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aDepartment of Air Conditioning and frigeration, National Taipei University of Technology, Taipei, bDepartment of Mechanical Engineering, Taiwan 106, ROC," National Taiwan University, Taipei, Taiwan 106, ROC; CFu Sheng Industrial Co. Ltd., No. 60, Sec. 2, Kwang-Fu Rd., Sanchung, Taiwan 241, ROC (ceived 15 October 1998," In jiânal form 6 August 1999) A theotical model is proposed in this paper in order to study the performance of oil-less and oil-injected . Based on this model, a computer simulation program is developed and the effects of diffent design parameters including rotor profile, geometric clearance, oil-injected angle, oil temperatu, oil flow rate, built-in volume ratio and other operation conditions on the performance of a investigated. The simulation program gives us output variables such as specific power, compssion ratio, compssion efficiency, volumetric efficiency, and discharge temperatu. Some of the above sults a then compad with experimentally measud data and good agement is found between the simulation sults and the measud data. Keywords. Twin scw compssor, Performance simulation, Compssion ratio, Specific power, Built-in volume ratio, Compssion efficiency INTRODUCTION The twin scw air compssor is a positive displacement compssor. It utilizes the contial variations of the space formed between rotor grooves and case of the compssor to compss gas. In the early stage of development, the compssion process of a twin scw compssor was usually estimated empirically. The drawbacks of the empirical process we time consuming, difficult to attain an optimal performance, and quiring al model tests. Many mathematical models for the performance analysis of a twin scw compssor have been proposed. Bein and Hamilton (1982) psented a theotical model for an oil-injected scw air compssor. They utilized a polytropic compssion process model to find the value of the polytropic constant that assud the highest consistency of the model sults as well as the experimental data. SAngfors (1982) studied both oil-less and oilinjected twin scw compssors. By taking into account the dynamic loss, leakage and heat transfer, he used R12 and air as the working fluid and Corsponding author. Tel.: 886-2-2771-2171, ext. 3515. Fax: 886-2-2731-4919. E-mail: fl0911 @ntut.edu.tw. developed a merical method to pdict variations of internal state of the compssor. Singh and Bowman (1986) have discussed the effects of parameters such as gear tooth form and gear tooth mber in a pair of female and male rotors on the performance of an oil-injected compssor. To make a model adaptable to assorted kinds of fluid, Xiao et al. (1986) has psented a theotical model taking into account the al gas effect, leakage, heat transfer and flow sistance of the discharged gas. Geometric profiles such as length of the gas-seal line and orifice aa of the discharge valve we also consided in the model. cently, Fujiwara and Osada (1995) have applied both merical method and experimental measument to study the performance of a twin scw air compssor. To evaluate the performance of , an effort has been made to develop a general theotical model accompanying with its computer simulation program in the psent study. The theotical model takes into consideration most merits of the above mentioned papers. In particular, the effects of geometric clearance, oilor water-injected angle, oil or water temperatu, gas leakage, heat transfer between oil and air, and mass transfer between water and air a consided. Aside from the theotical study, experimental data we measud in a test laboratory in order to verify the simulation program with measud data. Once the theotical model is justified, the optimum operation condition of a twin scw air compssor which is helpful in design can be well masted. rotational angle of the rotor when the motion is in steady state. It also indicates that when the compssion chambers rotate to an angle of the same deges, they all have the same temperatu, pssu, mass, etc. Based on this assumption, the following derivations a introduced within a compssion chamber. Governing Equations Since the intrinsic state and property of the two fluids contained in the control volume a not identical, it is quid to derive the governing equations for the gas and oil spectively. With garding to the gas, the following equations a derived by the consideration of conservation of mass and energy: d0m tom \k=l /4/gik /4/gok k=l cOmmgCv, g Tg nl tom d0m (/4/1 /4/li) -+- Z igik/4/gik -, thgik k-1 .V rn / pl mg mg OP m/01 Tg OP v k=l rhgok MATHEMATICAL MODEL hA(Tg T1) }, (2) As shown in Fig. 1, the compssion chamber of a twin scw air compssor is a space encompassed by the male rotor groove, female rotor groove and the case. In general, the a several compssion chambers between a pair of male and female rotors. In a particular compssion chamber, the state of gas is lated only to the whe nl and n2 psent the total mber of inlet and outlet channels including suction valve orifice, discharge valve orifice and leaking path. These channels allow gas to enter into and discharge from the control volume. With garding to the oil, the following equations can be derived based on conservation of Position of Oil-injection Case Clearance Female Rotor FIGU Male Rotor Geometry of a twin scw air compssor. mass and energy: dm d0m d0m flowing speed of gas, the pssu loss during gas flowing is negligible. Thefo, the mass flow rate of the suction gas can be consided as rhAviV/2p(Ps- P), in which Avi denotes aa of the suction orifice. Discharge Process whe n3 and n4 psent the total mber of inlet and outlet channels such as oil-injected orifice, discharge valve orifice and leaking path. Suction Process In the suction process, since aa of the suction valve orifice is large enough to slow down the The diffence between the gas pssu in a compssion chamber and the system back pssu is the major pssu loss when gas passes through an orifice. Thefo, the flow rate in the discharge process can be calculated as rh CDAvo V/2pl P Pdl, in which Avo indicates the orifice aa of a discharge valve. The total flow rate is the summation of all flow rates of the gas passing through various orifices with diffent ratios of opening aas. As gas and oil a discharged simultaneously, the mass flow rates of gas and oil, spectively, a -rh the coefficient of flow rate, /3 is the specific heat ratio, and Rm is the vised gas constant. The last two variables can be calculated as /4/g- + qSâ rh, >- Pd, (4) /3 Cp qt_ (Cv,1 P _ Pd, P<Pd, Heat Transfer between Oil and Gas Heat transfer is another vital factor that affects the performance of an air compssor. It is very difficult to determine the heat transfer coefficient and heat transfer aa between oil and gas. To simplify the analysis, the heat transfer between oil and gas is classified into two categories in the psent study. The first category assumes that the compssion room is in oil-injected process, and the heat transfer coefficient is the coefficient between the sphe and the flowing fluid. Based on this assumption, many approximate formulas can be used to describe such a heat transfer mode. Among them, the formula proposed by Ranz in 1952 is adopted in the psent paper. It says 05 indicates the mass, i.e., b ml/mg. Gas Leakage Model ratio of oil mass to gas Gas leakage is one of the vital factors that affect the performance of an air compssor. The calculation of leakage also affects the simulation accuracy of the compssor. In a twin scw air compssor, the a four major leaking paths, namely blowhole, clearance between two rotors, clearance between rotor tip and case, and clearance between end plate and case. In general, it is very difficult to obtain the pcise amount of leakage from experiments. The convergent nozzle model is thefo adopted hein to account for the leakage. It states that rh-- CDAcPh 2.0 + 0.6 SPr33. The heat transfer aa is the sum of surface aa of all sphes and can be calculated from A in which the size of oil drop diameter is calculated by using the mean Sauter diameter NprrDZp, fl--1RmTh when formula, /3+1 <hh P< (5) Dp 0.0366\----/ Np 6/1/1 p, TrD p and The mber of oil drops, Np, is calculated as CD A P h /R/3 Th / 2r- ( ) (/3+1)/(/3-1) whenO<hh in which (6) P1 and Ph a the lower pssu and higher pssu of the adjacent grooves, CD is The second category of heat transfer between oil and gas assumes that the compssion chamber is not in an oil-injected process. Under this circumstance, most of the oil is attached to the rotors as well as the case. The heat transfer mode in this (9) (lO) category is then diffent from the pviously described one. Fujiwara and Osada derived the lationship between the heat transfer coefficient and the volumetric efficiency in the suction process. The formula can be written as of a twin scw air compssor. Experimental measument work (Chen, 1996) has also been performed to test the accuracy of the analytical model. Some of the sults a shown in the psent section. whe 0.51 â74 (11) merical Algorithm Dzm (12) As indicated by Eqs. (1)-(3) as well as equations for the suction and discharge processes, the governing equations of a twin scw air compssor comprise four nonlinear equations. It is almost impossible to solve them analytically and merical solution is thefo needed. In the psent study, a fourth order Runge-Kutta method is employed. The flow chart of the merical algorithm is shown in Fig. 2. Start PERFORMANCE SIMULATION AND MODEL TESTING Based on the above theotical analysis, a computer program is written to simulate the performance ////Rotor geometric data & // working fluid properties Inputdata Calculate geometrical parameters Calculate suction, compssion and discharge processes Yes Calculate discharge temperatu Calculate volumetric efficiency Calculate compssion efficiency Lutput FIGU 2 Flow chart of merical simulation. Model Testing Some of the theotically obtained sults such as discharge temperatu, volumetric efficiency, compssion efficiency and specific power a compad with experimentally measud data (Xiao et al., 1986). Among these data, the volumetric efficiency is defined as the ratio of al amount of discharged gas to the groove volume in the twin scw compssor. The compssion efficiency is defined as the ratio of theotical power consumption to measud shaft horsepower. The specific power is defined as the ratio of the shaft horsepower to the calculated amount of discharged gas. A typical sult is shown in Fig. 3. It is found that the volumetric efficiency calculated theotically is very close to the experimental sult. The isothermal efficiency, however, is found to have certain diffence between the theotical and experimental sults. The diffence becomes significant when the compssor is running at high rotor speed. It may be attributed to the dynamic loss of friction that we did not consider in the theotical derivation. From viscous fluid dynamics viewpoint, the consumed energy of the dynamic loss should hold the lationship in which #m is the mean viscosity of the oil and gas mixtu, Vt is the velocity of the tooth tip, c is Fluid: R-729(air) Suc. Pss.: 1.033 bar Suc. Temp.: 303 K Dis. Pss.: 8 bar 8O az I< Simulationsults Experiment dat | )a 6.0-s.o4.0 10 20 Tip Speed (m/s) FIGU 3 Comparison between calculated and experimental sults. Flui d" R- 72 9(air) Suc. Pss." 1.033 bar Suc. Temp. 303 K Dis. Pss.â 4 bar 4OO 0 0.1 IE+0 5.0E-5 Volume (m 3) 1.0E-4 1.5E-4 FIGU 4 Effect of clearance on the performance of an oil-less twin scw air compssor. the clearance between the tooth tip and the case, and Ac is the characteristic aa. When the dynamic loss is consided in the psent model, the accuracy of the theotically pdicted compssion efficiency incases gatly as that shown in the third frame of Fig. 3. Finally, from the last frame of the same figu, it is found that the theotical values of the specific power a also very close to the experimental data. The accuracy of the above theotical model is thefo justified. Performance Analysis of an Oil-Less Compssor After justification of the theotical model, the same analysis is now applied to an oil-less twin scw air compssor in order to study the influence of diffent design parameters on the performance of the compssor. Among the sults, Fig. 4 indicates the influence of gear tooth clearance. It shows that in case that clearance is made wider, the external gas leakage in addition to the internal gas leakage occurs, and the incased amount of leakage makes the pssu distribution to be lower than that of the isentropic process. Figu 5 indicates the influence of rotor speed on the performance of the compssor. It shows that, when the rotor speed incases, the gas leakage decases and the sulting pssu distribution tends to closer to that of the isentropic process. Under this circumstance, the heating effect diminishes and the amount of gas leakage decases. As the heating effect and gas leakage a irvocable factors in the compssion process, it implies that the incase of Fluid R-729(air) Suc. Pss." 1.033 bar Sue. Temp. 303 K Dis. Pss. 4 bar "- â", 1",, 5.0E-5 Volume (m 3) e/D=0.0007 Vt=80m/s e/D=0.0035 Vt=80m/s e/D=0.0007 Vt=120m/s /D=0"0035 vt=120m/â 0.0E+0 1.0E-4 1.5E-4 FIGU 5 Effect of clearance and rotor speed on the performance of an oil-less twin scw air compssor. TABLE compssor Performance of an oil-less twin scw air Performance Analysis of an Oil-Injected Ratio of clearance to male rotor diameter (e/D) Compssor In an oil-injected twin scw air compssor, oil plays the role of cooling the compssed gas. It has gat influence on the performance of the compssor. Thefo, parameters such as oil flow rate, oil-injected temperatu and oil-injected angle have to be consided in the theotical analysis. Based on the pviously proposed model and under the assumption that the rotor speed, suction and discharge pssu, gas temperatu and geometric factors a all kept the same as befo. Figu 6 shows that the volumetric efficiency becomes higher as the injected temperatu becomes lower. This may be attributed to the duction of leakage driving force due to lower pssu distribution in the compssion process. As for the compssion Rotor tip speed (m/s) Volumetric efficiency (%) Compssion efficiency (%) Specific power (kW min/m 3) rotor speed can duce the effect of irvocable lectors that make the compssion process to be closer to that of the isotropic process. To summarize the sult, Table I is constructed and the efficiency as well as the energy consumption data a shown thein. Fluid" R-729 (air) Suc. Pss." 1.033 bar Suc. Temp." 303 K Dis Pss." 8 bar 9o Oil Inlet Temperatu (K) FIGU 6 Effect of oil inlet temperatu on the performance of an oil-injected twin scw air compssor. efficiency, it is found that higher efficiency is generally obtained as the injected temperatu becomes higher. The tendency, however, verses after the temperatu aches a certain value. The ason is that lower oil temperatu in general decases the work done by the compssed gas. However, it incases the dynamic loss due to higher oil viscosity. An optirnal oil-injected temperatu may thefo exist as that shown in Fig. 6. Similar situation occurs with gard to the specific power. The influence of oil flow rate on the compssor performance is shown in Fig. 7. It indicates that both the volumetric and compssion efficiency incases as the amount of injected oil incases. The specific power, however, decases as the amount of oil incases. The influence of oilinjected angle on the performance of the compssor is shown in Fig. 8. It indicates that the volumetric efficiency decases but the specific power incases as the oil-injected angle is moved closer to the discharge side. With gard to the compssion efficiency, the may be an optimal injected angle as shown in the figu. For twin scw compssors with fixed suction pssu, the performance of compssion is determined by built-in volume ratio and system discharge pssu. The built-in volume ratio of scw compssors is defined as the ratio of volume of the thad at the start of compssion process to the W.S. LEE Fluid" R-729 (air) Suc. Pss." 1.033 bar Suc. Temp." 303 K Dis. Pss. 8 bar et al. 7O Oil Injection Quantity Ratio (%) FIGU 7 Effect of oil-injection quantity on the performance of an oil-injected twin scw air compssor. volume of the same thad when it first begins to open the discharge port. For a fixed built-in volume ratio compssor, a mismatch between the internal and system discharge pssus may cause overcompssion or undercompssion with a sulting decase in capacity and an incase in power input. Overcompssion occurs when the internal pssu in the compssion chamber aches the system discharge pssu befo the compssed air arrives at the discharge port. On the other hand, undercompssion occurs when the internal pssu aches the discharge port prior to achieving system discharge pssu. The lationship between the compssion efficiency and compssion ratio of various built-in volume ratios is shown in Fig. 9. Four discharge temperatus we consided in the figu, in which the compssion ratio is defined as the ratio of the expected suction pssu and to the expected discharge pssu. From the figu it is calculated that, with a fixed built-in volume ratio, the compssion efficiency incases along with the incase of the compssion ratio. The compssor may have an optimal compssion ratio that gives the operation the maximum efficiency. Further incase of the compssion ratio will then decase the compssion efficiency. To be mo pcise, consider the case of Fig. 9(a) that operates at the discharge temperatu of 65C. If the built-in volume ratio is set to be 3, the optimal compssion ratio is found to be 4.0. In general, the compssors should be designed to match the above-mentioned optimal condition as possible. When the compssion ratio is selected to be lower than the optimum point, the compssor is in undercompssion condition. On the other hand, Fluid" R-729 (air) Suc. Pss." 1.033 bar Suc. Temp." 303 K Dis. Pss." 8 bar oâ" Oil Injection Angle (dege) FIGU 8 Effect of oil-injection angle on the performance of an oil-injected twin scw air compssor. when the compssion ratio is selected to be higher than the optimum point, the compssor then operated in overcompssion condition. To summarize the sult shown in Fig. 9, it is observed that when the compssion ratio is low, better efficiency is usually obtained for lower built-in volume ratios. On the contrary, when the compssion ratio is high, better efficiency is obtained at higher built-in volume ratios. Figu 10 demonstrates the lationship between the optimal compssion ratio and the built-in volume ratio for diffent discharge temperatus. The curves in this figu a plotted based on the optimal design conditions obtained in Fig. 9. The sult shows that the optimal compssion ratio is linearly proportional to the built-in volume ratio. Although the slopes of these curves a diffent, their variation is very small. It can be concluded that the optimal compssion ratio is insensitive to the discharge temperatus at the temperatus range of 65-80C. CONCLUSIONS The following conclusions can be drawn from the psent study: (1) The leakage caused by the clearance can be classified into two kinds, namely external and internal leakage, spectively. When the external leakage is higher than the internal leakage, the pssu distribution is lower than that of the isentropic process. Conversely, when the () (b) /"/â // [/ Tdis=65ocl/ / / vi=2 vi= Tdis=70C vi= j 10.0 40.00 2.0 4.0 6.0 8.0 Compssion Ratio / vi=4/ vi=2 vi=3 vi=4 | | vi=5J Compssion Ratio (d) ," ," ,"/ ,,â" ,/ ,/â/ _/" // // o.oo Iâ1 Tdis=75C vi=3 60.00 --,/ -// Tdis=80C vi=31 vi-4 | vi=5 / 8.0 6.0 Compssion Ratio 8.0 6.0 Compssion Ratio FIGU 9 Effect of compssion ratio on the percent efficiency at various built-in volume ratios; Td for (a) 65C, (b) 70C, (c) 75C, and (d) 80C. external leakage is lower than the internal leakage, the pssu distribution will be higher than that of the isentropic process. Higher rotor speed can also duce the leakage and make the compssion process move toward the isentropic process. (2) The volumetric efficiency of a scw compssor can be improved by raising the injected oil temperatu and making the discharge temperatu lower. As for the energy consumption, the may exist an optimal injected temperatu that sults in the highest compssion efficiency and the lowest specific power. (3) The oil-injected angle is irlevant to the discharge temperatu, but it has gat influence on the distribution of pssu and temperatu. Better volumetric efficiency can usually be achieved by the selection of an earlier oilinjected angle. (4) For a fixed volume ratio, the exists an optimal compssion ratio that sult in the maximum compssion efficiency. For a fixed compssion ratio, depending on the compssion ratio the maximum compssion efficiency may be higher or lower as the build-in volume ratio becomes higher. Optimal Disdrge Pssu Toffs=65 Tdi70 Tdi75 Tdis=80 Built-in Volume Ratio FIGU 10 lation of optimal compssion ratio and built-in volume ratio at various discharge temperatus. NOMENCLATU Aa (m 2) Characteristic aa (m 2) Valve aa (m 2) Coefficient of flow rate Specific heat at constant pssu P Pr R Pssu (Pa) Prandtl mber Universal gas constant (KJ/kg K) vised gas constant (KJ/kg K) ynolds mber Rotational ynolds mber Ac Av CD Cp Rm T V Op Ed,1 EH (KJ/kg K) Specific heat at constant volume (KJ/kg K) Mean Sauter diameter Dynamic loss (KJ) Heat transfer coefficient (W/m 2 K) Specific enthalpy (J/kg) Mass (kg) mber of oil drops sselt mber x,y,z Temperatu (K) Volume (m 3) Rotor tip velocity (m/s) Flow rate (m/s) System coordinates Gek Symbols Modified specific heat rate Clearance (m) Ratio of oil mass to gas mass Efficiency fences Bein, T.W. and Hamilton, J.F., 1982, Computer modeling of an oil flooded single scw air compssor, Proc. Int. Compssor Engineering Conf., Purdue, USA, pp. 127-134. Chen, S.L. and Wu, W.F., 1996, Performance Simulation and Experimental Testing of Twin Scw Compssors, Technical port, Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan. Fujiwara, M. and Osada, Y., 1995, Performance analysis of an oil-injected scw compssor and its application, Int. J. frig., 18(4), 220-227. Sngfors, B., 1982, Analytical model of helical scw machines for analysis and performance pdiction, Proc. Int. Compssor Engineering Conf., Purdue, USA, pp. 135-139. Singh, P.J. and Bowman, J.L., 1986, Effect of design parameters on oil-flooded scw compssor performance, Proc. Int. Compssor Engineering Conf., Purdue, USA, pp. 71-88. Xiao, D., Xiong, Z. and Yu, Y., 1986, The computer simulation of oil-flooded frigeration twin-scw compssors, Proc. Int. Compssor Engineering Conf, Purdue, USA, pp. 349-361. Pssu angle (rad) Dynamic viscosity (kg/m s) Density (kg/m 3) Surface tension (N/m) Rotor speed (tad/s) Superscript Physical quantity per unit time Subscript d f g m s Discharge side Female rotor Gas Oil, Low pssu side Male rotor, Mean value Suction side

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