- Email: [email protected]
Theoretical and experimental study of dry scroll vacuum pump
Vacuum 84 (2010) 415–421
Contents lists available at ScienceDirect
Vacuum journal homepage: www.elsevier.com/locate/vacuum
Theoretical and experimental study of dry scroll vacuum pump Zeyu Li, Liansheng Li*, Yuanyang Zhao, Gaoxuan Bu, Pengcheng Shu National Engineering Research Center of Fluid Machinery and Compressors, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 15 July 2009 Received in revised form 17 August 2009 Accepted 12 September 2009
The rarefied gas flows through suction port, scroll clearance and discharge port are treated as leakage of dry scroll vacuum pump (DSVP). The models for predicting the above-mentioned leakage rate were derived in this paper. The model for predicting the heat transfer rate between rarefied gas and working chamber wall was also developed. Then, a general model for describing the working process of DSVP was set up according to the energy and mass conservation principle. This model can be applied to predict the performance of DSVP. The pumping speed for different suction pressures was obtained. Furthermore, the ultimate pressure and power consumption for different speeds were gotten. A good agreement between the theoretical results and experimental data was obtained. Finally, the volume ratio of prototype was changed and its influence on the performance was studied by experiment. Ó 2009 Elsevier Ltd. All rights reserved.
Keywords: Dry scroll vacuum pump Leakage Heat transfer Performance Volume ratio
1. Introduction The dry scroll vacuum pump (DSVP) has so many advantages, such as simple structure, high performance, low cost, easy maintenance, low noise, small vibration and so on when compared with other kinds of dry vacuum pump. It is widely used in the semiconductor industry. The pumping mechanism of DSVP was studied by Su et al., in which a leakage model for predicting the flow rate of rarefied gas flows through the scroll clearance was given in Ref. [1]. Based on the leakage model, the pumping performance of DSVP can be predicted [2,3]. However, the model mentioned above does not consider the heat transfer in DSVP. So it cannot be used to predict the power consumption precisely under different conditions. The aim of this paper is to set up the general model for describing the working process of DSVP by taking into account leakage and heat transfer. Solving the model, the performance of DSVP can be predicted. Then, the influence of some important design parameters, such as speed and volume ratio, on the performance of DSVP was studied.
to the high demand of symmetry. So, the prototype of our research was designed and manufactured based on the single scroll type, its main construction can be seen in Fig. 1. The structure parameters of prototype are shown in Table 1. The geometry of scroll is shown in Fig. 2. The positive direction of orbiting angle q is also indicated in Fig. 2. The orbiting angle is regarded as zero when the suction chamber is just formed. The geometrical calculation of prototype, such as discharge angle, working chamber volume for different orbiting angles, can be referred to reference [4]. 3. Models for working process The model for describing the working process of DSVP is developed based on energy and mass conservation principle. Models of leakage and heat transfer are combined with the conservation equations. In order to solve the model easily, the models of leakage and heat transfer are set up by analytical method. 3.1. Leakage
2. Prototype The prototype studied by Sawada et al. is the double acting type dry scroll vacuum pump [1–3]. Although the double acting dry type scroll vacuum pump has the advantages of high pumping speed and better balance of axial force, it is more difficult to manufacture due
In the paper, the gas flows in DSVP are all treated as leakage, including: (1) rarefied gas flows through suction port, usually is called suction process; (2) rarefied gas flow through scroll clearance, usually is called internal leakage; (3) gas flows through discharge port, usually is called discharge process.
* Corresponding author. Tel.: þ86 29 82663792; fax: þ86 29 83237910/ 82663792. E-mail address: [email protected] (L. Li).
3.1.1. Leakage from suction port The schematic of suction port is shown in Fig. 3. The section of suction port is thought to be rectangle [5].
0042-207X/$ – see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.vacuum.2009.09.005
416
Z. Li et al. / Vacuum 84 (2010) 415–421
Nomenclature b Dh h ht Kn m _ m _ M n p P Pr Q Q_ Q R Rave Re Sm T
v Vc
slip coefficient proposed by Beskok hydraulic diameter height, specific enthalpy heat transfer coefficient Knudsen number mass mass flow rate per unit width mass flow rate rotation speed of prototype pressure power Prandlt number heat transfer rate volumetric flow rate per unit width non-dimensional flow rate gas constant average curve radius Reynolds number pumping speed temperature
Greek letter p Pi b rarefaction coefficient of viscosity sn tangential momentum accommodation coefficient m viscosity l thermal conductivity k adiabatic exponent r density Subscript 0 c FM h i, j l max suc vac
According to reference [6,7], the mass flow rate for flows of rarefied gas through an orifice in the free molecular regime can be expressed:
(1)
So, it can be supposed that the mass flow rate for flows of rarefied gas through an orifice in any flow regime has the following form:
_ ¼ f ðKnÞM _c M
(2)
Where f (Kn) is the correlation function and it should be fulfilled:
lim f ðKnÞ ¼
Kn/N
1 ffi kþ1 kþ1=2ðk1Þ pffiffiffiffiffiffi 2pk 2
1 PPhl
(3)
lim f ðKnÞ ¼ 1
Kn/0
Crank
Small counterbalance
Frame Fixed scroll Big counterbalance
Shaft
initial continuum, channel free molecular flow high loop number low maximal suction vacuum chamber
So, f(Kn) can be solved and has the following expression:
f ðKnÞ ¼
k þ 1 kþ1=2ðk1Þ 1 p _ _ M 1 l M c; max FM ¼ pffiffiffiffiffiffiffiffiffi ph 2 2pk
(
specific volume volume
1þ
h
1 ffi kþ1 kþ1=2ðk1Þ pffiffiffiffiffiffi 2pk 2
1 PPhl
i Kn (4)
1 þ Kn
The mass flow rate corresponding to the pressure ratio in contin_ c can be obtained by the nozzle model [5]. uous flow M 3.1.2. Leakage from scroll clearance The rarefied gas flows through the scroll clearance is usually called internal leakage. There are two kinds of internal leakage in DSVP: one is the gas flows through the axial clearance, called radial leakage; another is the gas flows through the radial clearance, called flank leakage. Generally, it is thought that the axial clearance is a constant value and the radial clearance changes with the involute angle. The characteristic of internal leakage channel can be seen in Fig. 4. The internal leakage in the DSVP is usually modeled as rarefied gas flows through an infinite channel [1]. This problem can be solved using the Navier–Stokes equation and the general slip boundary condition. However, the above results are valid in the viscous and slip flow regime. In the transition and the free molecular regime, the continuous flow hypothesis is not valid, and the collision between the gas molecular and the flow boundary gradually becomes important to the flow. The increased rarefaction effects can be taken into account by the correlation of viscosity, and the expression of flow rate is [8]:
h3 dp 2 sv 6Kn 1þ ð1 þ bKnÞ Q_ ¼ c sv 1 bKn 12m0 dx
Discharge
(5)
Table 1 Structure parameters of prototype.
Suction Motor Orbiting scroll Fig. 1. Construction of prototype.
Items
Value
Radius of basic circle Pitch of involute Initial angle of involute Thickness of scroll wrap Height of scroll wrap Loops of scroll Axial clearance Minimum radial clearance
3 mm 18.85 mm 40 4.19 mm 30 mm 5.25 0.3 mm 0.036 mm
Z. Li et al. / Vacuum 84 (2010) 415–421
417
y
Discharge port Orbiting scroll
Orbiting scroll ls
θ
h
e=10.5
o
x
ls Fixed scroll
Section of suction port
Fixed scroll
os=2
θ
Fig. 3. Schematic of suction port.
Fig. 2. Geometry of scroll.
seals between the fixed scroll and the orbiting scroll, it is equal to the axial clearance. While for flank leakage, it can be calculated by the simplified formula of Fanno’s flow [10]. In the scroll compressor, because the orbiting radius of scroll is very small, the relative wall velocity of scroll caused by orbiting is far less than the gas velocity of leakage. And it was shown that the influence of relative wall velocity on leakage of scroll compressor is very little by Chen [11]. The working principle of DSVP is similar to scroll compressor. So, the effect of relative wall velocity on leakage does not consider in the paper.
Using the similar method presented by Beskok [8], and the numerical data of Cercignani [9], the expression of b can be calculated by the following expression:
(
Kn ¼ t ð2c
b ¼ c1
t c2
A 2 Þð3c2 Þt
2c2
þ c1 c2 ð2cA2 Þð3c2 Þt
(6)
Here, t is the parametric variable. A, c1, c2 is equal to 0.15, 1.479952, 0.1551753, respectively. To verify Equation (6), a comparison of calculated values and numerical data of pCercignani [10] was carried out ffiffiffiffiffiffiffiffiffi (ðQ ¼ rQ_ =ðh2c ðdp=dxÞÞ 2RT Þ). It is shown that they agree well in Fig. 5. The volumetric flow rate of internal leakage is given by Equation (5) and (6). To obtain the mass flow rate, the pressure distribution along the internal leakage channel should be written:
_ dp 12m0 RT m ¼ 6Kn dx 3 phc 1 þ 1bKn ð1 þ bKnÞ
3.1.3. Leakage from discharge port In the discharge process, the gas flow is in the viscous flow regime. So, the corresponding mass flow rate can be directly calculated by the nozzle model [12]. 3.2. Heat transfer
(7)
In the operation of DSVP, there is temperature difference between gas and working chamber wall. And the heat transfer occurs due to this temperature difference. In the scroll compressor, the convective heat transfer coefficient between the gas and the scroll wraps is calculated by the correlation for a spiral heat exchanger [5]:
_ should be assumed. Then Equation (7) Firstly the mass flow rate m is solved by the 4-steps Runge–Kutta method in the known inlet pressure of leakage channel. And the outlet pressure of leakage channel can be gotten. If there is some difference between the _ should be modicalculated outlet pressure and the known one, m fied and the process will be repeated until they both get a good _ is mass flow rate of leakage agreement. And the corresponding m channel. It should be noticed that the calculation of internal leakage flow rate should take into account by the length of leakage line. The length of leakage line for different orbiting angles was derived by Wang et al. [4]. Finally, the height of internal leakage channel hc should be discussed. For radial leakage, because there are not tip
ht ¼ 0:023
D Re0:8 Pr 0:4 1 þ 1:77 h Dh Rave
l
(8)
Considering the increased rarefaction effect of gas in DSVP, the viscosity using in Equation (8) should be modified by the following relationship [8]:
m ¼ m0 =ð1 þ bKnÞ Fixed scroll
Fixed scroll
Orbiting scroll
Radial Orbiting scroll clearance
Fig. 4. Internal leakage of DSVP.
(9) Axial clearance
418
Z. Li et al. / Vacuum 84 (2010) 415–421
(3) Compare the initial guess values and the calculated values at q ¼ 2p, if convergent condition is satisfied, calculate the suction pressure of next step psuc,jþ1. It is equal to:
3.4
Nondimensional flow rate
3.2 3.0 2.8
psuc; jþ1
2.6
mvac; j Sm;j 60 n RTvac ¼ Vvac
(13)
and go to (1); otherwise, let the calculated values equal to the initial values and go back to (2).
2.4 2.2
(4) If the pumping speed is less than the set value (1 L/min), the corresponding suction pressure is thought to be reached the ultimate pressure of prototype, and the calculation is end. Otherwise, go back to (1) to star the calculation in the suction pressure psuc,jþ1.
2.0 1.8 Cercignani Model
1.6 1.4 20
0
40
60
80
100
Kn
In a certain suction pressure psuc,jþ1, if the calculations converge, the current pumping speed is [14]:
Fig. 5. Comparison of non-dimensional flow rate.
Sm;j ¼ The temperature distribution of fix scroll and orbiting scroll, the corresponding heat transfer area can be referred by Chen [5]. Then, the heat transfer rate between rarefied gas and working chamber wall can be calculated. 3.3. Conservation equations for working chamber The working chamber of DSVP is divided into suction chamber, compression chamber and discharge chamber. The ‘‘three chamber control volume’’ method [10] was applied to describe the working process of DSVP. For each working chamber, the energy change is governed by the first law of thermodynamics for an open control volume. The temperature change of gas as a function of the orbiting angle can be written as [13]:
h dT ¼ dq
idv 1 X dmi dQ ðhi hÞ þ dq dq Vc dq vp 1 vh v vT vT
vp 1 vh v vv T vv T
v
The current power of working process is
Pj ¼
m h i n X Vðqi Þ psuc; j ðqiþ1 Þ psuc; j ðqi Þ 60 i ¼ 1
(15)
5. Testing system The principle of testing system is shown in Fig. 6. The picture of testing system can be seen in Fig. 7. The range of gas flow meter is from 0 to 300 L/min, its precision is equal to: reading error 0.8% þ full scale error 0.2%. The range of vacuometer is from 0.01 Pa to 1.33 105 Pa; its relative error is 5%.
(10)
v
(11)
The performance of prototype can be predicted by solving the Equations (10) and (11) simultaneously for each working chamber. The pumping speed for different suction pressures, ultimate pressure and power consumption for different speed are then obtained theoretically. Comparison between theoretical results and experimental data is discussed in the following parts.
The air can be treated as ideal gas. The parameters of thermodynamic state satisfy the ideal gas state equation:
pv ¼ RT
(14)
6. Results and discussion
And the mass balance is equal to:
X dmi X dmo dm ¼ dq dq dq
nVvac psuc; j1 ln 60 psuc; j
Ambient
(12)
The mass transfer rate and the heat transfer rate in Equations (10) and (11) can be calculated by the leakage model and the heat transfer rate, respectively. 4. Model solution Equations (10) – (11) for each working chamber can be solved simultaneously by four-step Runge–Kutta method. The solving steps are shown as follows: (1) Let q ¼ 0 and initial guess values of each working chamber. The initial guess values of each working chamber can be obtained by the ideal adiabatic compression process. (2) Then, the thermodynamic parameter values of each working chamber can be calculated according to the 4-steps Runge– Kutta method, until the calculation process is finished.
Diaphragm valve
Needle valve
Gas flow meter Ambient Valve 1 Ambient
Valve 2 Valve 4
Vacuum chamber Valve 3
Secondary vacuum pump Vacuometer
DSVP
Power meter Power Motor
Fig. 6. Principle of testing system.
Z. Li et al. / Vacuum 84 (2010) 415–421
419
6.2. Ultimate pressure and power consumption for different speeds
Fig. 7. Picture of Testing system.
6.1. Pumping speed The pumping speed for different suction pressures is predicted by Equation (14). The comparison of theoretical and experimental results is shown in Fig. 8. A reasonable agreement is gotten. In the theoretical calculations, the resistance of suction pipe, valve and inside the prototype is ignored. This assumption results in the difference between the calculated results and the experimental data. When the suction pressure is in the relative higher range, the resistance has more influence on the pumping speed. As the suction pressure is near the ultimate pressure, the impact of resistance on pumping speed becomes less. The following tendency was also shown by Sawada [2]. The change trend of pumping speed with suction pressure is shown in Fig. 8. The pumping speed goes down quickly as the suction pressure is near the ultimate pressure. The average decreasing rate of pumping speed with suction pressure is 5.09 L/ (kPa min), 9.9 L/(kPa min), 26.6 L/(kPa min) when the speed is 1136 rpm, 1420 rpm, 1704 rpm, respectively. Near the ultimate pressure, it can be concluded that the average decreasing rate of pumping speed with suction pressure becomes larger with the increment of speed. While the suction pressure is in other ranges, the pumping speed decreases slowly with the decrement of suction pressure. The corresponding average decreasing rate of pumping speed with suction pressure is 0.5–0.85 L/(kPa min).
The theoretical ultimate pressure of prototype can be obtained when the solving process of differential Equations (10) and (11) for each working chamber is end. The theoretical and experimental results of ultimate pressure for different speeds are shown in Fig. 9. A good agreement is gotten. The ultimate pressure decreases by 72.6% as the speed increases form 568 rpm to 852 rpm. The ultimate pressure decreases by 73.4% when the speed increases form 852 rpm to 1420 rpm. The ultimate pressure decreases by 31.5% as the speed increases form 1420 rpm to 1704 rpm. The pumping speed goes up with the increment of speed. However, the deformation of scroll parts, which causes by the scroll temperature, also gets larger with the increment of speed. The deformation of scroll parts can increase the scroll clearance, and leads to more internal leakage. So, the ultimate pressure goes down rapidly with the increment of speed in a certain range. As the speed is large enough, the changing rate of ultimate pressure with speed becomes less and less. The power of working process can be predicted by Equation (15). The total input power can be obtained by considering the mechanical efficiency and the motor efficiency. It is known that the power of DSVP changes with suction pressure. It is thought that the DSVP has a maximal ideal adiabatic compression power in a certain suction pressure by the reference [15]. And the corresponding input power is called maximal one. This value directly shows the power consumption level of DSVP. The maximal input power for different speeds is shown in Fig. 10. The deviation of theory and experiment is less than 10%. The maximal input power in a certain speed goes up linearly with the increment of speed. Its changing rate with speed is 0.43 W/(r min). It is concluded that there is a relative ideal speed in the DSVP by the above analysis. The relative lower ultimate pressure and power consumption can be obtained under this speed.
6.3. Influence of volume ratio on the performance The volume ratio is one of the important geometrical parameters of DSVP. It is equal to the ratio of suction chamber volume to discharge chamber volume. In the design of scroll compressor, the volume ratio can be directly derived by pressure ratio. However, the volume ratio of DSVP can not be obtained by the same method because the pressure ratio is always increasing during the operation of DSVP. 100
150
100
1136r·min-1, experiment 1136r·min-1, theory 1420r·min-1, experiment 1420r·min-1, theory 1704r·min-1, experiment 1704r·min-1, theory
50
0
Ultimate pressure / kPa
Pumping speed / L·min-1
Experimental data Theoretical results
80
200
60
40
20
0 0
20
40
60
80
Suction pressure / kPa Fig. 8. Pumping speed for different suction pressures.
100
400
600
800
1000
1200
Speed / r·min-1 Fig. 9. Ultimate pressure.
1400
1600
1800
420
Z. Li et al. / Vacuum 84 (2010) 415–421
100
800
Volume ratio = 2.57 Volume ratio = 2.0 80
Ultimate pressure / kPa
Maximal input power / W
700 600 500 400 300
400
600
800
1000
1200
Speed /
1400
1600
40
20
Experimental data Calculated results
200
60
0
1800
400
r·min-1
600
800
1000
1200
Speed /
1400
1600
1800
r·min-1
Fig. 10. Maximal input power. Fig. 12. Comparison of ultimate pressure.
In order to study the influence of volume ratio on performance, the involute ending angle is fixed and the involute starting angle goes up from 2p to 3p, the corresponding volumetric ratio goes down from 2.57 to 2.0. The influence of volume ratio on performance is tested by experiment. The results of pumping speed are shown in Fig. 11. The pumping speed gains increment in the whole working process as the prototype operates in the relative lower speed (1136 rpm). While the prototype operates in the relative higher
a
speed (1420, 1704 rpm), the pumping speed just gets larger in a certain range of suction pressure. The pumping speed is directly determined by volumetric efficiency. So, the change of pumping speed can be explained by the change of volumetric efficiency. When the suction pressure is in the higher range, the prototype works in over-compression condition, the pressure distribution in working chamber becomes more uniform due to the decreasing of volumetric ratio. This factor leads
b
160
200
Pumping speed / L·min-1
120 100 80 60 40 20
Volume ratio is 2.57 Volume ratio is 2.0
0 0
20
40
60
80
150
100
50 Volume ratio is 2.0 Volume ratio is 2.57
0 0
100
20
Suction pressure / kPa
40
60
Suction pressure / kPa
c 250
Pumping speed / L·min-1
Pumping speed / L·min-1
140
200 150 100 50 Volume ratio is 2.0 Volume ratio is 2.57
0 0
20
40
60
80
100
Suction pressure / kPa Fig. 11. Influence of volume ratio on pumping speed. (a) 1136 rpm; (b) 1420 rpm; (c) 1704 rpm.
80
100
Z. Li et al. / Vacuum 84 (2010) 415–421
pressure when DSVP works in the relative higher speed. But, the choice of volume ratio should be also taken into account by the scroll geometry, discharge temperature, etc.
800
700
Maximal input power / W
421
600
7. Conclusions
500
A general model for describing the working process of DSVP was set up according to the energy and mass conservation principle. By solving this model, the performance of DSVP can be predicted. The influence of speed on pumping speed, ultimate pressure, and power consumption was studied. Furthermore, the influence of volume ratio on performance of DSVP was analyzed experimentally. The lowest ultimate pressure of prototype is 4.43 kPa as the speed is 1704 rpm, and the corresponding maximal Kn of leakage is 0.04. Some conclusions are summarized as follows:
400
300 Volume ratio = 2.57 Volume ratio = 2.0
200 400
600
800
1000
1200
1400
1600
1800
Speed / r·min-1 Fig. 13. Comparison of power.
to the decrement of leakage and the improvement of volumetric efficiency. When the suction pressure is in the lower range, the prototype works in under-compression condition, the decrement of volume ratio can cause the pressure distribution of working chamber to be non-uniform. So the leakage increases and the volumetric efficiency drops. It can be also seen that the trend of pumping speed with suction pressure does not change with the decrement of volume ratio. However, the average decreasing rate of pumping speed with suction pressure for the whole working process increases by 15.4%, 23.8%, 14.9% when the speed is 1136 rpm, 1420 rpm, 1704 rpm, respectively. It can be concluded that in the condition of same decrement of suction pressure, the decrement of pumping speed gets larger as the volume ratio decreases. The ultimate pressure is tested, and the results are shown in Fig. 12. The ultimate pressure goes down with the decrement of speed when the prototype works in low speed condition. This phenomenon can be also explained by the change of volumetric efficiency. For the low speed, the prototype works mainly under over-compression condition as the suction pressure is near the ultimate pressure. According to the analysis for influence of volume ratio on pumping speed, the volumetric efficiency gains improvement and makes the ultimate pressure decreases; and via versa. Moreover, the ultimate pressure goes down by 83.4% when the speed increases from 568 rpm to 852 rpm. While the ultimate pressure just decreases by 46.7% as the speed increases form 852 rpm to 1704 rpm. It can be concluded that the ultimate pressure goes down rapidly with the decrement of speed when the speed is less than 852 rpm. As the speed is higher than 852 rpm, the influence of speed on ultimate pressure becomes less. The maximal input power for different speeds is tested, and the results are shown in Fig. 13. It can be seen that the influence of volume ratio on the maximal input power is small. It is easy to find that the ideal adiabatic compression power is just determined by suction pressure and actual discharge pressure in calculation [15]. By considering the influence of volume ratio on ultimate pressure and power consumption, the ideal speed of prototype changes with the decrement of volume ratio. On the other hand, the increment of volume ratio is helpful to improve the ultimate
(1) Considering the leakage and heat transfer, a general model for predicting the performance of DSVP was presented by the energy and mass conservation principle. A good agreement between theoretical results and experimental data is obtained when Kn < 0.04. (2) The pumping speed goes down with the decrement of suction pressure. The pumping speed drops much more rapidly with the increment of speed as the suction pressure is near ultimate pressure. However, the pumping speed decreases slowly in other suction pressure range. (3) The ultimate pressure drops rapidly with the increment of speed. When the speed is large enough, its impact on ultimate pressure becomes less and less. The maximal input power goes up linearly with the increment of speed. There is a relative ideal speed in the DSVP. The relative lower ultimate pressure and power consumption can be obtained under this speed. (4) The average decreasing rate of pumping speed with suction pressure gets larger with the decrement of volume ratio. With the decrement of volume ratio, the ultimate pressure goes down in the low speed, while it goes up in the high speed. However, the maximal input power of prototype for different speeds does not change apparently.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
[12] [13] [14] [15]
Su Y, et al. Vacuum 1996;47:815. Sawada T, et al. JSME International Journal 1998;44:184. Sawada T, et al. Vacuum 1999;53:233. Wang BL, Li XT, Shi WX. International Journal of Refrigeration 2005;28: 958. Chen Y, et al. International Journal of Refrigeration 2002;25:731. Sharipov F. Journal of Fluid Mechanics 2004;518:35. Shen Q. Rarefied gas dynamics. Defense Industry Press; 2003 [in Chinese]. Beskok A, Trimmer W, Karniadakis GE. ASME, J Fludis Eng 1996;118:448. Cercignani C, Daneri A. J Appl Phys 1963;34:3509. Gu ZL, et al. Scroll compressor and other scroll machinery. Shan’xi Science and Technology Press; 1998 [in Chinese]. Chen ZM, Yuan XL, Zhou ZC. Mathematical modeling and experimental study on two dimensional leakage flow of gas through small gap. Xi’an: Xi’an Jiaotong University Press; 1991. Jing SR, Zhang MY. Fluid mechanics. Xi’an Jiaotong University Press; 2001 [in Chinese]. Li LS, et al. Scroll compressors. Mechanical Industry Press; 1998 [in Chinese]. Clausing P. Journal of Vacuum Science & Technology 1971;8:636. Meng DH, Ba DC. Study on the design theory and manufacture technology of dry scroll vacuum pump. Shenyang: Dongbei University Press; 2004.
Contents lists available at ScienceDirect
Vacuum journal homepage: www.elsevier.com/locate/vacuum
Theoretical and experimental study of dry scroll vacuum pump Zeyu Li, Liansheng Li*, Yuanyang Zhao, Gaoxuan Bu, Pengcheng Shu National Engineering Research Center of Fluid Machinery and Compressors, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 15 July 2009 Received in revised form 17 August 2009 Accepted 12 September 2009
The rarefied gas flows through suction port, scroll clearance and discharge port are treated as leakage of dry scroll vacuum pump (DSVP). The models for predicting the above-mentioned leakage rate were derived in this paper. The model for predicting the heat transfer rate between rarefied gas and working chamber wall was also developed. Then, a general model for describing the working process of DSVP was set up according to the energy and mass conservation principle. This model can be applied to predict the performance of DSVP. The pumping speed for different suction pressures was obtained. Furthermore, the ultimate pressure and power consumption for different speeds were gotten. A good agreement between the theoretical results and experimental data was obtained. Finally, the volume ratio of prototype was changed and its influence on the performance was studied by experiment. Ó 2009 Elsevier Ltd. All rights reserved.
Keywords: Dry scroll vacuum pump Leakage Heat transfer Performance Volume ratio
1. Introduction The dry scroll vacuum pump (DSVP) has so many advantages, such as simple structure, high performance, low cost, easy maintenance, low noise, small vibration and so on when compared with other kinds of dry vacuum pump. It is widely used in the semiconductor industry. The pumping mechanism of DSVP was studied by Su et al., in which a leakage model for predicting the flow rate of rarefied gas flows through the scroll clearance was given in Ref. [1]. Based on the leakage model, the pumping performance of DSVP can be predicted [2,3]. However, the model mentioned above does not consider the heat transfer in DSVP. So it cannot be used to predict the power consumption precisely under different conditions. The aim of this paper is to set up the general model for describing the working process of DSVP by taking into account leakage and heat transfer. Solving the model, the performance of DSVP can be predicted. Then, the influence of some important design parameters, such as speed and volume ratio, on the performance of DSVP was studied.
to the high demand of symmetry. So, the prototype of our research was designed and manufactured based on the single scroll type, its main construction can be seen in Fig. 1. The structure parameters of prototype are shown in Table 1. The geometry of scroll is shown in Fig. 2. The positive direction of orbiting angle q is also indicated in Fig. 2. The orbiting angle is regarded as zero when the suction chamber is just formed. The geometrical calculation of prototype, such as discharge angle, working chamber volume for different orbiting angles, can be referred to reference [4]. 3. Models for working process The model for describing the working process of DSVP is developed based on energy and mass conservation principle. Models of leakage and heat transfer are combined with the conservation equations. In order to solve the model easily, the models of leakage and heat transfer are set up by analytical method. 3.1. Leakage
2. Prototype The prototype studied by Sawada et al. is the double acting type dry scroll vacuum pump [1–3]. Although the double acting dry type scroll vacuum pump has the advantages of high pumping speed and better balance of axial force, it is more difficult to manufacture due
In the paper, the gas flows in DSVP are all treated as leakage, including: (1) rarefied gas flows through suction port, usually is called suction process; (2) rarefied gas flow through scroll clearance, usually is called internal leakage; (3) gas flows through discharge port, usually is called discharge process.
* Corresponding author. Tel.: þ86 29 82663792; fax: þ86 29 83237910/ 82663792. E-mail address: [email protected] (L. Li).
3.1.1. Leakage from suction port The schematic of suction port is shown in Fig. 3. The section of suction port is thought to be rectangle [5].
0042-207X/$ – see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.vacuum.2009.09.005
416
Z. Li et al. / Vacuum 84 (2010) 415–421
Nomenclature b Dh h ht Kn m _ m _ M n p P Pr Q Q_ Q R Rave Re Sm T
v Vc
slip coefficient proposed by Beskok hydraulic diameter height, specific enthalpy heat transfer coefficient Knudsen number mass mass flow rate per unit width mass flow rate rotation speed of prototype pressure power Prandlt number heat transfer rate volumetric flow rate per unit width non-dimensional flow rate gas constant average curve radius Reynolds number pumping speed temperature
Greek letter p Pi b rarefaction coefficient of viscosity sn tangential momentum accommodation coefficient m viscosity l thermal conductivity k adiabatic exponent r density Subscript 0 c FM h i, j l max suc vac
According to reference [6,7], the mass flow rate for flows of rarefied gas through an orifice in the free molecular regime can be expressed:
(1)
So, it can be supposed that the mass flow rate for flows of rarefied gas through an orifice in any flow regime has the following form:
_ ¼ f ðKnÞM _c M
(2)
Where f (Kn) is the correlation function and it should be fulfilled:
lim f ðKnÞ ¼
Kn/N
1 ffi kþ1 kþ1=2ðk1Þ pffiffiffiffiffiffi 2pk 2
1 PPhl
(3)
lim f ðKnÞ ¼ 1
Kn/0
Crank
Small counterbalance
Frame Fixed scroll Big counterbalance
Shaft
initial continuum, channel free molecular flow high loop number low maximal suction vacuum chamber
So, f(Kn) can be solved and has the following expression:
f ðKnÞ ¼
k þ 1 kþ1=2ðk1Þ 1 p _ _ M 1 l M c; max FM ¼ pffiffiffiffiffiffiffiffiffi ph 2 2pk
(
specific volume volume
1þ
h
1 ffi kþ1 kþ1=2ðk1Þ pffiffiffiffiffiffi 2pk 2
1 PPhl
i Kn (4)
1 þ Kn
The mass flow rate corresponding to the pressure ratio in contin_ c can be obtained by the nozzle model [5]. uous flow M 3.1.2. Leakage from scroll clearance The rarefied gas flows through the scroll clearance is usually called internal leakage. There are two kinds of internal leakage in DSVP: one is the gas flows through the axial clearance, called radial leakage; another is the gas flows through the radial clearance, called flank leakage. Generally, it is thought that the axial clearance is a constant value and the radial clearance changes with the involute angle. The characteristic of internal leakage channel can be seen in Fig. 4. The internal leakage in the DSVP is usually modeled as rarefied gas flows through an infinite channel [1]. This problem can be solved using the Navier–Stokes equation and the general slip boundary condition. However, the above results are valid in the viscous and slip flow regime. In the transition and the free molecular regime, the continuous flow hypothesis is not valid, and the collision between the gas molecular and the flow boundary gradually becomes important to the flow. The increased rarefaction effects can be taken into account by the correlation of viscosity, and the expression of flow rate is [8]:
h3 dp 2 sv 6Kn 1þ ð1 þ bKnÞ Q_ ¼ c sv 1 bKn 12m0 dx
Discharge
(5)
Table 1 Structure parameters of prototype.
Suction Motor Orbiting scroll Fig. 1. Construction of prototype.
Items
Value
Radius of basic circle Pitch of involute Initial angle of involute Thickness of scroll wrap Height of scroll wrap Loops of scroll Axial clearance Minimum radial clearance
3 mm 18.85 mm 40 4.19 mm 30 mm 5.25 0.3 mm 0.036 mm
Z. Li et al. / Vacuum 84 (2010) 415–421
417
y
Discharge port Orbiting scroll
Orbiting scroll ls
θ
h
e=10.5
o
x
ls Fixed scroll
Section of suction port
Fixed scroll
os=2
θ
Fig. 3. Schematic of suction port.
Fig. 2. Geometry of scroll.
seals between the fixed scroll and the orbiting scroll, it is equal to the axial clearance. While for flank leakage, it can be calculated by the simplified formula of Fanno’s flow [10]. In the scroll compressor, because the orbiting radius of scroll is very small, the relative wall velocity of scroll caused by orbiting is far less than the gas velocity of leakage. And it was shown that the influence of relative wall velocity on leakage of scroll compressor is very little by Chen [11]. The working principle of DSVP is similar to scroll compressor. So, the effect of relative wall velocity on leakage does not consider in the paper.
Using the similar method presented by Beskok [8], and the numerical data of Cercignani [9], the expression of b can be calculated by the following expression:
(
Kn ¼ t ð2c
b ¼ c1
t c2
A 2 Þð3c2 Þt
2c2
þ c1 c2 ð2cA2 Þð3c2 Þt
(6)
Here, t is the parametric variable. A, c1, c2 is equal to 0.15, 1.479952, 0.1551753, respectively. To verify Equation (6), a comparison of calculated values and numerical data of pCercignani [10] was carried out ffiffiffiffiffiffiffiffiffi (ðQ ¼ rQ_ =ðh2c ðdp=dxÞÞ 2RT Þ). It is shown that they agree well in Fig. 5. The volumetric flow rate of internal leakage is given by Equation (5) and (6). To obtain the mass flow rate, the pressure distribution along the internal leakage channel should be written:
_ dp 12m0 RT m ¼ 6Kn dx 3 phc 1 þ 1bKn ð1 þ bKnÞ
3.1.3. Leakage from discharge port In the discharge process, the gas flow is in the viscous flow regime. So, the corresponding mass flow rate can be directly calculated by the nozzle model [12]. 3.2. Heat transfer
(7)
In the operation of DSVP, there is temperature difference between gas and working chamber wall. And the heat transfer occurs due to this temperature difference. In the scroll compressor, the convective heat transfer coefficient between the gas and the scroll wraps is calculated by the correlation for a spiral heat exchanger [5]:
_ should be assumed. Then Equation (7) Firstly the mass flow rate m is solved by the 4-steps Runge–Kutta method in the known inlet pressure of leakage channel. And the outlet pressure of leakage channel can be gotten. If there is some difference between the _ should be modicalculated outlet pressure and the known one, m fied and the process will be repeated until they both get a good _ is mass flow rate of leakage agreement. And the corresponding m channel. It should be noticed that the calculation of internal leakage flow rate should take into account by the length of leakage line. The length of leakage line for different orbiting angles was derived by Wang et al. [4]. Finally, the height of internal leakage channel hc should be discussed. For radial leakage, because there are not tip
ht ¼ 0:023
D Re0:8 Pr 0:4 1 þ 1:77 h Dh Rave
l
(8)
Considering the increased rarefaction effect of gas in DSVP, the viscosity using in Equation (8) should be modified by the following relationship [8]:
m ¼ m0 =ð1 þ bKnÞ Fixed scroll
Fixed scroll
Orbiting scroll
Radial Orbiting scroll clearance
Fig. 4. Internal leakage of DSVP.
(9) Axial clearance
418
Z. Li et al. / Vacuum 84 (2010) 415–421
(3) Compare the initial guess values and the calculated values at q ¼ 2p, if convergent condition is satisfied, calculate the suction pressure of next step psuc,jþ1. It is equal to:
3.4
Nondimensional flow rate
3.2 3.0 2.8
psuc; jþ1
2.6
mvac; j Sm;j 60 n RTvac ¼ Vvac
(13)
and go to (1); otherwise, let the calculated values equal to the initial values and go back to (2).
2.4 2.2
(4) If the pumping speed is less than the set value (1 L/min), the corresponding suction pressure is thought to be reached the ultimate pressure of prototype, and the calculation is end. Otherwise, go back to (1) to star the calculation in the suction pressure psuc,jþ1.
2.0 1.8 Cercignani Model
1.6 1.4 20
0
40
60
80
100
Kn
In a certain suction pressure psuc,jþ1, if the calculations converge, the current pumping speed is [14]:
Fig. 5. Comparison of non-dimensional flow rate.
Sm;j ¼ The temperature distribution of fix scroll and orbiting scroll, the corresponding heat transfer area can be referred by Chen [5]. Then, the heat transfer rate between rarefied gas and working chamber wall can be calculated. 3.3. Conservation equations for working chamber The working chamber of DSVP is divided into suction chamber, compression chamber and discharge chamber. The ‘‘three chamber control volume’’ method [10] was applied to describe the working process of DSVP. For each working chamber, the energy change is governed by the first law of thermodynamics for an open control volume. The temperature change of gas as a function of the orbiting angle can be written as [13]:
h dT ¼ dq
idv 1 X dmi dQ ðhi hÞ þ dq dq Vc dq vp 1 vh v vT vT
vp 1 vh v vv T vv T
v
The current power of working process is
Pj ¼
m h i n X Vðqi Þ psuc; j ðqiþ1 Þ psuc; j ðqi Þ 60 i ¼ 1
(15)
5. Testing system The principle of testing system is shown in Fig. 6. The picture of testing system can be seen in Fig. 7. The range of gas flow meter is from 0 to 300 L/min, its precision is equal to: reading error 0.8% þ full scale error 0.2%. The range of vacuometer is from 0.01 Pa to 1.33 105 Pa; its relative error is 5%.
(10)
v
(11)
The performance of prototype can be predicted by solving the Equations (10) and (11) simultaneously for each working chamber. The pumping speed for different suction pressures, ultimate pressure and power consumption for different speed are then obtained theoretically. Comparison between theoretical results and experimental data is discussed in the following parts.
The air can be treated as ideal gas. The parameters of thermodynamic state satisfy the ideal gas state equation:
pv ¼ RT
(14)
6. Results and discussion
And the mass balance is equal to:
X dmi X dmo dm ¼ dq dq dq
nVvac psuc; j1 ln 60 psuc; j
Ambient
(12)
The mass transfer rate and the heat transfer rate in Equations (10) and (11) can be calculated by the leakage model and the heat transfer rate, respectively. 4. Model solution Equations (10) – (11) for each working chamber can be solved simultaneously by four-step Runge–Kutta method. The solving steps are shown as follows: (1) Let q ¼ 0 and initial guess values of each working chamber. The initial guess values of each working chamber can be obtained by the ideal adiabatic compression process. (2) Then, the thermodynamic parameter values of each working chamber can be calculated according to the 4-steps Runge– Kutta method, until the calculation process is finished.
Diaphragm valve
Needle valve
Gas flow meter Ambient Valve 1 Ambient
Valve 2 Valve 4
Vacuum chamber Valve 3
Secondary vacuum pump Vacuometer
DSVP
Power meter Power Motor
Fig. 6. Principle of testing system.
Z. Li et al. / Vacuum 84 (2010) 415–421
419
6.2. Ultimate pressure and power consumption for different speeds
Fig. 7. Picture of Testing system.
6.1. Pumping speed The pumping speed for different suction pressures is predicted by Equation (14). The comparison of theoretical and experimental results is shown in Fig. 8. A reasonable agreement is gotten. In the theoretical calculations, the resistance of suction pipe, valve and inside the prototype is ignored. This assumption results in the difference between the calculated results and the experimental data. When the suction pressure is in the relative higher range, the resistance has more influence on the pumping speed. As the suction pressure is near the ultimate pressure, the impact of resistance on pumping speed becomes less. The following tendency was also shown by Sawada [2]. The change trend of pumping speed with suction pressure is shown in Fig. 8. The pumping speed goes down quickly as the suction pressure is near the ultimate pressure. The average decreasing rate of pumping speed with suction pressure is 5.09 L/ (kPa min), 9.9 L/(kPa min), 26.6 L/(kPa min) when the speed is 1136 rpm, 1420 rpm, 1704 rpm, respectively. Near the ultimate pressure, it can be concluded that the average decreasing rate of pumping speed with suction pressure becomes larger with the increment of speed. While the suction pressure is in other ranges, the pumping speed decreases slowly with the decrement of suction pressure. The corresponding average decreasing rate of pumping speed with suction pressure is 0.5–0.85 L/(kPa min).
The theoretical ultimate pressure of prototype can be obtained when the solving process of differential Equations (10) and (11) for each working chamber is end. The theoretical and experimental results of ultimate pressure for different speeds are shown in Fig. 9. A good agreement is gotten. The ultimate pressure decreases by 72.6% as the speed increases form 568 rpm to 852 rpm. The ultimate pressure decreases by 73.4% when the speed increases form 852 rpm to 1420 rpm. The ultimate pressure decreases by 31.5% as the speed increases form 1420 rpm to 1704 rpm. The pumping speed goes up with the increment of speed. However, the deformation of scroll parts, which causes by the scroll temperature, also gets larger with the increment of speed. The deformation of scroll parts can increase the scroll clearance, and leads to more internal leakage. So, the ultimate pressure goes down rapidly with the increment of speed in a certain range. As the speed is large enough, the changing rate of ultimate pressure with speed becomes less and less. The power of working process can be predicted by Equation (15). The total input power can be obtained by considering the mechanical efficiency and the motor efficiency. It is known that the power of DSVP changes with suction pressure. It is thought that the DSVP has a maximal ideal adiabatic compression power in a certain suction pressure by the reference [15]. And the corresponding input power is called maximal one. This value directly shows the power consumption level of DSVP. The maximal input power for different speeds is shown in Fig. 10. The deviation of theory and experiment is less than 10%. The maximal input power in a certain speed goes up linearly with the increment of speed. Its changing rate with speed is 0.43 W/(r min). It is concluded that there is a relative ideal speed in the DSVP by the above analysis. The relative lower ultimate pressure and power consumption can be obtained under this speed.
6.3. Influence of volume ratio on the performance The volume ratio is one of the important geometrical parameters of DSVP. It is equal to the ratio of suction chamber volume to discharge chamber volume. In the design of scroll compressor, the volume ratio can be directly derived by pressure ratio. However, the volume ratio of DSVP can not be obtained by the same method because the pressure ratio is always increasing during the operation of DSVP. 100
150
100
1136r·min-1, experiment 1136r·min-1, theory 1420r·min-1, experiment 1420r·min-1, theory 1704r·min-1, experiment 1704r·min-1, theory
50
0
Ultimate pressure / kPa
Pumping speed / L·min-1
Experimental data Theoretical results
80
200
60
40
20
0 0
20
40
60
80
Suction pressure / kPa Fig. 8. Pumping speed for different suction pressures.
100
400
600
800
1000
1200
Speed / r·min-1 Fig. 9. Ultimate pressure.
1400
1600
1800
420
Z. Li et al. / Vacuum 84 (2010) 415–421
100
800
Volume ratio = 2.57 Volume ratio = 2.0 80
Ultimate pressure / kPa
Maximal input power / W
700 600 500 400 300
400
600
800
1000
1200
Speed /
1400
1600
40
20
Experimental data Calculated results
200
60
0
1800
400
r·min-1
600
800
1000
1200
Speed /
1400
1600
1800
r·min-1
Fig. 10. Maximal input power. Fig. 12. Comparison of ultimate pressure.
In order to study the influence of volume ratio on performance, the involute ending angle is fixed and the involute starting angle goes up from 2p to 3p, the corresponding volumetric ratio goes down from 2.57 to 2.0. The influence of volume ratio on performance is tested by experiment. The results of pumping speed are shown in Fig. 11. The pumping speed gains increment in the whole working process as the prototype operates in the relative lower speed (1136 rpm). While the prototype operates in the relative higher
a
speed (1420, 1704 rpm), the pumping speed just gets larger in a certain range of suction pressure. The pumping speed is directly determined by volumetric efficiency. So, the change of pumping speed can be explained by the change of volumetric efficiency. When the suction pressure is in the higher range, the prototype works in over-compression condition, the pressure distribution in working chamber becomes more uniform due to the decreasing of volumetric ratio. This factor leads
b
160
200
Pumping speed / L·min-1
120 100 80 60 40 20
Volume ratio is 2.57 Volume ratio is 2.0
0 0
20
40
60
80
150
100
50 Volume ratio is 2.0 Volume ratio is 2.57
0 0
100
20
Suction pressure / kPa
40
60
Suction pressure / kPa
c 250
Pumping speed / L·min-1
Pumping speed / L·min-1
140
200 150 100 50 Volume ratio is 2.0 Volume ratio is 2.57
0 0
20
40
60
80
100
Suction pressure / kPa Fig. 11. Influence of volume ratio on pumping speed. (a) 1136 rpm; (b) 1420 rpm; (c) 1704 rpm.
80
100
Z. Li et al. / Vacuum 84 (2010) 415–421
pressure when DSVP works in the relative higher speed. But, the choice of volume ratio should be also taken into account by the scroll geometry, discharge temperature, etc.
800
700
Maximal input power / W
421
600
7. Conclusions
500
A general model for describing the working process of DSVP was set up according to the energy and mass conservation principle. By solving this model, the performance of DSVP can be predicted. The influence of speed on pumping speed, ultimate pressure, and power consumption was studied. Furthermore, the influence of volume ratio on performance of DSVP was analyzed experimentally. The lowest ultimate pressure of prototype is 4.43 kPa as the speed is 1704 rpm, and the corresponding maximal Kn of leakage is 0.04. Some conclusions are summarized as follows:
400
300 Volume ratio = 2.57 Volume ratio = 2.0
200 400
600
800
1000
1200
1400
1600
1800
Speed / r·min-1 Fig. 13. Comparison of power.
to the decrement of leakage and the improvement of volumetric efficiency. When the suction pressure is in the lower range, the prototype works in under-compression condition, the decrement of volume ratio can cause the pressure distribution of working chamber to be non-uniform. So the leakage increases and the volumetric efficiency drops. It can be also seen that the trend of pumping speed with suction pressure does not change with the decrement of volume ratio. However, the average decreasing rate of pumping speed with suction pressure for the whole working process increases by 15.4%, 23.8%, 14.9% when the speed is 1136 rpm, 1420 rpm, 1704 rpm, respectively. It can be concluded that in the condition of same decrement of suction pressure, the decrement of pumping speed gets larger as the volume ratio decreases. The ultimate pressure is tested, and the results are shown in Fig. 12. The ultimate pressure goes down with the decrement of speed when the prototype works in low speed condition. This phenomenon can be also explained by the change of volumetric efficiency. For the low speed, the prototype works mainly under over-compression condition as the suction pressure is near the ultimate pressure. According to the analysis for influence of volume ratio on pumping speed, the volumetric efficiency gains improvement and makes the ultimate pressure decreases; and via versa. Moreover, the ultimate pressure goes down by 83.4% when the speed increases from 568 rpm to 852 rpm. While the ultimate pressure just decreases by 46.7% as the speed increases form 852 rpm to 1704 rpm. It can be concluded that the ultimate pressure goes down rapidly with the decrement of speed when the speed is less than 852 rpm. As the speed is higher than 852 rpm, the influence of speed on ultimate pressure becomes less. The maximal input power for different speeds is tested, and the results are shown in Fig. 13. It can be seen that the influence of volume ratio on the maximal input power is small. It is easy to find that the ideal adiabatic compression power is just determined by suction pressure and actual discharge pressure in calculation [15]. By considering the influence of volume ratio on ultimate pressure and power consumption, the ideal speed of prototype changes with the decrement of volume ratio. On the other hand, the increment of volume ratio is helpful to improve the ultimate
(1) Considering the leakage and heat transfer, a general model for predicting the performance of DSVP was presented by the energy and mass conservation principle. A good agreement between theoretical results and experimental data is obtained when Kn < 0.04. (2) The pumping speed goes down with the decrement of suction pressure. The pumping speed drops much more rapidly with the increment of speed as the suction pressure is near ultimate pressure. However, the pumping speed decreases slowly in other suction pressure range. (3) The ultimate pressure drops rapidly with the increment of speed. When the speed is large enough, its impact on ultimate pressure becomes less and less. The maximal input power goes up linearly with the increment of speed. There is a relative ideal speed in the DSVP. The relative lower ultimate pressure and power consumption can be obtained under this speed. (4) The average decreasing rate of pumping speed with suction pressure gets larger with the decrement of volume ratio. With the decrement of volume ratio, the ultimate pressure goes down in the low speed, while it goes up in the high speed. However, the maximal input power of prototype for different speeds does not change apparently.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
[12] [13] [14] [15]
Su Y, et al. Vacuum 1996;47:815. Sawada T, et al. JSME International Journal 1998;44:184. Sawada T, et al. Vacuum 1999;53:233. Wang BL, Li XT, Shi WX. International Journal of Refrigeration 2005;28: 958. Chen Y, et al. International Journal of Refrigeration 2002;25:731. Sharipov F. Journal of Fluid Mechanics 2004;518:35. Shen Q. Rarefied gas dynamics. Defense Industry Press; 2003 [in Chinese]. Beskok A, Trimmer W, Karniadakis GE. ASME, J Fludis Eng 1996;118:448. Cercignani C, Daneri A. J Appl Phys 1963;34:3509. Gu ZL, et al. Scroll compressor and other scroll machinery. Shan’xi Science and Technology Press; 1998 [in Chinese]. Chen ZM, Yuan XL, Zhou ZC. Mathematical modeling and experimental study on two dimensional leakage flow of gas through small gap. Xi’an: Xi’an Jiaotong University Press; 1991. Jing SR, Zhang MY. Fluid mechanics. Xi’an Jiaotong University Press; 2001 [in Chinese]. Li LS, et al. Scroll compressors. Mechanical Industry Press; 1998 [in Chinese]. Clausing P. Journal of Vacuum Science & Technology 1971;8:636. Meng DH, Ba DC. Study on the design theory and manufacture technology of dry scroll vacuum pump. Shenyang: Dongbei University Press; 2004.