Numerical Simulation and Performance Analysis of Twin Screw Air Compressors

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


INTRODUCTION
The twin screw air compressor is a positive dis- placement compressor.It utilizes the continual variations of the space formed between rotor grooves and case of the compressor to compress gas.In the early stage of development, the compres- sion process of a twin screw compressor was usually estimated empirically.The drawbacks of the em- pirical process were time consuming, difficult to attain an optimal performance, and requiring real model tests.
Many mathematical models for the performance analysis of a twin screw compressor have been proposed.Bein and Hamilton (1982) presented a theoretical model for an oil-injected screw air com- pressor.They utilized a polytropic compression process model to find the value of the polytropic constant that assured the highest consistency of the model results as well as the experimental data.SAngfors (1982) studied both oil-less and oil- injected twin screw compressors.By taking into account the dynamic loss, leakage and heat trans- fer, he used R12 and air as the working fluid and Corresponding author.Tel.: 886-2-2771Tel.: 886-2- -2171, ext. 3515, ext. 3515.Fax: 886-2-2731-4919.E-mail: fl0911 @ntut.edu.tw.developed a numerical method to predict varia- tions of internal state of the compressor.Singh and Bowman (1986) have discussed the effects of parameters such as gear tooth form and gear tooth number in a pair of female and male rotors on the performance of an oil-injected compressor.To make a model adaptable to assorted kinds of fluid, Xiao et al. ( 1986) has presented a theoreti- cal model taking into account the real gas effect, leakage, heat transfer and flow resistance of the discharged gas.Geometric profiles such as length of the gas-seal line and orifice area of the dis- charge valve were also considered in the model.
Recently, Fujiwara and Osada (1995) have applied both numerical method and experimental measure- ment to study the performance of a twin screw air compressor.
To evaluate the performance of twin screw air compressors, an effort has been made to develop a general theoretical model accompanying with its computer simulation program in the present study.The theoretical model takes into consideration most merits of the above mentioned papers.In particular, the effects of geometric clearance, oil- or water-injected angle, oil or water temperature, gas leakage, heat transfer between oil and air, and mass transfer between water and air are consid- ered.Aside from the theoretical study, experimen- tal data were measured in a test laboratory in order to verify the simulation program with measured data.Once the theoretical model is justified, the optimum operation condition of a twin screw air compressor which is helpful in design can be well mastered.

MATHEMATICAL MODEL
As shown in Fig. 1, the compression chamber of a twin screw air compressor is a space encom- passed by the male rotor groove, female rotor groove and the case.In general, there are several compression chambers between a pair of male and female rotors.In a particular compression chamber, the state of gas is related only to the rotational angle of the rotor when the motion is in steady state.It also indicates that when the compression chambers rotate to an angle of the same degrees, they all have the same temperature, pressure, mass, etc.Based on this assumption, the following derivations are introduced within a com- pression chamber.

Governing Equations
Since the intrinsic state and property of the two fluids contained in the control volume are not identical, it is required to derive the governing equations for the gas and oil respectively.With regarding to the gas, the following equations are derived by the consideration of conservation of mass and energy: where nl and n2 represent the total number of inlet and outlet channels including suction valve orifice, 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 where n3 and n4 represent the total number of inlet and outlet channels such as oil-injected orifice, dis- charge valve orifice and leaking path.

Suction Process
In the suction process, since area of the suction valve orifice is large enough to slow down the flowing speed of gas, the pressure loss during gas flowing is negligible.Therefore, the mass flow rate of the suction gas can be considered as rh- AviV/2p(Ps-P), in which Avi denotes area of the suction orifice.

Discharge Process
The difference between the gas pressure in a com- pression chamber and the system back pressure is the major pressure loss when gas passes through an 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 is the summation of all flow rates of the gas passing through various orifices with different ratios of opening areas.
As gas and oil are discharged simultaneously, the mass flow rates of gas and oil, respectively, are in which 05 indicates the ratio of oil mass to gas mass, i.e., b ml/mg.

Gas Leakage Model
Gas leakage is one of the vital factors that affect the performance of an air compressor.The calcu- lation of leakage also affects the simulation accu- racy of the compressor.In a twin screw air compressor, 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 of leakage from experiments.The convergent nozzle model is therefore adopted herein to account for the leakage.
in which P1 and Ph are the lower pressure and higher pressure of the adjacent grooves, CD is the coefficient of flow rate, /3 is the specific heat ratio, and Rm is the revised gas constant.The last two variables can be calculated as Cp qt_ (Cv,1 and Rm /3 Cv + Cv,, +---R. (7)at Transfer between Oil and Gas Heat transfer is another vital factor that affects the performance of an air compressor.It is very difficult to determine the heat transfer coefficient and heat transfer area between oil and gas.To simplify the analysis, the heat transfer between oil and gas is classified into two categories in the present study.The first category assumes that the com- pression room is in oil-injected process, and the heat transfer coefficient is the coefficient between the sphere and the flowing fluid.Based on this assump- tion, 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 present paper.It says Nu 2.0 + 0.6 ReSPr33.
The heat transfer area is the sum of surface area of all spheres and can be calculated from A NprrDZp, in which the size of oil drop diameter is calculated by using the mean Sauter diameter formula, Dp 0.0366\----/ + - (9) The number of oil drops, Np, is calculated as 6/1/1 Np p,TrD p (lO) The second category of heat transfer between oil and gas assumes that the compression chamber is not in an oil-injected process.Under this circum- stance, most of the oil is attached to the rotors as well as the case.The heat transfer mode in this category is then different from the previously described one.Fujiwara and Osada derived the relationship between the heat transfer coefficient and 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 AND MODEL 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 analytical model.Some of the results are shown in the present section.

Numerical Algorithm
As indicated by Eqs. ( 1)-(3) as well as equations for the suction and discharge processes, the governing equations of a twin screw air compressor comprise four nonlinear equations.It is almost impossible to solve them analytically and numerical solution is therefore needed.In the present study, a fourth order Runge-Kutta method is employed.
The flow chart of the numerical algorithm is shown in Fig. 2 Model Testing Some of the theoretically obtained results such as discharge temperature, volumetric efficiency, com- pression efficiency and specific power are compared with experimentally measured data (Xiao et al., 1986).Among these data, the volumetric efficiency is defined as the ratio of real amount of discharged gas to the groove volume in the twin screw com- pressor.The compression efficiency is defined as the ratio of theoretical power consumption to mea- sured shaft horsepower.The specific power is defined as the ratio of the shaft horsepower to the calculated amount of discharged gas.A typical result 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 experimental results.The difference becomes significant when the compressor is running at high rotor speed.It may be attributed to the dynamic loss of friction that we did not consider in the theoretical derivation.
From viscous fluid dynamics viewpoint, the con- sumed energy of the dynamic loss should hold the relationship in which #m is the mean viscosity of the oil and gas mixture, Vt is the velocity of the tooth tip, c is  the clearance between the tooth tip and the case, and Ac is the characteristic area.When the dynamic loss is considered in the present model, the accuracy of the theoretically predicted compression efficiency increases greatly as that shown in the third frame of Fig. 3. Finally, from the last frame of the same figure, it is found that the theoretical values of the specific power are also very close to the experimental data.The accuracy of the above theoretical model is therefore justified.

Performance Analysis of an
Oil-Less Compressor After justification of the theoretical model, the same analysis is now applied to an oil-less twin screw air compressor in order to study the influence of different design parameters on the performance of the compressor.Among the results, 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 increased amount of leakage makes the pressure distribution to be lower than that of the isentropic process.Figure 5 indi- cates the influence of rotor speed on the performance of the compressor.It shows that, when the rotor speed increases, the gas leakage decreases and the resulting pressure distribution tends to closer to that of the isentropic process.Under this cir- cumstance, the heating effect diminishes and the amount of gas leakage decreases.As the heating effect and gas leakage are irrevocable factors in the compression process, it implies that the increase of  (kW min/m3) rotor speed can reduce the effect of irrevocable lectors that make the compression process to be closer to that of the isotropic process.To sum- marize the result, Table I is constructed and the efficiency as well as the energy consumption data are shown therein.
Performance Analysis of an Oil-Injected

Compressor
In an oil-injected twin screw air compressor, oil plays the role of cooling the compressed gas.It has great influence on the performance of the compres- sor.Therefore, parameters such as oil flow rate, oil-injected temperature and oil-injected angle have to be considered in the theoretical analysis.Based on the previously proposed model and under the assumption that the rotor speed, suction and dis- charge pressure, gas temperature and geometric factors are all kept the same as before.Figure 6 shows that the volumetric efficiency becomes higher as the injected temperature becomes lower.This may be attributed to the reduction of leakage driv- ing force due to lower pressure distribution in the compression process.As for the compression 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 is generally obtained as the injected temperature becomes higher.The tendency, however, reverses after the temperature reaches a certain value.The reason is that lower oil temperature in general decreases the work done by the compressed gas.
However, it increases the dynamic loss due to higher oil viscosity.An optirnal oil-injected tem- perature may therefore exist as that shown in Fig. 6.Similar situation occurs with regard to the specific power.
The influence of oil flow rate on the compressor performance is shown in Fig. 7.It indicates that both the volumetric and compression efficiency increases as the amount of injected oil increases.
The specific power, however, decreases as the amount of oil increases.The influence of oil- injected angle on the performance of the compres- sor is shown in Fig. 8.It indicates that the volumetric efficiency decreases but the specific power increases as the oil-injected angle is moved closer to the discharge side.With regard to the compression efficiency, there may be an optimal injected angle as shown in the figure.
For twin screw compressors with fixed suction pressure, the performance of compression is deter- mined by built-in volume ratio and system dis- charge pressure.The built-in volume ratio of screw compressors is defined as the ratio of volume of the thread at the start of compression process to the 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 volume ratio compressor, a mismatch between the internal and system discharge pressures may cause over- compression or undercompression with a resulting decrease in capacity and an increase in power input.Overcompression occurs when the internal pressure in the compression chamber reaches the system discharge pressure before the compressed air arrives at the discharge port.On the other hand, under- compression occurs when the internal pressure reaches the discharge port prior to achieving system discharge pressure.
The relationship between the compression effi- ciency and compression ratio of various built-in volume ratios is shown in Fig. 9. Four discharge temperatures were considered in the figure, in which the compression ratio is defined as the ratio of the expected suction pressure and to the expected discharge pressure.From the figure it is calculated that, with a fixed built-in volume ratio, the compression efficiency increases along with the increase of the compression ratio.The compres- sor may have an optimal compression ratio that gives the operation the maximum efficiency.Further increase of the compression ratio will then decrease the compression efficiency.To be more precise, consider the case of Fig. 9(a) that operates at the discharge temperature of 65C.If the built-in volume ratio is set to be 3, the optimal compression ratio is found to be 4.0.In general, the compressors should be designed to match the above-mentioned optimal condition as possible.When the compression ratio is selected to be lower than the optimum point, the compressor is in undercompression condition.On the other hand, Oil Injection Angle (degree) FIGURE 8 Effect of oil-injection angle on the performance of an oil-injected twin screw air compressor.

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when the compression ratio is selected to be higher than the optimum point, the compressor then operated in overcompression condition.To sum- marize the result shown in Fig. 9, it is observed that when the compression ratio is low, better efficiency is usually obtained for lower built-in volume ratios.On the contrary, when the compression ratio is high, better efficiency is obtained at higher built-in volume ratios.
Figure 10 demonstrates the relationship between the optimal compression ratio and the built-in volume ratio for different discharge tempera- tures.The curves in this figure are plotted based on the optimal design conditions obtained in Fig. 9.The result shows that the optimal compression ratio is linearly proportional to the built-in volume ratio.Although the slopes of these curves are different, their variation is very small.It can be concluded that the optimal compression ratio is insensitive to the discharge temperatures at the tem- peratures range of 65-80C.

CONCLUSIONS
The following conclusions can be drawn from the present study: (1) The leakage caused by the clearance can be classified into two kinds, namely external and internal leakage, respectively.When the exter- nal leakage is higher than the internal leakage, the pressure distribution is lower than that of the isentropic process.Conversely, when the 10.00 8.00 6.00 4.00

FIGURE 3
FIGURE 3 Comparison between calculated and experimental results.

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

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

TABLE Performance
Relation of optimal compression ratio and built-in volume ratio at various discharge temperatures.
Ratio of oil mass to gas mass