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Design methodology of a new smooth rotor profile of the screw vacuum pump
Vacuum 159 (2019) 456–463
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Design methodology of a new smooth rotor profile of the screw vacuum pump
T
Jun Wang∗, Shuhong Wei, Rundong Sha, Hongjie Liu, Zengli Wang College of Chemical Engineering, China University of Petroleum (East China), Qingdao, 266580, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Twin-screw vacuum pump (TSVP) Cross-section profile (CSP) Screw rotor Conjugate curve Stress
The screw rotor and its cross-section profile primarily determine the performance and efficiency of the twinscrew vacuum pump (TSVP). However, the traditional cross-section profile (CSP) contains cusps or unsmooth points, so that screw rotor surfaces generated by this CSP contain unsmooth edges, which will lead to wear, deformation, and difficulty of coatings for corrosion protection in the unsmooth region. With the aim of solving this problem caused by cusps or unsmooth points of the CSP, we put forward a meshing model of a circular arc and an equidistant curve of an epicycloid (ECE), and then proposed a design methodology for generating a completely smooth CSP. Screw rotors generated by the proposed CSP were completely smooth, and the proposed screw rotors could correctly mesh. Moreover, a geometric model of the proposed CSP was established, and the influence of the geometric parameters on its shapes and characteristics were studied. In addition, the proposed screw rotor was compared with the traditional screw rotor in terms of stress and deformation. Results indicated that the proposed screw rotor had continuous meshing and contact lines, and small values of maximum stress and deformation. The proposed smooth screw rotor is superior to the traditional one.
1. Introduction As a kind of positive displacement pump, the twin-screw vacuum pump (TSVP) has many advantages of a dry-running mode, compact structures, high reliability, and small vibration. Therefore, it has been widely used in metallurgical, pharmaceutical, chemical, and semiconductor industries. The screw rotor, which is the form swept out by a cross-section profile (CSP) moving along a specific helix, is a key component of the TSVP. Therefore, the CSP of the screw rotor, or screw rotor profile is very important and has received much attention. The traditional CSP consists of four main curves: an extended epicycloid, a dedendum circular arc, a circular involute, and an addendum circular arc [1]. Kawamura et al. [2] modified the intersection between the circular arc and the epicycloid by arc chamfering. However, the modified CSP can not correctly mesh, which will led to large gas leakage. Hsieh et al. [3] proposed a novel profile, using two circular arcs to replace the Archimedes curve of the claw-type rotor, which promoted gas sealing and eliminated carryover. Lu et al. [4] proposed a new profile that contains a pseudo Archimedes curve and an epicycloid in order to obtain the self-conjugate curves. Then, three types of profiles whose tips were cycloid, circular arc, and cusp were compared in terms of the contact line length and area utilization coefficient. Wang et al. [5] presented a method for generating smooth claw rotor profiles that ∗
can correctly mesh. In order to improve the built-in compression and achieve a large internal volume ratio, volume reduction designs of working chambers formed by two intermeshed screw rotors from the suction side to the discharge side were used. Im [6] proposed a variable pitch screw rotor to produce internal compression of the gas being delivered. Wang et al. [7] designed screw rotors with a variable pitch and CSPs, and geometric relations of screw pitch, CSPs and the helical angle were derived. The trend of the screw pitch and screw width is opposite from one side to the other side of this screw rotor. Becher [8] presented a screw rotor consisting of three segments with different pitches to increase the internal volume ratio, and put forward a settlement for the balance of screw rotors. Inagaki et al. [9] presented a screw pump whose volume in the suction side is larger than that in the discharge side by changing the helical angle of the first segment. Pfaller et al. [10] proposed an algorithm to design the optimal rotor pitch from an energetic point of view. Zhang et al. [11] introduced a solution for the dynamic balance by means of mass removal applied to single-threaded screw rotors. North et al. [12] proposed a type of tapered screw rotor in order to increase the internal volume ratio. Stosic et al. [13] estimated the flow and thermodynamic performance of TSVP, and showed that a change of rotor pitch will affect performances of the TSVP. Zhao et al. [14] presented a model of the gradational lead screw rotor for the TSVP in order
Corresponding author. E-mail address: [email protected] (J. Wang).
https://doi.org/10.1016/j.vacuum.2018.10.064 Received 5 September 2018; Received in revised form 23 October 2018; Accepted 23 October 2018 Available online 26 October 2018 0042-207X/ © 2018 Elsevier Ltd. All rights reserved.
Vacuum 159 (2019) 456–463
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Nomenclature L M12 O R R1 R2 R3 Rb
r r1 r2 t α β γ η θ φ
length of the contact line (m) coordinate transformation matrix from O1x1y1 to O2x2y2 origin of the static coordinate system distance between points p and O1 (m) radius of the addendum circular arc (m) radius of the pitch circle (m) radius of the dedendum circular arc (m) radius of the generating circle (m)
radius of a circle (m) radius of the tip circular arc (m) radius of the back circular arc (m) variable parameter involute initial angle (rad) angle (rad) angle (rad) area utilization coefficient angle (rad) rotational angle (rad)
2.1.2. Conjugate curve of a circular involute Fig. 2 shows a circular involute whose center point of the generating circle coincides with the rotational center O1 in system O1x1y1 and its conjugate curve that is also a circular involute in system O2x2y2. In system O1x1y1, equations of the circular involute can be represented by
to analyze the working process in the working chambers. Ohbayashia et al. [15] put forward a method to forecast performances and presented a method to fulfill specific demands in the pump design. Zhang et al. [16] presented a new tilt form grinding method to solve the issue of undercut in the grinding concave curve, and the method was validated by three numerical examples. The CSPs mentioned in the above literatures contain cusps or unsmooth points, and therefore the generated screw rotor surfaces contain unsmooth edges that will cause large deformations, easy wear, and large gas leakage of screw rotors. Moreover, the unsmooth screw rotor is not suitable for surface coatings for corrosion protection. Aimed at solving the problem caused by unsmooth points or cusps in screw rotor profiles, we established a meshing model between a circular arc and an equidistant curve of an epicycloid (ECE), which was used instead of a point and an epicycloid for generating CSPs, and then a completely smooth profile was presented. Screw rotors generated by the proposed CSP are completely smooth. Equations of all profiles were derived, and influences of parameters on its shape and performance were studies. The new screw rotor was compared with the traditional one in terms of the contact line, area utilization coefficient, stress, and deformation. This work can be beneficial in the design of new screw rotors with high performances for TSVPs.
⎡ xInv1 (t ) ⎤ ⎡ Rb (cos(t − α ) + t sin(t − α )) ⎤ ⎢ yInv1 (t ) ⎥ = ⎢ Rb (sin(t − α ) − t cos(t − α )) ⎥ ⎥ ⎢ ⎢ ⎥ 1 ⎦ ⎣ 1 ⎦ ⎣
(4)
In system O2x2y2, equations of its conjugate curve are obtained by
⎡ xInv2 (t ) ⎤ ⎡ Rb (cos(t − α ) + t sin(t − α )) ⎤ ⎢ yInv2 (t ) ⎥ = ⎢− Rb (sin(t − α ) + t cos(t − α )) ⎥ ⎥ ⎢ ⎢ ⎥ 1 ⎦ ⎣ 1 ⎦ ⎣
(5)
2.2. A meshing model for a circular arc 2.2.1. Conjugate curve of a circular arc A circle whose center point p does not coincide with the rotational center O1 is shown in Fig. 3. In system O1x1y1, equations of the circle are represented by
2. An analytical method for calculating conjugate curves In this section, we analyze the meshing relationship of common conjugate curves used for generating CSPs: a point and an epicycloid, a circular involute and its self-conjugate curve. We establish a meshing model of a circular arc whose center point does not coincide with the rotational center and its conjugate curve, an ECE.
⎡ xArc (t ) ⎤ ⎡ R cos θ + r cos t ⎤ ⎢ yArc (t ) ⎥ = ⎢ R sin θ + r sin t ⎥ ⎥ ⎢ ⎢ ⎥ 1 ⎦ ⎣ 1 ⎦ ⎣
(6)
In the equivalent motion, the loci of the circle form a cluster of curves that can be given by
2.1. Analysis of common conjugate curves 2.1.1. A point and an extended epicycloid As shown in Fig. 1, point p of the left rotor in system O1x1y1 is given by
x ⎡ p ⎤ ⎡ R cos θ ⎤ ⎢ yp ⎥ = ⎢ R sin θ ⎥ ⎢ ⎣1⎥ ⎦ ⎣ 1 ⎦
(1)
In the equivalent motion, point p forms a locus that is derived by the coordinate transformation in system O2x2y2.
x ⎡ x Cyc (φ) ⎤ ⎡ p ⎤ ⎡− R cos(2φ − θ ) + 2R2 cos φ ⎤ ⎢ yCyc (φ) ⎥ = M12 ⎢ yp ⎥ = ⎢ − R sin(2φ − θ) + 2R2 sin φ ⎥ ⎢ ⎥ ⎥ ⎢ ⎢ 1 ⎣1⎥ ⎦ ⎣ ⎦ ⎣ 1 ⎦
(2)
⎡− cos 2 φ − sin 2 φ 2R2 cos φ ⎤ M12 = ⎢ − sin 2 φ cos 2 φ 2R2 sin φ ⎥ ⎥ ⎢ 0 0 1 ⎦ ⎣
(3)
Thus, the conjugate curve of a point located outside the pitch circle is an extended epicycloid for the twin-rotation motion of TSVPs.
Fig. 1. The meshing relationship of a point and an extended epicycloid. 457
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the equivalent motion. The equidistant curves are obtained by offsetting epicycloid Cyc1 the distance r in the concave or convex direction. The ECE in concave direction lr1 is obtained by ′Cyc1 ⎡ x Cyc (t ) − r ⎤ 1 2 2 ⎢ x′Cyc1 (t ) + y′Cyc1 (t ) ⎥ x lr1 (t ) ⎡ ⎤=⎢ ⎥ ⎢ ylr1 (t ) ⎥ ⎢⎢ yCyc1 (t ) + r 2 x′Cyc1 (t )2 ⎥⎥ x′Cyc1 (t ) + y′Cyc1 (t ) ⎣ 1 ⎦ ⎢ ⎥ 1 ⎣ ⎦ (t )
y
(10)
The ECE in convex direction l-r1 is obtained by ′Cyc1 ⎡ x Cyc (t ) + r ⎤ 1 2 2 ⎢ x′Cyc1 (t ) + y′Cyc (t ) ⎥ x l−r1 (t ) 1 ⎡ ⎤=⎢ ⎥ ⎢ yl−r1 (t ) ⎥ ⎢⎢ yCyc1 (t ) − r 2 x′Cyc1 (t )2 ⎥⎥ x′Cyc1 (t ) + y′Cyc1 (t ) ⎣ 1 ⎦ ⎢ ⎥ 1 ⎣ ⎦ y
(t )
(11)
Thus, the circle whose center point does not coincide with the rotational center meshes with both the concave equidistant curve lr1 and convex equidistant curve l-r1 of the epicycloid Cyc1.
Fig. 2. The conjugate curve of a circular involute.
2.2.2. Discussions of conjugate curves of the circle When the length from points p to O1 is equal to zero, i.e. R = 0, the center point p of the circle coincides with the rotational point O1. The conjugate curve of the circle whose center point is p is also a circle whose center point coincides with the other rotational point O2, and whose radius is 2R2-r. This pair of conjugate curves is used for generating the addendum and dedendum circular arcs of the CSP. When 0 < R < R2, point p is inside the pitch circle. The conjugate curve of point p is a shortened epicycloid Cyc2, as shown in Fig. 4. The conjugate curves of the circle whose center point is p are equidistant curves of the shortened epicycloid, l-r2 and lr2. When R > R2, point p is outside the pitch circle. The conjugate curve of point p is an extended epicycloid Cyc3, as shown in Fig. 5. The conjugate curves of the circle whose center point is p are equidistant curves of the extended epicycloid, l-r3 and lr3. 3. Design methodology of a smooth screw rotor In this section, we analyze disadvantages of the traditional screw rotor whose CSP contains cusps or unsmooth points. We put forward a completely smooth CSP based on the established meshing model. In the design process of new CSP, a pair of conjugate curves, a circular arc and an ECE, were used instead of a point and an epicycloid that were used in the traditional CSP. A geometric model of the proposed new CSP was established, and its meshing accuracy was validated. Influences of the radius of the back arc r2 on the shape are discussed. Ultimately, a
Fig. 3. A circle and its conjugate curve.
⎡ xArc (t ) ⎤ ⎡ xl (t , φ) ⎤ ⎢ yl (t , φ) ⎥ = M12 ⎢ yArc (t ) ⎥ ⎥ ⎢ ⎥ ⎢ ⎣ 1 ⎦ ⎣ 1 ⎦ ⎡− R cos(θ − 2φ) − r cos(t − 2φ) + 2R2 cos φ ⎤ = ⎢ R sin(θ − 2φ) + r sin(t − 2φ) + 2R2 sin φ ⎥ ⎢ ⎥ 1 ⎣ ⎦
(7)
where, r is the distance. The meshing equation of the conjugate curve is shown as
∂x ∂y ∂xl ∂yl − l⋅ l = 0 ⋅ ∂φ ∂t ∂t ∂φ
(8)
Eq. (7) is substituted into Eq. (8), and then the dependence of φ and t can be obtained by
(R + r )sin(θ − t ) + R2 sin(t − φ) = 0
(9)
Eq. (7) and Eq. (9) express the conjugate curve of a circle whose center point does not coincide with the rotational center, which is an ECE. The trajectory pp', which is an epicycloid Cyc1 represented by Eq. (2), is generated by point p that is the center point of the circle during
Fig. 4. The conjugate curve of a circle whose center point is inside the pitch circle. 458
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Fig. 5. The conjugate curve of a circle whose center point is outside the pitch circle.
Fig. 8. Locus of the new CSP.
Fig. 9. The new CSPs varying with r2. (1) When r2 = 0, as shown in Fig. 9(a), circular arc BC changes to point B and its conjugate curve changes to epicycloid FA. The adjacent curves at intersection B are not smooth. (2) When 0 < r2 < R1-R2, as shown in Fig. 9(b), center point n of back circular arc BC is outside the pitch circle. GA is a concave equidistant curve of extended epicycloid Cycb2. The CSP is completely smooth. (3) When R1-R2 < r2 < R1-Rb, as shown in Fig. 9(c), center point n of back circular arc BC is inside the pitch circle. GA is a convex equidistant curve of shortened epicycloid Cycb3. The CSP is completely smooth. (4) When r2 = R1-Rb, as shown in Fig. 9(d), the circular involute does not exist. Back circular arc AB whose center point is located at the generating circle smoothly connects with ECE FA. FA is a concave equidistant curve of extended epicycloid Cycb4.
Fig. 6. The traditional CSPs.
smooth screw rotor generated by the proposed CSP was obtained. 3.1. Disadvantages of the traditional screw rotor profile As shown in Fig. 6, the traditional CSP [4] of the TSVP consists of four curves: a circular involute AB, an addendum circular arc BC, an extended epicycloid curve CD, and a dedendum circular arc DA. This CSP contains a cusp A, and two unsmooth points B and C. When the screw rotor is generated by using this CSP, points A, B, and C will form unsmooth edges. This traditional screw rotor has the following
Fig. 7. The new screw rotor profile.
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⎡ xAB (t ) ⎤ = ⎡ Rb (cos(t − α ) + t sin(t − α )) ⎤ ⎢ yAB (t ) ⎥ ⎢ Rb (sin(t − α ) − t cos(t − α )) ⎥ ⎣ ⎦ ⎦ ⎣
(14)
⎡ xBC (t ) ⎤ = ⎡ (R1 − r2)cos(β ) + r2 cos(t ) ⎤ ⎢ yBC (t ) ⎥ ⎢ (R1 − r2)sin(β ) + r2 sin(t ) ⎥ ⎣ ⎦ ⎦ ⎣
(15)
EF is the external ECE Cyca, and equations of Cyca can be derived by Eq. (2), where R = R1-r1 and θ = π-γ. Angle γ is represented by
y (t γ ) 1 γ = − arctan 2 x (t γ )
where, x(tγ) and y (tγ) are the coordinate value of the intersection of the following two curves, respectively:
Fig. 10. The two intermeshed screw rotors.
disadvantages. (1) It is easy to produce stress concentration at the portion near the unsmooth edges, which can easily lead to wear and deformation. Accordingly, large gas leakage will occur and the efficiency of the pump will be reduced in the working process. (2) The CSP produces a carryover in the meshing process, which will reduce the volumetric efficiency. (3) The CSP contains two unmeshed curves: curve AA1 of circular involute AB and AA2 of the dedendum circular arc DA, which will produce discontinuous contact lines and result in a gas leakage path between adjacent working chambers formed by the two intermeshed screw rotors.
(17)
⎤ ⎡ − 2R2 cos(t ) + (R1 − r1)cos(2t ) ⎥ ⎢ −2R2 cos(t ) + 2(R1 − r 1)cos(2t ) ⎢ + r1 R 2 + R 2 + r 2 − 2R r + 2R (r − R )cos(t ) ⎥ ( ) x t 1 1 2 1 1 γ2 1 1 2 ⎡ ⎤ ⎢ ⎥ ⎢ yγ2 (t ) ⎥ = ⎢ ⎥ ⎣ ⎦ ⎢ − 2R2 sin(t ) + (R1 − r1)sin(2t ) ⎥ R t − R − r t 2 sin( ) 2( )sin(2 ) 2 1 1 ⎥ ⎢ −r 1 ⎢ R12 + R22 + r 12 − 2R1 r 1 + 2R2 (r 1 − R1)cos(t ) ⎥ ⎦ ⎣
(18)
⎡ xDE (t ) ⎤ = ⎡ (R1 − r1)cos(π − γ ) + r1 cos(t ) ⎤ ⎢ yDE (t ) ⎥ ⎢ (R1 − r1)sin(π − γ ) + r1 sin(t ) ⎥ ⎣ ⎦ ⎦ ⎣
(19)
′Cyca ⎤ ⎡ x Cyca (t ) + r1 2 2 ⎢ x (t ) ⎥ ′Cyca (t ) + y′Cyc a ⎡ xEF (t ) ⎤ = ⎢ ⎥ ⎢ yEF (t ) ⎥ ⎢ x′Cyca (t ) ⎥ ⎣ ⎦ yCyca (t ) − r1 2 2 ⎢ x′Cyca (t ) + y′Cyca (t ) ⎥ ⎦ ⎣
(20)
y
As shown in Fig. 7, the new CSP contains 7 curves: a circular involute AB, a back circular arc BC, an addendum circular arc CD, a tip circular arc DE, an ECE EF, a dedendum circular arc FG, and an ECE GA. In the design process, the tip circular arc DE and the ECE EF were used to connect the dedendum circular arc FG and the addendum circular arc CD. The back circular arc BC was used to connect the circular involute AB and the addendum circular arc CD. The ECE EF was used to connect the circular involute AB and the dedendum circular arc FG. All adjacent curves smoothly connect with each other, and no cusps or unsmooth points exist. Thus, the proposed CSP is completely smooth. There are 9 geometric parameters in the CSP: R1, R2, R3, Rb, r1, r2, α, β, and γ. Equations of the back circular arc BC can be derived by Eq. (6), where r = r2, R = R1-r2 and θ = β. Equations of α and β are obtained by
⎜
⎡ x γ1 (t ) ⎤ ⎡ (R1 − r1)cos(t ) ⎤ ⎢ yγ1 (t ) ⎥ = ⎢ (R1 − r1)sin(t ) ⎥ ⎦ ⎣ ⎦ ⎣
Eq. (18) expresses an equidistant curve of an epicycloid, the offsetting distance is r1 in the convex direction. Equations of the tip circular arc DE and the ECE EF are represented by
3.2. A geometric model of a new CSP
R R α = tan ⎛arccos b ⎞ − arccos b R2 ⎠ R2 ⎝
(16)
(t )
where,
⎡ x Cyca (t ) ⎤ = ⎡− 2R2 cos(t + γ ) + (R1 − r1)cos(2t + γ ) ⎤ ⎢ yCyca (t ) ⎥ ⎢⎣ − 2R2 sin(t + γ ) + (R1 − r1)sin(2t + γ ) ⎥⎦ ⎣ ⎦
(21)
GA is the ECE Cycb, and meshes with the back circular arc BC. Equations of Cycb can be derived by Eq. (2). Eq. (2) is substituted into Eq. (10), and equations of ECE GA can be represented by
⎡ x Cyc (t ) − r2 b ⎢ ⎡ x GA (t ) ⎤ = ⎢ ⎢ yGA (t ) ⎥ ⎢ ⎣ ⎦ y (t ) + r2 ⎢ Cycb ⎣
⎟
(12)
2 β = tan ⎛⎜arccosRb / Rb2 + ( (R1 − r2)2 − Rb2 + r2) ⎟⎞ − arccosRb /(R1 − r2) ⎝ ⎠ (13) −α
y′Cyc (t )
⎤ ⎥ ⎥ x′Cycb (t ) ⎥ 2 2 x′Cycb (t ) + y′Cyc (t ) ⎥ b ⎦ b
2
2
x′Cycb (t ) + y′Cyc (t ) b
(22)
where,
⎡ x Cycb (t ) ⎤ = ⎡2R2 cos(t − β ) − (R1 − r2)cos(2t − β ) ⎤ ⎢ yCycb (t ) ⎥ ⎢⎣ 2R2 sin(t − β ) − (R1 − r2)sin(2t − β ) ⎥⎦ ⎣ ⎦
Equations of the circular involute AB and the back circular arc BC are obtained by
(23)
Fig. 11. Completely smooth screw rotors and their contact lines varying with r2.
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Fig. 12. Effects of r2 on screw rotors.
Fig. 13. Meshing and contact lines of the traditional screw rotor. Table 1 Meshing relations between the traditional CSPs. Left CSP
Right CSP
Meshing lines
Contact lines
Circular involute AB Addendum circular arc BC Point C Extended epicycloid curve CD Dedendum circular arc DA
Circular involute ab Dedendum circular arc da Extended epicycloid curve cd Point c Addendum circular arc bc
lAB lBC lC lCD lDA
LAB LBC LC LCD LDA
Fig. 14. Meshing and contact lines of the proposed screw rotor.
screw rotor profile, which shows that the proposed screw rotor profiles can correctly mesh with each other. The accuracy of the geometric model of the proposed CSP was validated.
3.3. Validation of the meshing accuracy Computerized simulation of the meshing accuracy of the proposed CSP was performed, as shown in Fig. 8. It can be seen that the envelope of the motion trajectories of the one screw rotor profile is the other 461
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(1) The proposed rotor profile is completely smooth without cusps and unsmooth points. (2) The proposed rotor profile can achieve correct mesh in the working process. (3) Compared with the traditional rotor, the proposed rotor does not have carryover.
Table 2 Meshing relationships between the proposed CSPs. Left CSP
Right CSP
Meshing lines
Contact lines
Circular involute AB Back circular arc BC Addendum circular arc CD Tip circular arc DE ECE EF Dedendum circular arc FG ECE GA
Circular involute ab ECE ga Dedendum circular arc fg ECE ef Tip circular arc de Addendum circular arc cd Back circular arc bc
lAB lBC lCD
LAB LBC LCD
lDE lEF lFG
LDE LDE LFG
lGA
LGA
Meanwhile, the proposed rotor has the following disadvantages. (1) The meshing lines of the proposed screw rotor cannot pass through the intersection of pump holes, which will result in a blowhole in the tip part of tooth. (2) The proposed rotor profile, consisting of seven curves, is more complicated than the traditional one.
3.4. Discussions of the proposed CSP The value of radius r2 of the back circular arc BC of the new screw rotor profile affects its shape. Fig. 9 shows the proposed CSPs varying with r2. The range of r2 was 0 ≤ r2 ≤ R1-Rb. Fig. 9 (a), (b), (c) and (d) are the CSPs whose back circular arc radius r2 was 0 mm, 20 mm, 40 mm, and 55 mm, respectively. The other geometric parameters were constant: R1 = 90 mm, R3 = 40 mm, Rb = 35 mm, and r1 = 6 mm.
4.2. Stress and deformation During the working process of the TSVP, in order to compare the proposed and traditional rotors, the same boundary and initial conditions were set on the two rotors. The suction pressure was 3 kPa, and the discharge pressure was 100 kPa. Stresses and deformations produced by the pressure forces were calculated by using the finite element method [17]. Fig. 15 illustrates stress distributions of the two rotors. It can be seen that the maximum values of stress are located at the dedendum region near the discharge side of the two screw rotors. The maximum stress value of the new rotor was 24.096 MPa, and the maximum stress value of the traditional rotor was 25.714 MPa. The former was smaller than the latter, and was reduced by 6.2%. Taking an arbitrary point on the same positions of two rotors as an example, the stress value of the new rotor was 5.371 MPa, while the stress value of the traditional rotor was 7.398 MPa. Fig. 16 presents deformation values of the new and traditional rotors. The maximum value for the two rotors occurred in the addendum region near the discharge side. The maximum value of the new rotor, 0.0115 mm, was smaller than that of the traditional rotor, 0.0123 mm, and was reduced by 6.5%.
3.5. A new smooth screw rotor A screw rotor was generated by the proposed CSP performing a screw motion, and two intermeshed screw rotors are shown in Fig. 10. The surfaces of the proposed screw rotors are completely smooth; and the two screw rotors can correctly mesh. The proposed screw rotors and their contact lines with different r2 are shown in Fig. 11. Effects of r2 on area utilization coefficient η of the CSP and length of the contact line L of screw rotors are shown in Fig. 12. As r2 increases, η remains approximately constant, but L decreases. When r2 = R1Rb = 55 mm, the contact line is the shortest, which means that the leakage through the contact line is the smallest. 4. Comparisons between the proposed and traditional screw rotors
5. Conclusion A meshing model of a circular arc and an ECE was established, a novel CSP was proposed, and a novel screw rotor was formed by using the proposed CSP. The proposed CSP could correctly mesh. In the design process of CSPs, a circular arc and an ECE were used, and therefore, the meshing relations between two CSPs were curve and curve, which was superior to that of the traditional CSPs: point and curve. Thus, the problem caused by the existence of cusps or unsmooth points in the traditional screw rotor profile was solved. The established meshing model of a circular arc and an ECE is also suitable for other rotor profile designs of fluid machinery that consist of two rotors synchronously rotating in opposite directions on parallel axes, such as claw rotors and Roots rotors. The proposed CSP and the generated rotor are completely smooth, and the proposed screw rotor has continuous meshing lines and contact lines in one pitch period. When the same pressure loads act upon the
4.1. Geometric characteristics The traditional CSP is shown in Fig. 6, and its meshing lines and contact lines are shown in Fig. 13. The meshing relations between the two CSPs are listed in Table 1. The proposed CSP is shown in Fig. 7, and its meshing lines and contact lines are shown in Fig. 14. The meshing relations between the two CSPs are listed in Table 2. As shown in Fig. 14 (a), the meshing lines of the proposed screw rotor cannot pass through the intersection of pump holes, which will result in a blowhole in the tip part of tooth. However, the radius of the tip circular arc DE is very small in practical application. Thus, the leakage through the blowhole is far smaller than that through the carryover of traditional profile. The proposed rotor has the following advantages.
Fig. 15. Stresses of the two rotors. 462
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Fig. 16. Deformations of the two rotors.
two rotors, the maximum values of stress and deformation of the new rotor are smaller than those of the traditional rotor are. The new rotors have better performance than that of the traditional ones in terms of gas sealing, stress, and deformation.
[4] Y. Lu, B. Guo, M.F. Geng, Study on design of rotor profile for the twin screw vacuum pump with single thread tooth, IOP Conf. Ser. Mater. Sci. Eng. 90 (2015) 012007. [5] J. Wang, D. Cui, X.F. Pang, et al., Geometric design and performance analysis of a novel smooth rotor profile of claw vacuum pumps, Vacuum 143 (2017) 174–184. [6] Im Kisu, Screw Vacuum Pump Having a Decreasing Pitch for the Screw Members, (1997) US 5667370, Sep. 16. [7] J. Wang, F. Cui, S.H. Wei, et al., Study on a novel screw rotor with variable crosssection profiles for twin-screw vacuum pumps, Vacuum 145 (2017) 299–307. [8] Ulrich F. Becher, Twin Screw Rotors and Displacement Machines Containing the Same, (2004) US 6702558 B2, Mar. 9. [9] Masahiro Inagaki, Shinya Yamamoto, Makoto Yoshikawa, et al., Screw Pump with Increased Volume of Fluid to Be Transferred, (2009) US 7497672 B2, Mar. 3. [10] D. Pfaller, A. Brümmer, K. Kauder, Optimized rotor pitch distributions for screw spindle vacuum pumps, Vacuum 85 (2011) 1152–1155. [11] S.W. Zhang, Z.H. Gu, Z.J. Zhang, Dynamic balancing method for the singlethreaded, fixed-pith screw rotor, Vacuum 90 (2013) 44–49. [12] Michael Henry North, Neil Turner, Tristan Richard Ghislain Davenne, et al., Screw Pump Having Varying Pitches, (2014) US 8827669 B2, Sep. 9. [13] Nikola Stosic, Sham Rane, Ahmed Kovacevic, et al., Screw Rotor Profiles of Varible Lead Vacuum and Multiphase Machines and Their Calculation Models, IMECE201666314. [14] F. Zhao, S.W. Zhang, K. Sun, et al., Thermodynamic performance of multi-stage gradational lead screw vacuum pump, Appl. Surf. Sci. 432 (2018) 97–109. [15] T. Ohbayashi, T. Sawada, M. Hamaguchi, et al., Study on the performance prediction of screw vacuum pump, Appl. Surf. Sci. 169 (2001) 768–771. [16] Z.X. Zhang, Z.H. Fong, A novel tilt form grinding method for the rotor of dry vacuum pump, Mech. Mach. Theor. 90 (2015) 47–58. [17] W.C. Jiang, Y.C. Zhang, W.C. Woo, Using heat sink technology to decrease residual stress in 316L stainless steel welding joint: finite element simulation, Int. J. Pres. Ves. Pip. 92 (2012) 56–62.
Acknowledgements This work was supported by the National Natural Science Foundation of China [51706247], and the Fundamental Research Funds for the Central Universities [16CX05007A]. Thanks to Dr. Edward C. Mignot, Shandong University, for linguistic advice. Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.vacuum.2018.10.064. References [1] C.Q. Chen, Y. Jia, D.B. Zheng, et al., On the profile of single thread and fixed pitch screw for the screw vacuum pump, Vacuum 48 (3) (2011) 8–11 (in chinese).. [2] Takeshi Kawamura, Kiyoshi Yanagisawa, Screw Rotor and Method of Generating Tooth Profile Therefor, (1998) US 5800151, Sep. 1. [3] C.F. Hsieh, Y.W. Hwang, Z.H. Fong, Study on the tooth profile for the screw clawtype pump, Mech. Mach. Theor. 43 (7) (2008) 812–828.
463
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Design methodology of a new smooth rotor profile of the screw vacuum pump
T
Jun Wang∗, Shuhong Wei, Rundong Sha, Hongjie Liu, Zengli Wang College of Chemical Engineering, China University of Petroleum (East China), Qingdao, 266580, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Twin-screw vacuum pump (TSVP) Cross-section profile (CSP) Screw rotor Conjugate curve Stress
The screw rotor and its cross-section profile primarily determine the performance and efficiency of the twinscrew vacuum pump (TSVP). However, the traditional cross-section profile (CSP) contains cusps or unsmooth points, so that screw rotor surfaces generated by this CSP contain unsmooth edges, which will lead to wear, deformation, and difficulty of coatings for corrosion protection in the unsmooth region. With the aim of solving this problem caused by cusps or unsmooth points of the CSP, we put forward a meshing model of a circular arc and an equidistant curve of an epicycloid (ECE), and then proposed a design methodology for generating a completely smooth CSP. Screw rotors generated by the proposed CSP were completely smooth, and the proposed screw rotors could correctly mesh. Moreover, a geometric model of the proposed CSP was established, and the influence of the geometric parameters on its shapes and characteristics were studied. In addition, the proposed screw rotor was compared with the traditional screw rotor in terms of stress and deformation. Results indicated that the proposed screw rotor had continuous meshing and contact lines, and small values of maximum stress and deformation. The proposed smooth screw rotor is superior to the traditional one.
1. Introduction As a kind of positive displacement pump, the twin-screw vacuum pump (TSVP) has many advantages of a dry-running mode, compact structures, high reliability, and small vibration. Therefore, it has been widely used in metallurgical, pharmaceutical, chemical, and semiconductor industries. The screw rotor, which is the form swept out by a cross-section profile (CSP) moving along a specific helix, is a key component of the TSVP. Therefore, the CSP of the screw rotor, or screw rotor profile is very important and has received much attention. The traditional CSP consists of four main curves: an extended epicycloid, a dedendum circular arc, a circular involute, and an addendum circular arc [1]. Kawamura et al. [2] modified the intersection between the circular arc and the epicycloid by arc chamfering. However, the modified CSP can not correctly mesh, which will led to large gas leakage. Hsieh et al. [3] proposed a novel profile, using two circular arcs to replace the Archimedes curve of the claw-type rotor, which promoted gas sealing and eliminated carryover. Lu et al. [4] proposed a new profile that contains a pseudo Archimedes curve and an epicycloid in order to obtain the self-conjugate curves. Then, three types of profiles whose tips were cycloid, circular arc, and cusp were compared in terms of the contact line length and area utilization coefficient. Wang et al. [5] presented a method for generating smooth claw rotor profiles that ∗
can correctly mesh. In order to improve the built-in compression and achieve a large internal volume ratio, volume reduction designs of working chambers formed by two intermeshed screw rotors from the suction side to the discharge side were used. Im [6] proposed a variable pitch screw rotor to produce internal compression of the gas being delivered. Wang et al. [7] designed screw rotors with a variable pitch and CSPs, and geometric relations of screw pitch, CSPs and the helical angle were derived. The trend of the screw pitch and screw width is opposite from one side to the other side of this screw rotor. Becher [8] presented a screw rotor consisting of three segments with different pitches to increase the internal volume ratio, and put forward a settlement for the balance of screw rotors. Inagaki et al. [9] presented a screw pump whose volume in the suction side is larger than that in the discharge side by changing the helical angle of the first segment. Pfaller et al. [10] proposed an algorithm to design the optimal rotor pitch from an energetic point of view. Zhang et al. [11] introduced a solution for the dynamic balance by means of mass removal applied to single-threaded screw rotors. North et al. [12] proposed a type of tapered screw rotor in order to increase the internal volume ratio. Stosic et al. [13] estimated the flow and thermodynamic performance of TSVP, and showed that a change of rotor pitch will affect performances of the TSVP. Zhao et al. [14] presented a model of the gradational lead screw rotor for the TSVP in order
Corresponding author. E-mail address: [email protected] (J. Wang).
https://doi.org/10.1016/j.vacuum.2018.10.064 Received 5 September 2018; Received in revised form 23 October 2018; Accepted 23 October 2018 Available online 26 October 2018 0042-207X/ © 2018 Elsevier Ltd. All rights reserved.
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Nomenclature L M12 O R R1 R2 R3 Rb
r r1 r2 t α β γ η θ φ
length of the contact line (m) coordinate transformation matrix from O1x1y1 to O2x2y2 origin of the static coordinate system distance between points p and O1 (m) radius of the addendum circular arc (m) radius of the pitch circle (m) radius of the dedendum circular arc (m) radius of the generating circle (m)
radius of a circle (m) radius of the tip circular arc (m) radius of the back circular arc (m) variable parameter involute initial angle (rad) angle (rad) angle (rad) area utilization coefficient angle (rad) rotational angle (rad)
2.1.2. Conjugate curve of a circular involute Fig. 2 shows a circular involute whose center point of the generating circle coincides with the rotational center O1 in system O1x1y1 and its conjugate curve that is also a circular involute in system O2x2y2. In system O1x1y1, equations of the circular involute can be represented by
to analyze the working process in the working chambers. Ohbayashia et al. [15] put forward a method to forecast performances and presented a method to fulfill specific demands in the pump design. Zhang et al. [16] presented a new tilt form grinding method to solve the issue of undercut in the grinding concave curve, and the method was validated by three numerical examples. The CSPs mentioned in the above literatures contain cusps or unsmooth points, and therefore the generated screw rotor surfaces contain unsmooth edges that will cause large deformations, easy wear, and large gas leakage of screw rotors. Moreover, the unsmooth screw rotor is not suitable for surface coatings for corrosion protection. Aimed at solving the problem caused by unsmooth points or cusps in screw rotor profiles, we established a meshing model between a circular arc and an equidistant curve of an epicycloid (ECE), which was used instead of a point and an epicycloid for generating CSPs, and then a completely smooth profile was presented. Screw rotors generated by the proposed CSP are completely smooth. Equations of all profiles were derived, and influences of parameters on its shape and performance were studies. The new screw rotor was compared with the traditional one in terms of the contact line, area utilization coefficient, stress, and deformation. This work can be beneficial in the design of new screw rotors with high performances for TSVPs.
⎡ xInv1 (t ) ⎤ ⎡ Rb (cos(t − α ) + t sin(t − α )) ⎤ ⎢ yInv1 (t ) ⎥ = ⎢ Rb (sin(t − α ) − t cos(t − α )) ⎥ ⎥ ⎢ ⎢ ⎥ 1 ⎦ ⎣ 1 ⎦ ⎣
(4)
In system O2x2y2, equations of its conjugate curve are obtained by
⎡ xInv2 (t ) ⎤ ⎡ Rb (cos(t − α ) + t sin(t − α )) ⎤ ⎢ yInv2 (t ) ⎥ = ⎢− Rb (sin(t − α ) + t cos(t − α )) ⎥ ⎥ ⎢ ⎢ ⎥ 1 ⎦ ⎣ 1 ⎦ ⎣
(5)
2.2. A meshing model for a circular arc 2.2.1. Conjugate curve of a circular arc A circle whose center point p does not coincide with the rotational center O1 is shown in Fig. 3. In system O1x1y1, equations of the circle are represented by
2. An analytical method for calculating conjugate curves In this section, we analyze the meshing relationship of common conjugate curves used for generating CSPs: a point and an epicycloid, a circular involute and its self-conjugate curve. We establish a meshing model of a circular arc whose center point does not coincide with the rotational center and its conjugate curve, an ECE.
⎡ xArc (t ) ⎤ ⎡ R cos θ + r cos t ⎤ ⎢ yArc (t ) ⎥ = ⎢ R sin θ + r sin t ⎥ ⎥ ⎢ ⎢ ⎥ 1 ⎦ ⎣ 1 ⎦ ⎣
(6)
In the equivalent motion, the loci of the circle form a cluster of curves that can be given by
2.1. Analysis of common conjugate curves 2.1.1. A point and an extended epicycloid As shown in Fig. 1, point p of the left rotor in system O1x1y1 is given by
x ⎡ p ⎤ ⎡ R cos θ ⎤ ⎢ yp ⎥ = ⎢ R sin θ ⎥ ⎢ ⎣1⎥ ⎦ ⎣ 1 ⎦
(1)
In the equivalent motion, point p forms a locus that is derived by the coordinate transformation in system O2x2y2.
x ⎡ x Cyc (φ) ⎤ ⎡ p ⎤ ⎡− R cos(2φ − θ ) + 2R2 cos φ ⎤ ⎢ yCyc (φ) ⎥ = M12 ⎢ yp ⎥ = ⎢ − R sin(2φ − θ) + 2R2 sin φ ⎥ ⎢ ⎥ ⎥ ⎢ ⎢ 1 ⎣1⎥ ⎦ ⎣ ⎦ ⎣ 1 ⎦
(2)
⎡− cos 2 φ − sin 2 φ 2R2 cos φ ⎤ M12 = ⎢ − sin 2 φ cos 2 φ 2R2 sin φ ⎥ ⎥ ⎢ 0 0 1 ⎦ ⎣
(3)
Thus, the conjugate curve of a point located outside the pitch circle is an extended epicycloid for the twin-rotation motion of TSVPs.
Fig. 1. The meshing relationship of a point and an extended epicycloid. 457
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the equivalent motion. The equidistant curves are obtained by offsetting epicycloid Cyc1 the distance r in the concave or convex direction. The ECE in concave direction lr1 is obtained by ′Cyc1 ⎡ x Cyc (t ) − r ⎤ 1 2 2 ⎢ x′Cyc1 (t ) + y′Cyc1 (t ) ⎥ x lr1 (t ) ⎡ ⎤=⎢ ⎥ ⎢ ylr1 (t ) ⎥ ⎢⎢ yCyc1 (t ) + r 2 x′Cyc1 (t )2 ⎥⎥ x′Cyc1 (t ) + y′Cyc1 (t ) ⎣ 1 ⎦ ⎢ ⎥ 1 ⎣ ⎦ (t )
y
(10)
The ECE in convex direction l-r1 is obtained by ′Cyc1 ⎡ x Cyc (t ) + r ⎤ 1 2 2 ⎢ x′Cyc1 (t ) + y′Cyc (t ) ⎥ x l−r1 (t ) 1 ⎡ ⎤=⎢ ⎥ ⎢ yl−r1 (t ) ⎥ ⎢⎢ yCyc1 (t ) − r 2 x′Cyc1 (t )2 ⎥⎥ x′Cyc1 (t ) + y′Cyc1 (t ) ⎣ 1 ⎦ ⎢ ⎥ 1 ⎣ ⎦ y
(t )
(11)
Thus, the circle whose center point does not coincide with the rotational center meshes with both the concave equidistant curve lr1 and convex equidistant curve l-r1 of the epicycloid Cyc1.
Fig. 2. The conjugate curve of a circular involute.
2.2.2. Discussions of conjugate curves of the circle When the length from points p to O1 is equal to zero, i.e. R = 0, the center point p of the circle coincides with the rotational point O1. The conjugate curve of the circle whose center point is p is also a circle whose center point coincides with the other rotational point O2, and whose radius is 2R2-r. This pair of conjugate curves is used for generating the addendum and dedendum circular arcs of the CSP. When 0 < R < R2, point p is inside the pitch circle. The conjugate curve of point p is a shortened epicycloid Cyc2, as shown in Fig. 4. The conjugate curves of the circle whose center point is p are equidistant curves of the shortened epicycloid, l-r2 and lr2. When R > R2, point p is outside the pitch circle. The conjugate curve of point p is an extended epicycloid Cyc3, as shown in Fig. 5. The conjugate curves of the circle whose center point is p are equidistant curves of the extended epicycloid, l-r3 and lr3. 3. Design methodology of a smooth screw rotor In this section, we analyze disadvantages of the traditional screw rotor whose CSP contains cusps or unsmooth points. We put forward a completely smooth CSP based on the established meshing model. In the design process of new CSP, a pair of conjugate curves, a circular arc and an ECE, were used instead of a point and an epicycloid that were used in the traditional CSP. A geometric model of the proposed new CSP was established, and its meshing accuracy was validated. Influences of the radius of the back arc r2 on the shape are discussed. Ultimately, a
Fig. 3. A circle and its conjugate curve.
⎡ xArc (t ) ⎤ ⎡ xl (t , φ) ⎤ ⎢ yl (t , φ) ⎥ = M12 ⎢ yArc (t ) ⎥ ⎥ ⎢ ⎥ ⎢ ⎣ 1 ⎦ ⎣ 1 ⎦ ⎡− R cos(θ − 2φ) − r cos(t − 2φ) + 2R2 cos φ ⎤ = ⎢ R sin(θ − 2φ) + r sin(t − 2φ) + 2R2 sin φ ⎥ ⎢ ⎥ 1 ⎣ ⎦
(7)
where, r is the distance. The meshing equation of the conjugate curve is shown as
∂x ∂y ∂xl ∂yl − l⋅ l = 0 ⋅ ∂φ ∂t ∂t ∂φ
(8)
Eq. (7) is substituted into Eq. (8), and then the dependence of φ and t can be obtained by
(R + r )sin(θ − t ) + R2 sin(t − φ) = 0
(9)
Eq. (7) and Eq. (9) express the conjugate curve of a circle whose center point does not coincide with the rotational center, which is an ECE. The trajectory pp', which is an epicycloid Cyc1 represented by Eq. (2), is generated by point p that is the center point of the circle during
Fig. 4. The conjugate curve of a circle whose center point is inside the pitch circle. 458
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Fig. 5. The conjugate curve of a circle whose center point is outside the pitch circle.
Fig. 8. Locus of the new CSP.
Fig. 9. The new CSPs varying with r2. (1) When r2 = 0, as shown in Fig. 9(a), circular arc BC changes to point B and its conjugate curve changes to epicycloid FA. The adjacent curves at intersection B are not smooth. (2) When 0 < r2 < R1-R2, as shown in Fig. 9(b), center point n of back circular arc BC is outside the pitch circle. GA is a concave equidistant curve of extended epicycloid Cycb2. The CSP is completely smooth. (3) When R1-R2 < r2 < R1-Rb, as shown in Fig. 9(c), center point n of back circular arc BC is inside the pitch circle. GA is a convex equidistant curve of shortened epicycloid Cycb3. The CSP is completely smooth. (4) When r2 = R1-Rb, as shown in Fig. 9(d), the circular involute does not exist. Back circular arc AB whose center point is located at the generating circle smoothly connects with ECE FA. FA is a concave equidistant curve of extended epicycloid Cycb4.
Fig. 6. The traditional CSPs.
smooth screw rotor generated by the proposed CSP was obtained. 3.1. Disadvantages of the traditional screw rotor profile As shown in Fig. 6, the traditional CSP [4] of the TSVP consists of four curves: a circular involute AB, an addendum circular arc BC, an extended epicycloid curve CD, and a dedendum circular arc DA. This CSP contains a cusp A, and two unsmooth points B and C. When the screw rotor is generated by using this CSP, points A, B, and C will form unsmooth edges. This traditional screw rotor has the following
Fig. 7. The new screw rotor profile.
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⎡ xAB (t ) ⎤ = ⎡ Rb (cos(t − α ) + t sin(t − α )) ⎤ ⎢ yAB (t ) ⎥ ⎢ Rb (sin(t − α ) − t cos(t − α )) ⎥ ⎣ ⎦ ⎦ ⎣
(14)
⎡ xBC (t ) ⎤ = ⎡ (R1 − r2)cos(β ) + r2 cos(t ) ⎤ ⎢ yBC (t ) ⎥ ⎢ (R1 − r2)sin(β ) + r2 sin(t ) ⎥ ⎣ ⎦ ⎦ ⎣
(15)
EF is the external ECE Cyca, and equations of Cyca can be derived by Eq. (2), where R = R1-r1 and θ = π-γ. Angle γ is represented by
y (t γ ) 1 γ = − arctan 2 x (t γ )
where, x(tγ) and y (tγ) are the coordinate value of the intersection of the following two curves, respectively:
Fig. 10. The two intermeshed screw rotors.
disadvantages. (1) It is easy to produce stress concentration at the portion near the unsmooth edges, which can easily lead to wear and deformation. Accordingly, large gas leakage will occur and the efficiency of the pump will be reduced in the working process. (2) The CSP produces a carryover in the meshing process, which will reduce the volumetric efficiency. (3) The CSP contains two unmeshed curves: curve AA1 of circular involute AB and AA2 of the dedendum circular arc DA, which will produce discontinuous contact lines and result in a gas leakage path between adjacent working chambers formed by the two intermeshed screw rotors.
(17)
⎤ ⎡ − 2R2 cos(t ) + (R1 − r1)cos(2t ) ⎥ ⎢ −2R2 cos(t ) + 2(R1 − r 1)cos(2t ) ⎢ + r1 R 2 + R 2 + r 2 − 2R r + 2R (r − R )cos(t ) ⎥ ( ) x t 1 1 2 1 1 γ2 1 1 2 ⎡ ⎤ ⎢ ⎥ ⎢ yγ2 (t ) ⎥ = ⎢ ⎥ ⎣ ⎦ ⎢ − 2R2 sin(t ) + (R1 − r1)sin(2t ) ⎥ R t − R − r t 2 sin( ) 2( )sin(2 ) 2 1 1 ⎥ ⎢ −r 1 ⎢ R12 + R22 + r 12 − 2R1 r 1 + 2R2 (r 1 − R1)cos(t ) ⎥ ⎦ ⎣
(18)
⎡ xDE (t ) ⎤ = ⎡ (R1 − r1)cos(π − γ ) + r1 cos(t ) ⎤ ⎢ yDE (t ) ⎥ ⎢ (R1 − r1)sin(π − γ ) + r1 sin(t ) ⎥ ⎣ ⎦ ⎦ ⎣
(19)
′Cyca ⎤ ⎡ x Cyca (t ) + r1 2 2 ⎢ x (t ) ⎥ ′Cyca (t ) + y′Cyc a ⎡ xEF (t ) ⎤ = ⎢ ⎥ ⎢ yEF (t ) ⎥ ⎢ x′Cyca (t ) ⎥ ⎣ ⎦ yCyca (t ) − r1 2 2 ⎢ x′Cyca (t ) + y′Cyca (t ) ⎥ ⎦ ⎣
(20)
y
As shown in Fig. 7, the new CSP contains 7 curves: a circular involute AB, a back circular arc BC, an addendum circular arc CD, a tip circular arc DE, an ECE EF, a dedendum circular arc FG, and an ECE GA. In the design process, the tip circular arc DE and the ECE EF were used to connect the dedendum circular arc FG and the addendum circular arc CD. The back circular arc BC was used to connect the circular involute AB and the addendum circular arc CD. The ECE EF was used to connect the circular involute AB and the dedendum circular arc FG. All adjacent curves smoothly connect with each other, and no cusps or unsmooth points exist. Thus, the proposed CSP is completely smooth. There are 9 geometric parameters in the CSP: R1, R2, R3, Rb, r1, r2, α, β, and γ. Equations of the back circular arc BC can be derived by Eq. (6), where r = r2, R = R1-r2 and θ = β. Equations of α and β are obtained by
⎜
⎡ x γ1 (t ) ⎤ ⎡ (R1 − r1)cos(t ) ⎤ ⎢ yγ1 (t ) ⎥ = ⎢ (R1 − r1)sin(t ) ⎥ ⎦ ⎣ ⎦ ⎣
Eq. (18) expresses an equidistant curve of an epicycloid, the offsetting distance is r1 in the convex direction. Equations of the tip circular arc DE and the ECE EF are represented by
3.2. A geometric model of a new CSP
R R α = tan ⎛arccos b ⎞ − arccos b R2 ⎠ R2 ⎝
(16)
(t )
where,
⎡ x Cyca (t ) ⎤ = ⎡− 2R2 cos(t + γ ) + (R1 − r1)cos(2t + γ ) ⎤ ⎢ yCyca (t ) ⎥ ⎢⎣ − 2R2 sin(t + γ ) + (R1 − r1)sin(2t + γ ) ⎥⎦ ⎣ ⎦
(21)
GA is the ECE Cycb, and meshes with the back circular arc BC. Equations of Cycb can be derived by Eq. (2). Eq. (2) is substituted into Eq. (10), and equations of ECE GA can be represented by
⎡ x Cyc (t ) − r2 b ⎢ ⎡ x GA (t ) ⎤ = ⎢ ⎢ yGA (t ) ⎥ ⎢ ⎣ ⎦ y (t ) + r2 ⎢ Cycb ⎣
⎟
(12)
2 β = tan ⎛⎜arccosRb / Rb2 + ( (R1 − r2)2 − Rb2 + r2) ⎟⎞ − arccosRb /(R1 − r2) ⎝ ⎠ (13) −α
y′Cyc (t )
⎤ ⎥ ⎥ x′Cycb (t ) ⎥ 2 2 x′Cycb (t ) + y′Cyc (t ) ⎥ b ⎦ b
2
2
x′Cycb (t ) + y′Cyc (t ) b
(22)
where,
⎡ x Cycb (t ) ⎤ = ⎡2R2 cos(t − β ) − (R1 − r2)cos(2t − β ) ⎤ ⎢ yCycb (t ) ⎥ ⎢⎣ 2R2 sin(t − β ) − (R1 − r2)sin(2t − β ) ⎥⎦ ⎣ ⎦
Equations of the circular involute AB and the back circular arc BC are obtained by
(23)
Fig. 11. Completely smooth screw rotors and their contact lines varying with r2.
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Fig. 12. Effects of r2 on screw rotors.
Fig. 13. Meshing and contact lines of the traditional screw rotor. Table 1 Meshing relations between the traditional CSPs. Left CSP
Right CSP
Meshing lines
Contact lines
Circular involute AB Addendum circular arc BC Point C Extended epicycloid curve CD Dedendum circular arc DA
Circular involute ab Dedendum circular arc da Extended epicycloid curve cd Point c Addendum circular arc bc
lAB lBC lC lCD lDA
LAB LBC LC LCD LDA
Fig. 14. Meshing and contact lines of the proposed screw rotor.
screw rotor profile, which shows that the proposed screw rotor profiles can correctly mesh with each other. The accuracy of the geometric model of the proposed CSP was validated.
3.3. Validation of the meshing accuracy Computerized simulation of the meshing accuracy of the proposed CSP was performed, as shown in Fig. 8. It can be seen that the envelope of the motion trajectories of the one screw rotor profile is the other 461
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(1) The proposed rotor profile is completely smooth without cusps and unsmooth points. (2) The proposed rotor profile can achieve correct mesh in the working process. (3) Compared with the traditional rotor, the proposed rotor does not have carryover.
Table 2 Meshing relationships between the proposed CSPs. Left CSP
Right CSP
Meshing lines
Contact lines
Circular involute AB Back circular arc BC Addendum circular arc CD Tip circular arc DE ECE EF Dedendum circular arc FG ECE GA
Circular involute ab ECE ga Dedendum circular arc fg ECE ef Tip circular arc de Addendum circular arc cd Back circular arc bc
lAB lBC lCD
LAB LBC LCD
lDE lEF lFG
LDE LDE LFG
lGA
LGA
Meanwhile, the proposed rotor has the following disadvantages. (1) The meshing lines of the proposed screw rotor cannot pass through the intersection of pump holes, which will result in a blowhole in the tip part of tooth. (2) The proposed rotor profile, consisting of seven curves, is more complicated than the traditional one.
3.4. Discussions of the proposed CSP The value of radius r2 of the back circular arc BC of the new screw rotor profile affects its shape. Fig. 9 shows the proposed CSPs varying with r2. The range of r2 was 0 ≤ r2 ≤ R1-Rb. Fig. 9 (a), (b), (c) and (d) are the CSPs whose back circular arc radius r2 was 0 mm, 20 mm, 40 mm, and 55 mm, respectively. The other geometric parameters were constant: R1 = 90 mm, R3 = 40 mm, Rb = 35 mm, and r1 = 6 mm.
4.2. Stress and deformation During the working process of the TSVP, in order to compare the proposed and traditional rotors, the same boundary and initial conditions were set on the two rotors. The suction pressure was 3 kPa, and the discharge pressure was 100 kPa. Stresses and deformations produced by the pressure forces were calculated by using the finite element method [17]. Fig. 15 illustrates stress distributions of the two rotors. It can be seen that the maximum values of stress are located at the dedendum region near the discharge side of the two screw rotors. The maximum stress value of the new rotor was 24.096 MPa, and the maximum stress value of the traditional rotor was 25.714 MPa. The former was smaller than the latter, and was reduced by 6.2%. Taking an arbitrary point on the same positions of two rotors as an example, the stress value of the new rotor was 5.371 MPa, while the stress value of the traditional rotor was 7.398 MPa. Fig. 16 presents deformation values of the new and traditional rotors. The maximum value for the two rotors occurred in the addendum region near the discharge side. The maximum value of the new rotor, 0.0115 mm, was smaller than that of the traditional rotor, 0.0123 mm, and was reduced by 6.5%.
3.5. A new smooth screw rotor A screw rotor was generated by the proposed CSP performing a screw motion, and two intermeshed screw rotors are shown in Fig. 10. The surfaces of the proposed screw rotors are completely smooth; and the two screw rotors can correctly mesh. The proposed screw rotors and their contact lines with different r2 are shown in Fig. 11. Effects of r2 on area utilization coefficient η of the CSP and length of the contact line L of screw rotors are shown in Fig. 12. As r2 increases, η remains approximately constant, but L decreases. When r2 = R1Rb = 55 mm, the contact line is the shortest, which means that the leakage through the contact line is the smallest. 4. Comparisons between the proposed and traditional screw rotors
5. Conclusion A meshing model of a circular arc and an ECE was established, a novel CSP was proposed, and a novel screw rotor was formed by using the proposed CSP. The proposed CSP could correctly mesh. In the design process of CSPs, a circular arc and an ECE were used, and therefore, the meshing relations between two CSPs were curve and curve, which was superior to that of the traditional CSPs: point and curve. Thus, the problem caused by the existence of cusps or unsmooth points in the traditional screw rotor profile was solved. The established meshing model of a circular arc and an ECE is also suitable for other rotor profile designs of fluid machinery that consist of two rotors synchronously rotating in opposite directions on parallel axes, such as claw rotors and Roots rotors. The proposed CSP and the generated rotor are completely smooth, and the proposed screw rotor has continuous meshing lines and contact lines in one pitch period. When the same pressure loads act upon the
4.1. Geometric characteristics The traditional CSP is shown in Fig. 6, and its meshing lines and contact lines are shown in Fig. 13. The meshing relations between the two CSPs are listed in Table 1. The proposed CSP is shown in Fig. 7, and its meshing lines and contact lines are shown in Fig. 14. The meshing relations between the two CSPs are listed in Table 2. As shown in Fig. 14 (a), the meshing lines of the proposed screw rotor cannot pass through the intersection of pump holes, which will result in a blowhole in the tip part of tooth. However, the radius of the tip circular arc DE is very small in practical application. Thus, the leakage through the blowhole is far smaller than that through the carryover of traditional profile. The proposed rotor has the following advantages.
Fig. 15. Stresses of the two rotors. 462
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Fig. 16. Deformations of the two rotors.
two rotors, the maximum values of stress and deformation of the new rotor are smaller than those of the traditional rotor are. The new rotors have better performance than that of the traditional ones in terms of gas sealing, stress, and deformation.
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Acknowledgements This work was supported by the National Natural Science Foundation of China [51706247], and the Fundamental Research Funds for the Central Universities [16CX05007A]. Thanks to Dr. Edward C. Mignot, Shandong University, for linguistic advice. Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.vacuum.2018.10.064. References [1] C.Q. Chen, Y. Jia, D.B. Zheng, et al., On the profile of single thread and fixed pitch screw for the screw vacuum pump, Vacuum 48 (3) (2011) 8–11 (in chinese).. [2] Takeshi Kawamura, Kiyoshi Yanagisawa, Screw Rotor and Method of Generating Tooth Profile Therefor, (1998) US 5800151, Sep. 1. [3] C.F. Hsieh, Y.W. Hwang, Z.H. Fong, Study on the tooth profile for the screw clawtype pump, Mech. Mach. Theor. 43 (7) (2008) 812–828.
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