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Study on the performance prediction of screw vacuum pump
Abstract
Pumping characteristics of the screw vacuum pump were investigated. The aim of this study was to establish a method of the performance prediction and a way to design the pump that satisfies specific requirements. The performance was analysed by the balance among geometrical pumping speed, net throughput and leaks. The leaks flow through clearances between a screw rotor and a stator, and clearances between two meshing rotors. These leaks were estimated with the results based on the linearised BGK model and the flows through ideal labyrinthes. Experiments were carried out by rotors of 120?mm diameter, and pumping speed and ultimate pressure were measured. The comparison between the measurements and the predicted values shows that the present method predicts the performance of the screw pump with a sufficient accuracy for practical applications
1. Introduction
In recent years, screw vacuum pumps have become noticed, since the structure of the pump is simple and liquids or solids are hard to accumulate when sucked with gas or are condensed or solidified in the pump. An analytical model of screw vacuum pump will be useful to design a pump that satisfies specific requirements and to predict pumping characteristics under conditions which differ from the condition designed for. So, we propose an analytical model for the screw vacuum pump.
2. Outline of analytical modelling
Fig. 1 shows a pair of meshing rotors of the screw vacuum pump. The volume enclosed by a groove of screw, a crest of thread of another rotor and a stator traps gas and transfers it from inlet side to outlet side as the rotors rotate. The model is built by the balance among geometrical pumping speed, net throughput and leaks.
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Fig. 1. Configuration of the meshing rotors of screw vacuum pump.
2.1. Path of leaks inside the screw vacuum pump
There are three kinds of clearances inside the screw pump, i.e. the clearance between rotor and stator δO, the radial clearance between rotors δI, and the axial clearance between rotors Δ. In the case of single thread, paths of leaks which come into or out of the third transfer volume (appearing in Fig. 2A) for example, are as follows.
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Fig. 2. Clearance and flowing path of leak: (A) development of a pair of screws; (B) axial clearances.
The leak through the axial clearance is considered as a superposition of the major component (represented by the straight arrow in Fig. 2B) and minor component (curved arrow).
2.2. Evaluating method of the leaks
We evaluate the leaks by a compound method. The method is compounded of the flow rate derived from BGK equations and diffuse reflections, and the flow rate of ideal labyrinthes.
The leak through the clearance between rotor and stator, and the leak through the axial clearance between rotors are both given in the following form
(1)
and the leak through the radial clearance between rotors is given by
(2)
where MP is the mass flow rate of Poiseuille flow between parallel plates, MS mass flow rate through a slit, MC mass flow rate of Couette flow between parallel plates, MRp mass flow rate through a gap between two cylinders induced by pressure difference, MRr mass flow rate through a gap between two cylinders induced by rotation of the cylinders and ML mass flow rate of ideal labyrinthes.
We obtained precise information on MP and MS from the studies of Hasegawa and Sone, and Sone and Itakura [3 and 6], respectively. MC is determined by the fact that the dimensionless flow rate equals a half, because of the anti-symmetry of the velocity. MRp and MRr are determined by
(3)
MRr=ρδIUQRr
(4)
where RC=1/(1/R1+1/R2), U=(R1+R2)ω. QRp and QRr are obtained by solving the MGL equation [1] by parabolic film approximation [2]. The results are shown in Fig. 3.
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Fig. 3. Nondimensional mass flux through the gap between a cylinder and a plane.
The axial clearance has a non-uniform gap, as shown in Fig. 4, then we define mean or representative quantity to apply the above evaluation methods. For example,
(5)
defines mean square clearance, where SM is the area of lens-like domain appearing in Fig. 4. The other definitions appear in the studies of Ohbayashi et al. [4 and 5].
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Fig. 4. Contour of the gap width of the axial clearance.
2.3. Pumping characteristics
Assuming that the pressure changes isothermally, the pressure pi in the ith transfer volume is represented by
(6)
where V is the volume of one transfer volume, MO leak rate through δO, MI leak rate through δI, MMb major component of the leak through Δ, MMc minor component of the leak through Δ via the minimum gap and iO,iI,iMb,iMc represent the differences between the number of upstream and downstream transfer volumes corresponding to the clearance or path indexed by subscript.
Balance among geometrical pumping speed, net throughput and leaks leads to
(7)
where Tc represents one cycle of periodic pressure change. In the case that the screws are single threaded, Tc equals π/ω, because the ith transfer volume comes to the (i+1)th position after half rotation.
(6) and (7) can be solved under the following periodic condition:
pit=0=pi?1t=Tc, (i=1,2,3,…,nm)
(8)
3. Experiment
Experiments are carried out with a screw vacuum pump whose dimensions are shown in Table 1. Fig. 5a shows the comparison between measurements and analytical predictions relating to pumping speed. Fig. 5b shows the comparison relating to ultimate pressure as a function of rotating speed. The measurements and the predictions for ultimate pressure and pumping speed in the inlet pressure over the 100?Pa range are well agreed.
Table 1. Dimensions of an experimental screw vacuum pump
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Fig. 5. Comparison between experimental results and analytical predictions: (a) pumping speed vs. inlet pressure; (b) ultimate pressure vs. rotation speed.
4. Conclusions
The conclusions are summarised as follows:
1. the screw vacuum pump was analysed, and the analytical model of its pumping characteristics was proposed;
2. the analytical model was verified through the experiments. This model has satisfying accuracy for practical applications.
References
1. S. Fukui, R. Kaneko, Molecular gas film lubrication, in: Handbook of Micro/Nanotribology, CRC Press, Florida, 1995, Chapter 13, pp. 559–604.
2. W.A. Gross, L.A. Matsch, V. Castelli, Fluid Film Lubrication, Wiley, New York, 1980.
3. M. Hasegawa and Y. Sone. Phys. Fluids A 3 3 (1991), pp. 466–477. Full Text via CrossRef
4. T. Ohbayashi, T. Sawada and M. Hamaguchi. Trans. Jpn. Soc. Mech. Eng. B 64 621 (1998), pp. 1419–1425.
5. T. Ohbayashi, T. Sawada, H. Miyamura, Study on the screw vacuum pump with two piecewise constant lead angles, Trans. Jpn. Soc. Mech. Eng. B 65 (637) (1999) 3048–3053.
6. Y. Sone and E. Itakura. J. Vac. Soc. Jpn. 33 3 (1990), pp. 92–94.