direct simulation of rarefied gas flows in rotating spiral channels

Direct simulation of rarefied gas flows in rotating spiral channelsJoong-Sik Heo and Young-Kyu Hwang Citation: Journal of Vacuum Science & Technology A 20, 906 (2002); doi: 10.1116/1.1472418 View online: http://dx.doi.org/10.1116/1.1472418 View Table of Contents: http://scitation.aip.org/content/avs/journal/jvsta/20/3?ver=pdfcov Published by the AVS: Science & Technology of Materials, Interfaces, and Processing Articles you may be interested in 3D flow simulation of a spiral-grooved turbo-molecular pump AIP Conf. Proc. 585, 933 (2001); 10.1063/1.1407659 Molecular transition and slip flows in rotating helical channels of drag pump AIP Conf. Proc. 585, 893 (2001); 10.1063/1.1407653 Three-dimensional rarefied flows in rotating helical channels J. Vac. Sci. Technol. A 19, 662 (2001); 10.1116/1.1350979 Spiral channel flows in a disk-type drag pump J. Vac. Sci. Technol. A 19, 656 (2001); 10.1116/1.1342865 Molecular transition and slip flows in the pumping channels of drag pumps J. Vac. Sci. Technol. A 18, 1025 (2000); 10.1116/1.582294

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Direct simulation of rarefied gas flows in rotating spiral channelsJoong-Sik Heoa) and Young-Kyu Hwangb)

School of Mechanical Engineering, Sungkyunkwan University, 300 Chunchun-dong, Jangan-ku,Suwon 440-746, South Korea

~Received 14 January 2002; accepted 4 March 2002!

The pumping characteristics of a two-stage disk-type drag pump from the free molecular flow regionto the slip flow region are calculated by the direct simulation Monte Carlo method. Spiral channelsare cut on both upper and lower sides of rotating disks. For the molecular collision kinetics, thevariable hard-sphere molecular model with the no time counter technique is employed. TheBorgnakke–Larsen phenomenological model is adopted for calculation of the rotational energyexchange between the colliding molecules. Experiments are performed under the outlet pressurerange of 0.4–526 Pa. Numerical results show good quantitative agreement with the experimentaldata. © 2002 American Vacuum Society.@DOI: 10.1116/1.1472418#

I. INTRODUCTION

Recently, molecular drag pumps with a dry backing pumphave been widely used in high-density plasma and etchingprocesses because of their higher discharge pressure andlarger throughput capabilities. Also, the semiconductor in-dustry is getting into a new era by using 300 mm wafers.Therefore, a drag pump is urgently needed to provide anultraclean environment and productivity.1,2

Since Chu3 proposed a hybrid molecular pump combininga disk-type drag pump~DTDP! with a turbomolecular pump,considerable efforts have been invested in studying pumpingcharacteristics within pumping channels of a DTDP.

A new design of DTDP, in which the direction of momen-tum transferred to the gas molecules by a rotating disk isidentical to the pumping direction of the pump, was firstproposed by Tu, Zhu, and Wang.4 Murakami et al.5 devel-oped a turboviscous pump with ceramic rotors, which can beoperated in the wide pressure range from atmospheric pres-sure to 1023 Pa. The computational fluid dynamics method-ology using the Navier–Stokes~NS! equations with no-slipboundary conditions was adopted to investigate the rarefiedflow field of a single-stage DTDP by Chenget al.6 Finite-difference approximations were employed to solve the trans-port equations with a body-fitted grid system. Their numeri-cal results were in good agreement with the experimentaldata in the continuum flow regime. But, it was found that NSequations are not valid to predict the performance of a DTDPin the molecular transition regime.

Recently, Heo and Hwang7 numerically studied the mo-lecular transition and slip flows by using both the directsimulation Monte Carlo~DSMC! method and the NS equa-tions with second-order slip boundary conditions. Theyfound that the numerical results obtained by both methodsagree well with the experimental data for the Knudsen num-ber Kn<0.02. But, the results from the DSMC method wereonly reasonable in the molecular transition regime for Kn>0.02. They did not take into account the exchange of en-ergy between the translational and internal modes in theDSMC method.

With the exception of the study by Chenget al.,6 most ofthe previous numerical studies were limited to the simplecases of the single rotor or stator in a DTDP. The purpose ofthe present study is to analyze the rarefied flow field in atwo-stage DTDP. The velocity and pressure fields for a two-stage DTDP are predicted by using the DSMC method. Ex-periments are also performed under the outlet pressure rangeof 0.4–526 Pa. Our experimental results are compared withnumerical ones obtained by the DSMC method.

II. COMPUTATIONAL METHOD

Spiral channels of a two-stage DTDP are cut on both up-per and lower sides of rotating disks, as seen in Fig. 1~a!.The rotor has ten blades with an Archimedes’ spiral profile,with an exit angle of 10° relative to the tangent. At the inlet,the blade angle relative to the tangent is 30°. In Fig. 1~a!, ddenotes the radial gap between the rotor and casing wall,d isthe channel depth, andDd is the clearance between the rotorand stator, respectively. The dimensions of the DTDP arelisted in Table I. Figure 1~b! shows the surface grid of thesingle impeller.

The DSMC method used in this study is based on theprinciples described by Bird.8 The interaction between mol-ecules is modeled by variable hard-sphere~VHS! scatteringassuming an inverse-power interatomic potential. The notime counter method is used as a collision sampling tech-nique. The VHS exponentv of nitrogen is chosen to be 0.74with a reference molecular diameter of 4.17310210 m at ref-erence temperature 273 K. Also, chemical reactions and vi-brational mode are assumed to be frozen. For simulation ofdiatomic gas flows, the Borgnakke–Larsen phenomenologi-cal model9 is employed together with the energy exchangeprobability of Boyd.10,11

In order to limit the memory and CPU required to performthe calculations, the computational domain of the DTDP isrestricted to one blade passage. In this analysis, the rotatingframe of reference is used. Figure 2 shows a three-dimensional body-fitted grid system of the two-stage DTDP.The regions of blocks 2, 4, 6, and 8 are the upper and thelower blades of the rotating disks. Blocks 3 and 7 are theradial clearance between the rotor and casing wall. Blocks 1and 9 are the inlet and outlet part of the pumping channel.

a!Research Associate.b!Professor; electronic mail: [email protected]

906 906J. Vac. Sci. Technol. A 20 „3…, MayÕJun 2002 0734-2101 Õ2002Õ20„3…Õ906Õ5Õ$19.00 ©2002 American Vacuum Society


III. RESULTS AND DISCUSSION

DSMC simulations are carried out for a two-stage DTDP,and the computational grids are illustrated in Fig. 2. The testgas is nitrogen with a temperature of 273 K. Computationsare based on the fixed inlet and outlet pressure.

The pumping efficiencyw is calculated by

w5N122N21

N1, ~1!

in which N12 ~or N21! is the number of molecules to betransmitted through the channel from the inlet~or outlet! tothe outlet~or inlet!, andN1 is the number of molecules com-ing from the inlet during sampling time.

The pumping speedS ~l/s!, and throughputQ ~Pa l/s! arecalculated by

S5wA1A8RT/pK~s1!/4.0, ~2!

Q5P1S, ~3!

in which A1 is the area of the inlet,R is the ordinary gasconstant,T is the absolute temperature, and

[email protected]~s1!#, ~4!

in which erf is the error function,s1 (5u1 /A2RT) is thevelocity ratio, andu1 is the normal component of the bulkvelocity of molecules crossing the inlet boundary, respec-tively.

Experiments are performed for a two-stage DTDP in theoutlet pressure range of 0.4–526 Pa. Also, comparison be-tween the experimental data and the DSMC results are pre-sented.

A schematic diagram of the experimental apparatus isshown in Fig. 3. The flow-meter method is adopted here tocalculate the pumping speed. Details of the geometrical di-

FIG. 1. DTDP: ~a! side view and~b! blade surface computational grid.

FIG. 2. Computational grid system for a two-stage DTDP.

FIG. 3. Schematic diagram of experimental apparatus for performance test.

TABLE I. Dimensions of DTDP.

R1 ~inner radius! 39.0 mmR2 ~outer radius! 86.0 mmd ~clearance! 1.0 mmd ~depth! 3.0 mmDd ~clearance! 0.5 mm

907 J.-S. Heo and Y.-K. Hwang: Direct simulation of rarefied gas flows 907

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mensions of the rotor are given in Table I. The rotor is madeof a high-strength aluminum alloy. Before the experiments,the vacuum chamber is evacuated sufficiently and the ulti-mate pressure of 1.3231023 Pa is reached. The test pump isconnected to a two-stage oil rotary pump with the pumpingspeed of 970 l/min. The pressure in the high-vacuum side ismeasured with a Pirani~0.1–1000 Pa! and Penning (1.031025– 1 Pa) gauge, and the pressure in the fore-vacuumside is measured with a Pirani gauge. The rotational speed ofthe rotor is 24 000 rpm and is controlled by a frequencyconverter. Test gas is supplied through a mass flow controllerfrom a regulated high-pressure cylinder.

Experimental data at various flow rates are shown in Figs.4~a! and 4~b!. Figures 4~a! and 4~b! show that the compres-sion characteristics of the test pump are very different fromthose of the turbomolecular pump. For a Gaede-type dragpump, Helmer and Levi12 pointed out that the leak-limitedvalue of the compression ratio results in the sharp corner inthe transition from the molecular to the slip flow. By intro-ducing the aperture effect of a circular cylinder, they ob-tained the corrected numerical results corresponding to theexperimental data. As seen in Fig. 4~a!, the corner point inthe transition also exists for a DTDP. The maximum com-pression (5P2 /P1) ratio at throughputQ50 sccm is about1000 at aroundP2566 Pa. When the outlet pressures arelower than 131.6 Pa, the compression ratios decrease linearlywith the outlet pressures, as can be seen from Fig. 4~a!.

The effect of the outlet pressure on the inlet pressure isshown in Fig. 4~b!. The inlet pressure becomes higher as themass flow rate increases, and it depends on the outlet pres-sure. In the case ofQ55 – 50 sccm, the inlet pressure isnearly constant forP2<100 Pa.

The selected outlet pressure for the present numericalsimulations is 39.4 Pa. The Knudsen number, Kn5l/d,based on this outlet pressure and the channel depth is 0.035.The time step must be small compared to the mean collisiontime between molecules. Therefore, time stepDt51.031026 was chosen in this case.

FIG. 4. ~a! Compression ratio vs outlet pressure.~b! Inlet vs outlet pressure.

FIG. 5. Pressure difference vs throughput.

FIG. 6. Pressure distribution along the pumping passage.

TABLE II. Comparison between the measured and calculated flow rates.

Case No. P1(Pa) P2(Pa)Q ~sccm!

~measured!Q ~sccm!

~calculated!Relativeerror ~%!

1 1.32 39.4 10 13.87 38.72 3.95 39.4 40 42.04 5.13 5.26 39.4 50 52.87 5.7


J. Vac. Sci. Technol. A, Vol. 20, No. 3, May ÕJun 2002


The pressure differenceDP (5P22P1) as a function ofthroughputQ at P2539.4 Pa is shown in Fig. 5. The solidcircles connected by solid lines give the experimental data.The solid line indicates the DSMC results. The value ofQincreases linearly with pressure differenceDP, such as in thecase of the helical-type drag pump.13 Comparison betweenthe experimental data and the DSMC results shows goodagreement.

Comparison between the measured and calculated flowrates are shown in Table II. The relative error for case 1 isabout 40%. This is partly related to the unsuitable computa-tional cells corresponding to the high compression ratio(P2 /P1530). The relative error for both cases 2 and 3 isabout 5%, which is acceptable.

The average pressure profiles along the pumping passagefor three different inlet pressure conditions are shown in Fig.6, in whichx is the local coordinate along the pumping chan-

nel andL is the total length of the pumping channel. All ofthe simulations show similar tendencies for prediction of thepressure rise but different pressure levels are attained at theradial gap between the rotor and casing wall. The arrowsindicate the locations of the radial gap~blocks 3 and 7 in Fig.2!. At the radial gap, no further pressure rise takes placebecause of both the abrupt reduction in flow area and thehigher velocity. Most of the pressure rise occurs near theoutlet of the pump, such as in the case of the helical-typedrag pump.14

Velocity vectors and pressure contours atP151.32 Pa andP2539.4 Pa are shown in Figs. 7~a! and 7~b!. The flow fieldsare the time-averaged ones during 50 000 time steps. Thevelocity and pressure data are sampled after the flow fieldreaches steady state.

Figure 7~a! shows the relative velocity vectors near theclearance region between the rotor and the stator. In the up-permost blades, the inlet flow rushes into the radial gap with

FIG. 7. Flow fields atP151.32 Pa:~a! velocity vectors near the clearanceregion and~b! pressure contours near the clearance region.



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high velocity. The gas moves from the suction surface to thepressure surface. In the lower blades of the second stagerotor, most of the gas flows toward the channel exit along thepressure surface, and the gas near the suction surface decel-erates and flows back from the exit along the surface. Similarflow patterns were also obtained by Chenget al.6

Figure 7~b! shows the pressure contours. With the excep-tion of the lower blades of the second stage, the pressure isnearly constant from the inlet to the half section of the upperblades of the second stage. The pressure contours show thatthe pressure in the pressure surface of the flow channel islarger than that in the suction surface. Also, a large pressuregradient near the outlet of the channel can be seen.

Computed flow fields atP153.95 Pa andP155.26 Pa areshown in Figs. 8 and 9, respectively. Similar flow patternscan be observed.

IV. CONCLUSIONS

The rarefied gas flows in a two-stage DTDP from the freemolecular flow region to the slip flow region are studied bythe use of the DSMC method. The present experimental datain the outlet pressure range of 0.4–526 Pa were comparedwith the DSMC results. Numerical results show good quan-titative agreement with the experimental data.

The maximum compression ratio at zero throughput isabout 1000. When the outlet pressures are lower than 131.6Pa, the compression ratios decrease linearly with the outletpressures. In the present case of a two-stage DTDP, most ofthe pressure rise occurs near the outlet of the pump, such asthe case of the helical-type drag pump. At the clearance re-gion between the rotor and the casing wall, no further pres-sure rise takes place because of both the abrupt reduction inflow area and the higher velocity.

ACKNOWLEDGMENT

This work was supported by Grant No. 2000-1-30400-014-3 from the Basic Research Program of the Korea Sci-ence and Engineering Foundation.

NOMENCLATURE

A aread channel depthKn Knudsen numberN number of moleculesP pressureQ throughputR radius of rotor gas constants speed ratio,u/A2RTS pumping speedT absolute temperatureu mean velocityw pumping efficiency

Subscripts

1 inlet2 outlet

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direct simulation of rarefied gas flows in rotating spiral channels

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