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Francis Türbinli Tesislerde Dönen Vorteks Çekirdeğinin geçici rejimlere etkisinin rolü

An Investigation on the influence of rotating vortex rope on transiet working conditions for Francis turbine power plants

  1. Tez No: 39213
  2. Yazar: METİN GÜLEÇ
  3. Danışmanlar: PROF.DR. CAHİT ÖZGÜR
  4. Tez Türü: Doktora
  5. Konular: Makine Mühendisliği, Mechanical Engineering
  6. Anahtar Kelimeler: Belirtilmemiş.
  7. Yıl: 1993
  8. Dil: Türkçe
  9. Üniversite: İstanbul Teknik Üniversitesi
  10. Enstitü: Fen Bilimleri Enstitüsü
  11. Ana Bilim Dalı: Belirtilmemiş.
  12. Bilim Dalı: Belirtilmemiş.
  13. Sayfa Sayısı: 155

Özet

ÖZET Kısmi yükte çalışan Francis türbinli tesislerde rastlanan döner vorteks halatı üzerinde şimdiye kadar bir çok çalışma yapılmış olmasına rağmen henüz olayın tüm nedenlerini ve sonuçlarını içeren kapsamlı bir analiz mevcut değildir. Bu çalışmada mevcut teorik ve deneysel araştırmalar ele alınarak teorik tabana olası ölçüde açıklık getirilmiştir. Olayla ilgili tüm önemli çalışmalar gözden geçirilip karşılaştırılmış ve döner vorteks frekansının yaklaşık hesap şekli ortaya konmuştur. Tezde, son yıllarda Fanelli tarafından lineer olmayan dinamiğin kısmi yükteki Francis türbini yayıcıları basınç çalkantılarına uygulanması açıklanmış ve periyodik uyarmaya rağmen çoklu periyodik ve periyodik olmayan çözümlerin bulunduğu sistemin kaotik karakter alabileceği ve her zaman deterministik kabul edilemiyeceği gösterilmiştir. Hidrolik türbomakinaların daimi titreşimli hallerinin oldukça kompleks olduğu hesaba katılırsa bu durumda geçiş rejimlerinin etüdü için basitleştirilmiş modellerle işe başlamanın gerekeceği açıktır. Bu amaçla ilk olarak Dörtler, Maria ve arkadaşları tarafından ortaya konan döner vorteks ile birlikte mevcut olan kavitasyonlu gaz çekirdeğinin rolü derinliğine incelenmiştir. Ancak bu modelin geçici rejimlere uygulanması mümkün olmadığından yeni bir matematik model geliştirilmiştir. Bu modelde türbin doğrusal ve karesel etkili olarak ele alınabilmektedir. Bu iki model arasındaki farklar tezde ortaya konmuştur. Yeni model daimi titreşimli rejimde test edildikten sonra kısmi yükte döner vorteks uyarısındaki türbinin değişik şartlarda yük atma simüiasyonları yapılmıştır. Ani yük atmada türbin girişindeki basınç artmasının titreşimli rejimde düzgün çalışmalı hale göre gözle görülür bir fark yaratmadığı görülmüştür. Buna karşılık süreli yük atmada durum değişmektedir. Aynı sürede yük atılan titreşimli tesisin türbin girişindeki su darbesi basınç artması daimi rejimde çalışan türbindeki basınç artmasına göre bir kaç kat mertebesinde olabilmektedir. Bulunan diğer sonuç ta çalkantılı rejimin yarattığı istenmiyen fazla basınç etkisinin, yük atma süresi ile orantılı oluşudur. Yeni modelle kavitasyon yumuşaklığı, uyarı frekansı ve cebri borunun etkisi kolaylıkla incelenebilmektedir. VIII

Özet (Çeviri)

Summary An Investigation on the Influence of Rotating Vortex Rope on Transient Working Conditions for Francis Turbine Power Plants In the draft tube of a Francis turbine of a hydroelectric power plant, at partial and over loads, the flow acquires a non-axissymetrical character due the existence of a tangential velocity component at the exit of the runner, and as a result the rotating vortex rope phenomenon develops. The speed of rotation of the vortex rope is between 20 to 40 % of the speed of rotation of the runner. Cavitating vortex rope in the draft tube causes disturbing oscillations in the Turbine-Generator unit itself as well as in other parts of the hydraulic power plant. In some cases the oscillations become so severe that it may be impossible to continue to run the units at partial load where excitation frequency is in resonance with the hydraulic system natural frequency and serious damage may be expected on the particular parts. Most of the time mechanical power fluctuations induced by the presence of a rotating field together with a cavitating core may reach 2-4 % of the rated power, while electrical power oscillations may be higher than 10 % when the resonance occurs in the electrical system. Although a lot of research work have been carried out on the problems of rotating vortex rope phenomena encountered in the power plants equipped with Francis units working at partial and over load, a comprehensive analysis covering all causes and consequences is yet to be produced. Rheingas was the first who noticed and investigated the phenomenon in 1940's since than many prominent researchers like Campas, Dörtler, Maria, Fanelli and Raabe have devoted their time and efforts to the problem. The development of the rotating vortex core itself is the first field of interest in the investigation of the complex phenomena encountered in the draft tubes of Francis Turbines. There have been quite a number of experimental and theoretical works on that topic. Experiments show that, for high tangential velocities at the runner exit, a back flow develops at the centre just below the runner. It is also shown that the back flow region grows larger when the speed of rotation of runner is increased. J.J. Cassidy, through the experiments made with air, has concluded that, a rotating runner is not essential for the formation of a rotating vortex rope and in a swirling flow whenever the tangential velocities reach a certain level, vortex breakdown occurs resulting an unsteady pattern associated with the rotating vortex. IXThe above mentioned flow in the draft tubes of Francis turbines is the result of high tangential velocity components generated by off design conditions, whereas when the machine works near its optimum point the flow remains axissymmetric and steady, A strong rotating flow field originated by the Francis runner produces a screw like vortex rope, the centre of which is a cavitating core gradually narrowing toward the downstream of the draft tube. Direction of rotation of this rope is the same as the runner's for the partial loads, conversely for overloads the rope rotates in opposite direction. Experimental studies have shown that the vortex rope can acquire different forms and that it can appear in single, double and triple shapes. It is also observed that the rope change its character from one region of the hill chart to another. The effect of the cavitation on the rotating vortex phenomenon has only recently been subject of investigation. The most important contributions in this context were made by Dörfler, J. Tadel and D. Maria. Their experimental studies have shown that for relatively high cavitating number, that is when working away from the cavitating conditions, excitation disturbances caused by the rotating field have not been accentuated in the system. In another words pressure oscillation amplitudes at the turbine outlet remain the same as the excitation amplitudes. If the sigma is decreased and approaches to the critical value cavitating void influences to the response and fluctuations increases. For specific cavitation coefficient sigma, depending on the installation, the response is a maximum, corresponding to the resonance condition. Further decrease in sigma value result in decreasing the amplitude of fluctuations. Very small sigma values has a reversal, i.e. damping, effect on the fluctuations. For most of the Francis units it is necessary to inject air below the turbine runner in order to reduce vibration and noise produced by cavitation. Appropriate amount of air injection provides smooth and more stable operation without causing noticeable decrease in overall efficiency. Increased amount of injected air enlarges the volume of cavitating vortex core which produce an effect similar to that of decreasing the cavitation coefficient. For non cavitating operation conditions, high injection air rates may not be beneficial. In brief, the effect of air injection for the part load operations depends on the particularity of installation and on the turbine working condition with respect to cavitation. Although the velocity field produced by the rotating vortex corresponds to an unsteady flow in the draft tube, for the case of straight rectilinear draft tubes, while the pressure at any point of the outlet section varies periodically, the mean pressure at this section remain constant. On the other hand, if the unite has an elbow draft tube, periodic pressure oscillations can not be avoided, due to axial dissymmetry. In the course of the main objective of the present work all principal studies relating to the rotating vortex have been reviewed and the theoretical foundations of the problem has been tried to establish as clearly as possible. It is shown that frequency of the rotating vortex can readily be estimated with reasonable accuracy by using simple expressions based on some realistic assumptions. In this respect the works of Sayann and Raabe have to be sited. xIt was Fanelli who tried the first time a three dimensional approach and established a theoretical base demonstrating how the back flow and rotating vortex can simultaneously exist. He also gave the theoretical bases for the rotating vortex flow in elbow. Mention should also be made to another attempt made in recent years by Fanelli for the application of chaotic non linear dynamics to the draft tube surges for Francis unites at partial load. By this work it was shown that there may be multiple periodic or non periodic solutions and the chaotic character may exist although the driving force remains periodic. Consequently it is interesting to note that when the gas cavity presenting the cavitating vortex rope is under isotherm non linear transformation the system can not always be considered completely deterministic. The behaviour of the hydraulic machinery under steady oscillatory condition is already very complex. In order to study the transitions, one has to start with simplified models to be applicable. In this context the role of the cavitation bubble associated with the rotating vortex which is first demonstrated by Dörtler, Maria et al with linear theory, has been thoroughly investigated. These investigators have used the impedance method to analysis the dynamic response of the hydraulic system when it is excited by the rotating vortex. The complex response expression obtained from this linear frictionless model contains, in addition to the penstock, turbine and draft tube impedance a cavitation compliance factor which is defined by the ratio of vortex core volume variation to the bubble pressure variation. With the help of this model the influence of the cavitation coefficient, which is related to cavitation compliance, on the dynamic response and the resonance can easily studied. However the model has two drawbacks, namely it can only be applicable to the steady oscillatory flow and it is very difficult to account for friction. Furthermore non linear turbine characteristics can not be incorporated into the calculation. Above mentioned model not being applicable to transient conditions a new mathematical model was necessary and was developed by the author which allows to investigate the response of the hydraulic power plant to the rotating excitations. The new model not only useful for unsteady transient regimes but at the same time it can be applied to the cases of pulsating flows so that the response curves obtained by impedance method for the same installation can be compared and checked. HP(n.) / 4*- / m-1 ni HPT QP(nl) HPA QPT QPA Kr> ht t^HPCn. + l) QP(n. + l) \ «- 1 A A ^ m Schematic representation of the developed model XIThis mathematical model is based on the method of characteristics and the assumptions that were made in the calculation may be summarised as below; -The excitation takes place in the infinitely narrow region between the gas bubble representing the vortex core and entrance of the draft tube. - Rate of flow of water entering and leaving the turbine are equal at all times. - Elasticity of the cavitation volume (V) is represented by cavitation compliance factor Ct defined as the ratio of the partial increase of the volume (AV) to the corresponding increase in algebraic value of the pressure head (AH); that is; C = -^ (D AH v ; At the entrance and exit of the gas volume (V), the pressures are equal to each other for all times but the rate of flow values are different, that is; QPT*QPA = QP(ni + 1) The difference between the rate of flow of water entering and leaving the cavitation bubble is equal to the changes of the gas volume; that is, the difference between the rate of inflow and the rate of outflow gives the rate of contraction of the gas volume, which is expressed as, AV = -At QPT + QT QP(n. + l) + Q(n, + l)' Hence, change of volume at time step is; AV = -- [QPT - QP(n, + 1) + QT - 0(11, + 1)] (2) Relation for excitating pressure; HP(n, + l) = HPT + ht (3) The character of the excitation can be assumed as sinusoidal,“Dht”being its amplitude; ht = Dht Sincot The frequency of the excitation, can be calculated through the time step (DT) used for the application of the Method of Characteristics by the following expression; XIIft= 1 mDT where m is the number of time steps in a period. The increase in piezometer head at the exit of the turbine for a time step of DT is; AH= HPT-HT (4) The right hand characteristic (C~) can be used to obtain the rate of flow at the exit of the gas volume (point A);“, ”v _kl HP(ni + l) QP(ni + l) = CN K- - (5) B? B2 = ? °“ gS2 From equations (1) to (5), after eliminating the quantities of AV,AH1HP(ni + l),QP(n. + l); the relation between the pressure head and the rate of flow at the exit of the turbine at the end of the time step DT can be expressed as follows; HT + ^ f QPT + QT - Q(n, + 1) - CN - ^-) HPT = 2Ç4 B2I 1 + 2Q This equation is set up as the boundary condition of the cavitation core, as the turbine characteristics was not required for the determination of that condition, If the relation between the net head and the rate of flow of the turbine is known, this equation can be solved and rates of flow and pressure values can be determined, Two cases have been considered as the turbine boundary condition, The first one assumes that the amplitudes of the disturbances are not high and the net head and rate of flow changes are proportional with each other, i.e. linear model. The second assumes that net head changes are proportional with the square of the changes of the rate of flow. A computer programme has been developed for the evaluation of the responses of a Francis turbine system to the excitations of the rotating vortex rope of the draft tube as a parameter of the cavitation compliance in non dimensional form defined as; Ct”= Ct2.7tRTft where ft is the excitation frequency and Rt is the real part of the turbine impedance. XIIIDiffering responses are observed at various points of hydroelectric system excited with the frequency of“ft”. These agitations are frequently regarded very disturbing and they can even endanger the stability of the system due to the resulting large power oscillations. For a given system, amplitude of the disturbances varies with the a cavitation coefficient. Most dangerous condition corresponds neither to the cavitation-free running nor to the extremely cavitating operations. Depending on the degree of elasticity of the cavitating core, the same may excitation cause different disturbances at the runner exit. It is not physically possible to measure the degree of elasticity of the cavitation core nor it can be calculated theoretically. Realistic estimates that are made using the experimental results on the correlations between the a cavitation coefficient and Ct* are required for the determination of the value of the degree of elasticity of the cavitation core. To be able to increase the reliability of the a versus Ct* correlation cure, much more experimental data is required but for a crude evaluation the proposed correlation may be employed. After the establishment of the Ct* value, using the new dynamic model, the behaviour at transient conditions of a partially loaded Francis turbine power plant can be investigated with ease. As a first step in order to make a comparison and check the validity of the calculations, both the new model and linear model were applied to the same installation which is under synchronous excitations and running at constant load. The response curves obtained by two different methods agreed very well each other, snowing the reliability of the mathematical model. As for the transitory conditions several load simulations were carried out for Francis units at partial load, subjected to harmonic excitation due to rotating cavitating vortex core. It was observed that for sudden load rejections, water hammer pressure rise at the inlet of the turbine has not been noticeably affected by oscillatory conditions. Contrary to this situation, for slow load rejections that is a rejection takes substantial time, there are considerable increases in the pressure rise at the turbine inlet, compared to the unit working under the same condition and the load of which was similarly rejected. An another outcome of the work is that undesirable effect of the part load fluctuation varies in inverse proportion to the speed of wicket gate closure. The effect of cavitation compliance, excitation frequency and upstream conduit can also be investigated with the proposed model. XIV

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