster

@MISC{morwiki_thruster,
  author =       {{Oberwolfach Benchmark Collection}},
  title =        {Micropyros Thruster},
  howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
  url =          {http://modelreduction.org/index.php/Micropyros_Thruster},
  year =         2005
}

• For the background on the benchmark:

@InProceedings{RudBKetal02,
  author =       {E.B. Rudnyi and T. Bechtold and J.G. Korvink and
                 C. Rossi},
  title =        {Solid Propellant Microthruster: Theory of Operation
                 and Modelling Strategy},
  booktitle =    {Nanotech 2002 - At the Edge of Revolution, September
                 9--12, 2002, Houston (USA)},
  year =         2002,
  note =         {AIAA Paper 2002-5755},
  doi =          {10.2514/6.2002-5755}
}