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Numerical Simulation of Compressible Gas Flows in a Micro-nozzle Using Direct Simulation Monte Carlo Method

Vincent Lijoand Heuy-Dong Kim*

Numerical Simulation of Compressible Gas Flows in a Micro-nozzle Using Direct Simulation Monte Carlo Method

Vincent Lijoand Heuy-Dong Kim* Key Words : DSMC (), micro-nozzle (), rarefied gas (),shock waves (),supersonic flow ()

Abstract

In order to obtain insight into the physics of micro-nozzle flows, numerical simulations of rarefied flows in a convergent-divergent micro-nozzle is investigated by using the Direct Simulation Monte Carlo (DSMC) method. This method can be applied to a wide range of rarefied flows within regimes that neither Navier-Stokes nor collisionless Boltzmann equations are appropriate. In the present work, the molecular collision kinetics is modeled by the variable hard sphere model and energy exchange between kinetic and internal modes is controlled by the phenomenological Larsen- Borgnakke statistical model. Simulations are performed by considering a non-reacting gas model consisting of two chemical species, N2 and O2 for various back pressures and results are presented for the computed flow field quantities. Comparisons are made with the available experimental data, and the factors which affect the solutions are discussed. This study revealed that in micro-nozzles surface effects play the main role on the flow structure. Separate calculations are also performed for the macro-nozzle flows and detailed comparisons between typical rarefied and continuum behaviors are made.

1. INTRODUCTION

Micro propulsion system for the new generation micro-satellites is capable of delivering low thrust for orbital maintenance, small maneuvers to correct trajectories and to overcome drag present in the space navigation. Because of the low moment of inertia of small spacecraft, the thrust requirements are mostly in the micro-milli Newton (N-mN) range.

One of the simplest forms of micro propulsion system is a cold gas thruster (micro-nozzle) in which, ideally, the cold gas or a mixture of gases pressurized in a chamber is accelerated in the convergent section of the nozzle to sonic conditions and then further to supersonic in the divergent/expander section to the exit. These microthrusters can be applied individually or as array patterns to small satellite propulsive systems.

For micro sized devices, the Knudsen number (Kn)

which is defined as the ratio of the molecular mean free path () to a characteristic geometry length (Dh) determines the degree of rarefaction and the applicability of traditional flow models.

22h h

kTKnD pD

pi= = (1)

where T is the temperature, the molecular diameter, p the pressure and k the Boltzmann constant (1.38x10-23 m2kg/s2K).

For Kn < 10-3, the flow is continuum flow, and it can be accurately modeled by the compressible NavierStokes equations with classical no-slip boundary conditions [1].

Low pressure gas flows in micro-nozzles, can seldom be treated as fully continuum flow with no-slip boundary conditions. These micro-flows usually experience continuum regime from gas chamber to the convergent part of the nozzle, slip flow (10-2 < Kn 0.1 [3].For the slip flow regime, Navier-Stokes equations with discontinuous boundary conditions

FMTRC, Daejoo Machinery Co. Ltd., Daegu, Korea

*School of Mechanical Engineering, Andong National University, Korea E-mail: kimhd@ andong.ac.kr (Prof. H. D. Kim)

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of velocity slip and temperature jump must be taken into account.

For micro-nozzles with Kn < 10-2 , slip flow and rarefaction effects can be neglected. In this regime, micro-nozzle may be simulated based on the compressible Navier-Stokes equations [2-3]. Comparisons between numerical data and experiments have been provided by [4-5]. For micro-nozzles with 10-2 < Kn

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different approaches and the experimental values is fairly good and clearly demonstrates the proper behavior. In the experiments, the upstream pressure (P0) is kept constant (100 kpa) while the downstream pressure (pe) is varied.

Fig. 2. Numerical setup for simulations.

Fig. 3. Comparisons of predicted and experimental mass flow rates.

4. Results and discussion

Figure 4 shows the centerline Mach distributions for different values of pe. The maximum Mach number for the nozzle as per 1D theory is 2.0. But, for the micro-nozzle the maximum Mach number at the exit is around 1.5. The reduction is Mach number is due to the boundary layer blockage and the higher temperature in the micro-nozzle divergent part (Fig. 5). The higher temperature is due to the dominant viscous losses resulting from a larger surface-to-volume ratio when compared to that of the conventional scale nozzle.

The more intense changes in these properties occur for pe equal to 10 kPa, which induces steady supersonic flows in the divergent part. It can be seen that, the Mach distribution is over predicted by the 1D theory in the divergent part, due to the failure in accounting the viscous effects which are dominant in the micro-scale. Also, for the conventional macro-nozzles, viscous effects are virtually negligible and, as a consequence, these nozzles can be accurately designed and analyzed based on one-dimensional isentropic flow theory.

Fig. 4. Mach number distributions along the micro-nozzle axis.

Fig. 5 Centerline static temperature distributions.

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(a) Under-expanded flows (both nozzles) with shock waves appearing in the plume.

(b) Macro-nozzle with shock waves appearing near

the exit; perfectly-expanded micro-nozzle with no shockwaves.

Fig. 5. Pressure distributions along the axis.

Fig. 6. Mach contours in macro and micro nozzles.

Shock waves appearing near the throat of the macro-nozzle are a result of the choice of sharp throat geometry (c.f. Fig. 1). The flow is strongly deflected near the throat, which results in the formation of a weak shock along the divergent part of the macro-nozzle. However, for the micro-nozzle, the supersonic flow will never experience such an abrupt flow path change, since the sonic throat is further downstream from the actual sharp throat (c.f. Fig. 4). Hence, no shock waves are observed near the throat of the micro-nozzle.

When operated under the same pressure ratio, there is a strong shock occurring near the end of the macro-nozzle. However it can be seen that micro-nozzle flow are devoid of shockwaves. The mysterious disappearance of the shockwaves at the exit can be seen in Fig. 6.

From Fig. 5b, for a macro-nozzle, strong shocks are observed near the exit, since the flow is highly over-expanded. The thick boundary layer present along the divergent part of the micro-nozzle reduces the area ratio, such that the flow became perfectly-expanded. As the micro-nozzle flow is perfectly-expanded, no shockwaves are observed near the exit.

Nozzle Type Pe Thrust Specific Impulse (s) Micro

10 73.6 N 14.74 30 48.6 N 9.9

Macro 10 2.3 kN 52.5 30 1.5 kN 33.9 Table 1. Comparison of thrusts at different scales.

In micro-nozzles, the viscous boundary layer will have a larger thickness as compared to the macro-nozzles, which lowers the exit velocity considerably. Thrust is also reduced due to the shock waves inside (cf. Table 1). For micro-nozzles with thrusts in the N range, the molecular mean free path is no longer smaller compared to the dimensions of the nozzle and the DSMC method used properly accounted for the rarefaction effects.

5. Conclusions

In the present work, numerical simulations have been carried with compressible NavierStokes equations with classical no-slip boundary conditions, Maxwell velocity-slip/Smoluchowski temperature jump conditions and DSMC methods to understand the flows in micro-nozzles when downscaling from macro to micro scale. In micro-nozzle, rarefaction effects strongly influence the overall performance. Unlike the conventional nozzles, one-dimensional isentropic flow theory is inadequate for predicting the micro-nozzle flow physics. A comparison between isentropic theory, numerical simulations and experimental results has been presented for a micro-nozzle expanding into a low pressure ambient, showing a good agreement between the available data. Also, shockwaves occurring inside the macro-nozzle is

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compared with the shock-free flow in micro-nozzles and the possible reasons for this non-traditional physics are enumerated.

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