Numerical simulations on performance of MEMS-based nozzles at moderate or low temperatures
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- Wang, M.R. & Li, Z.X. Microfluid Nanofluid (2004) 1: 62. doi:10.1007/s10404-004-0008-5
Performance of microelectromechanical systems (MEMS)-based nozzles at moderate and low temperatures is numerically analyzed using the direct simulation Monte Carlo method. Considering the intermolecular attractive potential caused by low temperature, the generalized soft sphere collision model is introduced. The Larsen–Borgnakke model for the generalized sphere model is used to model the energy exchange between the translational and internal modes. The results for nozzle flows at an initial temperature of 300 K show that the temperature behind the throat is quite low and the intermolecular attractive potential cannot be ignored. Different working conditions in two-dimensional (2D) nozzles are simulated using the present method, including exit pressure, inlet pressure, initial temperature, nozzle geometry, and gas species. The effects on the nozzle performance are analyzed. Simulations on flows in a three-dimensional (3D) low aspect ratio flat nozzle show that the increased surface-to-volume ratio, which leads to high viscosity dissipation, causes a much lower flow characteristic and performance comparing with the 2D case.
KeywordsDSMCFlow fieldGeneralized soft sphere modelMEMS-based nozzle
miss-distance impact parameter, m
collision diameter, m
throat width, m
relative translational energy, J
thruster force, N
specific impulse, s
Boltzmann constant, J/K
Knudsen number at throat, defined by averaged values
molecular mass, kg
reduced mass, kg
gas number density, m−3
inlet pressure, Pa
exit pressure, Pa
Reynolds number at throat, defined by averaged values
gas temperature, K
dimensionless temperature kT/ε
wall temperature, K
inlet temperature, K
rotational relax number
soft sphere scattering law
- βj, ωj
parameters for the GSS model
depth of the potential well, J
dimensionless constant for polarization
gas viscosity, kg/m·s
low-velocity diameter, m
total collision cross-section, m2
rotational degree of freedom
translational degree of freedom
integral for the self-diffusion coefficient
integral for the viscosity
At present, microelectromechanical system (MEMS)-based “digital micro propulsion” systems have been developed to offer new possibilities of increased orbit and station-keeping or attitude-controlling capabilities at potentially lower cost to small satellites, microsatellites or even nanosatellites (Lewis 1999; Lewis 2000). The thrust must be very low (0.1~10 mN) for the small mass of a microspacecraft (such as <1 kg). To obtain these low thrust values, a microscale nozzle and low chamber pressures and temperatures are usually used. This leads to the throat Reynolds number within the range between 10 and 500. As a result of such a low level of Reynolds numbers, the viscous losses are significant in micronozzles.
A number of micronozzles have been developed and the performance has been experimentally studied using mass flow and thrust measurements (Kohler et al. 2002; Kerechanin et al. 2001; Reed et al. 2001; Bayt et al. 1998; Bayt. 1999). The previous performance test for micronozzles indicate that specific impulse efficiencies drop rather dramatically for Re<1000, compared with the ideal situations (Bayt 1999). Therefore, a comprehensive study is indispensable for understanding the special features of viscous flow within such micronozzles and for determining the optimal nozzle geometry, working parameters and gas species, which ensure high performances.
Performance evaluations of micro nozzles have been conducted using the Navier–Stokes (NS) solvers (Bayt 1999; Wang et al. 2001) and the direct simulation Monte Carlo (DSMC) method (Markelov et al. 2001; Hyakutake and Yamamoto 2003; Alexeenko et al. 2002; Alexeenko et al. 2003). It was shown that the uses of the 2D continuum approach led to overpredictions of the specific impulse even for Reynolds numbers Re~1000 (Bayt 1999). The standard DSMC of Bird’s (1994) or the DSMC-based software, such as the SMILE of Ivanov et al. (1998), were used for micro nozzle performance analysis. Better agreements with the experimental data were found for high-temperature gas flows expanding from micronozzles into a vacuum (Markelov et al. 2001; Hyakutake and Yamamoto 2003; Alexeenko et al. 2002; Alexeenko et al. 2003). In most of the previous simulations, the variable hard sphere (VHS) model (Markelov et al. 2001; Hyakutake and Yamamoto 2003) or the variable soft sphere (VSS) model (Alexeenko et al. 2002; Alexeenko et al. 2003) was used as the intermolecular potential, in which only the repulsive interaction between molecules is considered. However, in actual gases the force between two molecules is not only repulsive at small distances, but also weakly attractive at long distances. The attractive force effect is very weak at high temperatures in a pure gas, and becomes stronger when temperature is low (such as T<300 K). The previous results showed that the transitional temperature near the exit could also be low even if the inlet gas temperature was high in the nozzle exhausting into a vacuum. Therefore, the attractive potential should be considered to improve the nozzle flow simulations.
The first attempt at reproducing the effects of attraction was made by Kuscer (1989) who suggested a total cross that reproduced Sutherland’s viscosity formula. Hassan and Hash (1993) and Hash et al. (1994) proposed a generalized hard sphere (GHS) model for a Lennard–Jones fluid. In GHS, the total cross is a function of the gas transitional temperature, while the scattering deflection angle remains the hard sphere value. However, it is hard to fit one set of parameters for a good agreement with the standard values in all transport properties (Kunc et al. 1995). Fan (2002) developed the GHS model. He combined the GHS model and the VSS model, and then proposed a generalized soft sphere (GSS) model. Parameters were determined with least-square fitting and remarkable agreements were found in all transport properties with both theoretical and experimental values at moderate and low temperatures.
In this paper, a detailed performance analysis of two-dimensional (2D) MEMS-based nozzles at moderate or low temperatures is presented using the DSMC method in a GSS model. The influence of the pressure boundary condition, the temperature condition, the nozzle geometry, the throat size, and the gas species on the nozzle performance is examined. The current results are compared with other numerical results (e.g., VSS-DSMC and NS). Finally, the three-dimensional (3D) effect is also discussed.
2 Numerical methods
The DSMC method in the GSS collision model was applied in the present work for the low Reynolds number micronozzle flows at moderate and low temperatures. The GSS model is briefly introduced below.
It was satisfactorily found that the gas properties from the GSS model are in better agreement with the experimental data than other models when using this set of parameters (Fan 2002). In the present paper, they are also employed.
The walls are isothermal. The Maxwell model with full momentum and energy accommodation is used for the gas–surface interaction. In this model, the emission of the impinging molecules is not correlated with the pre-impingement state of the molecules. The outgoing velocity is randomly assigned according to a half-range Maxwellian distribution determined by the wall temperature.
The inlet pressure and temperature, as well as the outlet pressure when the outflow is not a vacuum, are specified. The pressure boundary condition of Fang and Liou (2002) is implemented in the present paper.
In current study, the inlet pressure ranges from 1 to 3 atm, the exit pressure from 0.5 atm to a vacuum, the upstream gas temperature from 300 to 1000 K, the expansion area ratio from 1.5 to 3.8, the throat width from 4 to 20 µm, and the gas species varies among N2, O2, CO2, and H2. Non-uniform rectangular cells with several sub-cells are used and the sub-cell sizes are smaller than the local molecular mean free path. The time step is also smaller than the local averaged collision time. All simulations run on Pentium III 550 MHz processors. The total sample size for each case is greater than 1×105, and the running time for each case is more than 100 h.
3 Model comparisons
Before the GSS model is used for simulations and analysis of nozzles, the model applicability is verified firstly. The nozzle performances predicted by the GSS model are compared with those for the VSS model.
Nozzle performance calculated in different models for SFC
4 Results and discussion
4.1 Exit pressure effects
4.2 Inlet pressure effects
Flow characteristics and the nozzle performance under different inlet pressures
4.3 Temperature effects
Flow conditions and nozzle performance under different temperature conditions
The Reynolds number at throat Reth and the thruster force Ft decrease with the temperature, while the specific impulse Isp increases nearly linearly with the temperature. It is shown that the influence of the temperature on the thruster force is not as strong as that on the specific impulse. Especially, when the temperature rises from 300 to 400 K, the specific impulse increases by 14.67%, while the thruster force decreases by 1.43%.
4.4 Geometry effects
Flow characteristics and nozzle performance at different throat width
The decrease in throat width leads to a decrease in Reynolds number, and then a decrease in thruster force. However, the specific impulse is hardly affected.
Flow characteristics and nozzle performance in different nozzle configurations
The results indicate that the nozzle configuration affects the flow characteristics greatly, however, has little effect on the nozzle performance.
4.5 Gas species effects
Flow characteristics and nozzle performance for different gas species
4.6 3D effects
5 Concluding remarks
Performance simulations and analyses of MEMS-based nozzles at moderate and low temperatures have been conducted using the DSMC method. The intermolecular attractive potential is considered because of the low temperature and the GSS collision model is introduced. The Larsen–Borgnakke model, which was suggested in the generalized hard sphere model, is used to model the energy exchange between the translational and internal modes. The initial temperature ranges from 300 to 1000 K and the flow Reynolds number at the throat ranges from 14 to 387.
For nozzle flows exhausting into a vacuum at an initial temperature of 300 K, the temperature in the downstream flow of the throat is quite low (far lower than 200 K). The comparisons between the VSS model and the GSS model show that both the temperature and Mach number distributions separate clearly behind the throat. Because the Knudsen number is low (<0.016), the GSS-DSMC method agrees well with the NS solver in the Mach number distributions along the centerline of the nozzle. The thruster force predicted by GSS-DSMC is a little lower than those by VSS-DSMC and NS solution.
Different working conditions in 2D nozzles are considered. The exit pressure value has a great effect on the nozzle performance and a lower exit pressure leads to a stronger effect. The nozzle thruster force and specific impulse decrease as the exit pressure increases. The inlet pressure affects the thruster force nearly proportionally however has little effect on the specific impulse. A higher initial temperature results in a smaller thruster force and a higher specific impulse. When the throat size is given, the nozzle configuration has a finite effect on the nozzle performance. However, when the nozzle configuration is given the throat size affects the thruster force greatly. Comparisons between different gas species show that a lighter gas can have a higher specific impulse and a smaller thruster force.
3D simulations on a low aspect ratio nozzle flow show that 3D effects cause lower performances compared to the 2D case. The main reason is the increased surface-to-volume ratio, which leads to a high viscous dissipation.
This work gives a solution of performance prediction of MEMS-based nozzles using the DSMC method at moderate and low temperatures. More work should be done to find the optimal nozzle geometries and working conditions in the future.
The present work was supported by the National Natural Science Foundation of China (Grant No. 59995550–2) and National Key Basic Research and Development Program of China (Grant No. G1999033106).