Rotor Load Analysis Method for Twin Screw Compressors with Considering Gaseous Pressure and Working Temperature

The 14th IFToMM World Congress, Taipei, Taiwan, October 25-30, 2015 DOI Number: 10.6567/IFToMM.14TH.WC.OS3.020 Rotor Load Analysis Method for Twin Screw Compressors with Considering and Working Temperature Ruei-Hung Hsu 1 Bachelor s Program in Precision System Design, Feng Chia University, Taichung, Taiwan u-ren Wu 2,* Department of Mechanical Engineering, National Central University, Taoyuan, Taiwan Wei-Te Wu 3 Department of Biomechatronics Engineering, National Pingtung University of Science and Technology, Pingtung, Taiwan Van-The Tran 4 Department of Mechanical Engineering, National Central University, Taoyuan, Taiwan Abstract: Screw rotors of a twin screw compressor often deform and interfere due to the cyclic gas pressure and working temperature, which lead to noise, vibration and abrasion of the screw rotors. Therefore, this paper proposed a simplified CAE (Computer Aided Engineering) method instead of the past massive and complex CFD (Computational Fluid Dynamics) method to simulate and estimate reasonably the screw rotor load and deformation with considering the gas pressure and working temperature applied on its operating process. The analysis results could provide the possible maximum rotor deformation and cyclic load acting on the rotors for design consideration. Keywords: Screw rotor, Twin screw compressor, CAE, Rotor deformation I. Introduction Nowadays, twin screw compressors have been widely used and performed important roles in the refrigeration and air-condition fields. As shown in Fig. 1, the gaseous refrigerant or air mixed with lubricant oil is usually compressed from the suction end to the discharge end in the compressor chamber, which is formed by a pair of screw rotor and housing wall, to exhaust the high pressure gas into the refrigerating or air-conditioning cycle system. However, the screw rotors generally bear the high gaseous pressure and working temperature that lead to rotor deformation and surface abrasion companying with the noise and vibration of twin screw compressors. Therefore, how to predict the rotor deformation and compensate back to rotor machining to avoid aforementioned problems with an acceptable compressor performance are always study subjects until now. Outlet Demister Angular Contact Ball Screw Rotor Housing Spacing Ring Motor Inlet Cylindrical Roller Fig. 1. Structure of oil-injected twin screw compressor 1 ray0722@gmail.com 2 yurenwu@ncu.edu.tw (*Corresponding Author) 3 weite@mail.npust.edu.tw 4 vanct4hut@gmail.com Geometry of a pair of rotors would greatly affect performance of a twin screw compressor. Stosic et al. [1] published a reference book to explain a novel rotor profile design method and its related applications, such as cutter profile and clearance design, optimization and experiments. Wu and Chi [2] proposed a numerical method to estimate the clearance between two screw rotors with a fixed assembly center distance. Further, Buckney et al. [12] illustrated an estimating method of geometric characteristics of screw compressor using the computer aided design techniques. The geometric characteristics are the basis factors to analyze the compressor performance and load. For rapidly developing and estimating the performance of arbitrary twin screw compressors, some research groups developed the design software packages that include the rotor profile design, geometric characteristics calculation, thermos-fluid performance simulation, load analysis, rotor deflection and clearance estimation and cutter design, et al. More well-known groups are ing et al. ([9], [10] and [13]) and Stosic et al. ([1], [6], [7] and [12]). Noteworthily, Kovacevic et al. [6] successfully established the complete CFD model to couple the solid, thermal and fluid fields to predict and analyze dynamic behaviors of the twin screw compressor. ing et al. [9] verified the compression loads in a twin screw compressor with an elaborated experiment arrangement and applied ANSS software to estimate the rotor deflection. Thermal transfer is the main reason that causes the rotor deformation in its operating process while gaseous pressure is the second one. Many researchers tried to establish models to simulate the thermodynamic process in the twin screw compressor and solve gas leakage and solid interference. Weathers et al. ([4], [8] and [11]) used thermodynamic simulation to define boundary conditions for finite element analyses and determine the rotor-to-housing and rotor-to-rotor clearances through the transient thermal analysis. Fleming et al. ([3], [5]) presented the development, applications and competitive position of twin screw compressors as well as the thermal interaction in a refrigeration twin screw compressor. These references offer useful information on studying the thermal deformation of rotors in this paper. In this paper, a simplified CAE analysis model was established by using finite element analysis software ANSS. This model can be used to estimate the rotor load and deformation with considering the working temperature and gaseous pressure during the compressor process. Only

1 0.9 0.8 0.7 0.6 0.5 the rotor information, working pressure and temperature variations were needed to construct the simplified models. The proposed modelling method considered not only the cyclic variation of the gas s pressure and temperature but also the degree rise of the bearing s freedom and temperature, which may also influence the rotor shaft and tooth deformation. The maximum clearance and interference results on the discharge end of rotor profiles were presented for predicting the abrasion areas on the rotor surfaces. In addition, the axial forces applied on the male and female rotors simulated from the proposed CAE models were also compared with the results calculated from the theoretical program. II. Modelling Method of Rotor Deformation Analysis The overview of the proposed modelling procedure for the rotor deformation analysis of twin screw compressor can be illustrated in Fig. 2. Set Basic Geometry Parameters of Rotors Generate Rotor Profiles Calculate Geometric Characteristics of Compressor Compose Working Sealing Curves Construct 3D CAD Model of Rotors Divide Load Area Using Working Sealing Curves Import Rotor CAD Model to ANSS Generate FE Model of Rotors Axial Forces (N) 6000 5500 700 600 0 Theoretical CAE Simulation 50 100 150 200 250 300 350 Compare Cyclic Axial Forces between Theoretical and CAE Simulation Results Superposition of Rotor Deformations due to and Working Temperature the finite element (FE) models of male and female rotors can be generated and the grid points can be constructed according to the working chamber sealing line. Fifthly, the obtained data of gaseous pressure and temperature variation from theoretical calculation (Hsieh et al. [17]) or experimental measurement in advance is applied to put specified pressure and temperature on to the corresponding load area by the rules. Besides, material properties, feasible constraints and other loads are also set on the analyzed rotor models. Sixthly, the rotor deformation, force, stress and strain distributions at different rotation angle can be obtained by using the finite element method integrated in ANSS. Finally, the axial force variations on male and female rotors can also be obtained from the simulated results and compared to those of theoretical values. Some assumptions are applied in the proposed modelling method: 1. The working conditions of twin screw compressor are independent and steady at every rotation angle of screw rotors. 2. The operating behavior of gas in every working compression chamber of screw compressor is cyclic. 3. The pressure and temperature loads are uniformly applied on the specified regions of the rotors. 4. The screw rotors are ideal elastic bodies. 5. The supports of bearings on rotors are assumed as the rigid bodies and no deformation at their support sections of screw rotors. 6. Only the rotor shafts and bearings are considered in analyzing process, the influence of other machine parts are obviated. The complementary information of the proposed analysis method has been presented as follows: (MPa) Variation Curve during One Cycle Rotation Angle of Male Rotor (degree) Gaseous Temperature (K) 450 400 350 Variation Curve during One Cycle Rotation Angle of Male Rotor (degree) Apply Specified and Temperature on the Corresponding Region Give Materials, Constraints and Other Loads to Rotors Fig. 2. Procedure of proposed rotor deformation analysis method for twin Screw Compressors A. Rotor Models with Different Rotation Angles As shown in Fig. 3, the orientation of rotors should be defined when constructing the rotor CAD models for the purpose to apply the corresponding gaseous pressure and working temperatures on the working chamber at every rotation angle of rotors. The initial contact position of rotor was defined at the interesting point of outer circles of two screw rotors. The rotor analysis models with different its rotation angles are separately constructed, then the different working loads are applied and cascaded as a varying process of rotors. In this study, only one-tooth-meshing process is considered, because the load process is assumptively cyclic and repeat in every tooth groove chamber. Firstly, the basic parameters for generating a pair of conjugated rotor profiles are given and calculated. Then the enveloping theory of gearing is used to generate the rotor profiles and shaft structure. Secondly, the geometric characteristics, including the blow-hole, rotor tip sealing line and contact line, are generated to compose the sealing line of single working chamber of compressor. Thirdly, the 3-D CAD (3-dimensional Computer Aided Design) models of male and female rotors are constructed for dividing load areas using the sealing line of single working chamber. Fourthly, after importing 3-D CAD models to ANSS, Outer Circle of Intersecting Point of Outer Circles Md Outer Circle of Female Rotor Contact Pair Fig. 3. Rotor models with different rotation angles 9 27 45 63

B. Generation of Sealing line of Single Working The rotor profiles used in this paper are patented by Chen [14] and generated according to the enveloping theory of gearing in the textbook proposed by Litvin and Fuentes [15]. The rotor generation is practiced according to the dissertation presented by Wu [16]. As shown in Fig. 4, the sealing line of single working chamber is a 3-D spatial curve and it is composed of the high and low pressure side of blow-holes, the tooth tip sealing lines and the contact line between male and female rotors. It will translate axially when two rotors rotate and mesh to each other. The methods to obtain these geometric characteristics have been explained and presented in many researches, such as references [1], [3], [12], [13], and [16]. Tip Cylinder Tip sealing line Blow-hole Contact line Blow-hole Contact line Tip sealing line D. Load Regions for Applying Both s and Working Temperatures For reflecting approximately the practical condition of twin screw compressors, the CAD models of rotors are divided into several load regions for applying the suction temperature, discharge temperature, and temperatures generated from different bearings and gaseous temperatures in working chambers, respectively, as shown in Fig. 6. Especially, the gaseous temperature is not only different at each pressure load regions of every tooth groove but also changes in the axial direction, so the load regions on rotor screw surfaces are laterally divided into more sections to approximately reflect the practical condition. The divided distance s can be decided according to the required solving accuracy by the following equation: where N 2 ( r p cot p ) s (2) N is the region number for applying loads. 1-2 High Pressure Blow-hole 2-3 Contact Line 3-4 High Pressure Blow-hole 4-5 Tip Sealing Line 5-6 Low Pressure Blow-hole 6-7 Contact Line 7-8 Low Pressure Blow-hole 8-1 Tip Sealing Line 4 1 2 3 Rotating Axis Intersecting Line of Outer Circles 5 Tip Cylinder Rotor Body 6 7 8 Temp. Temp. Higher Temp. Temp. Higher Temp. Temp. Temp. Lower Temp. s Temp. Temp. Fig. 4. Compositions of the single-working-chamber sealing line C. Load Regions for Applying s As shown in Fig. 5, the sealing line of single working chamber obtained in the previous section can be imported into the CAD software to divide the rotor screw surfaces by functions build-in the software. Every screw groove surface can be divided into two regions; one for applying the higher pressure, and another for applying the lower pressure. The axial displacement of each sealing line on every tooth groove can be defined by the following equation: 2 d ( r p cot p ) (1) z where r p is the pitch radius of rotor, p is the pitch helical angle of rotor, and is the tooth number of rotor. Besides, the regions for applying the bearing constraints are also defined in advance. z Temp. Temp. s Lower Temp. Fig. 6. Load regions for applying both gaseous pressures and working temperatures III. Load Analysis Models and Load Rules As shown in Fig. 7, considering the bearing supports, the displacement constraint in the -axis direction is given on the shaft end surface of the discharge side, and the displacement constraints in the - and -axes are given on the shaft cylindrical surfaces of both suction and discharge ends. The suction and discharge pressures were applied on the shaft cylindrical surfaces and rotor end surfaces on the respective ends. Similarly, the loads and constraints are applied on the female rotor with the relative rotation angle as shown in Fig. 8. The working pressure variation in single compression chamber of twin screw compressor is shown in Fig. 9. The gaseous pressures can be defined by the following rules: Higher Pressure Support Higher Pressure Support Support Lower Pressure Support Lower Pressure Support Fig. 5. Load regions for applying gaseous pressures q ( q 1), q 1, 2,, 2z (3) P P( ), q 1, 2,, 2z (4) q q where is the initial rotation angle of male rotor (see Fig. 3), is an increment of rotation angle, q is the specified rotation angle of corresponding gaseous pressure, P is the corresponding gaseous pressure applied on the q specified load region.

Pressure 9 8 7 6 5 Pressure Constraint 10 4 3 2 1 Fig. 7. Load analysis model of male rotor for applying gaseous pressure Pressure 5 6 7 8 9 Pressure Constraint 1 2 3 4 10 Fig. 8. Load analysis model of female rotor for applying gaseous pressure 1, P (MPa) 0.9 0.8 0.7 0.6 0.5 generated from bearing friction and compression process, the bearing temperatures estimated from Eqs. (5)~(7) of each bearings are applied on the corresponding shaft cylindrical sections, the suction and discharge temperature are applied on the rotor ends and shaft regions without bearing supports. Similarly, the loads and constraints are also applied on the female rotor with the relative rotation angle, as shown in Fig. 11. The load analysis model of male rotor for applying working temperatures as shown in Fig. 11. The gaseous temperatures can be calculated on the rotor surface according to the following rules: where q ( q 1), q 1~ N (8) tq 0.5 t( q 1) t( q ), q 1 ~ N (9) t q is the corresponding gaseous temperature for applying on the specified load region. Constraint Temperatures B14 B15 B16 Temperature B13 20 19 18 17 16 15 14 7 6 5 4 3 2 1 Temperature B12 Temperatures B11 Fig. 10 Load analysis model of male rotor for applying working temperatures Rotation Angle of, θ (degree) Fig. 9. Working pressure variation in single compression chamber of twin screw compressor Except the pressure loads on rotors, the heat convection and working temperatures should be considered in the rotor deformation analysis, wherein the contact friction and oil drag of bearings are important reasons of heat generation. According to the reference [18], the total friction torque M, the friction power loss N R, and the working temperature of bearing can be estimated as following equations: t A M M M M (5) NR t ish rs rr sj drag 4 1.05 10 Mn (6) N R A ti (7) WS where ish is the shear reduction coefficient, is the oil shortage coefficient, M rr is rolling friction torque, M sj is the sliding friction torque, M drag is the friction torque produced by the oil drag, n is the shaft rotation speed, W S is the cooling coefficient and, t I is the initial temperature of bearing. As shown in Fig. 10, considering the heat would be rs Temperature Temperature 14 15 16 17 18 19 20 Temperatures and Constraints B21 and Constraints 1 2 3 4 5 6 7 B22 Temperatures and Constraints B23 B24 B25 Fig. 11 Load analysis model of female rotor for applying working temperatures, t (K) 450 400 350 {{{{{{{{{{{{{{{{{{{{ Rotation Angle of, θ (degree) Fig. 12 Working temperature variation in single compression chamber of twin screw compressor Additionally, for considering the contact between male and female rotors, the male rotor is applied a fixed Constraint

torque M d on the motor driving section, as shown in Fig. 3, the female rotor is limited its axial rotation and the contact pair is set between two rotors in the ANSS model. The models and results obtained from the thermal analysis in the temperature field are transformed into the structure field with forces and pressures to obtain the coupling rotor deformation results. IV. Examples This section carried out a practical modelling example according to the aforementioned method. The geometric parameters of analyzed screw rotors are listed in Table 1. The ANSS model for applying working temperatures is firstly established to carry out the thermal analysis. The element types of Solid90 and Solid186 are selected to mesh the rotor models for the thermal analysis and for the structural analysis, respectively. And the other settings can be seen in Table 2. The working temperatures of bearings are estimated as listed in Table 3. Table 1 Geometric parameters of analyzed screw rotors Items Material S45C S45C Tooth Number (teeth) 5 6 Screw Length (mm) 162.453 162.453 Lead (mm) 228.128 273.754 Wrap Angle (degree) 258 215 Outer Diameter(mm) 173.840 138.990 Pitch Diameter (mm) 111.818 134.182 Table 2 Settings of thermal and structural analysis Items Values Element Type in Thermal Analysis Solid90 Element Type in Structural Analysis Solid186 oung s Modulus (Pa) 207 10 9 Poisson s Ratio 0.3 Density of Rotors (g/cm 3 ) 7.85 Mean Pressure (Pa) 344870 Mean Pressure (Pa) 1017170 Thermal Conductivity (W/m-K) 49.8 Expansion Coefficient ( C) 1.36 10-5 2 Coefficient of Heat Convection (W/m -K) 20 Initial Temperature ( C) 23 Temperature ( C) 5.9 Temperature ( C) 70 Rotation Speed (rpm) 3600 66.025 C on the female rotor tooth tip on the discharge end, respectively. The analysis results show that the temperature is higher gradually from the tooth root on the suction end to the tooth tip on the discharge end in every working chamber and the temperature on bearing will be the main heat source. t min = 94.493 C 108.852 C t max = 143.012 C t max = 67.702 C t min = 47.288 C 66.025 C Fig. 13 Temperature distribution on screw rotors Furthermore, at zero rotation angle, the maximum deflections are 0.127 mm at the rotor tooth tip on the discharge end for the male rotor and 0.061 mm for the female rotor, respectively. The deflection is larger radially from the rotor shaft to the rotor tooth tip. Especially, the deformation closed to the discharge chambers are much larger. These analysis results mentioned above are much closed to the thermal deformation results presented by Hsieh et al. [19]. Table 3 Estimated working temperatures of bearings ID Working Temperatures ( C) B11 143.012 B12 123.489 B13 113.02 B14 115.73 B21 58.5635 B22 67.0952 B23~B25 50.2828 As shown in Fig. 13, it is one of the working situations at zero rotation angle of rotors, the maximum temperature is 143.012 C on the bearing support region of discharge end of male rotor and 67.702 C on the bearing support region of discharge end of female rotor, respectively. Besides, there are a maximum temperature of 108.852 C on the male rotor tooth tip and a maximum temperature of Fig. 14 Distribution of displacement of screw rotors The maximum deflections of male rotor and female rotor at different rotation angles during one-tooth-spacing angle are shown in Fig. 15 and Fig. 16, where the values are quite different with the change of rotation angle of rotors. The varying range of maximum deflection at tooth tip is from 0.1247 mm to 0.127 mm for the male rotor and 0.0574 mm to 0.0614 mm for the female rotor, respectively. These results can be provided the design consideration of clearance between the rotor tooth tip and the compressor

housing and between two rotors. There may be given clearance of (0.1821~0.1911) mm or higher than this between two rotors to avoid the rotor interference. Maximum Deflection (mm) 0.1270 0.1265 0.1260 0.1255 (0.1273 mm) 10 20 30 40 50 60 63 (0.1247 mm) Fig. 15 Maximum deflections on male rotor at different rotation angles Maximum Deflection (mm) 0.061 0.060 0.059 9 (6.142 10-2 mm) 10 20 30 40 50 60 63 (5.738 10-2 mm) Fig. 16 Maximum deflections on female rotor at different rotation angles Additionally, the radial displacements of shaft axes of male and female rotors at different rotation angles are shown in Fig. 17(a) and Fig. 17(c). Wherein a maximum radial displacement about 40 at the discharge section of male rotor and about 9 at the discharge section of female rotor are produced when two rotors mesh at the location of zero rotation angle, and the deflection angle at this moment is about 35 for the male rotor and about 2 for the female rotor, as shown in Fig. 17(b) and Fig. 17(d). These values are much smaller than the rotor deflections but they may cause the non-uniform contact between rotors and induce the vibration of twin screw compressors. Radial Displacement, (µm) Radial Displacement, (µm) 40 30 20 10 8 6 4 2-2 100 200 300 400 500 (a) 150 200 250 300 350 400 450 (c) m m Deflection Angle, (degree) Deflection Angle, (degree) 50-50 50-50 100 200 300 400 500 (b) 150 200 250 300 350 400 450 (d) Fig. 17 Variations of radial deflections and deflection angles of male and female rotors with rotor lengths at different rotation angles operating male rotor and female rotor can be analyzed by using the proposed analysis models and estimated from the theoretical method proposed by Wu and Chi [20]. As shown in Fig. 18, the varying axial forces obtained from the proposed CAE simulation method is much close to that of result from theoretical calculation. When the male rotor rotates one cycle, the axial force acting on the male rotor will generate five fluctuations approximately from 5700 N to 6000 N. And when the male rotor rotates one cycle, the axial force acting on the female rotor will generate six fluctuations approximately from 600 N to 720 N, as shown in Fig. 19. Axial Force on (N) 6000 5500 5000 0 50 100 150 200 250 300 350 Theoretical CAE Simulation Fig. 18 Comparison of varying axial force acting on male rotor between theoretical and CAE results Axial Force on (N) 700 600 500 400 300 0 50 100 150 200 250 300 350 Theoretical CAE Simulation Fig. 19 Comparison of varying axial force acting on female rotor between theoretical and CAE results V. Conclusions This paper proposed a modelling method for analyzing the rotor deformation of twin screw compressor by using the CAE method instead of CFD method with considering the working temperature and gaseous pressure as well as the contact force between two screw rotors. The thermal-structural coupling method are applied to analyze the rotor deformations and loads. The analysis simulation results reveal that the proposed model may not only provide the clearance design consideration but also obtain the varying forces acting on two rotors. The trend of temperature distribution and rotor deflection are similar with the theoretical publication, and the varying axial forces acting on male and female rotors are much closed to the theoretical calculation results. Acknowledgement The authors would like to appreciate the financial supports of Ministry of Science and Technology Project (MOST 103-2221-E-008-120) and HANBELL Company in Taiwan. Moreover, the varying axial forces acting on the

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