Paper ID:63, Page 1 TORQUE RESEARCH OF SINGLE SCREW EXPANDERS ABSTRACT 1. INTRODUCTION

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1 Paper ID:63, Page 1 TORQUE RESEARCH OF SINGLE SCREW EXPANDERS Ruiping Zhi 1 *, Yuting Wu 1, Yeqiang Zhang 1, Biao Lei, Wei Wang 1, Guoqiang Li 1 an Chongfang Ma 1 Key Laboratory of Enhance Heat Transfer an Energy Conservation of Ministry of Eucation an 1 Key Laboratory of Heat Transfer an Energy Conversion of Beijing Municipality, College of Environmental an Energy Engineering, Beijing University of Technology, Beijing 10014, PR China * Zhiruiping@gmail.com ABSTRACT In recent years, people have pai much attention to single screw expaners. Many experiments have been carrie out on the performance of single screw expaners in funamental applications such as inustry waste heat recovery, compresse air power system, an pressure energy recovery. This paper presents a theoretical torque moel of single screw expaner uner the conitions of ieal aiabatic expansion an air as working flui. It was foun that the torque ratio is inepenent of inlet pressure of single screw expaner an is close to 1. It inicates that the single screw expaner runs smooth. Besies, the calculate output power base on the theoretical torque moel was close to the measure output power by experiments, if the shaft efficiency from experiments results an output work loss uner over-expansion process were consiere. An it emonstrates that this moel can be use to estimate the output power of single screw expaner before esigning it accoring to the iameter of main rotor, volume ratio, inlet an back pressure of single screw expaner. 1. INTRODUCTION In recent years, with heavy fog an haze ominating our life, people realize the ba effects of PM.5, such as coughs, asthma, sore throats an other respiratory illnesses. Many researchers an inustries have realize the importance of using low-grae energy so that we can contribute to a permanent Beijing Blue. Without oubt, ORC system is a great technology to save energy an keep the energy clean. A whole ORC system is basically compose of evaporator, expaner, conenser an working flui pump. Among these components, the selection of expaner type is crucial to this efficiency of the system because the expaner ecies how much pressure energy is converte to power. Among piston expaners, turbo expaners, scroll expaners an screw expaners, single screw expaners (SSEs) have receive much attention in recent years ue to its unique avantages, such as balance loas, long working life, simple structure, low vibration an so on. In 008, Ma Chongfang an his team [1] first reporte the SSEs with 10kW an 40kW in Science Times. He et al. [] carrie out the stuy on the power system of compresse air base on three-stage SSEs. By programming an mathematical moeling results, it is shown that the single screw expaner (SSE) has a goo potential to be use in power systems. Liu et al. [10] verifie the feasibility of SSE on ORC system with heat source of flue gas. Wang et al. [3] use compresse air as working flui to verify the performance of SSE prototype. The esigne flow rate of SSE is 1.1Nm³/min. The performance tests were conucte uner ifferent conitions incluing ifferent intake flow, ifferent humiity, constant torque an constant rotational spee. Accoring to the experimental results, it is shown that the output power is 5kW at rotary spee of 850r/min, ischarge temperature is -45, the maximum temperature rop was about 6, an the maximum of aiabatic efficiency an total efficiency were 59% an 3.5% respectively. An it also shows that this SSE prototype has goo part-loa performance. An lubrication may be a factor to cause low aiabatic efficiency. He et al. [4] carrie out experiments on the performance of the SSEs with compresse air as working flui uner ifferent intake pressures an showe that the measure torque has a large increase as the intake pressure increases an has a slight ecrease as the rotational spee increases. Desieri et al. [5] evaluate the SSE moifie from a stanar compressor an evelope a steay-state moel of the whole ORC unit. In this experiment, 3 r International Seminar on ORC Power Systems, October 1-14, 015, Brussels, Belgium

2 Paper ID:63, Page the working flui is Solkatherm (the azetropic mixture of HFC365mfc an YR-1800), the volume ratio of the expaner is 5 an the iameter of main rotor is 155mm. The results show that the maximum expaner isentropic efficiency an generate power are 64.78% an 7.8 kwe respectively. The whole cycle efficiency peake at 9.8% with the evaporating temperature of 108. Ziviani et al. [6] establishe a etaile moel of SSE base on the geometric parameters, heat transfer moel an governing equations. An then verifie the calculate ata using experiment ata between mass flow rates an output power. Although the performance of SSE have been carrie out in the ORC system an power system, little attention has been pai to the theoretical torque moel of SSE which can be use to preict if the SSE works in a goo state an also can be use to estimate how much output power are prouce. The paper presents a etaile torque moel of single screw expaner an gives out comparison of calculate torque an experimental ata.. Theoretical torque calculation of single screw expaner.1 Main Parameters of Single Screw Expaners The main rotor an a pair of gate rotors are the key components in SSE (shown in Figure 1). In this paper, the SSE is esigne by our laboratory team. The single screw expaner belongs to CP type. It means that the main rotor is machine by a cutting tool with straight line like a single tooth of gate rotor. The gate rotor is generate by envelope of main rotor. The main rotor an gate rotor are machine by special purpose machine evelope by our team (shown in Figure ). Their meshing relations are just like worm gears. The main structure parameters of meshing pair are the founation to calculate the basic volume, the volume ratio an the whole esign of SSE. The main parameters are illustrate in Figure 3. Figure 1: Single screw expaner Figure : Main rotor an gate rotor Figure 3: Structure of single screw meshing pair i θ z θ z1 π γ = z () r k0 = r1 (3) H k = r (4) 1 = = (1) 1 3 r International Seminar on ORC Power Systems, October 1-14, 015, Brussels, Belgium

3 . Swept Volume Equation Paper ID:63, Page 3 C = r(1 + k k) (5) 1 0 = r1 0 (6) l kk k a = arcsin( kk 0 k ) k0 (7) l a = arcsin( ) rk 1 0 (8) l = 0.7l (9) e ξ = r1 (10) γ γ b= r1[( k0 k)sin( ) ξ cos( )] (11) γ γ ( k0 k)sin( ) ξ cos( ) b b s = = (1) r k 0 b s δ = arcsin( ) (13) In this paper, base on the metho use by Sun Guangsan (1988), we calculate the whole single groove basic volume of the main rotor. The metho mainly refers to the area of the gate rotor an the centroi of the gate rotor when meshing with the main rotor. The whole single groove basic volume is ivie into two parts. One part is the volume of the gate rotor sweeping the groove of the main rotor from the front ege of the gate rotor just coming into the groove to the back ege of the gate rotor just coming to the groove (in Figure 4). The other part is the volume of the gate rotor sweeping the groove from the back ege of the gate rotor just coming into the groove to just beginning ischarge. However, there is some ifference between the volume equation inferre by Sun Guangsan an this volume equation in this paper. This volume equation of this paper is mainly expresse by non-imensional coefficients of k an k 0 an the raius (r 1 ) of main rotor. The formulae are as follows: a+ δ a δ 1 1 θ a δ a θ (14) V = V + V = A ir + A ir s Aa δ= k0r1 arcsin( bs) + bs 1 bs bkr s 01( k0 k)sec( a δ) (15) a + θ A = A a ; a θ a + (16) b A= b ( k0 r1 x ( kr 01 H)secθ + xtanθ) x; θb θ a (a) Cross-section (b) Primary view Figure 4: Movement relations of single screw meshing pair For the single screw expaner, the whole expansion process is ivie into three phases. It is suction, close expansion an ischarge respectively. The first part volume ( V 1 ) is an inherent suction volume for every main rotor when the angle of gate rotor is between α δ anα + δ. So when we calculate the expaner suction volume at any time uring the suction phase, the formula is as follows: 3 r International Seminar on ORC Power Systems, October 1-14, 015, Brussels, Belgium

4 V Paper ID:63, Page 4 α+ V1 = A ir1θ, α θ α + = α (17) a δ a δ V + 1 V = + A ir1θ + A irθ, θ α θ se θ < α When we calculate the closing expansion volume at any time uring the closing expansion phase, the formula is as follows: α+ θ θ θ 1, θ θ θ a δ b se V A ir A ir = + < a δ (18) When we calculate the ischarge volume at any time uring the ischarge phase, the formula is as follows: α θ θ + + < a δ V = α+ A ir1θ,θ b α θ < θ α b α +.3 Calculation of Instantaneous an Average Torque + b A ir1θ A irθ,θ a δ b α θ θ b (19) For the single screw expaner, the instantaneous torque of main rotor is equal to prouct of the force on the gate rotor tooth area an the istance from the centroi of the gate rotor tooth area to the center of main rotor. Assuming that the pressure on the lower teeth surface of gate rotor is equal to the ischarge pressure an the instantaneous torque is ivie into three phases accoring to the operation process of single screw expaner. Assuming that the expansion is a perfect aiabatic process an that the working flui is compresse air, the formulae of the instantaneous torque are as follows: F = ( pin pout ) A, α < θ α + F = ( pin pout ) A, θse < θ α k V F = pin A, θb < θ θse V1 F = ( pout pout ) A= 0, θ α θ θb T = ( pin pout ) AR1, α < θ α + T = ( pin pout ) AR, θse < θ α k V T = pin AR, θb < θ θse V1 T = ( p p ) AR = 0, θ α θ θ out out b b (0) (1) Therefore, the average torque is given by the following equation (): t Tt t1 Ta 1 = () t t1 The theoretical output power can be written as Ta 1n P e1 = (3) Output Power Loss Percentage of Over or Uner Expansion Process 3 r International Seminar on ORC Power Systems, October 1-14, 015, Brussels, Belgium

5 Paper ID:63, Page 5 It is known that the single screw expaner is a positive isplacement expaner without ischarge valve. As a result, there is a built-in volume ratio ( η iv ) for the single screw expaner. If the inlet pressure of expaner ( P in ) is certain, the internal outlet pressure ( P iout ) is a certain value when the expansion ens an the expaner begins ischarging. But the internal outlet pressure finally nees to be equal to the back pressure ( P ) of the ischarge pipe, so that the working flui can be ischarge. The internal an external expansion ratio ( ε i an ε ) can be obtaine in the following equation (4-5). If the internal outlet pressure P iout is equal to the external outlet pressure P, the expansion process is calle as full expansion (shown in Figure 5a). If the internal outlet pressure P iout is less than the external outlet pressure P, the expansion process is calle as over expansion (shown in figure 5b). If the internal outlet pressure P iout is greater than the external outlet pressure P, the expansion process is calle as uner expansion (shown in Figure 5c). For the over an uner expansion, there is aitional power consumption an it will ecrease the output power. The output work loss w is shown in the triangle with shaow part. The output power loss percentage η w can be calculate in equation 6. Hence, the output power e can be written as p in ε i = (4) piout p in ε = (5) p 1 1 κ 1 κ κ w κ κ κ ( ε εi ε + εi ) t w ηw = = = κ κ 1 κ w w κ t 1 ε (6) P after consiering the output power loss uner ifferent expansion type P (1 ) e = Pe 1 ηw (7) Furtherly consiering the shaft efficiencyη s, the output power P e3 is given by P (1 ) e3 = Pe 1 ηw ηs (8) (a) Full expansion (b) Over expansion (c) Uner expansion Figure 5: Expansion type 3.1 Experimental Results 3. Results an Discussions The single screw expaner esigne by our laboratory is installe in a compresse air power system. This whole system inclues five parts: air intake an exhaust circuit, oil lubrication circuit, power testing system, water cooling system an ata acquisition system. The air source comes from a compresse air storage tank. The rotary spee of single screw expaner is 3000r/min. An the inlet air measure pressure was varie from 4.97bar to 9.975bar. The main parameters of SSE are liste in 3 r International Seminar on ORC Power Systems, October 1-14, 015, Brussels, Belgium

6 Paper ID:63, Page 6 Table 1. The measure variables an their uncertainties are shown in Table. An the experimental results are shown in Table 3. The built-in volume ratio of single screw expaner is 5. This experiment uses the air as working flui. An the aiabatic exponent of air is assigne as an ieal value of 1.4. Assuming that the expansion process is an aiabatic process, accoring to the aiabatic equation of air, the ieal aiabatic expansion ratio is As can be seen from Table 3, the expansion ratios calculate by experiments ata are less than that of ieal aiabatic expansion ratio. So this single screw expaner was in an overexpane working state. This inlet pressure in this experiment just sets less than 10 bar, because there is an upper en of measure output power for the ey current ynamometer (0~10 kw). That s the reason that experiments with higher inlet pressure cannot be carrie out. Figure 6: General layout of experimental setup with air compressor Table 1: Main parameters of single screw expaner Parameters Diameter of main rotor 155mm Diameter of gate rotor 155mm Transmission ratio 11/6 Center istance 14mm Volume ratio 5 Tooth with of gate rotor 3.4mm Table : Measure variables an their uncertainties Variable Units Uncertainties Temperature 0.5% Pressure bar 0.5% flow m³/h 1.5% Torque N.m ±0.~0.4%FS Rotate spee r/min ±1% Table 3: Experimental results of single screw expaner Inlet Output Outlet External Shaft pressure Rotate spee Torque Power Pressure expansion efficiency P n (r/min) T in (bar) ex (N.m) P ex (kw) P (bar) ratioε η s (%) r International Seminar on ORC Power Systems, October 1-14, 015, Brussels, Belgium

7 3. Calculate Torque an Torque Ratio of Single Screw Expaner Paper ID:63, Page 7 Base on the equation presente in this paper, the Matlab coes are programme an evelope to calculate the instantaneous torque of single screw expaner with the rotary angle changes of main rotor. Assuming that the inlet pressure is equal to the measure values by experiments, the expansion process is a perfect aiabatic expansion, an that there is no mechanical an frictional loss. From the Figure 7, the output torque value appears again with perio 60 egrees. In a perio, no matter how much the inlet pressure is equal to, all the relative highest point an lowest point appears at the same egrees (46egrees an 30egrees respectively). In Figure 7, it is shown that the output torque of single screw expaner increases with the increase of inlet pressure. This tren is in accorance with that of experimental results. Torque ratio is efine as theoretical torque at ifferent angle of main rotor in to average torque of single screw expaner an can be calculate in equation (shown in equation 9). Torque ratio can reflect the operation performance of single screw expaner. The torque ratio is closer to 1, the single screw expaner runs smoother. It can be observe in Figure 8 that the torque ratio is inepenent of inlet pressure, an that there is a small fluctuation for torque ration aroun at the horizontal line (Torque ratio of the horizontal line is equal to 1). It coul be conclue that the single screw expaner runs steaily. T λ = T a1 3. Calculate an Measure Output Power of Single Screw Expaner (9) Accoring to the trapezoi rule of numerical integral formulas, the average torque in one perio is obtaine by equation (), an then by equation (3-8), the calculate output power can be calculate (shown in Table 5). The relative error can be obtaine by the following equation (30). As liste in table 5, it reveals that the over-expansion woul cause great output power loss an that the output power loss percentage will ecrease sharply with a slow increase of external expansion ratio. Pe 3 P Er = P ex ex (30) It can be observe in Figure 9 that the output power of single screw expaner increases linearly with growth of inlet pressure. The relative error of output power is given in Table 4. The ifference value of output power is aroun at 1 an there is almost no change for P, but the relative error is bigger compare to the change of ifference value P of it. Because the output power value itself is not a big value. Although the relative error is not much small, the output power by this mathematical moel still can be accepte. That s because in this moel, the leakage an friction loss are not consiere. Theoretical torque of single screw expaner(n.m) bar 5.976bar 7.033bar bar 8.975bar 9.975bar Rotary angle of main rotor( ) Torque ratio of main rotor bar 5.976bar 7.033bar 8.010bar 8.975bar 9.975bar Rotary angle of main rotor( ) Figure 7: Theoretical torque of main rotor at ifferent rotary angle of main rotor Figure 8: Torque ratio at ifferent angle of main rotor 3 r International Seminar on ORC Power Systems, October 1-14, 015, Brussels, Belgium

8 Paper ID:63, Page 8 Output power of single screw expaner(kw) P e1 P e3 P ex Inlet pressure of single screw expnaer (bar) Figure 9: Output power of single screw expaner with changes of inlet pressure Table 4: The relative error, the theoretical output power, the output power after consiering shaft efficiency an output power loss percentage an the measure output power Inlet P e1 (kw) η w (%) P e3 (kw) P ex (kw) Pressure(bar) P (kw) Er (%) Conclusions In this paper, base on the moifie mathematical moel of basic volume for main rotor, the theoretical moel torque moel of single screw expaner is establishe uner ieal aiabatic expansion process an air as working flui. An this paper presents the output power loss percentage equation uring uner or over expansion process. Accoring to the present analysis, the following results are conclue: (1) From the torque equation, it can be foun that the theoretical torque mathematical moel is inepenent of rotation spee of single screw expaner. The instantaneous torque an the torque ratio reflect that the single screw expaner runs steaily. () The output power loss percentage equation can also be applie to polytropic process in twin screw expaners an single screw expaners. The κ value in equation (6) is replace by the polytropic exponent n. (3) By comparison between calculate output power by mathematical moel an measure torque by experiments, this relative error is 10%~19% while the ifferential value P between calculate an measure output power is just aroun at 1. Hence, this moel can be use to estimate the output power of SSE uner given iameter of main rotor, inlet pressure, built-in volume ratio an back pressure when esign the SSE. An there is a rapi increase for η w with a slight ecrease of external expansion ratio. So the SSE shoul avoi over-expansion process when esign it in orer to lower great output power loss. This theoretical mathematical moel can just be use to estimate the operation state an output power of SSE uner ieal aiabatic expansion process an air as working flui, if the iameter of main rotor, inlet pressure, volume ratio an back pressure are given. But there is still some room to improve. On the one han, the inlet pressure loss shoul be inclue in the future stuy in orer to make the 3 r International Seminar on ORC Power Systems, October 1-14, 015, Brussels, Belgium

9 Paper ID:63, Page 9 calculate torque be closer to the measure torque. On the other han, in orer to calculate the torque of SSE in ORC system, the state equation of organic flui nee to be stuie. With the continuous improvement of this moel, this moel will be more accurate. NOMENCLATURE Variable efinition units i transmission ratio ( ) z number of grooves or teeth ( ) γ inexing angle of gate rotor (ra) r raius (mm) k 0 ratio of the main rotor raius in to the gate rotor raius ( ) k meshing epth coefficient ( ) H the maximum meshing epth (mm) C Center istance of single screw meshing pair (mm) l axial length of the ischarge sie (mm) α meshing angle of the ischarge sie ( ) α meshing angle of the suction sie ( ) l axial length of the suction sie ( ) b s tooth with coefficient ( ) b tooth with of the gate rotor (mm) δ half angle of the tooth with (ra) e the minimum with of the groove wall (mm) ξ coefficient of the groove wall ( ) A area of gate rotor tooth meshing with main rotor (mm ) V volume of main rotor groove (mm 3 ) θ rotary angle (ra) p inlet or outlet pressure of single screw expaner (Mpa) p back pressure (Mpa) p iout internal expansion pressure of single screw expaner (Mpa) T instantaneous output torque of single screw expaner (N.m) Ta1 theoretical average output torque (N.m) P output power of single screw expaner (kw) η s shaft efficiency of single screw expaner ( ) η w output power loss percentage of single screw expaner ( ) λ torque ratio Er relative error ( ) ε i internal expansion ratio ( ) ε external expansion ratio ( ) P ifferential value of output power (kw) Subscript 1 main rotor gate rotor e1 theoretical e after consiering output power loss percentage e3 after consiering shaft efficiency 3 r International Seminar on ORC Power Systems, October 1-14, 015, Brussels, Belgium

10 ex in out se b experimental results inlet outlet suction ening ischarge beginning Paper ID:63, Page 10 REFERENCES [1] (008) (in Chinese) [] HE, W., Wu, Y.T., Ma, C.F., Ma, G.Y., (010). Performance stuy on three-stage power system of compresse air vehicle base on single-screw expaner [J]. Science China Technological Sciences, 010, 53(8): [3] Wang, W., Wu, Y.T., Ma, C.F., Liu, L.D., an Yu, J., (011). Preliminary experimental stuy of single screw expaner prototype. Applie Thermal Engineering, 31: [4] He, W., Wu, Y.T., Peng, Y.H., Zhang, Y.Q., Ma, C.F., an Ma, G.Y., (013). Influence of intake pressure on the performance of single screw expaner working with compresse air. Applie Thermal Engineering, 51: [5] Desieri, A., van en Broek, M., Gusev, S., an Quoilin, S., (014). Experimental campaign an moeling of a low-capacity waste heat recovery system base on a single screw expaner. In International Compressor Engineering Conference. Paper [6] Ziviani, D., Bell, I., Paepe, D., an M., van en Broek, M., (014). Comprehensive moel of a single screw expaner for orc-systems applications. In 014 Purue Conferences: Compressor Engineering Refrigeration an air conitioning high performance builing. Paper [7] Lu, Y.W., He, W., Wu, Y.T., Ji, W.N., Ma, C.F., an Guo, H., (013). Performance stuy on the compresse air refrigeration system base on single screw expaner. Energy, 55: [8] Sun, G., (1988). The investigation of some basic geometric problems of the single screw co. In International Compressor Engineering Conference. Paper 630. [9] ZHANG, Y.Q., WU, Y.T., XIA, G.D., Ma, C.F., Ji, W.N., Liu, S.W., Yang, K., an Yang, F.B., (014). Development an experimental stuy on organic Rankine cycle system with single-screw expaner for waste heat recovery from exhaust of iesel engine [J]. Energy, 77: [10] Liu, L.D., (010). Research of the single screw expaner an organic Rankine cycle system [D]. (in Chinese) [11] Peng, Y.H., (013) Performance stuy of the compresse-air power system base on singlescrew expaners[d].(in Chinese) [1] Peng, Y.H., Wu, Y.T., He, W., Ji, W.N. An Ma, C.F., (014) Experimental stuy of single screw engine at ifferent intake pressure.(in Chinese) ACKNOWLEDGEMENTS The authors are grateful to acknowlege the financial support provie by the National Basic Research Program of China with Grant Numbers 011CB an 013CB8306,International S&T Cooperation Program of China with Grant Numbers 014DF Thanks for experimental ata supporte by Yeqiang Zhang, Weining Ji an Yanhai Peng. If there is no their har work, there will be no moel valiation of this paper. I am quite grateful for the guie from Biao Lei. Thanks for goo suggestions from Wei Wang. Thanks them very much for their help. 3 r International Seminar on ORC Power Systems, October 1-14, 015, Brussels, Belgium