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International Journal of Rotating Machinery2001, Vol. 7, No. l, pp. 65-78

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Numerical Simulation and Performance Analysis ofTwin Screw Air Compressors

W.S. LEE a’*, R.H. MAb, S.L. CHENb, W.F. WUb and H.W. HSIA

aDepartment of Air Conditioning and Refrigeration, National Taipei University of Technology, Taipei, Taiwan 106, ROC,"bDepartment of Mechanical Engineering, National Taiwan University, Taipei, Taiwan 106, ROC;

CFu Sheng Industrial Co. Ltd., No. 60, Sec. 2, Kwang-Fu Rd., Sanchung, Taiwan 241, ROC

(Received 15 October 1998," In ji’nalform 6 August 1999)

A theoretical model is proposed in this paper in order to study the performance of oil-less andoil-injected twin screw air compressors. Based on this model, a computer simulation programis developed and the effects of different design parameters including rotor profile, geometricclearance, oil-injected angle, oil temperature, oil flow rate, built-in volume ratio and otheroperation conditions on the performance of twin screw air compressors are investigated. Thesimulation program gives us output variables such as specific power, compression ratio,compression efficiency, volumetric efficiency, and discharge temperature. Some of the aboveresults are then compared with experimentally measured data and good agreement is foundbetween the simulation results and the measured data.

Keywords. Twin screw compressor, Performance simulation, Compression ratio,Specific power, Built-in volume ratio, Compression efficiency

INTRODUCTION

The twin screw air compressor is a positive dis-placement compressor. It utilizes the continualvariations of the space formed between rotorgrooves and case of the compressor to compressgas. In the early stage ofdevelopment, the compres-sion process of a twin screw compressor was usuallyestimated empirically. The drawbacks of the em-

pirical process were time consuming, difficult toattain an optimal performance, and requiring realmodel tests.

Many mathematical models for the performanceanalysis of a twin screw compressor have beenproposed. Bein and Hamilton (1982) presented atheoretical model for an oil-injected screw air com-pressor. They utilized a polytropic compressionprocess model to find the value of the polytropicconstant that assured the highest consistency ofthe model results as well as the experimentaldata. SAngfors (1982) studied both oil-less and oil-injected twin screw compressors. By taking intoaccount the dynamic loss, leakage and heat trans-

fer, he used R12 and air as the working fluid and

Corresponding author. Tel.: 886-2-2771-2171, ext. 3515. Fax: 886-2-2731-4919. E-mail: fl0911 @ntut.edu.tw.

65

66 W.S. LEE et al.

developed a numerical method to predict varia-tions of internal state of the compressor. Singhand Bowman (1986) have discussed the effects ofparameters such as gear tooth form and gear toothnumber in a pair of female and male rotors on theperformance of an oil-injected compressor. Tomake a model adaptable to assorted kinds offluid, Xiao et al. (1986) has presented a theoreti-cal model taking into account the real gas effect,leakage, heat transfer and flow resistance of thedischarged gas. Geometric profiles such as lengthof the gas-seal line and orifice area of the dis-charge valve were also considered in the model.Recently, Fujiwara and Osada (1995) have appliedboth numerical method and experimental measure-ment to study the performance of a twin screw aircompressor.To evaluate the performance of twin screw air

compressors, an effort has been made to developa general theoretical model accompanying with itscomputer simulation program in the present study.The theoretical model takes into considerationmost merits of the above mentioned papers. Inparticular, the effects of geometric clearance, oil-or water-injected angle, oil or water temperature,gas leakage, heat transfer between oil and air, andmass transfer between water and air are consid-ered. Aside from the theoretical study, experimen-tal data were measured in a test laboratory in orderto verify the simulation program with measureddata. Once the theoretical model is justified, theoptimum operation condition of a twin screw air

compressor which is helpful in design can be wellmastered.

MATHEMATICAL MODEL

As shown in Fig. 1, the compression chamber ofa twin screw air compressor is a space encom-passed by the male rotor groove, female rotorgroove and the case. In general, there are severalcompression chambers between a pair of maleand female rotors. In a particular compressionchamber, the state of gas is related only to the

rotational angle of the rotor when the motion isin steady state. It also indicates that when thecompression chambers rotate to an angle of thesame degrees, they all have the same temperature,pressure, mass, etc. Based on this assumption, thefollowing derivations are introduced within a com-

pression chamber.

Governing Equations

Since the intrinsic state and property of the twofluids contained in the control volume are notidentical, it is required to derive the governingequations for the gas and oil respectively. With

regarding to the gas, the following equations are

derived by the consideration of conservation ofmass and energy:

/4/gik /4/gokd0m tom \k=l k=l

cOmmgCv,gTg

vtom d0m

nl

(/4/1 /4/li) -+- Z igik/4/gik

.V rn/pl OP -, thgikmg v k-1

V m/01 TgOP

rhgokmg v k=l

hA(Tg T1) }, (2)

where nl and n2 represent the total number ofinlet and outlet channels including suction valveorifice, discharge valve orifice and leaking path.These channels allow gas to enter into and dis-charge from the control volume.With regarding to the oil, the following equa-

tions can be derived based on conservation of

TWIN SCREW AIR COMPRESSORS

Position ofOil-injection

Case

67

Clearance

Female Rotor Male Rotor

FIGURE Geometry of a twin screw air compressor.

mass and energy:

dm

d0m

d0m

where n3 and n4 represent the total number of inletand outlet channels such as oil-injected orifice, dis-charge valve orifice and leaking path.

Suction Process

In the suction process, since area of the suctionvalve orifice is large enough to slow down the

flowing speed of gas, the pressure loss during gasflowing is negligible. Therefore, the mass flow rateof the suction gas can be considered as rh-AviV/2p(Ps- P), in which Avi denotes area of thesuction orifice.

Discharge Process

The difference between the gas pressure in a com-pression chamber and the system back pressure isthe major pressure loss when gas passes throughan orifice. Therefore, the flow rate in the dis-charge process can be calculated as rh

CDAvo V/2plP Pdl, in which Avo indicates the ori-fice area of a discharge valve. The total flow rate isthe summation of all flow rates of the gas passingthrough various orifices with different ratios ofopening areas.

68 W.S. LEE et al.

As gas and oil are discharged simultaneously,the mass flow rates of gas and oil, respectively, are

-rh

/4/g- + qS’ P >- Pd,

rh, P<Pd,

rhl ----, P >_ Pd,

O, P<Pd,

(4)

in which 05 indicates the ratio of oil mass to gasmass, i.e., b ml/mg.

Gas Leakage Model

Gas leakage is one of the vital factors that affectthe performance of an air compressor. The calcu-lation of leakage also affects the simulation accu-

racy of the compressor. In a twin screw aircompressor, there are four major leaking paths,namely blowhole, clearance between two rotors,clearance between rotor tip and case, and clear-ance between end plate and case. In general, it is

very difficult to obtain the precise amount ofleakage from experiments. The convergent nozzlemodel is therefore adopted herein to account forthe leakage. It states that

rh-- CDAcPh

Xfl--1RmTh h h

2 ) Pwhen/3+1 <hh < (5)

and

/R/3 ( 2 )h CDA Phm Th / 2r-

(/3+1)/(/3-1)

whenO<hh <+1

(6)

in which P1 and Ph are the lower pressure andhigher pressure of the adjacent grooves, CD is

the coefficient of flow rate, /3 is the specific heatratio, and Rm is the revised gas constant. The lasttwo variables can be calculated as

Cp qt_ (Cv,1 and Rm/3Cv + Cv,, +--- R. (7)

Heat Transfer between Oil and Gas

Heat transfer is another vital factor that affectsthe performance of an air compressor. It is verydifficult to determine the heat transfer coefficientand heat transfer area between oil and gas. Tosimplify the analysis, the heat transfer between oiland gas is classified into two categories in the pre-sent study. The first category assumes that the com-pression room is in oil-injected process, and the heattransfer coefficient is the coefficient between thesphere and the flowing fluid. Based on this assump-tion, many approximate formulas can be used todescribe such a heat transfer mode. Among them,the formula proposed by Ranz in 1952 is adopted inthe present paper. It says

Nu 2.0 + 0.6 ReSPr33.

The heat transfer area is the sum of surface areaof all spheres and can be calculated from ANprrDZp, in which the size of oil drop diameteris calculated by using the mean Sauter diameter

formula,

Dp 0.0366\----/ + - (9)

The number of oil drops, Np, is calculated as

6/1/1Np

p,TrDp(lO)

The second category of heat transfer between oiland gas assumes that the compression chamber isnot in an oil-injected process. Under this circum-stance, most of the oil is attached to the rotorsas well as the case. The heat transfer mode in this

TWIN SCREW AIR COMPRESSORS 69

category is then different from the previouslydescribed one. Fujiwara and Osada derived therelationship between the heat transfer coefficientand the volumetric efficiency in the suction process.The formula can be written as

Nu 0.51Re’74 (11)

where

Re Dzm (12)g

PERFORMANCE SIMULATION ANDMODEL TESTING

Based on the above theoretical analysis, a compu-ter program is written to simulate the performance

of a twin screw air compressor. Experimental mea-surement work (Chen, 1996) has also been per-formed to test the accuracy of the analyticalmodel. Some of the results are shown in the presentsection.

Numerical Algorithm

As indicated by Eqs. (1)-(3) as well as equationsfor the suction and discharge processes, the govern-ing equations of a twin screw air compressorcomprise four nonlinear equations. It is almostimpossible to solve them analytically and numericalsolution is therefore needed. In the present study, afourth order Runge-Kutta method is employed.The flow chart of the numerical algorithm is shownin Fig. 2.

////Rotor geometric data & //working fluid propertiesStart

/ Inputdata /Calculate geometrical

parameters

Calculate suction,compression and discharge

processes

Yes

Calculate dischargetemperature

Calculate volumetricefficiency

Calculate compressionefficiency

LutputFIGURE 2 Flow chart of numerical simulation.

70 W.S. LEE et al.

Model Testing

Some of the theoretically obtained results such as

discharge temperature, volumetric efficiency, com-pression efficiency and specific power are comparedwith experimentally measured data (Xiao et al.,1986). Among these data, the volumetric efficiencyis defined as the ratio of real amount of dischargedgas to the groove volume in the twin screw com-

pressor. The compression efficiency is defined asthe ratio of theoretical power consumption to mea-sured shaft horsepower. The specific power isdefined as the ratio of the shaft horsepower to thecalculated amount of discharged gas. A typicalresult is shown in Fig. 3. It is found that the volu-metric efficiency calculated theoretically is very

close to the experimental result. The isothermal

efficiency, however, is found to have certain dif-ference between the theoretical and experimentalresults. The difference becomes significant when thecompressor is running at high rotor speed. It may beattributed to the dynamic loss offriction that we didnot consider in the theoretical derivation.From viscous fluid dynamics viewpoint, the con-

sumed energy of the dynamic loss should hold therelationship

in which #m is the mean viscosity of the oil and gasmixture, Vt is the velocity of the tooth tip, c is

100

90

>8O

90

80

az 70

60

m 90

80

Fluid: R-729(air)Suc. Press.: 1.033 barSuc. Temp.: 303 KDis. Press.: 8 bar

70 I <>

Simulationresults)aExperiment dat60 |

6.0--s.o-

4.010 20 30 40

Tip Speed (m/s)

FIGURE 3 Comparison between calculated and experimental results.

TWIN SCREW AIR COMPRESSORS 71

600

4OO

200

00.1

Fluid" R-729(air)Suc. Press." 1.033 bar

Suc. Temp. 303 KDis. Press.’ 4 bar

IE+0 5.0E-5 1.0E-4 1.5E-4

Volume (m3)

FIGURE 4 Effect of clearance on the performance of an oil-less twin screw air compressor.

the clearance between the tooth tip and the case,and Ac is the characteristic area. When thedynamic loss is considered in the present model,the accuracy of the theoretically predicted compres-sion efficiency increases greatly as that shown in thethird frame of Fig. 3. Finally, from the last frameof the same figure, it is found that the theoreticalvalues of the specific power are also very close tothe experimental data. The accuracy of the abovetheoretical model is therefore justified.

Performance Analysis of anOil-Less Compressor

After justification of the theoretical model, thesame analysis is now applied to an oil-less twinscrew air compressor in order to study the influence

of different design parameters on the performanceof the compressor. Among the results, Fig. 4indicates the influence of gear tooth clearance. Itshows that in case that clearance is made wider,the external gas leakage in addition to the internalgas leakage occurs, and the increased amount ofleakage makes the pressure distribution to be lowerthan that of the isentropic process. Figure 5 indi-cates the influence of rotor speed on the perfor-mance of the compressor. It shows that, when therotor speed increases, the gas leakage decreases andthe resulting pressure distribution tends to closerto that of the isentropic process. Under this cir-

cumstance, the heating effect diminishes and theamount of gas leakage decreases. As the heatingeffect and gas leakage are irrevocable factors in thecompression process, it implies that the increase of

72 W.S. LEE et al.

600Fluid R-729(air)

Suc. Press." 1.033 barSue. Temp. 303 KDis. Press. 4 bar

200

"- e/D=0.0007 Vt=80m/s

’ e/D=0.0035 Vt=80m/s

",, e/D=0.0007 Vt=120m/s

",,,,,,1- /D=0"0035 vt=120m/’

0.0E+0 5.0E-5 1.0E-4 1.5E-4

Volume (m3)

FIGURE 5 Effect of clearance and rotor speed on the performance of an oil-less twin screw air compressor.

TABLE Performance of an oil-less twin screw aircompressor

Ratio of clearance to male 0.0007 0.0035rotor diameter (e/D)Rotor tip 80 120 80 120speed (m/s)

Volumetric 96.36 97.61 85.52 89.95efficiency (%)Compression 95.89 94.67 99.81 97.51efficiency (%)

Specific power 3.80 3.80 4.11 4.00(kW min/m3)

rotor speed can reduce the effect of irrevocablelectors that make the compression process to becloser to that of the isotropic process. To sum-marize the result, Table I is constructed and theefficiency as well as the energy consumption dataare shown therein.

Performance Analysis of an Oil-InjectedCompressor

In an oil-injected twin screw air compressor, oil

plays the role of cooling the compressed gas. It hasgreat influence on the performance of the compres-sor. Therefore, parameters such as oil flow rate,oil-injected temperature and oil-injected angle haveto be considered in the theoretical analysis. Basedon the previously proposed model and under theassumption that the rotor speed, suction and dis-charge pressure, gas temperature and geometricfactors are all kept the same as before. Figure 6shows that the volumetric efficiency becomes higheras the injected temperature becomes lower. This

may be attributed to the reduction of leakage driv-

ing force due to lower pressure distribution inthe compression process. As for the compression


100

95

9o

80

70

6.0

Fluid" R-729 (air)Suc. Press." 1.033 barSuc. Temp." 303 KDis Press." 8 bar

5.0310 320 330 340 350 360

Oil Inlet Temperature (K)

FIGURE 6 Effect of oil inlet temperature on the performance of an oil-injected twin screw air compressor.

73

efficiency, it is found that higher efficiency isgenerally obtained as the injected temperaturebecomes higher. The tendency, however, reversesafter the temperature reaches a certain value. Thereason is that lower oil temperature in generaldecreases the work done by the compressed gas.However, it increases the dynamic loss due tohigher oil viscosity. An optirnal oil-injected tem-perature may therefore exist as that shown in Fig. 6.Similar situation occurs with regard to the specificpower.The influence of oil flow rate on the compressor

performance is shown in Fig. 7. It indicates thatboth the volumetric and compression efficiencyincreases as the amount of injected oil increases.

The specific power, however, decreases as theamount of oil increases. The influence of oil-

injected angle on the performance of the compres-sor is shown in Fig. 8. It indicates that thevolumetric efficiency decreases but the specificpower increases as the oil-injected angle is movedcloser to the discharge side. With regard to thecompression efficiency, there may be an optimalinjected angle as shown in the figure.

For twin screw compressors with fixed suctionpressure, the performance of compression is deter-mined by built-in volume ratio and system dis-charge pressure. The built-in volume ratio of screwcompressors is defined as the ratio ofvolume of thethread at the start of compression process to the

74 W.S. LEE et al.

100

90

80

7O

Fluid" R-729 (air)Suc. Press." 1.033 barSuc. Temp." 303 KDis. Press. 8 bar

5.060 80 100 120 140

Oil Injection Quantity Ratio (%)

FIGURE 7 Effect of oil-injection quantity on the performance of an oil-injected twin screw air compressor.

volume of the same thread when it first begins to

open the discharge port. For a fixed built-in volumeratio compressor, a mismatch between the internaland system discharge pressures may cause over-compression or undercompression with a resultingdecrease in capacity and an increase in power input.Overcompression occurs when the internal pressurein the compression chamber reaches the systemdischarge pressure before the compressed air arrivesat the discharge port. On the other hand, under-compression occurs when the internal pressurereaches the discharge port prior to achieving systemdischarge pressure.The relationship between the compression effi-

ciency and compression ratio of various built-involume ratios is shown in Fig. 9. Four dischargetemperatures were considered in the figure, inwhich the compression ratio is defined as the ratio

of the expected suction pressure and to theexpected discharge pressure. From the figure it iscalculated that, with a fixed built-in volume ratio,the compression efficiency increases along with theincrease of the compression ratio. The compres-sor may have an optimal compression ratio thatgives the operation the maximum efficiency. Fur-ther increase of the compression ratio will thendecrease the compression efficiency. To be more

precise, consider the case of Fig. 9(a) that operatesat the discharge temperature of 65C. If thebuilt-in volume ratio is set to be 3, the optimalcompression ratio is found to be 4.0. In general, thecompressors should be designed to match theabove-mentioned optimal condition as possible.When the compression ratio is selected to be lowerthan the optimum point, the compressor is inundercompression condition. On the other hand,


100

>90

80

70

o’" 6.0

Fluid" R-729 (air)Suc. Press." 1.033 barSuc. Temp." 303 KDis. Press." 8 bar

5.0300 400 500

Oil Injection Angle (degree)

FIGURE 8 Effect of oil-injection angle on the performance of an oil-injected twin screw air compressor.

75

when the compression ratio is selected to be higherthan the optimum point, the compressor thenoperated in overcompression condition. To sum-marize the result shown in Fig. 9, it is observed thatwhen the compression ratio is low, better efficiencyis usually obtained for lower built-in volume ratios.On the contrary, when the compression ratio ishigh, better efficiency is obtained at higher built-involume ratios.

Figure 10 demonstrates the relationship betweenthe optimal compression ratio and the built-involume ratio for different discharge tempera-tures. The curves in this figure are plotted basedon the optimal design conditions obtained in Fig. 9.The result shows that the optimal compressionratio is linearly proportional to the built-in volumeratio. Although the slopes of these curves are

different, their variation is very small. It can beconcluded that the optimal compression ratio isinsensitive to the discharge temperatures at the tem-peratures range of 65-80C.

CONCLUSIONS

The following conclusions can be drawn from thepresent study:

(1) The leakage caused by the clearance can beclassified into two kinds, namely external andinternal leakage, respectively. When the exter-nal leakage is higher than the internal leakage,the pressure distribution is lower than that ofthe isentropic process. Conversely, when the

76 W.S. LEE et al.

() 100.00

80.00

60.00

/"/’ // \

/ [ Tdis=65ocl// vi=2 // vi= // vi=4/L vi= j

40.002.0 4.0 6.0 8.0

Compression Ratio

90.00

80.00 ," ," /,"/ /

70.00 //

I’1o.oo

50.00

Tdis=75Cvi=3

10.0

(b) 90.00

80.00

70.00

60.00

50.00

40.002.0

(d) 90.00

80.00

70.00

60.00

50.00

4.0 6.0Compression Ratio

,,’" ,//,/’/

_/" /

////

--,/-//

Tdis=70C 1vi=2 |vi=3 |vi=4 /vi=5J

8.0

Tdis=80C

vi=31vi-4 |vi=5 /

)

40.002.0 40.004.0 6.0 8.0 10.0 2.0 4.0 6.0 8.0 10.0

Compression Ratio Compression Ratio

0.0

FIGURE 9 Effect of compression 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 internalleakage, the pressure distribution will be higherthan that of the isentropic process. Higherrotor speed can also reduce the leakage andmake the compression process move toward theisentropic process.

(2) The volumetric efficiency of a screw compres-sor can be improved by raising the injected oiltemperature and making the discharge tem-

perature lower. As for the energy consumption,there may exist an optimal injected temperaturethat results in the highest compression effi-ciency and the lowest specific power.

(3) The oil-injected angle is irrelevant to the dis-

charge temperature, but it has great influenceon the distribution of pressure and tempera-ture. Better volumetric efficiency can usuallybe achieved by the selection of an earlier oil-injected angle.

(4) For a fixed volume ratio, there exists an optimalcompression ratio that result in the maximumcompression efficiency. For a fixed compres-sion ratio, depending on the compression ratiothe maximum compression efficiency may behigher or lower as the build-in volume ratiobecomes higher.


10.00

8.00

6.00

4.00

Optimal Disdrge Pressure

Toffs=65

Tdi70

Tdi75

Tdis=80

2.002 3 4 5

Built-in Volume Ratio

FIGURE 10 Relation of optimal compression ratio and built-in volume ratio at various discharge temperatures.

77

NOMENCLATURE

AAcAvCDCp

OpEd,1h

EH

NNu

Area (m2)Characteristic area (m2)Valve area (m2)Coefficient of flow rate

Specific heat at constant pressure(KJ/kg K)Specific heat at constant volume(KJ/kg K)Mean Sauter diameterDynamic loss (KJ)Heat transfer coefficient (W/m2 K)Specific enthalpy (J/kg)Mass (kg)Number of oil dropsNusselt number

PPrRRmReReTVvw

x,y,z

Pressure (Pa)Prandtl numberUniversal gas constant (KJ/kg K)Revised gas constant (KJ/kg K)Reynolds numberRotational Reynolds numberTemperature (K)Volume (m3)Rotor tip velocity (m/s)Flow rate (m/s)System coordinates

Greek Symbols

Modified specific heat rateClearance (m)Ratio of oil mass to gas mass

78 W.S. LEE et al.

EfficiencyPressure angle (rad)Dynamic viscosity (kg/m s)Density (kg/m3)Surface tension (N/m)Rotor speed (tad/s)

Superscript

Physical quantity per unit time

Subscript

dfg

ms

Discharge sideFemale rotorGasOil, Low pressure sideMale rotor, Mean valueSuction side

References

Bein, T.W. and Hamilton, J.F., 1982, Computer modeling of anoil flooded single screw air compressor, Proc. Int. CompressorEngineering Conf., Purdue, USA, pp. 127-134.

Chen, S.L. and Wu, W.F., 1996, Performance Simulation andExperimental Testing of Twin Screw Compressors, TechnicalReport, Department of Mechanical Engineering, NationalTaiwan University, Taipei, Taiwan.

Fujiwara, M. and Osada, Y., 1995, Performance analysis ofan oil-injected screw compressor and its application, Int. J.Refrig., 18(4), 220-227.

Sngfors, B., 1982, Analytical model of helical screw machinesfor analysis and performance prediction, Proc. Int. Com-pressor Engineering Conf., Purdue, USA, pp. 135-139.

Singh, P.J. and Bowman, J.L., 1986, Effect of design parameterson oil-flooded screw compressor performance, Proc. Int. Com-pressor Engineering Conf., Purdue, USA, pp. 71-88.

Xiao, D., Xiong, Z. and Yu, Y., 1986, The computer simulationof oil-flooded refrigeration twin-screw compressors, Proc. Int.Compressor Engineering Conf, Purdue, USA, pp. 349-361.

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