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ScienceDirect Procedia CIRP 63 (2017) 483 – 487
The 50th CIRP Conference on Manufacturing Systems
A Numerical Method for Reversely Generating a Pair of Conjugated Rotor Profiles of Twin-screw Compressor from the Measured Profile Data Ruei-Hung Hsu a, Yu-Ren Wu b,*, Zi-An Chen b a
Bachelor’s Program in Precision System Design, Feng Chia University, Taiwan, R.O.C. Department of Mechanical Engineering, National Central University, Taiwan, R.O.C.
b
* E-mail address:
[email protected]
Abstract Reverse engineering technique is generally practiced to reconstitute the original conjugated rotor profiles from the measured point data with profile deviation for further evaluating performance of the competitor’s same-level products in the twin-screw compressor manufacturing industry. However, it is time-consuming when synthesizing and restoring the original rotor profiles via either trial or enveloping methods. This paper proposed a numerical method, named meshing clearance elimination (MCE) method, for approaching pairs of conjugated male and female rotor profiles from the discrete measured profile points. By using this method, the clearance between two screw rotors can be attained via the normal rack generation computation and then eliminated by proportionally distribute the normal compensation onto the measured rotor profiles to obtain the “ideal” rotor profiles to avoid numerical divergence in the following compressor performance computation. Two kinds of rotor profiles measured from CMM (Coordinates Measurement Machine) were demonstrated in the numerical examples to show the effectivity and advantages of the MCE method. © Authors. Published by Elsevier B.V. This ©2017 2017The The Authors. Published by Elsevier B.V.is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of The 50th CIRP Conference on Manufacturing Systems. Peer-review under responsibility of the scientific committee of The 50th CIRP Conference on Manufacturing Systems Keywords: twin-screw compressor ; rotor profile ; meshing clearance ;
Nomenclature
ck , x
coefficient for the curve equation in the x direction
ck ,y
coefficient for the curve equation in the y direction
k k M n n n ,i
order of the curve equation unit vector of z axis transformation matrix unit normal vector unit normal vector of normal rack
nr
unit normal vector of transverse rack
n3,i
unit normal vector of rotors
N Nr
normal vector normal vector of transverse rack
p
proportion of the clearance
r rp
position vector pitch radius
2212-8271 © 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of The 50th CIRP Conference on Manufacturing Systems
doi:10.1016/j.procir.2017.03.137
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rn ,i
normal rack profile equations
rr
profile data of single, center transverse rack
r3,i
rotor profile equations
u us
profile parameter of each curve maximum value of u in each curve
E
helix angle
G n ,2
normal clearance on each point data
Ii
rotation angle for transformation matrix
Subscripts 1 male rotor 2 female rotor 1. Introduction The twin-screw compressor is composed of a pair of paralleled male and female screw rotors. With its high reliability, ease of operation, low vibration, high adaptability and the ability to transport various mixtures of fluids, it’s widely used in a variety of industries such as food processing, medicine manufacturing and air conditioning. The operation of the twin-screw compressor is based on the change of the volume enclosed by the male and female rotor grooves and the housing (working volume). Fluids are being pushed along the rotor channel during operation. The male and female rotors mesh together and forms a working volume that keeps getting smaller and smaller as the rotors rotates. While the working volume becomes smaller, the pressure becomes higher according to the ideal gas equation of state. The working volume is enclosed by the male rotor, female rotor and the housing, therefore, the clearance between them is critical for the performance of the compressor. Each channel on the rotor has different pressure during operation, therefore, the fluids would leak through the clearances from a channel with higher pressure to a channel with lower pressure. There are several kinds of clearances that lead to the leakage, including: x The clearance on the contact line between male and female rotors. x The clearance between the rotor’s end surface and the housing. x The clearance between the tip of the rotor and the housing. x The blow hole enclosed by the male and female rotor tips and the housing. The clearance in twin screw compressor affects the performance of the compressor dramatically, therefore, the clearance should be as small as possible, however, it should also be large enough to ensure smooth operation. In the twin screw compressor manufacturing industry, reverse engineering technique is an important strategy for the researchers to evaluate and compare the performance of compressors between different models, it costs much less and takes much shorter time to obtain some performance indices
through numerical calculations rather than conducting experiments. However, the clearances distributed on the rotor profile lead to difficulties for further analyzing. By using the CMM (coordinate measuring machine), the approximate position of each point on the rotor surface can be obtained. In order to apply the point data to the most analysis, usually it is necessary to make the profiles to be meshed together without clearances. Traditionally, it takes a lot of time for researchers to approach a pair of conjugated rotors from the profile data, and further, the accuracy is usually pretty low. The meshing condition of the screw rotors is similar to the condition in meshing helical gears. Litvin [1] explained the relationship between gears and racks in his book. Here, the concept of gearing theory is applied to generation of the racks of rotors. Xing [2] also described the design geometry of twin screw rotors, and the methods of generating inter tooth clearance. In their papers, Wu and Fong [3,4] put the theories into actual implementation. These papers show how effective the normal-rack generation method is for generating optimized conjugated rotors. With some of the fundamental theories, it is possible to calculate the clearance distribution on the rotor surface through a series of calculations. In Wu and Chi’s paper [5], they described the process of evaluation of the meshing clearance through a numerical strategy in detail. Here, the meshing clearance evaluation method is adopted to evaluate the meshing clearance. The overall process of the calculation is shown in Fig. 1. In this paper, the profile data measured by the coordinate measuring machine is processed in order to calculate the meshing clearance along the contact line. The point data is used to generate the racks both for the male and female rotors, after that, the transverse racks are being projected on to the normal plane to obtain normal racks. The clearance can then be calculated in the form of vector based on each point data of the female normal rack. Then, by adding certain proportion of the distance of vectors to each corresponding point data, a new, single normal rack can be obtained. Finally, project the single normal rack back into the transverse plane and use the single transverse rack to generate the new pair of conjugated screw rotor without clearances. The results are further analyzed to proof the feasibility of the method.
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Fig. 1. Flowchart of the process
Where u is the profile parameter of each curve, and the range for u lies between 0 and us . ck , x and ck ,y are
2. Meshing clearance evaluation method Measured by the coordinate measuring machine, the profile of the rotors is measured from the groove of rotors in most cases, therefore, the profile data of male rotors must be recomposed. After pre-processing, the rotor profiles can be used to generate the transverse racks. In Wu’s [5] paper, the processes are described in detail. Here, cubic splines are used to fit the discrete point data. The fitted curves are expressed as follows: 3 ª 3 º r (u ) [ x(u ), y (u )] « ¦ ck , x u k , ¦ ck ,y u k , 0 d u d us » k 0 ¬k 0 ¼
(1)
coefficients for the curve equation. And k is the order of the equation. For generating the rack, some equations and the coordinate system must be defined:
ri (ui )
> xi (ui ), yi (ui ),1@ , i
N i (ui ) k u
wri ; ni (ui ) wui
1, 2
(2)
N i (ui ) N i (ui ) N i (ui )
, i 1, 2
(3)
In the above equations, r1 and r2 represents the male and female tooth profile. Also, N1 , N 2 and n1 , n 2 represent the normal vectors and unit normal vectors respectively, and k is the unit vector of z axis. nTr ,i (ui , Ii ) M r ,i (Ii ) nTi (ui ), i 1, 2 (5) where
Fig. 2. The coordinate systems of male and female rotors and the transverse rack
With the coordinate systems showed in Fig. 2., the locus equations and the unit normal vectors of racks can be derived as:
rrT,i (ui , Ii ) M r ,i (Ii ) riT (ui ), i 1, 2
M r ,1
ª1 0 rp1 º ª cos(S I1 ) sin(S I1 ) 0 º «0 1 r I » « sin(S I ) cos(S I ) 0 » p1 1 » « 1 1 « ». «¬0 0 » « 1 ¼ ¬ 0 0 1 »¼
M r ,2
ª1 0 rp 2 º ªcos I2 «0 1 r I » « sin I p2 2 » « 2 « 1 ¼» ¬« 0 ¬«0 0
sin I2 cos I2 0
0º 0 »» . 1 ¼»
Here, M r ,i is the transformation matrix that describe the relative motion between the rotor and the rack, rp1 and rp 2 are the pitch radius for male and female rotors. According to the gearing theory, the meshing equation can be written based on the above parameters:
(4)
f i (ui , Ii ) n r ,i wIi rr ,i
0, i 1, 2
(6)
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By solving the relationship between u and I , the transverse rack and its unit normal vectors can be obtained. In order to calculate the meshing clearance between the two rotors, transverse racks have to be projected on to the normal plane. The equation is shown as follows:
M1,r
ªcos(S I1 ) sin(S I1 ) 0 º ª1 0 rp1 º « sin(S I ) cos(S I ) 0 » «0 1 r I » 1 1 p1 1 » . « » « «¬ 0 0 1 »¼ «¬0 0 1 »¼
for the male rotor profile, and
ª¬ xt ,i , yt ,i cos E ,1º¼ , i 1, 2
rn ,i
N n ,i (ui ) k u
wrn ,i wui
(7)
N n ,i (ui )
; n n ,i (ui )
N n ,i (ui ) N n ,i (ui )
, i 1, 2
(8)
Where E is the helix angle, rn ,i is the normal rack profile equations and n n ,i is the unit normal vector of normal racks. Using the female normal rack as the datum rack, the normal clearance on each point data can be obtained through a series of searching and calculation. Then collect the data and save it as G n ,2 for further calculation.
M 2,r
ª cos I2 « sin I 2 « ¬« 0
sin I2 0 º ª1 0 rp 2 º cos I2 0 »» ««0 1 rp 2I2 »» . 0 1 ¼» ¬«0 0 1 ¼»
for the female rotor profile.
3. Conjugated rotor pair generation To generate a pair of conjugated rotor profile without meshing clearance, a single, center rack is required. The new rack can be obtained by adding certain proportion of the distance of the clearance to the female rack profile data, as shown in Fig. 3. Therefore, the single normal rack can be expressed as:
rn ,3
rn ,2 pG n ,2 n n ,2
(9)
Where p represents the proportion of the clearance, which controls where meshing clearance set on. It is set to be 0.5 in the examples. When the profile data of the new rack is determined, it can then be fitted with cubic splines and projected back to the transverse plane.
rr (u )
> xn (u ), yn (u ) / cos E ,1@
N r (u ) k u
wrr ; n r (u ) wu
N r (u ) N r (u ) N r (u )
(10)
(11)
With the new single, center rack, it can be used to generate both male and female rotor profile respectively. Using the meshing equation, the new rotor profiles and the rack have the relationship as follows:
r3,i (ui , Ii ) M i ,r (Ii ) rrT (ui ), i 1, 2
(12)
n3,i (ui , Ii ) M i ,r (Ii ) nTr (ui ), i 1, 2
(13)
f3,i (ui , Ii ) n3,i wIi r3,iT
(14)
Where
0, i 1, 2
Fig. 3. Generating of new single normal rack
4. Numerical examples For the example of this study, two kinds of screw rotors are tested in this paper to ensure the feasibility of this method. Both sets of screw rotor profiles are provided by a well-known twinscrew compressor manufacturer. Detailed parameters of the rotor profiles are shown in Table 1. Table 1. Specification of the rotors Items (units)
Case A
Male rotor tooth number
5
Case B 5
Female rotor tooth number
6
6
Center distance (mm)
123
125
Pitch helix angle (degree)
46
36
Male outer diameter (mm)
173.8
165.9
Female outer diameter (mm)
139
136.5
These rotors are measured profile data using coordinate measuring machine. The analysis shows that both of them have meshing clearances. The graph of the results are plotted on normal plane to show the actual clearance.
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(a) (a)
(b) Fig. 5. Result of Case B
(b) Fig. 4. Result of Case A
The results of Case A are shown in Fig. 4. The graph is plotted based on the female rack. It’s easy to see that the clearance distributed on the rack is massively reduced. In Case B, the effectiveness of the meshing clearance elimination method is also proved. However, the clearance in some segments rises a little bit, which shows the instability of the numerical solution. Currently, the problem remains unsolved, however, it is believed that the problem can be solved through some calculation optimization. Still, the result shows that the method is very effective. Detailed figures are collected in Table 2. 5. Conclusions Traditional methods of reconstruction of rotor tooth profile from the measured point data takes enormous time and effort while the results are inaccurate. This paper proposes a numerical method for reversely generating a pair of conjugated rotor profiles of twin-screw compressor from the measured profile data. By using the meshing clearance elimination method, measured profile data can be optimized, which ensures the accuracy of further analysis. The numerical examples show that the method is easy, fast and effective. With this method, the work of the researchers can be reduced, while the development and experiment process for twin screw compressors can be massively accelerated.
Table 2. Result of the process Items (units)
Case A
Case B
Original max. clearance (mm)
1.59369*10-2
6.79063*10-2
Optimized max. clearance (mm)
8.83621*10-3
1.59683*10-2
Original avg. clearance (mm)
6.01378*10-3
3.91937*10-2
Optimized avg. clearance (mm)
3.43824*10-5
4.02268*10-4
Max. clearance reduced (%)
44.56
76.48
Avg. clearance reduced (%)
99.43
98.97
Acknowledgements The authors would like to thank the Ministry of Science and Technology of Taiwan, R.O.C. (MOST 105-2221-E-008 -047) and the HANBELL Precise Machinery Co., Ltd in Taiwan for their supports. References [1] Faydor L. Litvin, Alfonso Fuentes. Theory of Gearing. 2nd ed. New York: Cambridge University Press; 2004. [2] Z.W. Xing. Screw Compressors: Theory, Design and Application. China Machine Press, Beijing, China; 2000. (in Chinese). [3] Y.R. Wu, Z.H. Fong. Rotor profile design for the twin-screw compressor based on the normal-rack generation method. Journal of Mechanical Design, ASME 130 (2008) 042601-1̢042601-8. [4] Y.R. Wu, Z.H. Fong. Optimization design of an explicitly defined rack for the generation of rotors for twin-screw compressors. Mechanism and Machine Theory 44 (2009) 66̢82. [5] Y.R. Wu, J.W. Chi. A numerical method for the evaluation of themeshing clearance for twin screw rotors with discrete tooth profile points. Mechanism and Machine Theory 70 (2013) 62̢73
