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Effects of rotor solidity and leakage flow on the unsteady flow in axial turbine
Applied Thermal Engineering 128 (2018) 926–939
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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Research Paper
Effects of rotor solidity and leakage flow on the unsteady flow in axial turbine Keke Gao a, Yonghui Xie a,⇑, Di Zhang b a b
Shaanxi Engineering Laboratory of Turbomachinery and Power Equipment, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China MOE Key Laboratory of Thermo-Fluid Science and Engineering, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China
h i g h l i g h t s The solidity effects on the unsteady flow both in full and partial admission turbines are revealed. The combined effects of leakage flow and partial admission flow are presented. The frequency excitation amplitude can be modified via solidity. Leakage flow is beneficial to the axial aerodynamic exciting force reduction.
a r t i c l e
i n f o
Article history: Received 21 March 2017 Revised 30 July 2017 Accepted 15 September 2017 Available online 18 September 2017 Keywords: Unsteady flow Axial turbine Partial admission Aerodynamic force Loss
a b s t r a c t Partial admission turbine plays an important role in power control, which strengthens the unsteady flow. The effects of rotor solidity at partial admission condition can be more complicated, and leakage flow deeply strengthens the unsteadiness. Hence, the investigations of rotor solidity and leakage flow on the unsteady flow are of great significance for turbine designs. Turbines including five kinds of solidity at full and partial admissions are modeled based on 3D viscous compressible NS equation. In addition, the leakage flow model including tip leakage, inlet and outlet cavity is investigated. The results show that the solidity effects on partial admission turbine are not exactly the same as that on full admission turbine. The differences are identified, especially the axial unsteady aerodynamic force. Moreover, the attack angle tends to be negative with rotor solidity increasing; meanwhile, the low pressure region varies with solidity due to flow separation or throat area reduction. Leakage flow model is more able to reveal the unsteady flow, and the comparative analysis of flow phenomenon under the combined effects of leakage flow and partial admission flow is conducted. The change of rotor inlet parameters is smoothed down for leakage model and the axial exciting force is relative lower. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction The output power is usually adjusted by three ways including sliding pressure, throat value and partial admission to meet the requirement of external load. The sliding pressure operation adjusts loads through the boiler. Hence, it takes a long time to cope with changes. The mass flow can also be controlled with throat value while the efficiency is relatively lower due to large pressure losses and short blades. Partial admission method can meet the load change through the intake area adjustment. Further, it is widely used due to no throttling loss, low secondary losses and the rapid load response. However, the unsteady flow in turbine is extremely complicated and then the partial admission construction ⇑ Corresponding author. E-mail address: [email protected] (Y. Xie). http://dx.doi.org/10.1016/j.applthermaleng.2017.09.078 1359-4311/Ó 2017 Elsevier Ltd. All rights reserved.
strengthens the unsteadiness. Compared to full admission turbines, windage loss and arc end loss are induced at the partial admission turbine as well as unsteady low frequency excitation of aerodynamic forces. In order to realize higher efficiency and reliability, the investigations of partial admission are of a great significance to the industry. Nowadays, the deep researches on the unsteady flow in partial admission turbines have been carried out. The performance of partial admission turbines with impulse design has been discussed by Ohlsson [1] through theoretical method based on incompressibility and non-friction assumption. The detailed flow parameter distributions of the partial admission control stage have been presented by He [2] through 2D unsteady numerical method. It also points out that the mixing process may benefit the turbine overall performance. Hushmandi [3] indicates that 3D influence on the turbine unsteady flow is very important based on two low reaction stages
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Nomenclature C 3e Mt k T c2 w2 n Cax,rotor FAT am m Cax,blade FAA w1 Dhast c1 u
variables which determines how dissipation rate affected by the buoyancy, [–] turbulent Mach number, [–] kinetic energy of turbulence, [kJ] torque, [N⁄m] absolute velocity at rotor outlet, [m⁄s1] relative velocity at rotor outlet, [m⁄s1] rotation speed, [r⁄min1] axial chord of rotor tangential force of all blades exponent coefficient exciting order axial chord of blade axial force of all blades relative velocity at rotor inlet, [m⁄s1] ideal enthalpy drop absolute velocity at rotor inlet, [m⁄s1] velocity components in a generalized coordinate system, [m⁄s1]
model. Unsteady flow and exciting force under four partial admission degrees are in detailed comparison based on 3D unsteady numerical investigation by Xie [4]. Bohn [5] and Fridh [6] conduct experimental investigations on the unsteady flow in turbines. The flow parameters non-uniformity caused by partial admission is also pointed out. Tousi [7] shows that the turbine efficiency is closely related to partial admission degree. And the optimum partial admission degree is obtained through numerical analyze. The flow parameters of the partial admission turbine at rated and offdesigned conditions are investigated by Song [8]. In addition, the partial admission turbine used in ORC are analyzed by Cho [9] and Martins [10]. Although the investigations on partial admission are very extensive, only a few studies focus on the effect of rotor solidity on turbine performance. The solidity plays an important role in blade design. As far back as 1945, Zweifel [11] has estimated optimum solidity for turbines with large angular deflection. However, Horlock [12] and Aungier [13] points out the Zweifel correlation limitation that the prediction of optimum solidity is also limited to the outlet flow angle. In addition, the effects of rotor solidity on the full admission stage efficiency have also been investigated by Simpson [14] and the optimum solidity is approximately 1.25 for the investigated model. Only little information regarding the optimum solidity for axial or radial turbine stages is available, while the important effects of solidity on wind turbine performance have been valued [15,16]. When it comes to the partial admission turbine investigations, the references can be much less. According to the open literature, Cho [17,18] has merely done preliminary research on the effects of rotor solidity on aerodynamic force and the results show that the tangential force of single rotor increases with the reduction of rotor solidity while the axial force decreases. Meanwhile, limited to the computing resource and the complexity of modeling, the researches of leakage flow effects are mainly focused on the turbine efficiency or heat transfer, which are with steady leakage models or unsteady periodic admission models. The research can be of a great benefit to understand the leakage flow on the turbine aerodynamic performance. The influences of the tip geometry on leakage flows are investigated by Krishnababu [19] through single passage with the periodic boundary. The result shows the leakage mass flow increases with the gap and the hollow structure contributes to the reduction of leakage
Cl _ m Gb
modified constant, [–] mass, [kg] generation of turbulence due to buoyancy, [–]
Greek
ak q e leff
b
a1 a2
b1 b2
g
f
inverse effective Prandtl numbers, [–] density, [kg⁄m3] dissipation rate, [–] effective dynamic viscosity coefficient, [Pa⁄s] coefficient of thermal expansion, [–] absolute flow angle at rotor inlet, [deg] absolute flow angle at rotor outlet, [deg] relative flow angle at rotor inlet, [deg] relative flow angle at rotor outlet, [deg] efficiency degree of admission
mass flow. The leakage flow and main flow interaction is studied by Anker [20] based on the numerical method, and it is found that the leakage flow along the pitch direction is not uniform. The tip leakage flow and secondary flow interaction is investigated through the unsteady 1.5 stage axial turbine model with shroud by Peters [21]. Pfau [22] obtains the unsteady flow interaction in the rotor inlet cavity based on experimental method for full admission. The vortex structures in the cavity are illustrated. The effects of inlet boundary condition and the relative motion of case on the tip leakage flow have been presented by Coull [23]. The flow structures under different group are also given. In addition, the tip leakage shapes effects on the aerodynamic performance are also investigated by Nho [24], Lee [25,26] and Silva [27]. Partial admission which plays an import role in the power control strengthens the unsteady effects and induces partial admission losses as well as external unsteady exciting forces. Considering the limited investigated model, it is not clear how the aerodynamic performance and unsteady exciting force change under the combined effects of leakage flow and partial admission flow. However, it is extremely important to obtain the accurate flow details and unsteady exciting forces to guarantee higher efficiency and reliability during the control of power output. As explained above, it is still not clear how rotor solidity affects partial admission aerodynamic parameters and exciting forces. Furthermore, there is hardly any information to illustrate the effects of leakage flow including tip clearance, inlet and outlet cavity on unsteady flow at partial admission. It is necessary to know how the solidity and leakage flow behave at partial admission in an attempt to obtain higher efficiency and reliability. For these reasons, the paper is divided into two parts. First, the turbine unsteady flow performances with 5 kinds of rotor solidity are obtained to reveal the effects of rotor solidity. Secondly, the leakage model including tip clearance, inlet and outlet cavity is modeled, and then the unsteady effects of leakage flow on the partial admission turbine can be obtained by comparing leakage model performance with the no leakage control stage model performance. According to general understanding, the optimum solidity exists due that the larger solidity will give the flow more guidance while larger viscous losses occur. The paper explains why the optimum solidity exists from the new perspective, and it finds that the solidity effects on partial admission turbines are not totally the same as that on full admission turbines. In addition, the paper presents
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the key flow parameters under the combined effects of the leakage flow and partial admission flow, such as aerodynamic forces. It is pointed out that it may be beneficial to allow appropriate leakage at inlet and outlet cavity considering the axial exciting force drop.
2. Numerical model 2.1. Computational method CFX commercial software is applied to solve the three dimensional unsteady flow. The turbulence stress term is closed with RNG k-e model which has higher precision and faster convergence speed. The detailed numerical validation will be conducted later in the paper.
@ @ @ @k þ Gk þ Gb qe Y M ðqkui Þ ¼ ak leff ðqkÞ þ @t @xi @xj @xj
ð1Þ
@ @ @ @e e þ C 1e ðGk þ C 3e Gb Þ ðqeÞ þ ðqeui Þ ¼ ae leff @t @xi @xj @xj k C 2e q
e2 k
ð2Þ
Re
where Gk means turbulence kinetic energy, YM means the influence of compressibility and Re is transport source term.
C 1e ¼ 1:42; C 2e ¼ 1:68; Y M ¼ 2qeM 2t
Re ¼
Gk ¼ u t ð
@ui @uj þ Þ; @xj @xi
C l qg3 ð1 g=g0 Þ e2 1 þ bg3 k
2.2. Computational model In order to obtain the accuracy numerical results, three times stator chord length are extended ahead of stator inlet and two times rotor chord length are extended behind rotor outlet as shown in Fig. 1. Different rotor solidity is achieved through the rotor numbers change and the relative geometry parameters are shown in Table 1. Five kinds of solidity at full admission have been investigated to understand the solidity effects on the unsteady flow when the turbine operates with full power output. Then, five kinds of solidity at 0.5 partial admission have also been investigated to reveal the solidity effects on the unsteady flow under the influence of partial admission effects. Considering the leakage effects, the flow mechanisms can be found well via the model with/without leakage keeping same geometry parameters. In addition, considering the computing resource, the effects of leakage are investigated based on one selected solidity. Total pressure, total temperature and turbulence intensity are set as inlet boundary. The average static pressure is set as outlet condition. The frozen rotor method is applied to deal with rotor-stator interface and the steady results are used as the initial value of unsteady computation. The transient rotor stator method is used for the rotor stator interface under unsteady condition. The accuracy flow information can be captured when 15 steps are set for a single stator passage. According to previous research [4], the H-type grid is used for stator, and H-type as well as O-type grid is used for rotor. And the near wall grid refinement has been conducted to improve the calculation accuracy and grid independence is needed to speed up the calculation. Finally, approximately 7 million grids are used to complete the research for no leakage model and 10 million grids are used for leakage model.
ð3Þ 2.3. Computational validation
Convection term discretization is solved by high order accurate schemes proposed by Barth [28] and time term discretization is solved by the dual time step implicit iteration method proposed by Jamson [29].
outlet
inlet
stator
The numerical investigations based on Matsunuma [30] model have been conducted to validate the numerical method through the data comparison in the paper. Takayuki Matsunuma’s
outlet cavity
inlet cavity
rotor
tip leakage
Fig. 1. Full numerical model (left: full admission with no leakage simplification; middle: partial admission with no leakage simplification; right: partial admission with leakage model).
Table 1 Cascade geometry parameters. Type
A
B
C
D
E
Rotor numbers Mean diameter (mm) Chord (c:mm) Blade height (h:mm) Area ratio (h/c) Pitch at mean diameter (s:mm) Solidity (c/s)
61 332 18.60 27 1.45 17.10 1.09
73 332 18.60 27 1.45 14.29 1.30
88 332 18.60 27 1.45 11.85 1.57
105 332 18.60 27 1.45 9.93 1.87
126 332 18.60 27 1.45 8.27 2.24
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numerical method can be sufficiently proved if the numerical results below 70% blade length have a good agreement with the test results. It can be found the numerical data of 5 models agree with experimental results well while RNG k-e turbulence model can be better, especially for axial velocities and relative flow angle. In addition, RNG k-e turbulence model is also adopted by Hushmandi [3] based on the comparison of his experimental data and numerical results. Finally, RNG k-e turbulence model is adopted in the paper and the numerical method is validated to be reliable.
3. Results and discussion Solidity has been widely studied as an important design parameter for wind turbine; however, there is less solidity research related to axial or radial turbine. Moreover, few researches on the leakage flow influence on partial admission turbine can be found. In order to reveal the mechanism of turbine performance, three crucial parameters including mass flow, torque and efficiency are presented first in the paper. Turbine mass flows of different model are shown as Fig. 4. It is obvious that the mass flow decreases with rotor solidity due that the throat area is reduced and mass flow tends to accelerate to decrease. Further, it can be seen that the mass flow ratio of 0.5 partial admission and full admission is over 50%, and the mass flow ratio increases with rotor solidity. It means full admission turbine is easier to be influenced by solidity. In addition, the mass flow of leakage model is higher than that of no leakage model.
Fig. 2. Annular turbine wind tunnel [30].
experimental model including the single axial stage is shown as Fig. 2. The axial velocity of 4.4 m/s and turbulent intensity of 0.5% are given as inlet boundary. The outlet Reynolds number based on outlet velocity and rotor chord is set as 3.5 ⁄ 104. The rotor speed is 402 rpm. Numerical results of 5 kinds of turbulence model including SST, k-e, k-w, BSL Reynold stress and RNG k-e are compared with experimental data. The rotor outlet axial velocity, tangential velocity and velocity angle are shown as Fig. 3. The simplified model with no tip clearance has been adopted for numerical simulation to reduce the computing cost. Hence, only the results related to 070% blade length are compared, which is less influenced by tip leakage. Then
0.7
0.4
Spanwise distance Y/H
0.5
0.6
SST kkBSL Reynold stress RNG kExperiment
0.3 0.2 0.1 0.0 0.0
0.5 0.4
SST kkBSL Reynold stress RNG kExperiment
0.3 0.2 0.1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.0 0.2
Axial velocities Vz/V2,Des
0.4
0.6
0.8
0.6 0.5 0.4
SST kkBSL Reynold stress RNG kExperiment
0.3 0.2 0.1 0.0 30
1.0
Tangential velocities Vx,Rel/V2,Des
0.7
Spanwise distance Y/H
Spanwise distance Y/H
0.6
0.7
40
50
60
70
80
Relative flow angle Fig. 3. Turbulence model verification with the comparison of experimental data [Takayuki Matsunuma model].
1.2
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1.8
1.0
=0.5 no leakage =1 no leakage =0.5 leakage
1.6
=0.5 no leakage =1 no leakage =0.5 leakage
0.9
Mass flow/kg·s
1.4
0.8
1.2 1.0
0.7
0.8
0.6
0.6 0.4 0.9
1.2
1.5
1.8
2.1
0.5 0.9
2.4
1.2
Solidity
Torque/N·m
35 25 15 5 0.9
1.2
1.5
2.1
2.4
Fig. 6. Efficiencies of different model.
=0.5 no leakage =1 no leakage =0.5 leakage
45
1.8
Solidity
Fig. 4. Mass flows of different model.
55
1.5
1.8
2.1
2.4
Solidity Fig. 5. Torques of different model.
Fig. 5 shows the turbine torques of different model. There exists optimum solidity of maximum torque. When rotor solidity is too small, the turbine torque is lower due to relatively less rotor numbers. And the turbine torque is also lower for large rotor solidity due to the power capability reduction of single rotor. It can be seen that the optimum torque of partial admission turbine is larger than that of full admission turbine. That is, the power reduction with rotor solidity increasing at partial admission is less. In addition, the torque of leakage model is lower than that of no leakage model. Efficiencies of different model are shown in Fig. 6. It can be seen that there exists optimum rotor solidity of efficiency. Further,
optimum rotor solidity of efficiency is higher than that of torque and optimum rotor solidity of partial admission efficiency is very close to that of full admission efficiency. The phenomenon above can be explained through Eqs. (4) and (5), in which efficiency is related to torque, rotating speed and ideal enthalpy drop. Further, there exists no mass flow term in Eq. (5) and mass flow accelerates to decrease with rotor solidity increasing as shown in Fig. 4. Combined with Eq. (4), it can be obtained that optimum rotor solidity of efficiency is higher than that of torque. Meanwhile, optimum rotor solidity of partial admission torque is slightly higher than that of full admission torque as shown in Fig. 5 and then mass flow term in efficiency is offset. Hence, the optimum rotor solidity of partial admission efficiency should be close to that of full admission efficiency. In addition, the efficiency of leakage model is lower than that of no leakage model. The fundamental performance parameters have been already presented now and the flow mechanism will be revealed through flow parameters later.
_ 1 cos a1 þ c2 cos a2 Þ ¼ mðw _ 1 cos b1 þ w2 cos b2 Þ T ¼ mðc
g¼
Tn n ¼ ðw cos b1 þ w2 cos b2 Þ _ Dh 9550Dh 1 9550m
ð4Þ ð5Þ
4. Flow parameters The effects of rotor solidity on flow parameters at full admission turbine are presented first to further study the cascade flow mechanisms. The distributions at 50% spanwise have been widely used as the representative cross to investigation the aerodynamic
Fig. 7. Pressure distributions at 50% blade height of different solidity at full admission (left: solidity 1.09; middle: solidity 1.57; right solidity 2.24).
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126000
solidity 1.09 solidity 1.30 solidity 1.57 solidity 1.87 solidity 2.24
Pressure/Pa
117600
Rotor inlet 109200
Rotor outlet 100800 92400 84000 0.0
.
Stator outlet 0.5
1.0
1.5
2.0
Streamwise location(x/Cax,blade )
Fig. 8. Pressure distribution along flow direction of different solidity at full admission.
characteristics of a turbine [31–33], which is also discussed as representative cross of the overall patterns in the paper. Pressure distributions at 50% blade height of different solidity at full admission are shown in Fig. 7. It can be found that rotor inlet pressure rises
with rotor solidity due that the flow resistance in rotor passage increases. Further, the rotor inlet flow angle (the angle between flow direction and axial direction) decreases with the rotor solidity increasing and then rotor attack angle has gradually turned positive value into negative value. Low pressure region of rotor are marked out with dash line for the convenience of observation. As for small rotor solidity, flow separation appears at the leading edge and trailing edge of rotor suction due to large positive attack angle and adverse pressure gradient. As for optimum rotor solidity, there is only flow separation at the trailing edge of rotor suction due to adverse pressure gradient and the low pressure region is not completely obvious. As rotor solidity increases, rotor inlet attack angle tends to be negative and then flow rushes forward to rotor suction. The low pressure region appears near the rotor throat. Different from the previous cause, the low pressure appears due that flow accelerates to pass through the throat region and the flow nears suction and pressure surface interacts strongly for close distance. Then, the power capability decreases. Pressure distribution along flow direction of different solidity at full admission is shown in Fig. 8. It is obvious that the pressure drop in stator passage gradually decreases with rotor solidity increasing. For small rotor solidity as 1.09 and 1.30, rotor inlet pressure is relatively lower due to large pressure drop in stator
B
B
A
A
Inlet section
Blockage
Inlet section
Blockage
Blockage
Inlet section
Blockage
Inlet section
A
A
B
B
(a)
(b) B
B
A
A
Inlet section
Blockage
Inlet section
Blockage
Blockage
Inlet section
Blockage
Inlet section
A
A
B
B
(c)
(d)
Fig. 9. Rotor inlet pressure distributions at 0.5 partial admission (a: solidity 1.57; b: solidity 2.24; c: solidity 1.30; d: solidity 1.30, leakage model).
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backflow appears at blockage. Axial flow velocity decreases with rotor solidity increasing, which is consistent with the mass flow variation rule. Serious backflow mainly appears at the transition region from inlet section to blockage as marked with B. Meanwhile, high pressure appears at B region as shown in Fig. 9. In addition, high axial flow velocity also can be found at the transition region from blockage to inlet section as marked with C. The position of B and C are mainly effected by the distribution of admission and blockage section, instead of rotor solidity. B and C region of leakage model is eased obviously, and the axial flow velocity in inlet cavity is relatively low. Rotor inlet static entropy distributions at 0.5 partial admission are shown as Fig. 11. The static entropy at inlet section is relatively lower than that at blockage and stator wakes can also be observed. The static entropy at blockage is larger mainly due to windage loss. The rotor inlet static entropy decreases with rotor solidity increasing due to the reaction changed. Meanwhile, aiming at leakage model, we can find that the static entropy at inlet cavity is large due that the cavity is full of low energy fluid and the static entropy at rotor passage inlet section also increases compared to no leakage model.
passage. Further, the pressure increases and then decreases in rotor passage. For large solidity as 2.24, the pressure drops sharply in rotor passage and then pressure increase due to the limitation of back pressure. Hence, both low solidity and large solidity are not beneficial to turbine flow. The effects of rotor solidity and leakage model on flow at partial admission are presented in the paper. Rotor inlet pressure increases with solidity due to the increasing of rotor passage resistance, which is consistent with that at full admission as shown in Fig. 9. It is found that low pressure region appears at the transition region from inlet section to blockage as marked with A due to suction effects. Meanwhile, high pressure region marked as B can be found near A region, which is probably related to backflow in rotor passage. Moreover, A and B region positions are not changed with rotor solidity. In addition, A and B region of leakage model is not as obvious as no leakage model, because the cavity and tip leakage contribute to the parameters uniformity to some extent. The pressure in inlet cavity is relatively low. Rotor inlet axial flow velocity distributions at 0.5 partial admission are shown as Fig. 10. It can be seen that the axial flow velocity is larger at inlet section and lower at blockage. Moreover, even the
B
B
C
Inlet section
C
Blockage
Inlet section
Blockage C
C Inlet section
Blockage
Inlet section
Blockage
B
B
(a)
(b) B
B
C
Inlet section
C
Blockage
Inlet section
Blockage C
C Inlet section
Blockage
B
Inlet section
Blockage
B
(c)
(d)
Fig. 10. Rotor inlet axial flow velocity distributions at 0.5 partial admission (a: solidity 1.57; b: solidity 2.24; c: solidity 1.30; d: solidity 1.30, leakage model).
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Arc end losses Stator wakes Inlet section
Blockage
Inlet section
Blockage
Blockage
Inlet section
Blockage
Inlet section
Windage loss
(a)
(b) Inlet cavity leakage loss
Inlet section
Blockage
Inlet section
Blockage
Blockage
Inlet section
Blockage
Inlet section
(c)
(d)
Fig. 11. Rotor inlet static entropy distributions at 0.5 partial admission (a: solidity 1.57; b: solidity 2.24; c: solidity 1.30; d: solidity 1.30, leakage model).
5. Rotor blade load
solidity 1.09 solidity 1.30 solidity 1.57 solidity 1.87 solidity 2.24
110000 104000
Pressure/Pa
The power capability depends on rotor blade loads and it can be further known through rotor blade pressure distributions. The rotor blade loads at full admission are first presented to understand the effects of rotor solidity on rotor blade load as shown in Fig. 12. Effects of rotor blade solidity are quite obvious. The largest pressure difference between pressure and suction surface comes to the smallest solidity and then the power capability of single rotor is the strongest. From small rotor solidity to optimum solidity, the pressure change on pressure surface is small and the pressure on suction surface at leading and middle part increases apparently. From optimum rotor solidity to large solidity, both pressure surface and suction surface have great changes. The pressure on suction surface at leading part still increase while pressure on other parts of pressure surface drops dramatically, and the phenomenon is relevant to the close throat distance as shown in Fig. 7. In addition, we can found that the pressure on suction surface tends to decrease and then increase for all rotor solidity. The point where pressure gradient alters is defined as the turning point in the paper. As we all know, flow separation occurs easily for large adverse pressure gradient and long adverse pressure distance. It can be
98000 92000
Turning point 86000 80000 0.0
0.2
0.4
0.6
0.8
1.0
Streamwise location(x/Cax,stator) Fig. 12. Rotor blade load at full admission.
seen that rotor with the optimum solidity has the minimum adverse pressure gradient and distance. In other words, the position as well as pressure difference between turning point and
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section, leading edge of rotor1 has completed positive work output due to the reduction of suction pressure. By contrast to rotor1, the pressure surface of rotor2 is at the blockage and the suction surface is at inlet section. Hence, it can be seen that pressure on pressure surface is lower than that on suction surface and then negative work has been done. The negative work region at leading edge decreases with rotor solidity. Rotor blade load comparisons between leakage mode and no leakage model at 0.5 partial admission have been presented in Fig. 15. The effects on pressure surface of rotor1is relatively small while suction surface of rotor1 has a great change. The leading edge and tailing edge of leakage model is larger than no leakage model. As for rotor2, the pressure of leakage is larger due that the inlet section flow is inhaled into the cavity. As for rotor3, the most part pressure of leakage model is also larger due that larger mass flow is obtained and cavity is filled with fluid. Rotor4 is at blockage and the pressure of leakage model is slightly lower than that of no leakage model.
rotor 1
rotor 2 rotor4
6. Unsteady flow aerodynamic force
rotor3 Fig. 13. Rotor positions schematic diagram.
trailing edge is smallest. It can also be found that negative work at rotor leading part has been done for large rotor solidity due to the negative attack angle. The selective analysis rotors at 0.5 partial admission are shown as Fig. 13, which marked as rotor1, rotor2, rotor3 and rotor4 respectively. Where, rotor1 is at the transition region from the inlet section to the blockage, rotor2 is at the transition region from the blockage to the inlet section, rotor3 is at the inlet section and rotor4 is at the blockage. Rotor blade load comparisons of different rotor solidity at 0.5 partial admission are presented at Fig. 14. Considering that rotor3 is similar to rotor blade at full admission and rotor4 has no obvious difference for solidity change due to almost no power output, hence only roto1 and rotor2 are shown. It can be seen that rotor1 has the strong ability of working due that pressure surface is at the inlet section and suction surface is at the blockage. Therefore, pressure on suction surface decreases while the pressure on pressure surface remains large. Further, compared with rotor at the inlet
solidity 1.09 solidity 1.30 solidity 1.57 solidity 1.87 solidity 2.24
110000
98000 92000
solidity 1.09 solidity 1.30 solidity 1.57 solidity 1.87 solidity 2.24
110000 104000
Pressure/Pa
Pressure/Pa
104000
98000 92000 86000
86000 80000 0.0
The unsteady axial and tangential forces of single rotor blade with different solidity at full admission are presented as Fig. 16. Periodic sine wave of aerodynamic force at full admission appears due to stator wakes and potential field. Axial force gradually increases with rotor solidity due to rotor passage resistance increasing. However, tangential force decreases with rotor solidity increasing, which is opposite to the variation tendency of axial force. In addition, pulsation amplitude of axial force decreases with rotor solidity increasing while there is no obvious change for tangential pulsation amplitude. The unsteady axial and tangential forces of single rotor blade with different solidity at 0.5 partial admission are presented as Fig. 17. It can be seen that the variation tendency of axial and tangential force at inlet section are approximately the same with that at full admission. The effects of stator wakes and potential field on amplitudes pulsating can also be found. It is obviously that axial force of small solidity at inlet section tends to decrease when single rotor is rotating to blockage and tangential force of small solidity at inlet section tends to increase at the same time. However, the change tendency of aerodynamic force of large solidity at inlet section remains relatively stable. Moreover, largest negative axial force appears at small rotor solidity and no obvious force difference of different solidity can be found at the blockage due to stagnant fluid.
0.2
0.4
0.6
0.8
Streamwise location(x/Cax,stator )
1.0
80000 0.0
0.2
0.4
0.6
0.8
Streamwise location(x/Cax,stator )
Fig. 14. Rotor blade load comparison of different rotor solidity at 0.5 partial admission (left: rotor1; right: rotor 2).
1.0
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106000
106000
102400
102400
98800
98800
Pressure/Pa
Pressure/Pa
leakage model no leakage model
95200
leakage model no leakage model
95200
91600
91600
88000 0.0
0.2
0.4
0.6
0.8
88000 0.0
1.0
0.2
(a)
0.6
0.8
1.0
(b) 106000
leakage model no leakage model
leakage model no leakage model
102400
102400
98800
98800
Pressure/Pa
Pressure/Pa
106000
0.4
Streamwise location(x/Cax,rotor )
Streamwise location(x/Cax,rotor )
95200
95200 91600
91600 88000 0.0
0.2
0.4
0.6
0.8
88000 0.0
1.0
0.2
0.4
0.6
0.8
1.0
Streamwise location(x/Cax,rotor )
Streamwise location(x/C ax,rotor )
(c)
(d)
Fig. 15. Rotor blade load comparison between leakage mode and no leakage model at 0.5 partial admission (a: rotor1; b: rotor 2; c: rotor3; d: rotor 4).
solidity 1.09 solidity 1.30 solidity 1.57 solidity 1.87 solidity 2.24
2.5 2.0
solidity 1.09 solidity 1.30 solidity 1.57 solidity 1.87 solidity 2.24
5
4
Amplitude/N
Amplitude/N
1.5 1.0 0.5
Mean value
3
Mean value
2 0.0
Pulsation -0.5
0
10
20 30 40 Circumferential angle/deg
Pulsation 50
1
0
10
20 30 40 Circumferential angle/deg
50
Fig. 16. Axial force (left) and tangential force (right) of single rotor blade with different solidity at full admission.
Unsteady axial force and tangential force of single rotor blade with different model at 0.5 partial admission is presented at Fig. 18. Axial force is obviously different between leakage model and no leakage model while tangential force not. As for no leakage model, the rotor axial force amplitude at the inlet section decreases gradually along circumferential direction and large negative amplitude appears at the transition region from inlet section to blockage. However, the flow parameters change more smoothly
for leakage model due to the inlet and outlet cavity. Moreover, the axial force drop at the transition from inlet section to blockage decreases obviously due that inlet cavity increases rotor inlet pressure and outlet cavity decreases rotor outlet pressure, as shown in Figs. 9 and 15. Turbine tangential force reflects the power capability while axial force is not beneficial to turbine safe operation. Hence, the ratio of axial force and tangential force of all blades can reflect
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solidity 1.09 solidity 1.30 solidity 1.57 solidity 1.87 solidity 2.24
3
Tendency 2
solidity 1.09 solidity 1.30 solidity 1.57 solidity 1.87 solidity 2.24
6 5
Tendency
Amplitude/N
Amplitude/N
4 1
0
Tendency
3 2 1
Tendency
-1
Negative force
Blockage
0 -1
-2 0
40
80
120
160
200
240
280
320
360
0
40
80
Circumferential angle/deg
120
160
200
240
280
320
360
Circumferential angle/deg
Fig. 17. Axial force (left) and tangential force (right) of single rotor blade with different solidity at 0.5 partial admission.
2
6
leakage model no leakage model
1
4
Amplitude/N
Amplitude/N
leakage model no leakage model
5
0
-1
3 2 1
Negative force 0
-2 0
40
80
120
160
200
240
280
320
360
-1 0
40
80
Circumferential angle/deg
120
160
200
240
280
320
360
Circumferential angle/deg
Fig. 18. Axial force (left) and tangential force (right) of single rotor blade with different model at 0.5 partial admission.
tangential force of all rotor blades has optimum value. Moreover, the ratio at full admission is relatively larger than that at partial admission, and ratio difference tends to increase with rotor solidity. It is necessary to be pointed out that the ratio of axial force and tangential force exceeds 50% when the rotor solidity is approximately 1.7, which is suggested to be considered for the turbine design. The discussion above is mainly based on time domain analysis while it is hardly to obtain the frequency of disturbances from the time domain. Hence, unsteady force is analyzed based on FFT method, and then the frequency domain data can be obtained. The rotor unsteady force, x(t), can be expressed as follows:
1.5
ε=0.5 no leakage ε=1 no leakage ε=0.5 leakage
FAA/FAT
1.2
0.9
0.6
0.3
0.0 0.9
1.2
1.5
1.8
2.1
2.4
Solidity Fig. 19. Ratios of axial force and tangential force of all blades.
xðtÞ ¼
am eimt
ð6Þ
m¼1
where x(t) is with a period of 2p and exponent coefficient can be shown as Eq. (7).
am ¼ the turbine performance to some extent. Negative effect of axial force is relatively weaker for small ratio of axial force and tangential force of all blades. Ratios of axial force and tangential force of all blades are presented as Fig. 19. The ratio increases with solidity due that axial force of all rotor blades increase with solidity while
þ1 X
1 2p
Z 2p
xðtÞeimt dt
ð7Þ
0
Frequency analysis of axial forces and tangential forces of solidity 1.30, 1.50 with no leakage model as well as solidity 1.30 with leakage model at 0.5 partial admission are presented in Figs. 20–22. The analyzed rotation speed is 5600 r/min and it can
937
1.0
3.0
0.8
2.4
Amplitude/N
Amplitude/N
K. Gao et al. / Applied Thermal Engineering 128 (2018) 926–939
0.6
0.4
0.2
0.0
1.8
1.2
0.6
0
1200
2400
3600
4800
0.0
6000
0
1200
Frequency/Hz
2400
3600
4800
6000
Frequency/Hz
1.0
3.0
0.8
2.4
Amplitude/N
Amplitude/N
Fig. 20. Frequency analysis of axial force (left) and tangential force (right) of solidity 1.57 with no leakage model at 0.5 partial amdission.
0.6
0.4
0.2
0.0
1.8
1.2
0.6
0
1200
2400
3600
4800
0.0
6000
0
1200
Frequency/Hz
2400
3600
4800
6000
Frequency/Hz
1.0
3.0
0.8
2.4
Amplitude/N
Amplitude/N
Fig. 21. Frequency analysis of axial force (left) and tangential force (right) of solidity 1.30 with no leakage model at 0.5 partial amdission.
0.6
0.4
0.2
0.0
1.8
1.2
0.6
0
1200
2400
3600
4800
6000
0.0
0
1200
Frequency/Hz
2400
3600
4800
6000
Frequency/Hz
Fig. 22. Frequency analysis of axial force (left) and tangential force (right) of solidity 1.30 with leakage model at 0.5 partial amdission.
Table 2 Aerodynamic exciting force of single rotor of different solidity at full admission. Solidity
Axial aerodynamic exciting force, N
Frequency, n/60
Tangential aerodynamic exciting force, N
Frequency, n/60
1.09 1.30 1.57 1.87 2.24
0.134 0.143 0.126 0.0965 0.0745
41 41 41 41 41
0.248 0.198 0.201 0.225 0.191
41 41 41 41 41
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Table 3 Aerodynamic exciting force of single rotor of different solidity at 0.5 partial admission. Solidity
Axial aerodynamic exciting force, N
Frequency, n/60
Tangential aerodynamic exciting force, N
Frequency, n/60
1.09 1.30 1.57 1.87 2.24
0.702, 0.719, 0.868, 1.018, 1.295,
2, 2, 2, 2, 2,
2.750, 2.377, 1.947, 1.561, 1.115,
2, 2, 2, 2, 2,
0.155, 0.230, 0.294, 0.320, 0.316,
0.260, 0.302, 0.312, 0.304, 0.301,
0.151 0.188 0.192 0.183 0.178
4, 4, 4, 4, 4,
6, 6, 6, 6, 6,
8 8 8 8 8
0.527 0.473 0.408 0.333 0.252
6 6 6 6 6
Table 4 Aerodynamic exciting force of single rotor of solidity 1.30 with different model at 0.5 partial admission.
No leakage model Leakage model
Axial aerodynamic exciting force, N
Frequency, n/60
Tangential aerodynamic exciting force, N
Frequency, n/60
0.719, 0.230, 0.302, 0.188 0.523, 0.059, 0.088, 0.058
2, 4, 6, 8 2, 4, 6, 8
2.377, 0.473 2.409, 0.546
2, 6 2, 6
be found that large amplitude appears at low frequency. Further, vibration amplitudes are larger for the even-order frequency due to two symmetric inlet sections and the largest amplitude appears at second order frequency (186.7 Hz). In addition, the high frequency exciting force caused by stator wakes is much less than the low frequency exciting force caused by partial admission. Hence, the high frequency exciting force is not obvious in the figure. The detailed comparisons of axial and tangential exciting force have been done based on FFT method. The aerodynamic exciting force at full admission, 0.5 partial admission and leakage model are shown in Tables 2–4, respectively. As for full admission, large amplitude of aerodynamic exciting force is mainly caused by stator wakes and the relative frequency is related to stator numbers. The largest amplitude of axial exciting forces increases and then decreases with rotor solidity, and there is no obvious variation tendency for that of tangential exciting forces. As for 0.5 partial admission, large amplitude is mainly caused by partial admission, which is different from full admission. Hence, the effects of solidity on aerodynamic exciting force changes. The axial force of inlet section increases with rotor solidity while the axial force of blockage remains low. Hence, the largest amplitude of axial exciting force, which appears at second order frequency, increases with rotor solidity. In contrast, the tangential force decreases with rotor solidity increasing and then the frequency excitation amplitude of tangential force decreases with rotor solidity increasing. As for leakage model, the largest excitation amplitude of axial force of leakage model is lower than that of no leakage model and the excitation amplitude of tangential force is less affected as explained before. That means the leakage flow in the inlet and outlet cavity contributes to the reduction of excitation amplitude of axial aerodynamic force. 7. Conclusion The effects of solidity and leakage on unsteady flow in axial turbine have been conducted to reveal the flow mechanism for higher efficiency and reliability. The effects of rotor solidity on unsteady flow have been investigated based on 5 kinds of rotor solidity with 2 kinds of admission degree and the effects of leakage on unsteady flow are obtained via the model with/without tip clearance, inlet and outlet cavity. Some of the important conclusions are presented as follows, in order to provide the guidance for the design of axial turbine especially at partial admission: (1) Due to the decreasing throat area with rotor solidity increasing, the mass flow accelerates to decrease. There exists the optimum solidity for efficiency and torque, and the optimum
solidity of efficiency is higher than that of torque due to the mass flow variation. According to our simulations, the optimum solidity of efficiency is near 1.57 for our model. In addition, the optimum solidity for torque of partial admission is larger than that of full admission. The torque and efficiency of leakage model is relatively lower. (2) The pressure distribution is influenced obviously by solidity, and it can be found that the positive angle and adverse pressure gradient are the main causes of low pressure regions for small solidity while high speed flow interactions for large solidity. For partial admission turbine, parameters which change at the transition region between admission and blockage section have been marked out, such as pressure, which is less affected by solidity. However, it can be eased obviously by leakage flow. (3) The pressure of the pressure surface remains similar from small rotor solidity to optimum value while the pressure at leading edge of the suction surface increases significantly. Moreover, pressure changes a lot both at the pressure surface and the suction surface when the rotor solidity is larger than optimum value, and the pressure near the middle part decreases even dramatically due to the high speed flow interactions. Due to the cavity interactions, the rotor blade load is affected by leakage model. Rotor pressure is slightly higher at admission region and lower at blockage than that of no leakage model. (4) The axial force of single blade at the inlet admission increases with the rotor solidity while the power capability decreases. In addition, high frequency excitation amplitudes of the axial force caused by stator wakes decrease with the rotor solidity increasing, however, low frequency excitation amplitudes of the axial force caused by partial admission increase. Therefore, we can reduce the frequency excitation amplitude via the modification of solidity considering our operation condition. It is found that excitation amplitudes of the axial force can be reduced obviously through leakage model; hence, it may be beneficial to allow appropriate leakage at inlet and outlet cavity to improve reliability. References [1] G.O. Ohlsson, Partial-Admission Turbines, J. Aero. Sci. 29 (1962) 1017–1028. [2] L. He, Computation of unsteady flow through steam turbine blade rows at partial admission, P. I. Mech. Eng. A-J. Pow. 211 (1997) 197–205. [3] N. Baagherzadeh Hushmandi, Numerical analysis of partial admission in axial turbines, KTH Industrial Engineering and Management. Stockholm, Sweden, 2010. [4] Y. Xie, K. Gao, J. Lan, G. Xie, Computational fluid dynamics modeling threedimensional unsteady turbulent flow and excitation force in partial admission air turbine, Math. Probl. Eng. 2013 (2013).
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[20] J.E. Anker, J.r.F. Mayer, Simulation of the interaction of labyrinth seal leakage flow and main flow in an axial turbine, in: ASME Turbo Expo 2002: Power for Land, Sea, and Air, American Society of Mechanical Engineers, 2002, pp. 217– 224. [21] P. Peters, V. Breisig, A. Giboni, C. Lerner, H. Pfost, The influence of the clearance of shrouded rotor blades on the development of the flowfield and losses in the subsequent stator, in: ASME Turbo Expo 2000: Power for Land, Sea, and Air, American Society of Mechanical Engineers, 2000, pp. V001T003A057– V001T003A057. [22] A. Pfau, J. Schlienger, D. Rusch, A. Kalfas, R. Abhari, Unsteady flow interactions within the inlet cavity of a turbine rotor tip labyrinth seal, J. Turbomach. 127 (2005) 679–688. [23] J.D. Coull, N.R. Atkins, The influence of boundary conditions on tip leakage flow, J. Turbomach. 137 (2015) 061005. [24] Y.C. Nho, J.S. Park, Y.J. Lee, J.S. Kwak, Effects of turbine blade tip shape on total pressure loss and secondary flow of a linear turbine cascade, Int. J. of Heat Fluid Flow. 33 (2012) 92–100. [25] S.W. Lee, J.H. Cheon, Q. Zhang, The effect of full coverage winglets on tip leakage aerodynamics over the plane tip in a turbine cascade, Int. J. of Heat Fluid Flow. 45 (2014) 23–32. [26] J.H. Cheon, S.W. Lee, Tip leakage aerodynamics over the cavity squealer tip equipped with full coverage winglets in a turbine cascade, Int. J. of Heat Fluid Flow. 56 (2015) 60–70. [27] L. Silva, J. Tomita, C. Bringhenti, Numerical investigation of a HPT with different rotor tip configurations in terms of pressure ratio and efficiency, Aerosp. Sci. Technol. 63 (2017) 33–40. [28] T.J. Barth, D.C. Jespersen, The design and application of upwind schemes on unstructured meshes, in: 27th Aerospace Sciences Meeting, Reno, NV, USA, 1989. [29] A. Jameson, Time dependent calculations using multigrid, with applications to unsteady flows past airfoils and wings, AIAA Paper 1596 (1991) 1991. [30] T. Matsunuma, Unsteady flow field of an axial-flow turbine rotor at a low Reynolds number, J. Turbomach. 129 (2007) 360–371. [31] K. Yamada, K. Funazaki, M. Kikuchi, H. Sato, Influences of Axial Gap Between Blade Rows on Secondary Flows and Aerodynamic Performance in a Turbine Stage, in: ASME Turbo Expo 2009: Power for Land, Sea, and Air, American Society of Mechanical Engineers, 2009, pp. 1039–1049. [32] D. Chang, S. Tavoularis, Effect of the axial spacing between vanes and blades on a transonic gas turbine performance and blade loading, Int. J. Turbo. Jet. Eng. 30 (2013) 15–31. [33] N.B. Hushmandi, T.H. Fransson, Effects of multiblocking and axial gap distance on performance of partial admission turbines: a numerical analysis, J. Turbomach. 133 (2011) 031028.
Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Research Paper
Effects of rotor solidity and leakage flow on the unsteady flow in axial turbine Keke Gao a, Yonghui Xie a,⇑, Di Zhang b a b
Shaanxi Engineering Laboratory of Turbomachinery and Power Equipment, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China MOE Key Laboratory of Thermo-Fluid Science and Engineering, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China
h i g h l i g h t s The solidity effects on the unsteady flow both in full and partial admission turbines are revealed. The combined effects of leakage flow and partial admission flow are presented. The frequency excitation amplitude can be modified via solidity. Leakage flow is beneficial to the axial aerodynamic exciting force reduction.
a r t i c l e
i n f o
Article history: Received 21 March 2017 Revised 30 July 2017 Accepted 15 September 2017 Available online 18 September 2017 Keywords: Unsteady flow Axial turbine Partial admission Aerodynamic force Loss
a b s t r a c t Partial admission turbine plays an important role in power control, which strengthens the unsteady flow. The effects of rotor solidity at partial admission condition can be more complicated, and leakage flow deeply strengthens the unsteadiness. Hence, the investigations of rotor solidity and leakage flow on the unsteady flow are of great significance for turbine designs. Turbines including five kinds of solidity at full and partial admissions are modeled based on 3D viscous compressible NS equation. In addition, the leakage flow model including tip leakage, inlet and outlet cavity is investigated. The results show that the solidity effects on partial admission turbine are not exactly the same as that on full admission turbine. The differences are identified, especially the axial unsteady aerodynamic force. Moreover, the attack angle tends to be negative with rotor solidity increasing; meanwhile, the low pressure region varies with solidity due to flow separation or throat area reduction. Leakage flow model is more able to reveal the unsteady flow, and the comparative analysis of flow phenomenon under the combined effects of leakage flow and partial admission flow is conducted. The change of rotor inlet parameters is smoothed down for leakage model and the axial exciting force is relative lower. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction The output power is usually adjusted by three ways including sliding pressure, throat value and partial admission to meet the requirement of external load. The sliding pressure operation adjusts loads through the boiler. Hence, it takes a long time to cope with changes. The mass flow can also be controlled with throat value while the efficiency is relatively lower due to large pressure losses and short blades. Partial admission method can meet the load change through the intake area adjustment. Further, it is widely used due to no throttling loss, low secondary losses and the rapid load response. However, the unsteady flow in turbine is extremely complicated and then the partial admission construction ⇑ Corresponding author. E-mail address: [email protected] (Y. Xie). http://dx.doi.org/10.1016/j.applthermaleng.2017.09.078 1359-4311/Ó 2017 Elsevier Ltd. All rights reserved.
strengthens the unsteadiness. Compared to full admission turbines, windage loss and arc end loss are induced at the partial admission turbine as well as unsteady low frequency excitation of aerodynamic forces. In order to realize higher efficiency and reliability, the investigations of partial admission are of a great significance to the industry. Nowadays, the deep researches on the unsteady flow in partial admission turbines have been carried out. The performance of partial admission turbines with impulse design has been discussed by Ohlsson [1] through theoretical method based on incompressibility and non-friction assumption. The detailed flow parameter distributions of the partial admission control stage have been presented by He [2] through 2D unsteady numerical method. It also points out that the mixing process may benefit the turbine overall performance. Hushmandi [3] indicates that 3D influence on the turbine unsteady flow is very important based on two low reaction stages
K. Gao et al. / Applied Thermal Engineering 128 (2018) 926–939
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Nomenclature C 3e Mt k T c2 w2 n Cax,rotor FAT am m Cax,blade FAA w1 Dhast c1 u
variables which determines how dissipation rate affected by the buoyancy, [–] turbulent Mach number, [–] kinetic energy of turbulence, [kJ] torque, [N⁄m] absolute velocity at rotor outlet, [m⁄s1] relative velocity at rotor outlet, [m⁄s1] rotation speed, [r⁄min1] axial chord of rotor tangential force of all blades exponent coefficient exciting order axial chord of blade axial force of all blades relative velocity at rotor inlet, [m⁄s1] ideal enthalpy drop absolute velocity at rotor inlet, [m⁄s1] velocity components in a generalized coordinate system, [m⁄s1]
model. Unsteady flow and exciting force under four partial admission degrees are in detailed comparison based on 3D unsteady numerical investigation by Xie [4]. Bohn [5] and Fridh [6] conduct experimental investigations on the unsteady flow in turbines. The flow parameters non-uniformity caused by partial admission is also pointed out. Tousi [7] shows that the turbine efficiency is closely related to partial admission degree. And the optimum partial admission degree is obtained through numerical analyze. The flow parameters of the partial admission turbine at rated and offdesigned conditions are investigated by Song [8]. In addition, the partial admission turbine used in ORC are analyzed by Cho [9] and Martins [10]. Although the investigations on partial admission are very extensive, only a few studies focus on the effect of rotor solidity on turbine performance. The solidity plays an important role in blade design. As far back as 1945, Zweifel [11] has estimated optimum solidity for turbines with large angular deflection. However, Horlock [12] and Aungier [13] points out the Zweifel correlation limitation that the prediction of optimum solidity is also limited to the outlet flow angle. In addition, the effects of rotor solidity on the full admission stage efficiency have also been investigated by Simpson [14] and the optimum solidity is approximately 1.25 for the investigated model. Only little information regarding the optimum solidity for axial or radial turbine stages is available, while the important effects of solidity on wind turbine performance have been valued [15,16]. When it comes to the partial admission turbine investigations, the references can be much less. According to the open literature, Cho [17,18] has merely done preliminary research on the effects of rotor solidity on aerodynamic force and the results show that the tangential force of single rotor increases with the reduction of rotor solidity while the axial force decreases. Meanwhile, limited to the computing resource and the complexity of modeling, the researches of leakage flow effects are mainly focused on the turbine efficiency or heat transfer, which are with steady leakage models or unsteady periodic admission models. The research can be of a great benefit to understand the leakage flow on the turbine aerodynamic performance. The influences of the tip geometry on leakage flows are investigated by Krishnababu [19] through single passage with the periodic boundary. The result shows the leakage mass flow increases with the gap and the hollow structure contributes to the reduction of leakage
Cl _ m Gb
modified constant, [–] mass, [kg] generation of turbulence due to buoyancy, [–]
Greek
ak q e leff
b
a1 a2
b1 b2
g
f
inverse effective Prandtl numbers, [–] density, [kg⁄m3] dissipation rate, [–] effective dynamic viscosity coefficient, [Pa⁄s] coefficient of thermal expansion, [–] absolute flow angle at rotor inlet, [deg] absolute flow angle at rotor outlet, [deg] relative flow angle at rotor inlet, [deg] relative flow angle at rotor outlet, [deg] efficiency degree of admission
mass flow. The leakage flow and main flow interaction is studied by Anker [20] based on the numerical method, and it is found that the leakage flow along the pitch direction is not uniform. The tip leakage flow and secondary flow interaction is investigated through the unsteady 1.5 stage axial turbine model with shroud by Peters [21]. Pfau [22] obtains the unsteady flow interaction in the rotor inlet cavity based on experimental method for full admission. The vortex structures in the cavity are illustrated. The effects of inlet boundary condition and the relative motion of case on the tip leakage flow have been presented by Coull [23]. The flow structures under different group are also given. In addition, the tip leakage shapes effects on the aerodynamic performance are also investigated by Nho [24], Lee [25,26] and Silva [27]. Partial admission which plays an import role in the power control strengthens the unsteady effects and induces partial admission losses as well as external unsteady exciting forces. Considering the limited investigated model, it is not clear how the aerodynamic performance and unsteady exciting force change under the combined effects of leakage flow and partial admission flow. However, it is extremely important to obtain the accurate flow details and unsteady exciting forces to guarantee higher efficiency and reliability during the control of power output. As explained above, it is still not clear how rotor solidity affects partial admission aerodynamic parameters and exciting forces. Furthermore, there is hardly any information to illustrate the effects of leakage flow including tip clearance, inlet and outlet cavity on unsteady flow at partial admission. It is necessary to know how the solidity and leakage flow behave at partial admission in an attempt to obtain higher efficiency and reliability. For these reasons, the paper is divided into two parts. First, the turbine unsteady flow performances with 5 kinds of rotor solidity are obtained to reveal the effects of rotor solidity. Secondly, the leakage model including tip clearance, inlet and outlet cavity is modeled, and then the unsteady effects of leakage flow on the partial admission turbine can be obtained by comparing leakage model performance with the no leakage control stage model performance. According to general understanding, the optimum solidity exists due that the larger solidity will give the flow more guidance while larger viscous losses occur. The paper explains why the optimum solidity exists from the new perspective, and it finds that the solidity effects on partial admission turbines are not totally the same as that on full admission turbines. In addition, the paper presents
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the key flow parameters under the combined effects of the leakage flow and partial admission flow, such as aerodynamic forces. It is pointed out that it may be beneficial to allow appropriate leakage at inlet and outlet cavity considering the axial exciting force drop.
2. Numerical model 2.1. Computational method CFX commercial software is applied to solve the three dimensional unsteady flow. The turbulence stress term is closed with RNG k-e model which has higher precision and faster convergence speed. The detailed numerical validation will be conducted later in the paper.
@ @ @ @k þ Gk þ Gb qe Y M ðqkui Þ ¼ ak leff ðqkÞ þ @t @xi @xj @xj
ð1Þ
@ @ @ @e e þ C 1e ðGk þ C 3e Gb Þ ðqeÞ þ ðqeui Þ ¼ ae leff @t @xi @xj @xj k C 2e q
e2 k
ð2Þ
Re
where Gk means turbulence kinetic energy, YM means the influence of compressibility and Re is transport source term.
C 1e ¼ 1:42; C 2e ¼ 1:68; Y M ¼ 2qeM 2t
Re ¼
Gk ¼ u t ð
@ui @uj þ Þ; @xj @xi
C l qg3 ð1 g=g0 Þ e2 1 þ bg3 k
2.2. Computational model In order to obtain the accuracy numerical results, three times stator chord length are extended ahead of stator inlet and two times rotor chord length are extended behind rotor outlet as shown in Fig. 1. Different rotor solidity is achieved through the rotor numbers change and the relative geometry parameters are shown in Table 1. Five kinds of solidity at full admission have been investigated to understand the solidity effects on the unsteady flow when the turbine operates with full power output. Then, five kinds of solidity at 0.5 partial admission have also been investigated to reveal the solidity effects on the unsteady flow under the influence of partial admission effects. Considering the leakage effects, the flow mechanisms can be found well via the model with/without leakage keeping same geometry parameters. In addition, considering the computing resource, the effects of leakage are investigated based on one selected solidity. Total pressure, total temperature and turbulence intensity are set as inlet boundary. The average static pressure is set as outlet condition. The frozen rotor method is applied to deal with rotor-stator interface and the steady results are used as the initial value of unsteady computation. The transient rotor stator method is used for the rotor stator interface under unsteady condition. The accuracy flow information can be captured when 15 steps are set for a single stator passage. According to previous research [4], the H-type grid is used for stator, and H-type as well as O-type grid is used for rotor. And the near wall grid refinement has been conducted to improve the calculation accuracy and grid independence is needed to speed up the calculation. Finally, approximately 7 million grids are used to complete the research for no leakage model and 10 million grids are used for leakage model.
ð3Þ 2.3. Computational validation
Convection term discretization is solved by high order accurate schemes proposed by Barth [28] and time term discretization is solved by the dual time step implicit iteration method proposed by Jamson [29].
outlet
inlet
stator
The numerical investigations based on Matsunuma [30] model have been conducted to validate the numerical method through the data comparison in the paper. Takayuki Matsunuma’s
outlet cavity
inlet cavity
rotor
tip leakage
Fig. 1. Full numerical model (left: full admission with no leakage simplification; middle: partial admission with no leakage simplification; right: partial admission with leakage model).
Table 1 Cascade geometry parameters. Type
A
B
C
D
E
Rotor numbers Mean diameter (mm) Chord (c:mm) Blade height (h:mm) Area ratio (h/c) Pitch at mean diameter (s:mm) Solidity (c/s)
61 332 18.60 27 1.45 17.10 1.09
73 332 18.60 27 1.45 14.29 1.30
88 332 18.60 27 1.45 11.85 1.57
105 332 18.60 27 1.45 9.93 1.87
126 332 18.60 27 1.45 8.27 2.24
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numerical method can be sufficiently proved if the numerical results below 70% blade length have a good agreement with the test results. It can be found the numerical data of 5 models agree with experimental results well while RNG k-e turbulence model can be better, especially for axial velocities and relative flow angle. In addition, RNG k-e turbulence model is also adopted by Hushmandi [3] based on the comparison of his experimental data and numerical results. Finally, RNG k-e turbulence model is adopted in the paper and the numerical method is validated to be reliable.
3. Results and discussion Solidity has been widely studied as an important design parameter for wind turbine; however, there is less solidity research related to axial or radial turbine. Moreover, few researches on the leakage flow influence on partial admission turbine can be found. In order to reveal the mechanism of turbine performance, three crucial parameters including mass flow, torque and efficiency are presented first in the paper. Turbine mass flows of different model are shown as Fig. 4. It is obvious that the mass flow decreases with rotor solidity due that the throat area is reduced and mass flow tends to accelerate to decrease. Further, it can be seen that the mass flow ratio of 0.5 partial admission and full admission is over 50%, and the mass flow ratio increases with rotor solidity. It means full admission turbine is easier to be influenced by solidity. In addition, the mass flow of leakage model is higher than that of no leakage model.
Fig. 2. Annular turbine wind tunnel [30].
experimental model including the single axial stage is shown as Fig. 2. The axial velocity of 4.4 m/s and turbulent intensity of 0.5% are given as inlet boundary. The outlet Reynolds number based on outlet velocity and rotor chord is set as 3.5 ⁄ 104. The rotor speed is 402 rpm. Numerical results of 5 kinds of turbulence model including SST, k-e, k-w, BSL Reynold stress and RNG k-e are compared with experimental data. The rotor outlet axial velocity, tangential velocity and velocity angle are shown as Fig. 3. The simplified model with no tip clearance has been adopted for numerical simulation to reduce the computing cost. Hence, only the results related to 070% blade length are compared, which is less influenced by tip leakage. Then
0.7
0.4
Spanwise distance Y/H
0.5
0.6
SST kkBSL Reynold stress RNG kExperiment
0.3 0.2 0.1 0.0 0.0
0.5 0.4
SST kkBSL Reynold stress RNG kExperiment
0.3 0.2 0.1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.0 0.2
Axial velocities Vz/V2,Des
0.4
0.6
0.8
0.6 0.5 0.4
SST kkBSL Reynold stress RNG kExperiment
0.3 0.2 0.1 0.0 30
1.0
Tangential velocities Vx,Rel/V2,Des
0.7
Spanwise distance Y/H
Spanwise distance Y/H
0.6
0.7
40
50
60
70
80
Relative flow angle Fig. 3. Turbulence model verification with the comparison of experimental data [Takayuki Matsunuma model].
1.2
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1.8
1.0
=0.5 no leakage =1 no leakage =0.5 leakage
1.6
=0.5 no leakage =1 no leakage =0.5 leakage
0.9
Mass flow/kg·s
1.4
0.8
1.2 1.0
0.7
0.8
0.6
0.6 0.4 0.9
1.2
1.5
1.8
2.1
0.5 0.9
2.4
1.2
Solidity
Torque/N·m
35 25 15 5 0.9
1.2
1.5
2.1
2.4
Fig. 6. Efficiencies of different model.
=0.5 no leakage =1 no leakage =0.5 leakage
45
1.8
Solidity
Fig. 4. Mass flows of different model.
55
1.5
1.8
2.1
2.4
Solidity Fig. 5. Torques of different model.
Fig. 5 shows the turbine torques of different model. There exists optimum solidity of maximum torque. When rotor solidity is too small, the turbine torque is lower due to relatively less rotor numbers. And the turbine torque is also lower for large rotor solidity due to the power capability reduction of single rotor. It can be seen that the optimum torque of partial admission turbine is larger than that of full admission turbine. That is, the power reduction with rotor solidity increasing at partial admission is less. In addition, the torque of leakage model is lower than that of no leakage model. Efficiencies of different model are shown in Fig. 6. It can be seen that there exists optimum rotor solidity of efficiency. Further,
optimum rotor solidity of efficiency is higher than that of torque and optimum rotor solidity of partial admission efficiency is very close to that of full admission efficiency. The phenomenon above can be explained through Eqs. (4) and (5), in which efficiency is related to torque, rotating speed and ideal enthalpy drop. Further, there exists no mass flow term in Eq. (5) and mass flow accelerates to decrease with rotor solidity increasing as shown in Fig. 4. Combined with Eq. (4), it can be obtained that optimum rotor solidity of efficiency is higher than that of torque. Meanwhile, optimum rotor solidity of partial admission torque is slightly higher than that of full admission torque as shown in Fig. 5 and then mass flow term in efficiency is offset. Hence, the optimum rotor solidity of partial admission efficiency should be close to that of full admission efficiency. In addition, the efficiency of leakage model is lower than that of no leakage model. The fundamental performance parameters have been already presented now and the flow mechanism will be revealed through flow parameters later.
_ 1 cos a1 þ c2 cos a2 Þ ¼ mðw _ 1 cos b1 þ w2 cos b2 Þ T ¼ mðc
g¼
Tn n ¼ ðw cos b1 þ w2 cos b2 Þ _ Dh 9550Dh 1 9550m
ð4Þ ð5Þ
4. Flow parameters The effects of rotor solidity on flow parameters at full admission turbine are presented first to further study the cascade flow mechanisms. The distributions at 50% spanwise have been widely used as the representative cross to investigation the aerodynamic
Fig. 7. Pressure distributions at 50% blade height of different solidity at full admission (left: solidity 1.09; middle: solidity 1.57; right solidity 2.24).
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126000
solidity 1.09 solidity 1.30 solidity 1.57 solidity 1.87 solidity 2.24
Pressure/Pa
117600
Rotor inlet 109200
Rotor outlet 100800 92400 84000 0.0
.
Stator outlet 0.5
1.0
1.5
2.0
Streamwise location(x/Cax,blade )
Fig. 8. Pressure distribution along flow direction of different solidity at full admission.
characteristics of a turbine [31–33], which is also discussed as representative cross of the overall patterns in the paper. Pressure distributions at 50% blade height of different solidity at full admission are shown in Fig. 7. It can be found that rotor inlet pressure rises
with rotor solidity due that the flow resistance in rotor passage increases. Further, the rotor inlet flow angle (the angle between flow direction and axial direction) decreases with the rotor solidity increasing and then rotor attack angle has gradually turned positive value into negative value. Low pressure region of rotor are marked out with dash line for the convenience of observation. As for small rotor solidity, flow separation appears at the leading edge and trailing edge of rotor suction due to large positive attack angle and adverse pressure gradient. As for optimum rotor solidity, there is only flow separation at the trailing edge of rotor suction due to adverse pressure gradient and the low pressure region is not completely obvious. As rotor solidity increases, rotor inlet attack angle tends to be negative and then flow rushes forward to rotor suction. The low pressure region appears near the rotor throat. Different from the previous cause, the low pressure appears due that flow accelerates to pass through the throat region and the flow nears suction and pressure surface interacts strongly for close distance. Then, the power capability decreases. Pressure distribution along flow direction of different solidity at full admission is shown in Fig. 8. It is obvious that the pressure drop in stator passage gradually decreases with rotor solidity increasing. For small rotor solidity as 1.09 and 1.30, rotor inlet pressure is relatively lower due to large pressure drop in stator
B
B
A
A
Inlet section
Blockage
Inlet section
Blockage
Blockage
Inlet section
Blockage
Inlet section
A
A
B
B
(a)
(b) B
B
A
A
Inlet section
Blockage
Inlet section
Blockage
Blockage
Inlet section
Blockage
Inlet section
A
A
B
B
(c)
(d)
Fig. 9. Rotor inlet pressure distributions at 0.5 partial admission (a: solidity 1.57; b: solidity 2.24; c: solidity 1.30; d: solidity 1.30, leakage model).
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backflow appears at blockage. Axial flow velocity decreases with rotor solidity increasing, which is consistent with the mass flow variation rule. Serious backflow mainly appears at the transition region from inlet section to blockage as marked with B. Meanwhile, high pressure appears at B region as shown in Fig. 9. In addition, high axial flow velocity also can be found at the transition region from blockage to inlet section as marked with C. The position of B and C are mainly effected by the distribution of admission and blockage section, instead of rotor solidity. B and C region of leakage model is eased obviously, and the axial flow velocity in inlet cavity is relatively low. Rotor inlet static entropy distributions at 0.5 partial admission are shown as Fig. 11. The static entropy at inlet section is relatively lower than that at blockage and stator wakes can also be observed. The static entropy at blockage is larger mainly due to windage loss. The rotor inlet static entropy decreases with rotor solidity increasing due to the reaction changed. Meanwhile, aiming at leakage model, we can find that the static entropy at inlet cavity is large due that the cavity is full of low energy fluid and the static entropy at rotor passage inlet section also increases compared to no leakage model.
passage. Further, the pressure increases and then decreases in rotor passage. For large solidity as 2.24, the pressure drops sharply in rotor passage and then pressure increase due to the limitation of back pressure. Hence, both low solidity and large solidity are not beneficial to turbine flow. The effects of rotor solidity and leakage model on flow at partial admission are presented in the paper. Rotor inlet pressure increases with solidity due to the increasing of rotor passage resistance, which is consistent with that at full admission as shown in Fig. 9. It is found that low pressure region appears at the transition region from inlet section to blockage as marked with A due to suction effects. Meanwhile, high pressure region marked as B can be found near A region, which is probably related to backflow in rotor passage. Moreover, A and B region positions are not changed with rotor solidity. In addition, A and B region of leakage model is not as obvious as no leakage model, because the cavity and tip leakage contribute to the parameters uniformity to some extent. The pressure in inlet cavity is relatively low. Rotor inlet axial flow velocity distributions at 0.5 partial admission are shown as Fig. 10. It can be seen that the axial flow velocity is larger at inlet section and lower at blockage. Moreover, even the
B
B
C
Inlet section
C
Blockage
Inlet section
Blockage C
C Inlet section
Blockage
Inlet section
Blockage
B
B
(a)
(b) B
B
C
Inlet section
C
Blockage
Inlet section
Blockage C
C Inlet section
Blockage
B
Inlet section
Blockage
B
(c)
(d)
Fig. 10. Rotor inlet axial flow velocity distributions at 0.5 partial admission (a: solidity 1.57; b: solidity 2.24; c: solidity 1.30; d: solidity 1.30, leakage model).
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Arc end losses Stator wakes Inlet section
Blockage
Inlet section
Blockage
Blockage
Inlet section
Blockage
Inlet section
Windage loss
(a)
(b) Inlet cavity leakage loss
Inlet section
Blockage
Inlet section
Blockage
Blockage
Inlet section
Blockage
Inlet section
(c)
(d)
Fig. 11. Rotor inlet static entropy distributions at 0.5 partial admission (a: solidity 1.57; b: solidity 2.24; c: solidity 1.30; d: solidity 1.30, leakage model).
5. Rotor blade load
solidity 1.09 solidity 1.30 solidity 1.57 solidity 1.87 solidity 2.24
110000 104000
Pressure/Pa
The power capability depends on rotor blade loads and it can be further known through rotor blade pressure distributions. The rotor blade loads at full admission are first presented to understand the effects of rotor solidity on rotor blade load as shown in Fig. 12. Effects of rotor blade solidity are quite obvious. The largest pressure difference between pressure and suction surface comes to the smallest solidity and then the power capability of single rotor is the strongest. From small rotor solidity to optimum solidity, the pressure change on pressure surface is small and the pressure on suction surface at leading and middle part increases apparently. From optimum rotor solidity to large solidity, both pressure surface and suction surface have great changes. The pressure on suction surface at leading part still increase while pressure on other parts of pressure surface drops dramatically, and the phenomenon is relevant to the close throat distance as shown in Fig. 7. In addition, we can found that the pressure on suction surface tends to decrease and then increase for all rotor solidity. The point where pressure gradient alters is defined as the turning point in the paper. As we all know, flow separation occurs easily for large adverse pressure gradient and long adverse pressure distance. It can be
98000 92000
Turning point 86000 80000 0.0
0.2
0.4
0.6
0.8
1.0
Streamwise location(x/Cax,stator) Fig. 12. Rotor blade load at full admission.
seen that rotor with the optimum solidity has the minimum adverse pressure gradient and distance. In other words, the position as well as pressure difference between turning point and
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section, leading edge of rotor1 has completed positive work output due to the reduction of suction pressure. By contrast to rotor1, the pressure surface of rotor2 is at the blockage and the suction surface is at inlet section. Hence, it can be seen that pressure on pressure surface is lower than that on suction surface and then negative work has been done. The negative work region at leading edge decreases with rotor solidity. Rotor blade load comparisons between leakage mode and no leakage model at 0.5 partial admission have been presented in Fig. 15. The effects on pressure surface of rotor1is relatively small while suction surface of rotor1 has a great change. The leading edge and tailing edge of leakage model is larger than no leakage model. As for rotor2, the pressure of leakage is larger due that the inlet section flow is inhaled into the cavity. As for rotor3, the most part pressure of leakage model is also larger due that larger mass flow is obtained and cavity is filled with fluid. Rotor4 is at blockage and the pressure of leakage model is slightly lower than that of no leakage model.
rotor 1
rotor 2 rotor4
6. Unsteady flow aerodynamic force
rotor3 Fig. 13. Rotor positions schematic diagram.
trailing edge is smallest. It can also be found that negative work at rotor leading part has been done for large rotor solidity due to the negative attack angle. The selective analysis rotors at 0.5 partial admission are shown as Fig. 13, which marked as rotor1, rotor2, rotor3 and rotor4 respectively. Where, rotor1 is at the transition region from the inlet section to the blockage, rotor2 is at the transition region from the blockage to the inlet section, rotor3 is at the inlet section and rotor4 is at the blockage. Rotor blade load comparisons of different rotor solidity at 0.5 partial admission are presented at Fig. 14. Considering that rotor3 is similar to rotor blade at full admission and rotor4 has no obvious difference for solidity change due to almost no power output, hence only roto1 and rotor2 are shown. It can be seen that rotor1 has the strong ability of working due that pressure surface is at the inlet section and suction surface is at the blockage. Therefore, pressure on suction surface decreases while the pressure on pressure surface remains large. Further, compared with rotor at the inlet
solidity 1.09 solidity 1.30 solidity 1.57 solidity 1.87 solidity 2.24
110000
98000 92000
solidity 1.09 solidity 1.30 solidity 1.57 solidity 1.87 solidity 2.24
110000 104000
Pressure/Pa
Pressure/Pa
104000
98000 92000 86000
86000 80000 0.0
The unsteady axial and tangential forces of single rotor blade with different solidity at full admission are presented as Fig. 16. Periodic sine wave of aerodynamic force at full admission appears due to stator wakes and potential field. Axial force gradually increases with rotor solidity due to rotor passage resistance increasing. However, tangential force decreases with rotor solidity increasing, which is opposite to the variation tendency of axial force. In addition, pulsation amplitude of axial force decreases with rotor solidity increasing while there is no obvious change for tangential pulsation amplitude. The unsteady axial and tangential forces of single rotor blade with different solidity at 0.5 partial admission are presented as Fig. 17. It can be seen that the variation tendency of axial and tangential force at inlet section are approximately the same with that at full admission. The effects of stator wakes and potential field on amplitudes pulsating can also be found. It is obviously that axial force of small solidity at inlet section tends to decrease when single rotor is rotating to blockage and tangential force of small solidity at inlet section tends to increase at the same time. However, the change tendency of aerodynamic force of large solidity at inlet section remains relatively stable. Moreover, largest negative axial force appears at small rotor solidity and no obvious force difference of different solidity can be found at the blockage due to stagnant fluid.
0.2
0.4
0.6
0.8
Streamwise location(x/Cax,stator )
1.0
80000 0.0
0.2
0.4
0.6
0.8
Streamwise location(x/Cax,stator )
Fig. 14. Rotor blade load comparison of different rotor solidity at 0.5 partial admission (left: rotor1; right: rotor 2).
1.0
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106000
106000
102400
102400
98800
98800
Pressure/Pa
Pressure/Pa
leakage model no leakage model
95200
leakage model no leakage model
95200
91600
91600
88000 0.0
0.2
0.4
0.6
0.8
88000 0.0
1.0
0.2
(a)
0.6
0.8
1.0
(b) 106000
leakage model no leakage model
leakage model no leakage model
102400
102400
98800
98800
Pressure/Pa
Pressure/Pa
106000
0.4
Streamwise location(x/Cax,rotor )
Streamwise location(x/Cax,rotor )
95200
95200 91600
91600 88000 0.0
0.2
0.4
0.6
0.8
88000 0.0
1.0
0.2
0.4
0.6
0.8
1.0
Streamwise location(x/Cax,rotor )
Streamwise location(x/C ax,rotor )
(c)
(d)
Fig. 15. Rotor blade load comparison between leakage mode and no leakage model at 0.5 partial admission (a: rotor1; b: rotor 2; c: rotor3; d: rotor 4).
solidity 1.09 solidity 1.30 solidity 1.57 solidity 1.87 solidity 2.24
2.5 2.0
solidity 1.09 solidity 1.30 solidity 1.57 solidity 1.87 solidity 2.24
5
4
Amplitude/N
Amplitude/N
1.5 1.0 0.5
Mean value
3
Mean value
2 0.0
Pulsation -0.5
0
10
20 30 40 Circumferential angle/deg
Pulsation 50
1
0
10
20 30 40 Circumferential angle/deg
50
Fig. 16. Axial force (left) and tangential force (right) of single rotor blade with different solidity at full admission.
Unsteady axial force and tangential force of single rotor blade with different model at 0.5 partial admission is presented at Fig. 18. Axial force is obviously different between leakage model and no leakage model while tangential force not. As for no leakage model, the rotor axial force amplitude at the inlet section decreases gradually along circumferential direction and large negative amplitude appears at the transition region from inlet section to blockage. However, the flow parameters change more smoothly
for leakage model due to the inlet and outlet cavity. Moreover, the axial force drop at the transition from inlet section to blockage decreases obviously due that inlet cavity increases rotor inlet pressure and outlet cavity decreases rotor outlet pressure, as shown in Figs. 9 and 15. Turbine tangential force reflects the power capability while axial force is not beneficial to turbine safe operation. Hence, the ratio of axial force and tangential force of all blades can reflect
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solidity 1.09 solidity 1.30 solidity 1.57 solidity 1.87 solidity 2.24
3
Tendency 2
solidity 1.09 solidity 1.30 solidity 1.57 solidity 1.87 solidity 2.24
6 5
Tendency
Amplitude/N
Amplitude/N
4 1
0
Tendency
3 2 1
Tendency
-1
Negative force
Blockage
0 -1
-2 0
40
80
120
160
200
240
280
320
360
0
40
80
Circumferential angle/deg
120
160
200
240
280
320
360
Circumferential angle/deg
Fig. 17. Axial force (left) and tangential force (right) of single rotor blade with different solidity at 0.5 partial admission.
2
6
leakage model no leakage model
1
4
Amplitude/N
Amplitude/N
leakage model no leakage model
5
0
-1
3 2 1
Negative force 0
-2 0
40
80
120
160
200
240
280
320
360
-1 0
40
80
Circumferential angle/deg
120
160
200
240
280
320
360
Circumferential angle/deg
Fig. 18. Axial force (left) and tangential force (right) of single rotor blade with different model at 0.5 partial admission.
tangential force of all rotor blades has optimum value. Moreover, the ratio at full admission is relatively larger than that at partial admission, and ratio difference tends to increase with rotor solidity. It is necessary to be pointed out that the ratio of axial force and tangential force exceeds 50% when the rotor solidity is approximately 1.7, which is suggested to be considered for the turbine design. The discussion above is mainly based on time domain analysis while it is hardly to obtain the frequency of disturbances from the time domain. Hence, unsteady force is analyzed based on FFT method, and then the frequency domain data can be obtained. The rotor unsteady force, x(t), can be expressed as follows:
1.5
ε=0.5 no leakage ε=1 no leakage ε=0.5 leakage
FAA/FAT
1.2
0.9
0.6
0.3
0.0 0.9
1.2
1.5
1.8
2.1
2.4
Solidity Fig. 19. Ratios of axial force and tangential force of all blades.
xðtÞ ¼
am eimt
ð6Þ
m¼1
where x(t) is with a period of 2p and exponent coefficient can be shown as Eq. (7).
am ¼ the turbine performance to some extent. Negative effect of axial force is relatively weaker for small ratio of axial force and tangential force of all blades. Ratios of axial force and tangential force of all blades are presented as Fig. 19. The ratio increases with solidity due that axial force of all rotor blades increase with solidity while
þ1 X
1 2p
Z 2p
xðtÞeimt dt
ð7Þ
0
Frequency analysis of axial forces and tangential forces of solidity 1.30, 1.50 with no leakage model as well as solidity 1.30 with leakage model at 0.5 partial admission are presented in Figs. 20–22. The analyzed rotation speed is 5600 r/min and it can
937
1.0
3.0
0.8
2.4
Amplitude/N
Amplitude/N
K. Gao et al. / Applied Thermal Engineering 128 (2018) 926–939
0.6
0.4
0.2
0.0
1.8
1.2
0.6
0
1200
2400
3600
4800
0.0
6000
0
1200
Frequency/Hz
2400
3600
4800
6000
Frequency/Hz
1.0
3.0
0.8
2.4
Amplitude/N
Amplitude/N
Fig. 20. Frequency analysis of axial force (left) and tangential force (right) of solidity 1.57 with no leakage model at 0.5 partial amdission.
0.6
0.4
0.2
0.0
1.8
1.2
0.6
0
1200
2400
3600
4800
0.0
6000
0
1200
Frequency/Hz
2400
3600
4800
6000
Frequency/Hz
1.0
3.0
0.8
2.4
Amplitude/N
Amplitude/N
Fig. 21. Frequency analysis of axial force (left) and tangential force (right) of solidity 1.30 with no leakage model at 0.5 partial amdission.
0.6
0.4
0.2
0.0
1.8
1.2
0.6
0
1200
2400
3600
4800
6000
0.0
0
1200
Frequency/Hz
2400
3600
4800
6000
Frequency/Hz
Fig. 22. Frequency analysis of axial force (left) and tangential force (right) of solidity 1.30 with leakage model at 0.5 partial amdission.
Table 2 Aerodynamic exciting force of single rotor of different solidity at full admission. Solidity
Axial aerodynamic exciting force, N
Frequency, n/60
Tangential aerodynamic exciting force, N
Frequency, n/60
1.09 1.30 1.57 1.87 2.24
0.134 0.143 0.126 0.0965 0.0745
41 41 41 41 41
0.248 0.198 0.201 0.225 0.191
41 41 41 41 41
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Table 3 Aerodynamic exciting force of single rotor of different solidity at 0.5 partial admission. Solidity
Axial aerodynamic exciting force, N
Frequency, n/60
Tangential aerodynamic exciting force, N
Frequency, n/60
1.09 1.30 1.57 1.87 2.24
0.702, 0.719, 0.868, 1.018, 1.295,
2, 2, 2, 2, 2,
2.750, 2.377, 1.947, 1.561, 1.115,
2, 2, 2, 2, 2,
0.155, 0.230, 0.294, 0.320, 0.316,
0.260, 0.302, 0.312, 0.304, 0.301,
0.151 0.188 0.192 0.183 0.178
4, 4, 4, 4, 4,
6, 6, 6, 6, 6,
8 8 8 8 8
0.527 0.473 0.408 0.333 0.252
6 6 6 6 6
Table 4 Aerodynamic exciting force of single rotor of solidity 1.30 with different model at 0.5 partial admission.
No leakage model Leakage model
Axial aerodynamic exciting force, N
Frequency, n/60
Tangential aerodynamic exciting force, N
Frequency, n/60
0.719, 0.230, 0.302, 0.188 0.523, 0.059, 0.088, 0.058
2, 4, 6, 8 2, 4, 6, 8
2.377, 0.473 2.409, 0.546
2, 6 2, 6
be found that large amplitude appears at low frequency. Further, vibration amplitudes are larger for the even-order frequency due to two symmetric inlet sections and the largest amplitude appears at second order frequency (186.7 Hz). In addition, the high frequency exciting force caused by stator wakes is much less than the low frequency exciting force caused by partial admission. Hence, the high frequency exciting force is not obvious in the figure. The detailed comparisons of axial and tangential exciting force have been done based on FFT method. The aerodynamic exciting force at full admission, 0.5 partial admission and leakage model are shown in Tables 2–4, respectively. As for full admission, large amplitude of aerodynamic exciting force is mainly caused by stator wakes and the relative frequency is related to stator numbers. The largest amplitude of axial exciting forces increases and then decreases with rotor solidity, and there is no obvious variation tendency for that of tangential exciting forces. As for 0.5 partial admission, large amplitude is mainly caused by partial admission, which is different from full admission. Hence, the effects of solidity on aerodynamic exciting force changes. The axial force of inlet section increases with rotor solidity while the axial force of blockage remains low. Hence, the largest amplitude of axial exciting force, which appears at second order frequency, increases with rotor solidity. In contrast, the tangential force decreases with rotor solidity increasing and then the frequency excitation amplitude of tangential force decreases with rotor solidity increasing. As for leakage model, the largest excitation amplitude of axial force of leakage model is lower than that of no leakage model and the excitation amplitude of tangential force is less affected as explained before. That means the leakage flow in the inlet and outlet cavity contributes to the reduction of excitation amplitude of axial aerodynamic force. 7. Conclusion The effects of solidity and leakage on unsteady flow in axial turbine have been conducted to reveal the flow mechanism for higher efficiency and reliability. The effects of rotor solidity on unsteady flow have been investigated based on 5 kinds of rotor solidity with 2 kinds of admission degree and the effects of leakage on unsteady flow are obtained via the model with/without tip clearance, inlet and outlet cavity. Some of the important conclusions are presented as follows, in order to provide the guidance for the design of axial turbine especially at partial admission: (1) Due to the decreasing throat area with rotor solidity increasing, the mass flow accelerates to decrease. There exists the optimum solidity for efficiency and torque, and the optimum
solidity of efficiency is higher than that of torque due to the mass flow variation. According to our simulations, the optimum solidity of efficiency is near 1.57 for our model. In addition, the optimum solidity for torque of partial admission is larger than that of full admission. The torque and efficiency of leakage model is relatively lower. (2) The pressure distribution is influenced obviously by solidity, and it can be found that the positive angle and adverse pressure gradient are the main causes of low pressure regions for small solidity while high speed flow interactions for large solidity. For partial admission turbine, parameters which change at the transition region between admission and blockage section have been marked out, such as pressure, which is less affected by solidity. However, it can be eased obviously by leakage flow. (3) The pressure of the pressure surface remains similar from small rotor solidity to optimum value while the pressure at leading edge of the suction surface increases significantly. Moreover, pressure changes a lot both at the pressure surface and the suction surface when the rotor solidity is larger than optimum value, and the pressure near the middle part decreases even dramatically due to the high speed flow interactions. Due to the cavity interactions, the rotor blade load is affected by leakage model. Rotor pressure is slightly higher at admission region and lower at blockage than that of no leakage model. (4) The axial force of single blade at the inlet admission increases with the rotor solidity while the power capability decreases. In addition, high frequency excitation amplitudes of the axial force caused by stator wakes decrease with the rotor solidity increasing, however, low frequency excitation amplitudes of the axial force caused by partial admission increase. Therefore, we can reduce the frequency excitation amplitude via the modification of solidity considering our operation condition. It is found that excitation amplitudes of the axial force can be reduced obviously through leakage model; hence, it may be beneficial to allow appropriate leakage at inlet and outlet cavity to improve reliability. References [1] G.O. Ohlsson, Partial-Admission Turbines, J. Aero. Sci. 29 (1962) 1017–1028. [2] L. He, Computation of unsteady flow through steam turbine blade rows at partial admission, P. I. Mech. Eng. A-J. Pow. 211 (1997) 197–205. [3] N. Baagherzadeh Hushmandi, Numerical analysis of partial admission in axial turbines, KTH Industrial Engineering and Management. Stockholm, Sweden, 2010. [4] Y. Xie, K. Gao, J. Lan, G. Xie, Computational fluid dynamics modeling threedimensional unsteady turbulent flow and excitation force in partial admission air turbine, Math. Probl. Eng. 2013 (2013).
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