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Vietnam Journal of Science and Technology 57 (2) (2019) 249-260 doi:10.15625/2525-2518/57/2/13094 A NOVEL DESIGN OF THE ROOTS BLOWER Tran Ngoc Tien, Nguyen Hong Thai* School of Mechanical Engineering, Hanoi University of Science and Technology No 1 Dai Co Viet, Ha Noi * Email: thai.nguyenhong@hust.edu.vn Received: 13 September 2018; Accepted for publication: 13 February 2019 Abstract. This paper reports a novel curve developed from non - circular gearing theory, which can be applied in rotor profile design of the two - lobe Roots blower. The formulas for calculating the volumetric efficiency and specific flow rate of the blower have also been established. To evaluate this type of the Roots blower, the volumetric efficiency and specific flow rate are being compared with those parameters of the two traditional designs and one recent variant. The results show that with the new design, the specific flow rate significantly increases for 20% to 37%, and the transverse dimension decreases for 4% to 15%. All these changes confirm usefulness and advantages of this new design. Keywords: hydraulic machine, Roots blower, profile, flow. Classification numbers: 5.5.1, 5.6.1, 5.10.1. 1. INTRODUCTION Roots blower belongs to the non - contact hydraulic machines with external mating gears [1]. This kind of blowers with the rotor profile generated by circular arches was invented by the Roots brothers yet in 1860 [2]. In 1875, Palmer and Knox applied cycloidal curves in profile designing process, in which the addendum rotor was epicycloidal and the dedendum rotor was hypocycloidal [3]. Litvin in [4] simplified the rotor profile with the circular addendum and the dedendum created by the curve meshing with the circular addendum in order to guarantee continuous matching process. During the history of 150 years of research and development of the Roots blower, this type of pump has improved from the structure and design aspect, as well as been applied in more industrial fields. One of the improvement methods is to redesign the rotor profile, which can increase the blower flow without enlarging the blower size. In 2008, Hsieh and Hwang developed the blower presented in [3] by changing a trochoid rate to increase the inlet chamber and outlet chamber volumes, and by that to increase the blower flow [5]. In 2015, Hsieh proposed a novel rotor profile generated by the locus of the fixed point of ellipse rolling on the pitch circle [6]. In this paper, the optimal ratio between the semi - minor and semi-major axes of the rolling ellipse was proven to equal 0.6. In [7], Cai et al. used a conjugated curve in designing three - lobe rotor, in which he combined circular dedendum and cycloidal addendum (2016). In the same year, Shinde also proposed another profile with involute addendum and circular dedendum [8]. With those conjugated profiles, the specifications of the Tran Ngoc Tien, Nguyen Hong Thai blower, especially the flow, are distinctly improved. All these works showed us the importance of finding new and better profile in order to achieve larger flow of the blowers. In the curent paper, with the purpose of increasing volumes of the inlet and outlet chambers, we propose a novel design of the Roots blower with the rotor profile { } generated by non - circular gearing theory. Figure 1 describes this method of profile generation: i) the rotor addendum { d} is a curve generated by point M fixed on the circle { S} when { S} rolling outside on the elliptical pitch { E}; ii) the rotor dedendum { c} is a curve generated by point M fixed on the circle { S} when { S} rolling inside on the elliptical pitch { E}. Rotor dedendumr c { Rotor addendum d } { } E2 E3 b E { } r a E M O0 E { S } E1≡M0 E4 Figure 1. Principle of the generation of rotor profile. 2. MATHEMATICAL MODEL OF A NEW ROTOR FOR BLOWER 2.1. Mathematical model of the addendum rotor To establish the equation of addendum rotor following the method of profile generation presented in section 1, in Figure 2 we have: 0{O0 x0 y0 } 1{O1x1 y1 } 2{O2 x2 y2 } : the fixed reference coordinate system with origin E O0 of { }. : the local coordinate system with origin O1 Pi (contact point between { S} and { E}) and axis O1x1 O0 Pi . : the local coordinate system connected to the rolling cirle { S}, and with origin located at the centre of { S}. {Γ d } y1 x2 Pi+1 M y2 b φ O2 Pi≡O1 ξ θ O0 ψ P0 a γ x1 M0 x 0 r S E { } { } Figure 2. Principle of the generation of rotor addendum profile. : rotation angle of 2 in relation to 1 during relative motion of { 250 O2,i+1 y0 S } when rolling on { E }. A novel design of the Roots blower : rotation angle of 1 in relation to to Pi (random position). : rotation angle of 2 in when { 0 relation to S } rolling from P0 (initial position) during relative motion. 0 S r : radius of { }. a ,b : semi-major and semi-minor axes of { : angle parameter of { }, respectively. S }. S Let M be the point fixed on { E S }, when { } is rolling outside {  rM  rPi  rO2  rO2 M E }, from figure 2 we have: (1) By transforming equation (1) into the algebraic form, we have: 0 0 where: d rM ( , , ) r Pi ( ) , 1 r O2 , 2 r M ( ) 0 0 r Pi ( ) 1 R1 ( z, ) r O2 0 2 (2) R 2 ( z, ) r M ( ) are the vectors determining positions of Pi , O2 , M in the coordinate systems 0 , 1 , 2 ; 0 R1 ( z, ) and 0 R 2 ( z, ) are the rotation matrix presenting counterclockwise revolution around z axis the angles , . Developing equation (2), we have: { d }: 0 r dM ( , , ) d xM ( , , ) d yM ( , , ) r cos r sin r cos r sin xP ( ) yP ( ) (3) equation (3) is presenting profile of the rotor addendum. This equation contains three parameters , , , and it is necessary to find the relation between them. Determining ( ) From figure 2, it is clearly shown that is the angle between the common normal of { and { S} on the contact point Pi and the axis O0 x0 . Therefore is given by: ( ) tan xP ( ) / yP ( ) / 1 E } (4) where: xP ( ) with P( P yP ( ) T P( ( ) is a distance from arbitrary point Pi on { ) cos P( ) sin T (5) E } to the origin O0 of 0 . According to [9], ) is expressed by: P( Determining ) 2ab a b (a b) cos(2 ) (6) ( ) From 2, using relation between the angles of triangle, therefore ( ) is presented by: ( ) ( ) ( ) (7) 251 Tran Ngoc Tien, Nguyen Hong Thai S from equation (7), it is clear that we need to find ( ) before calculating ( ) . When { } is rolling outside { E}, the length of elliptical arc on { E} need to be equal to the length of the circular arc on { S}, therefore: 1 r ( ) 2 xP ( ) 1 2 2 yP ( ) (8) d 0 y0 2.2. Mathematical model of the dedendum rotor Pi+1 y1 x2 O2,i+1 M0≡P0 φ ψ γ { S} With the same nomenclature already stated in section 2.1, but let { S} is only rolling clockwise inside { E} as described in Figure 3. Similarly, we have the equation of the rotor dedendum profile: x1 Pi≡O1 O2 b M r y2 {Γ } E } ξ θ c { O0 x0 a Figure 3. Principle of generation of the rotor dedendum profile. xMc ( , , ) yMc ( , , ) { c}: 0 r cM ( , , ) r cos r sin r cos r sin xP ( ) yP ( ) (9) Merging equations (3) and (9), the rotor profile equation { } is presented in the universal form as below: x ( , , ) y ( , , ) { }: 0 r ( , , ) r cos r sin r cos r sin xP ( ) yP ( ) (10) Since now, the symbol is used to denote the rotor profile. On the other hand, in equation (10) sign “ ” is used as follows: Using “ ” when the angle Using “ - ” when the angle with and E E 0 E ( E ( E E) ) ( ( E E) ) (2 (2 E E) ) 2 . . is the angle determining limits of the addendum rotor and dedendum rotor (Figure 1), is given by: E 1 cos 2 1 a b . a b (11) 2.3. Designing dimensional parameters of the blower As presented in section 1, the blower designed by the non - circular gears matching theory in general, and with elliptical gear especially, will have the analytical profile formed in sections 2.1 and 2.2. Based on continuous mating process of this elliptical gear pair, formation of the inlet chamber and outlet chamber is described in Figure 4. 252 A novel design of the Roots blower { S stator } inlet chamber Pockets O1 1 R Rotor 1 { } Rotor 1 Rotor 2 Timing Gears Bearings 2 b O2 r Rotor 2 Electric motor a outlet chamber { d E } E A a) Cross section of the blower b) Struxture of the blower Figure 4. Principle of blower design. Let: E , R , A , d be distance of the shafts, radial dimension, transverse dimension of the blower, axial dimension of the inlet and outlet chamber, from figure 4 we have: E a b (12) R a 2r (13) A E 2R (14) Therefore, parameters E , R , A , d are the design parameters of the blower, and a , b , r are the characteristic design parameters of the rotor as well as the blower itself. 2.3. Condition of the characteristic design parameters for generation profile of the blower There is a problem that not every set of the characteristic design parameters a , b , r can help to generate the rotor profile. Practically, the rotor profile can not be generated by some sets of parameters, and interference between profiles or undercutting phenomenon can happen with the other sets. It raises the need to set condition for those parameters. Let C E , C S be circumference of { E} và { S}, respectively: 2 CE 2 xP ( ) 1 2 2 yP ( ) (15) d 0 CS (16) 2 r S following the principle of profile generation presented in section 2, { } is only rolling on { The symmetricity of the rotor profile is also taken into consideration, we have: 2 CE 4C S or xP ( ) 2 yP ( ) 2 E }. 1 2 d 8 r (17) 0 253 Tran Ngoc Tien, Nguyen Hong Thai on the other hand, from figures 1 and 4, if the profile is symmetrical and satisfies (17), then radius of the rolling cirle { S} need to fulfil following condition: r b 2 (18) Let b / a be the characteristic design parameter of the blower. By substituting and (18), we have: 0.5 into (17) (19) 1 Formulas (18) và (19) are boundary condition of the characteristic design parameters for generating the rotor profile of the blower. 3. DESIGN AND EVALUATION 3.1. Theoretical specific flow rate Stator Specific flow rate of blower is defined by the volume amount discharged out while the driving shaft accomplish one revolution. According to [10], the specific flow rate Qr is given by: Qr O1 Sd 1 S 2 O2 (20) 2ZSd Sc S where: d is the radial dimension of the blower; Z is the number of the rotor teeth; S is the area of perpendicular cross section (see Figure 5). From Figure 5, we have: S 1 R2 2 Figure 5. Calculation of the specific flow rate. (21) S with S is the area of the rotor cross section perpendicular to the blower shaft, and S is given by: S 4(S d Sc) (22) where S d and S c are the areas of the sub cross - section bordered by the profiles of addendum rotor and dedendum rotor in relation to the origin O2 (figure 5). S d and S c are calculated as follows: Sd E xMd ( , , ) y d ( , , )d (23) xMc ( , , ) y c ( , , )d . (24) 0 Sc /2 E Case study Given are the set of characteristic design parameters: radius of { S} r 11.6704 mm ; rolling ellipse { E} with major semiaxis a 56.6591mm , minor semiaxis b 33.9955 mm ; radial 254 A novel design of the Roots blower dimension d 130mm ; number of rotor teeth Z 2 . The specific flow rate of the blower will be equal Qr 3.37 106 mm3/rev. 3.2. Volumetric efficiency of the blower According to [6], to access the theoretical flow of the blower, the volumetric efficiency is being used. This parameter is defined by the ratio between the volume blown from inlet to outlet of the blower in one revolution of the driving shaft and the total volume measured inside of the blower stator. From Figure 5, we have: 2ZS 100 % S stator (25) where: Sstator is the area of the inside blower chamber (See figure 4), and Sstator is given by: Sstator (a 2r) (a 2r) 2(a b) . (26) 3.3. Evaluation To prove the advantages of this design, the authors have carried out comparison of the volumetric efficiency, specific flow rate and blower dimension between the proposed design and two traditional variants - type 1 [3], type 2 [4], which were presented in [6, 11, 12] (Those two traditional designs had already been applied in manufacturing real product). A new design of Hsieh [6] called type 3 is also added to comparing process. a) Type 1: Traditional design (Palmer and Knox [3] in 1875) A { S} R RL r S { L} RL r rs RL rs rs sin rs { }: r ( RL { } E Figure 6. Roots blower type 1. Profile equation: According to [3], the rotor profile { } consist of design epicycloidal addendum { d} and hypocycloidal dedendum {  rs cos RL c }: rs ) cos (27) ( RL rs ) sin where: RL , r , are radius of the circle pitch { L}, radius of the rolling circle { S} and parametric angle { L}, respectively (Figure 6). The signs “ ” and “  ” are chosen by the following rules: the upper sign for { d}, the lower sign for { c}. Dimensional design parameters: from figure 6, those parameters are given by: E R A 2 RL RL 2r E 2R (28) Condition for profile generation: to generate the profile, according to [10], the characteristic design parameters need to satisfy: RL 2Zr (29) 255 Tran Ngoc Tien, Nguyen Hong Thai b) Type 2: Traditional design (Litvin [4] in 1956) Profile equation: In [4], the rotor addendum profile { 7), with equation: { d cos sin }: r d c d } are drawn by circular arch (Figure A (30) R RL The rotor dedendum profile { c} is the meshing curve with { d} (Figure 7) presented by following equation: { c }: r c cos( sin( 2 ) c cos 2 2 ) c sin 2 2RL cos 2RL sin c O2 O1 S { } (31) E L { } where: RL , c , are radius of { L}, distance from origin Figure 7. Roots blower type 2. O1 to the centre of the rotor circular arch, radius of the rotor addendum { d}, respectively. And , are the parametric angle of { L} and rotation angle of the driving shaft. Dimensional design parameters: from Figure 7, the design parameters of the blower are: E R A 2 RL c E 2R (32) Condition for profile generation: According to [4], c / RL is the characteristic design parameter of the rotor profile. Condition for profile generation with no singularities is: 0.5 (33) 0.9288 A { ES } R 2b1 RL RL 2a1 S O1 O2 c) Type 3: new design in 2015 (Hsieh [6]) { } Profile equation: In this type of blower, { d} is the locus E { } ES ES of a point fixed on the rolling ellipse { }, when { } is only rolling outside on { L} of the gear. { c} is the Figure 8. Roots blower type 3. locus of a point fixed on { ES}, when { ES} is only rolling inside on { L} (Figure 8). The equation of { } is given by: L { }: r  a1 (1 cos ) cos( ) b1 sin( ) sin RL cos a1 (1 cos ) sin( ) b1 cos( ) sin RL sin (34) where: RL , a1 , b1 are radius of { L}, major semi-axis of { ES} and minor semi-axis of { ES}, respectively. And , are the parametric angle of { L} and { ES}. is the angle between the common normal of { L} and { ES} on the contact point, and the line connecting the centers of the rotor 1 and rotor 2. Dimensional design parameters: from Figure 8, those parameters are presented as below: E R A 256 2 RL RL 2a1 E 2R (35) A novel design of the Roots blower Condition for profile generation: As stated in [6], the characteristic design parameter is calculated by formula a1 / b1 . Condition for profile generation with no self-intersection: 0.4 1. (36) d) Analysis and evaluation Let take two case studies into consideration: Case study 1: Determine characteristic design parameters of the variants of the blower, of which the radial dimension R = 72 mm and the axial dimension d = 150 mm. The characteristic parameters are shown in Table 1. Table 1. Characteristic design parameters. Characteristic design parameters Parameters Value R [mm] 48.0000 Type 1 (Fig. 6) L (1875) r [mm] 12.0000 RL [mm] 44.0921 Type 2 (Fig. 7) c [mm] 39.6829 (1956) [mm] 32.3171 RL [mm] 43.6754 Type 3 (Fig. 8) a1 [mm] 14.1623 (2015) b1 [mm] 7.0811 [mm] 51.6393 Roots blower with a new design b [mm] 25.8196 (Fig. 5) r [mm] 10.1803 Blower type A [mm] Volumetric efficiency 240.0000 = 54.09% 232.1842 = 63.66% 231.3508 = 64.49% 221.4587 = 81.21% From Table 1 we have: (i) The volumetric efficiency is increasing, and the transverse dimension is decreasing in the history of blower development, which matches well with our notice made in section 1. (ii) The volumetric efficiency of the newly proposed blower is distinctly higher than efficiency of the older designs: 27.12 % higher in comparison with type 1, 18.15 % higher than type 2, and 16.72 % higher than type 3. It means that the specific flow rate is the largest one, while dimension of the novel blower is smallest, which leads to consideration that the novel design is approximately optimal. For thorough evaluation we are going to examine case number 2 below. Case study 2: In order to generally evaluate, we are going to determine the characteristic design parameter of each type of blower based on parameter ( needs to satisfy conditions 19, 29, 33, 36 for each type of blower to generate profile), with constraint of radial dimesion R = 72 mm and axial dimension d = 150 mm for all of the evaluated blowers. On the other hand, from conditions (19 and 36), when = 1, both { S} and { ES} become a circle, which means that blower of type 3 and actually proposed blower will transform to type 1. Therefore, if the increment = 0.1 to satisfy conditions (19, 29, 33, 36 ), then [ 0.5 1]. From the values of , the relation of the characteristic design parameters for each type will be found. By 257 Tran Ngoc Tien, Nguyen Hong Thai substitution to formulas (12, 13, 17, 29, 32, 23), and solve this repeating problem in Matlab, the characteristic design parameters will be listed in the Table 2 below. Table 2. Characteristic design parameters of each type of blower based on parameter . (2015) (1956) Characteristic design parameters Roots blower with new design Type 3 Type 2 Blower type Note: when = 0.5 = 0.6 = 0.7 = 0.8 = 0.9 = 1.0 RL [mm] 63.0654 59.3078 55.4314 51.5470 47.7664 - c [mm] 31.5327 35.5847 38.8020 41.2376 42.9898 - [mm] 46.4673 42.4153 39.1980 36.7624 35.0102 - RL [mm] 47.3150 48.2868 49.2524 50.1980 51.1156 52.0000 a1 [mm] 15.3425 14.8566 14.3738 13.9010 13.4422 13.0000 b1 [mm] 7.6713 a [mm] 55.9426 55.2426 54.4895 53.6912 52.8578 52.0000 b [mm] 27.9713 33.1456 38.1426 42.9529 47.5720 52.0000 r [mm] 11.0287 11.3787 11.7552 12.1544 12.5711 13.0000 8.9139 10.0617 11.1208 12.0979 13.0000 = 1 blower of type 3 and new design of blower will transform to type 1. A [mm] Applying equations (14, 28, 32, 35) for the data calculated in Table 2, we can obtain the graph in Figure 9. 290 280 270 260 250 New design 240 Type 3 230 Type 2 220 Type 1 ( =1.0) 210 =0.5 =0.6 =0.7 =0.8 =0.9 Figure 9. Transverse dimension of the blower types. 258 =1.0