Unsteady Computations of Rarefied Gas Flows Induced by a
Rotor-Stator Interaction in a Disk-Type Drag Pump
Young-Kyu Hwang, Joong-Sik Heo, and Myoung-Keun Kwon
School of Mechanical Engineering, Sungkyunkwan University, 300 Chunchun-dong,
Jangan-ku, Suwon 440-746, S. Korea
Abstract. The direct simulation Monte Carlo (DSMC) method to predict the steady and unsteady flow fields in a singlestage disk-type drag pump is presented. Two different kinds of pumps are considered: the first one is a rotor-rotor
combination, and the second one is a rotor-stator combination. In the present DSMC method, the variable hard sphere
molecular model with the no time counter technique is employed to simulate the molecular collision kinetics. For
simulation of diatomic gas flows, the Borgnakke-Larsen phenomenological model is adopted to redistribute the
translational and internal energies. Some experimental works are carried out to verify the numerical results. Compression
ratio and pumping speed for the nitrogen gas are measured under the outlet pressure range of 1∼526 Pa. The flowmeter
method is adopted to calculate the pumping speed. The present experimental data show the leak-limited value of the
compression ratio in the molecular transition region.
INTRODUCTION
The vacuum pumping system composed of a drag pump with a dry backing pump is widely used in the etching
and high-density plasma processes. Recently, the semiconductor industry is getting into a new era by using 300 mm
wafers. Therefore, there is a growing need for a drag pump with higher discharge pressure and larger throughput
capabilities to provide an ultraclean environment and productivity.1,2
In the present study, the pumping performance of a disk-type drag pump (DTDP) is analyzed. Since Siegbahn
proposed a DTDP, considerable efforts have already been invested in studying rarefied gas flows within pumping
channels of a DTDP. Chu3 proposed a hybrid molecular pump combined a DTDP with a turbomolecular pump.
Mase et al.4 developed a new type of turbo pump, which could attain an inlet pressure of the order of 10-2 Pa under
an outlet pressure of atmosphere.
By using the matrix-probability method, Liu and Pang5 studied the pumping performance of a DTDP in free
molecular flow. Shi et al.6,7 calculated the transmission probability of molecules by the test particle Monte Carlo
method. They found that the performance is strongly dependent on the rotational speed and geometrical parameters.
Cheng et al.8 studied the continuum flows in a single-stage (rotor-rotor combination) DTDP by the
computational fluid dynamics (CFD) methodology, which employs the Navier-Stokes (N-S) equations with no-slip
boundary conditions and energy equation. Finite difference approximations were employed to solve the transport
equations with a body-fitted grid system. Recently, Heo and Hwang9 numerically studied the molecular transition
and slip flows by using both the direct simulation Monte Carlo (DSMC) method and the N-S equations with secondorder slip boundary conditions. They found that the numerical results obtained by both methods agree well with the
experimental data for the Knudsen number Kn ≤ 0.02 . With the exception of the study by Cheng et al.8, most of the
previous numerical studies considered only half-stage (i.e., single rotor or single stator) DTDP. The objectives of the
present study are to analyze the complex molecular transition flow field and to investigate the effects of the rotorstator (RS) and the rotor-rotor (RR) interaction on the flow field in a single-stage DTDP.
First, for a half-stage DTDP, the velocity and pressure fields are predicted by using the DSMC method. Second,
for a single-stage DTDP, both RS and RR configurations are considered. Experiments are performed for a singlestage DTDP under the outlet pressure range of 1∼526 Pa. Our experimental results are also compared with
numerical ones obtained by the DSMC method.
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
1
PUMP GEOMETRY AND NUMERICAL METHOD
A typical geometry of a single-stage DTDP is illustrated in Fig. 1. In the present study, both RS and RR
configurations are considered. In the RS configuration, spiral channels are cut on a rotor and stator, while, in the
case of the RR configuration, those are cut on both upper and lower sides of a rotating disk. In each configuration,
both the rotor and stator have ten blades with Archimedes' spiral profile, each with an exit angle of 10 deg relative to
the tangent. At the inlet, the blade angle relative to the tangent is 30 deg.
Figure 1(a) shows the RR configuration and the surface grid of the impeller. The system of blade passage
coordinates is given in Fig. 1(b), where Wr and Wt denote the radial velocity and its normal component,
respectively.
Wr
Suction
surface
Wt
Ω
Pressure
surface
r
R2
Ω
(a)
(b)
FIGURE 1. Disk-type drag pump: (a) rotor-rotor configuration and blade surface computational grid; (b) system of blade
passage coordinates.
The DSMC method used in this study is based on the principles described by Bird.10 The interaction between
molecules is modelled by the variable hard sphere scattering assuming an inverse-power interatomic potential. The
no time counter method is used as a collision sampling technique. For the calculation of the rotational energy
exchange between the colliding molecules, the Borgnakke-Larsen phenomenological model11 is employed.
In order to limit the memory and CPU required to perform the calculations, the computational domain of the
DTDP is restricted to one blade passage. In the analysis of the RR configuration, the rotating frame of reference is
used. Figure 2(a) shows a 3D body-fitted grid system of the RR. However, since the relative position of non-moving
and moving parts of the RS configuration changes, computations of flows in such systems require a special
treatment involving some grid movement. To analyze the RS, an interface between rotating and stationary
components has to be defined.
Figure 2(b) shows the block structure of the RS configuration. Due to the relative movement of stationary and
rotating blocks, the grid system becomes time dependent, and has to be updated after each time step. The
circumferential faces of the grid are supplied as periodic boundary conditions. At each successive time after the
initial condition, the rotating zones rotate by an increment Ωt . At the end of each time step the grid coordinates of
the rotating part are recomputed according to the rotational velocity. The velocity vectors in each cell are adapted to
the grid rotation. Therefore, for every time step, a new frozen rotor interface is defined. For the period of a time step,
the interface appears as a steady-state condition. The numerical procedure can be considered as a series of snapshots
leading to a transient simulation. In the present implementation of the method, the grid movement has to be in such a
way that the grid lines match at the interfaces between stationary and rotating blocks. This means that the time step
must be related to the azimuthal grid size and the rotational velocity. The present DSMC method couples a threedimensional unsteady flow calculation in the rotor with a three-dimensional time-averaged flow calculation in both
2
the radial gap and the stator. This method has been used to calculate unsteady flows in a centrifugal compressor with
a volute.12
Rotor
Rotor
Stator
Rotor
(a)
(b)
FIGURE 2. Computational grid system for a single-stage DTDP: (a) rotor-rotor; (b) rotor-stator.
EXPERIMENTAL APPARATUS
Experiments on the pumping characteristics of the DTDP are carried out by using the test pump to verify the
present numerical results. A schematic diagram of the experimental apparatus is shown in Fig. 3. The test pump is
connected to a two-stage oil rotary pump (970 l/min). The pressure in the high-vacuum side is measured with a
Pirani (0.1∼1000 Pa) and Penning (1.0×10-5∼1 Pa) gauge, and the pressure in the fore-vacuum side is measured
with a Pirani gauge. Experiments are performed by varying the outlet pressure P2 (location f in Fig. 4) in the range
of 1 Pa ≤ P2 ≤ 526 Pa. The rotational speed of the rotor is 24,000 rpm and is controlled by a frequency converter.
Test gas ( N 2 ) is supplied through a mass flow controller from a regulated high-pressure cylinder.
A cross-section of the DTDP is illustrated in Fig. 4 for the case of the RS configuration. The test pump consists
of a plane disk (without blades), rotor and stator. Dimensions and shape of the spiral channel for rotor and stator are
all the same. The pressure is measured at the inlet side (c), the rotor outlet (d), the stator outlet (e), and the
discharge side (f), respectively.
Variable
leak valve Mass flow
controller
Pirani
gauge
Penning gauge
Inlet
Pirani gauge
1
Test pump
N2
Frequency
converter
Rotor
Stator
Gas supply
Pirani
gauge
Gauge controller
Block
valve
Pump
2
Pirani gauge
Pirani gauge
3
Water
4
To rotary pump
Rotary pump
Gas outlet
FIGURE 3. Experimental apparatus.
FIGURE 4. Cross-section of a DTDP.
3
RESULTS AND DISCUSSION
Experiments are performed for a single-stage DTDP with RS and RR combinations in the outlet pressure range
of 1∼526 Pa. Also, comparison between the experimental data and the DSMC results are presented. In the previous
paper9 for the RS case, the DSMC results were compared with the CFD solutions for the single stage DTDP. For the
helical-type drag pump, the DSMC results were compared with both the simple analytical Couette-Poiseuille
solutions by the diffusion equation and the CFD solutions, respectively.13 However, in the present case of RS and
RR configurations, it is not easy to get the simple analytical solutions because of the complex flow field and
geometry. Further studies should be concentrated on the analytical solutions for the RS and RR configurations.
Experimental data at two different flow rates are shown in Fig. 5. This figure shows that the compression
characteristics of the test pump are very different from those of the turbomolecular pump. The maximum
compression (= P2 / P1 ) ratio at Q = 0 is about 100 at around P2 =31 Pa for the RR and at around P2 =10 Pa for the
RS, respectively. When the outlet pressure is lower than 12 Pa, the combination of RS gives higher compression
ratio than that of RR. In the RS case, the function of stator for compression is remarkably deteriorated as the outlet
pressure increases. But, in the RR case, the pumping performance is relatively higher because the rotating channels
impart angular momentum directly to the gas molecules.
The selected outlet pressure for the present numerical simulations is 131.6 Pa. The Knudsen number based on
this outlet pressure and the channel depth is 0.006 for the RS and 0.01 for the RR. The time step must be small
compared to the mean collision time between molecules. Therefore, the time step, ∆t = 5.0 ×10−7 , was chosen in the
case of the RR. In the case of the RS, it is also related to the azimuthal grid size and the rotational velocity. A value
of ∆t = 2.5 ×10−7 was chosen for this case so that a point-to-point matching was maintained at the interface between
the moving and stationary zones where the grid spacing is uniform in the azimuthal direction.
The pressure difference ∆P (= P2 − P1 ) as a function of the throughput Q at P2 =131.6 Pa is shown in Fig. 6.
Square symbols connected by solid lines give the experimental data. Circles indicate the DSMC results. The value of
Q increases as ∆P decreases. Comparison between the experimental data and the DSMC results shows good
agreement.
The average pressure profiles along the pumping passage for different inlet pressure conditions are shown in Fig.
7. All of the simulations show similar tendencies for the prediction of the pressure rise but different pressure levels
are attained at the radial gap between rotor and casing wall. For comparison, the experimental data are also included
in Fig. 7(a). It is seen from this figure that the DSMC results agree well with the experimental data in the case of the
RS. At the radial gap, no further pressure rise takes place because of both the abrupt reduction in flow area and the
higher velocity. This phenomenon is similar to the case of the RR, as seen in Fig. 7(b). In the RS configuration, most
of the pressure rise occurs near the outlet of the upper rotor. On the other hand, in the case of the RR, the rate of the
pressure rise in the lower rotor is larger than that in the upper rotor.
10 3
Rotor-Rotor
150
Rotor-Stator
Rotor-Stator
Experiment
0 sccm
DSMC
200 sccm
10 2
100
∆ P (Pa)
Compression ratio
Rotor-Rotor
10 1
10 0
10 0
10 1
10 2
50
0
10 3
0
100
200
300
400
500
600
Q ( Pa l/s )
Outlet pressure (Pa)
FIGURE 5. Compression ratio vs outlet pressure.
FIGURE 6. Pressure difference vs throughput.
4
P2=131.6 Pa
P1=59.2 Pa
P1=98.7
P1=59.2 (Exp.)
300
P 1=26.3 Pa
150
P 1=46.1
Pressure (Pa)
Pressure (Pa)
250
P 2=131.6 Pa
200
150
100
P 1=65.8
100
50
50
0
0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
x/L t
x/Lt
(a)
(b)
FIGURE 7. Pressure distribution along pumping channel: (a) rotor-stator; (b) rotor-rotor.
Computed flow fields for the RR configuration at P1 =59.2 Pa and P2 =131.6 Pa are shown in Fig. 8. Figure 8(a)
shows relative velocity vectors near the clearance region between the rotor and the stator. In the upper blades, the
inlet flow rushes into the radial gap with a high velocity. The gas moves from the suction surface (SS) to the
pressure surface. In the lower blades, most of the gas flows toward the channel exit along the pressure surface (PS),
and the gas near the suction surface decelerates and flows back from the exit along the surface. Similar flow patterns
were also obtained by Cheng et al.9 Figure 8(b) shows pressure contours. In the upper blades, the pressure is nearly
constant from the inlet to the half section of the channel, as can be seen in Figs. 7(b) and 8(b). The pressure contours
show that the pressure in the pressure surface of the flow channel is larger than that in the suction surface. Also, a
large pressure gradient near the outlet of the lower channel can be seen.
SS
50
60
50
70
80
100
PS
50
150 m/s
Unit: Pa
60
4.5 mm
rotor
SS
70
4.5 mm
80
90
stator
120
140
150
PS
(a)
100
110
130
(b)
FIGURE 8. Flow fields for steady computations: (a) velocity vectors; (b) pressure contours.
5
SS
PS
150 m/s
4.5 mm
(a)
150 m/s
(b)
FIGURE 9. Velocity vectors for unsteady computations.
SS
80
70
110
120
60
50
90
PS
160
180
(a)
70
80
90
110
120 160
180
60
50
(b)
FIGURE 10. Pressure contours for unsteady computations.
6
Unsteady flow fields for the RS configuration using the moving grid technique are presented in Figs. 9 and 10.
The unsteady computation was not started from a fluid at rest but with initial values taken from the solution of the
steady-state computation at the first grid position. The solution is then advanced by one time step, which initiates a
rotation of the rotating grid zones by an angular movement equal to Ω∆t . Experience has shown that about four
impeller rotations are needed before a periodic impeller flow is obtained.
Figure 9 illustrates velocity vectors for different values of t after startup from initial conditions at P1 =59.2 Pa.
All of the simulations show similar flow patterns. The pressure contours, as seen in Fig. 10, show that, at the same
radial distance r (see Fig. 1(b)), the pressure in the pressure surface of flow channel is larger than that in the suction
surface. The pressure also increases with flow along the channel, and most of the pressure rise occurs near the outlet
of the rotor. This also can be seen in Fig. 7(b). The pressure gradient near the outlet of the RR is larger than that of
the RS (see Fig. 7).
To clearly show the reverse flow near the outlet, contours of the radial velocity, Wr , in the impeller at two
different radial locations along the flow passage are shown in Fig. 11. Figure 11 illustrates the radial velocity
distributions for the lower blades in the case of the RS configuration. The flow field shows a primary region with a
high velocity core near the pressure surface. Flow field development also shows a secondary region with a vortex
(i.e., negative radial velocity region) near the suction surface. This secondary flow, which affects flow features, is
caused by rotation and curvature of the impeller passage. The distortion of the flow field by the secondary flow
vortices is often referred to as the jet-wake structure of the exit flow. In the lower blades for each case, as the flow
goes to the outlet of the channel, the negative velocity region near the suction surface becomes large and the peak
values of the radial velocity in the primary regime increase.
rotor side
-7.7
-4.8
-2.9
-1.0
pressure
surface
suction
surface
0.9
2.9
4.8 6.7 8.6
10.6 12.5 13.5
stator side
(a)
-43.5
-35.7
-30.5
-22.8
-15.0
-7.2
-4.6
0.6
3.2
10.9
5.8
-2.0
8.3
13.5
16.1
18.7
(b)
FIGURE 11. Radial velocity (m/s) distributions at P1 =59.2 Pa: (a) r / R1 =1.2; (b) r / R1 =1.0.
7
CONCLUSIONS
A three-dimensional DSMC method for the analysis of steady and unsteady rarefied flows in a single-stage
DTDP has been developed. The unsteady calculation is based on a splitting of the computational domain into
stationary and rotating parts. The present experimental data in the outlet pressure range of 1∼526 Pa were
compared with the DSMC results.
In the case of the RS configuration, a recirculation zone arises near the hub side of the lower stator. However, in
the RR configuration, this secondary flow patterns cannot be seen. Also, for exit flow patterns, a secondary region
with a negative radial velocity of the RS is larger than that of the RR. This secondary flow significantly affects flow
features. Consequently, the pumping performance of a DTDP with the RS is lower than that with the RR. In the
lower blades for each configuration, as the flow goes to the outlet of the channel, the negative radial velocity region
near the suction surface becomes larger and the corresponding peak values in the primary regime increase.
At the radial gap between the rotor and the casing wall, no further pressure rise takes place because of both the
abrupt reduction in flow area and the higher velocity. It was found that the jet-wake structure of the exit flow in the
conventional centrifugal compressors also exists in a DTDP.
ACKNOWLEDGMENT
This work was supported by grant No. R05-2000-000-00311-0 from the Basic Research Program of the Korea
Science and Engineering Foundation.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
Akutsu, I. and Ohmi, T., “Innovation of the Fore Pump and Roughing Pump for High-Gas-Flow Semiconductor
Processing”, J. Vac. Sci. Technol. A, vol. 17, no. 6, pp. 3505-3508, 1999.
Jou, R. Y., Cheng, H. P., Chang, Y. W., Chen, F. Z., and Iwane, M., “Designs, Analyses, and Tests of a SpiralGrooved Turbobooster Pump”, J. Vac. Sci. Technol. A, vol. 18, no. 3, pp. 1016-1024, 2000.
Chu, J. G., “A New Hybrid Molecular Pump with Large Throughput”, J. Vac. Sci. Technol. A, vol. 6, no. 3, pp. 12021204, 1988.
Mase,M., Gyoubu, I., Nagaoka, T., and Taniyama, M., “Development of a New Type of Oil-Free Turbo Vacuum
Pump”, J. Vac. Sci. Technol. A, vol. 6, no. 4, pp. 2518-2521, 1988.
Liu, N. and Pang, S. J., “Microscopic Theory of Drag Molecular Dynamics in the Range of Free Molecular Flow”,
Vacuum, vol. 41, no. 7-9, pp. 2015-2017, 1990.
Shi, L., Wang, X. Z., Zhu, Y., and Pang, S. J., “Design of Disk Molecular Pumps for Hybrid Molecular Pumps”, J. Vac.
Sci. Technol. A, vol. 11, no. 2, pp. 426-431, 1993.
Shi, L., Zhu, Y., Wang, X. Z., and Pang, S. J., “Influence of Clearance on the Pumping Performance of a Molecular
Drag Pump”, J. Vac. Sci. Technol. A, vol. 11, no. 3, pp. 704-710, 1993.
Cheng, H. P., Jou, R. Y., Chen, F. Z., Chang, Y. W., Iwane, M., and Hanaoka, T., “Flow Investigation of Siegbahn
Vacuum Pump by CFD Methodology”, Vacuum, vol. 53, pp.227-231, 1999.
Heo, J. S. and Hwang, Y. K., “Spiral Channel Flows in a Disk-Type Drag Pump”, J. Vac. Sci. Technol. A, vol. 19, no.
2, pp. 656-661, 2001.
Bird, G. A., Molecular Gas Dynamics and the Direct Simulation of Gas Flows, pp. 218-256, Clarendon, Oxford, 1994.
Borgnakke, C. and Larsen, P. S., “Statistical Collision Model for Monte Carlo Simulation of Polyatomic Gas Mixture”,
J. Comput. Phys., vol. 18, pp. 405-420, 1975.
Hillewaert, K. and Van den Braembussche, R. A., “Numerical Simulation of Impeller-Volute Interaction in Centrifugal
Compressors”, J. of Turbomachinery, vol. 121, pp. 603-608, 1999.
Hwang, Y. K. and Heo, J. S., “Three-Dimensional Rarefied Flows in Rotating Helical Channels”, J. Vac. Sci. Technol.
A, vol. 19, no. 2, pp. 662-672, 2001.
8