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Evaluating the inter-lobe clearance of twin-screw compressor by the iso-clearance contour diagram (ICCD)
Mechanism and Machine Theory 36 (2001) 725±742
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Evaluating the inter-lobe clearance of twin-screw compressor by the iso-clearance contour diagram (ICCD) Z.H. Fong a,*, F.C. Huang a, H.S. Fang b a
Department of Mechanical Engineering, National Chung Cheng University, 160 San-Hsing, Ming-Hsiung, Chia-Yi 621, Taiwan, ROC b N500 Mechanical Industry Research Laboratories, Industrial Technology Research Institute, Bldg. 21, 195-3 Chung-Hsing Road, Section 4, Chu-Tung, Hsin-Chu 310, Taiwan, ROC Received 19 January 2000; received in revised form 6 November 2000; accepted 18 January 2001
Abstract A mathematical procedure is proposed to calculate the inter-lobe clearance between two mating screw rotors, and then represent the clearance ®eld by iso-clearance contour diagram (ICCD). The theory of gearing and the tooth contact analysis (TCA) have been applied to solve the geometrical and kinematic relations of mating rotors. The contact line and the approximate blowhole can be obtained by TCA. However, it is insucient to describe the inter-lobe leakage. By inspecting the ICCD, we can ®nd all possible inter-lobe leakage paths, not just the contact line and the blowhole. The bene®t of utilizing ICCD is that it can also be modeled by a single mathematical model such as the cubic spline interpolation instead of the mathematical model of the multi-segment tooth pro®les. Hence, this method avoids the problem of discontinuity and divergence in optimal programming. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Screw compressor; Inter-lobe clearance; Iso-clearance contour diagram
1. Introduction The theory of gearing has been applied to obtain the tooth pro®les of the twin-screw compressor while the contact line and the blowhole between mating rotors are solved by tooth contact analysis (TCA). The pro®les of two mating rotors of the conventional twin-screw compressor such as S.R.M. and Atlas are dierent and may be conjugate to each other. Recently, twin-screw compressor with square-threaded rotor (Kashiyama, 1995) is developed for the vacuum system *
Corresponding author. Tel.: +886-5272-0411 ext. 7340; fax: +886-5272-0589. E-mail address: [email protected] (Z.H. Fong).
0094-114X/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 4 - 1 1 4 X ( 0 1 ) 0 0 0 1 8 - 0
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Nomenclature _ e parameter that determines the length of segment D2 E2 fj equation of meshing of segment j and the conjugate one A2 P , i.e., radius of arc A2 B2 of female with center at pitch point P h1 A1 Oa , i.e., parameter that determines the position of point A1 h2 i gear ratio lead of rotor j `j submatrix; transform from coordinate system Sj to coordinate system Sk Lkj Mkj coordinate transformation matrix; transform from coordinate system Sj to coordinate system Sk nkj component of unit normal vector in the direction of k-axis in coordinate system Sj ; j 1; 2 and k X ; Y ; Z f component of unit normal vector nkj represented in coordinate system nkj Sf ; j 1; 2 and k X ; Y ; Z unit normal vector represented in coordinate system Sk nk Nk normal vector represented in coordinate system Sk f Nk normal vector Nk represented in coordinate system Sf Oj origin of the coordinate system Sj ; j 1; 2; a; b; c; d; f position vector of point jk ptjk pj screw parameter of rotor j; pj `j =2p P centrode point as shown in Figs.1 and 6 cavity j de®ned in Figs. 8, 10, 12, 13 and 14 Pj _ rjgk position vector of segment jk gk rk position vector of rotor k represented in coordinate system Sk ; k a; b radius of pitch circle, j 1; 2 rj centrode of rotor j rpj position vector of rotor surface formed by rk represented in coordinate system Rk Sk ; k a; b f position vector Rk represented in coordinate system Sf ; k a; b Rk position vector for the rotor surface formed by rjgk Rjgk f Rjgk position vector Rjgk represented in coordinate system Sf Sj xj ; yj ; zj coordinate system j where j 1; 2; a; b; c; d; f denotes the coordinate systems de®ned in Figs. 1 and 2 variable parameter of rotor j _ uj variable parameter of segment jk gk for rotor k ujgk upper bound of the parameter ujgk for rotor k uup jgk component of segment j in the direction of X-axis Xj component of segment j in the direction of Y-axis Yj component of segment j in the direction of Z-axis Zj b angular parameter that determines the position of point A1 /j rotation angle of rotor j; j a; b rotation angle of rotor j when generate the surfaces, j 1; 2 hj
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H1 H2 Rk
727
_
angular parameter that determines the position of segment B2 C2 _ angular parameter that determines the position of segment D2 E2 surface k represented in coordinate system Sk ; k a; b
and distinguished from the conventional one by the extra large wrap angle except the tooth pro®le. The axial pro®les of rotors of twin square-threaded screw compressor such as Kashiyama [5] are exactly the same and may not be conjugate at all. The wrap angle of the conventional twin-screw compressor is always less than 360° while the wrap angle of the twin square-threaded screw compressor is several times of 360°. The conventional twin-screw compressor is designed to form a sealing line (instant contact line) between mating rotors; however, the sealing line is usually incomplete between rotor tips and the housing cusp. This critical section of this incomplete sealing is called a blowhole, and the ¯uid will leak back from one cavity to the following mating cavity through the blowhole. In the conventional twin-screw compressor, the sealing line divides the cavity of the screw rotor into two portions, the low- and high-pressure portions. The ¯uid leaks back mainly through the blowhole in the high-pressure side. Therefore, the tooth pro®les of the conventional twin-screw compressor are asymmetric to reduce the blowhole in the high-pressure side while the size of blowhole in the low-pressure side is neglected. However, in case of the twin square-threaded screw compressor, the ¯uid is ®rst trapped in the mating cavities from the low-pressure inlet then transferred to the high-pressure outlet. During the transfer process, the ¯uid will leak back not only in the ``high-pressure'' side but also in the ``lowpressure'' side. The leakage paths are complicated because of the extra large wrap angle, blowholes in the low- and high-pressure sides and the incomplete adjacent sealing lines. The results of TCA are no longer sucient to describe all possible leakage paths; therefore, we propose a procedure to calculate the inter-lobe clearance between mating rotors and then represent the clearance topography by the iso-clearance contour diagram (ICCD). Although the techniques of analysis and design for the geometrical pro®les and the thermodynamics of the mating rotors in several ®elds have been developed decades ago, a great many are still unpublished yet. Deng and Shu [1] collated the working process, rotor design and feature calculations for several types of rotational compressor. Singh and Onuschak [6] presented some results about how the geometrical parameters of pro®les aect the performance of screw compressors by a computerized model. Singh and Schwartz [7], and Singh and Bowman [8] introduced an exact method based on vector analysis and two approximate methods for blowhole area calculation. They selected the third vortex, which forms the bounded curvilinear triangle and minimizes the amplitude of the vector area by calculating it one by one. Zhou [10] derived the housing crossing line of the blowhole area by the height at dierent angle of rotation while a point on pro®le of one rotor moves a distance in the direction of Z-axis. Zhang and Hamilton [9] proposed the calculation of the main geometric characteristics including the contact line and blowhole on the assumption that the shape of the blowhole is an irregular three-dimensional surface and this surface is approximated by a plane. Recently, Fleming et al. [2] presented pictorial and schematic representations for several internal leakage paths through clearances within the housing.
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The tooth pro®le of the rotor can be modeled by the transverse section and the coordinate transfer matrix, which performs the screw motion. If the transverse sections of the mating screw rotors are known, we can calculate the axial clearance by the screw parameters and other related data. The ®eld of the normal clearance is thus represented by ICCD. By inspecting the proposed mathematical procedure of ICCD, we can ®nd all the possible leakage paths, not only the contact line and blowhole. The bene®t of applying ICCD is that the ICCD can also be modeled by a single mathematical model such as the cubic spline interpolation instead of the mathematical model of the multisegment tooth pro®les. The reason is that the trouble solving the inter-relationship between parameters and writing the relationships in the problem's constraints when using exact mathematical model in optimization is avoided with cubic spline interpolation. And there is no limitation of usage of cubic spline with TCA results from the dierentiation operation in solving the basic meshing condition anymore and the problem of discontinuity and divergence in optimal programming if we apply the ICCD method to calculate inter-lobe clearance. By utilizing the ICCD, the tooth pro®les of the two mating rotors are not necessarily conjugated to each other. The ¯uid sealing is evaluated directly by the clearance, not the exact contact line and blowhole. This feature increases more ¯exibility in geometrical design and manufacturing. The result of the proposed analysis model provides the basis of the exact evaluation of the ¯uid leakage. More realistic estimation of the ¯uid leakage in the vacuum system can be calculated in collaboration with the computational ¯uid dynamics (CFD) software.
2. Mathematical modeling As shown in Fig. 1, two mating screw rotors rotate with constant gear ratio in the opposite directions about parallel axes. The coordinate systems, Sa xa ; ya ; za , Sb xb ; yb ; zb , and Sf xf ; yf ; zf , are rigidly attached to rotational axes of screw rotors 1, 2 and the frame, respectively. At least one of the transverse sections of screw rotors is known either in the form of mathematical equations or the discrete measured data points. If axial section pro®le is known, the transverse section can be
Fig. 1. Coordinate systems applied to the transverse sections of twin-screw rotors.
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derived after the operation of screw motion. In the method of ICCD, the transverse sections are not necessary to satisfy the conjugate conditions, but it is essential that pro®les should not interfered with each other. There are two approaches to utilize the ICCD: (a) the exact mathematical model of the rotor pro®le is known, and (b) the discrete measured data points of the transverse section are known. The details of the procedure of the ICCD are described in the following section. 2.1. Exact mathematical model of the rotor pro®le is known Assume that transverse section of the rotor pro®les of screw rotors can be represented in the coordinate systems, Sa and Sb , respectively, by the following equations: rj rj uj X uj
Y uj
0 1
T
j a; b:
1
If one of the transverse sections, say rb , of two mating screw rotors is known, then the tooth pro®le of the other mating rotor, say ra , can be derived according to the theorem of planar gearing. The locus of known tooth pro®le, rb , represented in coordinate system Sa by the operation of coordinate transformation is ra Maf Mfb rb ; where Maf
2
2
cos /a sin /a 6 sin /a cos /a 6 4 0 0 0 0 2
cos /b 6 sin /b Mfb 6 4 0 0
sin /b cos /b 0 0
0 0 1 0
3 0 07 7; 05 1
0 0 1 0
3 r1 r2 0 7 7: 0 5 1
3
Since the mating rotors rotate with constant gear ratio in the opposite directions, the gear ratio i r1 =r2 constant and /b i/a . The common normal at point of tangency must pass through the centrode point P as shown in Fig. 1. The equation of meshing between mating rotors are derived as follows: fa u; /
XP xa Nxa
YP ya 0; Nya
4
where XP ; YP are the Cartesian coordinates of centrode point P . The xa ; ya and the Nxa ; Nya are the Cartesian coordinates and the normal vector of instant contact point on the transverse tooth pro®le of rotor 1, respectively. Based on the theory of gearing [3,4], by solving the equation of meshing, fa u; / 0, together with the locus of rotor 2, rb , in coordinate system Sa , yield the envelope of the locus, i.e., the transverse tooth pro®le of rotor 1, ra . The surface equations of mating rotors could be obtained by means of screw motion of the pro®les along the respective axes of rotors as shown in Fig. 2. The mating rotors rotate with
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Fig. 2. Coordinate systems of right- and left-handed screw surfaces.
constant gear ratio between parallel axes in the opposite directions. Coordinate systems Sa Xa ; Ya ; Za and Sb Xb ; Yb ; Zb are used to de®ne the cross section of the screw rotors while coordinate systems Sc Xc ; Yc ; Zc and Sd Xd ; Yd ; Zd are used to de®ne the cross-section that is elevated from the reference datum plane. The relationship between rotors might be dierent when the lead of the screw is dierent from the theoretical value. The setting of coordinate systems like above could increase the ability to simulate the manufacturing error of the rotor. The tooth surfaces, R1 and R2 , of mating rotors are expressed by the following equations: R1 M1c h1 ra ua ;
R2 M2d h2 rb ub ;
5
where M1c and M2d are the coordinate transformation matrices for the right-handed and lefthanded screw motions, respectively, 2 3 2 3 cos h1 cos h2 sin h1 0 0 sin h2 0 0 6 sin h1 cos h1 0 6 0 7 0 7 7; M2d 6 sin h2 cos h2 0 7; 6 M1c 6 4 0 5 4 0 1 p1 h1 0 0 1 p2 h2 5 0 0 0 1 0 0 0 1 p1 ip2 and h2 ih1 . The normal vectors of surfaces R1 and R2 are represented in S1 and S2 by N1
oR1 oR1 ; oua oh1
N2
oR2 oR2 oub oh2
7
and the unit vectors can be determined by nj
Nj nxj i nyj j nzj k jNj j
j 1; 2:
8
Then the surfaces expressed in the ®xed coordinate system Sf may be determined by the operation of coordinate transformation:
Z.H. Fong et al. / Mechanism and Machine Theory 36 (2001) 725±742 f
R1 Mfa /a R1 ua ; h1 ;
f
R2 Mfb /b R2 ub ; h2 :
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9
The normal vectors of surfaces R1 and R2 represented in coordinate system Sf is obtained by the following matrix equation: f
N1 Lfa /a N1 ; where
2
cos /a 4 Lfa sin /a 0
f
N2 Lfb /b N2 ;
sin /a cos /a 0
3 0 0 5; 1
10 2
cos /b 4 Lfb sin /b 0
sin /b cos /b 0
3 0 0 5: 1
11
The overlapped cavity is the chamber formed by the adjacent surfaces of two dierent rotors as shown as the hatched region in Figs. 3 and 4. The working ¯uid leaks mainly through the overlapped cavities. Therefore, the shape and size of the overlapped cavity between the two rotors are required to describe the leakage paths between the mating rotors. The overlapped cavities translate parallel to the rotation axes during the working process. The shape of the overlapped cavity is the same through the working process. However, the calculation for the shape of the
Fig. 3. De®nitions of the overlapped cavity.
Fig. 4. Projected area of overlapped cavity from the measured data points for twin square-threaded screw compressor.
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overlapped cavities is quite complex in most cases. There are many two-dimensional simpli®ed method proposed to describe the leakage paths. Following, we propose a three-dimensional methodology to represent the size and the shape of the overlapped cavities. Since the size and the shape of the overlapped cavities will not change through the working process, the projection of the overlapped cavities to the transverse plane is the same all the time. The projected area can be simply determined by the outer diameter of two mating rotors as shown in Figs. 3 and 4. Therefore, we use the projected area as the datum plane for further discussion. The tooth surfaces of mating rotors are derived through the screw motion of the transverse sections, the axial height of the overlapped cavity is the dierence of the axial distances resulted from the respective rotation angle of screw motion. For an arbitrary point J X ; Y in the projected overlapped area, we can ®nd two points, point U XU ; YU and V XV ; YV on the respective transverse sections as shown in Fig. 3. Points U and V are in the same transverse plane and belong to the mating tooth pro®les, R1 and R2 , respectively. After the operation of screw motion, the x- and y-coordinates reach the same point J X ; Y . Therefore, Ob J Ob V and Oa J Oa U . The z-coordinates of points U and V are the same. The position vectors of tooth surfaces of rotors 1 and 2 are denoted by R1 and R2 , respectively, as expressed in Eq. (5). The position vectors R1 and R2 are transformed to the ®xed coordinate system Sf as follows: f
R1 Mfa /a R1 ua ; h1 XU ua ; h1
YU ua ; h1
T
ZU ua ; h1 1 ;
f
R2 Mfb /b R2 ub ; h2 XV ub ; h2
YV ub ; h2
ZV ub ; h2
12
1T ;
where /a and /b is the rigid body rotation angle of rotor 1 and rotor 2, respectively. /a and /b are determined by the non-interference checking. As shown in Fig. 3, coordinates of points U and V are solved by the following simultaneous equations: XU ua ; h1 X ; 13 YU ua ; h1 Y ;
XV ub ; h2 X ; YV ub ; h2 Y :
14
Substitute the values of ua ; h1 and ub ; h2 back to Eq. (5) to determine the position vectors of points U and V . The axial dierence, Dz p2 h2 p1 h1 , is the axial height of the overlapped cavity at projected point J X ; Y . Therefore, the normal height at point J is f
H Dz nz2 :
15
By applying the above-mentioned calculation, the shape of the overlapped cavity can be described by the projected area and the corresponding normal height. The results of the calculation are represented by the iso-height contour plot. Since the height of the plot is the normal clearance between rotors, the plot is called as the ``ICCD'', as shown in Fig. 8.
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2.2. Measured data points of the transverse section are known When the mating rotors are available, the transverse or axial section of rotors can be obtained by CNC coordinate-measuring machine. The measured tooth pro®le is represented in the form of a series of data points. The mathematical equation for the tooth pro®les can be derived by curve ®tting or interpolation from the discrete measured data points. Once the equations for the tooth
Fig. 5. Flowchart of the method of ICCD.
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Fig. 6. Unsymmetrical cyclode-arc pro®le of Example 1 for conventional twin-screw compressor.
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pro®les are known, we can follow the above-mentioned procedures to construct the ICCD. The ¯owchart for the calculation of ICCD is shown in Fig. 5.
3. Examples and discussions 3.1. Example 1: Tooth pro®le of unsymmetrical cycloid-arc for the conventional twin-screw compressor A conventional twin-screw compressor with unsymmetrical cycloid-arc pro®les is shown in Fig. 6. The lobe combination is 4 + 6. The transverse section of female rotor is given in the coordinate system Sb as shown in Table 1. The tooth pro®le of male rotor is derived in the coordinate system Sa by the theory of planar gearing according to Table 2. The coordinate systems Sa and Sb are attached to male and female rotors, respectively. Parameters required to input are r2 ; i; h1 ; uup AB2 (i.e., the upper bound of uAB2 ) and e as shown in Fig. 6. The transverse sections of the mating rotors are de®ned by the equations shown in Tables 1 and 2. Apply Eqs. (1)±(14), and the normal clearance can be calculated. The overlapped cavity is partitioned into two regions by the topland of the male rotor. As shown in Fig. 7, the forward side of the overlapped cavity is the clearance between lower tooth _ _ side of the overlapped surfaces, i.e., between arc B1 C1 of male rotor and arc B2 C2 . The backward _ E of male rotor and arc cavity is the clearance between upper tooth surfaces, i.e., between arc A 1 1 _ A2 E2 . Therefore, the ICCD should be calculated in both forward and backward sides. The result of normal clearance is exhibited in Fig. 8 in the form of contour diagram while the values of input are r2 30; i 2=3; h1 14; uup AB2 10°; e 1 and choosing the screw Table 1 Pro®les of female rotor Type Arc
Notation _ A2 B2
Equations
Notes
rAB2 r2
h1 cos uAB2 uup AB2
where Degenerate hypocycloid Epicycloid
_ B2 C2
_
A2 E2
_
D2 E2 _
C2 D2
h1 sin uAB2 0 1 ;
6 uAB2 6 p=2
0 1T ;
ptA1 h2 cos b h2 sin b 0 T ; rAE2 Mbf Mfa ptA1 q rAE2 uAE2 ; where h2 h21 r12 2h1 r1 cos p uup AB2 and 1
Given
1
sin h1 =r2
rBC2 uBC2 cos H1 uBC2 sin H1 q where r22 h21 6 uBC2 6 r2
b sin Degenerate hypocycloid Arc
T
Given
Given point A1
h1 sin uup AB2 h2
rDE2 uDE2 cos H2 uDE2 sin H2 0 1T ; where r2 e 6 uDE2 6 r2 rCD2 r2 cos uCD2 r2 sin uCD2 0 1T ; where H2 6 uCD2 6 p3 H1
Given Given
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Table 2 Pro®les of male rotor Type
Notation
Arc
_ A1 B1
Epicycloid
B1 C1
Epicycloid
A1 E1
Epicycloid
D1 E1
Arc
C 1 D1
_
_
_
_
Equations
Notes
rAB1 Maf Mfb rAB2 fAB h1 sin uAB2 sin uAB2 uAB1 r2 0
Derived
rBC1 Maf Mfb rBC2 fBC uBC2 cos uBC1
Derived
H1 r2 0
ptE2 r2 e cos H2 r2 rAE1 Maf Mfb ptE2
e sin H2 0 T
Given point E2
rDE1 Maf Mfb rDE2 fDE uDE2 cos uDE1 H2 r2 0
Derived
rCD1 r1 cos uCD1 r1 sin uCD1 0 1T ; H2 p H1 where 6 uCD1 6 2 i i
Given
Fig. 7. Determination of the two partitioned regions of overlapped area for conventional twin-screw compressor.
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Fig. 8. ICCD of Example 1 for unsymmetrical cycloid-arc pro®le.
parameter p2 11:4592. The ICCD shown in Fig. 8 indicates that the leakage path is relatively small between cavities Pj1 and Pi1 . The working ¯uid trapped in the cavity will be pumped from suction port to the discharge port without large pressure losses. However, the leakage path between discharge port and cavity Pi1 is relatively wide. Therefore, the wrap angle of the rotors of conventional twin-screw pump should be less than 360°, otherwise, the working ¯uid will leak back from cavity Pi to Pi1 . The positions of the zero clearance regions in the ICCD consistent with the contact lines solved by the TCA. However, additional information on the sealing eect is observed by inspecting the gradient of the iso-clearance lines. Large zero-clearance region indicates good eect on sealing. By inspecting the gradient of the iso-clearance lines shown in Fig. 8, the normal clearance between the housing cusp and the nearest contact point is not a triangle. Therefore, the two-dimensional triangle simpli®cation to estimate the size of blowhole is not accurate. 3.2. Example 2: Pro®le of discrete measured data points for twin square-threaded screw compressor As shown in Fig. 9, the axial section of the rotor of twin square-threaded screw compressor is measured on the CNC coordinate-measuring machine. The axial sections of mating rotors are the same while the hand of rotation are opposite to each other. The transverse section can be obtained after the operation of screw motion as shown in Fig. 2. The pro®le of transverse section is shown in Fig. 4, and the hatched region is the projected area of overlapped cavity. The ICCD is exhibited in Fig. 10 with the screw parameter, p1 p2 5:7296.
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Fig. 9. The measured data points of Example 2 for twin square-threaded screw compressor.
The tooth pro®les of mating rotors shown in Example 2 are not conjugate to each other. The TCA fails to ®nd the contact lines. By utilizing the ICCD, the pseudo-contact lines are approximated by zero-clearance contour lines. Additional 0.2 mm normal backlash is required to avoid interference. However, the ICCD results show that the leakage paths between cavities are relatively small. 3.3. Example 3: Conjugate tooth pro®les of cycloid-involute for twin square-threaded screw compressor The pro®les of transverse sections are formed by cycloid and involute curves as shown in Fig. 11. The tooth pro®les in Example 3 are conjugate to each other and can be ground on the gear grinding machine or thread grinding machine. The ICCD is exhibited in Fig. 12 while the value of screw parameter is the same as Example 2, p1 p2 5:7296. By means of the ICCD proposed here, we can easily determine the position of blowhole and obtain the contact line to describe the contact relations of mating rotors. Moreover, inspecting the ICCD, we may ®nd all the possible leakage paths to evaluate the inter-lobe clearance.
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Fig. 10. ICCD of Example 2 for the twin square-threaded screw compressor.
Fig. 11. Tooth pro®le of Example 3 for twin square-threaded screw compressor.
In detail, blowhole is located within the high-pressure area and formed by the housing cusp and the nearest contact point as shown in Figs. 8, 10 and 12. The contact line is where the normal height of overlapped cavity is equal to zero. The possible leakage paths are demonstrated in Figs. 13 and 14, including the contact line, blowhole, the suction and discharge port, and the axial leakage between the rotor tips and housing, etc.
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Fig. 12. ICCD of Example 3 for twin square-threaded screw compressor.
Fig. 13. Transverse view of possible leakage paths for twin-screw compressor.
According to the results of the ICCD as shown in Figs. 8 and 10, there exists leakage path as long as the normal height of overlapped cavity is not zero. In other words, the ¯uid ¯ow may leak back through the same groove or along the axial direction till over next two stages, not only the adjacent one.
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Fig. 14. Axial view of possible leakage paths for twin square-threaded screw compressor.
The leakage paths are quite dierent between the twin square-threaded screw compressor and the conventional one. All possible leakage paths are shown in Fig. 13. The unsymmetrical cycloidarc pro®le for the conventional twin-screw compressor as shown in Fig. 13, the cavity Pj is open to the suction port while the cavity Pj2 is open to discharge port. The leakage paths B2 ±B3 and D will not produce the direct leakage of working ¯uid from discharge port to suction port. Therefore, the sealing at leakage paths B2 ±B3 and D can be ignored in the conventional twin-screw compressor. For the pro®le of the twin square-threaded screw compressor, the contact line as referred to Fig. 10 is almost a slanting straight line and divides the overlapped cavity into two parts. With the characteristic of extra large wrap angle, there is no distinctive separation from the low- and highpressure side. All possible leakage paths should be taken into consideration and calculated during the design phase. 4. Conclusion A mathematical procedure is proposed to calculate the inter-lobe clearance between two mating screw rotors and then represent the clearance topography by ICCD. The ICCD proposed here may be used to determine the size and shape of the overlapped cavity and the complete inter-rotor clearance. By means of the ICCD, we can easily determine the position of blowhole and obtain the contact line to describe the contact relations of mating rotors. Moreover, inspecting the ICCD, we may ®nd all the possible leakage paths to evaluate the inter-lobe clearance. According to the examples and discussions, following conclusions are drawn: (a) The two-dimensional triangle simpli®cation to estimate the size of blowhole is not accurate. The estimated size by the triangle simpli®cation is larger than true blow whole. (b) The ICCD provides better description on the leakage paths. Not only the position of the blowhole but also the shape of critical leakage path are presented in the ICCD. The ¯ow rate of the leakage can be accurately calculated by the CFD software.
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(c) The ICCD can also be modeled by a single mathematical model such as the cubic spline interpolation instead of the mathematical model of the multi-segment tooth pro®les. Hence, this method avoids the problem of discontinuity and divergence in optimal programming. (d) By utilizing the ICCD, the tooth pro®les of the two mating rotors are not necessarily conjugated to each other. The pseudo-contact lines are approximated by zero-clearance contour lines. (e) The ¯uid sealing is evaluated directly by the clearance, not the exact contact line and blowhole, and this feature increases more ¯exibility in geometrical design and manufacturing. References [1] D.G. Deng, P.C. Shu, Rotational Compressor, Mechanical Engineering Press, China, 1985, pp. 91±96. [2] J.S. Fleming, Y. Tang, G. Cook, The twin helical screw compressor, part 1: Development applications and competitive position, J. Mech. Eng. Sci. 212 (1998) 355. [3] F.L. Litvin, Theory of Gearing, NASA Reference Publication 1212 (AVSCOM 88-C-035), Washington, DC, 1989. [4] F.L. Litvin, in: Gear Geometry and Applied Theory, Prentice-Hall, New York, 1994. [5] Kashiyama Industry Co., Ltd., Osamu Ozawa, Nagano, Gas Exhaust System and Pump Cleaning System for a Semiconductor Manufacturing Apparatus, US Patent 5443644, 1995. [6] P.J. Singh, A.D. Onuschak, A comprehensive computerized method for twin-screw rotor pro®le generation and analysis, in: Proceedings of the 1984 International Compressor Engineering Conference at Purdue, West Lafayette, 1984, pp. 519±527. [7] P.J. Singh, P.R. Schwartz, Exact analytical representation of screw compressor rotor geometry, in: Proceedings of the 1990 International Compressor Engineering Conference at Purdue, West Lafayette, 1990, pp. 925±937. [8] P.J. Singh, J.L. Bowman, Calculation of blow-hole area for screw compressors, in: Proceedings of the 1990 International Compressor Engineering Conference at Purdue, West Lafayette, 1990, pp. 938±948. [9] L. Zhang, J.F. Hamilton, Main geometric characteristics of the twin screw compressor, in: Proceedings of the 1992 International Compressor Engineering Conference at Purdue, West Lafayette, 1992, pp. 449±456. [10] Z. Zhou, Computer aided design of a twin-rotor screw refrigerant compressor, in: Proceedings of the 1992 International Compressor Engineering Conference at Purdue, West Lafayette, 1992, pp. 457±466.
www.elsevier.com/locate/mechmt
Evaluating the inter-lobe clearance of twin-screw compressor by the iso-clearance contour diagram (ICCD) Z.H. Fong a,*, F.C. Huang a, H.S. Fang b a
Department of Mechanical Engineering, National Chung Cheng University, 160 San-Hsing, Ming-Hsiung, Chia-Yi 621, Taiwan, ROC b N500 Mechanical Industry Research Laboratories, Industrial Technology Research Institute, Bldg. 21, 195-3 Chung-Hsing Road, Section 4, Chu-Tung, Hsin-Chu 310, Taiwan, ROC Received 19 January 2000; received in revised form 6 November 2000; accepted 18 January 2001
Abstract A mathematical procedure is proposed to calculate the inter-lobe clearance between two mating screw rotors, and then represent the clearance ®eld by iso-clearance contour diagram (ICCD). The theory of gearing and the tooth contact analysis (TCA) have been applied to solve the geometrical and kinematic relations of mating rotors. The contact line and the approximate blowhole can be obtained by TCA. However, it is insucient to describe the inter-lobe leakage. By inspecting the ICCD, we can ®nd all possible inter-lobe leakage paths, not just the contact line and the blowhole. The bene®t of utilizing ICCD is that it can also be modeled by a single mathematical model such as the cubic spline interpolation instead of the mathematical model of the multi-segment tooth pro®les. Hence, this method avoids the problem of discontinuity and divergence in optimal programming. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Screw compressor; Inter-lobe clearance; Iso-clearance contour diagram
1. Introduction The theory of gearing has been applied to obtain the tooth pro®les of the twin-screw compressor while the contact line and the blowhole between mating rotors are solved by tooth contact analysis (TCA). The pro®les of two mating rotors of the conventional twin-screw compressor such as S.R.M. and Atlas are dierent and may be conjugate to each other. Recently, twin-screw compressor with square-threaded rotor (Kashiyama, 1995) is developed for the vacuum system *
Corresponding author. Tel.: +886-5272-0411 ext. 7340; fax: +886-5272-0589. E-mail address: [email protected] (Z.H. Fong).
0094-114X/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 4 - 1 1 4 X ( 0 1 ) 0 0 0 1 8 - 0
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Nomenclature _ e parameter that determines the length of segment D2 E2 fj equation of meshing of segment j and the conjugate one A2 P , i.e., radius of arc A2 B2 of female with center at pitch point P h1 A1 Oa , i.e., parameter that determines the position of point A1 h2 i gear ratio lead of rotor j `j submatrix; transform from coordinate system Sj to coordinate system Sk Lkj Mkj coordinate transformation matrix; transform from coordinate system Sj to coordinate system Sk nkj component of unit normal vector in the direction of k-axis in coordinate system Sj ; j 1; 2 and k X ; Y ; Z f component of unit normal vector nkj represented in coordinate system nkj Sf ; j 1; 2 and k X ; Y ; Z unit normal vector represented in coordinate system Sk nk Nk normal vector represented in coordinate system Sk f Nk normal vector Nk represented in coordinate system Sf Oj origin of the coordinate system Sj ; j 1; 2; a; b; c; d; f position vector of point jk ptjk pj screw parameter of rotor j; pj `j =2p P centrode point as shown in Figs.1 and 6 cavity j de®ned in Figs. 8, 10, 12, 13 and 14 Pj _ rjgk position vector of segment jk gk rk position vector of rotor k represented in coordinate system Sk ; k a; b radius of pitch circle, j 1; 2 rj centrode of rotor j rpj position vector of rotor surface formed by rk represented in coordinate system Rk Sk ; k a; b f position vector Rk represented in coordinate system Sf ; k a; b Rk position vector for the rotor surface formed by rjgk Rjgk f Rjgk position vector Rjgk represented in coordinate system Sf Sj xj ; yj ; zj coordinate system j where j 1; 2; a; b; c; d; f denotes the coordinate systems de®ned in Figs. 1 and 2 variable parameter of rotor j _ uj variable parameter of segment jk gk for rotor k ujgk upper bound of the parameter ujgk for rotor k uup jgk component of segment j in the direction of X-axis Xj component of segment j in the direction of Y-axis Yj component of segment j in the direction of Z-axis Zj b angular parameter that determines the position of point A1 /j rotation angle of rotor j; j a; b rotation angle of rotor j when generate the surfaces, j 1; 2 hj
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H1 H2 Rk
727
_
angular parameter that determines the position of segment B2 C2 _ angular parameter that determines the position of segment D2 E2 surface k represented in coordinate system Sk ; k a; b
and distinguished from the conventional one by the extra large wrap angle except the tooth pro®le. The axial pro®les of rotors of twin square-threaded screw compressor such as Kashiyama [5] are exactly the same and may not be conjugate at all. The wrap angle of the conventional twin-screw compressor is always less than 360° while the wrap angle of the twin square-threaded screw compressor is several times of 360°. The conventional twin-screw compressor is designed to form a sealing line (instant contact line) between mating rotors; however, the sealing line is usually incomplete between rotor tips and the housing cusp. This critical section of this incomplete sealing is called a blowhole, and the ¯uid will leak back from one cavity to the following mating cavity through the blowhole. In the conventional twin-screw compressor, the sealing line divides the cavity of the screw rotor into two portions, the low- and high-pressure portions. The ¯uid leaks back mainly through the blowhole in the high-pressure side. Therefore, the tooth pro®les of the conventional twin-screw compressor are asymmetric to reduce the blowhole in the high-pressure side while the size of blowhole in the low-pressure side is neglected. However, in case of the twin square-threaded screw compressor, the ¯uid is ®rst trapped in the mating cavities from the low-pressure inlet then transferred to the high-pressure outlet. During the transfer process, the ¯uid will leak back not only in the ``high-pressure'' side but also in the ``lowpressure'' side. The leakage paths are complicated because of the extra large wrap angle, blowholes in the low- and high-pressure sides and the incomplete adjacent sealing lines. The results of TCA are no longer sucient to describe all possible leakage paths; therefore, we propose a procedure to calculate the inter-lobe clearance between mating rotors and then represent the clearance topography by the iso-clearance contour diagram (ICCD). Although the techniques of analysis and design for the geometrical pro®les and the thermodynamics of the mating rotors in several ®elds have been developed decades ago, a great many are still unpublished yet. Deng and Shu [1] collated the working process, rotor design and feature calculations for several types of rotational compressor. Singh and Onuschak [6] presented some results about how the geometrical parameters of pro®les aect the performance of screw compressors by a computerized model. Singh and Schwartz [7], and Singh and Bowman [8] introduced an exact method based on vector analysis and two approximate methods for blowhole area calculation. They selected the third vortex, which forms the bounded curvilinear triangle and minimizes the amplitude of the vector area by calculating it one by one. Zhou [10] derived the housing crossing line of the blowhole area by the height at dierent angle of rotation while a point on pro®le of one rotor moves a distance in the direction of Z-axis. Zhang and Hamilton [9] proposed the calculation of the main geometric characteristics including the contact line and blowhole on the assumption that the shape of the blowhole is an irregular three-dimensional surface and this surface is approximated by a plane. Recently, Fleming et al. [2] presented pictorial and schematic representations for several internal leakage paths through clearances within the housing.
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The tooth pro®le of the rotor can be modeled by the transverse section and the coordinate transfer matrix, which performs the screw motion. If the transverse sections of the mating screw rotors are known, we can calculate the axial clearance by the screw parameters and other related data. The ®eld of the normal clearance is thus represented by ICCD. By inspecting the proposed mathematical procedure of ICCD, we can ®nd all the possible leakage paths, not only the contact line and blowhole. The bene®t of applying ICCD is that the ICCD can also be modeled by a single mathematical model such as the cubic spline interpolation instead of the mathematical model of the multisegment tooth pro®les. The reason is that the trouble solving the inter-relationship between parameters and writing the relationships in the problem's constraints when using exact mathematical model in optimization is avoided with cubic spline interpolation. And there is no limitation of usage of cubic spline with TCA results from the dierentiation operation in solving the basic meshing condition anymore and the problem of discontinuity and divergence in optimal programming if we apply the ICCD method to calculate inter-lobe clearance. By utilizing the ICCD, the tooth pro®les of the two mating rotors are not necessarily conjugated to each other. The ¯uid sealing is evaluated directly by the clearance, not the exact contact line and blowhole. This feature increases more ¯exibility in geometrical design and manufacturing. The result of the proposed analysis model provides the basis of the exact evaluation of the ¯uid leakage. More realistic estimation of the ¯uid leakage in the vacuum system can be calculated in collaboration with the computational ¯uid dynamics (CFD) software.
2. Mathematical modeling As shown in Fig. 1, two mating screw rotors rotate with constant gear ratio in the opposite directions about parallel axes. The coordinate systems, Sa xa ; ya ; za , Sb xb ; yb ; zb , and Sf xf ; yf ; zf , are rigidly attached to rotational axes of screw rotors 1, 2 and the frame, respectively. At least one of the transverse sections of screw rotors is known either in the form of mathematical equations or the discrete measured data points. If axial section pro®le is known, the transverse section can be
Fig. 1. Coordinate systems applied to the transverse sections of twin-screw rotors.
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derived after the operation of screw motion. In the method of ICCD, the transverse sections are not necessary to satisfy the conjugate conditions, but it is essential that pro®les should not interfered with each other. There are two approaches to utilize the ICCD: (a) the exact mathematical model of the rotor pro®le is known, and (b) the discrete measured data points of the transverse section are known. The details of the procedure of the ICCD are described in the following section. 2.1. Exact mathematical model of the rotor pro®le is known Assume that transverse section of the rotor pro®les of screw rotors can be represented in the coordinate systems, Sa and Sb , respectively, by the following equations: rj rj uj X uj
Y uj
0 1
T
j a; b:
1
If one of the transverse sections, say rb , of two mating screw rotors is known, then the tooth pro®le of the other mating rotor, say ra , can be derived according to the theorem of planar gearing. The locus of known tooth pro®le, rb , represented in coordinate system Sa by the operation of coordinate transformation is ra Maf Mfb rb ; where Maf
2
2
cos /a sin /a 6 sin /a cos /a 6 4 0 0 0 0 2
cos /b 6 sin /b Mfb 6 4 0 0
sin /b cos /b 0 0
0 0 1 0
3 0 07 7; 05 1
0 0 1 0
3 r1 r2 0 7 7: 0 5 1
3
Since the mating rotors rotate with constant gear ratio in the opposite directions, the gear ratio i r1 =r2 constant and /b i/a . The common normal at point of tangency must pass through the centrode point P as shown in Fig. 1. The equation of meshing between mating rotors are derived as follows: fa u; /
XP xa Nxa
YP ya 0; Nya
4
where XP ; YP are the Cartesian coordinates of centrode point P . The xa ; ya and the Nxa ; Nya are the Cartesian coordinates and the normal vector of instant contact point on the transverse tooth pro®le of rotor 1, respectively. Based on the theory of gearing [3,4], by solving the equation of meshing, fa u; / 0, together with the locus of rotor 2, rb , in coordinate system Sa , yield the envelope of the locus, i.e., the transverse tooth pro®le of rotor 1, ra . The surface equations of mating rotors could be obtained by means of screw motion of the pro®les along the respective axes of rotors as shown in Fig. 2. The mating rotors rotate with
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Fig. 2. Coordinate systems of right- and left-handed screw surfaces.
constant gear ratio between parallel axes in the opposite directions. Coordinate systems Sa Xa ; Ya ; Za and Sb Xb ; Yb ; Zb are used to de®ne the cross section of the screw rotors while coordinate systems Sc Xc ; Yc ; Zc and Sd Xd ; Yd ; Zd are used to de®ne the cross-section that is elevated from the reference datum plane. The relationship between rotors might be dierent when the lead of the screw is dierent from the theoretical value. The setting of coordinate systems like above could increase the ability to simulate the manufacturing error of the rotor. The tooth surfaces, R1 and R2 , of mating rotors are expressed by the following equations: R1 M1c h1 ra ua ;
R2 M2d h2 rb ub ;
5
where M1c and M2d are the coordinate transformation matrices for the right-handed and lefthanded screw motions, respectively, 2 3 2 3 cos h1 cos h2 sin h1 0 0 sin h2 0 0 6 sin h1 cos h1 0 6 0 7 0 7 7; M2d 6 sin h2 cos h2 0 7; 6 M1c 6 4 0 5 4 0 1 p1 h1 0 0 1 p2 h2 5 0 0 0 1 0 0 0 1 p1 ip2 and h2 ih1 . The normal vectors of surfaces R1 and R2 are represented in S1 and S2 by N1
oR1 oR1 ; oua oh1
N2
oR2 oR2 oub oh2
7
and the unit vectors can be determined by nj
Nj nxj i nyj j nzj k jNj j
j 1; 2:
8
Then the surfaces expressed in the ®xed coordinate system Sf may be determined by the operation of coordinate transformation:
Z.H. Fong et al. / Mechanism and Machine Theory 36 (2001) 725±742 f
R1 Mfa /a R1 ua ; h1 ;
f
R2 Mfb /b R2 ub ; h2 :
731
9
The normal vectors of surfaces R1 and R2 represented in coordinate system Sf is obtained by the following matrix equation: f
N1 Lfa /a N1 ; where
2
cos /a 4 Lfa sin /a 0
f
N2 Lfb /b N2 ;
sin /a cos /a 0
3 0 0 5; 1
10 2
cos /b 4 Lfb sin /b 0
sin /b cos /b 0
3 0 0 5: 1
11
The overlapped cavity is the chamber formed by the adjacent surfaces of two dierent rotors as shown as the hatched region in Figs. 3 and 4. The working ¯uid leaks mainly through the overlapped cavities. Therefore, the shape and size of the overlapped cavity between the two rotors are required to describe the leakage paths between the mating rotors. The overlapped cavities translate parallel to the rotation axes during the working process. The shape of the overlapped cavity is the same through the working process. However, the calculation for the shape of the
Fig. 3. De®nitions of the overlapped cavity.
Fig. 4. Projected area of overlapped cavity from the measured data points for twin square-threaded screw compressor.
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overlapped cavities is quite complex in most cases. There are many two-dimensional simpli®ed method proposed to describe the leakage paths. Following, we propose a three-dimensional methodology to represent the size and the shape of the overlapped cavities. Since the size and the shape of the overlapped cavities will not change through the working process, the projection of the overlapped cavities to the transverse plane is the same all the time. The projected area can be simply determined by the outer diameter of two mating rotors as shown in Figs. 3 and 4. Therefore, we use the projected area as the datum plane for further discussion. The tooth surfaces of mating rotors are derived through the screw motion of the transverse sections, the axial height of the overlapped cavity is the dierence of the axial distances resulted from the respective rotation angle of screw motion. For an arbitrary point J X ; Y in the projected overlapped area, we can ®nd two points, point U XU ; YU and V XV ; YV on the respective transverse sections as shown in Fig. 3. Points U and V are in the same transverse plane and belong to the mating tooth pro®les, R1 and R2 , respectively. After the operation of screw motion, the x- and y-coordinates reach the same point J X ; Y . Therefore, Ob J Ob V and Oa J Oa U . The z-coordinates of points U and V are the same. The position vectors of tooth surfaces of rotors 1 and 2 are denoted by R1 and R2 , respectively, as expressed in Eq. (5). The position vectors R1 and R2 are transformed to the ®xed coordinate system Sf as follows: f
R1 Mfa /a R1 ua ; h1 XU ua ; h1
YU ua ; h1
T
ZU ua ; h1 1 ;
f
R2 Mfb /b R2 ub ; h2 XV ub ; h2
YV ub ; h2
ZV ub ; h2
12
1T ;
where /a and /b is the rigid body rotation angle of rotor 1 and rotor 2, respectively. /a and /b are determined by the non-interference checking. As shown in Fig. 3, coordinates of points U and V are solved by the following simultaneous equations: XU ua ; h1 X ; 13 YU ua ; h1 Y ;
XV ub ; h2 X ; YV ub ; h2 Y :
14
Substitute the values of ua ; h1 and ub ; h2 back to Eq. (5) to determine the position vectors of points U and V . The axial dierence, Dz p2 h2 p1 h1 , is the axial height of the overlapped cavity at projected point J X ; Y . Therefore, the normal height at point J is f
H Dz nz2 :
15
By applying the above-mentioned calculation, the shape of the overlapped cavity can be described by the projected area and the corresponding normal height. The results of the calculation are represented by the iso-height contour plot. Since the height of the plot is the normal clearance between rotors, the plot is called as the ``ICCD'', as shown in Fig. 8.
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2.2. Measured data points of the transverse section are known When the mating rotors are available, the transverse or axial section of rotors can be obtained by CNC coordinate-measuring machine. The measured tooth pro®le is represented in the form of a series of data points. The mathematical equation for the tooth pro®les can be derived by curve ®tting or interpolation from the discrete measured data points. Once the equations for the tooth
Fig. 5. Flowchart of the method of ICCD.
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Fig. 6. Unsymmetrical cyclode-arc pro®le of Example 1 for conventional twin-screw compressor.
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pro®les are known, we can follow the above-mentioned procedures to construct the ICCD. The ¯owchart for the calculation of ICCD is shown in Fig. 5.
3. Examples and discussions 3.1. Example 1: Tooth pro®le of unsymmetrical cycloid-arc for the conventional twin-screw compressor A conventional twin-screw compressor with unsymmetrical cycloid-arc pro®les is shown in Fig. 6. The lobe combination is 4 + 6. The transverse section of female rotor is given in the coordinate system Sb as shown in Table 1. The tooth pro®le of male rotor is derived in the coordinate system Sa by the theory of planar gearing according to Table 2. The coordinate systems Sa and Sb are attached to male and female rotors, respectively. Parameters required to input are r2 ; i; h1 ; uup AB2 (i.e., the upper bound of uAB2 ) and e as shown in Fig. 6. The transverse sections of the mating rotors are de®ned by the equations shown in Tables 1 and 2. Apply Eqs. (1)±(14), and the normal clearance can be calculated. The overlapped cavity is partitioned into two regions by the topland of the male rotor. As shown in Fig. 7, the forward side of the overlapped cavity is the clearance between lower tooth _ _ side of the overlapped surfaces, i.e., between arc B1 C1 of male rotor and arc B2 C2 . The backward _ E of male rotor and arc cavity is the clearance between upper tooth surfaces, i.e., between arc A 1 1 _ A2 E2 . Therefore, the ICCD should be calculated in both forward and backward sides. The result of normal clearance is exhibited in Fig. 8 in the form of contour diagram while the values of input are r2 30; i 2=3; h1 14; uup AB2 10°; e 1 and choosing the screw Table 1 Pro®les of female rotor Type Arc
Notation _ A2 B2
Equations
Notes
rAB2 r2
h1 cos uAB2 uup AB2
where Degenerate hypocycloid Epicycloid
_ B2 C2
_
A2 E2
_
D2 E2 _
C2 D2
h1 sin uAB2 0 1 ;
6 uAB2 6 p=2
0 1T ;
ptA1 h2 cos b h2 sin b 0 T ; rAE2 Mbf Mfa ptA1 q rAE2 uAE2 ; where h2 h21 r12 2h1 r1 cos p uup AB2 and 1
Given
1
sin h1 =r2
rBC2 uBC2 cos H1 uBC2 sin H1 q where r22 h21 6 uBC2 6 r2
b sin Degenerate hypocycloid Arc
T
Given
Given point A1
h1 sin uup AB2 h2
rDE2 uDE2 cos H2 uDE2 sin H2 0 1T ; where r2 e 6 uDE2 6 r2 rCD2 r2 cos uCD2 r2 sin uCD2 0 1T ; where H2 6 uCD2 6 p3 H1
Given Given
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Table 2 Pro®les of male rotor Type
Notation
Arc
_ A1 B1
Epicycloid
B1 C1
Epicycloid
A1 E1
Epicycloid
D1 E1
Arc
C 1 D1
_
_
_
_
Equations
Notes
rAB1 Maf Mfb rAB2 fAB h1 sin uAB2 sin uAB2 uAB1 r2 0
Derived
rBC1 Maf Mfb rBC2 fBC uBC2 cos uBC1
Derived
H1 r2 0
ptE2 r2 e cos H2 r2 rAE1 Maf Mfb ptE2
e sin H2 0 T
Given point E2
rDE1 Maf Mfb rDE2 fDE uDE2 cos uDE1 H2 r2 0
Derived
rCD1 r1 cos uCD1 r1 sin uCD1 0 1T ; H2 p H1 where 6 uCD1 6 2 i i
Given
Fig. 7. Determination of the two partitioned regions of overlapped area for conventional twin-screw compressor.
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Fig. 8. ICCD of Example 1 for unsymmetrical cycloid-arc pro®le.
parameter p2 11:4592. The ICCD shown in Fig. 8 indicates that the leakage path is relatively small between cavities Pj1 and Pi1 . The working ¯uid trapped in the cavity will be pumped from suction port to the discharge port without large pressure losses. However, the leakage path between discharge port and cavity Pi1 is relatively wide. Therefore, the wrap angle of the rotors of conventional twin-screw pump should be less than 360°, otherwise, the working ¯uid will leak back from cavity Pi to Pi1 . The positions of the zero clearance regions in the ICCD consistent with the contact lines solved by the TCA. However, additional information on the sealing eect is observed by inspecting the gradient of the iso-clearance lines. Large zero-clearance region indicates good eect on sealing. By inspecting the gradient of the iso-clearance lines shown in Fig. 8, the normal clearance between the housing cusp and the nearest contact point is not a triangle. Therefore, the two-dimensional triangle simpli®cation to estimate the size of blowhole is not accurate. 3.2. Example 2: Pro®le of discrete measured data points for twin square-threaded screw compressor As shown in Fig. 9, the axial section of the rotor of twin square-threaded screw compressor is measured on the CNC coordinate-measuring machine. The axial sections of mating rotors are the same while the hand of rotation are opposite to each other. The transverse section can be obtained after the operation of screw motion as shown in Fig. 2. The pro®le of transverse section is shown in Fig. 4, and the hatched region is the projected area of overlapped cavity. The ICCD is exhibited in Fig. 10 with the screw parameter, p1 p2 5:7296.
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Fig. 9. The measured data points of Example 2 for twin square-threaded screw compressor.
The tooth pro®les of mating rotors shown in Example 2 are not conjugate to each other. The TCA fails to ®nd the contact lines. By utilizing the ICCD, the pseudo-contact lines are approximated by zero-clearance contour lines. Additional 0.2 mm normal backlash is required to avoid interference. However, the ICCD results show that the leakage paths between cavities are relatively small. 3.3. Example 3: Conjugate tooth pro®les of cycloid-involute for twin square-threaded screw compressor The pro®les of transverse sections are formed by cycloid and involute curves as shown in Fig. 11. The tooth pro®les in Example 3 are conjugate to each other and can be ground on the gear grinding machine or thread grinding machine. The ICCD is exhibited in Fig. 12 while the value of screw parameter is the same as Example 2, p1 p2 5:7296. By means of the ICCD proposed here, we can easily determine the position of blowhole and obtain the contact line to describe the contact relations of mating rotors. Moreover, inspecting the ICCD, we may ®nd all the possible leakage paths to evaluate the inter-lobe clearance.
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Fig. 10. ICCD of Example 2 for the twin square-threaded screw compressor.
Fig. 11. Tooth pro®le of Example 3 for twin square-threaded screw compressor.
In detail, blowhole is located within the high-pressure area and formed by the housing cusp and the nearest contact point as shown in Figs. 8, 10 and 12. The contact line is where the normal height of overlapped cavity is equal to zero. The possible leakage paths are demonstrated in Figs. 13 and 14, including the contact line, blowhole, the suction and discharge port, and the axial leakage between the rotor tips and housing, etc.
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Fig. 12. ICCD of Example 3 for twin square-threaded screw compressor.
Fig. 13. Transverse view of possible leakage paths for twin-screw compressor.
According to the results of the ICCD as shown in Figs. 8 and 10, there exists leakage path as long as the normal height of overlapped cavity is not zero. In other words, the ¯uid ¯ow may leak back through the same groove or along the axial direction till over next two stages, not only the adjacent one.
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Fig. 14. Axial view of possible leakage paths for twin square-threaded screw compressor.
The leakage paths are quite dierent between the twin square-threaded screw compressor and the conventional one. All possible leakage paths are shown in Fig. 13. The unsymmetrical cycloidarc pro®le for the conventional twin-screw compressor as shown in Fig. 13, the cavity Pj is open to the suction port while the cavity Pj2 is open to discharge port. The leakage paths B2 ±B3 and D will not produce the direct leakage of working ¯uid from discharge port to suction port. Therefore, the sealing at leakage paths B2 ±B3 and D can be ignored in the conventional twin-screw compressor. For the pro®le of the twin square-threaded screw compressor, the contact line as referred to Fig. 10 is almost a slanting straight line and divides the overlapped cavity into two parts. With the characteristic of extra large wrap angle, there is no distinctive separation from the low- and highpressure side. All possible leakage paths should be taken into consideration and calculated during the design phase. 4. Conclusion A mathematical procedure is proposed to calculate the inter-lobe clearance between two mating screw rotors and then represent the clearance topography by ICCD. The ICCD proposed here may be used to determine the size and shape of the overlapped cavity and the complete inter-rotor clearance. By means of the ICCD, we can easily determine the position of blowhole and obtain the contact line to describe the contact relations of mating rotors. Moreover, inspecting the ICCD, we may ®nd all the possible leakage paths to evaluate the inter-lobe clearance. According to the examples and discussions, following conclusions are drawn: (a) The two-dimensional triangle simpli®cation to estimate the size of blowhole is not accurate. The estimated size by the triangle simpli®cation is larger than true blow whole. (b) The ICCD provides better description on the leakage paths. Not only the position of the blowhole but also the shape of critical leakage path are presented in the ICCD. The ¯ow rate of the leakage can be accurately calculated by the CFD software.
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(c) The ICCD can also be modeled by a single mathematical model such as the cubic spline interpolation instead of the mathematical model of the multi-segment tooth pro®les. Hence, this method avoids the problem of discontinuity and divergence in optimal programming. (d) By utilizing the ICCD, the tooth pro®les of the two mating rotors are not necessarily conjugated to each other. The pseudo-contact lines are approximated by zero-clearance contour lines. (e) The ¯uid sealing is evaluated directly by the clearance, not the exact contact line and blowhole, and this feature increases more ¯exibility in geometrical design and manufacturing. References [1] D.G. Deng, P.C. Shu, Rotational Compressor, Mechanical Engineering Press, China, 1985, pp. 91±96. [2] J.S. Fleming, Y. Tang, G. Cook, The twin helical screw compressor, part 1: Development applications and competitive position, J. Mech. Eng. Sci. 212 (1998) 355. [3] F.L. Litvin, Theory of Gearing, NASA Reference Publication 1212 (AVSCOM 88-C-035), Washington, DC, 1989. [4] F.L. Litvin, in: Gear Geometry and Applied Theory, Prentice-Hall, New York, 1994. [5] Kashiyama Industry Co., Ltd., Osamu Ozawa, Nagano, Gas Exhaust System and Pump Cleaning System for a Semiconductor Manufacturing Apparatus, US Patent 5443644, 1995. [6] P.J. Singh, A.D. Onuschak, A comprehensive computerized method for twin-screw rotor pro®le generation and analysis, in: Proceedings of the 1984 International Compressor Engineering Conference at Purdue, West Lafayette, 1984, pp. 519±527. [7] P.J. Singh, P.R. Schwartz, Exact analytical representation of screw compressor rotor geometry, in: Proceedings of the 1990 International Compressor Engineering Conference at Purdue, West Lafayette, 1990, pp. 925±937. [8] P.J. Singh, J.L. Bowman, Calculation of blow-hole area for screw compressors, in: Proceedings of the 1990 International Compressor Engineering Conference at Purdue, West Lafayette, 1990, pp. 938±948. [9] L. Zhang, J.F. Hamilton, Main geometric characteristics of the twin screw compressor, in: Proceedings of the 1992 International Compressor Engineering Conference at Purdue, West Lafayette, 1992, pp. 449±456. [10] Z. Zhou, Computer aided design of a twin-rotor screw refrigerant compressor, in: Proceedings of the 1992 International Compressor Engineering Conference at Purdue, West Lafayette, 1992, pp. 457±466.