Self-validated calculation of characteristics of a Francis turbine and the mechanism of the S-shape operational instability

Transcription

1 IOP Conference Series: Earth an Environmental Science Self-valiate calculation of characteristics of a Francis turbine an the mechanism of the S-shape operational instability To cite this article: Z Zhang an M Titzschau IOP Conf. Ser.: Earth Environ. Sci. 5 6 Relate content - Rotating stall mechanism an stability control in the pump flows Z Zhang - Numerical stuy of vortex rope uring loa rejection of a prototype pump-turbine J T Liu, S H Liu, Y K Sun et al. - Effect of inner guie on performances of cross flow turbine K Koubu, K Yamasai, H Hona et al. View the article online for upates an enhancements. Recent citations - Master equation an runaway spee of the Francis turbine Zh. Zhang - Streamline similarity metho for flow istributions an shoc losses at the impeller inlet of the centrifugal pump Zh. Zhang - Mechanism of the S-Shape Characteristics an the Runaway Instability of Pump-Turbines Linsheng Xia et al This content was ownloae from IP aress on //9 at 5:55

2 Self-valiate calculation of characteristics of a Francis turbine an the mechanism of the S-shape operational instability Z Zhang an M Titzschau Rütschi Flui Lt. Brugg, Switzerlan Oberhasli Hyroelectric Power Company Lt, Innertirchen, Switzerlan zhengji.zhang@hotmail.com bstract. calculation metho has been presente to accurately estimate the characteristics of a Francis turbine. Both the shoc loss at the impeller inlet an the swirling flow loss at the Impeller exit have been confirme to ominantly influence the turbine characteristics an particularly the hyraulic efficiency. Both together totally govern the through flow of water through the impeller being at the rest. Calculations have been performe for the flow rate, the shaft torque an the hyraulic efficiency an compare with the available measurements on a moel turbine. Excellent agreements have been achieve. Some other interesting properties of the turbine characteristics coul also be erive from the calculations an verifie by experiments. For this reason an because of not using any unreliable assumptions the calculation metho has been confirme to be self-valiate. The operational instability in the upper range of the rotational spee, nown as the S-shape instability, is ascribe to the total flow separation an stagnation at the impeller inlet. In that range of the rotational spee, the operation of the Francis turbine oscillates between pump an turbine moe.. Introuction Hyraulic characteristics of a Francis turbine are unerstoo as the flow rate an the mechanical moment in function of the hyraulic hea, the guie vane angle an the rotational spee of the turbine. Because such hyraulic characteristics are strongly relate with flow losses of iverse types, especially at part loas, they coul not yet be calculate with sufficient accuracy. Experimental measurements on a moel turbine therefore still remain as the unique metho to establish those characteristics. This circumstance strongly restricts the further estimation of relate flow ynamic behaviours of a Francis turbine while getting starte or stoppe for instance. It also affects the accurate calculation of the transient flow that is relate to the change of the guie vane angle or/an the rotational spee for the purpose of regulating the power output. It seems therefore to be quite inispensable to wor out a reliable calculation metho to reveal the complex characteristics of a Francis turbine. In performing this activity, all types of flow losses that are relate to both the flow rate an the rotational spee nee to be quantifie. These inclue the shoc loss at the runner inlet (ominant at part flow rate), the exit flow loss associate with the axial an the tangential velocity component an the mechanical loss relate to the rotational spee. special phenomenon with the operational instability is the so calle S-shape instability, when the flow rate is close to zero, as in the case of getting starte an the case of reaching the runaway spee. Publishe uner licence by Lt

3 Despite of lots of investigations [-] the mechanism of such an operational instability coul still not be reveale satisfactorily. The present stuy tries to mae a comprehensible explanation of the bacgroun of such an instability base on a flow mechanical moel an the relate characteristic calculations.. Flow analysis.. Euler equation for the specific wor ccoring to the Euler equation for escribing the energy exchange in a rotating flui machine the specific wor conucte at a Francis turbine is given by Y Sch η gh u c u c () The respective parameters in this equation are, accoring to figure, as follows: c uniform absolute flow velocity at the runner inlet c u uniform circumferential component of the velocity vector c at the runner inlet c u circumferential component of the velocity vector c at the runner exit H available hyraulic hea u constant circumferential spee of the runner perimeter (inlet) u circumferential spee of the runner trailing ege, in function of the raius η hyraulic efficiency of the Francis turbine Hy Hy u u Inlet u Exit u (a) α c c m β w c m α c w β (b) Figure. Velocity triangles at the runner inlet an exit of a Francis turbine (α as the given flow angle) Stream line Draft tube a η c m R s λ ε i r Figure. Exit flow out of a Francis turbine The average term u cu which is relate to the exit flow out of the impeller has to be calculate from the integration of the non-uniform flow istribution along the trailing ege of the impeller vanes from s to ss (figure ): u S cu ( ucu ) ( ucu ) πrcmsin( ε + λ) s ()

4 the meriian velocity component at the impeller exit is esignate by c m. t the normal operation point at which the flow out of the impeller oes not possess any circumferential velocity component, there is u cu. The flow out of the impeller flow channel can be consiere to be the potential flow. This consieration relies on the approximation of the straight an uniform flow istribution (potential flow) in the ownstream raft tube. For such an outlet flow the meriian velocity c m can be calculate in function of the coorinate s along the trailing ege of the impeller vanes [4]: () s N cm,n () I both () s an Ι are geometrical parameters that are etermine from the geometrical layout of the impeller exit. ccoring to [4] they are calculate in imensionless form as ln s ε η sin ε r () s + cos( ε + λ) + sin( ε + λ) R s (4) S π I r s + () sin( λ ε ) In the first equation, R is the streamline coverture raius, which changes along the trailing ege s an can be approximate through the linear interpolation between i an a. lso the flow angle ε which specifies the streamline irection nees to be interpolate. The coorinate η is perpenicular to the streamline. Both () s an I can be calculate by means of the spreasheet metho. The normal flow rate Q N is relate to the rotational spee. For changeable rotational spee for instance in the case of changeable hyraulic hea, the normal flow rate is also changeable. The further relationship at the normal operation point, accoring to figure bfor α 9, is given by c m, N u tan β (6) s (5) Obviously the normal flow rate Q N of a Francis turbine, that is proportional to the meriian velocity c m,n, is etermine by the geometrical inclination angle β of the impeller vanes. t the part loa as well as at the overloa operation, the exit flow out of the impeller possesses the circumferential velocity component so that there is generally u cu. The meriian velocity istribution can, however, be still consiere to have the similar istribution as in the case uner the normal operational conition. Thus () s cm (7) The similarity of the meriian velocity istribution can be confirme for instance, where the flow is almost axial. For the Francis turbine with large specific numbers the similar meriian flow can thus be assume. Thus Eq. () can be written as Ι

5 with respect to c ( u c tanβ ) an cm c m,n ucu ( ucu ) π r( s) sin( ε + λ) s (8) Ι S u m an u π rn as well as c m, N u,n tanβ Q Q from Eqs. () an (7),the above equation is further calculate as N u n N n N S 8π n cu r () s sin( ε + λ) Ι obviously the integration again represents a geometrical parameter. It is enote by imensionless form as s II (9) in thus one obtains u ΙI π r () s sin( ε + λ)s 4 () S n 4 π () ΙI ( ) N n n cu Ι N.. Hyraulic efficiency In using u πn an c u cm tanα, that are relate to the impeller inlet, an with respect to Eq. (), the hyraulic efficiency is calculate from Eq. ()to π ( ) n cm 4 π n II n N η Hy gh tanα gh Ι n () N for further calculations the flow rate number, the rotational spee number an the torque number are introuce in imensionless form as π n M, n, M () gh gh ρ gh with as the raft tube iameter. With respect to Q c π b it follows then from Eq. () m cm n η II,N Hy nq 8 n πb tanα n (4) Ι,N the normal flow rate number Q is relate to the normal flow rate Q N an the currently available, N hyraulic hea H. The same is applie to n, Nan n... Estimation of flow losses The hyraulic efficiency of a Francis turbine can be calculate if all flow losses are nown. Because the exit flow out of the raft tube usually possesses very low spee, the associate loss can be neglecte. The losses in a Francis turbine then are mainly etermine by the shoc loss at the impeller inlet, the loss relate to the swirling flow at the impeller exit an the mechanical loss. Thus it can be written in general as Q 4

6 η Hy ΔηMech ΔηShoc ΔηSwirl (5) the respective losses are consiere below. To be mentione is the alterative that the mechanical loss an other unnown losses can be replace by η Hy, N with η Hy, N as the hyraulic efficiency at the normal operation at which both Δ η Shoc an ηswirl Δ vanish. Thus Eq. (5) can also be expresse by η Hy η Δη Δη (6) Hy,N Shoc as will be shown, this last equation leas to the calculation simplification.... Mechanical loss. The mechanical loss is usually proportional to the thir power of the rotational spee, so that it can be formulate as ( π n) ζ ρ π n gh n Δ η Mech ζ ζ (7) ( ρgh ) gh the coefficient ζ can be etermine from the normal operational conition, at which the mechanical loss is given by n,n Δ η Mech,N ζ (8) for instance for n. 56, Q. 8 an Δη. 5 there isζ.5., N, N Mech, N... Shoc Loss. The shoc loss is relate to the flow at the impeller inlet an arises from the abrupt change of the flow irection an flow separation there. Usually the impeller vane angle is esigne for normal flow rate Q N. t part loas as well as at the overloas the flow unergoes an abrupt change in the flow irection while getting into the impeller. In the relative system an by applying the relative velocity ( w ), the shoc loss is calculate as ( β β ),N sin S w w ηshoc μ μ sin β sin βs gh tan βs tan β Δ (9) gh Herein β S represents the geometrical vane angle at the impeller inlet. Only when the flow angle β (see figure a) is equal to the vane angle β S, the shoc loss becomes negligible. The coefficient μ consiers the part of the shoc loss in the theoretical maximum. ccoring to the efinition of the flow rate number Q the relative velocity w is expresse as follows 4 w cm () gh gh sin β sin β Swirl the flow angle β at the impeller inlet is further represente as c tan β u m cu c u m cm tanα π n b tan α () 5

7 thus Eq. (9) becomes Δη Shoc μ + tan S tan n () β α πb except for Q an n all other parameters in this equation are geometrical parameters. For Δη Shoc there is u +, which inee can also be irectly obtaine from the tan βs tanα cm velocity triangle at the impeller inlet (figure a), when assuming β S β as the conition for the inlet flow without shoc loss.... Swirling flow loss at the runner exit. The swirling flow loss is associate with the inetic energy involve in the rotational flow at the impeller exit. The specific inetic energy is given by cu g. Thus the relate efficiency rop is calculate by cu Δη Swirl () gh To the average specific inetic energy the following integration has to be carrie out c u S c m ( + ) u c u πrc sin ε λ Following the same calculation proceure as in Sect.. one obtains ΙI nn c u 4( π n) Ι n Q (5) N With respect to Eq. () for Q an n Eq. () is then written as s (4) n ΙI,N Q Δ η Swirl 4 n (6) n Ι,N s in Eq. () all parameters except Q an n are nown quantities. It is worth to note that from Eq. () an (5) the following relationship generally exists: c u c u nn Q (7) u n N.4. Determination of characteristics of the Francis turbine.4.. Solution with explicit mechanical loss. Combining Eq. (4) an (5) with respect to each calculate losses one obtains, after a rearrangement (8) with B C D 6

8 4 ΙI Ι n,n,n + μ tan β S + tan α 4 B μ μ tanα tan β S n πb C μ II 4 n Ι D ζn by solving the specific flow rate Q for each given guie vane angle an the rotational spee the Q f α have been thus obtaine. characteristics of a Francis turbine in form of ( ),n t n it eals with a through flow of water through the turbine. The flow rate number is calculate to Q, (9) On the other sie this flow rate number can also be irectly obtaine from Eq. () for shoc loss an Eq. (6) for swirling flow loss by setting n an subsequently Δη Shoc, + ΔηSwirl,. Thus it is clear that the through flow at n is completely governe by the shoc an the swirling flow loss. Eq. (8) represents an equation of thir orer an is thus associate with some ifficulty in solving it. n alternative solution is presente below..4.. Solution with implicit mechanical loss. Combining Eq. (4) an (6) yiels + + () with B C 4 ΙI Ι n,n,n + μ tan β S + tan α 4 B μ μ tanα tan β S n πb II C μ 4 n ηhy,n Ι It eals with a polynomial equation of secon orer, so that it can be solve much easier than Eq. (8). Both an B are the same as in Eq. (8). There is, however, a small moification in C..4.. Mechanical moment. The power output of the Francis turbine can be written as P η gh Hy ρ () 7

9 on the other han, the power output is represente by the prouct of the mechanical torque(moment) exerte on the shaft an the shaft angular spee: P πnm () by equalising these two equations an applying Eq. () for respective imensionless numbers one obtains ηhy M () n. pplication example The calculation algorithms presente above has been applie to an existing Francis turbine at the OberhasliHyroelectric Power Company (KWO). It is a turbine with nown characteristics in form of f ( α,n) base on the moel test (nritz). Figure shows the comparison between measurements an calculations. To be mentione is that in the complete calculations an appropriate coefficient μ const has been chosen for the best agreement with measurements (here μ ). Such an agreement precisely confirms the applicability of the flow mechanical moels use in the present stuies for calculating the characteristics of a Francis turbine. While performing calculations of the turbine characteristics both the hyraulic efficiency an iverse losses coul also be obtaine. For the guie vane angle equal to 7.5 for instance the calculate hyraulic efficiency has been compare with measurements, as shown in figure 4again with excellent agreement. In figure 5, both the shoc an the swirling flow losses from calculations are shown. Obviously the great eficit in the hyraulic efficiency of a Francis turbine at operation points out of the esign point ( n. 564) mainly arises from the shoc loss. The corresponing flow pattern can be imagine as that with total flow separation at the impeller inlet because of the totally wrong flow angle. This will be iscusse in more etails in the paragraph 5. In applying Eq. (), the moment exerte on the shaft has also been calculate an compare with measurements, as this is shown in figure 6. Excellent agreement between calculation an experiments has been expecte because of the excellent agreements both for the flow rate number (figure ) an the hyraulic efficiency (figure 4). Figure. Turbine characteristics an comparison between calculations an measurements (KWO) Figure 4.Hyraulic efficiency of a Francis turbine (KWO, r) 8

10 Figure 5. Shoc an swirling flow losses of the Francis turbine (KWO, r) Figure 6. Torque number of the Francis turbine (KWO, r) 4. Self-valiation of the calculation moel an the Zhang s equation Calculations shown above have been valiate by comparing the calculation results with experiments base on the moel test. lthough the satisfactory agreement has been achieve by choosing the parameter μ, calculations themselves precisely reveal what happens in a Francis turbine with respect to the shoc, swirling an mechanical losses an how they influence the hyraulic efficiency of the turbine. In reality, the calculations are self-valiate, as shown here by consiering figure an 4. ccoring to figure at the starting point n the flow rate number Q harly changes with the change of the rotational spee number. This circumstance can be confirme by Eq. (8) with D because of the negligible mechanical loss. The equation then becomes a polynomial of secon orer from which n will be calculate. By subsequently setting n an μ one obtains: n tanβ S πb with respect to Eq. (9) the above equation can also be expresse as n (4) Q, tanβs πb (5) because this property of the turbine characteristics highly agrees to the measurements, it can be consiere to be a in of self-valiation of the calculation moels applie in the present stuy. In the consiere example with α 7.5 for instance it is obtaine Q n.. Usually it is of a very small value. Figure 4 is again concerne. The hyraulic efficiency, starte from n, increases with the rotational spee number n. The slope of this increase can be calculate from Eq. (5) with regar to respective losses an subsequently by setting n. The following approximation can then be foun: η n Hy It is almost inepenent of the guie vane angle of the Francis turbine. To the mentione Francis turbine with.7 an. one obtains η Hy n. 6. It highly agrees to the measurement, as shown in figure 4. This incient provies an aitional reason of that the flow moel use in the present stuy is self-valiate. ccoring to its finer Eq. (6) is calle the Zhang s equation. (6) 9

11 5. Discussion to the S-shape of turbine characteristics From figure 4 the hyraulic efficiency of the Francis turbine become zero an negative, as long as the rotational spee number get beyon the critical value of about n. 9 in the present example. This means that for n >. 9 the turbine operates as a pump. No woner that the operational instability occurs at n about n. 9. Beyon this value calculations o not run correctly because the flow mechanical moels use in this stuy is only for the turbine operation. From the analysis an the application example, as show above, it has been emonstrate that the shoc loss in the upper range of the rotational spee results in the main efficiency rop in a Francis turbine. Because the associate flow rate is significantly low, the total flow separation at the impeller inlet has to be expecte. It eals with the non-stationary flow separation. While some vane channels are totally bloce by the flow separation, some other channels serve as the flow channel. This type of the flow separation is from the same mechanism as the rotating stalls that are often confirme at the pump an compressor flows at the part loa. The turbine characteristics in the upper range of the rotational spee are often confirme to possess a non-stationary S-shape. In the practical applications, it is inee not interesting at all why it is just an S-shape. It is, however, much crucial to now that because of the non-stationary flow separation the characteristics cannot be etermine accurately, neither experimentally norby calculations. Calculation results shown in figure were obtaine by assuming regular an stationary flow separations. 6. Summary flow mechanical moel has been introuce to calculate complete characteristics of a Francis turbine. The flow moel accounts for the most sensible an reasonable losses that occur at operation points out of the esign point. These losses are shoc loss at the impeller inlet, the swirling flow loss relate to the circumferential velocity component at the impeller exit an the mechanical loss. The former two losses ominate in influencing the characteristics an the hyraulic efficiency of a Francis turbine. Both together completely etermine the through flow rate of water through the impeller at rest (n). Calculations have been performe for the flow rate number, torque number (shaft torque) an the hyraulic efficiency an compare with measurements. Excellent agreements have been achieve. The calculation metho has been thus confirme to be self-valiate. The operational instability in the upper range of the rotational spee, that possesses the S-shape, arises from the total flow separation an stagnation at the impeller inlet. Because the hyraulic efficiency tens to be zero, the operation of the Francis turbine oscillates between the pump an the turbine moe. References [] Hasmatuchi V, Roth S, Botero F, vellan F an Farhat M High-spee flow visualization in a pump-turbine uner off-esign operating conitions 5th IHR Symposium on Hyraulic Machinery an Systems (Timisoara, Romania) [] Staubli T, Senn F an Sallaberger S 8 Instability of Pump-Turbines uring Start-up in Turbine Moe, Hyro 8 ( Ljubljana, Slovenia) [] Martin C Instability of Pump-Turbines with S-Shape Characteristics,th IHR Symposium on Hyraulic Machinery an Systems (Charlotte, US [4] Zhang Z Private communication