Introduction to Turbine Machinery

Marko Hočevar

Introduction to turbine machinery

Ljubljana, 2018

Naslov publikacije: Avtor: Strokovna recenzenta: Lektoriranje besedila: Izdelava slik in diagramov: Prelom in priprava za tisk: Založba: Izdaja: Leto izida: Naslovnica knjige: Avtorske pravice so pridržane. Gradiva iz publikacije ni dovoljeno kopirati, objavljati ali prevajati v drug jezike brez pisnega dovoljenja založbe.

CIP - Kataložni zapis o publikaciji Narodna in univerzitetna knjižnica, Ljubljana dopolniti

Contents

Contents ...... 3 Nomenclature ...... 7 Foreword and acknowledgments ...... 8 Topics for seminars ...... 9 Dictionary ...... 10 1 Introduction ...... 13

1.1 WORKING PRINCIPLE ...... 13 1.2 TYPES OF TURBINE MACHINERY ...... 15 1.2.1 Classification to enclosed or open turbine machines ...... 15 1.2.2 Classification regarding direction of the flow ...... 16 1.2.3 Classification regarding energy conversion direction ...... 18 1.2.4 Classification regarding application type ...... 19 1.2.5 Classification regarding physical action ...... 20 1.3 TURBINE MACHINERY PRODUCTION IN SLOVENIA ...... 20 1.4 APPROACHES TO TURBINE MACHINERY OPERATION ...... 22 1.5 NOMENCLATURE ...... 23 2. Thermodynamics and efficiencies ...... 24

2.1 FIRST LAW OF THERMODYNAMICS ...... 24 2.2 EFFICIENCY...... 28 2.2.1 Efficiency in steam and gas turbines ...... 30 2.2.2 Efficiency in water turbines...... 32 2.2.3 Efficiency of pumps and compressors ...... 33 2.2.4 Local losses within the turbine machine runner ...... 34 3 Control volume approach to turbine machinery ...... 38

3.1 CONTROL VOLUME ...... 38 3.2 STATIONARY CASE OF OPERATION ...... 41 4 Differential approach to turbine machinery ...... 43 5 Euler approach to fluid flow in turbine machinery ...... 46

5.1 ENERGY TRANSFER AND TRIANGLES OF VELOCITY ...... 46 3

5.1.1 Triangles of velocity in the runner ...... 46 5.2 ENTHALPY AND PRESSURE CHANGES IN TURBINE MACHINERY ...... 48 5.2.1 Enthalpy and pressure rise in radial compressor ...... 48 5.2.2 Entalpy and pressure reduction in radial turbine...... 51 5.2.3 Enthalpy and pressure rise in axial compressor ...... 53 5.2.4 Enthalpy and pressure reduction in axial turbine ...... 55 5.3 TORQUE ON THE BLADES OF THE TURBINE MACHINE RUNNER ...... 57 5.4 EULER EQUATION FOR TURBINE MACHINERY ...... 60 5.5 RADIAL FLOW COMPRESSORS AND PUMPS ...... 62 5.5.1 Triangles of velocity in radial compressors and pumps ...... 62 5.5.2 Second form of Euler equation ...... 66 5.5.3 Reactivity ...... 68 5.5.4 Radial runner characteristics ...... 72 5.5.5 Radial runner power and torque ...... 74 5.5.6 Radial runner static pressure ...... 75 5.5.7 Radial runner real pressure characteristics ...... 75 5.5.7 Cordier diagram ...... 95 5.6 AXIAL FLOW VENTILATORS, FANS, COMPRESORS AND PUMPS...... 81 5.6.1 Euler equation for axial turbine machinery ...... 82 6 Similarity theory and dimensional analysis of turbine machinery ...... 87

6.1 SIMPLE APPROACH TO OF SELECTION OF DIMENSIONLESS NUMBERS ...... 88 6.2 USE OF DIMENSIONLESS NUMBERS ...... 91 6.3 FUNDAMENTAL APPROACH TO SELECTION OF DIMENSIONLESS NUMBERS ...... 93 7 Water turbines ...... 100

7.1 WATER TURBINE CLASSIFICATION ...... 102

7.2 CRITERIA FOR WATER TURBINE CLASSIFICATION WITH RESPECT TO THE SPECIFIC SPEED (NS) ...... 97 7.3 PELTON TURBINE ...... 103 7.4 FRANCIS WATER TURBINES ...... 106 7.5 KAPLAN WATER TURBINES ...... 112 7.6 TUBULAR TURBINES ...... 115 7.6.1 Bulb turbine ...... 118 7.6.2 Pit turbine ...... 119 7.6.3 Tubular turbine S ...... 119 7.6.4 Axial turbine with a vertical shaft – Saxo turbine ...... 120 7.7 OTHER TYPE OF WATER TURBINES ...... 121 8 Other elements of hydro power plants ...... 124

8.1 WATER INTAKE SYSTEM ...... 124 8.1.1 Dams ...... 124 8.1.2 Teeth and trash racks ...... 125 4

8.2 WATER SUPPLY SYSTEM ...... 127 8.2.1 Headrace channels and tunnels ...... 127 8.2.2 Surge tank ...... 127 8.2.3 Penstock ...... 129 8.2.4 Bypass valve or pressure regulator ...... 129 8.2.5 Ball valve or butterfly valve in and bypass pipeline ...... 130 8.2.6 Gates ...... 131 8.3 POWERHOUSE EQUIPMENT ...... 132 8.3.1 Spiral casing ...... 133 8.3.2 Stay vanes and guide vanes ...... 133 8.3.3 Turbine runner, turbine cover, anti-lifting plate and system for air blowing... 134 8.3.4 Shaft ...... 136 8.3.5 Ležaji ...... 136 8.3.6 Shaft seal ...... 139 8.3.7 Creep detector ...... 139 8.3.8 Governing of the angle of guide vanes and runner blades ...... 140 8.3.9 Inverter ...... 142 8.3.10 Brakes ...... 143 8.3.11 Generator and its electric equipment ...... 143 8.4 OTHER SYSTEMS IN THE POWERPLANT ...... 145 8.4.1 Spillways and spillway gates ...... 146 8.4.2 Sump pumps ...... 148 8.4.3 Fish passages ...... 148 9 Manufacture of runners of water turbines...... 151

9.1 MANUFACTURE OF PELTON TURBINE RUNNERS ...... 152 9.2 MANUFACTURE OF FRANCIS TURBINE RUNNERS...... 154 9.3 MANUFACTURE OF RUNNERS OF KAPLAN TURBINES ...... 158 9.4 PRE ASSEMBLY, ASSEMBLY AND INSTALLATION OF WATER TURBINES ...... 159 10 The properties of operation of water power plants ...... 161

10.1 CHARACTERISTICS AND HILL DIAGRAM OF TURBINES...... 163 10.1.1 Characteristics of water turbines ...... 163 10.1.2 Hill diagram ...... 166 10.2 PUMP/TURBINE OPERATION IN FOUR QUADRANTS (EXTENDED RANGE OF OPERATION) ...... 170 10 Cavitation in water turbines and pumps ...... 172

10.1 INTRODUCTION TO CAVITATION ...... 172 10.2 RELATION TO THE OPERATING POINT OF THE TURBINE MACHINE...... 174 10.2.1 In Kaplan turbine ...... 174 10.2.2 In Francis turbine ...... 176 10.2.3 IN CENTRIFUGAL PUMP ...... 177 5

10.3 CAVITATION AND NPSH/NPSE CHARACTERISTICS ...... 178 10.3.1 In pumps ...... 179 10.3.2 In turbines ...... 181 10.4 PREVENTION OF CAVITATION EFFECTS IN WATER TURBINES ...... 183 10.5 INFLUENCE OF PARTICLES IN THE FLOW ON CAVITATION IN WATER TURBINES ...... 184 10.6 RUNNER EROSION DUE TO SOLID DISPERSED PARTICLES ...... 185 References ...... 190

Nomenclature

simbol pomen enota

indeksi pomen

okrajšave pomen

Foreword and acknowledgments

Topics for seminars

We encourage students to make seminars and present them to colleagues. Possible topics for seminars are: - history of turbine machinery, - manufacture of hydraulic turbine machinery, - acceptance test of turbine machinery, - wind turbines, - direct design oof pumps, turbines or fans (centrifugal, axial), - various other topics, related to turbine machinery, suggested by students.

Dictionary slovenski izraz angleški izraz Banki turbina cross-flow turbine bruto padec gross head cevna turbina s hruško bulb turbine cevna turbina v jašku pit turbine čep vodilne lopatice guide vane pivot pin čistilni stroj cleaning machine črpalna vodna elektrarna pump turbine delna obremenitev partial load dovodni kanal head race channel dovodni rov head race tunnel drenažna črpalka sump pump gonilna lopatica runner blade gonilnik runner gred shaft iglasti ventil needle valve iztočni kanal tailrace channel iztočni rov tailrace tunnel jašek shaft jez dam kavitacija cavitation kavitacijski vrtinec cavitation vortex ležaj bearing lopatica turbine turbine blade, pri Peltonovih turbinah se lopatice imenujejo tudi buckets navitje rotorja rotor winding navitje statorja stator winding neto padec net head nosilni ležaj support bearing ali thrust bearing neto pozitivna sesalna višina neto positive suction head obvod bypass

10 odklonilo deflector peskolov sand trap pesto hub, boss pobežna vrtilna frekvenca runaway speed pokrov generatorja generator cover potopitev submersion predturbinski ventil ball valve ali butterfly valve predvodilna lopatica stay vane predvodilnik stay ring prelivno polje spilway pretočno število discharge coefficient pretok discharge prevzemni preizkus acceptance test propeler propeller razbremenilni ventil bypass valve ali pressure regulator razbremenitev turbine load rejection rešetka trash rack ribja steza fish passage ribam prijazna turbina fish friendly turbine roka bracket sesalna cev draft tube sistem krmiljenja governor specifična hitrost specific speed spiralno ohišje spiral casing, scroll casing spodnji akumulacijski zbiralnik lower reservoir školjčni diagram hill diagram ali performance diagram šoba nozzle ali injector tesnilka seal tesnilka gredi shaft seal tlačni cevovod penstock tlačni udar water hammer tlačno število pressure coefficient turbinski pokrov turbine cover venec rim ali tip vodilna lopatica guide vane vodilni ležaj guide bearing vodilni obroč guide vane ring ali guide vane wheel vodilnik wicket gate vodni padec head

11 vodostan surge tank ali surge chamber vstop vode water inlet vtočni sistem water intake system vzbujanje excitation vztrajnik flywheel začetna kavitacija incipient cavitation zapornica gate zavore brakes zaznavalo premikanja creep detection sensor zgornji akumulacijski zbiralnik upper reservoir udarne izgube trenjske izgube optimalna delovna točka best efficiency point notranje izgube inner losses, internal losses vijak (ladijski) screw (ship), marine propeller

1 Introduction dodaj: - načrtovanje gonilnikov Turbine machinery is a subject of great importance today. People's life is greatly influenced by use of electric energy. Extensive use of electric energy has dramatically changed everyone's life. We are almost unable to imagine life without use of electric motors, domestic appliances and consumer electronics. Clothes washing and drying machines, refridgerators, microwave ovens, television sets, audio and video electronics, air conditioning and computers fall in this category. Almost all the available electric energy nowadays is produced with the use of turbine machinery. Turbine machinery is also used for aero and naval propulsion and in internal combustion engines featuring turbochargers. Turbine machinery is a machine, where a continuous energy transfer is present between fluid and an element of the machine, which rotates around its axis. This element of the turbine machine is common for all turbine machines and is called an impeller, also turbine wheel, rotor, screw or runner. Turbine machines are pumps, fans, compressors, water turbines, gas turbines, steam turbines, wind turbines, aircraft and marine propellers, torque converters, hydrodynamic clutches etc. This textbook is about principles of turbine machinery. Only in the section about water turbines we will review technical aspects of the machinery more in detail.

1.1 Working principle Turbine machinery and piston machinery perform similar task, but they differ in the type of energy conversion. The operation of a reciprocating piston engine is immediately apparent to understand. With Fp we denote the force on the piston, for which we can write the equation

F푝 = pS . (1)

Here p is pressure and S surface of the piston. With the movement of the piston by dl, the work performed by the piston may be described by

푑A = F푝푑l = pS푑l = p푑V . (2)

In agreement with the above equation, work performed by piston relates to the change of the fluid volume. 13

In the turbine machinery the energy conversion is indirect and always takes place through the change of kinetic energy of the fluid. For this we may consider Fig. below. In the turbine the fluid flows from the pressure penstock into the turbine machine and first flows through a cascade of guide vanes. In the cascade of guide vanes flow velocity and with it connected kinetic energy increases at the expense of reduction of pressure or potential energy. At the same time the shape and direction of guide vanes direct the flow in the tangential direction of the runner. However, only tangential velocity can increase, the axial velocity can not increase due to consersvation of the mass flow rate. The cross section can not change much due to problems of flow separation. In the runner, the fluid transfers its kinetic energy to the runner by changing its direction. This decreases the tangential velocity, while again axial velocity does not change much. The flow exits the turbine with reduced energy ob the suction side of the turbine. In pumps, ventilators, fans and comporessors, the procedure is the opposite. We will consider the radial pump from Fig. below. The fluid flows in the runner with negligible tangential velocity. In the runner the tangential velocity is increased due to the torque of the shaft. However, due to conservation of the mass flow rate, radial velocity is related to the axial velocity from the inlet considering ratio of both cross sections. At the exit from the runner, the flow has the energy, given by the shaft, mostly in the form of tangential velocity. To decrease the flow and change the velocity into pressure, in the vaned difussor, the tangential flow is slowed down, and the shaft torque is converted into pressure.

Fig. 1: Turbine machinery working principle, left: axial turbine and right: radial pump, above: turbine machine diagram, below: runner and guide vanes

1.2 Types of turbine machinery All turbine machinery works on the above-mentioned principle of energy transfer through velocity. However, many variants exist, and several different classifications of turbine machines are possible. Turbine machines can be classified according to several principles as follows: - enclosed or open, - direction of the fluid flow, - energy conversion direction, - application type and - physical action. Above mentioned classification will be explained below.

1.2.1 Classification to enclosed or open turbine machines Regarding whether fluid is enclosed within the turbine machine casing or not (Fig. below) we distinguish: - enclosed (water turbines, pumps, gas and steam turbines, ducted fans, compressors) or - open (propellers, screws, windmills, unshrouded fans, propfan engines, wind turbines) Open machines act on an infinite extent of fluid, whereas closed machines operate on a finite quantity of fluid as it passes through a housing or casing. Fig. below (left) shows an offshore 5 MW wind turbine. With wind turbines, not all the energy of blowing wind can be used. Conservation of mass requires that the amount of air entering and exiting a turbine must be equal. Accordingly, Betz's law gives the maximal achievable extraction of wind

15 power by a wind turbine as 16/27 (59,3 %) of the total kinetic energy of the air flowing through the turbine. In Fig. below (right) a Francis water turbine is shown. Water is contained in housing throughout the turbine.

Fig. 2: Example of open (left, 5 MW offshore wind turbine Senvion, North Sea) and enclosed turbine machinery (right, Francis turbine, HPP Hubelj, Slovenia), source: Wikipedia.

1.2.2 Classification regarding direction of the flow Regarding direction of the flow (Fig. below) turbine machines are - axial, - diagonal, - mixed flow, - radial, - tangential and - cross flow turbine machines.

Fig. 3: Flow through turbine machine. Left: pressure type (from top radial flow turbine, diagonal turbine, axial turbine and cross flow fan), right: impulse Pelton turbine (tangential flow) When the path of the through-flow is wholly or mainly parallel to the axis of rotation, the device is termed an axial flow turbomachine. The axial water turbine shown in Fig. below (left) is a bulb turbine Dubrava, Croatia. HPP Dubrava has max. head 18,8 m, max. flow rate 250 m3/s and max. power per unit 43,5 MW. Bulb turbines are suitable for low head applications for lower courses of rivers. When the path of the through-flow is wholly or mainly in a plane perpendicular to the rotation axis, the device is termed a radial flow turbomachine. When axial and radial flows are both present and neither is negligible, the device is termed a mixed flow turbomachine. It combines flow and force components of both radial and axial types. The Francis and Kaplan turbine may be viewed as radial or mixed flow turbine, in Fig. below (right) is show Francis turbine Doblar I. In literature, there is often confusion about the direction of the flow within the machine. Sometimes direction of the flow is taken with regard to the machine

(turbine machine comprises among others of casing, guide vanes and runner) or with regard to the runner only. For instance, inflow into the Kaplan turbine is radial, while inflow onto the Kaplan turbine runner is axial. Outflow from Kaplan turbine is for both cases axial.

Fig. 4: Example of axial and mixed flow water turbine. Left: flow tract of axial water turbine HPP Dubrava, Croatia. The man standing in front of turbine shows axial arrangement of elements of bulb turbine. Right: Kaplan axial/mixed/radial flow water turbine HPP Doblar I, Slovenia.

1.2.3 Classification regarding energy conversion direction Regarding purpose whether the runner gives energy to the fluid (machinery) or takes energy from the fluid (engine) one may classify turbine machines as: - machinery, runner gives energy to the fluid (pumps, propellers, ventilators, fans, compressors, etc.), - engine, runner takes energy from the fluid (water turbines, steam turbines, gas turbines), - combination of engines and machinery, runner takes energy from the fluid and returns it to the fluid (fluid couplings, transmissions, hydrodynamic clutchs and torque converters). Fig. below (left) shows a low pressure 175 kW compressor HV Turbo for air compression to provide aeration to aeration panels for secondary treatment of municipal and industrial wastewater with up to 0,7 bar pressure difference. The rotational speed gearbox multiplicator is used between the electric motor and compressor runner. The air enters the compressor through the rectangular filter, flows through the guide vane ring, through the runner, 18 diffuser and volute chamber and exits the machine at the end of compressor and leaves through the diffuser directed downwards in front of the compressor. The runner gives the air tangential kinetic energy, a velocity that is later in the diffuser and volute chamber reduced and pressure recovered.

Fig. 5: Turbine machines can provide energy to the flow or get energy from the flow. Left: 175 kW compressor HV Turbo, Danemark (Now Siemens Energy), wastewater treatment plant Domžale Kamnik. Right: 42 MW gas turbine SGT 800, Siemens, TPP Šoštanj, Slovenia. The SGT 800 turbine is designed for combined heat and power and combined- cycle applications. The airflow enters turbine through filter and flows through multistage axial compressor. Each stage of compressor comprises of a runner and guide vane, acting as diffuser directing and slowing down the flow. At the exit from compressor compressed air enters the combustion chamber. In combustion chamber the fuel/air mix is burned. The exhaust gases exit the turbine through the multistage axial turbine. Each turbine stage is comprised of runner and guide vane, similar like in compressor. The useful torque of turbine blades is converted to electric power.

1.2.4 Classification regarding application type Installation type may be on of the following - marine propulsion, - aircraft propulsion, - land vehicle installation, - energy production and - industrial applications. Other installations are also possible.

1.2.5 Classification regarding physical action Turbine machines can be classified according the relative magnitude of the pressure changes across each stage. According to this classification we distinguish impulse and pressure types of turbine machines Impulse turbine machines operate by accelerating and possibly changing the flow direction of fluid through a stationary nozzle onto the runner blade. In such configuration, the nozzle functions as a stator or guide vane blade. In the nozzle the incoming pressure is transformed into velocity, or the enthalpy of the fluid decreases as the velocity increases. Pressure and enthalpy drop over the runner blades is minimal, while velocity will decrease while in contact with the runner almost to standstill. Impulse turbomachines do not necessarly require a casing around the nozzle and runner. Usually they still have a casing to prevent water spilling around the turbine machinery building. In most cases, air pressure within the casing is atmospheric pressure. A Pelton turbine is an example of an impulse turbine. Reaction turbine machine operates by reacting to the flow of fluid through runner and stator blade stages. Pressure and enthalpy consistently decrease through the stages.

Fig. 6: Classification of turbine machines according to physical action. Left: 46,2 MW impulse Pelton turbine spining in open, view from below, Lünnersee, Austria, right: reaction 600 MW Alstom axial steam turbine unit 6, Šoštanj, Slovenia

1.3 Turbine machinery production in Slovenia The whole engineering field of turbine machinery in Slovenia is well developed. In year 1897 started operation the first hydraulic power plant on Ljubljanica river, supplying electricity to paper mill company Vevče. HE Završnica was built in 1915 as first public hydraulic power plant in Slovenia. First very large power plant in Slovenia was HE Fala, built in 1918.

The main companies in Slovenia in the field of manufacture of water turbines are: - Litostroj Power d. o. o. (Already supplied more than 500 water turbines in more than 50 countries, over 18 GW of installed power) - Kolektor Turboinštitut d. o. o., - Andino d. o. o., - Vodosil d. o. o., Laško - Hidropower d. o. o., Idrija, - Ligres d. o. o., - Hydro Hit Gameljne, Ljubljana, - TPS Turbine, Celje, - Siapro d. o. o., Most na Soci, - S.var d. o. o., Godovič, - Tinck Engineering d. o. o., Cerkno, - MBM energy d. o. o., Šentjanž. The main companies in Slovenia in the field of manufacture of pumps are: - IMP Pumps, - Litostroj Power d. o. o., - Kolektor Turboinštitut d. o. o., - Elko Elektrokovina d.o.o., - Ydria motors. The main companies in Slovenia in the field of manufacture of fans are: - Hidria Rotomatika, - Domel, - Tehnovent, - Systemair, - Lindab IMP Klima, - Ebm Papst Slovenija, - Klima Celje, - Gros, - Rotometal, - Agromehanika, - Zupan sprayers, - Pimso Urbanč, - Prislan, - LF AIR, itd. There are no producers of gas or steam turbines in Slovenia.

1.4 Approaches to turbine machinery operation Several approaches (i.e. theories) of turbine machinery operation exist. Among them are: - thermodynamic (chapter 2), - control volume (chapter 3), - differential CFD (chapter 4), - Euler (chapter 5), - linear momentum, - blade element and - blade element momentum approach. Thermodynamic approach is useful if we treat the turbine machine as a black box and want to evaluate efficiency of turbine machine. In linear momentum approach (also called actuator disk approach), for instance for a propeller, we consider axial flow velocity far away before and far away after the propeller. From the change of axial velocity, we calculate force 퐹 = 푚푣̇ by which a propeller acts on the fluid and vice versa. In blade momentum approach for instance for an axial fan, we calculate lift and drag forces on the runner blades in a similar way as for Blade element momentum theory is a combination of momentum and blade element theories. We will in most cases in this book use Euler approach, where in contrast to the blade element theory we follow the fluid flow. In Euler approcah we consider flow velocity and angular momentum at the entrance and exit from the turbine machinery runner. From the change of tangential velocity, we calculate the power, pressure rise etc. We can think of the change of velocity in tangential direction because of the runner acting with force on the fluid flow (see connection to the blade element theory).

Fig. 7: Approaches to turbine machinery For differential CFD approach we calculate Navier stokes equations of the flow inside turbine machinery. For this we divide the flow tract in the very large number of cells. From pressures, calculated from cells on runner surfaces, among others we calculate torque and power on the runner and shaft. Allthough all aproaches are valid and useful, for teaching purposes the Euler approach is most general for all turbine machinery and provides good understanding to students.

1.5 Nomenclature Thoughout the textbook we will use terms turbine, turbine machine, aerodynamic machine and hydraulic machine. Turbine machinery will be a term generally used. With the term aerodynamic machines, we will name low pressure fans, and the term hydraulic machines we will use for pumps and water turbines, in all these cases no heat exchange in the turbine will accur and in most case flow will be non-compressible. In the following we will first review types of turbine machinery, and then we will consider ideal performance laws. In te last part, we will consider different types of turbine machinery. Througt the textbook we will use nomenclature, as explained in separate section about nomenclature.

2. Thermodynamics and efficiencies

In this section we will review thermodynamic properties of turbine machinery. We will present first law of thermodynamics, adapted for the use in turbine machinery. Later, we will present thermodynamic view on efficiency of turbine machinery. In the third subsection we will later present mechanic equations of turbine machinery and will merge both representations in section 4. For additional topics about thermodynamics, available textbooks about turbine machinery often review equation of continuity, first and second law of thermodynamics, steady flow energy equation, momentum equation, Euler equation, Bernoulli equation, compressible flow equations etc. During your study, this was covered in previous courses. In this textbook we will consider that students are to some extent familiar with topics about thermodynamics. Those students who want to study it again, are invited to consider a very good textbook about energy machinery with good sections about thermodynamics and turbine machinery [Tuma and Sekavčnik, 2005].

2.1 First law of thermodynamics The first law of thermodynamics describes conservation of energy for the case of thermodynamic systems. The law of conservation of energy states that the total energy of a thermodynamic isolated system is constant; energy can be transformed from one form to another but cannot be created or destroyed. Total energy change of the system E is a sum of all work 퐴 and heat 푄 supplied to the system

E = Q − A . (3)

For the sign of work 퐴 we will use Clausius convention. By Clausius convention, heat transfer in the system is positive and work done by the system (in our case turbine machine) is also positive. IUPAC convention is however different. As the flow through the turbine machine is continuous, we may also use time derivatives. According to figure below 24

Fig. 8: First law of thermodynamics (left) and the same adapted for the case of turbine machinery (right). Shown are flows into and out of the turbine machine. Here we consider entire turbine machine, not just runner like in the case of Euler equations. we have: - fluid flow into the turbine machine (inlet, index 1), - fluid flow out of the turbine machine (outlet, index 2), - heat flow in or out of the turbine machine and - work flow (power) in or out from the turbine machine (shaft). The first law of thermodynamics for the turbine machine is therefore

E = Q − A = Efluid power out 2 − Efluid power in 1 . (4)

We write expressions for energy 퐸 for the fluid flow flowing into and out of the turbine machine, composed of the following parts: - internal specific energy u, - pressure p, - velocity c and - elevation z. In the following we will often use small letters for variables per unit mass, for instance we write here specific internal energy as u, which is internal energy U per unit mass. The total specific fluid energy comprises of specific 1 internal energy 푢, specific flow work 푝푣, specific kinetic energy 푐2 and 2 specific potential energy 푔푧

1 e = u + pv + c2 + gz . (5) 2 For a steady uniform flow, the first law of thermodynamics has a form 25

1 ṁ (u + p v + c2 + gz ) + Q̇ 1 1 1 2 1 1 1 (6) = ṁ (u + p v + c2 + gz ) + Ȧ 2 2 2 2 2 2

In the above equation we have considered the entire turbine machine, while later we will write equations that consider only runners (Euler equations). In the parenthesis on the left and on the right, we have specific internal energy 1 u, pressure part pv, specific kinetic energy 푐2 and specific potential energy 2 gz. v is specific volume per unit mass. The term pv represents the work done by the fluid inside the control volume to move the fluid in or out out of the control volume. The equation can be extended to multiple inlets and outlets. Remember that both parentheses correspond to the fluid flow in and out of the turbine machine, as shown in Fig. 8 above. The sum of specific internal energy and specific flow work is specific enthalpy 푚2 [ ] 푠2

h = u + pv . (7)

We can further complicate the issue with specific variables and divide the equation above with 푚̇ . The first law of thermodynamics for turbine machinery becomes

1 1 h + c2 + gz + q = h + c2 + gz + a . (8) 1 2 1 1 2 2 2 2

푄̇ 퐴̇ where 푞 = and 푎 = . 푚̇ 푚̇ Sum of specific internal energy, specific flow work and specific kinetic energy and specific potential energy is called stagnation enthalpy or total enthalpy

1 h = u + pv + c2 . (9) 0 2 The first law of thermodynamics can be written as

h01 + gz1 + q = h02 + gz2 + a . (10) q and a are specific heat and a specific work (per unit mass). In turbomachinery, flows are nearly adiabatic and temperature inside the turbine machine is for stationary operation nearly constant, thus q = 0 and we 26 can write first law of thermodynamics for turbine (푎 > 0), remember, work done by the system is positive. In the similar was as total enthalpy, we may also introduce total pressure. Total pressure is a static pressure, increased for a velocity term, which we in this case call dynamic pressure (total pressure = static pressure + dynamic pressure). We can elso neglect potential energy terms to get

a = h01 − h02 . (11) and for compressors and pumps (푎 < 0), but we can also write

a = h02 − h01 . (12)

In the equation above we have ignored the convention of thermodynamics of denoting work out from a system as positive and work in as negative, and the equations are written in a form that gives a positive value for work, for both a turbine and a compressor (it is easier for the lecturer and students also...). Later we will very much rely on the first law of thermodynamics, when we will discuss efficiency of turbine machinery. In this textbook we will in most cases consider systems, where the heat is not supplied or extracted from the system like water turbines, water pumps and low pressure fans. This is because within the Slovenian industry there are many manufacturers with long history in mentioned fields but much less for thermal turbine machines. For such case we substitute enthalpy with pressure and we write first law of thermodynamics for various machines in the following ways: 1. For general case of turbine machines (energy per unit mass)

E p2 1 p1 1 e = = Y = gH = + c2 + gz − − c2 − gz (13) m ρ 2 2 2 ρ 2 1 1

This representation is useful for instance to asses specific hydraulic energy of water in an upper accumulation of the hydraulic water turbine. 2. For low pressure fans with no compressibility effects (energy per unit volume) we prefer to operate with pressures

E 1 1 = p + ρc2 + ρgz − p − ρc2 − ρgz (14) V 2 2 2 2 1 2 1 1 3. For pumps (energy per unit weight) we prefer to operate with heights (elevations above a selected level) 푧

2 2 E Y p2 1 c p1 1 c = = H = + 2 + z − − 1 − z (15) mg g ρg 2 g 2 ρg 2 g 1

The reason why we do so is that the pumping height (usually called head) is for the case of turbine pumps the same regardless of the fluid. For instance, pump pumping water with the head 123 m will pump oil with the same head 123 m. The pumping pressure will be however different, because densities of fluids are different.

2.2 Efficiency Efficiency is often the most important parameter of turbine machinery, think of a water turbine with 600 MW power operating continuously, how much additional energy and income is generated in 50 years of operation if efficiency is improved by 0.1 %. In this subsection we will write the efficiency in a thermodynamic way, without any consideration of flows inside the runner and stator. This falls within the thermodynamic approach as discussed in section 1.4. Efficiencies in turbine machinery are shown in Fig. below. Turbines are designed to convert the available power of a flowing fluid 푃푎푣푎𝑖푙 into useful mechanical power delivered at the coupling of the output shaft Pelect. Pumps and fans operate in the opposite way; they transform useful mechanical power delivered at the coupling of the input shaft Pelect into power of the fluid P.

Fig. 9: Efficiencies in turbine machinery The overall efficiency 휂 of such energy transformation is for the turbines set with eq. 16

Pelect η = (16) Pavail 28 and for pumps and fans with eq. 17

Pavail η = . (17) Pelect

Thermal turbine machines feature in addition the thermal efficiency 휂푡

Pavail η = . (18) t Q̇

Heat flow 푄̇ is supplied to thermal turbine machines. Thermal efficiency 휂푡 considers irreversible losses because of energy conversions in thermal cycle. Thermal efficiencies have approximate value from 0.5 to 0.6 (Tuma and Sekavčnik, 2005). Thermal efficiency 휂푡 is set as a ratio or power of ideally reversible machine working with ideal thermal cycle and the heat supplied to this machine. The highest thermal efficiency 휂푡 has Carnot cycle. The term thermal efficiency only applies to thermal processes and is not relevant at all with hydraulic and aerodynamic machines. To avoid thermal efficiency, we therefore in eq. 16 in denominator write available power 푃푎푣푎𝑖푙. The mechanical power at output shaft coupling is usually power available to the electric generator 푃푒푙푒푐푡. To cover losses in the flow tract including the runner and stator, we introduce internal or isentropic or hydraulic (Dixon and Hall, 2010) efficiency 휂𝑖 (slo.: notranji izkoristek), which is set by eq. 17 for turbines as the ratio of mechanical power of the runner and maximum power possible for the fluid

Pshaft η𝑖 = (19) Pavail and with eq. 20 for fans and pumps

Pavail η𝑖 = . (20) Pshaft

We will study more in details the internal losses in the following chapters. Mechanical losses are represented by mechanical efficiency 휂푚푒푐ℎ. Mechanical losses occur between the turbine runner and the generator/electric motor shaft coupling. This is the result of the work done against friction at the glands, bearings, seals, auxiliary equipment, radiation losses etc. The amount of mechanical losses is for small machines up to 5 %, while for large machines it is well below 1 % (Dixon and Hall, 2010). For turbines we may write mechanical efficiency 휂푚 as a ratio of electric power to the power of the shaft 29

P푒푙푒푐푡 η푚 = , (21) Pshaft while eq. 22 is used for fans and pumps

Pshaft η푚 = . (22) P푒푙푒푐푡

One may separately introduce also other types of efficiencies, for instance due to thermal losses, radiation losses, volumetric losses, bearing cooling loses, draft tube losses, aerodynamic losses etc. In this textbook we will not go into details regarding efficiency.

2.2.1 Efficiency in steam and gas turbines We will consider efficiency in steam and gas turbines based on Mollier diagram shown in Fig. 10. Line 1-2 shows actual expansion and line 1-2s shows ideal adiabatic reversible expansion. We may notice that for any useful installation of steam and gas turbines potential energy terms may be negligible. The actual work rate of a steam or gas turbine may be written using this simplification as

Ȧ 1 = h − h = (h − h ) + (c2 − c2) . (23) ṁ 01 02 1 2 2 1 2 On the right hand side of eq. 23 we have have written stagnation enthalpy as a sum of enthalpy and velocity terms. This is because fluid velocities at entry and exit from the turbine are high. Turbines differ depending on the use of exit kinetic energy. In some turbines exit kinetic velocities is lost, while at others it is not. In the case that it is not lost, ideal work rate output is obtained between states 01 and 02s according to eq. 24 (Dixon and Hall, 2010)

Ȧ max 1 = h − h = (h − h ) + (c2 − c2 ) . (24) ṁ 01 02s 1 2s 2 1 2s We may now write internal efficiency (eq. 25) using eq. 23 and eq. 24 (remember we discussed earlier that internal efficiency represents losses inside the flow tract)

a h01 − h02 휂i = = . (25) a푚푎푥 h01 − h02s

If the difference in inlet and exit velocity is not significant (c1~c2), we may write eq. 25 without kinetic energy terms with enthalpies instead of stagnation enthalpies

a h1 − h2 휂i = = . (26) a푚푎푥 h1 − h2s

Kinetic energy at the exit from turbine is not wasted in aircraft engines, where exhaust gas from last turbine stage contributes to overall engine thrust. Also, kinetic energy from one stage is available in the next stage of multistage turbine. If the exhaust kinetic energy can not be used and is entirely wasted, we have the case, where the ideal expansion is to the same static pressure as the actual process, but with zero exit kinetic energy. The maximum possible work is in this case obtained between 01 and 2s

Ȧ max 1 = h − h = (h − h ) + c2 (27) ṁ 01 2s 1 2s 2 1 and corresponding internal efficiency is total to static effiency

a h01 − h02 휂i = = . (28) a푚푎푥 h01 − h2s

Like before, if the difference in inlet and exit velocity is not significant (c1~c2), we may write the above equation for internal efficiency

a h1 − h2 휂 = = . i 1 (29) a푚푎푥 h − h + c2 1 2s 2 1 Total to static internal efficiency is always lower than total to total internal efficiency. Total to total internal efficiency relates to internal losses in the runner due to creation of vortices, etc., while in addition total to static internal efficiency includes additional loss of kinetic energy at the exit of the turbine.

Fig. 10: Enthalpy in adiabatic turbine and adiabatic compressor. Variables having zeros in the index represent stagnation specific enthalpies, while those without represent specific enthalpies. Dashed line corersponds to total to total and full line to static to static efficiency.

2.2.2 Efficiency in water turbines Water turbines are an example of turbine machines, where the fluid is not compressible. The other examples are low pressure fans and ventilators. Modern water turbines are very efficient machines. They feature large draft tubes, which serve as difusors. Within draft tubes, flow is decelerated to near zero velocity. Before, we have argued, that gas and steam turbines due to the properties of steam or gas cycle and thermal efficiency 휂푡 are only available to use energy rate 푄̇ ∙ 휂푡. Something similar may be written for water turbines, however it does not have same root cause as with steam or gas turbines. The water turbines use water from upper accumulation and release it to the lower accumulation. Mechanical part of water turbines starts at the end of penstock with the turbine valve, if the water turbine has one. Some energy is lost in the water conduit from the upper accumulation to the end of penstock, resulting from friction in the upper accumulation intake, trash rack, channel, tunnel, gates, surge tank itd. Losses from similar causes appear also in the flow tract from the turbine to the lower accumulation. These losses are somehow equivalent to the thermal efficiency of the gas or steam turbine, as they are not related to the mechanical engineering part of the (water) power plant, instead

32 they are civil engineering part of the water turbine design and very much cost related. Additional losses of course appear in the flow tract (internal efficiency 휂𝑖) of the mechanical part of the water turbine (from the end of penstock to the end of draft tube) and due to friction of mechanical parts of the water turbine (mechanical efficiency 휂푚). We will in the relevant section about water turbines go more in detail into efficiency of water turbines.

2.2.3 Efficiency of pumps and compressors For the efficiency of pumps and compressors we follow the similar reasoning as for steam and gas turbines. For a complete adiabatic compression process (Fig. above), we write equation for the specific work

Ȧ = (h − h ) + g(z − z ) . (30) ṁ 02 01 2 1 Fig. above shows real compression process (going from state 1 to state 2) and ideal (which we may also call minimum) compression process (going from state 1 to state 2s). Based on the Mollier diagram in Fig. above, for a compressor the height term on the right of Eq. above is usually negligible and we may write internal efficiency as

amin h02s − h01 휂i = = . (31) a h02 − h01

Similar as with turbines, if the difference in inlet and exit velocity is not significant (c1~c2), we may write Eq. 24 without kinetic energy terms with enthalpies instead of stagnation enthalpies

amin h2s − h1 휂i = = . (32) a h2 − h1

For low pressure ventilators, fans and water pumps the flow is incompressible, therefore ideal specific work may be written as

̇ A (p2 − p1) 1 2 2 = + (c2 − c1 ) + g(z2 − z1) = g(H2 − H1) ṁ ρ 2 (33) = gH .

With H we denote pump head. Because 푝 = 휌푔퐻, using head H instead of pressure is a very useful for pumps. A pump, rotating at the same rotational speed, delivers same head regardless of the pumping medium. For instance, pump pumping water will pump water, oil or air to the same height, however pressures will be different. With ventilators and fans we usually denote performance instead of with head with total pressure. The internal efficiency of a pump pumping incompressible medium is thus

amin gH 휂 = = . (34) i a a In the case, that we are only able to measure electric power, for the case of fans, we write effieciency in the following way

∆p0V̇ 휂i = , (35) Pelect in the above equation, if we neglect any increase in height (z1~z2) of the pumped fluid, the aerodynamic power is

Paero = ∆p0V̇ . (36)

Fig. 11: Characteristics and efficiency of turbine machinery.

2.2.4 Local losses within the turbine machine runner Thermodynamic approach can not provide and information about the local flow properties within the turbine machine runner. However, this is also true

34 with other approaches like Euler approach except to some extend CFD modeling using Navier Stokes equations. The losses discussed in this subsection correspond to hydraulic losses or inner losses according to Figure 9. The losses are in general a consequence of: - pressure gradient perpedicular to the flow, while the pressure gradient itself is a consequence of variation of velocity within the cross section and Bernoulli equation, - flow in the boundary layer (slo: mejna plast) from high to low pressure and corresponding adverse pressure gradients, - presence of solid bodies, etc. Some of the local phenomena, associated with losses within the turbine runner are depicted in Figure below and are discussed here: - Tip leakage flow (A). The pressure on pressure side of the blade is higher then on the suction side. The flow therefore flows over the blade. This loss is present at axial runners and centrifugal runners without the cover. Axial runners can have winglets to mitigate this loss. - Corner separation or corner stall (B). Corner separation is an inherent flow feature of the corner formed by the blade suction surface and the endwall, is a high loss structure in axial compressors. Also called corner stall, it is typically a three-dimensional phenomenon whose main cause is the reverse flow on both the endwall and the suction surface in the boundary layer. If the reverse flow increases to some extent this may lead to serious consequences, such as passage blockage, a reduction in the static pressure rise, a considerable total pressure loss and reduction in efficiency. - Leading edge separation or shock losses or impact losses (C) (slo: udarne izgube). For low and high-volume flow rates, triangles of velocity are very inadequate and relative velocity is not parallel with the blade. Leading edge separation may form on either pressure (high flow rate) or suction side of the blade (low flow rate). In water turbine machinery leading edge separation is also related to cavitation occurence. Leading edge separation loss is less pronounced for thick blades with round inlet edge. This loss is only present in operationg points away from optimal one. - Horseshoe vortex and passage vortex (D) (slo: vrtinec v obliki podkve). The incoming end-wall boundary layer rolls up into a horseshoe vortex, which then wraps around the leading edge. One leg of this vortex stays close to the suction-side corner as it is swept downstream, while the pressure-side leg is driven across the passage by the pressure difference, as it moves downstream, and becomes part of the passage vortex.

- Inlet back flow vortex (E) (slo: povratni tok). In centrifugal runners with large inlet breadth (height of blades at the entrance of the runner for the centrifugal runner), an inlet vortex is formed near the shroud at blade leading edge. The vortex fills approximately 1/4 of flow cross section and increase velocity in the remaining part of the runner. This loss is only present in centrifugal runners with large inlet breadth. - Entry losses (F) (slo: vstopne izgube). Losses due to change of direction and flow acceleration and decelleration because of change of flow tract cross section are represented by formation of vortices. - Collision loss (G) (slo: zastojne izgube). Collision losses are due to the velocity reduction of the fluid at the tip of the blade at it's stagnation point. For thick blades, this loss is significant. - Runner wake (H). Blades have finite thickness. Directly behind this wake velocity is lower. This uneven velocity distribution propagates downstream. Vortices form on the location of high shear stresses. - Wall losses (I). Due to friction between all wals and fluid flow, wall losses appear. - Leakage flows in runner (J) (slo: puščanje v gonilniku). Between front wall and back wall in centrifugal runners, leakage flow flows from the exit to the inlet in the opposite direction of the main flow. - Leakage flows in sealings (K) (slo: puščanje v tesnilih). In water turbines and large pumps, we often allow leakage from sealings. The energy of leaked water is lost. Similar losses as discussed above appear in guide vanes and draft tube/diffusor if the turbine machine has one. We will not consider stall and surge because they only appear at very small volume flow rate and may be also viewed as a special case of leading edge separation losses.

Fig. 12: Losses in turbomachinery runner depicted for the centrifugal fan. Tip leakage flow (A) for the case of a runner without a shroud, corner separation or corner stall (B), Leading edge separation or shock losses or impact losses (C) for the case of low volume flow rate, horseshoe vortex and passage vortex (D), inlet back flow vortex (E), entry losses (F), collision loss (G), runner wake (H), wall losses (I) and leakage flows in runner and sealings (J).

3 Control volume approach to turbine machinery

In the previous section we have studied the turbine machine in a black box approach and without much insight to the flow inside the machine. Now we want to investigate it deeper and we will consider the runner of the turbine machine. In fluid mechanics we can approach the turbine machine runner in an integral way. By doing so we will consider a control volume (Fig. below), which is set around the turbine machine. Only the shaft of the turbine machine extends out of the control volume. All equations, describing the turbine machine with the control volume approach, are based on Reynolds transport equation [Laksmiranayana, 1996]. The control volume approach is also applied to basic laws of conservation of mass, momentum and energy. In the following of this section we will deliberately not be mathematically concise and will trade concisity for som additional clarity.

3.1 Control volume The control volume approach does not allow analysis of flow properties inside the control volume; however, it provides basic understanding of turbine machinery operation. The control volume approach is limited to simple geometries, that may lead to simplified analytical solutions. In comparison with approach from previous chapter, runner operation is addressed. We consider a turbine machine, which is enclosed in a control volume (Fig. below). The control volume is fixed around the turbine machine and does not move. Fluid and heat flow across the control volume surface is allowed, while work is suplied only through the shaft, penetrating accross the control colume to the turbine runner, enclosed within the control volume. Fig. below shows an imaginary control volume, while on it's right is more realistic implementation of the control volume, adapted for the use with turbine machinery.

Fig. 13: Control volume around a turbine machine. Only the shaft of a turbine extends out of the turbine machine. Fluid and heat flow across the control volume surface is allowed. The generalized equation of laws of fluid mechanics for such case is given for arbitrary property N. The N is an extensive property (magnitude is dependent of the size of the system, or additive) like mass, linear momentum, angular momentum or energy. The generalized equation is

dN ∂ = ∬ nρc⃗ ∙ dS⃗⃗ + ∭ nρdV . dt S ∂t V (37) rate of flux of rate of change ( change of N ) = ( N over the ) + ( of N inside ) for the system control surface control volume

On the left-hand side of equation above, there is a rate of change of property N for the system. The first term on the right-hand side of equation above is flux of property N through the control surface. The last term in equation above is rate of change of property N inside the control volume. We denote integration over the closed surface with two integration signs and integration over closed volume with three integration signs. With n we denote specific property of N by unit mass. The equations for mass (continuity), linear momentum, angular momentum and energy are based on equation above. To get continuity equation we must substitute the property N with m and n with n = m/m = 1 and we get

∂ 0 = ∬ ρc⃗ ∙ dS⃗⃗ + ∭ ρdV . (38) S ∂t V

the net outflow rate the production rate 0 = ( of mass through ) + ( of mass in the ) the control surface control volume On the left-hand side of equation we get zero because of conservation of mass. The unit of continuity equation is [kg/s]. For linear momentum, we substitute n for velocity 푐⃗

n → c⃗ . (39)

On the left handside of equation above we must in addition of substitution above also multiply with mass, while on the right handside we directly substitute equation above into equation above to get equation of linear momentum for control volume

⃗⃗ ⃗⃗⃗ ∬S FdS + ∭V BρdV =

∂ ∬ c⃗(ρc⃗ ∙ dS⃗⃗) + ∭ c⃗(ρdV) S ∂t V (40) surface forces body forces ( acting on the matter ) + ( acting on the matter ) in the control volume in the control volume net outflow rate time rate of change = ( of momentum ) + ( of momentum ) through control surface in control volume

On the left hand side, we have a total derivative of velocity, acceleration which is related to the force acting upon the fluid. In this was we may see the above linear momentum equation as an application of Newton's second law of motion. Total derivative in fluid mechanics is for time derivation of a variable, following motion of the parcel, while partial derivative is for the time derivative in fixed point in space. In the above linear momentum equation for control volume we have surface force 퐹⃗ (normalized per unit area) acting per unit area dS, which represent normal and tangential surface forces, acting on the surface of the control volume. As usual for turbine machines, normal forces correspond to pressure forces and tangential forces correspond to viscous and turbulent stresses, acting on the control surface surface. 퐵⃗⃗ are all body forces (normalized per unit mass, also called mass forces), acting on the fluid inside the measurement volume. Examples of these forces are gravitational force and electromagnetic forces. Turbine machinery gives energy to the fluid by rotation or gets it from the fluid. Using linear momentum may therefore be awkward and it is better to use angular momentum instead. We use 40

n → r⃗ × c⃗ . (41)

The 푟⃗ is radius vector from the origin of coordinate system. We get angular momentum equation for control volume

∬ r⃗ × F⃗⃗dS + ∭ (r⃗ × B⃗⃗⃗)ρdV = S V

∂ ∬ (r⃗ × c⃗)(ρc⃗ ∙ dS⃗⃗) + ∭ (r⃗ × c⃗)(ρdV) S ∂t V (42)

surface torque body torque ( acting on the matter ) + ( acting on the matter ) in the control volume in the control volume net outflow rate time rate of change = ( of angular momentum ) + (of angular momentum) through control surface in control volume

On the left-hand side of the equation above same principles apply as mentioned regarding momentum equation. First term on the left can be replaced by shaft torque. For energy equation we substitute n → e and we get energy equation for the control volume. In this was enery equation is similar to the first law of thermodynamics that we introduced in the previous chapter. We will therefore not write the energy equation again here. The work transferred to the fluid (or from) may be of many forms, but in turbine machinery the most dominant form is through the turbine shaft. The rate of work on the turbine shaft is power Ps and is a product of torque M and rotational speed 

Ps = Mω . (43)

The second way, how work can be transferred to the fluid inside control volume, is by a normal stress. The third way, how work is transferred to the fluid, is by tangential stresses. Work done by tangential stresses is low in comparison with work done by a shaft and by normal stresses. In the following, we will write equations for stationary case.

3.2 Stationary case of operation In this section we will skip linear momentum equations, because the angular momentum equation is more suitable for turbine machinery. This is because the turbine machine rotor is rotating along it's axis.

We will also consider stationary operation of turbine machinery. This is usual operation of turbine machinery. Terms involving partial time derivative are then equal to 0. The stationary form of continuity equation is

0 = ∬ ρc⃗ ∙ dS⃗⃗ . (44) S

The stationary case of angular momentum equation is

∬ r⃗ × F⃗⃗dS + ∭ (r⃗ × B⃗⃗⃗)ρdV = ∬ (r⃗ × c⃗)(ρc⃗ ∙ dS⃗⃗) . (45) S V S

Forces 퐹⃗ acting per unit area dS represent normal and tangential surface forces, acting on the surface of the control volume. As usual for turbine machines, normal forces correspond to pressure forces and tangential forces correspond to viscous and turbulent stresses, acting on the control surface. The first term on the left side in the above equation represents angular momentum/torque of all surface forces acting on control surface. The second term on the left-hand side of equation above represents total angular momentum/torque of all body forces and the term on the right represents total flux of angular momentum across control surfaces. Usually, only the torque with respect to shaft rotation is important. The left term involving surface integration of 푟⃗ × 퐹⃗ is a momentum of force on the shaft of turbine machine. The left term can be thus for turbine machines replaced by torque

Mshaft.

4 Differential approach to turbine machinery

As discussed above, control volume approach is unable to provide information about flow properties inside the control volume. For a more in-depth analysis, we must switch to differential approach. Differential approach leads to general equations of the flow, which are Navier Stokes equations. Differential approach enables calculation of properties of the flow in every single point inside control volume. Such approach forms the core of any commercial computer fluid dynamics CFD software solution. Continuity, momentum equations and energy equation form a system of differential equations. This system is not closed, and additional assumptions are required. Such systems can only be solved if suitable initial and boundary conditions are available. Solved problems are accurate solutions for variables like velocity, pressure, temperature etc. in every point inside turbine machine. The differential approach enables analysis of: - local velocity fields, - local pressure fields, - local temperature fields, - forces on the solid boundaries, - local entropy generation, - etc. The differential equations may be derived from control volume equations using Gauss divergence theorem, that relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface. By using Gauss divergence theorem, we get for instance in linear momentum equation all terms with volume integrals. In this textbook we will not perform derivation of equations of motion, reader should consider books about fluid mechanics [Schlichting, 1979]. The equations of motion for compressible viscous flow in differential form are continuity equation [Lakshmiranayana, 1996]

∂ρ + 훻⃗⃗ ∙ (ρc⃗) = 0 (46) ∂t

43 and three linear momentum equations for each direction [Lakshmiranayana, 1996]

dρc⃗ ∂ρc⃗ = (c⃗ ∙ 훻⃗⃗)ρc⃗ + = −훻p + ρ⃗g⃗ + 훻⃗⃗ ∙ τ⃗⃗ dt ∂t (47) (1) (2) (3) (4) (5) (6)

The 6 terms in linear momentum equations are: - (1) total acceleration, - (2) convective acceleration, - (3) local acceleration, - (4) pressure force term, - (5) gravitational force term and - (6) viscous stress tensor. For incompressible, laminar flow, the density and viscosity are constant. For inviscid flow, last term is zero. Under such assumptions, these equations represent a complete set of equations for the four unknowns (pressure and three velocity components), also meaning that fourth energy equation is not required. In most situations, the fifth term 휌푔⃗ is neglected. For further discussion about linear momentum equations in various forms, energy equation and rate of work done by shear stresses, readers should see [Lakshmiranayana, 1996]. The diferential governing equations of the flow are most often simplified to such extent, that they enable engineering analysis of turbine machinery. Often, the flow is: - incompressible, - viscosity gradients are small and - gravitational effects can be neglected. Density is constant in the case of water turbines and low-pressure ventilators and fans. The assumption of constant density can not be used for steam and gas turbines and high-pressure compressors. In the mentioned case of constant density and low viscous gradients the following simplifications: - density is constant and may be omitted from equations where applicable, - viscous terms in momentum equations are simplified and - energy equation is not needed because of density is treated as constant and we need one equation less. The continuity equation becomes

훻⃗⃗ ∙ c⃗ = 0 , (48)

44 telling us that divergence of velocity is zero for such case of the flow (constant density, simplified viscous terms). Linear momentum equations become

dc⃗ ρ = −훻p + μ훻2c⃗ dt (49) (1) (2) (3)

The above equations are called thin layer Navier stokes equations for incompressible flow. In the above equation on the left we have acceleration while on the right-hand side of equation we have forces acting on the fluid. In this case we may see the above simplified linear momentum equation as a development of Newton second law of motion 푚 ∙ 푎 = 퐹, density 휌 being used 푑푐⃗ instead of mass m, being used instead of acceleration a and −∇푝 + 휇∇2푐⃗ 푑푡 being used instead of force F. Individual terms in the above equation represent - (1) local and convective acceleration, - (2) pressure gradient and - (3) viscous forces. Simplification of viscous terms is already incorporated in the above equation. In most cases of development and use of turbine machines the thickness of the boundary layer is small compared to the overall characteristic dimension of turbine machine. In such case, viscosity perpendicular to the element of volume (y in equation below) is much more important than viscosity in the two other directions (x and z in equation below)

휕2c⃗ 휕2c⃗ 휕2c⃗ μ ≫ μ ( + ) (50) 휕y2 휕x2 휕z2

This assumption was already incorporated in linear momentum equation above. We can make further simplifications, from which we can derive Bernoulli equation, potential flow equations etc. [Lakshminarayana, 1996].

5 Euler approach to fluid flow in turbine machinery

In this section we will discuss turbine machinery performance, pressure characteristics, Euler equation, etc. We will assume, that the flow is ideal and no energy is lost due to friction, turbulence decay etc. By considering the ideal turbine machines we will for instance neglect any losses in guide vanes, diffusor or spiral casing. In this chapter, we will consider energy conversions in radial compressors, radial turbines, axial comporessors and axial turbines.

5.1 Energy transfer and triangles of velocity In turbomachinery energy transfer takes place across one or more blade row in a continuously flowing fluid. Energy is transferred to (pumps, compressors) or from a fluid (turbines). During this process energy and angular momentum of the fluid is changed.

5.1.1 Triangles of velocity in the runner Velocities of the flow in turbine machines we will denote with the following symbols and indexes: - c is absolute velocity of the flow in the stationary absolute coordinate system,

- ca is axial velocity of the flow in the direction of the axis of the turbine machine in the stationary absolute coordinate system,

- cr is radial velocity of the flow in the direction of the radius vector in the stationary absolute coordinate system, - u is circumferential velocity of the runner in the stationary absolute coordinate system,

- cu is projection of absolute velocity onto the circumferential velocity of the runner in the stationary absolute coordinate system,

- cm is meridional velocity of the flow and is vector sum of axial and radial velocity in the stationary absolute coordinate system, - w is relative velocity of the flow in relative rotating coordinate system. For the meridional velocity 푐m of the flow we can write the following: - for radial turbine machine 푐m ≈ 푐r, 46

- for axial turbine machine 푐m ≈ 푐a and 2 2 - for diagonal turbine machine 푐m ≈ √푐a + 푐r . Meridional velocity cm is a measure of the volume flow rate of fluid, therefore it is for instance comprised almost entirely on the radial velocity for the radial machine. For meridional velocity and volume flow rate we can write the following equation

V̇ = c푚 S . (51) Here S is a cross section of the flow channel of the turbine machine in the location where the meridional velocity is measured. For the circumferential velocity we write the basic equation of physics, here 휔 is rotational frequency and r radius, where circumferential speed is determined 푢 = 휔푟. For the relation between absolute, relative and circumferential speed we can write the following equation (52), which is the basic kinematic equation of the fluid flow inside turbine machinery

c⃗ = w⃗⃗⃗⃗ + u⃗⃗ . (52)

Absolute velocity of the flow 푐⃗ is vector sum of relative velocity of the flow 푤⃗⃗⃗ and circumferential velocity of the runner 푢⃗⃗. With angle  we will later denote angles of the absolute flow and with the angle  we will denote angles of relative fluid flow (also angle of blades for best efficiency point). For the indexes we will use the notation from Figure below. That means that entrance to the runner will be denoted by index 1 and exit with index 2. Other authors may use different notation. For instance, standard about acceptance tests for water turbine and pump models [IEC 600193, 1999] uses index 1 for high pressure part (for turbine inlet, for compressors exit) and index 2 for low pressure part (for turbine exit, for compressors inlet). Sometimes also indexes 0 and 3 are used, denoting flow conditions just before the runner (index 0) and 3 just outside the runner. We denote angle  as an angle between absolute velocity of the flow 푐⃗ and circumferential velocity of the runner 푢⃗⃗. Angle β is an angle between relative velocity of the flow 푤⃗⃗⃗ and negative circumferential velocity of the runner −푢⃗⃗ (Fig. below). Angle β corresponds to the angle of the blade, it is therefore called also blade angle.

Fig. 14: Triangles of velocity for a cascade. Index 1 relates to entrance to the runner and index 2 to the exit.

5.2 Enthalpy and pressure changes in turbine machinery In this subsection we will show enthalpy changes in both axial and radial compressors and turbines.

5.2.1 Enthalpy and pressure rise in radial compressor Radial compressor is a turbine machine, in which fluid medium flows in radial or in axial direction and exits in radial direction. They are also called centrifugal compressors. They achieve relatively high pressure rise in a single stage. Further stages can be added in series if a higher pressure is required. For instance, compressors of aircraft engines feature more than 10 stages. A three-stage radial compressor is shown in Figure below.

Fig. 15: Radial air compressor of producer Man with three stages. Each runner stage is followed by a radial difussor. Each runner stage has thinner flow channels; this is because the air is compressible. Succesive stages have slightly smaller inlet diameter. Last stage (on the left) has larger exit diameter for high exit speed and higher pressure, also diffuser is longer. Mechanical work, supplied to the runner by the turbine shaft, is converted into stagnation increase of pressure, angular momentum and enthalpy. Radial compressors have usually much higher pressure rise than axial compressors for the reasons which we will discuss later. Let's describe flow properties inside the compressor shown in Fig. below. At the entrance to the runner with inducer, flow direction is mostly oriented axially. At this point, the flow enters the inducer. The inducer is build as the first part of compressor runner and extends to the point, where the flow starts to flow predominanty in radial direction. Some runners have inducer only on for instance every second blade and not all compressors have an inducer. Mechanical work is transferred from the internal combustion engine or electric motor through shaft on the compressor blades. This work is converted into angular momentum, stagnation pressure, head, and enthalpy rise. Throughout the compressor runner blade passage these variables gradually rise as shown in Fig. below (left). Runners without inducer are noisy because flow separation occurs, causing turbulent mixing and vortices generation near the leading edge. In the runner, the static enthalpy, total enthalpy and pressure gradually rise. This is because work is transferred from the runner to the fluid. Tangential velocity of the flow is increased. The reader should remember, that in the introduction we mentioned, that in turbine machines, energy transforms takes

49 place through changes of velocity. Meridional velocity (a vector sum of axial and radial velocity, both contributing to the volume flow rate) depends on the cross sections changes but does not change much. This is because continuity of mass flow rate applies. Rapid changes of velocity are also not desirable because we want to prevent transverse pressure gradients, leading to flow separation and vortices generation. Cross sections of flow channels in turbine machinery therefore do not change much and not suddenly. At the exit from the runner, the flow has substantial amount of the kinetic energy in the form of tangential velocity. Much less kinetic energy is related to the radial, axial or meridional velocity. In the difusser, which is usually designed as a stator with a cascade of stator vanes, but may also be vaneless radial diffuser, the flow is decelerated in tangential direction. Therefore, in the diffuser both static enthalpy and static pressure rise. Stagnation enthalpy does not rise, because no work is done in the difusser (contrary to the runner). In the volute casing or spiral casing, the flow is further decelerated, while static enthalpy rises further. The stagnation enthalpy does not change. The most important task of volute casing is to direct the flow from the circumference of the runner to the flange and pressure piping, if the compresor delivers the compressed fluid into the pipe. Many applications exist, where there is no need to pump the fluid in the pipe, instead the fluid is pumped in the volute.

Fig. 16: Enthalpy and stagnation enthalpy rise in runner, diffusor and volute casing of centrifugal compressor. Shown is the runner with inducer.

5.2.2 Entalpy and pressure reduction in radial turbine The opposite process as described in subsection above is happening in the radial turbine. Among them, we will consider Francis water turbine. Francis turbine is water turbine and the water as a medium is not compressible. In the Francis turbine, water flows from the upper reservoir through the pressure penstock to the turbine. In the upper reservoir, the water has potential energy. At the end of penstock, the potential energy is transformed to pressure energy. The water at the end of the penstock (end of the pressure penstock and beginning of turbine is a turbine valve for Pelton and Francis turbines) flows into spiral casing, stay vanes and guide vanes, where pressure energy from the penstock is transferred into tangential velocity. At the entrance to the runner (at the exit from guide vanes) the radial velocity is comparably low to conform to the countinuity equation.

Fig. 17: Enthalpy and stagnation enthalpy decrease in runner, diffusor and volute casing of centrifugal turbine. The high tangential velocity is in the runner reduced to approximately zero tangential velocity. Thus, at the exit from the Francis runner, the only velocity that the water posess, is axial velocity. This is required to empty the runner and allow influx of new water and to satisfy the continuity equation. Such situation is however only present in the best efficiency point. Away from this point, exiting flow posesses various amounts of tangentional velocity, thus lowering the efficiency.

Fig. 18: Runners of Francis turbines. Left runner is for high heads and

right runner for low heads. Runner for high heads (low ns) have longer blades to efficiently convert the available tangential velocity into shaft torque. They also have more runner blades to

guide the flow better. Runners for low heads (high ns) usually have high blades for high volume flow rate, Fig. adapted according to Giesecke and Mosonyi [2009] Francis turbines, which have high heads, feature high tangential velocities at the entrance to the runner. Such Francis runners have many long blades to be able to efficiently transform all available tangential water velocity into mechanical shaft torque. After the runner, the fluid flows in the difussor or draft tube for water turbines. In the draft tube, water velocity is further reduced. We may apply Bernoulli equation for the lower reservoir (end of draft tube) and exit from the runner (beginning of the draft tube). The reduction of velocity in the draft tube causes pressure reduction at the low-pressure side of the runner. Thus, reaction type water turbines can fully make use of the water energy. This is however not the case for impulse turbines like Pelton turbine.

5.2.3 Enthalpy and pressure rise in axial compressor Enthalpy rise in axial compressor follows similar principles as in the centrifugal compressor. The main difference is that the exit radius is approximately the same as the entrance radius. In the axial compressor fluid flows in axial direction and at the exit from the compressor also in axial direction. Cross section of multi stage axial compressor with cover open is shown in Fig. below.

Fig. 19: Siemens axial compressor with 11 stages for industrial use. First stages are larger and last stages smaller because of the compression of air. In all stages, runner blades are followed by stator blades. At the entrance to the runner the fluid flow is in axial direction. In the runner, the static enthalpy, total enthalpy and pressure gradually rise, like the case in radial compressor. Here the increase is in general smaller than in radial compressor, however it depends also much on the rotational speed. Flow channel cross section gently decrease due to the compression of fluid in the runner and later in the diffusor. At the exit from the runner, the flow enters the diffusor. Here flow has substantial amount of the tangential kinetic energy, while axial kinetic energy is such as required to satisfy continuity equation. In the diffusor (stator), the tangential kinetic energy is transferred to pressure rise. Here no work is done to the fluid, but enthalpy is increased, while stagnation enthalpy stays the same. Also, static pressure is increased, total pressure does not increase. Axial compressors do not require spiral casings.

Fig. 20: Enthalpy and stagnation enthalpy rise in runner, diffusor and volute casing of four stages axial compressor. Only two stages are shown on the right, but the process continues. The inlet guide vane produces a weak tangential flow at the entrance to the first runner stage.

5.2.4 Enthalpy and pressure reduction in axial turbine In an axial turbine, water at high pressure (for instance bulb turbine) or gas at high pressure and temperature (gas or steam turbine) flows onto stationary vanes and rotating runner blades. In water turbines, stationary blades are called wicked gates and in gas turbines nozzles. The axial steam turbine is shown in Fig. below. In the water bulb turbine, the water flows through the pressure channel onto the wicket gate. In the following we will follow the flow in a water bulb turbine.

Fig. 21: Axial steam turbine Siemens SST 800 for an output range of up to 250 MW, n = 3000 /min. Consecutive stages are larger, because steam expands (pressure after each stage is reduced). First stages have shrouded rotors. All stages are attached to the same shaft and rotate at the same rotational speed. In the wicket gate, the water gets tangential direction of the flow. The more head (pressure) the water turbine must recover; the wicked gate blades turn the flow in the tangential direction. Such turbines, intended to recover higher heads, also have longer wicked gate blades and higher number of blades. In the axial direction, the flow does not accelerate or decelerate much inside wicked gate. The stagnation enthalpy here does not change, but enthalpy and pressure descrease because of increase of velocity in tangential direction. The flow then enters the runner (rotor for water turbines is called runner). Inside the runner the energy transfer is performed. The energy of the fluid is transferred trough the runner blades to the shaft and to electric generator. Here head, static and stagnation enthalpy decrease. The rotation of the fluid in the form of tangential velocity is in the optimum operating point fully removed and at the exit from runner the fluid does not have any tangential velocity. Axial velocity of the flow does not change in the runner. In water turbines the cross section of the flow channel does not expand when enthalpy and pressure decrease, while this is not the case with gas and steam turbines.

Fig. 22: Enthalpy and stagnation enthalpy changes in wicket gate and runner of bulb water turbine. Situation is similar in gas and steam turbines. In steam turbines, stator blades are usually called nozzles. The temperature and pressure decrease; therefore, the flow channel cross section gradually increase.

5.3 Torque on the blades of the turbine machine runner The turbomachinery runner of centrifugal compressor is schematically shown in Fig. below. Using the enclosing of the blade (greyed part) the control volume approach is applied. Both upper and lower surfaces are axisymmetric, and we say that in our idealized approach no mass, momentum or energy transfer takes place there (I- inner surface and O-outer surface).

The inlet velocity axial component ca and tangential component ct form a vector of velocity c1. The inlet and outlet surfaces are away from the runner. The reason of doing so is to ensure that all variables are indeed axissymetric and uniform along the runner. Other variables are also indicated on the Fig. below (index 1 corresponds to inlet and index 2 to outlet).

Fig. 23: Energy and momentum transfer in turbomachinery compressor runner. The energy transfer takes place through shaft work, which is due to axisymetricity in the form of torque M. Input shaft power is transmitted to the control volume as torque on the blade element enclosed within the control volume. The runner blade exerts a tangential force Ft on the fluid. The runner blade force then increases for this compressor the fluid angular momentum.

The ct2 is due to the action of the runner larger then ct1. The flow passage in Fig. above acts like a diffusor and thus changes static and stagnation properties of the flow across the runner.

Because of our previous assumption that in our idealized approach no mass, momentum or energy transfer takes place across lower and upper control volume surfaces. What is transmitted is the shaft torque through the blade being submerged in the control volume. We therefore perform integration at the inlet and outlet only (rings A and D). The continuity equation is

∬ ρ1c⃗1 dS⃗⃗1 = ∬ ρ2c⃗2 dS⃗⃗2 (53) ring A ring D

The angular momentum equation may be written based on previous equation. The surface forces in the tangential direction on the control surfaces I and O are very small and can be neglected. As we mentioned before in Eq. above, surface and volume forces in the control volume drive the fluid in the turbine machine. Volume forces like gravity do not contribute to the net torque. Surface forces however contribute to the net torque. Surface forces are composed of those perpendicular to the surface (pressure) and parallel with the surface (shear). This translates equation, written for the control volume to

∬ rFBdA = ∬ (r2cu2) (ρ2c⃗2d⃗S⃗2) − ∬ (r1cu1) (ρ1c⃗1dS⃗⃗1) (54) S ring D ring A

As before here blade force 퐹퐵 is force per unit area. The thickness of the stream integration layer is assumed very small. We can therefore use the assumption of uniform entry and exit velocity and thus the above equation reduces to

dM = rFBdA = dṁ (r2cu2 − r1cu1) (55)

The differential torque 푑푀 on the left side of equation above 퐹퐵 is a surface force per unit area, exerted by blades. The integration is sum of all moments of all forces acting in the control volume, being in total to the shaft torque. 푚̇ is the mass flow through the control volume. Right hand side of equation above represents the change of angular momentum of the flow. In such a way we have linked the shaft torque with the change of angular momentum of the flow. We need to discuss here why velocities 푐푢1 and 푐푢2 appear in the equation above. Equation above is written without a vector product, hence we have replaced term 푟⃗ × 푐⃗ with product of scalar radius and velocity 푟 ∙ 푐. The

59 velocity component, perpendicular to 푟⃗ and 푀⃗⃗⃗ is projection of actual velocity 푐⃗ to the circumferential velocity 푢⃗⃗. This projection we denote with 푐푢. If we multiply the above equation with angular frequency, we get shaft power

−dPshaft = ω(dṁ )(r2cu2 − r1cu1) (56)

We are not concerned a lot with the sign of the shaft power. Usually the power is positive, when the power is done by the fluid (turbines) and negative when the power is given to the fluid (fans, compressors, pumps).

5.4 Euler equation for turbine machinery We may integrate the above equation and we get the Euler equation of turbine machinery

−Pshaft = ṁ ω(r2cu2 − r1cu1) . (57)

The above Euler equation is valid only for a two-dimensional strip shown in the Figure above. The equation above is valid for the cases, when the velocities 푐푢1 and 푐푢2 are constant across the entire inlet and outlet cross sections. This is however usually not the case and we use averaged values or perform the integration of the equation above. For the axial turbine machine, both radii 푟1 and 푟2 are equal and the equation above becomes

−Pshaft = ṁ c(cu2 − cu1) (58)

In the above equation we have used a known relation 푐 = 휔푟. For a fan, compressor or pump the exit tangential velocity 푐푢2 is higher than inlet velocity 푐푢1. In this way the axial turbine machine (fan, compressor or pump) gives power to the fluid. The situation is reversed for an axial turbine, here exit tangential velocity 푐푢2 is lower than inlet velocity 푐푢1. Same logic applies also for radial turbines, fans, compressors and pumps. For a centrifugal machine we have

−Pshaft = ṁ (u2cu2 − u1cu1) (59)

푢1 and 푢2 are exit and inlet blade velocities. For a centrifugal machine 푢2>푢1, therefore for the same change of tangential velocities 푐푢1 and 푐푢2 the centrifugal machine has higher input and output shaft power. This results in higher head or pressure drop or rise. Also, higher blade velocities result in higher input and output shaft power. The change in angular momentum or tangential velocity are directly proportional to the shaft power. 60

The energy equation, applied to the control volume from Figure above is

q̇ = Pshaft + Pshear + ṁ (h02 − h01) (60)

If we consider that not heat is transferred through machine walls (adiabatic process) and that 푃푠ℎ푒푎푟 = 0 we get

−Pshaft = ṁ (h02 − h01) (61)

This equation is valid for viscous, compressible flows and includes all losses associated with viscous flow. Shear stresses due to shear forces of shear layers in radial direction are neglected. If upper and lower control surfaces are the same as upper and lower walls of the turbine machine, where velocity is zero 푐 = 0, the above equation is valid for all flows. The shaft power input in fan, pump and compressor is directly related to the change of total enthalpy of the fluid. Again, for the fan, compressor or pump, ℎ02 > ℎ01, therefore shaft power is negative. The situation is refersed for a turbine mode of operation. Equation above (for angular momentum) and equation above (for energy), relate 푃푠ℎ푎푓푡 to the flow and thermodynamic properties. Both give for general turbine machine handling liquids and compressible fluids

Pshaft − = (h − h ) = (u c − u c ) (62) ṁ 02 01 2 u2 1 u1 for a liquid (pump, water turbine) or low-pressure air handling (fan) turbine machine we write

Pshaft 1 − = (p − p ) = g(h − h ) (63) ṁ ρ 02 01 2 1

Here h is head or height of delivered fluid in the units of height. Again, it is evident, that stagnation pressure or enthalpy rise or drop in a turbine machine is directly related to the change of tangential velocity and blade velocity. This Euler equation is the most basic equation of turbine machinery. In the following we will discuss the flow properties in radial and axial compressors. We will consider idealized situation and will therefore make few assumptions that: - blades are very thin, - runner has infinite number of blades - blades ideally guide the flow, - no shadow of velocity exists behind the blade, 61

- the turbine machine operates at the nominal operating point of highest efficiency (BEP, best efficiency point), - no gap losses are present, - the flow in the entire turbine machine is frictionless, - etc.

5.5 Radial flow compressors and pumps Radial flow compressors and pumps are used in applications, where high pressure is required. Radial compressors are simple, reliable, cheap and therefore often preferred choice of compressors. On the other side, the radial compressors have a rather high cross section area and are suited for aircraft applications only for low to moderate power. Radial flow compressors are widely used in industrial, automotive, rocket, air conditioning, irrigation systems, power generation sytems, and other applications. Pressure ratios of over 10 may be achieved in single stages, while rotational speed over 10000 rpm are usual, up to almost 200000 rpm in some applications. Radial pumps are the most used type of pumps for industrial applications. In centrifugal turbine machines, on the fluid in the runner act centrifugal, Coriolis and tangential forces. These forces are the consequence of rotation and change of radius of the element of the fluid, while rotation of the runner is responsible for them. In radial turbine machines fluid enter the machine in axial direction and exits in radial direction. However, in much of radial runner designs, the flow changes direction from axial to radial even before it enters the runner. Such radial turbine machines have axial entrance to the turbine machine, but radial entrance to the runner. Exit from both runner and turbine machine is radial.

5.5.1 Triangles of velocity in radial compressors and pumps To explore the principle of operation of radial turbine machinery, we must consider Euler equation first (Eq. above). The Euler equation governs enthalpy, pressure and temperature rise in the turbine machines. Velocity triangles are shown in Fig. below for a radial turbine.

Fig. 24: Triangles of velocity in radial turbine in best efficiency point Velocity triangles are shown for the case away from best efficiency point in

Fig. below. In both cases the runner experiences impact losses. At 푉̇ < 푉̇표푝푡 a vortice forms on the pressure side of the blade, at 푉̇ = 푉̇표푝푡 a no vortice is present and at 푉̇ < 푉̇표푝푡 a vortice forms on the suction side.

Fig. 25: Triangles of velocity in radial turbine in best efficiency point Velocity triangles are shown in Fig. below for a radial compressor.

Fig. 26: Triangles of velocity in radial compressor in best efficiency point

In fully axial machine, 푢1 = 푢2 and the flow enters and exits the turbine machine at the same radius. The total enthalpy rise is achieved in the runner stage, when the fluid is accelerated in tangential direction and fluid's 64 tangential momentum increases. In addition to this, the centrifugal compressor achieves part of the total enthalpy rise from centrifugal and Coriolis forces due to change of radius. This offers radial compressors additional increase of total enthalpy or pressure rise in comparison with axial compressors, resulting in a widespread use of these machines. The increased total enthalpy and pressure rise has also adverse effects. Among them are formation of adverse pressure gradient, flow instabilities, flow separation and secondary flow. Due to this and the need to turn the flow direction in the case of multistage design, centrifugal compressors have in general lower efficiency than axial compressors. The flow in radial compressor is shown in the Fig. above. The figure above considers two cases, one with the axial entrance to the runner and flow inducer and the other with radial entrance to the runner. The role of the inducer is to turn the flow from axial to radial direction. The inducer helps to decrease flow separation near the leading edge, increase efficiency and decrease noise. We will consider the case, when the flow after leaving the inducer continues in purely radial direction. If the flow is in radial and axial direction, such machines are called diagonal compressors. At the exit from the runner, the flow has large velocity components in radial and tangentional direction. The component in radial direction contributes to the volume flow rate, while the component in the tangentional direction must be slowed down to gain enthalpy and pressure rise. For this case, the flow is diffused or slowed down. This is performed in radial direction in radial diffusor or in axial diffusor, in both cases the flow must be slowed down slowly to prevent flow separation and generation of turbulent vortices that decrease efficiency of diffusors. Axial diffusors often start with a scroll or spiral casing, extending into a pipe or axial diffusor within the piping. In radial compressors the cross section of runner passages increases with radius. This may lead to compressibility problems. To mitigate this, flow channel cross section is decreased such that runner's thickness is smaller at the exit (distance between both shrouds). In many instances, especially for small pumps and compressors, diffusor or scroll are omitted. In such case, the runner discharges directly in the volute. In such case, stall and surge margins are greatly reduced, but the flow is not well guided and much of the pressure rise (due to stagnation pressure loss) is lost due to turbulent dissipation. However, pressure rise in the runner is not lost. In the following we will examine the flow in the radial compressor. The flow enters the inducer in position 1. The relative flow to the runner is described by the vector 푤⃗⃗⃗1 in the relative rotating coordinate system of the blade. The

65 relative velocity 푤⃗⃗⃗1 is directed in the same direction as the blade leading edge to prevent losses at the entrance to the runner. However, the blade rotates with tangential velocity 푢⃗⃗1 and the absolute velocity of the flow at the entrance to the runner is according to the equation of the velocity triangle equal to 푐⃗1. The product 푢1푐푢1 is angular momentum that the flow has at the entrance to the runner. With purely radial entrance as shown in Fig. above, the product 푢1푐푢1 equals zero. We do a very serious simplification here regarding how the flow accelerates just before it enters the runner. Basically, we allow that the flow accelerated just before it enters the runner. This is not the case for instance with the compressors with inlet guide vanes, which direct the flow to the runner or with for instance Francis water turbines, which have a guide vane at the entrance to the runner. The runner of the centrifugal compressor accelerates the flow in the tangential direction. In radial direction the flow can not be accelerated, because this would violate the continuity equation. By giving the flow a component of velocity in tangential direction, the runer increases the angular momentum of the flow and gives energy to the fluid. At the exit from the runner at position 2, the radius has increased to 푟2. The velocity exits the runner at relative velocity vector of the flow in the relative rotating coordinate system 푤⃗⃗⃗2. The flow wants to continue the flow in the direction of the blade, however in the relative coordinate system of the blade. For the observer in the stationary coordinate system, the flow has a velocity 푐⃗2, as it is the sum of velocities 푤⃗⃗⃗2 and 푢⃗⃗2. The flow then enters the diffuser or volute. For a vaneless radial difussor or volute, the position 2 is the final position, which needs discussion. But if the compressor is equipped with a vaned diffusor, the vanes leading edges at position 3 should be oriented in the direction of exit velocity 푐⃗2 in absolute coordinate system. The flow thus enters the vaned diffusor at such angle, that no losses appear due to transition from the runner to the vaned difussor. The diffusor directs the flow in the desired direction and increases pressure to use all available velocity of the stagnation pressure.

5.5.2 Second form of Euler equation The Fig. below shows velocity triangles at the exit from the centrifugal runner. The flow direction corresponds to fans, compressors and pumps. We want to write second form of Euler equation without terms 푐푢1 and 푐푢2.

Fig. 27: Velocity triangle at the exit from the centrifugal runner, used to write second form of Euler equation The Euler equation can be re-written using the cosinus theorem (law of cosines), written for the case of Fig. above

2 2 2 2 2 w2 = c2 + u2 − 2u2c2 푐표푠(α2) = c2 + u2 − 2u2cu2 (64)

With the use of Equation above, we get for the entrance

1 u c = (c2 + u2 − w2) (65) 1 u1 2 1 1 1 and for the exit

1 u c = (c2 + u2 − w2) (66) 2 u2 2 2 2 2 and Euler equation can be rewritten in a so called second form of Euler equation as

P − shaft = (h − h ) = (u c − u c ) ṁ 02 01 2 u2 1 u1 1 (67) = [(c2 − c2) + (u2 − u2) − (w2 − w2)] 2 2 1 2 1 2 1

This form of Euler equation provides additional understanding of energy transfer in the turbine machinery. We will consider the case of radial compressor: 2 2 - The first term (푐2 − 푐1 ) corersponds to the increase (Eck, Ventilatoren, 2003) of kinetic energy. The pressure available from this term is at the exit from the runner not readily available. According to the Bernoulli equation can this velocity be slowed down and transferred to the pressure in diffusors, spiral casings etc.

2 2 - The second term (푢2 − 푢1) corresponds to the increase of static pressure (Eck, Ventilatoren, 2003) in the runner. It is important to stress this very much, this is the most important term in the entire Euler equation, because there are no losses connected with this term. This term is not available with axial turbine machines (inlet and outlet radii being the same), meaning that radial turbine machines are able to achieve higher pressure increase then axial turbine machines. - The third term includes relative velocity change. In the turbine machines runner, the flow is decelerated because of cross section increase and therefore 푤2 < 푤1. This corresponds according to Bernoulli equation if we 휌 assume flow without losses to static pressure increase of (푤2 − 푤2). 2 1 2 We can write the above Equation with enthalpies instead of stagnation enthalpies and we get

1 (h − h ) = [(u2 − u2) − (w2 − w2)] (68) 2 1 2 2 1 2 1 As above, the part of the enthalpy (or pressure or temperature) rise is due to change of radius (centrifugal effect, first term on the right) and part due to change of relative velocity (reduction of velocity, second term on the right). In comparison with radial compressors, axial compressors do not have the centrifugal term The Euler equation can be written for the case of compressor with radial inlet as (we assume purely radial inlet without any tangential velocity or pre-swirl)

Pshaft − = u c (69) ṁ 2 u2 This is because we assume that in the perpendicular velocity triangle at the inlet the projection of velocity is 푐푢1 = 0. The first form of Euler equation reads now

Pshaft 1 − = [c2 + u2 − w2] (70) ṁ 2 2 2 2

5.5.3 Reactivity For the characterisation of fans, compressors and pumps, it is important to discuss, how high is the pressure immediately after the runner and how much pressure can be gained from the transformation of velocity to pressure. The 휌 answer is again provided in Eq. above. The last two terms ∆푝 = ((푢2 − 푢2) − 2 2 1

2 2 (푤2 − 푤1 )) set the static pressure after the runner, also called gap presure. As 휌 discussed above, the first term (푐2 − 푐2) can be transformed to pressure if 2 2 1 the fluid flow velocity is decreased. Because transformation of velocity into pressure is always related to significant losses, we want to minimise this term. This means that we want to maximise the term R, which we call reactivity

∆pstat ∆p R = = (71) ∆ptotal ∆p0

Reactivity R is therefore the ratio of pressure increase in the runner versus total pressure increase. We can further relate static pressure increase with reactivity in the followng discussion. Static pressure increase is (we do not consider total velocity for the static pressure increase):

∆p 1 = [(u2 − u2) − (w2 − w2)] (72) ρ 2 2 1 2 1

If we assume, that 푐푢1 = 0 (radial entrance to the radial runner), then the inlet velocity triangle is rectangular

2 2 2 w1 − u1 = c1 (73) and

∆p 1 = [u2 − w2 + c2] (74) ρ 2 2 2 1 we get the equation for reactivity

2 2 2 ∆pstat u − w + c R = = 2 2 1 . (75) ∆ptotal 2u2cu2

We make further simplification now and we assume that actual velocity may be replaced by meridional velocity 푐1 = 푐1푚 = 푐2푚. It is true for axial turbine machinery, but only approximate for a radial machine, in everyday's design of radial fans however this is a valid assumption. We get therefore

2 u2 − w2 + c2 u2 − (u − c ) R = 2 2 2m = 2 2 u2 (76) 2u2cu2 2u2cu2

We have used in the above equation 69

2 2 2 w2 − c2m = (u2 − cu2) (77)

The equation above transforms to

−c2 + 2u c 1 c R = u2 2 u2 = 1 − u2 (78) 2u2cu2 2 u2

We compare runners with the same circumferential speeds, that at the same radius and same thickness deliver the same volume flow rate. Therefore, the velocity 푐2푚 is constant. For different blade angles at the exit 훽2 from the runner different velocity triangles are formed according to the Figure below. For the presentation purposes we include the dimensionless pressure ratio

휓푡ℎ, while we have Δ푝푡ℎ = ρ푢2푐푢2 and in denominator the pressure as in Bernoulli equation written with circumferential velocity

훥pth cu2 ψth = ρ = 2 (79) u2 u2 2 2 and in the Figure below we include both pressure ratio 휓푡ℎ and static pressure ratio 휓푠푡푎푡 푡ℎ over the velocity triangles. The absolute value of static pressure ratio may be found from the product below

c ( u2) 2 cu2 u2 cu2 cu2 ψstat th = ψth ∙ R = 2 ( ) 1 − = 2 ( ) − ( ) u2 2 u2 u2 ( ) (80) 2 c c = 2 ( u2) − ( u2) u2 u2

휓푠푡푎푡 푡ℎ is parabola, that crosses the abscissa at 푐푢2 = 0 and 푐푢2 = 2푢2. The highest value is achieved at 푐푢2 = 푢2 with the value 1. At the blade angle 90 °, that is with radial blades, one half of the total pressure is static, and the other half is provided in the form of kinetic energy. At 푐푢2 = 2푢2 is static pressure is 0 and the total pressure the highest, thus offering the highest possible total pressure, if the velocity can be transformed later in the static pressure. In this case 푐푢2 = 2푢2 the runner has produced only kinetic energy. Such runners are known as impulse runners (Slo.: impulzni gonilniki, Ger.: Gleichdruckräder). A 70 very well-known example are Pelton turbine runners. The runners, in which pressure in the runner increases, are called reactive runners (Überdruckräder). The most fans in everydays life use the at least some of the reaction principle of operation.

Fig. 28: Kinetic and static pressure for different blade angles The flow conditions from the above picture can be achieved by: 1 - forward curved blades (푅 < and 푐 > 푢 ), 2 푢2 2 1 - radial blades and (푅 = and 푐 > 0) 2 푢2 1 - backward curved blades (푅 > and 푐 < 푢 ). 2 푢2 2 Figure above shows forms of blades for the mentioned three cases. The inlet angle is for all types in the Figure below the same, because the same inlet angle is required for a selected volume flow rate and rotational frequency. Fans in everydays use all three forms of curvature of blades. 71

The reactivity is a very important parameter of turbine machinery design. Nevertheless, reactivity is related to the internal operation of a turbine machine, that is for the operator not important, with other words, it is an internal matter of turbine machine developer. For the operator of a turbine machine, the reactivity is of little interest, because he is only concerned about overall performance of a turbine machine. What is however important for the end user in relation to reactivits, is related to the cross-section area of outlet. Some fans and pumps deliver very high- volume flow rates. If such machines deliver the flow at very high velocities, it is difficicult for a user to use very large cross sections, allowing for gradual decrease of velocity. We may consider a fan that sucks air from an infinite space and releases it in the pipe or channel of selected cross section. Very large fans can not have forward curved blades due to the very large required size of the pressure channel. On the other hand, small fans may benefit from forward curved blades if the installation is such that it allows for good recuperation of velocity into pressure. This will be shown in next subsection in the Cordier diagram. Small radial fans with forward curved blades are used where aerodynamic performance is more important than efficiency.

Fig. 29: Types of runners with backward curved blades, radial blades and forward curved blades (vir: EBM Papst in Continental Fans).

5.5.4 Radial runner characteristics The pressure characteristics in relation to for the operation of a turbine machine. The Fig. below shows the effect of backward, radial and forward curved blades. By using the geometric relation, we want to link the velocity projection on tangential velocity 푐푢2, velocity in meridional direction 푐푚2 (to be later related to volume flow rate) and blade angle 훽2 using equation

cm2 = 푡푎푛 β2 (81) u2 − cu2

72 from the above equation follows

cm2 cu2 = u2 − (82) 푡푎푛 β2 text

∆p0 1 = u2cu2 = u2 (u2 − cm2 ) (83) ρ 푡푎푛 β2

By using the continuity equation for the volume flow rate, the meridional velocity, being the same as radial velocity for purely radial machines, is at constant rotational frequency proportional to volume flow rate

V̇ cm2 = . (84) πD2b

For the pressure we get from first form of Euler equation for the case of radial inlet 푢1푐푢1 = 0

∆p0 V̇ 1 = u2cu2 = u2 (u2 − ∙ ) (85) ρ πD2b 푡푎푛(β2)

With the increase of angle of relative velocity 훽2 at constant tangential velocity 푢2 the absolute 푐2 and relative 푤2 velocity of the flow both increase. This change theoretical characteristics of the turbine machine according to the modified Euler equation

̇ ∆p0 2 V 2 = u2 (1 − 푐표푡(β2)) = u2 − u2k2V̇ (86) ρ πD2bu2

∆푝 The relation 0 = 푓(푉̇ ) results in a straight line with the slope related to the 휌 blade angle 훽2: - with the forvard curved blades the total pressure increases with the volume flow rate, - with radial blades the total pressure remains constant and - with backward curved blades the total pressure decreases with the volume flow rate. The relation is shown in Fig. below.

Fig. 30: Radial flow compressor and it's velocity triangles

5.5.5 Radial runner power and torque We can write the above equation in terms of power instead of specific power, enthalpy or pressure

P = ∆p0 ∙ V̇ (87)

here ∆푝0 is total or stagnation pressure rise across the centrifugal compressor. The theoretical power of the centrifugal turbine machine is

2 u2 P = ρV̇ (u2 − V̇ ∙ ) . (88) πD2b2 푡푎푛(β2)

The equation above results in the required power consumed by the centrifugal compressor according to the type of blades. The power of the centrifugal compressor according to the type of blades is shown in the Figure above. The power: - increases as a parabola with volume flow rate for forward curved blades (above the line of radial blades), - increases a line with flow rate for radial blades and - inreases as a parabola with volume flow rate for backward curved blades (below the line of radial blades). 74

Because of equation 푀 = 푃 ∙ 휔 the torque characteristics features same behavior as pressure curves.

5.5.6 Radial runner static pressure In the section above, we have discussed static and total pressure increase in relation to the runner blades angle. We have set constant flow rate and meridional velocity. Now we want to derive the equation for the variation of static pressure regarding the volume flow rate. Static pressure increase is of high importance as dynamic pressure is difficult to transform to pressure without turbulent losses. For this, we revert to second form of Euler Equation above

∆p 1 = [u2 − w2 + c2] (89) ρ 2 2 2 1 and assume 푐1 = 푐2푚 ∆p 1 = [u2 − w2 + c2 ] (90) ρ 2 2 2 m2 and

cm2 = w2 푠𝑖푛 β2 (91) to get

̇ 2 ∆p 1 2 2 1 1 2 V = (u2 − cm2 ( 2 )) = (u2 − 2 2 2 2 ) . (92) ρ 2 푡푎푛 β2 2 π d2b2 푡푎푛 β2

∆푝 The function = 푓(푉̇ ) is a parabola. Because of the function tan2 훽 there are 휌 2 no difefrencies between forward and backward curved blades. For radial ∆푝 1 blades with 훽 = 90 ° the is equal 푢2. At zero flow rate the static pressure 2 휌 2 2 is one half of total pressure.

5.5.7 Radial runner real pressure characteristics The power consumption and pressure or enthalpy rise in the Figure above is only theoretical. The combined result of most important sources of losses mentioned effects is shown in the Figure below about the theoretical pressure characteristics:

- A: Losses occur because flow does not follow exactly the runner blades curvature and angle. This is a consequence of finite number of blades. Irrespectible of the blade angle 훽2 the projetion 푐푢2 will be smaller, reducing the teoretically achievable pressure rise of the runner. One can not necessarily consider this as a loss of energy, because the energy is not provided by the runner. - B: Losses because of friction in the runner, these are proportional to the square of velocity (∝ 푐2). - C: Impact losses (Slo.: udarne izgube) from the impact of the flow to the blades at a non-ideal angle. At the volume flow rate, higher that the nominal (optimal) the triangles of velocity change and relative velicity angles β are not in the direction of the blade (higher or lower). The consequences are impact of the flow to the blades walls, increase of stagnation pressure, vortices, separation of the boundary layer etc. Impact losses are present in the runner and in the stator. - D: Volumetric losses (also called gap losses, Ger. Spaltverluste) are a consequence of fluid flow past the runner between the wall and the runner in the direction of the pressure gradient, for pumps and fans the volume flow rate decreases, and operating point also decreases.

Fig. 31: Influence of finite number of blades, friction and non-ideal inflow in the runner, guide vane or difussor. Left: Radial fan with forward curved blades, right: radial fan with backward curved blades. A: losses due to finite number of blades, B: friction losses, C: impact losses and D: characteristics improvement at low flow rates for fans with forward curved blades.

5.5.7.1 Finite number of blades Finite number of blades leads to suboptimal guidance of the flow. In such case, the flow at the exit from the runner 'slips' backwards in the opposite direction of the runner rotation (Figure below). We notice: - the relative discharge angle 훽3 is in all cases less than blade angle 훽2, - the relative discharge velocity 훽3changes, for backward curved and radial blades it increases, while for forward curved blades it decreases, - for all cases the average absolute velocity 푐3 is smaller than 푐2 and corresponding projections onto 푢2 are also smaller, - for all cases the angle 훼3 of the absolute velocity is sharper, a very important fact for the design of vaned diffusors, - the velocity loss coefficient ε is proportional to

휌푢2(푐푢2 − ∆푐푢) 푐푢3 ε = = . (93) 휌푢2푐푢2 푐푢2

Fig. 32: Influence of finite number of blades on discharge angle β3. (zamenjaj oznake ' s 3) Energy losses due to the finite number of blades appear as relative airflow circulation in the interblade channel, while the friction of the working fluid is here not considered. The calculation was performed by Stodola [1924]. Stodola based the calculation on the estimation, that the relative circulation (Figure below) at the exit from the blade is responsible for the reduction in output.

Fig. 33: Vortice in the interblade channel Vortices are produced in the blade passages such that velocity on the suction side is more then on the pressure side (pressure on the pressure side is higher then on the suction side). According to Stodola, therefore at the exit from the blade passage, the back flow along the circumference is identical to the reduction of the cu2 component. We write the result only for the velocity loss coefficient ε

푢 휋 푠𝑖푛 훽 휀 = 1 − 2 ∙ 2 . (94) 푐푢2 푧 z represents the number of blades. The approach is limited to the exit from the runner and does not consider blade curvature. High number of blades reduces velocity loss coefficient 휀. The Stodola approach is still very useful for long blades, where 'history' from the runner inlet is not anymore important.

5.5.7.2 Friction losses Friction losses are proportional to 푉̇ 2.

5.5.7.3 Impact losses The shaded surface shown with letter C denotes energy losses due to the impact of the working fluid on the blade. This occurs due to the inadequate direction of the airflow at the runners's inlet and can be analitically expressed as [Eck, 1973] pressure drop due to impact losses ∆푝′′:

2 ̇ 2 휌 2 푟1 푉x ∆푝′′ = 휇 푢2 ( ) [ − 1] . (95) 2 푟2 푉̇

휇 is coefficient with value in the interval from 0,5 to 0,9. The equation was derived from the velocity triangles at the inlet of the runner and the introduction of the flow parameters 푉푥̇ (volume flow rate away from the best efficiency point) and the flow at the optimum point of operation 푉̇ . The impact loss curve has the form of a parabola with a minimum at optimum point of operation. 2 The influence of term 푢2 is also important, it is for instance beneficial for fans with forward curved blades this value is low in comparison with overall pressure increase. 푟 2 The value of impact losses falls with the ratio ( 1) . Small impact losses can be r2 achieved for small inlet radius 푟1 where inlet and angular velocities are both low.

5.5.7.4 Characteristics improvement at low flow rates for fans with forward curved blades In the pressure characteristic of the fan with forward curved blades, we notice that at low volume flow rates the pressure rises (Fig. above - D). Eck [1973] interprets the phenomenon with partial clogging of the airflow in the interblade channel at flows below the optimum operating point. The phenonemon differs significantly for the runners with backward and foward curved blades and is schematically shown in Fig. below. Due to the separation of the airflow on the suction side of the blade, the cross section of the channel 79 indicated by 푎′ is apparently reduced. This is the reason for increase of the ′ relative velocity 푤2, which can be written with the expression 푎 푤′ = 푤 ( ) . (96) 2 2 푎′ Due to the increase in relative speed, the velocity triangle at the runner's exit is modified as shown in the Figure below. The dashed lines consider the modification of the triangle of velocity to consider the final number of blades. In the case of a runner with backward curved blades and partiall blocked flow ′ channel, a decrease in the peripheral absolute velocity component 푐푢3 is observed. Due to this pressure loss at very low and volume flow rates is higher than expected.

Fig. 34: Comparison of the airflow clogging in the centrifugal runner's channel with forward (left) and rearward (right) curved blades and the corresponding change in the velocity triangle at the exit, situation for very low volume flow rate. In the case of forward curved runner blades, the value of the absolute velocity ′ projection in the apparently reduced cross section 푐푢3 exceeds the value 푐푢3 in a properly filled interblade channel. Consequently, the value of the expression ′ 휌푢2(푐푢3 − 푐푢3), which describes energy losses due to the apparent reduction of the cross section, is positive at very low volume flow rates well below the

80 optimal operating point. At the transition between the partially clogged and properly filled interblade channel, an unstable velocity field appears and, a decrease in the characteristic curve is observed. 5.6 Axial flow ventilators, fans, compresors and pumps Like a centrifugal turbine machinery, also axial turbine machinery got its name from the prevailing direction of the flow. In the axial turbine machinery, the flow direction is in the axis of rotation. For this, the axial turbine machine consists of a hub, on which blades are attached. Ideally, all fluid elements receive same specific energy, regardless where on the radius they are. Often the axial turbine machinery is enclosed in a duct. Implementation of axial turbine machine in such cases is simple, as such machine forms only a part of the duct. Axial turbine machines are widely used for ventilation, propulsion and power generation. For ventilation, they are used whenever low pressure increase and large volume flow rates are required. The axial compressors are the preferred choice for aviation applications, because they have lower frontal cross section and lower drag. Also, they are better from the viewpoint of flow characteristics. Real fluid effects in centrifugal compressors are much more pronounced. On the other hand, the advantage of centrifugal compressor is its simplicity. This is sometimes preffered for industrial applications, where reliability is more important than high efficiency. Axial water turbines are widely used for water power generation for low heads and wind turbines are used as a source of green energy. Installation in ducts also dictates the use of a stator, because vortices in the duct are in a large part converted into heat. The stator directs the flow in the axial direction and by the same time enabling increase in pressure. For presentation, we often cut the blades of both runner and stator and represent them in a plane as a runner and stator cascade (see Figure above). A cascade is an infinite series of runner and stator blades. The cascade is set for a selected radius, as angles of runner and stator blades change with radius. Flow through cascades has been in the past researched a lot. In the figure below, we write triangles of velocity at the inlet and trailing edge of blades of axial cascade. To write triangles of velocity, we follow similar principles as for centrifugal machines. Fans in everydays use always have only a single stage, while compressors may have several stages. A stage is comprised of a runner (rotating part) and stator (stationary part). The purpose of inlet guide vane is to direct the flow smoothly onto the runner. The axial runner is sensitive to changes of incidence angle, thus use of inlet 81 guide vane may help. Inlet guide vanes also may prevent ingestion of debris and break large turbulent structures, present in the atmosphere.

Fig. 35: Axial flow compressor and it's velocity triangles. The triangles of velocity of an axial turbine cascade are shown in Fig. below.

Fig. 36: Axial flow turbine and it's velocity triangles

5.6.1 Euler equation for axial turbine machinery The Euler equation is in the simplest form and without considering the predznak

Pshaft = ṁ (u2cu2 − u1cu1) . (97)

For axial turbine machinery, we assume that flow enters and exits the runner at the same radius, the tangential velocity therefore does not change. The

82 radial velocity we assume is 0. Therefore 푢1 = 푢2 = 푢 and Euler equation, written for the pressure is

∆p = u(c − c ) . (98) ρ u2 u1

For the axial inlet the 푐푢1 = 0 and Euler equation simplifies to ∆p = uc . (99) ρ u2

For axial fans and compressors, we usualy consider four main cases: - stator, followed by the runner, - runner, folowed by the stator, - stator followed by runner and another stator, - stator, followed by a runner in a configuration, where both receive the flow at an angle. In the first case (stator, followed by the runner) the stator increases velocity of the flow and creates a vortice before the runner. At the exit from the runner the flow has only the axial velocity and no tangential velocity. This case looks promising because velocity in the stator increases and losses in the stator are low. However, the losses in the runner are high and such configuration is not very common. The pressure increase for this first case is

∆p = uc . (100) ρ u1

Reactivity in the first case is R > 1, because after the guide vane, pressure is lower than ambiental pressure. The runner thus must compensate for this pressure decrease and in addition also generate presure as expected from the fan. u and cu1 are in opposite directions, thus Euler equation is still valid (ucu1 is positive and ucu2 is 0). In the second case the runner gives rotation to the fluid and the stator downstream of the runner directs the flow in the axial direction. This is the most common case of axial turbine machinery. The pressure increase for this second case is

∆p = uc . (101) ρ u2

In the second case reactivity is R < 1.

In the third case the stator is provided in front and after the runner. The inlet and exit velocities to and from the runner are mirrored. The absolute inlet velocity and exit velocities are the same. This means that the runner produces only static pressure. The pressure increase for this third case is

∆p = 2uc = 2uc . (102) ρ u1 u2

In the third case reactivity R = 1, this is because cu1 = cu2. In the fourth case the stator is followed by a runner. In this case the relative and absolute velocities have the same amplitude. They are directed in mirrored directions. That means that both the stator and runner receive flow at an angle. This means, that velocities are lower than in the above three cases. This case is often applied when velocities in the flow channels are high and friction losses and compressibility losses high. Due to non-axial inlet this case is used in designs with several stages. In the third case reactivity R = 1/2.

5.6.2 Stall and surge Stall and surge are unwanted flow pehonomena in axial compressors. Stalling is the separation of flow from the compressor blade surface as shown in the Figure below. At low flow rates the angle of attack increases over the critical or maximum angle that the aerodynamic profile can sustain, and due to this there occurs the flow separation on the suction side of the blades which is known as positive stalling. The flow separation may occur also at very high- volume flow rates on the pressure side of the blade, this is known as negative stalling. At low volume flow rates angle of attack exceeds its stalling value and stall cells appear (which are the regions where fluid starts to whirl at a location and doesn't move forward). The size of these cells increases with decreasing flow rate and eventually these cells grow larger and appear on whole blade. This causes and significant drop in the delivered pressure. At very low flow rate, flow reversal occurs which is known as surge and discussed later. Stall also results in drop in efficiency.

Fig. 37: Stalling process. Above: normal operation, below: stall. Surging is the complete breakdown of steady flow in the compressor which occurs at low flow rate. Surging takes place when compressor is operated off the design point and it affects the whole machine. Surge is aerodynamically and mechanically undesirable. It can damage the rotor bearings, rotor seals, compressor driver and affect the whole operation. In gas turbines it results in high temperature, high vibration and leads to flow reversal. Due to flow reversal, pressure in the outlet pipe falls and the compressor regains its normal stable operation. Surge points are the peak points on the characteristic curves (as in Figure below) left of which the pressure generated by the compressor is less than the outlet pipe pressure and these points initiate the surge cycle. Surge line is the line which connects the surge points on each characteristic curve corresponding to different constant speeds. The stable range of operation for the compressor is on the right-hand side of the surge line.

Fig. 38: Compressor diagram with surge line

6 Similarity theory and dimensional analysis of turbine machinery

Similitude is a concept applicable to the testing of engineering models. A model is said to have similitude with the real application if the two-share geometric similarity, kinematic similarity and dynamic similarity. Similarity and similitude are interchangeable in this context. With the similarity theory, students may gain a deep and for their future work in industry very useful understanding of the operation of turbomachinery. Similarity theory is a formal process where a group of variables that describe the selected physical phenomena, is reduced or changed to a smaller number of dimensionless groups of variables. In practice, we often use terms model and prototype. For the hydraulic machines, the model is a water turbine, which is being developed at the institute, and the prototype is called the water turbine, which is installed in the power plant. The prototype may alo be called application. For conversion from model to prototype it it is necessary to provide hydraulic similarity. Hydraulic similarity is guaranteed if the turbomachinery: - has similar geometric dimensions (this similarity is also called geometric similarity) - both turbine machines have equal ratios of different forces acting between the fluid and the components of the turbine machine (this similar is called dynamic similarity) and - both turbine machines have equal ratios of velocity components in each of the corresponding points of the model and of the prototype (this similarity is called kinematic similarity or similarity of movement, this also means that the dimensional similarity is necessary for the existance of the kinematic similarity), this also means that both machines have the same velocity triangles. Theory of similarity turbomachinery has several major areas of application: - to predict the performance of prototype turbine machine from experiments carried out on a model turbine machine, that is for changed size of the turbine machine,

- to determinine the most suitable type of turbine machine based on maximum efficiency, pressure, flow rate and rotational speed, and - to predict the performance of turbomachinery for changed rotational speed or fluid density density. Similar turbomachines are two turbomachines, which differ only in size. This means that the ratio of all local dimensions between the two turbine machines is the same. For two similar turbine machines, all angles of the turbine blades in respective points are the same. Also, fluid flow in both similar turboine machines is directed in the same direction. In this chapter we will focus on hydraulic machinery. For research and application of hydraulic machinery (pumps and turbines), engineers use the similarity theory with great success for decades. Basic principles of similarity theory are well known for more than a century There are several methods to select the dimensionless variables. Based on logical considerations and using Bernoulli's equation dimensionless variables n, d and  and their exponents can be determined for flow, pressure and hydraulic power (see also Table below). In the following we will first use a simple approach (subsection 4.1) of selection of dimensionless numbers, while later will in subsection 4.2 make a more fundamental approach. Both will lead to the same selection of dimensionless numbers.

6.1 Simple approach to of selection of dimensionless numbers For rotational speed of the turbine machine we may make the following engineering considerations: - the volume flow rate is proportional to the rotational speed of the turbine machine (for instance: for 2 x increase of rotational speed, the volume of pumped fluid is 2 x larger, hence n  푉̇ ), - to establish a dependence between rotational speed and pressure, we should first consider Bernouli equation, in the Bernoulli equation, pressure is proportional to square of velocity, while velocity is proportional to angular speed (according to equation of rotational speed, 푣 = 휔푟, hence n2  p), - if we gain limit our simple approach to hydraulic machines, turbine machine power is a product of volume flow rate and pressure, hence n3  P, For dimension (size, diameter) we may make the following considerations: - the volume of the element of turbine machine is proportional to third power of dimension (for instance: for 2 x increase of characteristic dimesion, the volume of pumped fluid is 8 x larger, hence d3  푉̇ ),

- to establish a dependence between dimension and pressure, we again use Bernoulli equation, pressure is proportional to square of velocity, while velocity is proportional to angular speed (according to equation of rotational speed, 푢 = 휔푟  휔푑, hence d2  p). - if we again limit the simple approach to hydraulic machines, turbine machine power is a product of volume flow rate and pressure, hence d5  P (one of the highest powers in engineering). For density, we make the following considerations: - volume flow rate is not dependent on the density, the compressor runner delivers the same volume for a single rotation, the mass flow rate however increases with density - pressure is linearly dependent on density, this can be seen from Bernoulli ∆푝 equation, for instance 훼푐2 휌 The equations can be rewritten so that the specific speed v is dimensionless or has a dimension (min-1), includes the gravitational acceleration g, etc. In this way, we get the equation for the pressure coefficient (slo.: tlačno število) and the discharge coefficient (slo.: pretočno število) which are non-dimensional variables for volume flow rate and pressure. Pressure coefficient can be written with the gravitational acceleration or withoutn it. If one uses representation without gravitational acceleration, pressure number is not dimensionless number and therefore has an unit. However, such a format is useful only when comparing two similar turbine machines with each other. Discharge coefficient (Equation below) can be written in two different ways

V̇ V̇ φ = , φ = . n d3 n π2 (103) d3 60 4 We prefer the first version in the Equation above according to [IEC 00193]. Similarly, we can write the equation below for the pressure coefficient. Such representation is dimensionless variable of the pressure. Pressure coefficient can be written in three different ways

H g H g H ψ = , ψ = , ψ = . n2 d2 n2 d2 n 2 π2 (104) ( ) d2 60 2 We may also call this coefficient an energy coefficient according to [IEC 00193]. Later in the chapter x.y. we will also provide an equation for the power coefficient. From the above equations and follows (compare with the list of engineering considerations above): 89

- the flow rate is proportional to rotational speed, - the pressure is proportional to the square of the rotational speed, - the hydraulic turbine power is proportional to the third power of the rotational speed (hydraulic turbine power is equal to the product of pressure and volume flow rate) - the turbine flow rate is proportional to the third power of the diameter of the turbine machine, - the turbine pressure is proportional to the square of the diameter of the turbine machine, - the turbine power is proportional to the fifth power of the diameter of the turbine machine (such exponentation is rare in the field of mechanical engineering). The above findings are sumarized in Table below.

Table xyz: Similarity theory in turbine machinery, exponents of variables n, d and  for flow rate, pressure and hydraulic power variable flow rate pressure hydraulic power n  n  n2  n3 d  d3  d2  d5   1    

Dynamic and kinematic similarity conditions can not always be satisfied, but it is possible to come close to these requirements. The greater the departure from the application's operating conditions, the more difficult achieving similarity is. The kinematic similarity and dynamic similarity for both model and protype can not be satisfied because efficiency of two similar turbomachines is not the same. We use use empirical equations for correcting efficiency when going from a model to the prototype or vice versa. In the development of a water turbine, the efficiency of the actual prototype is always higher than the efficiency of the model. The difference in efficiency is mainly the result of a relatively thin boundary layer on a prototype compared to boundary layer on the model. The ratios of different forces acting between the fluid and the components of the turbine are determined by the various similarity numbers (in parentheses are forces, characteristic for each similarity number): - Reynolds number (inertia / viscosity), - Euler's number (pressure / inertia), - Thoma number (net positive suction head / specific hydraulic energy), - Froude number (inertia / gravity) and

- Weber number (inertia / surface tension). There are a lot of different models how the conversion of efficiency from the model to the prototype is estimated. Among these are empirical models, based on the measurement and the theoretical models. Frequently used is the model represented in equation (16) [IEC 60193, 1999]

α Re푚 ∆η = (1 − η푚) V (1 − ( ) ) . (105) Re푝

Equation (above) is an empirical equation for the conversion of efficiency from a motel to a prototype, which was derived for radial water turbines (among them most important Francis turbines). In equation (above) is Δη difference between the efficiency of the model and prototype. ηm is the efficiency of the model. 푉 is a share of losses, which can be estimated by the theory of similarity of turbine machinery, 푅푒m is Reynolds number of the model, 푅푒p is the Reynolds number of a prototype and α is empirical exponent. To estimate losses, which can be estimated by the theory of similarity in turbine machinery, 푉 is estimated to approximately 0.7 [Kjølle 2001] and the coefficient α estimated to approximately 0.16 [Kjølle 2001]. It is usually impossible to provide test conditions, which would satisfy all the different similarity numbers at a time. Therefore, usually the geometric similarity is satisfied and similarity number that has the greatest impact is considered for conversion. For hydrauclic turbine machinery the similarity number with the greatest impacti is the Reynolds number.

6.2 Use of dimensionless numbers Two similar turbine machines, may operate in similar operating points and have in it same pressure and discharge coefficients. If we compare two similar turbine machines, we denote them as model and prototype: 1 or m (model) and 2 or p (prototip). We can also denote with for instance index 1 water turbine, rotating at rotational frequency n = 10 Hz and with index 2 the same water turbine, rotating at rotational frequency n = 11 Hz. In the case of comparison of both model and prototype we have

V̇푚 V̇푝 φ푚 = 3 = φ푝 = 3 (106) n푚 d푚 n푝 d푝 and

g H푚 g H푝 ψ푚 = 2 2 = ψ푝 = 2 2 . (107) n푚 d푚 n푝 d푝

Models in Kolektor Turboinštitut have all outside diameter 푑m = 350 mm, while prototypes have diameters up to 푑p = 8 m. Since water turbines can not be tested at any diameter or diameter of the prototype, they are tested at a selected diameter in the institute. The dimensions of the model determine the diameter of the measuring station in which the model runner is installed, as the measuring station can not be changed for each measured model separately. Then the characteristics are converted to the diameter of the prototype. If the characteristics of a model water turbine are measured in dimensionless form (this means that, for example, the discharge number φ and the pressure number ψ are on the x and y axes), the characteristic for the model and the prototype is the same. Also, for a dimensionless diagram, the hill diagram is the same for the model and for the prototype. Pressure and discharge coefficients are useful for comparing the operating point of a turbine engine operating at two different speeds. The same turbine machine is like itself, which is why it is applied to the theory of similarity. We can write it down

V̇1 V̇2 φ1 = φ2 , = (108) n1 n2 and

H1 H2 ψ1 = ψ2 , 2 = 2 . (109) n1 n2

In equations above, because d1 = d2, d can be deleted. In Equation (108) we also delete gravitational acceleration g. Pressure and discharge coefficients are also useful for comparing the operating point of similar turbine machines operating at the same speeds. For a flow number, we can write down

V̇1 V̇2 φ1 = φ2 , 3 = 3 . (110) d1 d2

In the equation (108) n was deleted, because n1 = n2. For the pressure coefficient we write

H1 H2 ψ1 = ψ2 , 2 = 2 . (111) d1 d2 If the turbine machine changes both the size and the rotational speed at the same time, and if the turbine machines are similar, the Equations below can be written for operation in similar operating points

V̇1 V̇2 φ1 = φ2 , 3 = 3 (112) n1 d1 n2 d2 and

H1 H2 ψ1 = ψ2 , 2 2 = 2 2 . (113) n1 d1 n2 d2

6.3 Fundamental approach to selection of dimensionless numbers We assume that we have the following variables and assume the next function among variables

f(ρ, d, n, V̇ , H, μ, P, K) = 0 . (114)

In the above equation (114) ρ is fluid density, d diameter, n rotational speed, 푉̇ volume flow rate, H height, μ dynamic viscosity of water, P power in K bulk modulus. In total we have in equation 114 in our case 8 variables. Buckingham π theorem of dimensional analysis tells us [Buckingham, 1914], that if we have 3 basic dimensions (mass, length, time), then we can set (8 - 3 = 5) exactly 5 non-dimensional numbers. We therefore select ρ, d in n as the variables, that include 3 basic dimensions, and combine them with the rest of 5 variables 푉̇ , 퐻, 휇, 푃, 퐾. Although we have currently chosen and selected the specified variables, this is not the only possible choice of variables, but it is beneficial because in this way we derive the pressure and discharge coefficients, as we wrote at the beginning of the chapter. Therefore, we can write down the following findings. 푎 푏 푐 푑 (a) If ρ, d and n is combined with μ, we write 휋1 = 휇 휌 푑 푛 . This has a 휌 푛 푑2 solution 휋 = , being Reynolds number. 1 휇

푎 푏 푐 푑 (b) If ρ, d and n is combined with K, we write 휋2 = 퐾 휌 푑 푛 . This has a 휌 푛2 푑2 solution 휋 = , being Mach number, the ratio of the peripheral velocity 2 퐾 퐾 and velocity of the sound, considering the relationship 푐 = √ . sound 휌 푎 푏 푐 푑 (c) If ρ, d and n is combined with P, we write 휋3 = 푃 휌 푑 푛 . This has a 푃 solution 휋 = = 휆, being power coefficient. 3 휌 푛3 푑5 푎 푏 푐 푑 (č) If we combine ρ, d and n with 푉̇ , we write 휋4 = 푉̇ 휌 푑 푛 . This has a 푉̇ solution 휋 = = 휑, being discharge coefficient. 4 푛 푑3 푎 푏 푐 푑 (d) Če ρ, d in n kombiniramo s H, we write 휋5 = 퐻 휌 푑 푛 . This has a solution 퐻 휋 = = 휓, being pressure coefficient. 5 푛2 푑2 In this way, we also reduced the number of variables, on which the individual dimensionless number depends. For example,

φ = f′(Re, M, λ, ψ) (115) and

ψ = f ′′(Re, M, λ, φ) . (116)

In a similar way it is also possible to derive a dimensionless number specific speed of ns, which is used to classify water turbines on Pelton, Francis, Kaplan and tubular turbines.

6.4. List of most important non-dimensional numbers

Non-dimensional numbers are often written in various forms, depending on application (water turbines, pumps, fans, gas and steam turbines) and on origin (English and German notation). The table below summarizes the most important non-dimensional numbers. Table: Overview of dimensionless numbers, comparison of English and German notation.

Non-dimensional English notation German notation coefficients

푄 푉̇ 휙 = 3 휙퐶 = 2 Flow coefficient 휔 ∙ 푑2 휋 ∙ 푢2 ∙ 푑2

∆푝푡 2 ∙ ∆푝푡 Pressure coefficient – total 휓푡 = 휓푡,퐶 = 휔2 ∙ 푑 2 ∙ 휌 푢2 ∙ 푑 2 ∙ 휌 pressure 2 2

∆푝푠/휌 2 ∙ ∆푝푠 Pressure coefficient – static 휓푠 = 휓푠,퐶 = 휔2 ∙ 푑 2 푢2 ∙ 푑 2 ∙ 휌 pressure 2 2

푃 8 ∙ 푃 휆 = 휆 = 3 5 퐶 2 3 Power number 휌 ∙ 휔 ∙ 푑2 휋 ∙ 휌 ∙ 푑2 ∙ 푢2

푉̇ ∙ ∆푝푡 휙 ∙ 휓푡 휙퐶 ∙ 휓푡,퐶 휂푡 = = = Total efficiency coefficient 푃 휆 휆퐶

푉̇ ∙ ∆푝푠 휙 ∙ 휓푠 휙퐶 ∙ 휓푠,퐶 휂푠 = = = Static efficiency coefficient 푃 휆 휆퐶

1 1 1 1 2 휔 ∙ 푄2 휙2 2 ∙ √휋 ∙ 푛 ∙ 푉̇ 2 휙퐶2 Ω = 3 = 3 σ = 3 = 3 Speed coefficient ∆푝 4 휓 4 2 ∙ ∆푝 4 휓 4 ( ) 푡 ( ) 푡,퐶 휌 휌

1 2 2 ∙ ∆푝 4 푑 ∙ 휋 ∙ ( ) 1 2 √ 휌 ∆푝 4 1 δ = 1 푑2 ∙ ( ) 휌 휓푡4 2 ∙ 푉̇ 2 Diameter coefficient Δ = 1 = 1 1 푄2 휙2 휓푡,퐶4 = 1 휙퐶2 There are also other dimensionless coefficients like specific speed, whilch will be discussed later.

6.5 Cordier diagram In 1953, O. Cordier introduced an experimental study [5] in which he first presented a diagram with characteristic rotational frequencies of turbine machines as a function of typical diameters at the name of the operating point (BEP - best efficiency point):

휎BEP = 푓(훿BEP) . (117)

It turns out that the values lie in the narrow band along the curve, known as the Cordier curve, as shown in the figure below. Axial machines lie in the high value range of characteristic speed coefficients and low values of diameter coefficients, radial machines in the field of high diameter coefficients and low speed coefficients, and diagonal machines in the intermediate field. Even though the runner shape is not detailed in the diagram, it has a highly useful 95 value when planning or choosing a runner. For a known operating point, the type of machine that will operate at a given operating point with high efficiency may be selected based on the rotational speed specified by the choice of machine or given runner diameter, which is set by the installation site (for instance, head and flow rates available).

Fig. 39: Cordier diagram (Eck, 1973). Based on Cordier's study, several authors attempted to analytically derive and perform market survey for functional dependence 휎퐵퐸푃(훿퐵퐸푃). It turns out that fans with forward curved blades typically deviate from the remaining types. The reason for this is the outlet blade angle 훽2 > 90° and high ratio of 푏 height and diameter of the runner 2. This ratio is for fans with forward 푑2 curved blades around 0,4, while for fans with backward curved blades it is around 0,25. Because of these characteristics, the fans with forward wraparound blades operate with higher characteristic rotational frequencies at lower characteristic diameters. We analyzed the state of the art based on available catalog data for 580 backward curved blades and 260 forward curved fan blades from three different manufacturers. For each fan, the external diameter and rotational frequency of the runner were considered together with flow and pressure rise at the operating point with highest efficiency. Based on these parameters, we 96 have determined the dimensionless numbers from Table above and have shown the data in the Cordier diagram in the Figure below. Points for fans with forward curved blades are found on the graph at lower diameter coefficients. For the example of backward-curved fans, a good matching of the value with the curve presented by Eck in [1] is shown.

Fig. 40: Cordier diagram for centrifugal fans with backward, forward curved blades and Cordier curve according to Eck (Eck, Ventilatoren, 2003).

6.6 Criteria for water turbine classification with respect to the specific speed (ns) and specific speed diagram Different kinds of turbines are used for different volume flow rates and heads. Specific speed ns is a dimensional and dimensionless parameter used to evaluate the speed of turbine machines (Fig. below). Historically, the specific speed was defined as the rotational speed with which the water turbine should rotate to develop power of one horsepower while using head of 1 m and volume flow rate 1 m3/s. Later, this definition was later updated, but in different ways. With the aid of variable ns (specific speed) the turbines can be roughly classified into three different groups: Pelton, Francis and Kaplan turbines, with the order on appearance based on proportional increase of the specific speed. Pelton turbines cover the area of large heads and low volumetric flow rates, Francis turbines the area of medium heads and flow rates, and Kaplan turbines the area of low heads and large flow rates. The turbine's rotational frequency n, volumetric flow rate 푉̇ available head H (height differenc between the upper and lower water level) are project parameters, which allow the selection of the turbine type. Since there are several different definitons of ns, let us provide the definition after [IEC 60193, 1999] 97

n √V̇ n √V̇ n푠 = 3 = 3 . (118) E ⁄4 (g H) ⁄4

In the Equation above ns is specific speed, E specific hydraulic energy of the machine, n rotational frequency, 푉̇ volumetric flow rate and H head (the height difference of water levels). The values of 푉̇ , H and n used for calculation should be the ones defining the point of maximum turbine efficiency (i.e. the expected most common values of these parameters). In accordance to the upper definition ns is a dimensionless parameter. Despite the standard ISO 60193 [IEC 60193, 1999], which requires the operating point of maximum efficiency to be used for calculation of ns, some manufacturers use the point of nominal or maximum power instead. Some definitions of the specific speed omit the gravitational acceleration since it is practically constant anywhere on Earth. In this case, the definition is not dimensionless, and the calculated values vary depending on the units used. The problem occurs with American and UK manufacturers, which use imperial units such as gallons and feet. Also used is a similar form of equation above, in Slovenia (Turboinštitut, Litostroj) the equation below is used, where the rotational frequency must be input in / (min-1), while the head and the flow rate are given in SI units (the operating point with the maximum effciency is taken) [Krivchenko, 1993]

3.652 n √V̇ 576 √φ n푠 = 3 , n푠 = 3 . (119) (H) ⁄4 (ψ) ⁄4

Specific speed ns of a turbine sets the turbine's shape in a way so that it is independent of the turbine's size.  is the flow number and  the pressure number. The unit at the left side of the equation above is [1/min], while the right side is without a unit. The specific speed parameter also alows that the turbine is resized with respect to the base design with known properties. The specific speed is also the main criterion for suitability of a installation location with the turbine type. According to Equation above, the cassification of turbines is as follows (Turboinštitut, Litostroj) approximately:

- below ns = 70, Pelton turbines are used,

- from ns = 70 to ns = 350, Francis turbines are used,

- from ns = 350 to ns = 600, Kaplan turbines are used,

- above ns =600, bulb turbines are used

The Figure below shows selection of turbine runners for different specific speeds ns. The Figure below shows also different shapes of Francis runners, that corresponds to indicates specific speeds ns.

Fig. 41: Selection of water turbines for different flow volume rates and heads depending on specific speed ns [adapted according to Krivchenko, 1993]

7 Water turbines

Production of electricity in hydroelectric power plants is the most commonly used form of renewable energy. In 2010, it accounted for around 21 % of global electricity production or 3000 TW h [Dixon and Hall, 2010]. The annual increase in production in hydro power plants in recent years stands at around 3 %.

Fig. 42. Production of electric energy in hydro power plants in five largest producing countries [International Energy Statistics, 2015] Electricity is worldwide produced in hydroelectric power plants in 150 countries; the Asia-Pacific region in 2010 generated 32 % of its electricity from hydropower. Figure above shows the production of electricity from hydropower plants in the five major producing countries in last 30 years. China is the largest producer of over 800 TW h in 2012, which represents around 17% of Chinese domestic electricity consumption [International 100

Energy Statistics, 2015]. The greatest potential for growth in the production of electricity from hydroelectric power plants is in China, Latin America and Africa. The cost of electricity production in hydroelectric power plants is relatively low, thereby hydropower is cost-effective source of renewable energy. The average cost of electricity in large hydro power plants ranges from 3 cents to 5 cents per kilowatt hour. Hydropower is also a flexible source of electricity production, since the electric power production can be during energy production rapidly increased or decreased, whereby the water power plant operation is changed. However, damming of the stream flow which is required for dammed hydroelectric power plants, may disrupt the water flow of the stream and can alter the local ecosystem. In some cases, construction of dams and reservoirs include displacement of the population. When the hydro power plant is built, it produces no direct waste and has a greatly reduced production of the greenhouse gas carbon dioxide CO2 compared to fossil fuel power plants. Slovenia produced from hydropower approximately 1/3 of all energy produced [HSE in obnovljivi viri energije v Sloveniji, 2010]. In 2014, which was very rich with precipitation, the production of electricity in hydroelectric power plants amounted to approximately 40% in Slovenia produced electricity. The largest producers of water turbines in the world are Andritz, Voith hydro, Alstom power, General Electric, Mitsubishi, Hitachi, Wasserkraft Volk, Stellba, ČKD, Hydrolink, Turab, Ossberger, FARAB, Gilkes, itd. Largest hydro power plants in the world are shown in the Table below. Table 1: List of largest hydro power plants in the world, all runners are of Francis type name state flow installed yearly power production / MW / (TW h) Three Gorges China Jangce 22500 98,1 Itaipu Paraguai and Parana 14000 98,3 Brasil Xiluodu China Jinsha 13860 57,1 Guri Venezuela Caroni 10200 53 Tucurui Brasil Tocantins 8370 41 Grand Coulee USA Columbia 6809 20 Xiangjiaba China Jinsha 6448 30,7

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The biggest hydro power plant in Europe is Volgogradskaya, Russia with 2500 MW. The largest hydroelectric power plants in Slovenia are indicated in the table below. Table 2: The largest hydroelectric power plants in Slovenia name flow installed yearly type of power production water / MW /(GW h) turbine Avče Soča 185 in turbine 426 Francis (pumped regime storage) Zlatoličje Drava 126 577 Kaplan Formin Drava 116 548 Kaplan

The most powerfull in Slovenia produced water turbine has installed power 375 MW, produced by Litostroj Power, installed in China.

7.1 Water turbine classification In the following we will overview different types of turbines: Pelton, Francis, Kaplan, tube/bulb and other kinds of turbines. Water turbines we classify regarding the type of the turbine: - Pelton, - Francis, - Kaplan, - tube/bulb and other turbines. Hydroelectric power plants are classified by their operation: - run-of-the-river (slo. pretočne), - dammed/accumulation (slo. zajezne/akumulacijske), - pumped-storage (slo. črpalno-zajezne) Fig. below shows run-of-the-river and accumulation water power plant HPP Krško and HPP Moste.

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Fig. 43: Run-of-the-river and accumulation hydro power plants; left: run- of-the-river HPP Krško [source: www.hse.si] right: accumulation HPP Moste [source: SEL d.o.o.]

7.3 Pelton turbine Historically, Pelton turbines (Figure below) have evolved from water wheels. A Pelton turbine (also known as the Pelton wheel) is an impulse (slo. enakotlačna/impulzna) turbine with tangential water flow to the turbine blades. The term impulse turbine means that the pressure in the turbine casing is equal to the ambient pressure. Installation is vertical or horizontal.

Fig. 44: Pelton turbine; left: Pelton turbine with open turbine casing [source: http://www.hydrolink.cz]; right: nozzle of the Pelton turbine with needle [source: http://www.canyonhydro.com] A traditional Pelton turbine design is such that the rotor rotates with one half of the velocity of the water jet. Water exits the rotor blades (also known as buckets) with a very low velocity, so that the energy conversion efficiency is maximized. In practice, the water jet velocity is always slightly higher so that the water is removed from the rotor area. The peak of efficiency is narrower as with Francis or Kaplan turbines [Dixon and Hall, 2010].

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Pelton turbines are suitable for installation when the flow rate is low and the head is large, i.e. from 50 m to 2000 m. For the operation of Pelton turbines, it is not advised that the water level changes significantly. The reason for this is that the speed of the water flow from the nozzle depends on the pressure or the head of water. The rotational frequency of the runner is synchronous with the electrical network, and the rotation speed of the runner is expected to be only slightly larger than the velocity of the water jet. Elements of Pelton turbines are shown in Figure below. At the end of the penstock is turbine valve, later water flows into the distributor (slo.: razdelilni cevovod). Water is ejected from the nozzles or injectors (Figure below) (slo.: šoba). In the nozzle, the pressure energy is converted into velocity energy. The water jet is directed tangentially onto the blades. Upon impact of the jet, kinetic energy is transferred to the blade. For design it is important that the jet is deflected back from the blade but does not hit the next blade. The runner is spinning, which is why the central part of blades is partly cut out (see Figure below), forming a splitter (divider) structure which separates the buckets in two symmetrical compartments. The larger the number of nozzles, the more blades are exposed to the water jet simultaneously, resulting in larger power of the Pelton turbine. Turbine operation is regulated by moving the needle in the nozzle. The needle, which is usually bulb shaped, is moved by the rod to which it is attached. The system can also include a deflector, which prevents pressure surges in case of the rapid closing of the water flow through the turbine (e.g. due to emergency shutdown). The deflector deflects the water jet for so long that the spear completely closes the nozzle. There is a quantity regulation, meaning that the quantity (flow rate) of water is changed. Such regulation is efficient starting from very low turbine loads upwards (i.e. from 1/4 of the nominal load). For this reason, Pelton turbines are used when the flow rate of water changes significantly. The blades can be manufactured as a single part together with the turbine disk (blisk = blade + disk). This solution has been commonly used in recent years, as nowadays rotors are mostly manufactured by CNC machines. To close the water flow on the supplying pressure pipeline (penstock), a valve is used, just like with other turbine types, mostly Francis water turbines. Closing the water flow from the upper reservoir on the penstock, allows for a relative quick shutdown of the turbine (load rejection) and also for turbine maintenance or service.

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Fig. 45: Pelton water turbine schematics, here Pelton turbine with vertical axis and 6 nozzles is shown, 1: valve, 2: water distributor, 3: nozzle, 4: deflector, 5: runner, 6: shaft, 7: outflow, 8: lower water level, 9: generator, 10: turbine cover, 11: tailrace tunnel

Fig. 46: Runners of Pelton water turbines; left: old runner of Pelton turbine of PHHP Walchensee, Germany [vir: https://en.wikipedia.org]; right: runner details [z dovoljenjem: Andino d.o.o.] Pelton turbines can have one nozzle, but they may have more, up to 6 [Kjølle, 2001]. Pelton turbines with horizontal axis have up to two shafts, Pelton water turbines with vertical axis have 4-6 nozzles, if there is a need for greater volume flow rate. The reason for this is the fact that with the horizontal axis, with more than two nozzles water can not leave the space around the runner soon enough. 105

Fig. 47: Flow in Pelton turbine runner (a) and triangles of velocity (b) 7.4 Francis water turbines A Francis turbine is a back-pressure/reactive (slo. nadtlačna) turbine of a radial - axial design. It is used for medium flow rates and heads. Francis turbine is shown in Figure below, while cross section of flow field for PHPP Avče is shown in Figure below. Francis turbines have single regulation, meaning that only guide (stay) vanes have an adjustable position for regulation, while the rotor blades are fixed. Francis turbines are reactive turbines. This means that the water in the spiral casing and the guide vanes increases its circumferential velocity. Water with a large circumferential velocity component then passes through the runner blades, wherein the water circumferential velocity is reduced. In this way, the water transfer energy and the angular momentum to the runner and a generator, as it was described by the equation (above). In some rare cases, Francis turbines have the rotational frequency control. Such turbines are mostly pump storage water turbines. This means that the rotational frequency can be varied, usually by a few % of the nominal rotational frequency. In Slovenia, the only power plant with such regulation is HE Avče (regulation of nominal rotational frequency from -2% to +4%). Francis turbines are used for heads from 20 to 500 m. Typical diameters are from 1 m to 10 m. Almost all Francis turbines are installed so that the axis of rotation is vertical. In general, Francis water turbines are of horizontal or vertical type, depending on the orientation of axis. Smaller Francis water turbines have horizontal axis. They are easier to maintain, because the axial bearings are only lightly loaded in comparison with vertical arrangement, which also means that the cooling of the bearings is performed more easily. Francis turbines have very high efficiency, often more than 95 % for large designs [Dixon and Hall, 2010]. For Francis turbines with sharp or pointy characteristics, it approaches 96 %. The Francis turbines with broad characteristics, have efficiency slightly above 95 %. For these reasons, Francis turbines are the most commonly used turbine type among the water turbines. 106

The example of the very moder Francis turbine runner is shown in Fig. below at Krasnoyarsk HPP (manufacturer Siloviye Mashiny - Silmash from Saint Petersburg, Russia). In comparison with the old runner from the same power plant, we may notice, that blades are S shaped at the junction with the hub. At the rim, S shape is not peresent. Also, the entire leading edge is curved. This is a usual Francis turbine runner development and is the result of switch from old design techniques, based on Euler and potential flow equations, to modern CFD approach. The main components of a Francis turbine are: (1) spiral casing, (2) fixed inlet stay vanes, (3) guide vanes, (4) runner, (5) draft tube, (6) shaft, (7) bearings, etc. The water turbines usually also include a generator, but correctly in this case the whole system is called hydroaggregate. Most of these components are also used in Kaplan turbines. In the following part of this chapter, individual parts of a Francisos turbine will be discussed. Later (in chapter below) other parts necessary for safe and reliable operation of hydroelectric power plants will also be described.

Fig. 48: Schematics of Francis water turbine The spiral casing is installed around the turbine stay and guide vanes and connected to it through by an entire length of perimeter. At the inlet, the spiral casing is connected to the penstock or closing valve. Through the perimeter,

107 water flows onto the stay vanes and guide vanes and then in the turbine. The spiral casing is designed so that the fluid velocity is constant in all transversal segments of the aperture. For this reason, the diameter of the spiral casing is gradually reduced away from the inlet. Water flows through the opening in the stay vanes, guide vanes and continues later through the runner. Francis turbines have spiral casings with circular cross-section of welded steel and mostly molded in the concrete. The water guiding part consists of stay vanes and guide vanes. The guide vanes have a double function of converting the water pressure to circumferential kinetic energy and directing the flow onto the turbine blades. The guide vanes are usually distributed in two rows as fixed stay vanes and adjustable guide vanes in the second row. Stay vanes are not adjustable, they roughly guide water to the guide vanes and increase the strength of the turbine (strength of the spiral casing) such that the high pressure in the turbine casing can not to open the entire turbine. The guide vanes are adjustable with their rotation. Guide vanes are rotated (moved) around a pivot pin of the blade, which is also an integral part of the guide vane itself. By closing the flow of water on the runner of Francis water turbines, the volume flow rate is increased or reduced and thus the turbine power is regulated. The guide vanes are connected to a guide ring, which allows for all the vanes to be rotated at the same time and by the same angle. The guide vanes and the rings are rotated by hydraulic or servo motor propulsion. The guide vanes also allow to fully stop the water flow onto the turbine rotor (during normal or forced shutdown, maintenance etc.). The water supply to the runner can not be completely shut off with the guide vanes due to gap, acting above and below the guide vanes. Resulting volume flow rate through the gaps is too small to allow for even a slow rotation of the Francis water turbine runner.

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Fig. 49: PHPP Avče, cross section of flow tract, with water intake system and tailrace channel. From the figure the submergence of the turbine may be seen, implemented to prevent cavitation. The gap is necessary for the rotation of the guide vanes. The water flow is shut of with guide vanes for normal shutdown or emergency shutdown. During repairs, dates are used to shut of the flow through the turbine. The runner (note that the terms stator and rotor are usually used for the fixed and moving part of a generator, while for the rotating part of the turbine we use the term runner) rotates and converts the energy of water to the mechanical energy of the shaft rotation. The rotor is mechanically linked to the shaft, which is attached to the generator at the other end. The Francis runner has fixed blades, meaning that the Francis water turbine has single regulation (regulation is only possible by rotation of guide vanes).

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Fig. 50: Francis turbine runners, Left: Francis turbine runner, Three Gorges HPP, China [source: https://en.wikipedia.org]; right: PSHPP Avče runner - measurement of cavitation erosion with measurement hand, view of the runner from suction side [source: SENG d.o.o.] The draft tube is the element installed below the runner with a function of slowing the water flow and leading the water towards the tailrace tunnel (in pumped storage power plants, the inlet-outlet tailrace tunnel). For the Francis water turbine to convert all the available water flow energy (according to the Bernoulli equation water flow contains the energy according to the pressure, kinetic and potential component), the water flow should slow down in the draft tube and exit the Francis turbines with almost no kinetic energy. The function of the draft tube is therefore to enable that all available pressure energy is converted into useful work. The kinetic energy of the flow, which leaves the water draft tube, is lost energy, but a well-made and properly sized draft tube can reduce such losses to a minimum. In the case of Pelton turbines this is not so, with Pelton water turbines some pressure is always lost. Pelton runner must always be installed at some height above the lower water reservoir. The installation height of Pelton runner above the lower water is needed to enable the water from Pelton turbines to drain the turbine casing and this presure is lost with Pelton turbines. The other elements of hydroelectric power plants will be presented in chapter below. At this point, let us mention only the shaft, generator and bearings. The shaft is an element, linking the rotor to the generator. The bearings hold the runner, the shaft and the generator in a horizontal and vertical direction and enable rotation with minimal frictional losses. Normally, a turbine has at least one thrust bearing (slo.: nosilni ležaj) (supports the turbine in the vertical direction) and at least one guide bearing (slo.: vodilni ležaj).

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Fig. 51: HPP Krasnoyarsk, Russsia, 18x250 MW, the largest hydro power plant, where a Slovenian company performed acceptance tests, Kolektor Turboinštitut d.o.o. Left: new runner, middle: old runner, right: machinery [source: Kolektor Turboinštitut d.o.o.]. Apart from electricity production, a Francis water turbine can also be used for pumped storage power plants. In this case, we use the term pump storage hydro power plant (PSHPP), pump storage turbine, pumping hydro power plant or even pump turbine. The pump storage water turbine PSHPP Avče, Slovenija, is shown in Figure below. In the pumping mode of operation, the turbine acts as a pump, pumping the water from the lower reservoir to the upper reservoir. In the pumping mode, when a sufficient amount of cheap electric energy is available, nuclear power plants operate in base load mode (Slo: pasovno obratovanje), the generator operates as an electric motor. This is mostly in the night time, when nuclear power plants produce most of the electricity and the demand is low. By accumulating water, the lower and upper accumulation serve as large sources of storing the unneeded electric energy. This is one of several methods to temporarily store the surplus of electric power for later use. All water turbine runners, but especially the Francis pump storage turbine runners, must be sufficiently submerged (slo.: potopitev). Therefore, for all water turbines, the runner is installed at the lowest part of the flow tract. Submergion in the case of pump storage turbine Avče is shown in Figure above. High submergion must be provided for the runner to operate without the occurrence of cavitation, this problem is pronounced during the pumping mode of operation. Cavitation is a physical phenomenon in which the water vapor bubbles occur when the surrounding pressure is lower than the evaporation pressure of water. Cavitation can cause damage to the mechanical parts with the cavitation erosion or through the increased vibrations of the water turbine machine. When the cavitation bubbles implode near the surface of the runner, this may cause erosion of the surfaces of the runner. Cavitation often also increases the vibrations of water turbine, which leads to increased wear of mechanical parts. 111

Fig. 52: Building of PHPP Avče, lowering of the runner in the machinery hall shaft [source: SENG d.o.o.] To prevent cavitation, the runner must be installed low enough below the surface of the lower reservoir. With sufficient submersion of the turbine during the operation of the water turbine machine, high enough energy is achieved on the suction side of the runner. In Chapter below (cavitation in water turbines) we will later write numbers NPSE and NPSH, which are called the net positive suction energy and the net positive suction head. NPSE and NPSH are the sum of two contributions from pressure and velocity components. If the numbers NPSE and NPSH on the suction side and therefore the submergence of the water turbine runner are large enough, the cavitation is avoided. By submergence of the turbine high enough pressure on the suction side of the runner is provided and the cavitation is prevented. Cavitation does not occur in all operating points of Francis turbines. By selecting the operating point, the operator can influence the occurrence of cavitation, if the electricity market situation and the needs of the transmission electricity grid permit so. For the prevention of adverse effects of cavitation erosion and vibreations, in addition to the submergence of the Francis turbines, appropriate measures for cavitation occurence mitigation are applied. These are the control the Francis turbines with inverter (if provided) and blowing of air through the hollow shaft to the suction cone is performed.

7.5 Kaplan water turbines A Kaplan turbine (Figure below) is a reaction water turbine with adjustable rotor blades. It is a turbine with double regulation, as the angles of both guide vanes and runner blades can be adjusted. Historically speaking the Kaplan turbine is the evolution of the Francis turbine for watercourses with low heads and high flow rates. Kaplan turbines are normally used for heads between 10 m and 50 m and for power of up to 200 MW. The nominal

112 efficiency exceeds 92 % but can be lower in the case of very low heads and small flow rates. The inflow of water is carried out in the same way as with Francis turbines, i.e. through the pressure pipeline (penstock) into the spiral casing (Kaplan turbines because of the large size in most cases lack the turbine valve), through the fixed stay vanes and adjustable guide vanes. Kaplan turbines for high heads have spiral casing manufactured from steel, while those for low heads have simplified spiral case made of concrete (Figure below). After the stay and guide vanes, the water turns downwards, before reaching the rotor. This means bouth the inflow and outflow to/from the runner are axial, while the turbine has radial inflow and axial outflow (Figures below). The runner blades have adjustable angles. Blade angle adjustment is performed hydraulically, with hydraulic oil flowing through the center of the shaft. The inflow of oil into the shaft is on the upper side above the generator cover. The sealing of the oil system in the shaft must be well made to prevent oil leakages into the river. Similarly, to Francis turbines, the water exits the rotor into the suction tube to the outlet. The double regulation allows the operation of Kaplan turbines in a wide range of operating points. A special version of the Kaplan turbine is the propeller turbine, which has a similar design, but with fixed runner blades. Propeller turbines have single regulation with guide vanes, like Francis turbines. Due to the simpler design and a higher rotational frequency, propeller turbines are used to replace older Francis turbines installed in power plants with lower heads (up to approximately 10 m). A larger rotational frequancy allows for a smaller, lighter and cheaper generator. Higher rotational speed allows for lower number of poles of the generator. Usually with small agregates of propeller turbines, where water level changes are small, a multiplying gear mechanism is used, further raising the rotation frequency of the generator shaft. An example of such propeller turbine with multiplying gear in Slovenia is SHPP Planina, where the muliplication ratio is 10x.

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Fig. 53: Schematics of Kaplan turbine; 1: stay vanes, 2: upper/lower stay vane ring, 3: guide vanes, 4: upper/lower guide vane ring, 5: turbine cover, 6: impeller ring jacket, 7: draft tube wall, 8: turbine blade, 9: hydraulic motor for rotation of guide vane ring, 10: guide/thrust bearing combination

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Fig. 54: Kaplan turbine; left: spiral casing of Kaplan turbine during annual servicing check, HPP Solkan; in the middle: view of the runner from the spiral casing, HE Solkan; right: old runner of HPP Plave I, seen are pivot pins of runner blades [source: SENG d.o.o.]

0 Fig. 55: Flow field of Kaplan turbine, example of HE Solkan

7.6 Tubular turbines The term tubular (slo.: cevna turbina) turbines denotes a group of several turbine subtypes: bulb turbine (slo.: cevna turbina s hruško), pit turbine (slo.: cevna turbina v jašku), S turbine and Saxo turbine. Tubular turbines are a proper solution when the head is lower than 30 m and have in the recent

115 years almost completely replaced the low-head Kaplan turbines. Tube turbines can operate reversibly in e.g. tidal power plants. Some sources and authors treat tube turbines as a variation of Kaplan turbines, while the others consider them as an independent water turbine type. In this textbook we will consider tubular turbines as a separyte type of water turbines. The cross-section of flow filed of the bulb tubular turbine is shown in Figure below. Tubular turbines are appropriate choice of the water turbine where there are no large fluctuations of the net head and discharge, and they are often used as aggregates of biological minimum. Such aggregates exploit the biological minimum of water discharge through a riverbed, while the majority of the flow is diverted through the derivation channel or tunnel, that is away from the riverbed. Another application of such aggregates is the biological minimum discharge into the fish hatchery (slo.: ribje drstišče). In Slovenia, for example. such hydro power plant is SHPP Vrhovo.

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Fig. 56: Bulb turbine - cross section of flow field of the bulb turbine Among the tubular turbines, we will describe more in details bulb turbine, others will be only briefly discussed. Tubular turbines have in comparison with Kaplan turbines two most important advantages: - because it is not required that the flow inside the tubular turbine to make a turn, is the efficiency slightly higher in comparison with Kaplan turbines and - lower manufacturing and construction costs due to compact size of tubular turbines, because deep digging and construction of deep pits is not required (flow channel is horizontal), and because of suitability for very low heads it is also not reqzured to have large flooding area because of damming and upper reservoir. Main drawbacks of tubular turbines are: - in the bulb space is limited, therefore equipment in bulbs is of small size and may be therefore less durable, life span shorter and servicing more challenging and difficult and

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- smaller tubular turbines must use multiplying gear mechanism to increase rotational frequency of the generator shaft in relation to turbine shaft.

7.6.1 Bulb turbine A tubular turbine with a bulb (slo.: cevna turbina s hruško) is an axial turbine with a horizontal shaft and axial water inlet onto the runner. A tubular turbine with a bulb is shown in Fig. below. It is equipped with a flat conical draft tube. Bub turbines allow for a large flow rate and consequently large power even for low heads. A direct drive generator is installed in the watertight bulb, which is attached to the turbine's inlet stay vanes. The bulbous shape of the generator casing gives the name to this particular turbine type.

Fig. 57: A tubular turbine with a bulb, [source: http://alstomenergy.gepower.com] The difference between a Kaplan and a bulb turbine is in the inflow of water onto the turbine. Kaplan tubines have a radial inflow of water, while the inflow in bulb turbines is axial. Both types of turbines have an axial outflow. Due to such installation the water flow direction does not change significantly, allowing for a good efficiency and compact size. The compact installation size considerably lowers the costs of construcion works and allows for a flexible installation. The rotor of a bulb turbines has blades with adjustable angle, meaning that the turbine regulation is double (regulation of guide vanes and turbine blades), similar to the Kaplan turbine. Bulb turbines in comparison with Kaplan and Francis turbines lack the spiral casing and have a draft tube significantly different in shape from draft tubes of Kaplan and Francis turbines.

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7.6.2 Pit turbine Tubular pit turbine (slo. cevna turbina v jašku) design is similar to bulb turbines, with a notable difference that the generator is installed in the shaft within the flow tract. Tubular pit turbine is shown in Figure below. In the case of smaller power plamts it is impossible to have the generator enclosed in a watertight bulb within the flow, because the space is limited. The generator is usually connected to the turbine shaft through a multiplying gear mechanism, which is installed in the turbine shaft and enables generator rotation with a sufficiently high frequency despite the slow turbine rotation. This way, the costs of manufacturing the generator are reduced, making possible that even power plants with very low heads can operate profitably. A design with direct transmission of torque between the turbine and the generator is also possible.

Fig. 58: Tubular pit turbine [source: http://turab.com]

7.6.3 Tubular turbine S Tubular S turbine is a variation of the bulb turbine. Tubular S turbine is shown in Figure below. This type of turbine has a horizontal axis and axial inflow of water onto the runner. It is equiped with an S-shaped draft tube with one or two bends. The shaft runs through the bend of the draft tube out of flow tract. The generator is located outside of the flow tract. This turbine type is suitable for smaller hydroelectric plants with up to 10 MW power, which do not allow for installation of the generator in a watertight bulb within the flow.

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Fig. 59: Tubular S turbine, generator is located outside of flow tract of the water turbine, [source: http://www.cchpe.net]

7.6.4 Axial turbine with a vertical shaft – Saxo turbine Tubular Saxo turbine is a variation of the bulb turbine (Figure below). In recent years a tubular Saxo turbine design has been successfuly implemented by Litostroj in Canada. This is a vertical axial turbine, which in its upper part between the inlet and the runner is similar to a tube turbine with an inlet bend and a semi-axial stay and guide vanes. In the lower part between the rotor and the end of the draft tube, it is like conventional Kaplan turbines. The water flows onto the rotor in the axial direction. The generator is installed above the turbine, with the shaft running trough the inlet bend. The suction tube can either be straight or with a bend. Saxo turbines can cover the complete operating range of bulb and Kaplan turbines.

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Fig. 60: Schematics of tubular Saxo turbine [source: Litostroj]

7.7 Other type of water turbines Other types of water turbines will be mentioned below. We will separate them according to reaction and impulse type of operation. Reaction types of water turbines are: Deriaz turbine, Tyson turbine and Gorlov turbine. Deriaz turbines have adjustable runner blades, water flow in the runner is diagonal (Figure below). Deriaz turbine is like Kaplan turbine, only

121 that runner blades are inclined, what is useful for higher heads from 20 to 100 m. Tyson turbine does not need the casing, it is put directly in the flowing water of the watercourse. It is composed of propeller, attached to the raft. The runner drives the generator, usualy on the raft, connected to the runner with the belt. Tyson turbine is pulled in the middle of the watercourse, where the water flow speed is the highest. Gorlov turbine is water turbine, developed from Darriusove turbine, only that it has bent blades (Figure below). Water flow acts on the blades of Gorlov turbine with the torque, causing the turbine to spin. The fluid flow direction is perpendicular to the rotating shaft of the turbine.

Fig. 61: Runner of Deriaz turbine [source: http://dic.academic.ru] Impulse water turbines are: water wheel, Turgo turbine, Banki turbine (also called cross-flow turbine), Jonval turbine in Arhimed screw turbine. Water wheel is an old turbine machine, which in the past used to drive mills and water sawmills but was also used for irrigation and drainage (Figure below). The water whell was first mentioned BC, the usage has spread from ancient Greece to Roman empire to other parts of the world.

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Fig. 62. Other types of water turbines. Left: Gorlov turbine, [source: https://en.wikipedia.org/wiki/Gorlov_helical_turbine]. Right: Turgo turbine, [source: www.dtlhydro.com]. Turgo turbine is like Pelton turbine, only that it has only a half of the runner. The runner is therefore more easily to manufacture (Figure below). Water flows to the turbine from the side, while the turbine enables higher discharges as the Pelton turbine. Turgo turbine operates in the range og heads, where Pelton and Francis turbine overlap. Banki turbine is interesting, because it is the only turbine, through which the water flows twice. Jonval turbine is like water wheel with the difference, that water flows to the turbine from the top. Arhimed screw was traditionally used for water pumping, while today it is mostly used for wastewater pumping in wastewater treatment plants. in the other direction Arhimed wheel works as a turbine. Archimed screw water turbine is fish friendly turbine (Figure below).

Fig. 63: Other types of water turbines; left: water wheel; right: Arhimed screw turbine [source: https://en.wikipedia.org]

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8 Other elements of hydro power plants

In this chapter we will present main elements of hydro power plants. The main groups of elements are water intake system, equipment in machinery hall and other systems.

8.1 Water intake system Water intake system (slo.: vtočni sistem) consists of the upper accumulation, dams and trash racks.

8.1.1 Dams Dams (slo.: jezovi) are structures which hold the water for different purposes, including electricity production. There are several types of dams: concrete, gravity, arch, pillar, embankment or a combination thereof. Arch dam of HPP Moste and dam of HPP Medvode of combined pillar - embankment dam are shown in Figure below.

Fig. 64: Hydro power plant dams; left: concrete arch dam of HPP Moste is the highest dam in Slovenia; right: dam building of HPP Medvode of of combined pillar - embankment type, behind is Zbiljsko lake [source: SEL d.o.o.]. Embankment dams (also known as rock-fill dams) are constructed by piling of rock material around the central waterproof wall, which nowadays is usually made of concrete. These are used in cases when a wide valley must be

124 dammed. Bulk dams are gravitational and are held in place by their own weight. To dam deep and narrow gorges, a concrete dam is needed as only the concrete structure is strong enough to withstand the pressure of water. The highest concrete dams exceed 300 m in height. The dam cross-section is usually triangular. Construction of large concrete walls is complicated and slow, as the structure must be cooled during construction. Concrete dams are gravitational (held in place by their own weight), arched (curved in the shape of an arch and leaning on the sides to the valley banks which holds them in place; used for damming high and narrow gorges) or pillar (the pillars have deep and strong foundations which are supported by the ground), or a combination of differen types. Apart from the main dam structure, the surrounding supporting area must also me strenghtened. For example, the dam of HE Medvode (60 m high, highest in Slovenia) is located in an area where the river Sava created rapids in dolomite, which is mostly cracked an full of cavities. To assure good foundations, the ground was stabilized by an injection curtain. The unfilded width of the curtain is 190 m and extends until the imperviuos base made of shale and sandstone, which lies in the depth between 27 m and 45 m. Dams posses risks for accidents for humans and environment. Accidents due to dams of hydro power plants are very uncommon. The worst accident which happened due to ignoring the possibility of such landslide happened at Vajont dam near Longarone, Italy. A part of the nearby hill collapsed into the accumulation lake Vajont, causing a 200 m high tsunami wave. Nowadays, larger dams usually have an equipment for detection of landsliding.

8.1.2 Teeth and trash racks The inlet from dams to the power plant channel or tunnel can be designed in different ways, located separately or as a part of the dam or the dam building. Different elements are used at the inlet: teeth, trash racks etc. (Figure below). The teeth (slo.: zobje) reject large floating or sinking debris. Due to the tooth, the inlet channel is located a few meters below water level and above the bottom, which reduces the possibility that larger pieces of wood would enter the power plant. The debris collecting at the tooth must be cleaned with a cleaning machine. Trash racks (slo.: rešetke) hold the dirt (tree leaves, branches, stones, sand, gravel etc.) from travelling through the turbine and damaging it. A sample of the trash rack for a small hydrop power plant is shown in Figure below. Some trash racks have a differential pressure meter. If the differential pressure is

125 too high, the rack must be cleaned with the cleaning machines. On large water power plants trash racks are usually not visible.

Fig. 65: Teeth and trash racks of water hydro power plants; left: the tooth of HPP Medvode and the trash rack, the tooth holds the debris while the rack is not visible (under the metal guardrail in the lower left part of the image). Due to the tooth, the inlet channel is located about 3 m below the water level and above the bottom, which reduces the possibility that larger pieces of wood would enter the power plant. In the middle and right: the intake-outlet system at the pumped storage power plant Avče during construction and operation. The teeth are vertical and installed in a way wich prevents larger pices of floating wood from being sucked into the power plant, [source: SEL d.o.o. in SENG d.o.o.].

Fig. 66: An example of a trash rack on a small hydro power plant, which at the same time acts as a spilway in the case of flooding

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[source: http://www.tps.si].

8.2 Water supply system The water supply system (Figure below) from the upper accumulation to the turbine consists of supply (head race) channels, head race tunnels, sand trap, gates, surge tanks, penstocks (pressure pipelines) and closing elements (valves). The water supply system is a continuation of the inlet system and can be very large/long if the damming is far away from the powerhouse. In run-of- the-river/accumulation power plants the water supply system is short and incorporated in the dam structure.

8.2.1 Headrace channels and tunnels Headrace channels and tunnels (slo.: dovodni kanali in dovodni tuneli) supply the water to the penstock (slo.: tlačni cevovod), because in most cases the dam is not located directly above the powerhouse, but some distance away from it. The channels are open and are usually desiged as excavated asphalt structures with low inclination. The channels are made by drilling or blasting the rock and then covered with concrete. Near the end of the head race channel/tunnel there is usually a sand trap (slo.: peskolov), an expanded section where flow velocity is reduced and the particles heavier than water settle on the floor. Near the end of the head race channel or head race tunnel there is a surge tank (slo.: vodostan).

Fig. 67: Headrace channels and tunnels for water supply to the water turbine; left: headrace channel HPP Obernach, Germany; in the middle: inside penstock of PSHPP Avče [source SENG d.o.o.]; right, concrete headrace tunnel of HPP Hubelj [source SENG d.o.o.]

8.2.2 Surge tank The surge tank task (also surge tank, slo.: vodostan) is to reduce pressure and mass flow fluctuations in the penstock and head race tunnel caused by 127 changes in load, to within acceptable limits. Surge tanks of PSHPP Walchensee (Germany) and HPP Završnica are shown in Figure below. A surge tank is an expanded part of the head race tunnel. When pressure and mass fluctuations occur, water spills from the penstock into the surge tank, where the water level is momentarily increased. The surge tank prevents the pressure wave from propagating into the head race channel/tunnel, thus preventing damage and spilling.

Fig. 68: Surge tank; left: surge tank of PSHPP Walchensee in Germany; right: surge tank of HPP Završnica Behind the surge tank there is usually a gate chamber, which ends the relatively flat part of the water supply system. Inside the chamber a gate is installed for closing the channel when the penstock must be drained for inspection and maintenance, without the need of draining the head race channel. Behind the gate chamber, the penstock begins. The schematics of surge tanks of HPP Doblar II and Plave II are shown in section xyz. The surge tank may be built as a simple surge tank, which is just a horizontal cilinder, or may have in addition an upper and lower surge tank chambers, a choke and acess tunnel. Upper and lower surge tank chambers increase useful volume of the surge tank, while they also enable choking of water flow energy during inflow and outflow from the surge tank. During opening of turbine valve, when the water in penstock is stationary, the penstock is gradially filled from water from surge tank and lower surge tank chamber. Lower surge tank chamber may be built to suppkly water in the penstock, but when long enough, it may also serve to provide choking and replace the choking element.

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The choking element mitigates fluctuations of water discharge into and out of the surge tank. It is built as a constriction. The acess tunnel is used for inspections, service and air venting of the surge tank and both surge tank chambers.

8.2.3 Penstock Penstock (slo.: tlačni cevovod) is a pressure pipeline linking the gate chamber and the power plant. It ends with a ball valve, in case the power plant has one. The ball valve is usually used for high heads, while for low heads butterfly valves are more common. Penstocks of PSHPP Walchensee (Germany) and HPP Hubelj are shown in the Figure below.

Fig. 69: Pressure penstock leads water form upper accumulation to the water turbine; left: pressure penstock HPP Wachensee, Germany; in the middle: pressure penstock HPP Hubelj; right: pressure penstock HPP Hubelj, joint and dilatation [source: SENG d.o.o.] Longer penstocks have steel walls capable of sustaining high water pressure within it. Apart from the static pressure, a penstock must also withstand the additional pressure caused by quick shutdown of the power plant. If the penstock is vertical, it is named a shaft (slo.: jašek). Penstocks are usually mounted on pods, which can be fixed or sliding. Due to the (thermal) expansion penstocks include expansion joints, where two pipes slide one inside another. Penstock of HPP Solkan with kaplan turbine is shown in Figure xyz, located between trash rack (4) and turbine (8). Short penstocks (e.g. in run-of-the-river power plants) are often made of concrete.

8.2.4 Bypass valve or pressure regulator Bypass valve or pressure regulator (slo.: razbremenilni ali varnostni ventil) is a valve used with some Francis turbines with a large head (Figure below). It is installed at the turbine inlet and intended to divert some water from the

129 penstock past and downstream of the turbine. In an event of quick (emergency) turbine shutdown the rate of bypass valve opening is determined by the rate of guide vane closing, and reduces pressure loads in the penstock due to formation of a pressure wave. The typical time of bypass valve opening is a few seconds in an emergency shutdown event. Larger power plants are usually without bypass valve.

8.2.5 Ball valve or butterfly valve in and bypass pipeline A ball valve or butterfly valve (slo.: predturbinski ventil) is and element of water supply system, used for closing the penstock in a hydro power plant. It is installed just before the turbine's spiral casing inlet. The ball valve always closes when the power plant is shut down, and its closing is slow. Ball valves are hydraulically operated, but also have a weight for emergency closing in an event of a more serious malfunction of the power plant and its auxilliary systems. In the case of the emergency shutwown, the ball valve starts to close together with the stator (guide vanes), to prevent the turbine from accelerating to excessively high rotational frequencies (i.e. runaway conditions). Ball valves of PSHPP Avče and HPP Hubelj are shown in Figure below.

Fig. 70: Turbine valve and bypass for the valve. Left: turbine valve on HPP Avče. Bypass is bright pipe above the turbine valve. Right: turbine valve and bypass on HPP Hubelj, again bypass is horizontal pipe above the turbine valve. A relief valve is also visible left form the turbine valve [source: SENG]. When the ball valve is closed, the water may be pumped out of the water turbine for inspection or service. To open the ball valve, pressure across it must first be equalized. For this purpose, a bypass (slo.: obvod) is opened before opening the guide vanes, which results in equalization of pressure before and after the valve.

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Ball and butterfly valves are used. From the viewpoint of efficiency ball valves are better (no obstacle to the water flow), but they are more expensive. Ball valves are used for pressuresabove approximately 4∙105 Pa. Run-of-the-river power plants lack the ball valve and its bypass.

8.2.6 Gates The purpose of the gate (slo.: zapornica) is to close the water flow onto the turbine. Gates are not used for closing the water flow during normal shutdown events in everyday's operation, but for longer shutdowns like for inspection and overhaul In the case of a run-of-a-river power plant, the gate is lowered in the opening with a crane, usually in multiple parts. Such gates are named segment gates. In the case of a dammed power plant, the gate is usually installed at the end of the head race channel/tunnel behind the surge tank, while another gate is located at the end of the tailrace tunnel. Location of gates in the system of hydro power plant is shown in Figures below (check all Figures in the textbook), while location within the spilway is shown in Figure below below (check all Figures in the textbook). If gates are located inside the hill, they are arranged in a room that we call gate chamber.

Fig. 71: Installed gates at the outflow from the turbine in HPP Dubrava, Croatia. Turbine inlet gates, together with turbine outlet gates, enable the draining of the turbine compartment. A case of installation of turbine inlet gares for run- of-the-river power plant HPP Medvode is shown in Figure below. Apart from gates in turbine fields, the run of the river power plants also have the gates on the spillways (slo.: prelivno polje). For repair and inspection of gates are in some cases used temprary gates called xyz (slo.: zagatnice). Xyz are installed in front of the gate and do not allow water to reach gates. Later, water between xyz and gates are pumped out and inspection or repair is performed.

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Fig. 72: Gates of turbine inlet on HPP Medvode; left: gates storage; right: location where gates are installed in the turbine flow tract [source: SEL d.o.o.]

8.3 Powerhouse equipment Powerhouse is the facility where the turbines are installed. It can be a part of the dam structure, or in a separate building. For instance, the powerhouse of the pumped storage power plant Avče (Figure below) is 80 m deep, to assure sufficient suction head of the pump-turbine unit. In the following part of this chapter, the equipment installed in powerhouses will be presented.

Fig. 73: Powerhouse of the hydro power plant. Left: the powerhouse of the pumped storage power plant Avče, view from the generator casing upwards, yellow pipes contain power lines from the generator to the transformer. Grey pipes are for power lines leading down to the generator excitation. Three black pipes are for water drainage (water flowing up) and cooling of the generator (water going up and down) [source: SENG d.o.o.]. Right: the powerhouse of HE Dubrava, Croatia, the turbine is installed in the hole below the elevator on the left side of the

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image.

8.3.1 Spiral casing Spiral casing, spiral or scroll casing (slo.: spiralno ohišje, spirala) is an element for supplying the water to the inlet guide vanes and adjustable guide vanes. It is designed so that the wate outlet velocity is constant along its perimeter. Therefore the cross-section of the spiral casing is gradually reduced along the perimeter as the water is gradually directed through the stator and onto the runner. Francis and Pelton turbines have spiral casing, while Pelton and tubular turbines do not have the spiral casing. In the case of Pelton turbines, a water flow distributor is used instead of spiral casing. In spiral casings of some turbines, pressure measurement outlets are installed to measure the flow rate by the Winter-Kennedy method. The method is based on measurements of the pressure difference on two locations in the spiral casing, which rises proportionally with the flow rate. The method is usually calibrated during acceptance tests, when the flow rate is measured by and array of vane anemometers. In most of the larger power plants, the spiral casing is poured in concrete. In smaller power plants such as SHPP Hubelj, it is usually visible.

8.3.2 Stay vanes and guide vanes Stay vanes or stayring vanes (slo.: predvodilne lopatice) The inlet guide vanes direct the flow from the spiral casing towards the turbine and have a fixed position (Figure below). With their direction they direct the water flow at the most suitable angle to guide vanes (slo.: vodilne lopate). Stay vanes have an important function of providing mechanical strenght as they link the upper and the lower part of the inner side of the spiral casing.

Fig. 74: Stay vanes and guide vanes of water turbines; left: stay vanes and guide vanes of HPP Solkan (Kaplan turbine) [z dovoljenjem: SENG d.o.o.]; right: guide vanes of tubular bulb turbine HPP

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Dubrava, Croatia Outside of the flow tract, the guide vanes are connected by a guide vane ring or guide vane wheel (slo.: vodilni obroč), which moves all the vanes simultaneously and is steered by a hydraulic arm. Guide vane ring is shown in Figure below. Guide vane ring is moved by a hydraulic arm. In some cases, the guide vanes are soft mounted and equipped with micro switches. The microswitches detect if a vane did not close completely during the shutdown procedure (e.g. due to jamming by a tree branch). In this case, the operator can reopen and close the guide vanes, possibly removing the jammed objects to be washed away by the water flow. If required, this procedure must be performed several times. If not sucessfull, in a very unusual event, turbine flow tract must be emptied, and the part removed.

Fig. 75: Guide vane ring; left: guide vane ring on bulb tubular turbine HPP Dubrava, Croatia; right: guide vane ring on Kaplan turbine HPP Solkan, visible is the hydraulic arm for guide vane ring rotation, and the soft mounting of the vanes with micro switches [source: SENG d.o.o.]

8.3.3 Turbine runner, turbine cover, anti-lifting plate and system for air blowing Turbine runners (slo.: gonilnik) can be of different types (Pelton, Francis, Kaplan etc.) The shaft can be horizontal or vertical, the latter design more suitable for higher power ratings. Turbine cover (slo.: turbinski pokrov) of the runner is (Figure below) a thick steel plate installed above the rotor. Large thickness is required because the turbine casing is exposed to high pressures., for instance on PSHPP Avče approximately 30 cm. Within the turbine casing, the sealing system is also installed (slo.: tesnilka), which serves the purpose to prevent water flowing from the flow tract around the runner around the shaft into the powerhouse.

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The air blowing system has several different functions: - It allows starting up large pumped storage power plants (Figure below) in the pumping regime. To avoid excessive startup current, the compressors starts pumping in air, forcedly blown in the turbine compartment, which is only blown out when the turbine reaches the desired rotational frequency. In this case, the air is blown from the side. - It dampens the pressure pulsations in operation of Francis turbines at partial loads, when a cavitation vortex appears in the draft tube (Figure below). The vortex swings around with approximately 1/3 of the rotational frequency of the runner and causes large pressure, torque and electric power fluctuations as well as bearing vibrations. Introduction of air (compressible medium) into water reduces the rigidity of the mixture. The valve is located on the top of hollow shaft, above the generator cover. When the valve is opened, air flows from machinery room through the hollow shaft through the runner into the draft tube (Figure below). The pressure in the draft tube is lower than in the machinery room. - it dampens the water hammer effect during an emergency shutdown event. The valve lets the air from the machine room into the suction part of the flow tract (draft tube), reducing the pressure fluctuations in the penstock In this case the air enters the flow tract from the runner.

Fig. 76: Water turbines runners; left: runner of Kaplan turbine, view from below from suction tube (HPP Doblar I); in the middle: lowering of the runner, holes in the runner are visible, used for air blowing, PSHPP Avče; right: turbine cover PSHPP Avče [source: SENG d.o.o.] Turbine cover with Kaplan turbines includes also anti lifting plate (slo.: protidvižna plošča). Anti lifting plate is usually made from copper. During emergency shutdowns, with Kaplan turbines due to pressure, acting on the runner at eleveted rotational frequencies, lifting force higher than the rotating parts of the entire water turbine. We can sy, that the turbine runner "glisira" at elevated rotational frequency.

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Suport bearing , holding the turbine in place during normal operation, does not prevent lifting of the turbine, this role is taken over by the anti lifting plate. During normal operation the lifting plate is not used, because it is not required due to low lifting force. Anti lifting plate is composed of several parts, such that it can be replaced during service. Francis turbines do not need anti lifting plate.

8.3.4 Shaft Shaft (slo.: gred) connects the runner with the generator (Figure below). The shaft is usually made of two or three parts and as such, it can be divided to the turbine shaft (attached to the turbine's runner) and the generator shaft (attached to the generator's rotor). Older turbines have, for instance HPP Doblar I aor HPP Plave I have three very long shafts. In the past this was commonly used for the case of emergency of water breaking into machinery room. With such arrangement of very long shafts, water could only fill water turbine, but not generator and other electric equipment. Such turbines have beside turbine and generator shaft also intermediate shaft.

Fig. 77: Shafts of water turbines; left: Turbine shaft (PSHPP Avče). On the upper part is a flange, which is a connection between turbine and generator shaft. Right: turbine ane generator shaft of HPP Solkan [source: SENG d.o.o.].

8.3.5 Ležaji Ležaji (angl. bearings) on water turbines are rolling contact bearings (slo.: kotalni) on smaller turbines and sliding contact bearings on large turbines.

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Bearings on small turbines are rolling bearings, for instance ball or cylinder bearings. Each power plant has at least one turbine bearing and one generator bearing. The guide bearings hold the turbine in place in the radial direction and the support bearings support it in the axial direction. The bearings are made of segments with oil up to about the half of the segment height. The function of oil is both cooling and lubrication of bearings. The red-colored blocking elements in the image below (from HPP Solkan) have a function of stabilizing the position of each bearing segment in the axial direction. Besides, there is also a steel reinforcement in the radial direction (shown in dark color between the brighter blocking elements) which also serves for setting the air gap, adjustable by a screw on the outer side of the bearing, In the image below, the holes in the axial direction on the bearing segment No. 1 are used or disassembling the segment. The segment No. 2 has another hole in the center, where the bearing temperature probe is installed (for monitoring and protection purposes). Also visible from the below image are different bearing materials. Adjacent to the shaft there is a thin layer of white metal, which protects the shaft in the case of surface contact. Only the white metal is damaged because it is softer and has a lower melting point, meaning that the segment can be easily repaired. Away from the shaft (Figure below), without the requirement to lift the entire turbine.

Fig. 78: Turbine guide bearing during inspection; left: parts of turbine guide bearing of HPP Solkan; right: Details of turbine bearing of HPP Solkan. In dark color a reinforcement is shown to fix the bearing in the axial direction. Picture was taken with open bearing cover [source: SENG d.o.o.]. During operation an oil film is formed in the gap between the shaft and the bearing, and is able to sustain itself without forced lubrication. On larger and more modern turbines, the bearings are equipped with an oil lubrication pump for initial lubrication. This lubrication pump (Figure below) assures

137 sufficent bearing lubrication, while the turbine is stationary or starting to spin. The lubrication pump is of a great value for the support bearing, which lifts the turbine and enables smooth start. If the lubrication pump is in operation, the turbine can be rotated by hand if there is no water in the flow tract. When the turbine reaches the normal operating conditions, the lubrication pump stops. Older turbines without the system for initial lubrication are subject to greater wear during starts. The lubrication oil is cooled in a water-cooled oil cooler, which is usually installed near the bearing (Figure below).

Fig. 79: Cooling system and system for initial bearing lubrication; left: heat exchanger oil/water on HPP Medvode [source: SEL d.o.o.]; right: high pressure pump for initial bearing lubrication of HPP Solkan, left turbine shaft [source: SENG d.o.o.] Smaller hydro power plants have simpler bearings, such as HE Hubelj pictured below. Even smaller turbines have rolling bearings, which are easier for maintenance in comparison with sliding contact bearings.

Fig. 80: The bearings on small HPP are simple, in the case of HPP Hubelj are of sliding contact type. Also visible is the flywheel (slo.: vztrajnik), which is used when the powerplant runs in island mode of operation (slo.: otočno obratovanje) separated from 138

the remaining electricicty producers in the electric network [source: SENG d.o.o.].

8.3.6 Shaft seal Shaft seal (slo.: tesnilka gredi) is an element of water turbines, which prevents the water to flow from the region of high pressure of the flow tract into the powerhouse, except in for this envisaged place. The shaft seal is susceptible to the dirt from the watercourse from the turbine tract. It is therefore required to prevent the dirty water to flow into the seal. To do so, a cleaned sealing water is therefore pumped into the turbine tract near the sealing. This clean sealing water later flows through the seal and later into the drainage. System of the sealing water must be well designed. This includes filtering of water form the river, maintaining of pressure and volume flow rate from the turbine tract into the seal , because it is important to prevent the dirty water to enter the seal even, when the turbine is not in operation. When the turbine is not in operation, a small volume flow rate of clean water through the seal is required. The filtered wate helps to cool the bearing. For sealing a system of sealing ropes (slo.: tesnilna vrvica) and slip rings (slo.: drsni obroč) is used. Sealing ropes are usually of specially woven type with rectangular cross section. To protect the shaft, in the region of sealing, a special steel jacket is mounted on the shaft. This steel jacket can be replaced. Sometimes also labyrint seals are used or for very low head also cuffs (slo.: manšete). A pneumostop is also often used wuith seals. A pmneumostop is a rubber element of trapeze cross section, featuring a groove on the outside. This rubber element is then on the outside non-moving part of the seal mounted near the sealing. When the turbine is not in operation, the pressure is applient to the inside of a pneumostop. Then the turbine is not in operation, the pneumostop rubber touches the shaft and prevents entrance of water. The seal for good operation requires shaft rotation. Pneumostop is only used, when the shaft does not rotate, during daily shutdowns and during maintenance and servicing of the turbine.

8.3.7 Creep detector Creep detector (slo.: zaznavalo premikanja). Creep detector is shown in Figure below. Creep detector usually works on the principle of inductive detection of pulses. Creep detection device can be a part of the rotational frequency measurement system or a standalone instrument. If the turbine is rotating very slowly even when the guide vanes are closed, this usually means that one or more vanes 139 are leaking (due to dirt, branches etc.), being stuck betwwen the guide vanes. When the system senses a slow rotation, auxiliary systems (bearing pumps etc.) are turned on to prevent damage of the moving parts. In such case, must the operator of the turbine several times open and close the guide vanes, that the blocing branch is flushed away. In very rare cases, when the foreign object can not be flushed away, turbine mist be closed, water pumped out and the object removed manually. Some water turbines have installed one more protection device to prevent rotation with too high rotational frequency. The protection is composed of tho pins located on the shaft, which at the rotational speed of around 120% of nomoinal rotational frequency, move out of their original place and press the switch (in Figure below in the front of the creep detector). The switch starts the emergency shutdown procedure.

Fig. 81. Creep detector in HPP Solkan, located in rectangular casing, in the front is mechanical systems with pins, protecting from increase of rotational frequency of the turbine [source: SENG d.o.o.].

8.3.8 Governing of the angle of guide vanes and runner blades The governor (slo.: sistem krmiljenja) is the system of regulation of the angle of guide vanes and in the case of Kaplana and turbular turbine also runner blades. Old governor of SHPP Hubelj and governor of HPP Solkan are shown in Figure below. The governor in the phase of startup of the turbine, when the turbine is not synchronised with the electric grid (the electric grid does not brake the turbine yet), takes care of the proper rotational frequency of the turbine. The governor sets proper angle o guide vanes and runner blades. When the turbine is synchronized with the electric grid and producing electricity, also must regulate the angle og guide vanes and runner blades. This is done so, that at momentary net head the required power of the turbine is 140 achieved at the minimum required volume flow rate of water through the turbine, that is with optimal setting of angles of guide vanes and runner blades.

Fig. 82: Goeverning of angle of guide vanes and runner blades; left: old hydraulic governor on SHPP Hubelj (not in use anymore); right: hydraulic aggregate with mechanical governor of rotational speed of HPP Solkan [source: SENG d.o.o.] Several types of governors exist, from purely mechanic types, over electrohydrauliy types to electronic and digitaly types. Governors are connected to hydraulic aggregate, which if required performs changes of angles of guide vanes and runner blades. Very long time ago hydro power plants were operated manually and there were no governors. Now, electronic or digital governors are used (Figure below), that perform governing tasks more accurate than older governors. This relates to less mechanical stresses to turbine during synchronisation to electric grid. Hydraulic aggregate is connected with pressurized reservoir of oil, which is only partially filled with oil, rest being air. The air is pressurized at the same pressure as hydraulic oil and has several functions. It prevents frequent turn on of the hydraulic pump. At the same time it damps sudden pressure fluctuations in the oil system. In the case of failure of the hydraulic aggregate the compressed air contains enough energy to perform most important task to turn the turbine off, close the guide vanes and open the turbine blades. Hydraulic aggregate of HPP Medvode is shown in Figure below. The quality of oil in the hydraulic system is yearly checked for presence of metals. For instance, presence of copper indicates a copepr seal with heavy wear and possible leakage of hydraulic oil.

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Fig. 83: Regulation of HPP Medvode; left: digital turbine governor; right: the governor controls also the hydraulic aggregate for regulation of guide vanes and rotor blades of Kaplan turbine [source: SEL d.o.o.]

8.3.9 Inverter Few powerplants are equipped with an inverter (in Slovenia, only the pumped storage power plant Avče, Figure below). The inverter allows rotational frequency variation in a certain range, for example from -4 % to +6 % of the nominal rotational frequency, called also varspeed. This allows to reach a better efficiency and more flexibility for adjusting the operation to current conditions in the electroenergetic system and to the available quantity of water. In PSHPP Avče the motor /generator is a DFIM doubly fed induction machine (slo.: dvojno napajan asinhronski stroj) with variable rotational frequency. The benefit of variable rotational frequency is mainly better efficiency. The required power and available head can be adapted such that the efficiency of the system is as high as possible. Benefits are also short response time, synchronisation and poer regulation in the pump mode of operation. Excitation system is specific to PSHPP Avče and enables varspeed operation (Figure below). It is composed of three level VSI Voltage Source Inverter (slo.: trinivojski inverterski pretvorniški sistem), composed of rectifier/inverter, connected into direct current circuit. Main excitation system elements are thyristors, being among the most advanced versions of power converters. Excitation system sets voltage, frequency and slip to the generator rotor. The turbine can operate in wide interval of operating points. In the pump mode of operation the PSHPP Avče regulates the water volume flow rate solely by changing it's rotational frequency, without using guide vanes. regulation with guide vanes is possible, but efficiency is low.

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Fig. 84: Excitation system og PSHPP Avče; left in the container are control part, cooling part and power part, right is excitation transformer [source: SENG d.o.o.].

8.3.10 Brakes Brakes (slo.: zavore) are mechanical elements used to stop the turbine when it rotates very slowly. Before that point, the turbine is braked hydraulically by closing the guide vanes, or also adjusting the rotor blade angle in the case of Kaplan turbines. When the turbine stops, the brakes are released. After that, the turbine must remain still, which is monitored by the creep detection system. If the runner starts to rotate by itsef from the standstill, this is usually due to the dirt jammed in the guide vanes, which prevents full closing. The turbine must not rotate when not in operation, as such rotation can damage it. The brakes are not intended to be used for permanent braking. Brakes are usual mechanical with pneumatic excitation, driven by low pressure compressor.

8.3.11 Generator and its electric equipment The generator is a device where mechanical energy from the shaft is converted to electric energy. A generator consist of a rotor winding and a stator winding. Generators of HPP Doblar II and HPP Solkan are shown in Figure below. All the electric generators in powerplants operate on the principle of electric induction, where voltage is generated as a wire passes the magnetic field lines. In smaller generators, the magnetic field is produced by permanent magnets, while in bigger units electromagnets, which require additional source of current for excitation, are more common. Generators can be of synchronous or asynchronous type. In modern synchronous AC generators, the excitation current is produced from a separated external source. Since the excitation current is much lower than the current in the induced winding the excitation circuit is usually

143 installed to the rotor of the generator because the slip rings are not suitable for conducting very large currents. The excitation current represent an important part of plant's own use of energy (slo.: lastna raba energije). Most of generators is induced by DC current. With DC current excited generator can only operate sinchronous with the electric grid, which in Europe has frequency 50,0 Hz. Rotational speed of the generator and the entire poer plant is therefore set by the number of poles of generator staror winding and rotor winding. The only water power plant in Slovenia (and first of such type in Europe), which used AC current for excitation, is PSHPP Avče. A part of electric energy is taken from the electric grid and is fed over the excitation systen to the rotor of the generator. In PSHPP Avče this enables varspeed operation, as explained in the subsection above. Varspeed operation is useful in pumping mode of operation, where regulation is performed weth changes of rotational frequency instead of with the guide vanes. Excitation system is composed of control, cooling and power electropnics parts. For the purpose of cooling, water is filtered and demineralised to avoid electrical short circuiting. generator is very heavy and big, therefore in the case of large power plants it can not be transported in one piece. In PSHPP AVče stator and rotor each weigh more than 200 t. Generators are therefore assembled in water power plants and rotors must be therefore balanced before first use. This is performed at very low rotational frequency at first with measurement dial indicators, later with measurement of vibrations at various inreasing rotational frequencies for instance 25%, 50% etc. and with adding the mass on required places of the generator rotor. In large water power plants the generator is installed in generator casing or barrel, filled with gas and equiped with systems for gas conditioning and fire protection. Generator is with supports or brackets (slo.: kovinske podpore) connected to walls, supporting the generator bearing. The largest loads of supports are during startup and shutdown, when temperature variations and resulting stresses are the highest. Other electric equipment includes the switchyard, distribution switchard, transformers, generator breakers or fuses, diesel power unit, batteries etc. The diesel power unit (Figure below) is constantly heated to be always prepared for startup, when a backup power source is required for the plant's own energy consumption. The own consumption of the power plant includes the electric energy requirad for running the plant's systems, inclusing those running when the plant is out of operation. These are mostly the drainage

144 pumps, pumps for the needs of seal of the turbine shaft, for cooling of main transformer and for excitation system.

Fig. 85: Generators in water power plants; left: powerhouse and generator cover of HPP Doblar II; in the middle: rotor of the generator HPP Solkan during servicing; right: stator of the generator HPP Solkan during servicing [source: SENG d.o.o.]. When not in operation, water power plant uses electric energy for auxiliary systems from the electric grid. When this is not possible, a diesel generator is used. Fpor instance, amount of water needed during operation of PSHPP Avče is around 300 l/s, and while in not in operation it is 11 l/h.

Fig. 86. Diesel aggregate in HPP Medvode [source: SEL d.o.o.].

8.4 Other systems in the powerplant The other systems include spilways, the drainage system, different cooling systems, the system for oil supply and cleaning etc. The drainage pumps operate constantly, as water usually slowly breaks into the power plant building. The drainage pumps are a large source of the plant's own consumption and must also operate when the power plant is offline and not producing electricity.

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8.4.1 Spillways and spillway gates Spillways (slo.: prelivna polja) release water form watercourse, when upper resorvoir is full and volume flow rate of the river is too large, to be entirely directed through turbine fields. This is the case at high water and flooding. The spilway of HPP Solkan is shown in Figure below. The case, when the main spilway gate is open, is shown in the figure below in the middle. Gates (slo.: zapornice) of spilways are of two types, they may be attached to the sides of spilways or to the crown of of the dam. Gates, attached to the sides of spilways are for instance gredne, tablaste, segmentne, valjčne in kavljaste gates. Gates, attached to the dam's crown are for instance sektorske, preklopne ali krožne gates. In addition gates may be distinguished regarding operation to main or auxiliary gates. Main gates enable operation , when the spilway is active. Auxiliary gates are used dureing revisiona and servicing of main gates and the spilway. Auxiliary gates are similar in ooperation to the gates of the flow field. Depending on the type of water power plant, a spilway may have two auxiliary gates, one above and one below the main gate. Auxiliary segment gates are for instance put into place with a crane and are used, when water around the main gate must be drained. Main gates are from one or several segments. A sample of the main table gate of HPP Medvode is shown in Fig below (right). For instance, main table gate closes the flow through the spilway, when the flow rate of the river is low and all water flows through turbines. In the case of increased flow rate, when the turbines cannot take all the waterthe table gate first drops (low flow rate across the gate), then rises (high flow rate under the gate) or is fully removed (the whole spillway is closed). The problem with flooding waters is that the river flow carries large branches or even trees which can become stuck in the spillway, greatly reducing the flow rate. Fully open spillways in run-of-the-river power plants should not represent any resistance to the flow of the watercourse. The watercourse should flow through the open spilways full unrestricted, meaning that there should be no head lost on the spilways. This means, that in such case the head of turbine fields is almost zero and turbines on run-of-the-river power plants can not operate. With derivation power plants like HPP Doblar and HPP Plave operation during floods is possible. With derivation power plants the height (slo.: kota) of the upper water is almost the same as during normal operation, while height of lower water may be higher than usual.

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Fig. 87. Cross section of the spilway of HPP Solkan.

Fig. 88: Spilways on run-of-the-river water power plants, left: HPP Solkan with two spilways and three turbine fileds [source: SENG d.o.o.]; in the middle: flow fields of HPP Solkan during floods in year 2012 [source: SENG d.o.o.]; right: table gate of hook type on the spilway of HPP Medvode [source: SEL d.o.o.] The water from turbine or spillways flows into the lower accumulation, where some power plants have floating gates. These are hollow gates lowered by the crane into the water of the lower accumulation, then they float to the place of installation, where they are are filled with water and being sunk as a result.

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Floating gates are used for the maintenance of the outflow facility downstream of the power plant or under the waterfall (slo.: podslapje) etc.. Water power plants, that must alwas release some water to maintain a biological minimum of watre flow rate, may not close all gates. They therefor do not have gats on all spilways and turbine fields. Beside this, revisions and servicing are only performed on one turbine field at a time.

8.4.2 Sump pumps Sump or drainage pumps (slo.: drenažne črpalke) pump water from the machinery hall. Because turbines are usually mounted very low to achieve proper turbine submersion, water is ingressed in the machinery hall. This water is inside the machinery hall collected in channels into the sump, located in the lowest part of the water power plant. In th esump pump, sump pumps are located. They are operated from time to time to prevent flooding of the machinery hall. In the cases, when machinery hall is large and deep, walls of the machinery hall have side pressure relief caverns, from where the water is also pumped out. This is the case of PSHPP Avče. Sump pumps are also used to pump water from the turbine flow tract. For such case, a bypass is built, linking flow trach and sump pumps. Water first flows by gravity from turbine flow tract to the sump, from where sump pumps pum the water out of the power plant. By doing so, inspections and servicing of turbine flow tract are possible. Similar case is for gates of the spilway, if they are located near the sump.

8.4.3 Fish passages Fish passages (slo.: ribji prehodi oz. ribje steze) are technical mitigation measures to reduce negative impact of water power plants and dams on fishes. The selected type of fish passage is selected (passage, elevator, side channel, etc.) and conducted in the way, that suits fish species present in the watercourse. The construction of hydroelectric power plants is associated with a series of interventions in waterourse. One of the most negative effects of the destruction of habitats and disruption of migratory routes of fish on the spawning grounds and pasišča. The first notes on the installation of fish passages date back to the 17th century, their number has significantly increased until about 1850, with the building of the first hydroelectric plants. Previously, the dams were used for agricultural purposes. The first documented fish passes were built in the years 1852-1854 on stream Ballisodare Ireland [Kolman et al, 2010].

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Reservoirs affect river hydrology and seasonal variability of flow. Due removed riparian vegetation are subjected to heating embankment, resulting in higher water temperatures and lower oxygen content, which are fish stress factors. On the other hand colder water negatively affects the reproduction, increased by a relatively large number of specimens and adult fish [Kolman et al, 2010]. At the site of release of warmer water in the stream with normal cold water some fish may die. Such a change affects reproduction and food production for the rest of the fish population. Dams can also become a focus of a variety of diseases. In anaerobic processes at the bottom of the riverbed methane is formed, which can cause fish kills [Kolman et al, 2010]. The reasons for the migration of river fish species are in search of various habitats, allowing for the survival of fish species. Many mature freshwater fish migrate upstream to spawning grounds or in the stream itself or in its tributaries, young fish are migrating downstream with the flow [Kolman et al, 2010]. Under the dams fish species collect due to greater opportunities for pillaging even during seasonal migration. It has been shown that after setting the dam, the number of species and population size is very quickly reduced, leaving only those species that in the new circumstances can survive [Kolman et al, 2010]. The dam is often an impenetrable barrier that can completely alter the natural flow regime and thus prevent permanent migratory of fish species such as the transition between different mountainous and downstream habitats. An important building block of fish passages is an input and the ability to attract fish to migration [Kolman et al, 2010]. Also crucial is the flow rate of the watercourse during fish migration and behavior patterns of different species of fish. The plan for fish passage includes mechanical and hydraulic solutions to discourage fish going into the turbine flow tract for example with grids, and guidance into the fish passage. All power plants do not have have fish passages, in such cases it is necessary to migrat juvenile fish by other means. The issue of fish passage is mostly mentioned only in large hydroelectric plants, while on small hydropower plants and small streams the problem is too often forgotten. Example of fish passes in water power plant is shown in Figures below. Fish Bowling is usually carried out in such a way that the drop in head is divided into several segments, or small pools, through which water flows. In each pool regions of high and low velocity of water are present. Regions of low water velocity allow the young and weak fishes to rest.

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In Slovenia HPP Mavčiče and HPP Vrhovo have an artificial spawning as compensation for interrupted migration of fish species, however only artificial spawning in Mavčiče is in operation [Kolman et al, 2010]. On the river Drava more fish passages were built, but many have been abandoned in the past because of dysfunction.

Fig. 89: Fish passage of HPP Krško; left: upstream connection with Sava river, right: fish migrate into the fish passage only at a certain height, it is therefore required to make several entrances height wise. All entrances are then merged in a single fish passage. In the case of HPP Krško two entrances were built [Source: Hidroelektrarne na Spodnji Savi, d.o.o.]. In Slovenia, the most modern fish pasage was built on HPP Blanca, where on the side of the plant enough space for the fish passage was present. Some water power plants in Slovenia do not have fish passages, for example. HPP Solkan and HPP Moste. In the case of HPP Solkan local fishing association (family) cares for transportation of fish upstream and downstream.

Fig. 90: Fish passage of HPP Krško, individual basins contain regions of relatively calm water, allowing the fishes to rest [Source: Hidroelektrarne na Spodnji Savi, d.o.o.].

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9 Manufacture of runners of water turbines

Manufacture of runners of water turbines differs according to the type and size of the runner. In this chapter we will consider manufacture of runners, while we will not discuss manufacture of other mechanical equipment of water turbines. Usually for the manufacture of runners of water turbines two types of materials are used. In the first group are usual stainless steels like X6 CrNi 18- 10. In the other group are harder steels, often used are for instance X5 CrNi 18-10 or X3 CrNiMo 13-4. Selection of material depends on the type of runner and experience of the manufacturer. Austenitic stailess steels belong to the first group, they are used for the manufacture of simple and small runners up to the diameter approximately 500 mm. Using this material, manufacture is relatively easy, for instance before welding preheating is not required. Also, not required is the tempering. Because steel is not magnetic, for the control of quality of the material the method of magnetoflux is not used. Welding of the large Francis runner is shown in Figure below.

Fig. 91: Welding of large Francis runner [source: https://www.ewm- group.com] Steels in the second group are not stainless steels, however they have better cavitation resistance properties, they are much more resistant to cavitation erosion damage. For instance, material X3 CrNiMo 13-4 is hard, difficult to treat, availability is difficult and requires special treatment during welding and manufacture. For manufacture of runners from such materials a preheting 151 is required for instance runner must be preheated (with heaters to approximately 100 °C) and annealing must be performed for reduce inner stresses in the material. Using the mentioned material after welding, grinding, annealing, tempering etc. runner's geometrical shape is altered, therefore additional turning is required before installation. In general, several types of manufacturing processes may be used to manufacture runners of water turbines, they depend on size, type and complexity of the runner and technical abilities of the manufacturer. Processes differ to some extent among diffrent manufacturers. Manufacturing processes include: - manufacture with CNC machines from one piece, - manufacture with CNC machines from two pieces and welding of both halves, - manufacture with welding of blades on the hub (slo. pesto) and rim (also shroud, ring, slo. venec). With Kaplan turbines blades and hub are manufactured separately. Blades are attached to the hub later using screws. Pelton turbines are manufactured in one peace or with welding of blades on the hub of the water turbine.

9.1 Manufacture of Pelton turbine runners Runners of Pelton turbines are manufactured from one piece or with welding of blades on the hub. Pelton turbines are not manufactured from usual stainless steels like CrNi 18-10. Manufacture of runners of Pelton turbines with welding is shown in Figure below, while manufacture from one piece is shown in the Figure below.

Fig. 92: Manufacture of Pelton turbine runner, left: grinding of the runner before welding of blades; right: welding of blades [source: Andino d.o.o.] The hub of Pelton turbine is manufactured such, that steel casting is turned on the outer surface. Blades may be manufactured in one piece together with the 152 hub or separately. In the second case blades are later welded to the hub of the Pelton turbine. In the past, blades were always welded or even bolted to the hub, because the possibility of manufacture in one piece did not exist. All pieces, that is hub and all blades, are checked before further manufacture with the method of penetration and magnetic flux leakage.

Fig. 93: Manufacture of Pelton water turbine runners. Left: manufacture of Pelton turbine blades on milling machine, runner is manufactured from one piece; in the middle and right: grinding [source: Siapro d.o.o., www.siapro.eu] In the case, that blades ale welded to the Pelton turbine hub, all castings are processed before welding using milling procedure. Welding surface may be at the blade root, but may also be at some other radius, for instance in the middle of the blade, as shown in the Figure below.

Fig. 94: Pelton turbine runner with almost all blades welded [source: Andino d.o.o.] After the welding, all welds are grinded until the requested surface quality is achieved, and suitable radius is manufactured. For welding and further processing the runner is mounted such that a worker welds and grinds blades 153 in the suitable working height. After grinding, welds are checked with penetration, magnetic flux leakage method and ultrasound, and with Xrays imaging, if the buyer requests so. After welding, annealing is required as with Francis runners to reduce internal strasses. After annealing, Pelton turbine runners are again turned to achieve required outer dimensions of the runner. After turning, the runner is polished and balanced and such ready for preinstallation and installation.

9.2 Manufacture of Francis turbine runners Among all types of water turbine runners, manufacture of Francis turbine runners is the most difficult. Francis turbine runners may be manufactured using 3D milling method (small Francis turbine runners) from one piece or two pieces or with welding of blades between the hub and rim. The blades are manufactured using bending of the sheet metal (middle size of Francis turbine runners, Figure below) or with casting (large Francis turbine runners). Hub and rim are manufactured from castings, forging or sheet metal. All pieces are checked before further manufacture using penetration and magnetic flux leakage. If the runner is manufactured using welding of blades between the hub and rim, bended or casted blades are processed using 3D milling. Surfaces of blades are during this phase manufactured such, that later only grinding, and polishing is required, while geometric shape is not changed anymore. Using CNC milling also welded surfaces are processed.

Fig. 95: Manufacture of Francis water turbine runners; left: manufacture of small runner with milling; right: manufacture of blades using sheet metal bending [source: Andino d.o.o.] Hub and rim are before insertion of blades set in appropriate position in relation to each other. Hub and rim are then held together by additional welded steel supports (Figure below). Steel supports are positioned on the side of the runner at it's circumference. The hub is usually located on the 154 ground on the montage plateau, while the rim is lifted and supported by the machine, which enables measurements of oplet of the rim and it's rotation around it's horizontal axis. By doing so the runner is turned outside down. When the appropriate positioning of the rim is set and it's oplet is low enough, the rim is connected with the hub with steel supports and thus fixed. The steel supports remain attached until the annealing. Before the blades are welded in position, blades must be inserted in appropriate positions, inclined at appropriate angles in space and arranged at appropriate distances among each other. Because blades are after casting or bending processed using 3D milling, they usually fit well in the space between the hub and the rim and they fit only at an appropriate angle. In addition, during turning of hub and rim, marking lines are turned on the rim and hug, further facilitating insertion of blades in height. The arrangement at appropriate distances among each other is performed by dividing the circumference for the set number of blades (Figure below).

Fig. 96: Manufacture of Francis turbine runner, arrangement of hub and the rim in appropriate relative position. They are held in place by steel supports on the side of the runner [source: Andino d.o.o.] In the next step, blades must be welded in the position between the hub and the rim. Before welding, for the selected steels, welded parts must be heated to approximately 100 °C. This is checked by thermocouples, evenly arranged on welded parts of the runner (Figure below). First, a root of the weld is welded, while later this is grinded from the other side, because the first weld was welded such that a gap was present between both welded parts, the blade and rim or hub. Later, the weld is welded to it's full extent. Also, analysis samples are welded. Welding of largest runners is performed by certified companies with certified and highly skilled personnel. After welding, welds are grinded to the desired surface quality and radii are formed as desired. The above desribed work is very demanding for the welding personell, especially 155 during summers, in part due to the welding itself, but also because of very high temperature of the heated runner, poor access etc. After grinding, quality of welds is chequed for quality with penetration (Figure below), magnetic flux leakage method or if required by the buyer also with X- rays method. In the next phase of the manufacture of the runner, annealing is performed. By doings so, internal material stresses are reduced. Internal matterial stresses arised in previous steps of manufacture, especially during welding. This is performed in the oven at appropriate temperature and time. Annealing is performed according to selected programme, usually a slow increasing of temperature is required to around 600 °C, temperature is maintained for the selected period, later follows slow decrease of temperature. Test samples are also annealed.

Fig. 97: Placement of blades among the hub and rim. Blades after casting or bending are machined by 3D milling and fit well between the hub and rim over their entire surface, only they need to be equidistantly placed over the circumference [source: Andino d.o.o.]. After annealing steel supports at the side of the runner are removed. Again, turning is required to give the runner the final outside dimensions. Because welding and annealing insert addition stresses in the material, the runner's shape is changed slightly. Because of the, in the first phase of the manufacture of the runner, the runner is manufactured with few mm larger outside dimensions of the rim and hub and with slihtly smaller inner dimensions. After welding and annealing, the runner is turned again and during this final turning the runner gets it's final outside dimensions (Figure below).

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Fig. 98: Manufacture of the Francis water turbine runner, welding of blades onto the hub and rim. During welding the runner must be heated to around 100 °C, which is checked by temperature measurements using thermocouples as seen in the left picture [source: Andino d.o.o.].

Fig. 99: Manufacture of Francis water turbine runner, left: grinding [source: Siapro d.o.o., www.siapro.eu]; right: penetration of the runner for evaliation of material and quality of welds [source: Andino d.o.o.] After final turning the Francis turbine runner is grinded (Figure above), polished and balanced. Balancing is usually performed statically; only in few cased dynamic balancing is required.

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Fig. 100: Turning of the rim and hub of the Francis water turbine on the final outer dimension [source: Siapro d.o.o., www.siapro.eu] To detect similarity between the model and prototype a standars on measurements of properties performance of water turbine models provides information [IEC 60191, 1999] abouts the procedure of checking of similarity. Similarity of outlet openings, height of inlet opening and how evenly blades are arranged around the circumference, are cheched manually, but also sometimes shapes of individual blades are measured with the jig, measurement arm or 3D scanner. Among mentioned parameters, the outlet opening is the most important. This is the shortest distance between trailing edge of a blade and surface of the next blade. This is because this dimension has the most pronounced influence on the performance and hill diagram, most notably on the maximum volume flow rate. Recently, small and medium sized Francis turbines are manufactured from two halves as shown in the Figure below. In this case, welding surface a single plane located is in the middle of the runner in vertical dimension and not as described above between the hub, rim and the blades.

9.3 Manufacture of runners of Kaplan turbines Manufacture of runners of Kaplan turbines is simpler than manufacture of Francis turbine runners. Blades are manufactured from one piece. The blade is then bolted to the hub of the turbine runner. Blades of Kaplan turbines are usually casted. This is because blades of Kaplan turbines are usually very large to enable high volume flow rates. For manufacture with forging or sheet metal bending Kaplan runner blades are usually too large and too thick. Kaplan turbine blades are casted together with it's pivot pin. Later blades are checked for quality using penetration and magnetic flux leakage method. Separately also the hub is casted.

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Fig. 101: Manufacture of blades of Kaplan water turbine runners [source: Siapro d.o.o., www.siapro.eu] After casting of Kaplan turbine runner blades together with the pivot pin processed using CNC milling method (Figure above). After the milling, blade surface is grinded. In a similar was also the hub is milled with a CNC machine and later grinded. As with Francis turbines, blades must be annealed and checked for defects using penetration, magnetic flux leakage method or possibly xrays method. After the manufacture of Kaplan turbine blades are fully manufactured, pivot pins are attached to the hub during pre-assembly and assembly.

9.4 Pre-assembly, assembly and installation of water turbines Pre-assembly is assembly, performed at the location of the manufacturer of the water turbine. Figure below shows pre-assembly of Kaplan turbine. The final assembly and installation are performed at the final location of the power plant. During the pre-assembly the turbine manufacturer checks for matching of individual elements of water turbine, spiral casing, turbine coverguide vanes, bearings, runner, shaft, draft tube etc.

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Fig. 102: Pre-assembly of Kaplan water turbine [source: Siapro d.o.o., www.siapro.eu] Transportation of water turbines is often difficult because of large size and vey high weight of water turbines. If the water turbine is small enough, it is transportet assembled. In the case that this is not possible, it is transported in several pieces and partly assembled. With the largest turbines, all large elements are transported separately, for instance the runner.

Fig. 103: Installation of Kaplan turbine [source: Siapro d.o.o., www.siapro.eu (left) and SEL d.o.o. (in the middle and right)] During assembly at the location of power plant the pre-assembly is repeated (Figure below). For instance, spiral casing is usually within concrete, during concrete pouring and hardening the spiral casing is deformed slightly. In this case relevant elements of the flow tract must be turned before installation of the runner. These are all large elements, where a very small clearance must be achieved. This is made by a specially adapted turbing machine, which is lowered in the flow tract of the Francis or Kaplan turbine. With large and very large Francis turbines with vertical axis individual elements of water turbines are consecutively lowered in the water turbine opening, first runner with the runner shaft, later turbine casing, bearings, generator shaft, generator, etc.

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10 The properties of operation of water power plants

This section will present some properties of water turbines, namely the operating characteristics, hill diagram and operation of pumps/turbines, etc. For a better understending let us consider two variables which are a part of the characteristics, namely the specific hydraulic energy and flow rate. Indices 1 and 2 mark the pressure and suction part of the machine, respectively, where specific hydraulic energy is determined (Figure below).

Fig. 104: Schematic representation of a water turbine. The flow flows in the direction of the arrow for the pump or the turbine. With index 1 we denote pressure side, while with index 2 we denote suction side. The specific hydraulic energy is a variable which gives the amount of specific energy that water can transfer to the turbine [IEC 60193, 1999]

p − p c2 − c2 E = 푎푏푠1 푎푏푠2 + 1 2 + g̅ (z − z ) . (120) ρ̅ 2 1 2

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In above equation, 푝abs1 and 푝abs2 are the absolute pressures at measurement planes 1 and 2 and consist of two parts: - overpressure in the pipeline and - atmospheric pressure. 푐1 and 푐2 are corresponding velocities in measurement planes 1 and 2. 푔̅ g is the average gravitational acceleration. Difference (푧1 − 푧2) is the height geodethic difference between both measurement planes. 휌̅ is the average density of water. Let us try to write the above equation in a simplified form. For the sake of simplicity or historical reasons, often heights/heads are used instead of the specific hydraulic energy. Under the assumption that the water flow velocity difference 푐1 and 푐2 is insignificant, that the system has two free surfaces (1 and 2), and that atmospheric pressures 푝abs1 in 푝abs2 on the free surface of water are approximately the same, the equation above can be rewritten as the dependence of the specific energy on the geodetic height difference 퐻st the difference between the upper and lower water level:

E ≈ g̅ (z1 − z2) ≈ g H푠푡 . (121) The geodetic height difference between the upper and lower water level in hydroelectric power plants is called static height difference 퐻st. The total pressure difference of a hydro power plan 퐻b is the gross head, which follows from 퐻st and the kinetic energy difference. Therefore, 퐻b je torej bruto padec, is the static height difference, reduced by the fraction of energy, represented by velocity increase from 1 to 2

c2 c2 H = H + 1 − 2 . (122) 푏 푠푡 2 g 2 g

The net head 퐻n of a hydro power plant is obtained, if the gross head 퐻b is subtracted the hydrodinamic losses ∑ 퐻izg in the inlet part until the turbine and the outlet part until the lower accumulation

c2 c2 H = H + 1 − 2 − ∑ H . (123) 푛 푠푡 2 g 2 g 𝑖푧푔

The flow rate is the quantity of water, passing through planes 1 in 2. The flow rate through both planes is assumed to be the same. In the case of some power plants such as HE Plave and HE Doblar (Figure below), the upper accumulation is far from the powerhouse of the plant. Water flows through a long supply tunnel (penstock). In this case, the sum of 162 losses is relatively large and the water level in the penstock is not the same on the inlet and outlet, it is different by the sum of head losses from the equation above. Often, instead of the specific hydraulic energy, head or flow rate, dimensionless variables are used for instance 휑 (flow number), 휓 (pressure number) etc. Similarly, the dimensionless power number 휆, can be defined, but many other dimensionless numbers exist as well.

Fig. 105: Flow system of HPP Plave from the dam Ajba to the outflow into watercourse. Due to losses in the penstock, not all gross head is available to the turbine.

10.1 Characteristics and hill diagram of turbines Properties of turbines are presented by characteristic diagrams and hill diagrams. Depending on the type of regulation, three different types of turbines exist:: - turbines without regulation, - turbines with single regulation and - turbines with double regulation. In diagrams of characteristics and hill diagrams manufacturers of water turbines usually provides several regions, these are regions of of allowed operation, regions of operation with limitations, non allowed region, reference region for the purpose of acceptance tests and test region. In the following we will review characteristics and hill digrams and their regulation.

10.1.1 Characteristics of water turbines In the case of an unregulated tubine (Figure below) power, flow rate and efficiency are given at a selected rotational frequency. Measurements are conducted by variation of the specific hydraulic energy and measurement of other variables. 163

For the unregulated turbine, both axis are swithched usually in comparison with compressors, pumps and fans. With turbine pumps characteristics of pressure, head or specific hydraulic energy decreases with increasing flow rate, while with water turbines it is exactly the opposite. With water turbines the characteristics of pressure, head or specific hydraulic energy rises with the volume flow rate. The reason for this in a unregulated turbine (Figure below) is, that an operator of the turbine can not select the specific hydraulc energy, it is provided by the heights of upper and lower accumulation and losses related to the current volume flow rate. The specific hydraulic energy is therefore independent variable, while volume flow rate or flow number are dependent variables.

Fig. 106: Characteristics of non-regulated water turbine A characteristic diagram or shurtly turbine characteristics for single regulation turbines is shown in the Figure below. The difeference with unregulated turbine is for a single regulated turbine, that discharge through the turbine may be regulated even though the operator still can not influence the geodetic heights of upper na lower accumulation. we therefore with single and later also double regulated water turbines for the x axis select independent variable flow number or volume flow rate, while on the y axis we select specific hydraulic energy, head, pressure number or pressure. In the case of a single regulated turbine, measurements must be taken in a sufficient number of operating points for every selected dimensionless specific energy E, such that the curves of constant efficiency, guide vanes opening and

164 power can be drawn. Measurements on the measuring station are conducted by establishing the selected dimensionless specific energy E. on the boundaries of the turbine with the aid of an auxiliary pump. Then, the guide vanes are opened and closed and for each guide vane angle  we measure the flow rate (presented as the flow number on the diagram), power and efficiency. Measurements are usually performed for a single rotational frequency. Normally, the range of flow rates in which the turbine can operate is prescribed. A diagram in the image below can be used to obtain a chart with a three- dimensional surface, also known as the hill diagram (slo.: hill diagram). This is done such that the characteristics are sliced in planes, horizontal to the volume fklow rate and specific hydraulic energy.

Fig. 107: Characteristics of a single regulated water turbine, for instance Francis turbine Turbines with double regulation allow in addtion to regulation of guide vanes angle also variation of the runner blade angle . The characteristics of a double regulation turbine is presented in the Figure below. In the process of determining the characteristics, turbine power and efficiency are measured at different rotor blade angles. This is due to the fact that on model turbines, it is easier to change the stator opening than the rotor blade angle (in the latter case, the turbine must be disassembled). To disassemble the model turbine, it must be emptied of water, remove the draft tube and a runner, with templates 165 set the new angle of runner blades, attach the blades at the new angle, mount the runnaer, draft tube and fill the turbine with water again. Measurements on the measuring station are conducted by establishing the selected dimensionless specific energy E on the boundaries of the turbine with the air of an auxiliary pump. Then, the guide vanes/stator are opened and closed and for each stguide vane angle  we measure the flow rate (presented as the flow number on the diagram), power and efficiency. Efficiency curves turn out to be fairly steep at a constant angle of the rotor blade rotation  e. Over the peaks of these partial curves a common curve of constant efficiency is drawn as an envelope (presented by a dashed line in the Figure below). Measurements are usually performed for a single rotational frequency.

Fig. 108: Characteristcs of double regulated water turbine, for instance Kaplan turbine. With water turbine with double regulation we must for each measurement point change also angle of runner blades .

10.1.2 Hill diagram Characteristic – capability of a turbine as a hydraulic motor for driving the generator are obtained by model measurements on test rigs, as presented above in the previous subsection. Measured are basic characteristics: volume flow rate 푉̇ , torque M, efficiency 휂, specific hydraulic energy E and power P From characteristic diagrams (such as the one presented in the above

166 subsection), complex turbine characteristics named the hill diagram can be obtained (slo.: školjčni diagram). Hill diagram for single regulated turbine (Figure below) and for double regulated turbine (Figure below) is made such, that curves of partial efficiencies are cut horizontally with the plane of specific hydraulic energy E and volume flow rate 푉̇ . In the hill diagram there are lines of constant of guide vanes angle , lines of constant angle of runner blades , lines of constant effieciency 휂 (these have the shape of the hill isohypses, hence the name of the hill diagram) and lines of constant power. The following turbine operational limits are evident from the hill diagram: - maximum allowed flow rate, limited by the emergence of cavitation [Avellan, 2004], while cavitation is mitigated by slightly smaller opening of guide vanes, - maximum power, which is mostly limited by the generator power, - runaway curve at efficiency  = 0, - maximum allowed specific hydraulic energy as a consequence of cavitation on the leading edge of runner blades [Avellan, 2004] on the suction side or the highest possible denivelation of upper and lowest possible denivelation of lower accumulation, - lowest possible head or specific hydraulic energy an a consequence of cavitation on the runner leading edge [Avellan, 2004] on pressure side or lowest allowed denivelation of the upper accumulation and highest possible denivelation of the lower accumulation, - lowest allowed volume flow rate, set by manufacturer of the turbine, because at very low volume flow rates turbine experiences high vibrations and related turbine wear, with Francis turbines this is mainly due to the phenomenon of cavitation vortex in the draft tube and cavitation vortices in the inter blade space of the runner [Širok et al., 2006]. The fourth and fifth bullet points are limitations because of the flow properties on the inlet edge of the runner. This is especiall true for Francis turbines, because they lack possibility of regulation of angle of runner blades. This is much less relevant for Kaplan turbines. Limitation because of highest and lowest allowed denivelation in a limitation of the power plant and not of the turbine and is valid for both Francis and Pelton turbines. As stated in the sixth bullet point, with Francis turbines operating at partial loads and volume flow rates, that is to the left from the point of max. efficiency, two penomena occur, which influence operation of Francis turbines. These are occurence of cavitation vortex in the draft tube and occurence of

167 cavitation vortexes in inter blade space of the runner. Approximately 90 % from the nominal volume flow rate of Francis turbine down to the 60 % of the nominal flow rate, there is a region of marked occurence of cavitation vortex in the suction tube. This occurence is mitigated by injection of compressed air into the draft tube through the hollow shaft. The cavitation vortex does not limit operation of Francis turbines, because Francis turbines can be regulated down to even much lower volume flow rates. Later at much smaller part loads and volume flow rates, the remaining angular momentum is too high and cavitation vortex can not form, therefore the cavitation vortex in the draft tube dissapears. In the interval of around 15 to 30 % of nominal flow rate, cavitation vortices in inter blade space of the runner are poresent. In every, every second or every third inter blade space from the upper hub wall down the cavitation vortex winds down into the runner. This limitation is set by every manufacturer for each produced turbine. Cavitation vortex can form also to the right of the max. efficiency point. In this case the cavitation vortex os of different shape, flatter than left from the max. efficiency point and is limited to the middle of the draft tube. The maximum power is specified due to the properties of the generator, which is only capable of generating power up to the maximum allowed power. Exceeding the maximum power limit would lead to a generator failure. The runaway speed is the turbine speed at full flow rate.

Fig. 109: Hill diagram of water turbine with single regulation (Francis

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turbine). Shown are lines of constant efficiency  and lines of constant guide vanes opening. The runaway speed (slo.: pobežna vrtilna frekvenca) is the turbine rotational speed at full flow rate and zero load. In the case of a runaway event the turbine operation shifts very rapidly along the curve of the constant stator opening (as set in the moment of losing the generator load) down until the runaway curve. This means that both dimensional and dimensionless pressure across the turbine is singificantly reduced. In the case of Francis turbines, this means a transition to lower flow rates, while in the case of Kaplan turbines, the flow rates may even increase, depending on the direction of constant stator opening curves. Hill diagrams can be either dimensional or dimensionless (on x axis is flow number 휑 and on y axis pressure number 휓) In the case of dimensionless presentation, some parts of the diagram are either relatively compressed or expanded with respect to the others, which is why some customers require turbine manufacturers to provide both types of the hill diagram. Hill diagrams, available to the public show relative lalues of efficiency, max efficiency is usualy 1 or 100 %. Manufacturers of tubines do not want to disclose real efficiencies of their products.

Fig. 110: Hill diagram of water turbine with double regulation (Kaplan turbine). In addition to hill diagram of single regulated turbine we mark curves of constant runner blade angles . With the aid of hill diagrams shown in Figures above, the operator can easily find and select the regions of optimal operation of water turbine.

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10.2 Pump/turbine operation in four quadrants (extended range of operation) Turbines and pumps can operate in four quadrants depending on the specific 푛 퐷 푉̇ nominal speed 푛 and the nominal flow rate 푉̇ (푛 = in 푉̇ = ). ED ED ED √퐸 ED 퐷2 √퐸 The four quadrants are defined with respect to the value (positive or negative) of flow rate and rotational frequency. In the image below, (a) marks the oparation at the highest specified power and (c) the operation at the lowest allowed power. The individual parts (quadrants) are: - bottom left - pumping quadrant, - bottom right - braking pumping quadrant, - top left - braking turbine quadrant, - top right - turbine quadrant. Some of the quadrants are further divided, for example the top left quadrant is divided to the turbine part and turbine dissipative (brake) part with the runaway curve (zero shaft torque, generator is out of operation / disconnected from the power grid) separating them. The runaway curve denotes a region, where the the torque is zero and generator does not brake the turbine, that means is either malfunctioning or not connected to the electric grid. Above the runaway curve there is a pure turbine area, where turbines operate most of the time. In this region the volume flow rate is positive, rotational frequency positive and torque positive. Below the runaway curve is an area of turbine dissipation (brake) with positive flow rate and rotational frequency, but negative torque. Such operating conditions are rare thoug possible with some turbines, but only occur in transitional flow regimes. Such operation regime can not be calle operating point, because the water turbine operates in such regime only transiently. During the operation in the turbine regime, the turbine operates on one of the constant guide vanes opening  curves between the vertical lines, which mark the minimal and maximal allowed power. In the case of emergency shutdown, the operating point slides along the constant efficiency curve to the runaway curve (e.g., until the extreme upper-right point in the graph). Then, as the guide vanes are closing, the operating point moves along the runaway curve to the center of the coordinate system (zero flow rate and rotational frequency).

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Fig. 111: Operation of water turbines in four quadrants. With (a) we denote operation at the highest allowed power and with (b) operation with the smallest allowed power.

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10 Cavitation in water turbines and pumps

In the discussion about turbine in the introductory sections we did not distinguish between different fluids. However, water turbine machinery (pumps and turbines) are faced with a phenomenon of cavitation, which puts serious limits to water turbine machinery operation.

10.1 Introduction to cavitation Boiling can be characterized as a vaporization of a liquid at constant pressure, due to increase of a liquid’s temperature (Figure below). Vaporization of a liquid can also be achived at constant temperature if the pressure drops below vaporization pressure. Cavitation as a physical phenomenon, related to boiling, can be characterized as a transition from liquid to vapour phase and back to homogenious liquid at approximately constant pressure (Figure below).

Fig. 112: Boiling and cavitation in p-T and p- diagrams. Cavitation can be usually seen as a crowd of small vapour bubbles; which size can vary from nanometers to few milimeters in diameter. The energy collected in a cavitation bubble during the growth phase can be released in the collapse phase in various forms. Cavitation bubble collapse can cause very high- pressure pulsations – up to several 100 bars and extreme temperatures, up to 1000°C can occurs in the center of the collapsed bubble. If the bubble collapse asymmetrical a so called micro jet can be formed, which velocity can be reached up to several 100 m/s.

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Fig. 113: Cavitation bubbles in a Venturi section Cavitation probability and intensity is characterised by a cavitation number 휎, it is defined as

p0 − pv σ = 2 . ρc0 (124) 2 High values of cavitation number relate to high risks and low cavitation numbers with high risks of cavitation. Unfavourable cavitation effects in turbine machinery are: - increased noise, - increased vibrations, - damages to bearings and sealings, - cavitation erosion (Figure below), - reduction in efficiency etc.

Fig. 114: Kavitacijska erozija in kavitacija v gonilnikih Francisovih turbin; levo: kavitacijska erozija v gonilniku Francisove turbine; desno: kavitacija v modelu gonilnika Francisove turbine [Širok in sod., 2006] By knowing and understanding the phenomenon of cavitation, one can use it as a tool in various technical processes. Nowadays cavitation is used also for surface cleaning for surfaces with difficult acess, in wastewater for bacteria 173 killing and homogenisation, in medicine for kidney stone crushing, in food industry for milk homogenization etc. Types of cavitation according to how they are formed are - hydrodynamic (caused by flow passing the submerged body), - acoustic (caused by high intensity pressure waves, traveling through the liquid), - laser induced (caused by high intensity laser light, vaporization is caused by local temperature increase of a liquid – not regular growth mechanism), etc. Types of cavitation according to the shape - attached/steady cavitation (leading edge – pressure or suction side) - unsteady cavitation cloud shedding - sheet cavitation, - root cavitation, - individual bubble cavitation - tip vortex cavitation, - hub vortex cavitation - supercavitation and - etc.

Fig. 115. Different types of cavitation vortices in Francis turbine [source: Kolektor Turboinštitut d.o.o.].

10.2 Relation to the operating point of the turbine machine We will discuss cavitation in Kaplan and Francis turbines. For leading edge cavitation, we will provide an explanation, at what flow rates such cavitation is formed.

10.2.1 In Kaplan turbine The flow properties in Kaplan turbine runner are shown in the Figure below. The cavitation is present in several locations: - (a) Sheet cavitation is caused by local pressure drop due to high velocity flow 174

- (b) Cavitation on the leading edge / suction side is caused by non-optimal angle of attack - (c) Cavitation on the leading edge / pressure side is caused by non- optimal angle of attack - (d) Cavitation near the root of the runner is caused by local non-optimal hydrodynamic geometry of the runner - (e) Cavitation at the tip of the blade is caused by highest circumferential velocity.

Fig. 116. Triangles of velocities and pressures on blades in Kaplan turbine (PS - pressure side and SS - suction side) Locations where cavitation is found in Kaplan turbines is shown in Fig. below

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Fig. 117. Locations of cavitation occurence in Kaplan turbines pretvori v svg In Kaplan turbines cavitation vortex very rarely occurs. . Vortex occurs when the turbine operates at very low discharges and low pressures.

10.2.2 In Francis turbine Locations where cavitation is found in Francis turbines is shown in Fig. below: - (a) cavitation at the trailing edge, due to high velocity flow, which can not follow the blades, - (b) cavitation on the suction side at the leading edge, due to non-optimal angle of attack, - (c) cavitation on the pressure side at the leading edge, due to non-optimal angle of attack, - (d) vortex cavitation between the blades, due to flow separation (low discharge) and - (e) cavitating vortex in the draft tube. Cavitationg vortex in the draft tube is very common in Francis turbines.

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Fig. 118. Locations of cavitation occurence in Francis turbines

10.2.3 In centrifugal pump The flow properties in centrifugal pump are shown in the Figure below. The cavitation is present in several locations:

- (a) cavitation on the leading edge on pressure side (푉̇ > 푉̇표푝푡),

- (b) cavitation on leading edge on suction side (푉̇ < 푉̇표푝푡), - (c) cavitation by the sealing, - (d) cavitation on the holes – drilled to reduce axial forces and - (e) cavitation at the tongue.

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Fig. 119. Triangles of velocities and pressures on blades in centrifugal pump (PS - pressure side and SS - suction side) Locations where cavitation is found in centrifugal pumps are shown in Fig. below.

Fig. 120. Locations of cavitation occurence in centrifugal pumps

10.3 Cavitation and NPSH/NPSE characteristics The pump operates on the intersection of operating characteristics and resistance characteristics of the system. In addition, producer of pumps provides in addition also a NPSE characteristics for the pump. Fans don't need it, because they operate with air. The NPSE characteristics is a NPSH characteristics, multiplied by gravitationa acceleration 푁푃푆퐸 = 푔 ∙ 푁푃푆퐻. NPSE characteristics provides information how much energy is available to the pump on the suction side. Although E and NPSE have the same unit m2/s2, 178 there is a fundamental difference between the two. The specific hydraulic energy E represents the change of specific energy of water when pumped from suction to pressure side and is also according to the Euler equation to 푢2푐푢2 − 푢1푐푢1. The NPSE on the other hand only considers the flow conditions on one side, that is on the suction side.

10.3.1 In pumps The NPSE characteristics is provided by the manufacturer of the pump and is measured on the test stand. The measurement of NPSE is performed as shown in the Figure below. For every operationg point on the NPSE characteristics (right) an entire σ - η diagram is measured for turbines and an entire σ - H diagram is measured for pumps. The entire σ - η (or σ - H) or diagram corresponds to only one operating point, and different specific energies and heads are measured on the measuring station such that the vacuum is achieved in the measuring station using a vacuum pump. The 푉̇ and E values are in most of the σ - η (or σ - H) diagram the same, until (on the left of the diagram) the efficiency or head falls by 1 %. The cavitation is present for sure much earlier (at lower vacuum), the 1 % limit is used for convenience. The NPSE value, at which the efficiency or head falls by 1 % is indicated on the NPSE diagram on the right of the Figure below.

The above discussed NPSE value we also denote as required NPSE or NPSEr.

The name required NPSEr is from the requirement (Slo: "NPSE črpalke ali

NPSEč) of the turbine of the pump runner to operate without cavitation. The

NPSEr characteristic almost always increases with volume flow rate, as the available pressure decreases with increasing volume flow rate (velocity inside the runner also increases). However, there is another NPSE value to consider, which is provided by the system, NPSEa or available. The available NPSEa must be for cavitation free operation always higher than required NPSEr. With other words, the hydraulc system must be able to provide at the suction side of the turbine machine higher available energy than required by the machine for cavitation free operation. The pump is therefore able to operate left from the intersection of both NPSEa and NPSEr characteristics. Usually at the intersection the cavitation is already severe and for long term operation it is required to operate at even lower volume flow rates.

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Fig. 121: NPSEa and NPSEr characteristics and pressure characteristics of the turbine machine (right). Every single operationg point is derived from a diagram σ - η (or σ - H ) (left).

To estimate the available NPSEa it is the most convenient to consider the case of usual installation of pumps from the Figure below. According to the Figure below the pump gets water from the sump and pushes it in the pressure pipeline, upper open reservoir or pressure reservoir. Per definition the NPSEa is

p − p 푐2 NPSE = 1abs v + 1 . (125) a ρ 2

Available NPSEa is the total energy available on the suction side of the turbine machine, reduced for the contribution of the vapour pressure. This reduction may be very small for low water temperatures, but when the water temperature approaches boiling temperature, it may become significant. Usually it is impossible to measure the absolute pressure at the location of the inlet to the pump due to vortices and other turbulent flow phenomena near the runner entrance. We therefore measure absolute pressure somewhere further upstream in the location indicated as 푝0. For both location we write the Bernouilli equation and assume no flow losses between both locations 0 and 1. The location 0 is H above the location 1

p c2 p c2 0abs + 0 + gH = 1abs + 1 . (126) ρ 2 ρ 2

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We want to eliminate 푝0푎푏푠 from the Equation above, as we argued that it is impossible to measure pressure there. By also assuming that velocity in cross section 0 is negligible due to a large size of the cross section, we get

p p c2 1abs = 0abs + gH − 1 . (127) ρ ρ 2

With insertion of Equation above to Equation above we get

p0abs − pv NPSE = + gH. (128) a ρ

It is convenient to include losses and we get

p0abs − pv NPSE = + gH − E . (129) a ρ LS

The Equation above explains, why available NPSEa decreases with volume flow 2 2 rate 퐸퐿푆훼푉̇ 훼푐 . With the increase of volume flow rate less and less energy is available at the inlet of the pump, hence the pump is more prone to cavitation.

Fig. 122: Situation for the understanding of the available NPSEa

10.3.2 In turbines Figure below shows the installation of the water turbine. In agreement with Fig. below the plane 2 is located at the end of draft tube and not in location of the runner, here producer of mechanical equipment guarantees for specifications to the buyer. The second reason, why plane 2 is located away from the runner location 푧1 = 푧ref is, that we are unable to measure pressure directly at the runner die to high tangential velocity which may influence the validity of wall pressure measurements. Therefore, we must recalculate the flow conditions from the plane 2 to the reference level of the runner. Fo water turbines we modify Equation above for the change of specific hydraulic energy in the plane 2 and reference level zref. The reference level

181 usually corresponds to the middle of the runner and equation for neto positive specific energy is

2 p푎푏푠2 − p푣 c2 NPSE = + − g (z푟푒푓 − z2) . (130) ρ2 2 If pressure measurements in the plane 2 are possible and pressure tappings are available and accessible, we directly use the above equation. Very often water turbines have pre-installed pressure tappings enabling such measurements and acceptance measurements.

Fig. 123: Water levels and heads in the water turbine for estimation of NPSE and NPSH. The case is shown, that does not enable pressure measurements in plane 2 and losses from planes 2 and 2' are not negligible. If pressure measurement is not possible, we may estimate NPSE from water levels at the low accumulation. We use equation below, where losses ±퐸LS represent losses from plane 2 to the lower accumulation level and sign + is for turbine mode of operation and sign - for pump mode of operation

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NPSE = g NPSH 2 (p푎푚푏2′′ − p푣) c2′ = + − g (z푟푒푓 − z2′′ ) ± E퐿푆 ρ2 2 2 (131) (p푎푚푏2′′ − p푣) c2′ = + − g Z푆 ± E퐿푆 . ρ2 2

Water height in plane 2' is lower than stationary water in lower reservoir, because flow here still has some kinetic energy. Later, when the flow comes to a standstill and its kinetic energy is 0, the velocity term becomes zero and height may be calculyted to stationary lower water level far away from the turbine outflow. Equation above is modified into equation below and ±퐸LS represent losses from plane 2 to the outflow and sign + is used for turbine mode of operation and - for pump mode of operation.

(p푎푚푏2′′ − p푣) NPSE = g NPSH = − g (푧푟푒푓 − z푠푣) ± E퐿푆 . (132) ρ2

푧sv in the equation above is the level of water in lower reservoir. 10.4 Prevention of cavitation effects in water turbines From the viewpoint of available head and characteristics it is not necessary for a water turbine to be installed at the lowest point in the flow path. Installation at the lowest location is recommended and in fact also required for economic reasons due to cavitation. Cavitation occurs when the pressure in the liquid drops below the vapor pressure and cavitation bubbles develop. Cavitation and cavitation bubbles reduce the efficiency of the plant and damage the runner. In the case of implosions of cavitation bubbles near a solid surface, usually on a suction side of the runner, a high pressure is generated that erodes the material from the surface. The water turbine must operate at sufficiently large NPSH, that is, a sufficiently large net positive suction height. An example of the immersion of a water turbine is shown in Figure below, which show the upper accumulation, the supply tunnel, the pressure line, the cross-section of the flow field, the engine room and the inlet of the PHE Avče. The mechanical shaft in the case of the Avče Pumping Station is about 80 m deep, with the water turbine located on its bottom, approximately 60 m below the lower water level, that is, the accumulation of Ajba. In practice, water turbines are installed as high as possible to reduce the civil engineering construction costs, which represent the largest part of costs in the construction of a hydroelectric power plant. Installation of the plant as high as

183 possible can still mean that the immersion (the height of the installation of the water turbine below the lower water level) should be up to 60 m.

Fig. 124: Horizontal profile of ČHE Avče. Machinery hall is fully on the right and below. The vertical 7 and horizontal penstock 8 were built because of steepnes of the hill at that location and possibility of landslide in the case of horizontal penstock. 10.5 Influence of particles in the flow on cavitation in water turbines It is known that the appearance of cavitation bubbles and the resulting cavitation characteristics can be largely influenced by impurities and nuclei in the fluid (an invisible proportion of air or bubbles of gas with a radius of less than 50 μm). The content of the nuclei affects not only the beginning of the appearance of cavitation, but also the development of a traveling bubble cavitation. A precise determination of the required minimum values of the content of nuclei and dissolved gas is not yet possible, as these data are not available in accessible scientific literature [Dular, 2014]. The influence of individual parameters such as the type of water turbine machine, specific hydraulic energy, etc. it is not yet sufficiently explored. The appearance of the cavitation and the possibility of its observation depends on the type of cavitation associated with the type of water turbine. Especially for cavitation tests in the Francis model turbines of medium and high specific speeds, where cavitation appears at the exit of the runner, it is important that the water contains sufficient nuclei that can be activated and grown in areas where the local pressure is equal to or lower than vapour 184 pressure. Prototype measurements indicate that there are usually enough cores in the water to accelerate cavitation in areas of low pressure. On test sites where we have a closed water circuit, on the other hand, the number of particles is reduced due to the degassing of water during cavitation tests. As a result, a small number of cores are activated for each selected σ value (eg. for

σpl), which to some extent reduces the extent of visible cavitation. Therefore, the quality of water with respect to cavitation nuclei is similar to the conditions in the prototype, if the content of the nuclei in the flow path of the model measuring station is sufficient. This ensures the proper development of cavitation in all areas where the local pressure equals or decreases below the vapour pressure of the water. An adequate number of active cavity nuclei in model measurements can be provided by increasing the head, by introducing nuclei into water, or by using non-degassed water. By increasing the head in the measurements, it is necessary to be careful because it can lead to distorted results due to the reduced similarity.

10.6 Runner erosion due to solid dispersed particles The erosion of the driver may also result beside due to a cavitation also due to impurities in the water. Impurities or dispersed particles in water damage the runner by abrasion. Abrasion is different around the world, the most problematic are watercourses with a high amount of deposits, among them for instance watercourses in Southeast Asia and some watercourses in the Dolomites in Italy. To reduce the erosion of the runner due to abrasion with impurities in water, a few studies are carried out tregarding hydraulic shaping of blades and material selection, to reduce abrasion and maintain good hydraulic properties of the water turbine.

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11 Sample design of a Francis turbine runner

In the following we will provide the guideline how to design a Francis turbine runner. The design procedure is rudimentary and nowadays runners and turbines are not anymore designed in such way. The design procedure is based on determining inlet and outlet runner blades and circumferential velocity. The exact shape and length of the blade may be found using equations of potential flow (not discussed here) as it was the case around years 1940- 1975. Later, design of turbine blades was performed and still is using CFD computations. The hydraulic design of a water turbine normally starts with the suction side due to cavitation. The important geometrical parameters here are the outlet angle of the blade at the maximum diameter and the circuferential velocity of the runner at its hub. To avoid cavitation, the local velocity must be limited so the pressure stays above vapour pressure. Due to this a limit must be set on the circumferential speed of the runner as well as the meridional velocity component of the flow. This limitation depends also on the outlet angle of the blades, the number of blades and the curvature of the blades and if rotation of the draft tube flow is accepted at best efficiency flow. The inlet conditions of a runner for a Francis or other reaction turbine will be based on the required fraction of the specific energy should be converted into kinetic energy (velocity head) at the inlet of the runner. According to Euler equation the tangential component of the absolute velocity 푐푢1 can be found and then the selected meridional velocity at the runner inlet together with 푐푢1 gives the inlet water flow velocity. 11.1 Flow velocities, blade angles, radii and rotational speed In the following we will consider a case where the high head Francis turbine operates at volume flow rate 30 m3/s and head 400 m. For these parameters we want to optimise the turbine to operate in best efficiency point. We start the design at th exit from turbine runner (Brekke, 2001). We may select the flow angle at maximum outlet radius r2 to be β2 = 17.5 °, because normally the blade angle is in the interval from 13 ° to 19 °. The flow angle is 1 ° to 2 ° larger because the flow does not ideally follow the blade angle because the number of blades is not infinite. We may also select the maximum circumferential speed u2 of the runner at the exit. u2 at the shroud is normally from 40 m/s to 55 m/s for high head Francis 186 turbines. We select u2 = 41 m/s, this is a moderate setting. Higher circumferential speed may result in increased vibrations and noise. The meridional velocity cm2 is

푐푚2 = 푢2 ∙ tan 훽2 = 12,927 m/s . (133) Now we can calculate the diameter of the turbine at the outlet

푉̇ 푟2 = √ = 0,859 m . (134) 휋푐푚2 Rotational frequency of the turbine is thus set according to selected variables u2 and 훽2.

60 ∙ 푢2 푛 = = 455,5 rpm . (135) 2휋푟2 The calculated rotational frequency can not be a syncronous speed and it must be adjusted by adjusting the curcumferential velocity 푢2 and radius 푟2. The blade angle at the exit must also be adjusted, but this is nomally not the case, as the manufacturer wants to maintain the same blade angle 훽2 as other designs, manufactured in the past. The synchronous speed is 3000 푛 = , (136) 푍 where Z is number of pairs of poles in the generator for the case of electric grid operating at 50 Hz. In order not to increase the necessary submergence of the turbine, the speed should be reduced (!) to nearest synchronous speed. Choosing Z = 7 gives n = 428,6 rpm. If the outlet angle 훽2 and volume flow rate in the best efficiency point 푉̇ remain the same as above, we get the set of three equations with three unknowns

2 푟2 푐푚2 = 9,553

푐푚2 = 0,315 푢2 (137)

휋푛퐷2 푢 = 2 60 with the solution: - 푟2 = 0,978 m, - 푢2 = 39,38 m/s and - 푐푚2 = 12,42 m/s. The inlet dimensions may now be found by means of the Euler equation, in this case we add to the equation also hydraulic 휂ℎ efficiency 187

푃 = 푢 푐 − 푢 푐 = 휂 푔퐻 , (138) 푚̇ 1 푢1 2 푢2 ℎ

If we take for hydraulic efficiency 휂ℎ = 96 %, we must also assume how much of the pressure energy is converted to kinetic energy and how much of it remains as a pressure, the ratio being known as reactivity R. We may assume for simplicity that R = 0,5 or 푢1 = 푐푢1. We also assume 푐푢2 = 0 and

푢1 = 푐푢1 = 61,345 m/s . (139)

In this case 훽1 = 90 °. More common choice nowadays is to use backward curved blades at the inlet, in this case R < 0,5. The inlet radius r1 of the runner is

60 ∙ 푢1 푟 = = 1,367 m. (140) 1 2휋 ∙ 푛 The meridional velocity at the inlet may from experience be chosen approximately 10 % lower than at the outlet of the runner to obtain a slight acceleration of the meridional flow. The turbine outlet diameter is for instance approx. 10 % smaller also in the case of PSH plant Avče, although the flow direction varies according to the type of operation. However, choice of the smaller size of inlet opening will be different among manufacturers due to the philosophy of blade shape etc. m 푐 = 0,9 ∙ 푐 = 11,634 . (141) 푚1 푚2 s

The height of the blade at the inlet b1 can now be found by continuity equation as follows

푉̇ 푏1 = = 0,300 m . (142) 2휋푟1푐푚1

The turbine maximum power is 112 MW (푃 = ∆푝 ∙ 푉̇ ) and ns = 95. 11.2 Cavitation at the exit of the runner In this subsection we want to estimate how much the turbine runner must be submerged. This may vary a lot, for instance HPP Doblar 1 runner is above the lower reservoir while PSH Avče is submerged by 60 m. In the above subsection we have calculated meridional 푐푚2 and tangential 푢2 velocity at the exit from the runner. The NPSHr for this situation is given by the standard (IEC, 1995)

푐2 푢2 푁푃푆퐻 = 푎 푚2 + 푏 2 . (143) 푟 2푔 2푔

188 where constants a and in particulary b are depending on the specific speed and shape of the runner. Usually we use 1 < a < 1.2 and 0.05 < b < 0.11. In the equation above is required neto positive suction head NPSHr, which must always be by a safety limit lower then available NPSHa. To estimate available NPSHa we must know absolute pressure and velocity at the location of the runner. The absolute pressure very much depends on the submergence of the turbine. This was already discussed above in Section 10.

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References

Brekke, H., Hydraulic turbines - design, erection and operation, https://www.ntnu.no/documents/381182060/1267681377/HYDRA ULIC+TURBINES_Hermod+Brekke+-+2015.pdf/656e691a-f52f-4c0d- a6b1-eaf069c08ef5, retrieved 22.1.2019, 2001 Dixon, S. L., Hall, C. A., Fluid Mechanics and thermodynamics of turbomachinery, Elsevier, 2010 Schlichting, H., Boundary layer theory, 7th ed., McGraw-Hill, New York, 1979 Tuma, M., Sekavčnik, M., E, Energetski stroji in naprave, osnove in uporaba, druga izdaja, Univerza v Ljubljani, 2005 Korpela, S., A., Principles of Turbomachinery, John Wiley & sons, New Jersey, 2011 Lakshminarayana, B., Fluid dynamics and heat transfer of Turbomachinery, John Wiley & sons, New York, 1996 Stodola, Dampf- und Gasturbinen, 6th edn., Berlin, Springer. 1924 Eck, Bruno. Fans. 1st English ed., Pergamon Press, Oxford, 1973. Giesecke, J. and Mosonyi, E., Wasserkraftanlagen, 5th Edition, Berlin Heidelberg, Springer, 2009. IEC Standard, International standard, IEC 60041. “Field acceptance test to determine the hydraulic performance if hydraulic turbines, storage pumps and pump turbines”, Third edition 1991-11.

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