turbomachinery.education
9.

WATER TURBINES

–    page 9.3    –
Classification and specifications
–    Problem 1:    –
Calculation of water wheel dimensions
–    Problem 2:    –
Calculation of Francis turbine and draft tube dimensions
–    Problem 3:    –
Selection of type of turbomachine according to calculated specific speed [Problem 1 in art. 6]
–    Problem 4:    –
Calculation of force on water turbine blade [Problem 3 in art. 2]
–    Problem 5:    –
Calculation of reaction of Francis turbine [Problem 7 in art. 2]
–    Problem 6:    –
Velocity and pressure distribution in Kaplan turbine [Problem 8 in art. 2]
–    Problem 7:    –
Calculation of aerodynamic quantities of profile cascade [Problem 3 in art. 4]
–    page 9.9    –
Energy balance
–    Problem 8:    –
Calculation of power output of turboset [Problem 3 in art. 1]
–    page 9.10    –
Draft tube
–    page 9.11    –
Cavitation
–    page 9.13    –
References
–  pages 9.14-9.26  –
Appendices
View PDF
turbomachinery.education 
page 9.2
–    author:    –
ŠKORPÍK, Jiří (LinkedIn.com/in/jiri-skorpik)
–      issue date:      –
August 2025
–    title:    –
Water turbines
–    proceedings:    –
turbomachinery.education
–    provenance:    –
Brno (Czech Republic)
–    email:    –
skorpik.jiri@email.cz

Copyright©Jiří Škorpík, 2025
All rights reserved.

 WATER TURBINES
page 9.3

Classification and specifications

Currently, only four types of water turbines and their modifications prevail, see the subchapter Historical Notes. These four types include the Pelton turbine, Francis turbine, Kaplan turbine, and Deriaz turbine. The suitable type for an application is determined by specific speed.

Historical notes
Water wheels were first used in Mesopotamia in the 6th century BC and were designed to power irrigation systems. The first water wheels for driving mills and hammer mills (Figure 1) in the Czech Republic began to be used in the 12th century, with millers and woodcutters being the bearers of technical skill and progress in our territory for centuries [Štěpán and Křivanová, 2000]. At that time, mills were one of the few mechanical devices that were systematically improved, see Problem 1.
– 1: –
Water wheel on lower water of Velkopřevor mill on Čertovka river
Water wheel of Velkopřevor mill on Čertovka river
–    Problem 1:    –
Design a water wheel for lower water if you have a weir with a head of 0,6 m and a volumetric flow of 0,7 m3·s-1. Use the empirical knowledge of woodcutters and millers [Štěpán, and Křivanová, 2000] and knowledge of hydrodynamics from the late 18th century to perform the calculation.
The solution to the problem is shown in Appendix 1.
Water wheel for lower water
 WATER TURBINES
page 9.4
Water turbines were developed from Segner wheel theory
The principle of water wheels and their construction depended on the intuition and experience of their builders, but modern water turbines were preceded by a description of the general theory of turbomachines. The foundations of the theory of turbomachines were laid by physicist and mathematician Leonhard Euler (1707-1774) when he wanted to describe the function of the Segner wheel constructed by Ján Segner (1704-1777) around 1750. Based on this theory, Euler designed the first modern turbine, which already contained classic parts such as distributor and rotor blades. This concept was reportedly lost in the reports of the Academy of Sciences and was not implemented [Hoch, 1941, p. 66-67], but we still use his theoretical description today, see the article Essential equations of turbomachines.
Water turbine development in data
A practical water turbine was constructed in 1827 by Benoît Fourneyron (1802-1867), which had an output of 4.47 kW at 80% efficiency. This was followed by the invention of other turbines, three of which are significant for today's energy industry: the Francis turbine, built in 1849 by James Francis (1815-1892); the Pelton turbine, built by Lester Pelton (1829-1908) in 1884; and the Kaplan turbine, built in 1912 by Viktor Kaplan (1876-1934). Each of these turbines is suitable for a specific range of heads and volume flows. In 1957, the Sir Adam Beck pumped storage power plant began operating Deriaz turbines (Paul Dériaze (1895-1987)), which are diagonal type turbines capable of alternating between turbine and pump operation.
Birth of first hydroelectric power plants
Water turbines are usually part of a turboset with an electric generator. The first hydroelectric power plants for electricity generation were launched in 1881 with outputs of less than 1 kW to power light bulbs [Jílek et al., 1980, p. 144]. At that time, direct current was produced, and the first power plant for the production of alternating current was put into commercial operation on August 26, 1896 at Niagara Falls. The output of this hydroelectric power plant was 2x5000 horsepower [Jonnes, 2009, p. 340].
 WATER TURBINES
page 9.5
Use of water turbines to regenerate pumping work
Water turbines can be installed in the traditional manner between the upper and lower reservoirs in hydroelectric power plants (Figure 7), or in industrial plants where there is high consumption of high-pressure liquid (e.g., for cleaning, flushing, or cooling) with the possibility of using pump work regeneration. Pump work regeneration can look like Figure 2, which shows a Pelton turbine-electric motor-pump turbine turboset. Such an turboset reduces the consumption of electrical energy for pumping liquid by transforming the pressure energy of the liquid back into work in the turbine. A 2,5 MW regenerative hydraulic turbine is also installed on the TAL oil pipeline at the northern foot of the Alps, which the pipeline crosses from the south, where pumps are installed to pump oil to higher altitudes. Pumped storage hydroelectric power plants are also regenerative.
– 2: –
Principle of pumping work regeneration
Principle of pumping work regeneration: 1-rotodynamic pump; 2-electric motor; 3-Pelton turbine; 4-pressure tank in which high-pressure water is used, with no significant loss of water pressure during this process; 5-non-pressure waste channel.
Peltonova turbína
Pelton turbines do not exceed 400 MW of power, which is due to the fact that they are advantageous for processing very high water columns, and such locations do not have large mass flows.
Description of operation of Pelton turbine
The nozzle with a control needle (Figure 3) is the main control element of Pelton turbines. Moving the control needle allows for a wide range of flow rate changes while maintaining hydraulic efficiency, see Figure 5. The Pelton turbine can also have several nozzles around its circumference, which increases the power of the turbine. There can be two rotors on a common shaft with an electric generator; such turbosets achieve maximum power. A deviator behind the nozzle is used to divert the flow when a quick stop of the rotor is required, because the main shut-off valve for the water supply to the nozzle cannot be closed immediately without causing a water hammer in the pipeline.
 WATER TURBINES
page 9.6
– 3: –
Main parts of Pelton turbine
(a) horizontal Pelton turbine; (b) velocity triangle of a Pelton turbine according to [Shepherd, 1965, p. 351]. 1-water inlets via ball valve; 2-control needle; 3-water jet deviator; 4-water jet; 5-blades mounted on impeller disc; 6-braking nozzle (reduces coasting time of turbine during shutdown); 7-water outlet. Ød [m] mean diameter of blades; ω [rad·s-1] angular velocity; V [m·s-1] absolute velocity; U [m·s-1] blade speed; W [m·s-1] relative velocity. θ-denotes tangential direction, a-denotes axial direction (rotor shaft axis). Index 1 denotes the state in front of the rotor, index 2 denotes the state behind the rotor.
Calculation of mean diameter of blades
The approximate equality of velocities W1=W2≈U can be seen in Figure 3b. This means that the approximate equality 2U≈V1 also applies. The velocity V1 is given by the height of the water column. These velocities and the required rotational speed of the rotor are sufficient to calculate the mean diameter of the blades Ød. These relationships between velocities were already known and used by builders of water wheels, which are also tangential stages, see Problem 1.
Francis turbine
Francis turbines are used in a very wide range of flow rates and water levels and for power outputs exceeding 1000 MW. These are radial turbines, usually with a constant reaction along the length of the blades (see Problem 2). The radial design also allows Francis pump turbines to be constructed for pumped storage power plants, but a change in the direction of rotor rotation is necessary to change the purpose.
Reaction of Francis turbines
The reaction of Francis turbines is such that internal losses in the stator and rotor are approximately equal at the mean radius of the blades.
Inlet radius of Francis turbines
The inlet radius of the rotor r1 increases with the required Euler work of the turbine (U1 increases) and decreases with increasing rotational speed, see Figure 4. The not pronounced axial part of the rotor blades is another characteristic feature of Francis turbines (for more information, see the article Shapes of blades and flow parts of turbomachines and Problem 2).
 WATER TURBINES
page 9.7
– 4: –
Influence of specific speed on optimal rotor shape of Francis turbines
NS [min-1] specific speed; r [m] radius. The index m denotes parameters at the mean square radius of the blades, the index t at the tip of the blades, and the index h at the root of the blades.
–    Problem 2:    –
Carry out the first iteration of the calculation of the main dimensions of the blade part of the Francis turbine at the mean square radius, blade tips, and blade roots. The volume flow through the turbine must be 3,12 m3·s-1. Other entered parameters are: ΔzR-T=46,3 m; Δz2-T=1,6 m, V2=4 m·s-1; Δz2-e=2 m. The turbine rotational speed must be 750 min-1. The reservoir level is at an altitude of 500 m, which corresponds to a pressure of 92,8 kPa (consider the pressure at the tailwater level to be the same). In the first iteration, estimate the value of the loss coefficient of the turbine, including the draft tube, at 0,1, without taking into account the loss in kinetic energy at the outlet of the draft tube. Discuss the influence of the water column in front of the turbine on the dimensions of the turbine.
The solution to the problem is shown in Appendix 2.
Main dimensions of the Francis turbine
The dimensions in figure (b) are in mm.
 WATER TURBINES
page 9.8
Kaplan turbine
Kaplan turbines are only suitable for certain water levels and, in particular, flow rates, because the length of the blades increases with the flow rate. For these reasons, the largest Kaplan turbines do not exceed a capacity of 150 MW. The main advantage of Kaplan turbines is their ability to maintain high internal efficiency (Figure 5) over a wide range of flow rates by turning the stator and rotor blades – in particular, maintaining a small or zero value of the tangential component of the outlet velocity V2θ. Stator blades can be radial or axial – axial especially in small propeller turbines that do not have turning rotor blades.
– 5: –
Internal efficiency of water turbines during flow changes
Internal efficiency of water turbines during flow changes: a-Pelton turbine; b-Kaplan turbine; c-Francis turbine; d-Francis turbine; e-propeller turbine. ηi [1] internal efficiency, which, is referred to as hydraulic efficiency; Q [m3·s-1] volume flow; Qn [m3·s-1] nominal volume flow. Data source [Miller et al. 1972, p. 1237].
Deriaz pump turbines
Deriaz turbine with turning rotor blades is mixed flow type of turbines (Figure 6), is capable of pump operation without changing the direction of rotation, so it is mainly used in pumped storage power plants. Figure 6 shows an example of a Deriaz turbine with a radial stator blade cascade, but they are also used in diagonal stator blade cascades. In pump turbines, it is also necessary to take into account the change in load, especially on the radial bearings, after a change in the operating mode.
– 6: –
Deriaz turbine
 WATER TURBINES
page 9.9
Procedure for proposal of basic parameters of water turbine
Water turbines must be designed in the context of the entire waterworks, but the decisive factor in selecting the most suitable type of water turbine is its specific speed. The direct or indirect design of the meridional velocity and reaction is usually the next step, see Problem 2. The meridional velocity ranges from ~8 to 12 m·s-1, exceptionally even higher, but at the cost of higher losses. Subsequently, other turbine parameters can be determined. The values of key water turbine parameters for the first iteration of the calculation are given, for example, in [Pfleiderer and Petermann, 2005], [Gallano et al., 1998].

Energy balance

A water turbine can be part of various hydraulic systems, usually located between two pressure tanks or atmospheric tanks, as shown in Figure 7. The energy balance of the turbine is calculated between the inlet branch (index i) and the outlet branch (index e), whereby the outlet branche in Francis and Kaplan turbines acts as a draft tube, see the next chapter.

Equation of internal work and losses of water turbine
The internal work of the water turbine wi can be calculated using Equation 7a, which shows that the internal work is greater when the stagnation pressure at the outlet pe and internal losses Lw are lower. The internal losses of water turbines Lw do not exceed 5-7% of the internal work of the ideal water turbine, however, the calculation of the internal efficiency of water turbines often refers to the entire system (section R-T, or i-T), rather than just the section i-e, in which case the losses are higher.
– 7: –
Internal work of water turbine
wi [J·kg-1] internal work; z [m] geodetic height; g [m·s-2] gravitational acceleration; p [Pa] pressure. The index R denotes the reservoir level, the index T denotes the tailwater level, the index s denotes the stagnation pressure, the index i denotes the condition at the turbine inlet, the index 2 denotes the condition at the rotor outlet, and the index e denotes the condition at the turbine outlet respectively the draft tube outlet. The figure shows the dam and powerhouse of the Vranov Dam (Czech Republic). The derivation of the equations is shown in Appendix 3.
 WATER TURBINES
page 9.10

Draft tube

The draft tube is a gradually widening channel in which the stagnation pressure is reduced to the pressure at the outlet ps, e (Figure 8c-d), thereby increasing the internal work of the turbine.

The installation of draft tube allows turbine to be installed above tailwater level
The draft tube also allows the turbine to be placed at a certain height above the tailwater level, thus providing access to the turbine and preventing it from being flooded in the event of flooding and a rise in the tailwater level. The turbine located above the tailwater level without a suction pipe (Figure 8b) reduces its usable available head by the height Δz2-T. By inserting a fully flooded draft tube (Figure 8c), we achieve a reduction in pressure behind the turbine to p2, which, according to the U-tube principle, is lower than the pressure above the tailwater level – so, ideally, the output of the turbine will be the same as if it were located just above the tailwater level.
– 8: –
Description of draft tube operation
(a) turbine is close to tailwater level; (b) turbine is above tailwater level; (c) draft tube reduces pressure behind turbine (case pT=pe); (d) elbow draft tube allows reduction of outlet velocity at turbines (pT<pe). DT-Draft tube. Index 1 indicates the condition in front of the rotor, index 2 behind the rotor.
Decrease in Outlet velocity in elbow Draft tube
The lowest stagnation pressure ps,e, can be achieved with elbow draft tubes (Figure 8d), because the static pressure does not change with the length of the tube (in a straight draft tube, it increases as the draft tube submerges further below the water level) and decreases dynamically as the draft tube widens further.
Draft tubes at Overpressure stages
The condition for using the draft tube is a turbine design that can also use the pressure gradient between the inlet and outlet of the rotor, which can only be achieved by overpressure stages such as Francis and Kaplan turbines. In contrast, draft tube cannot be used with Pelton turbines, and the pressure in the rotor chamber is slightly higher than above the tailwater, see Figure 3.
 WATER TURBINES
page 9.11
Definition of draft tubes efficiency
The efficiency of draft tubes is defined as the efficiency of transformation of kinetic energy into pressure or potential energy (Formula 9); any other transformation is considered a loss. The lower the efficiency of the draft tube, the higher the pressure p2 must be, and thus the lower the internal work of the turbine wi will be.
– 9: –
Definition of draft tubes efficiency
ηDT [1] efficiency of draft tube; LDT [J·kg-1] internal losses in draft tube; V [m·s-1] absolute velocity.
Optimal parameters for draft tube
Internal losses in the draft tube are also influenced by its angle of divergence (10° to 11°). The mouth of the draft tube is always below the water level to prevent suction of air, which means that the pressure pe will always be higher than the pressure pT by the hydrostatic difference between the geodetic heights of points T and e. When assembling the anatomy of flow in the draft tube, one can start from the theory of diffusers, see the article Flow of gases and steam through diffusers [Škorpík, 2023]. The efficiency of draft tubes ranges from 0,7 to 0,8 for conical tubes and from 0,6 to 0,73 for elbow tubes [Kadrnožka, 2003, p. 146].

Cavitation

Cavitation in water turbines occurs where the pressure in the boundary layer of the blades falls below the saturated vapor pressure. The critical point for cavitation is at the trailing edges of the rotor blades, where the velocity is highest and the pressure is lowest, i.e., at radius r2,t. The height between the trailing edges of the blades and the tailwater level Δz2-T has a significant influence on the probability of cavitation occurring in a water turbine.

Methodologies for cavitation prediction
The methodology for predicting cavitation in the water turbine is described in detail in [Pfleiderer and Petermann, 2005, p. 95] – here the authors recommend (p. 113) that the difference 180-β2 should be around 20°. However, the basic condition is that the pressure behind the rotor p2 must be significantly higher than the saturated vapor pressure – in such a case, the water column in the draft tube would also break. The theoretical height of the suction pipe at which, with lossless flow, the saturated vapor pressure is set at the inlet to the draft tube is referred to as the Net Positive Suction Head (NPSH), where the difference between NPSH and the actual height of the draft tube above the water level Δz2-T to the available water column ΔzR-T is referred to as Thoma coefficient σ (Formula 10a), which is used to evaluate the risk of cavitation in water turbines.
 WATER TURBINES
page 9.12
– 10: –
Equation for evaluating cavitation
NPSH [m] net positive suction head of turbine, see Problem 2; NPSHR [m] required net positive suction head of turbine (compared to NPSH, influence of blade wrapping and losses is also taken into account); σ [1] Thoma coefficient; σc [1] critical Thoma coefficients; ωSP [rad·s-1] specific power speed; Pi [W] internal turbine power; ρ [kg·m-3] density.
The Thoma coefficient should be greater than its critical value.
The pressure p2 must be significantly higher than the saturated vapor pressure, because the pressure near the trailing edges of the blades is lower than the pressure p2, see the article Aerodynamics of airfoils [Škorpík, 2022]. This means that the actual or required suction head of the turbine to prevent cavitation must be lower than NPSH. The required suction head is referred to as NPSHR and includes the influence of turbine speed and its losses and is defined by Formula 10b, where σc is referred to as the critical Thoma coefficient. The inequality σ>σc should ensure cavitation free operation of the turbine. Critical Thoma coefficients (Table 11) are a function of the power specific speed ωSP (Formula 11c), which includes the influence of rotational speed and losses in the turbine.
– 11: –
Francis turbines Kaplan turbines
ωSP   σc   ωSP   σc ωSP   σc ωSP   σc
0,4   0,031 0,8   0,089 1,5   0,195 5   2
0,5   0,045 1   0,13 2   0,31 6   3,15
0,6   0,058 1,5   0,26 3   0,67  
0,7   0,071 2   0,41 4   1,15  
Data from [Dixon and Hall, 2010, p. 332].
Cavitation vs. Elbow draft tube
Note that when evaluating the probability of cavitation, the height of the turbine above the tailwater level Δz2-T is very important, which can be reduced by using the elbow draft tube shown in Figure 8d.
 WATER TURBINES
page 9.13

References

ŠKORPÍK, Jiří, 2022, Aerodynamika profilů, fluid-dynamics.education, Brno, https://fluid-dynamics.education/aerodynamika-profilu.html.
ŠKORPÍK, Jiří, 2023, Proudění plynů a par difuzory, fluid-dynamics.education, Brno, https://fluid-dynamics.education/proudeni-plynu-a-par-difuzory.html.
DIXON, S., HALL, C., 2010, Fluid Mechanics and Thermodynamics of Turbomachinery, Elsevier, Oxford, ISBN 978-1-85617-793-1.
GALLANO, Fernando, VEIGA DE OLIVEIRA, Ernesto, PEREIRA, Benjamin, 1998, Layman's handbook, on how to develop a small hydro site, 1998, A handbook prepared under contract for the Commission of the European Communities, Directorate-General for Energy by European Small Hydropower Association (ESHA), DG XVII – 97/010. Dostupné online z http://ec.europa.eu/energy/library/hydro/layman2.pdf.
HOCH, A., 1941, Vynálezy, které změnily svět, Orbis, Praha.
JAPIKSE, David, 1997, Introduction to turbomachinery, Oxford University Press, Oxford, ISBN 0–933283-10-5.
JÍLEK, František, KUBA, Josef, JÍLKOVÁ, Jaroslava, 1980 Světové vynálezy v datech, Mladá fronta, Praha.
JONNES, Jill, 2009, Empires of Light (Říše světla), Academia, Praha, ISBN 978-80-200-1664-5.
KADRNOŽKA, Jaroslav, 2003, Lopatkové stroje, Akademické nakladatelství CERM, s.r.o., Brno, ISBN 80-7204-297-1.
MILLER, Rudolf, HOCHRAINER, A., LÖHNER, K., PETERMANN, H., 1972, Energietechnik und Kraftmaschinen, Rowohlt taschenbuch verlag GmbH, Hamburg, ISBN 3-499-19042-7.
NECHLEBA, Miroslav, HUŠEK, Josef, 1966, Hydraulické stroje, Státní nakladatelství technické literatury, Praha.
PFLEIDERER, Carl, PETERMANN, Hartwig, 2005, Strömungsmaschinen, Springer Verlag Berlin, Heidelberg, New York, ISBN 3-540-22173-5.
SHEPHERD, D., 1965, Principles of turbomachinery, The Macmillab Company, New York.
ŠTĚPÁN, Luděk, KŘIVANOVÁ, Magda, 2000, Dílo a život mlynářů a sekerníků v Čechách, Argo, Praha, ISBN 80-7203-254-2.
©Jiří Škorpík, LICENCE