Roots Blower with Gradually Expanding Outlet Gap: Mathematical Modelling and Performance Simulation Yingjie Cai 1, a, Ligang Yao 2, b

Transcription

1 2nd International Conference on Advances in Mechanical Engineering and Industrial Informatics (AMEII 216) Roots Bloer ith Gradually Exanding Outlet Ga: Mathematical Modelling and Performance Simulation Yingjie Cai 1, a, Ligang Yao 2, b 1 School of Mechanical Engineering and Automation, Fuzhou University, Fuzhou, 35116, China 2 School of Mechanical Engineering and Automation, Fuzhou University, Fuzhou, 35116, China a cyj@fzu.edu.cn, b ylgyao@fzu.edu.cn Keyords: Gradually Exanding Ga; Roots bloer; Basic inter-sace; Simulation Abstract. Gradually exanding ga in the outlet of the Roots bloer can effectively reduce the backflo imact and the noise of the bloer. The orking rincile for the Roots bloer ith the gradually exanding ga is investigated in this aer. And the mathematical modelling for backflo is roosed. Numerical methods are given to solve the outlet ga size. The erformance arameters of different dimension ga are comared. The outlet flo field in Roots bloer ith gradually exanding ga is simulated by means of CFD simulation. Finally, the correctness of the numerical solution is verified. Introduction The Roots bloer ith the characteristics of large range of outut ressure, higher seed, and large exhaust air rate has been used idely [1]. As shon in Fig.1, to rotors are driven by a air of synchronous gears rotating in oosite direction at the same seed. While in orking rocess, a closed sace is formed beteen the rotor rofile and the inner alls of the housing, and the closed sace is called the basic inter-sace. With the rotation of a air of rotors, gas is ushed from the inlet to the outlet in the basic inter-saces. Without taking into account the leakages, there are 6 basic inter-saces of the gas hich are delivered ith a revolution of the rotors. Because of the large ressure differentials beteen the inlet and the outlet gas, the high ressure gas of the outlet ill have a greater imact hen the basic inter-sace is transferred to the outlet. In addition, the traed gas hen the rotors engaging can cause the local ressure to rise raidly. And all the imact ill roduce vibrations and noises. Roots bloer noise reduction has been a research focus of scholars. Xiong ut the muffler into the Roots bloer shell, and reduced the noise by 15~25 db [2]. Li studied the countercurrent cooling structure and gradually exanding ga of Roots bloer, and verified the effectiveness of them ith theoretical analysis and numerical simulation [3]. Wang analyzed the vibration energy of Roots bloer equiments and noise roagation both qualitatively and quantitatively ith the sectral characteristics of vibration and ideband noise taken into account [4] Fu solved the gradually exanding ga of Roots bloer by curve fitting method [5]. Many studies sho that the gradually exanding ga of the Roots bloer is one of the effective methods reducing noises. But there is no calculation method for the exanding ga. The gradually exanding ga of Roots bloer is analyzed, and the gas flo equation is established in this aer. Each of linear and quadratic function is chosen as the changing rule of the gradually exanding ga. The backflo rocess is modeled by mathematics. The otimal exit size of the gradually exanding ga is solved ith numerical method. Finally, the Roots bloer outlet flo field is simulated by CFD softare. And the correctness of the numerical solution is verified. The Theoretical Analysis and Mathematical Modeling of the Gradually Exanding Ga One of the effective ays to reduce the backflo imact is to use the helical rotors similar to the helical gear. The basic inter-sace in the Roots bloer ith helical rotors is gradually oened at the outlet, hich can effectively reduce the imact of the backflo at the outlet. Hoever, due to the characteristics of the structure, the manufacturing of the helical rotors is difficult, and the 216. The authors - Published by Atlantis Press 653

2 roduction cost is higher. So, it is rarely used in ractical Roots bloer roduction. And another feasible but more economical means to reduce the imact is to use the outlet gradually exanding ga [3]. As shon in Fig. 2, the inner housing all near the outlet of the bloer is gradually exanding; a edge-shaed sace beteen the outer circle of the rotor tooth rofile and the all of the bloer housing is formed. So, the ga beteen the rotors and the alls is gradually increasing ith the rotation of the rotors. The ressure of basic inter-sace is gradually increasing hen the backflo of the gas through the ga hich are gradually increasing. When the basic inter-sace connects ith the outlet sace of the bloer, the ressure of the basic inter-sace and the outlet sace gets to balance. The gradually exanding ga of the bloer can effectively reduce the imact of the backflo and the noise of the bloer. Rotor 1 Inlet Inlet Rotor 1 Rotor 2 Rotor 2 Shell ω1 O1 Shell ω1 O1 ω2 O2 O2 ω2 Basic inter-sace Gradually exanding gas Outlet Fig.1 Roots bloer Outlet Fig. 2 The gradually exanding gas of the outlet In order to avoid the collision beteen the to rotors, beteen the rotors and the inner housing all, the ga is set beteen them. The ga beteen rotor and the inner housing all can be calculated by the folloing formula [1], (1) δ r = (.5 ~.1) D. Where in Eq. (1), the δ r is the idth of ga beteen the rotor and the inner housing all; D is the addendum circle diameter of the rotor tooth rofile. O1 α M M2 M4 M6 Gradually exanding ga ω1 A h O2 α hω M2 Air M2 Fig. 3 The basic inter-sace and the gradually exanding ga Even if the Roots bloer doesn t have gradually exanding ga, there are backflo and leakages through the ga mentioned above. But, the backflo and leakage rate is basically unchanged and smaller. So, e can neglect the ga mentioned above hen solving the gradually exanding ga. As shon in Fig. 3, hen the oint A on the rotors runs from M to M 2, the ga idth h beteen the rotor tooth rofile and the housing all increases gradually from to h. And the ressure in the basic inter-saces increases gradually; meanhile the ressure differentials reduce gradually. At the exit, the velocities of backflo decrease to zero, and the ressure differential is to zero too. So, the backflo imact is greatly reduced. In engineering alications, the design of the ga value is often based on the exeriences [5]. Hoever, too small ga can t reduce the imact effectively, and too large ga ill enlarge the size 654

3 of the bloer. In this aer, calculation model of the flo-back rocess hen the basic inter-sace gets through the gradually exanding ga is established. And the otimal values of the gradually exanding ga are obtained. The change la of backflo arameters is studied. In the backflo rocesses, the air inut of the basic inter-sace is given as [6], t2 Q = ρ vhdt (2) In Eq. (2), Q is the intake air quality, t 2 is the time hen oint A gets through the gradually exanding ga, ρ is the air density in outlet, v is the velocity of backflo, h is the ga idth. The values of h are from to h h, and change by linear or quadratic function la. = ht () (3) h The velocities of backflo is, v = ϕ 2( ) / ρ (5) 2 = ht ( ) (4) n Where in Eq. (5), ϕ is the velocity coefficient, and its value is.64 [4]; and n are ressures in the exit and basic inter-sace resectively. The gauge ressure in outlet of Roots bloer is usually no more than 8 kpa,and the efficiency of the bloer reaches the maximum value hen gauge ressure is 4 kpa or so. In order to simlify calculating, e build a gas model aroximately by ideal gas. The air ressure in the basic inter-sace ill increases ith the increasing of the backflo air quality. Clayron equation of ideal gases: V = nrt (6) Where in Eq. (6), reresents the gas ressure; V is the gas volume; n is the amount of matter; R reresents the gas constant; T is the absolute temerature. From Eq. (6), it can be seen that in the case of constant temerature gives, n/ V ρ (7) Because a basic inter-sace volume of the rotor is fixed, so: Q + Q n = (8) Q In Eq. (8), Q is the initial air quality of the basic inter-saces, is the initial air ressure of the basic inter-sace. Outlet air density: ρ = ρ (9) Where, ρ is the air density ith one standard atmosheric ressure, h is solved by Eqs (2), (5) and (8). The boundary condition is given as = n =, hen t = t2. Because of the difficulty of solving differential equations, the equation variables are discreted to deal ith. The gradual exansion area is divided into 2 arts, as shon in Fig. 3. Let i = 1 : 2, then: Qi+ 1 = Qi + t vi hi ρ vi = ϕ 2( ni) / ρ (1) Q i 1 ni = ni + Q ni i 655

4 An Examle of Solving Gradually Exanding Ga Assuming that center distance of to rotors is a = 127mm, diameter of rotor addendum circle is D = 24mm, the rotors rofiles are designed by the eccentric arcs and cycloidal curves tooth rofile [9], the initial ressure in the basic inter-sace is = 11325Pa, the initial angle of the gradually exanding ga area is α = 9 o, the hole angle of the gradually exanding ga area is α = 1 o, the rotor seed is n = 146rm, the gauge ressure of outlet is 4kPa, the total ressure of the outlet is = Pa. The gradual exansion area is divided into 2 arts, as shon in Fig. 3. Let i = 1 : 2. According to the rotor seed and the hole angle of the gradual exansion area, the interval time can be solved and given as t =.571s. The boundary condition is given as 2 =, v 2 =. The gradually exanding ga is calculated by linear and quadratic function changing rule resectively by solving the Eq. (1), and the calculation results are shon in Tab. 1 and Tab. 2. Tab.1. Calculation result of backflo in the gradually exanding ga ith linear function changing rule No. i Air qualities in Air ressures in Pressure Velocities of Ga idths basic inter-sace basic inter-sace differentials backflo h( ) mi ( kg) ni( Pa) i ( Pa) vi ( m/ s) i m

5 Tab.2. Calculation result of backflo in the gradually exanding ga ith quadratic function changing rule No. i Air qualities in Air ressures in Pressure Velocities of Ga idths basic inter-sace basic inter-sace differentials backflo h( ) mi ( kg) ni( Pa) i ( Pa) vi ( m/ s) i m From the Tab. 1 and Tab. 2, it can be seen that both ressure differentials and velocities of backflo reduce to zero hen oint A on the rotor reaches to the exit of the gradual exansion, and the backflo is disaeared this moment. The ga idth of exit of linear function is 4.28mm, and the quadratic function is 6.32mm. The linear function is better than the quadratic function. From the change trend of ressure differentials in Fig. 4, e can see that the ressure differential of the linear function shoing smaller overall trend. The linear function is better than the quadratic function. As shon in Fig. 5, the backflo velocities of linear function are loer than that of quadratic function, and change more smoothly. The linear function is better than the quadratic function. The ressure differentials (kpa ) The rotating angle( ) The gradually exanding gas ith quadratic function changing rule The gradually exanding gas ith linear function changing rule Fig.4 The ressure differentials changing curves Comarison and Verification of Roots Bloer Pressure Pulsation Based on CFD To verify the effect of the gradually exanding ga, taking linear function changing rule gradually exanding ga of Roots bloer and non-gradually exanding ga of Roots bloer into comared. In CFD simulation, the unsteady flo field is used, the dynamic mesh is adoted, and the rotation seed of the rotor is given as 146rm. The fluid ressure and velocity of to kinds of bloer, hich are in a revolution of rotor, the normal ga and the linear function changing ga, are solved. Due to limitations on sace, the rocess of modeling and solving rocess is omitted. Based on the comarison results above, the erformance of Roots bloer ith gradually exanding ga in linear function changing rule are better than that of quadratic function. Due to 657

6 limitations on sace, the other changing rule of the gradually exanding ga isn t discussed here. Considering the convenience of design and manufacture, it is aroriate to choose the linear function rule. Fig. 6 and Fig. 7 are the flo velocity distribution in the Roots bloer at the time hen the ga is rotating near to the exit. The large flo rate is in the ga beteen the to rotors of the backflo, this kind of local high seed backflo cannot be avoided. Near to the exit of the normal ga Roots bloer has a intense backflo, as shon in Fig. 8. And the backflo velocity of the Roots bloer ith gradually exanding ga is obviously decreased, as shon in Fig. 9. The velocities of backflo v(m/s) The rotating angle( ) The gradually exanding gas ith quadratic function changing rule The gradually exanding gas ith linear function changing rule Fig.5 The velocity of backflo changing curves Fig.6 The flo velocity distribution in Roots bloer ith normal ga Fig.7 The flo velocity distribution in Roots bloer ith gradually exanding ga From Fig. 8, e can see that the velocity of the backflo reaches to 24 m/ s, and is much higher than that of the flo field nearby. When the basic inter-sace oens to the exit, the high-ressure airs ill imact the basic inter-sace of airs intensely, accomanied by flo ulsation and noise. And the backflo velocity of the Roots bloer ith gradually exanding ga is only about 3 m/ s, hich is consistent ith the theoretical calculation results, as shon in Tab. 1. Obviously, using the gradually exanding ga can significantly slo don the backflo rate and reduce the air ulsation and noise. The effects are obviously. 658

7 Fig.8 The backflo velocity near the exit of Roots bloer ith normal ga Fig. 9 The backflo velocity near the exit of Roots bloer ith gradually exanding ga Before simulation, setting a monitor oint nearby the outlet of bloer, monitoring the change of flo dynamic ressure. As shon in Fig. 1, the maximum value of the dynamic ressure of the bloer ith gradually exanding ga declines by 42% than that of bloer ith normal ga, and the fluctuation range declines by 62%. The effects of reducing backflo imact and noise are obviously. Conclusions The backflo equation of the gradually exanding ga is established. The otimal value of the ga is solved by the numerical method. And the erformances of the bloer ith the solved arameters are simulated and verified. The results sho that the air ulsation and noise can be reduced effectively by using the gradually exanding ga. In addition, the numerical method is used to solve the otimal value of the gradually exanding ga, hich rovides a theoretical basis for the engineering design. Acknoledgement In this aer, the research as sonsored by the National Natural Science Foundation of China (Project No and No ) and Fujian Provincial Research and Develoment Platform for Industrial Robotic Fundamental Comonents Technology (214H24). 659

8 The dynamic ressure of the outlet(pa) The rotating angle( ) The normal gas Fig.1. The dynamic ressure of the outlet Reference [1] Su Chunmo. Roots bloer and its use [M]. South industrial university ress [2] Xiong Weide. The design and manufacture of integrated lo noise bloer [J]. The research and alication of machinery, 214, 27 (2): [3] Li Zhibao. Roots suerchargers for gas ulsation and reduction methods of research [D]. Central south university master's degree thesis [4] Wang Zhong, Chen Changzheng, Kong Xiangjun. Isolation system design and noise control of Roots bloer room [J]. Advanced materials research. 214: [5] Fu Ming. Circular leaf bloer noise reduction of gradually exanding ga [J]. Wind turbine technology, 212, (1). [6] Liu Yuanquan, Zhang Baohua. Fluid mechanics [M]. Peking university ress, [7] He Zhicheng. The rincile of chemical engineering (Second edition) [M]. Chinese medical science and technology ress [8] Chen Li. Fluid mechanics and um and fan [M]. Higher education ress [9] Yao Ligang, et al. A ne bloer rotor [P]. The invention atent ( ), [1] Long Tianyu, Su Yaxin, et al. Comutational fluid mechanics [M]. Chongqing university ress, [11] Zhang Qiang. Aeroacoustics [M]. Defense industry ress