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MASTER'S THESIS
Uncertainties in Kaplan Cam Curve
Hanna Isaksson2015
Master of Science in Engineering TechnologySustainable Energy Technology
Luleå University of TechnologyDepartment of Engineering Sciences and Mathematics
Abstract The aim of this project was to investigate the sensitivity of the index test with focus on the behavior
of the guide vane and runner blade angles. This was achieved by comparing the cam curves from an
index test with a design of experiment. Index tests are a popular method for determining cam curves
for Kaplan turbines. Index tests, as they are performed today, do not take into account a suspected
hysteresis phenomenon in the guide vane and runner blade mechanism. If and how this affects the
cam curve is today unknown which this project has investigated.
An experiment was designed based on the “design of experiment” method. The difference from a
traditional index testing is that the measurements were performed in a random order and only two
propeller curves were plotted, but with eight replicates. Ten different guide vane angles were used;
five for each runner blade angle and a total of 80 measurements were performed over the span of
two days. Then, the measurements were compared with the results from an index test performed
the day before on the same machine, a fullscale turbine in the Lule Älvriver using the same
equipment.
When analyzing the distribution of the efficiency, it has a larger spread on the left hand side of the
propeller curve compared to the best efficiency point and negligible variation around the best
efficiency point. The difference in the cam curves between the two methods was smaller than the
estimated measurement error at the best efficiency point. The peak efficiency in the experiment was
found for a slightly higher flow rate at one of the runner blade angles but this point was between two
settings at the index test. The variations in efficiency are at their largest before the peak and may
vary between a high efficiency and significantly lower efficiency at this point. Additional
measurements may reduce the risk for choosing a guide vane angle at which such behavior occurs.
A constant difference in the readings between the fixed scale and the station sensor were observed
for the runner blade mechanism. This difference varied with the runner blade angle and amounted to
0.26° for 1° and 0.70° for 4° runner blade angle.
For the guide vane mechanism no hysteresis could be validated. A difference between the fixed scale
and the station sensor which varied depending on the direction was spotted and could be validated
with 95% significance. The difference in the readings from the fixed scale and the station sensor was
in average 0.15° larger when increasing the guide vane angle than when decreasing the angle.
Sammanfattning Målet med detta projekt var att undersöka känsligheten hos indexprov med fokus på löphjulets och
ledskenornas beteende. Detta skulle uppnås genom att jämföra kombineringskurvorna från ett
indexprov och från ett ”design of experiment”. Indexprov är en populär metod för att bestämma
kombineringskurvor till Kaplan turbiner i Sverige. Det sätt som indexprov utförs på idag tar inte
hänsyn till eventuell hysteres i maskineriet för löphjulet och ledskenorna. Om och hur detta påverkar
kombineringskurvan är idag okänt vilket detta projekt undersökt.
Ett experiment utformades efter ”design of experiment” metoden och det skiljde sig från ett
traditionellt indexprov i att mätningarna planerades i en slumpvis ordning och att endast två
propellerkurvor, men med åtta repetitioner istället uppmättes. Tio olika ledskenevinklar användes,
fem för varje löphjulsvinkel och totalt uppmättes 80 mätpunkter under loppet av två dagar. Dessa
jämfördes därefter med mätvärdena från ett indexprov som utfördes dagen innan experimentet på
samma maskin, en fullskalig turbin i Luleälven och med samma utrustning.
Vid analys av fördelningen av mätpunkterna för verkningsgraden så konstaterades en större
spridning vid de låga volymflödena på propellerkurvorna och försumbar variation vid bästa
verkningsgradspunkten. Skillnaden i kombineringskurvorna mellan de två genomförandemetoderna
var vid bästa kombineringspunkten mindre än det normala mätfelet. Toppen på en av
propellerkurvorna återfanns för ett lite högre flöde än vid indexpunkten men denna punkt befann sig
mellan två mätpunkter för indexprovet. Repetitionspunkter för bästa verkningsgradpunkterna på en
propellerkurva bör tas för att undvika att välja en för låg ledskenevinkel då variationen hos
verkningsgraden är som störst före toppen av propellerkurvan.
En konstant skillnad i avläsningarna mellan fasta skalan och stationens givare återfanns för
löphjulsmekanismen. Denna skillnad varierade med löphjulsvinkeln och uppgick till 0,26° för 1° och
0,70° för 4° löphjulsvinkel.
För ledskenorna kunde ingen hysteres valideras. Mellan fasta skalan och stationens sensor
upptäcktes en skillnad som varierade med förändringsriktningen, som kunde valideras med 95 %
signifikans. Skillnaden i avläsningarna från fasta skalan och stationens givare var i snitt 0.15° större
när ledskenevinkeln ökades mot när den minskades.
Preface The work within this report presents my Master Thesis in Sustainable Energy Engineering with
specialization in Energy Optimization and Biofuels at Luleå University of Technology (LTU). The
project was done for Vattenfall with assistance from LTU and Sweco Energuide.
I would like to thank my main supervisors; Prof. Michel Cervantes at LTU, Mr. Mikael Sendelius at
Sweco, Mr. Stefan Sandgren at Vattenfall, and Mr. Magnus Lövgren at Vattenfall. Special thanks to
Prof. Michel Cervantes for introducing me to hydropower and this project. I would also like to
express my gratitude to Ms. Jennie Molin at Sweco for the invaluable help during the experiments
and for practical information about Index tests. Great thanks also to the operators at the station at
which the experiment was performed, and to all employees of Vattenfall, Sweco and LTU that I have
met on different occasions for their friendly reception.
I would also like to thank my family and my friends for their support.
Luleå, September 2015
Hanna Isaksson
Content 1 Introduction ..................................................................................................................................... 1
1.1 Problem description ................................................................................................................ 3
1.2 State of the Art ........................................................................................................................ 3
1.3 Aim of thesis ............................................................................................................................ 3
2 Kaplan turbines ................................................................................................................................ 4
1.1 Cam curves .............................................................................................................................. 9
1.2 Index tests ............................................................................................................................. 13
1.3 WinterKennedy flow rate measurement ............................................................................. 16
3 Design of Experiment .................................................................................................................... 18
3.1 Basic statistical terms ............................................................................................................ 18
3.1.1 Theory of “design of experiment” – factorials .............................................................. 20
3.2 Choice of sample size ............................................................................................................ 22
3.3 Measurement error ............................................................................................................... 23
4 Materials and methods ................................................................................................................. 24
4.1 Experimental Setup ............................................................................................................... 24
4.2 Measurement program ......................................................................................................... 26
4.2.1 Index test ....................................................................................................................... 27
4.2.2 Experiment .................................................................................................................... 27
4.3 Measurement procedure ...................................................................................................... 28
5 Results ........................................................................................................................................... 29
5.1 Statistical analysis .................................................................................................................. 29
5.1.1 Comparability of the results and measurement error .................................................. 29
5.1.2 Sensor comparison ........................................................................................................ 30
5.1.3 Sensitivity to deviations in guide vane and runner blade angles ................................. 36
5.2 Comparison with index test................................................................................................... 39
5.2.1 Index test curve ............................................................................................................. 39
5.2.2 Experiment average curve compared to index test curve ............................................ 40
6 Recommendations......................................................................................................................... 44
7 Conclusions .................................................................................................................................... 45
8 References ..................................................................................................................................... 46
Variable list Variable Symbol Unit
Static head [m]
Headwater level [möh]1
Tailwater level [möh]
Head [m]
Hydraulic losses [m]
Gravity acceleration [m/s2]
Velocity at a point i in the turbine waterways
[m/s]
Correction factor for kinetic energy

The power removed from the water by the turbine
[W]
Density (of water) [kg/m3]
Flow rate [m3/s]
Efficiency 
Net capacity of the turbine [W]
Specific speed [rev/s]
Rotational speed of runner [rev/s]
Power in watt for one horsepower
[W]
Drag force [N]
Drag coefficient 
Average absolute velocity over the runner blade
[m/s]
Length of runner blade [m]
Derivative of the radial position in comparison to the turbine
Radius of the turbine
Lift coefficient 
Lift force [N]
Resulting force acting upon the runner blade
[N]
Pi 
The angle between the average relative velocity and the peripheral velocity over the runner blade
[°]
Peripheral velocity [m/s]
Power developed by the runner blade
[N]
Axial velocity [m/s]
Number of runner blades 
Relationship between number of runner blades and turbine radius
[m]
Attack angle of water flow relative to runner blade
[°]
1 Swedish for meters above sea level
Relative efficiency 
Relative flow rate [m3/s]
Crosssection area at a point i in the waterways
[m2]
Generator power [W]
Generator losses [W]
no load losses [W]
Generator load losses [W]
Generator current [A]
Designed generator current [A]
Magnetic no load losses [W]
Magnetic losses depending on load
[W]
Rotational speeds, two similar turbines or same turbine.
[rev/s]
Turbine (Runner) diameter, if same turbine equal to 1
[m]
Head, same turbine, different heads
[m]
Efficiency 
Flow rate [m3/s]
Power output [W]
Flow coefficient [m7/2/kg1/2]
Differential pressure [mmvp]2
Pressure exponent 
Tangential velocity [m/s]
Radial position relative to spiral casing center
[m]
Pressure [Pa]
Pressure at a position i in the spiral case
[Pa]
Random sample variable 
Mean value 
Variance 
Degrees of freedom DOF 
Sample mean 
Sample variance 
Estimated sample standard deviation

Sum of squares 
Null hypothesis 
Alternative hypothesis 
Type I error 
Type II error 
Power of a test 
Number of replicates 
Number of levels for factor A 
Number of levels for factor B
2 1 mmvp = 9.81 Pa (Alvarez, 2006)
Mean sum of squares for factor i 
Impact of factor divided by the mean sum of squares for the error

Absolute difference between two sample means (twofactor factorial)

Variable used for estimating risk for β error in a twosided ttest

Mean standard deviation 
1
1 Introduction Hydropower or hydroelectric power plants utilizes the kinetic and potential energy in flowing water
to generate electricity through the use of hydraulic turbines. The water makes the turbine rotate and
this energy is then converted into electricity by a generator.
A sketch of a typical hydropower plant is presented in Figure 1.1.
Figure 1.1 – Turbine installation at hydroelectric plant (Krivchenko, 2.1 Turbine head, 1993).
The water enters the plant from the headwater through the intake gate, goes through the pressure
conduit, through the turbine and is then discharged through the draft tube after which it flows out to
the tailwater (Krivchenko, 2.1 Turbine head, 1993).
The difference in elevation between the headwater and the tailwater area is called the static head
and is calculated through equation 1.1, same notation as in Figure 1.1.
1.1
Where and denotes the elevation at the headwater and tailwater water level (Krivchenko,
2.1 Turbine head, 1993) .
The head of a turbine however is the difference in mechanical energy between the inlet and outlet. It
is also called the net head and is specified by equation 1.2, same notation as in Figure 1.1
(Krivchenko, 2.1 Turbine head, 1993).
2
1.2
Where the static head, are the hydraulic losses, the free fall acceleration the velocity
at a point , and is the correction factor for kinetic energy (Krivchenko, 1.2 Energy of fluid (water),
1993)at a point , “0” denotes the intake (Krivchenko, 2.1 Turbine head, 1993).
The power that the turbine removes from the moving water, is expressed by
1.3
Where Q is the volume flow and H is the head (Krivchenko, 2.2 Turbine capacity (Power), 1993).
There are losses between in the turbine accounted in an efficiency η, defined in equation 1.4
1.4
Where is the net capacity of the turbine. By combining equations 1.3 and 1.4, an expression for the
net capacity can be defined as seen in 1.5 (Krivchenko, 2.2 Turbine capacity (Power), 1993)
1.5
The specific speed is used to evaluate the kind of turbine suited for a head and volume flow rate. It is
independent of the turbine diameter and is a characteristic of the turbine type. It is calculated by
equation 1.6 (Krivchenko, 5.8 Specific speed, 1993)
√
√ 1.6
This means, the larger the head the lower the specific speed.
In Sweden no new hydropower plants are built due to regulations and the environmental impact of
establishing new hydropower plants (Vattenfall AB, 2013). Sweden and Norway also have a joint
system to stimulate renewable electricity production consisting in awarding “electricity certificates”
to producers of renewable electricity and by declaring that energy intensive industries and energy
producers have an obligation quota. They must then procure a quota of their
production/consumption in renewable electricity. A new plant or a plant that has increased its
production is eligible for green certificates for 15 years. However not longer than until 2035 when
the system will expire (Energimyndigheten; Norges vassdrags og energidirektorat, 2013).
In order for a hydropower plant in Sweden or Norway to be eligible for a green certificate, the
production increase has to be due to an increase of the average flow rate through the station,
reduced losses in the waterways excluding the turbine, and having reduced the losses in the energy
transformation system. Efficiency improvements due to routine maintenance are not eligible for
certificates (Statens energimyndighets författningssamling, 2011). Therefore, it is of importance to
3
verify that at least one of the three requirements previously mentioned is fulfilled. The verification is
performed with index tests.
1.1 Problem description The Swedish hydropower is to a large extent used in the frequency control of the powergrid (Svensk
Energi, 2014). When a turbine is used for frequency control it is constantly making small changes to
the power output by changing the flow rate of the water going through the turbine, both increasing
and decreasing the flow. The index test performed today only measures in one direction, with
increasing flow in order to avoid hysteresis in the mechanism affecting the result (Bard, Att utföra
indexprov och utvärdera resultaten, 1993).
Kaplan turbines are often used for this frequency regulation in Sweden since they have a high
efficiency over a larger interval compared to other turbine types. The constant change of angles due
to frequency regulation creates wear and tear which may either create or increase the hysteresis
phenomenon (JR. & KAWASAKA, 2014).
Kaplan turbines are run according to a cam curve (Krivchenko, 8.3. Plotting the characteristics of the
adjustable blade turbines, 1994). The question is how the sensitive is the cam curve to the guide
vane and runner blade angles.
1.2 State of the Art Research about uncertainties in efficiency measurements for Kaplan turbines is being and has been
performed during the last couple of years; several articles have been made regarding the scope of
the measurement error for different relative flow measurement methods.
Cervantes, Andrée, Klason and Sundström have investigated available flow rate measurement
methods for low head turbines and their advantages and disadvantages (Cervantes, Andrée, Klason,
& Sundström, 2012).
Nicolle and Proulx have made investigations into the uncertainties of the WinterKennedy flow rate
measurement method, where it was showed that the kcoefficient varies with the guide vane angle
and that the measurements are sensitive to changes in the inlet boundary conditions (Nicolle &
Proulx, 2010).
Directives regarding how to perform index tests to obtain the best results exist as complements to
IEC 60041. One such example is “Index test and best cam curves design procedure for Kaplan
turbines” (JR. & KAWASAKA, 2014).
1.3 Aim of thesis This thesis aims to investigate the sensitivity of the index test with focus on the behavior of the guide
vanes and runner blades. This is to be done by comparing the cam curves from an index test and
from an experiment with measurements performed in a random order. The opening and closing of
the guide vanes and runner blades are also investigated closely.
4
2 Kaplan turbines There are two main types of turbines used for hydroelectric power plants, reaction and impulse
(active) turbines. Differences are aside from the way they work the heads and specific speeds at
which they operate. The most common types are Pelton, Francis, Kaplan and Mixedflow turbines
(Krivchenko, 2.3 Principal turbine kinds, 1993).
Table 2.1 and Table 2.2 indicate the head and specific speed range for different turbines.
Table 2.1 Heads for different types of hydraulic turbines. For turbines with low capacity (100 – 3000 kW) the limits are not corresponding to the data in this table (Krivchenko, 2.4 Applications of turbines of various kinds, 1993).
Turbine type Head (m)
Active Pelton 4001500
Reaction Axial – flow 165 Mixed – flow 40200 Radial – Axial – flow (Francis) 40700
Table 2.2 – Specific speeds for selected turbine types, specific speed calculated with equation 1.6 (Krivchenko, 5.8 Specific speed, 1993).
Turbine type Specific speed, [rev/min]
Adjustable blade axial flow (Kaplan) 1200 – 450
Adjustable blade mixed flow 500 – 300
Radialaxial flow (Francis) 400 – 80
Impulse (Pelton) 50 – 10
Radialaxial, axial, impulse and mixed flow are designation describing how the water enters and
leaves the turbine in question. Figure 2.1 presents the different types.
Figure 2.1 – How water enters and leaves different turbines, a) is an axial flow turbine, b) is a mixed (diagonal) flow turbine, c) radialaxial flow turbine and d) an impulse turbine (Pelton) (Krivchenko, 2.4 Applications of turbines of various kinds, 1993).
A Kaplan turbine is a doubleregulated axialflow turbine, working at heads between 1 and 60 m
(Alvarez, 2006) and at specific speeds between 4501200 rev/s. Kaplan machines are used at low
heads, their popularity stems from that they maintain a high efficiency over a wide operation
compared to singleregulated turbines. In a Francis turbine, only the guide vane opening is
adjustable. In a Kaplan turbine, both the guide vane opening and the runner blade angle are
5
adjustable; doubly regulated (Krivchenko, 2.3 Principal turbine kinds, 1993). The main components of
a Kaplan turbine are viewed in Figure 2.2.
Figure 2.2 – Main components of a Kaplan turbine (Krivchenko, 2.3 Principal turbine kinds, 1993).
1. Runner blades
2. Runner cone (casing for the runner blade)
3. Shaft
4. Turbine case
5. Stayring
6. Wicket gate, guide vanes
7. Top support band of the stayring
8. Lower support band of the stayring
9. Draft tube
The runner diameter characterizes the size of the turbine. There are usually four to eight runner
blades, depending on the head. Larger head leads to more blades. Figure 2.3 shows a picture of a
section of a runner from above. The stayrings purpose is to strength the structure. The wicket gate
consists of 2032 guide vanes arranged in an annular cascade, as can be seen in the BB section in
Figure 2.2. The guide vanes direct the water flow; the velocity vector v0 has the same angle relative
to the radius as the guide vanes. In Figure 2.3, guide vanes and runner blades installed at a
hydropower plant are presented.
6
Figure 2.3 – To the left: Photography of the runner and runner blades of a Kaplan turbine. To the right: Photography of a section of the guide vanes and stayring vanes in a Kaplan turbine. Both pictures are taken at Laxede kraftstation in the Lule Älv River by the author.
The operating gear of the guide vanes serves to adjust the angle of the guide vanes and make sure
that they all have the same angle . There are different technical solutions, the most common
(Krivchenko, 4.1 Guide vane operating gear, 1993) is through a regulation ring, see Figure 2.4.
Figure 2.4  Common operating gear for guide vanes in hydropower plants, regulating ring. (1) is levers set upon the upper pivot point of the guide vanes, (2) is the shackles and pull rods connecting the levers (1) to the regulating ring (3) (Krivchenko, 4.1 Guide vane operating gear, 1993).
The regulation ring ensures that all the levers are turned the same angle. To avoid solid objects in the
waterways damaging the guide vanes or the operating gear, weakened links are incorporated in the
construction. The regulation ring is controlled by hydraulic servomotors that are capable of
developing the forces required to move the wicket gate at smooth and even pace to the right
position (Krivchenko, 4.1 Guide vane operating gear, 1993).
7
Individual servomotors for each guide vane are occasionally used for large highpower Francis
turbines. When using this method, the regulator ring and the weakened links are not necessary, the
construction is thus lighter. The control of the guide vanes becomes more complicated and special
synchronization devices are necessary (Krivchenko, 4.1 Guide vane operating gear, 1993).
One of the most common servomotors used in combination with a regulating ring can be seen in
Figure 2.5
Figure 2.5 – Two servomotor solution for operation of guide vane regulation ring (Krivchenko, 4.1 Guide vane operating gear, 1993).
Servomotors use high pressure oil to create the desired movement by supplying highpressure oil to
either pipe A or pipe B (see Figure 2.5) and connecting the other pipe to the return line. The supplied
oil forces the rods and pistons in the servomotor to move which forces the regulating ring to turn. To
turn the regulating ring in Figure 2.5 clockwise, pipe A should be connected to the highpressure oil
while pipe B should be connected to the return line.
Other configurations are four servomotors; see Figure 2.6 a) and anchorring servomotors Figure 2.6
b). The four servomotors configuration, or “two twinned” as it is also called, requires smaller
diameter on the servomotors and shorter pullrods compared to the two servomotors solution.
The anchorring configurations works in a slightly different fashion from the two previously described
solutions; the moving element has the shape of an anchor ring. The center of the anchor ring
coincides with the regulation ring’s axis of revolution which necessitates a large size. No intermediate
links are necessary to connect the servomotor to the regulation ring (Krivchenko, 4.1 Guide vane
operating gear, 1993).
8
Figure 2.6 – Guide vane operating gear solutions, in a) using four hydraulic servomotors in a twotwinned configuration. b) anchorring servomotors (Krivchenko, 4.1 Guide vane operating gear, 1993).
The operating gear of the runner blades is responsible for setting and maintaining the runner blade
angles which varies in a 3040° interval. The operating rings are the mechanism responsible for this
action. The specifications that the operating rings must meet are high; the blades must be able to
operate despite water pressure, centrifugal and frictional forces, the space in the runner hub is
limited. As repairs require disassembling all of the component units, high reliability is expected.
Figure 2.7 – Runner blade operating gear (Krivchenko, 4.2 Runner blade operating gear of adjustable blade turbines, 1993).
The main components of the operating ring can be seen in Figure 2.7, the components are the
following;
1. Blade flange 2. Blade pivot 3. Housing 4. Housing
5. Lever 6. Pull rod 7. Piston connected to runner servomotor
9
The blade pivot is supported by the two housings. It is moved by the lever and the pull rod by the
piston connected to the runner servomotor. A downward motion of the piston results in a greater
blade angle. A highpressure oil system is responsible for moving and maintaining the desired
position (Krivchenko, 4.2 Runner blade operating gear of adjustable blade turbines, 1993).
Figure 2.8 – Detailed schematic of a solution for the operating gear of the runner blades in a lowhead machine (Krivchenko, 4.2 Runner blade operating gear of adjustable blade turbines, 1993).
There are many different solutions for how the operating gear is constructed; Figure 2.8 shows one
such solution used for lowhead machines. The numbering and the different parts are explained in
the following sentences. (1) is the runner hub to which the runner blades (2) are connected by the
pivot flange (3). The pivot flange (3) is in turn connected to the bearings (4) and (5). Hinge blande
levers (6) and pullrods (9) are connected to the piston (7), that all the blades are connected to the
same piston ensures that all the blades will be turned equally. (8) are cups rigidly connected to the
servomotor piston (7). (10) and (11) are oil supply pipes, (10) to the rod end and (11) to the head end
(Krivchenko, 4.2 Runner blade operating gear of adjustable blade turbines, 1993).
2.1 Cam curves In order for Kaplan turbines to have a high efficiency, the adjustable vanes and blades needs to be
run according a cam curve (Krivchenko, 8.3. Plotting the characteristics of the adjustable blade
turbines, 1994). The cam curve shows the efficiency depending on the flow rate and the
corresponding guide vane angle for each flow rate.
10
Figure 2.9 – Cam curve for a Kaplan turbine. The upper curves are the efficiency depending on the flow rate. The lower curves are the guide vane angle depending on the flow. (Krivchenko, 8.3. Plotting the characteristics of the adjustable blade turbines, 1994)
The cam curve is derived by interpolating between the best efficiency points for propeller curves; see
Figure 2.9, which are explained closely in the following chapter.
A propeller curve is the plot of the efficiency for a certain runner blade angle as a function of the flow
rate, changing with the guide vane opening. It is derived by holding the runner blade angle ϕ at a
constant level while testing different guide vane openings a0 until the combination with the best
efficiency is found (Krivchenko, 8.3. Plotting the characteristics of the adjustable blade turbines,
1994). More about how such measurements is presented later on and in “Field acceptance tests to
determine the hydraulic performance of hydraulic turbines, storage pumps and pumpturbines” (IEC
41, 1991).
The efficiency for a certain runner blade angle varies with the guide vane opening. As a matter of
fact, the circulation between the guide vanes trailing edge and the runner blades leading edge is
created by the wicket gate (Krivchenko, 5.1 Water flow created by the wicket gate of the reaction
turbines, 1993). The shape of the propeller curve is explained by analyzing the forces affecting a
runner blade. In Figure 2.10, the forces acting upon a runner blade are presented.
11
Figure 2.10 – Vector components of forces acting on a runner blade (Cervantes, Propellerkurvor, 2015).
The drag force is defined as
2.1
And the lift force as
2.2
Where is the average relative velocity of and , the average velocity of the water at the
leading and trailing edge of the blade. and are the drag respectively lift coefficients. l is the
length of the blade and is the angle between and the peripheral velocity , see Figure 2.10
(Cervantes, Propellerkurvor, 2015).
The angle at which the sum of the forces and , is directed at is then given by
2.3
The portion of the force acting in the rotational direction is
(
) ( ) 2.4
Where
( )
( ) 2.5
12
The power then developed by the runner blade is
( )
( ) 2.6
The maximum power available per airfoil of thickness is given by a version of Equation 1.5
( ) 2.7
The efficiency can then be expressed as follows
( )
( ) 2.8
2.9
is the number of runner blades, is the axial velocity, is the radius of the turbine and is the
blade pitch (Cervantes, Propellerkurvor, 2015).
The lift coefficient varies with the attack angle according to
The relation is valid for attack angles of which can be verified by looking at Figure 2.11.
Figure 2.11  and derived from experimental data (Cervantes, Propellerkurvor, 2015).
In Figure 2.11, is at its lowest for attack angles in the range ; .
From Equation 2.8 it can be read that CL multiplied with a function of the quota between the drag
and lift coefficient affects the efficiency, an estimation of the combined effect can be seen in Figure
2.12.
13
Figure 2.12 – Estimation of the combined effect of the drag and lift coefficient depending on the attack angle.
The combined effects curve in Figure 2.12 displays a clear peak, just as the propeller curve
(Cervantes, Propellerkurvor, 2015).
2.2 Index tests Kaplan turbines require experiments to determine and/or verify the best combination of guide vane
opening and runner blade angle. There is also a need to verify that a turbine fulfills the specification
given by the manufacturer. These tests are generally called “Field acceptance tests to determine the
hydraulic performance of hydraulic turbines, storage pumps and pumpturbines” and the method for
performing these tests is described in detail in IEC 60041 (International Standard IEC 41, 1991).
An index method test is such a test performed using a relative method for measuring the discharge.
It only gives relative values of the discharge and the efficiency. Their main use is during the
commissioning and operation of the machines. In order for an index test to be deemed acceptable to
use in a field acceptance test, the relative discharge needs to be calibrated by a method accepted in
IEC 60041. They are however accepted for usage in finding the correct relationship between runner
blade angle and guide vane opening in doubleregulated machines (IEC 41, 1991).
In Sweden, index tests are used for
I Improvement quantification after refurbishment
Cam curve adjustment due to wear and tear of the machine.
Finding the cam curve for the operation of the turbine
An index test on a Kaplan turbine usually requires three days; one day for installing and controlling
measurement equipment, one day for the test, and one day for dismantling the equipment. The
workforce needed is one technician for setting up the equipment, collecting and treating the data, an
operator for changing the angles and someone from the power plant to control that prescribed
water levels are kept and to maintain communications with the operations center. A schedule has to
be prepared beforehand with the planned discharges and the time to plan the river discharge (Bard,
Att utföra indexprov och utvärdera resultaten, 1993).
0,6
0,4
0,2
0
0,2
0,4
0,6
0,8
1
5 0 5 10 15
CL·f(λ)
Attack angle [°]
14
The settings should be made based on the fixed scales or other fixed references. This since dials and
displays may change over time and it is important to use the same angles when comparing tests
(Bard, Att utföra indexprov och utvärdera resultaten, 1993).
The changes of the guide vane angle should be made consequently in one direction to avoid
hysteresis. Four to six guide vane angles should be tested for each runner blade angle. Usually the
direction is increasing the guide vane angle. The angle should also be controlled to regulate any drift
before moving on to the next setting. Five to eight runner blade angles should be tested. (Bard, Att
utföra indexprov och utvärdera resultaten, 1993)
The following variables need to be measured according to an “old” Swedish instruction (Bard, Att
utföra indexprov och utvärdera resultaten, 1993);
The relative flow measured with an accepted method, the Winter Kennedy method is one
such method. The use of transparent tubes is recommended so that possible bubbles of air
can be observed and eliminated.
The head, the sensors should be placed at the same place as station sensors.
o Headwater level with a submersible sensor, if only one sensor, after trash rack.
o Tailwater level, if possible with four pressure taps in the draft tube, otherwise with a
submersible sensor placed somewhere where the level is as close as possible to the
level in the draft tube outlet.
Generator power, make sure to measure only the power from the turbine
In practice index test today includes the following measurements (Sendelius, 2015);
Guide vane angle, fixed scales
Guide vane angle, station sensor
Runner blade angle, fixed scales
Runner blade angle, station sensor
Headwater level before trash rack
Headwater level after trash rack
Headwater level, station sensor
Tailwater level
Tailwater level, station sensor
Differential Pressure
Generator Power
Generator Current
Generator Voltage
The station displays for power, current and voltage are also noted along with reactive power
and adjacent turbines power.
The relative flow estimated is described in section 2.3 and thus will not be explained here.
The relative efficiency is calculated by combining Equations 1.4 and 1.5
2.10
15
The net head is calculated by simplifying Equation 1.2 to
2.11
Where zhw is the headwater level after trash rack and ztw is the tailwater level according to either
station sensors or placed sensors.
Using where Ai is the crosssection area of the turbine inlet, respectively the outlet. The
gravity acceleration g is assumed constant. Sometimes, the gross head is used. The only difference
from the net head is that the hydraulic losses in the inlet and the outlet of the turbine are not taken
into account.
2.12
The turbine power is calculated as
2.13
Where Pgen is the generator power and Pgenf the generator losses which in turn is calculated as
(
)
2.14
Where Ptom is the idle running losses, Pbel the operation losses, I the generator current and Im the
generator marked current. However, if there is a magnetic generator on the turbine or the
generator axis the magnetic loss has to be taken into account too (Bard, Att utföra indexprov och
utvärdera resultaten, 1993). Then equation 2.14 becomes as below
( ) (
)
2.15
Where the subtext “mag” denotes the magnetic losses during operation and during idle running.
The magnetic losses are usually stated as one number and then 1/3 of this number is the magnetic
idle running losses and 2/3 magnetic operation losses (Bard, Att utföra indexprov och utvärdera
resultaten, 1993).
In order to compare the results, the flow, the guide vane opening and the power are normalized to
the same head using the similarity laws; (Krivchenko, Turbines of one type and similarity laws for
their modes of operation, 1994)
√
√ √
√ 2.16
(
)
√
√ √
√ 2.17
(
)
√
√ √
√ 2.18
16
Traditionally, the efficiency is assumed unchanged η1/η2 = 1
For statistical calculations a significance level of 95 % should be assumed (IEC 41, 1991).
2.3 WinterKennedy flow rate measurement The WinterKennedy method measures the discharge with the differential pressure in the turbine
spiral case. The discharge is proportional to the differential pressure, see equation 2.19 (IEC 41,
1991).
2.19
Where is the flow constant, is the reading from a manometer, or differential pressure sensor,
and (IEC 41, 1991) n is between 0.4852 depending on the shape of the spiral case (Lövgren, 2015).
The location of the taps is important. In a concrete semispiral, they are placed in the same radial
section. In Figure 2.13 the location of the pressure taps can be seen, the numbers in the following
section refers to this figure. (IEC 41, 1991)
The outer tap, “1” or “1’”needs to be positioned sufficiently far from the corners to avoid
disturbances. The inner tap “2” or “2’” should be located at the inner side, outside the stay vanes and
midway between two adjacent stay vanes and in a flow line. A third tap “3” can also be placed on
one of the stay vanes at the same elevation as the guide vane centerline or at the roof between two
stay vanes. Another set of taps is recommended to be placed in another radial section (IEC 41, 1991).
If the spiral case is made of steel, the same directives apply. However, the taps should not be placed
close to a geometry variation or welded joints. For horizontal spiral cases the taps should be located
in the upper half to simplify the pipes purging (IEC 41, 1991).
17
Figure 2.13Placement of pressure taps in a concrete semispiral case for discharge measurement in turbines using the WinterKennedy method (IEC 41, 1991).
The Winter Kennedy method is based upon the conservation of momentum. When a fluid is flowing
through a curved pipe it is subject to a centrifugal force. This force can, when assuming negligible
radial (with respect to the runner) and vertical components, laminar, steady and nonviscous flow, be
expressed as below
2.20
Where uθ, r, p and ρ are the tangential velocity, the radial position relative to the spiral casing center,
the pressure, and the density, respectively.
Expressing the flow as
2.21
Inserting 2.21 into 2.20 and assuming uθ constant over the radial section the resulting equation is
( )
2.22
Then by integrating equation 2.22 with respect to P and r from point “1” to point “2” in the spiral the
flow rate is given by
√
( )
2.23
And thus the k and h in equation 2.19 are
18
√ ( )
2.24
( ) √ 2.25
The flow coefficient k is typically determined by a model test. By scaling up the model efficiency at
the best efficiency point, the BEP, the prototype efficiency at BEP is determined. The flow rate is
estimated based upon the prototype efficiency, the prototype power output and head at BEP. The k
coefficient can then be determined from the differential pressure at BEP and the theoretical flow
rate. It is not necessary to use the BEP to determine k. Any point can be used, for example the
maximum power output (Cervantes, Andrée, Klason, & Sundström, 2012).
The WinterKennedy method is the most widely used method for flow rate measurements in low
head machines in Sweden because it is simple and cheap.
However, the results may present variation and the reasons are somewhat unclear. The results may
for example show that the efficiency of a turbine has dropped after a refurbishment. This may be due
to old pressure taps, the surfaces of the guide vanes and runner which have been changed between
the measurements. The geometry in the spiral case may be modified, the neighboring turbines
operated at different conditions during the tests. But no systematic error analysis of the method
exists (Cervantes, Andrée, Klason, & Sundström, 2012).
3 Design of Experiment The purpose of this work is to estimate the uncertainty in the determination of a combination curve
on a Kaplan. Special attention will be given to hysteresis phenomenon. The results will be put in
perspective with the conventional index method. The following chapter describes the theoretical
basis for the planning of the measurement procedure.
The first step is to specify the object of the investigation and the factors of interest to be controlled.
In the present case, the efficiency and difference between sensors and fixed scales are the responses
under investigation. The factors affecting the efficiency to be investigated are the runner blade angle,
guide vane angle, and their respective direction of movement, i.e., decreasing or increasing angle to
achieve the desired value. The last parameter in the following sections is referred as “direction of
movement”.
3.1 Basic statistical terms The design of experiment is a method to plan experiments using statistics. A description of statistical
terms used in this report follows in this section, more information may be found in “Design and
Analysis of Experiments (Montgomery, Design and Analysis of Experiments, 2005).
One of the main elements for a properly designed experiment is randomization. This is a necessary
condition in order to be able to make the assumption that the measurement errors are
independently distributed random variables. This can be achieved by using a computer program with
a random number generator to plan the order of the measurements (Montgomery, 23 Sampling and
sampling distributions, 2005).
19
Table 3.1 – Short descriptions of certain statistical terms and associated relationships. The letter “y” denotes a random variable (Montgomery, Simple comparative experiments; 22 Basic statistical concepts, 23 Sampling and distributions, 2005).
Term Symbol Formula Description Equation number
Mean µ
{
∫ ( )
∑ ( )
A measure of a probability distributions central tendency or location.
3.1
Variance
{
∫ ( ) ( )
∑( ) ( )
The variability or dispersion of a probability distribution
3.2
Degrees of Freedom
DOF
The number of independent elements in a sum of squares, n is the number of samples
3.3
Sample mean
∑
The average value of the random variables
3.4
Sample variance
∑ ( )
Dispersion of sample with n measurements
3.5
Sample standard deviation
√
The standard deviation of the sample, more commonly used since it has the same units as y
3.6
Corrected sum of squares
SS ∑( )
Unit used in ANOVA tables.
3.7
From Table 3.1 it can be derived that and are unbiased estimations of µ and , meaning that
they are the average value that the point estimator will assume in the long run. For more information
see (Montgomery, Simple comparative experiments; 22 Basic statistical concepts, 23 Sampling and
distributions, 2005).
One method used for proving or discarding a theory regarding a certain relationship between test
parameters is “hypothesis testing”. The “null hypothesis”, is tested against the alternative
hypothesis . The difference in the mean efficiency of a machine between two different days is an
example.
20
3.8
The hypothesis is tested through a random sample and then rejected or confirmed by computing the
test statistic. The set values in the test statistic for rejecting or failing to reject the null hypothesis are
called the critical region or rejection region for the test. There are two types of error related to
hypothesis testing. Type I error, when the null hypothesis is rejected even though it is true. The
second is type II error, when the null hypothesis is not rejected when it is false. The chance that each
of these errors were to be made is denoted as in equations 3.9 and 3.10
( ) (  ) 3.9
( ) (  ) 3.10
Usually, the “power” of a test is utilized to decide whether to reject or not, see equation 3.11
(  ) 3.11
The normal procedure is to specify the probability for type I error α, the significance level. Then
design the test procedure in a manner such as that the probability of type II error β is suitably small.
Thus if you have a value larger than your selected you cannot reject the null hypothesis . A
common level for α is 0.05, thus giving a significance level of 95 %. The value is typically calculated
using statistical software (Montgomery, Simple comparative experiments; 24.1 Hypothesis testing,
2005).
The excel addin “Analysis Toolpak” can be used for statistical analysis of data in Excel, for example to
generate Analysisofvariance tables, or ANOVAtables as the short name is. The commands that
have been used in this project are AnovaTwo factors with replicates and Anova – One factor.
3.1.1 Theory of “design of experiment” – factorials
For experiments where the effect of several parameters and the interaction between them is of
interest thus a factorial experiment is advised. The parameters are then called factors (Montgomery,
Introduction to Factorial Designs, 2005).
In a factorial design of an experiment, all possible combinations of factors are tested. In the present
study, A is the runner blade angle and a is the number of levels for the blade angle, two. B denotes
the guide vanes angle and b the number of levels for the angle. One replicate of a factorial contains
ab treatments combinations see Table 3.2. The number of replicates is denoted by the letter n.
21
Table 3.2  Possible combinations for a 2 factor experiment with 5 and 2 levels each with n replicates.
A, a=2
B, b=5 a1 a2
b1 a1b11, a1b12, …, a1b1n
a2b11, a2b12, …, a2b1n
b2 a1b21, a1b22, …, a1b2n
a2b21, a2b22, …, a2b2n
b3 a1b31, a1b32, …, a1b3n
a2b31, a2b32, …, a2b3n
b4 a1b41, a1b42, …, a1b4n
a2b41, a2b42, …, a2b4n
b5 a1b51, a1b52, …, a1b5n
a2b51, a2b52, …, a2b5n
How much each factor affects the response is defined as the change in the response when the factor
is changed. Interaction is when the level of one factor affects the result when changing the other
factor. Meaning that if we change A we get different magnitude of the change depending on whether
B is on a low or a high level (Montgomery, Introduction to Factorial Designs, 2005).
There are several advantages with factorials;
They require fewer experiments to determine the same thing as an experiment where each
factor is changed one at the time,
When interactions are present they are considered as necessary in order to prevent
misleading conclusions
Since each factor can be used at several levels the range of the experimental conditions for
which valid conclusions about the effects of a factor can be made is increased.
The analysis of variance table for a twofactor factorial is shown in Table 3.3.
Table 3.3 – Analysis of variance table ANOVA, for a twofactor factorial with the fixed effects model (Montgomery, Introduction to Factorial Designs, 2005).
Source of Variation
Sum of Squares Degrees of Freedom
Mean Square F0
A treatments
∑
B treatments
∑
AB Interaction
∑∑
( )( )
( )( )
Error ( )
( )
Total ∑∑∑
22
Experiments are run sometimes in “blocks” to mitigate the effect of nuisance factors such as
different batches of raw material, different operators or different days. The differences from a
regular factorial are both in how the experiment is designed and in how the data is treated
(Montgomery, Introduction to Factorial Designs, 2005).
The ANOVA in Table 3.4 is for the case of randomized complete block. In a randomized complete
block design, each block contains one complete replicate; the experiment is designed with 4
replicates per block.
Table 3.4  ANOVA for a twofactor factorial in a randomized complete block (Montgomery, Introduction to Factorial Designs, 2005).
Source of Variation
Sum of Squares Degrees of Freedom
Expected Mean Square
F0
Blocks
∑
A treatments
∑
∑
B treatments
∑
∑
AB Interaction
∑∑
( )( )
∑∑ ( )( )
Error ( )( )
Total ∑∑∑
For more information on factorials see for example “Design of Experiments” by Douglas C.
Montgomery.
3.2 Choice of sample size The choice of sample size, i.e., how many measurements and the probability of failing to reject a
false nullhypothesis are closely connected. In most cases, the error decreases as the sample size
increases. The difference between the two samples also affects the sample size, the larger the
difference, the less samples are needed to detect it. (Montgomery, 24.2 Choice of sample size, 2005)
By using “Operating Characteristic Curves for the Fixed Effects Model Analysis of Variance”
(Montgomery, Appendix V. Operating Characteristic Curves for the Fixed Effects Model Analysis of
Variance, 2005), assuming that the two populations under investigation have equal variance and
sample size equation 3.12, the necessary number of samples to achieve a specific β can be
estimated. This method is used for results with more than one variable and multiple levels.
3.12
In equation 3.12, D is the difference in treatments, i.e., between two treatments.
23
  3.13
For comparing a single variable with two levels figure 212 in (Montgomery, 24.2 Choice of sample
size, 2005), the operating characteristic curves for the twosided ttest with α = 0.05 can be used.
The probability for β error is read on the vertical axis and the parameter d on the horizontal axis, see
Equation 3.14. The number of replicates for a specified difference or the reversed can then be read
from where the two conditions intersect. This method also assumes equal variance and sample size
(Montgomery, 24.2 Choice of sample size, 2005).
 
3.14
When several aspects are under investigation, the necessary amount of samples for each aspect
under investigation needs to be calculated separately if they have different means and/or variance.
If the sample variances were to differ the following equation has been used to calculate the average
estimated standard deviation
√
3.15
3.3 Measurement error Index test measurements typically have a measurement error around 0.2  0.3 % on the efficiency
(Sendelius, 2015) (Bard, Att utföra indexprov och utvärdera resultaten, 1993). This error is calculated
according to
√(
√ )
( ) (
√ )
( ) (
( )
)
( ( ))
3.16
For the experiment the measurement error is as follows; by randomizing the experiment the error
can be approximated as random, and it is then calculated according to the Error row in Table 3.3 or
Table 3.4 depending on if blocking was deemed necessary or not. The error is then the mean square
of the sum of squares error for a fixed effects factorial,
( ) 3.17
For a randomized block design the error then becomes
3.18
24
4 Materials and methods
4.1 Experimental Setup The experiment was performed at one turbine at a fullscale hydropower plant in the Lule Älvriver
with the assistance of Mikael Sendelius and Jennie Molin from Sweco Energuide and their
equipment. The changes in guide vane and runner blade angles were performed by two operators at
the station.
The measurement equipment was connected and controlled according to standard procedures
during index tests. The same equipment and placement of the sensors was used during both the
experiment and the index test. The data acquisition system was a 16channel measurement
computer/data logger from Damill AB, the sensors are listed in Table 4.1.
Table 4.1 – The sensors used and their name, serial number and accuracy (Sendelius, 2015).
Measurement parameter
Product Serial number Range
Headwater level before trash rack
JUMO 4 APT26242 4404 4121 0 – 10 bar
Headwater level after trash rack
DRUCK PTX1730 3657131 0 – 7.0 mvp
Tailwater level DRUCK PTX1730 2159721 0 – 7.0 mvp
Differential pressure in the spiral case
Rosemount 3051S1CD2A2A12A1BD1L4M5Q4
9591060 0 – 4000 mmvp, 4 20 mA
Generator power Yokogawa WT 230
(760503C2F/DA12/EC)
91H62988 0 – 1125 W, 1 – 5 V
Generator current 0 – 2.5 A, 1 – 5 V
Generator voltage 0 – 150 V, 1 – 5 V
The Yokogawa power meter had variable settings and the range stated in Table 4.1 were the settings used during the test. The relation between primary and secondary current and voltage were
and
√
√
respectively.
Table 4.2 – Station signals (Sendelius, 2015).
Signal Measurement range Range signal
Station sensor headwater level 74.5 – 78.5 möh 0 – 10 mA
Station sensor tailwater level 43 – 47 möh 4 – 20 mA
Runner blade angle 13° – +11° 4 – 20 mA
Guide vane opening 0 – 100 % 4 – 20 mA
The sampling frequency were 2500 Hz during the test, each 250th sample were registered and saved
in a log file. In order to have stable measuring values a low pas filer of 0.500 Hz l were used.
Two types of cables were used, for longer distances screened cables and for shorter distances
laboratory cables. (Sendelius, 2015).
25
The headwater level sensor before the trash rack was placed next to the fixed scale. The same
location of the sensor was used during the index test made the day before. (Sendelius, 2015).
The headwater level sensor after the trash rack was placed on a ladder in a maintenance pit/shaft of
the intake floor. This location was chosen to measure the head water level according to IEC standard
and also to be able to measure the head losses of the trash rack. The positions of both the headwater
level sensors can be seen in Figure 4.1.
Figure 4.1 – To the left: Location of headwater level sensor before the trash rack, placed next to the fixed scale. Red arrow points to the sensor, orange arrow towards the fixed scale that is partly hidden by the ladder. To the right: Headwater after trash racksensor, visible on the picture is the hatch, the ladder and the cable for the sensor (Sendelius, 2015).
The sensor for the tailwater level was placed at the outlet of the turbine, behind a breakwater pillar,
see Figure 4.2. The sensor was weighted down with an anchor which in turn was secured with two
lines. The sensor was placed behind the breakwater to minimize fluctuations in the readings due to
surface oscillations.
Figure 4.2 – Placement of tailwater sensor (Sendelius, 2015).
In order to estimate the discharge using the WinterKennedy method, the differential pressure across
the spiral needs to be measured. This was done using two pressure taps in the turbine spiral case.
The same taps were used for the index test performed at the station. The pressure taps, the
differential pressure sensor and some control equipment are presented in Figure 4.3 below.
26
Figure 4.3 – The equipment used for measuring the differential pressure in the turbine spiral case. The pressure taps are marked with the blue arrow, the differential pressure sensor with the red arrow and the control equipment with the orange arrow (Sendelius, 2015).
The data acquisition system was connected to the plant/unit governing system to register the
stations reading of the generator power, head and tailwater level, generator current, generator
voltage, runner blade angle and guide vane opening, see Figure 4.4.
Figure 4.4 – Connections to station readings of generator power, generator voltage, generator current, water levels, runner blade angle and guide vane opening (Sendelius, 2015).
4.2 Measurement program This section presents the measurement program for both the experiment and the index test.
Before starting the measurements, the sensor signals were checked and adjusted. The water level
readings from the sensors were first controlled and checked with respect to fixed scales and cross
checked with station readings.
A reference pressure from the differential pressure transducer was taken with the turbine stopped;
this reading was used in the calculation as an offset. The pressure transducer was crosschecked with
known inputs.
27
The station readings of the guide vane opening and runner blade angle were compared with the
values at the operation control center, this to see if there were any fault in the communication
between the power plant and the control center. This type of check was primarily made for
evaluating the index test.
The measurement time was determined by performing a measurement of 10 min. The mean value of
the differential pressure was then computed after 1 min, 2 min etc. When the mean value did not
change by more than 0.1% between two calculations it was assumed that the measuring time was
sufficient. It was observed that after 4 min the results of the differential pressure did not change by
more than 0.1%. This is the reason why 4 min was chosen as a measuring time.
A deviation between the station readings of the runner blade angle and fixed scales was discerned.
Difference was also observed between the station power readings, the dial in the control room and
the power reading from the operations center. The control room dial showed the highest reading,
followed by the stations sensor and the operations center with the lowest reading. An attempt at
adjusting the station display was made before the start of the index test.
After all measurements were completed for the index test and the experiment, the turbine was
stopped. The differential pressure sensor reference pressure was checked. The water levels and the
dryvalue for the sensors were also checked.
4.2.1 Index test
Two of the runner blade openings from the index test performed the day before were chosen. The
guide vane angles were chosen to roughly match the guide vane angles from the previously
performed index test.
The following order was used for the settings during the test:
1) Start with the lowest runner blade angle and the lowest guide vane angle
2) Increase the guide vane angle by 1.53 % until you have a clear peak.
3) Then change the runner blade angle to the second lowest angle and repeat.
4) If the peak is “missed” the guide vane angle is first lowered and then increased again. This was
performed for 6 runner blade angles during the index test.
4.2.2 Experiment
The experiment consisted of 80 measurements with 10 different settings varied according a pre
planned schedule, see Appendix A. The ten different settings consisted of two runner blade angles
and 5 guide vane angles for each runner blade angle.
A decision for not using the “direction of movement” as a factor in the planning of the experiment
was made due to the increased measurement time.
The 10 different settings were chosen based on the index test and wishes from headquarters. The
two runner blade angles were chosen to match angles from the index test. Five guide vane angles
were chosen for each runner blade angle. The best efficiency point and two points “before” and two
points after with roughly 11.5% difference between them. The difference between the guide vane
angles was 23 % in the index test; a smaller interval was chosen for the experiments.
28
Using different guide vane angles for each of the blade angles increases the random error. However,
10 guide vane angles would have resulted in uninteresting combinations and doubling the number of
measurements.
The experiment was planned in two blocks; one for each day with 40 measurements, or 4 replicates.
The design was made to avoid the risk for nuisance factors between the days.
Using two runner blade angles was necessary in order to investigate joint effects for the guide vane
and the runner blade mechanism
The changes were made manually. Some measurement points were supposed to have a positive
change in the angle and ended up with a negative change and vice versa. The precision to set the
angles affects the analysis of the results.
The measurement schedules for day 1 and day 2 can be seen Appendix A
4.3 Measurement procedure The measurements for the index test and the experiment were performed as follows.
1. The operator sets the guide vane or runner blade angle
2. Wait 4 minutes for the flow to settle and reach steady state
3. Start the measurements lasting for 4 minutes (240 seconds). The data acquisition system
measured the following: Water levels from sensors; headwater before trash rack, after trash
rack and tailwater, water levels from station sensors; headwater after trash rack and
tailwater level. Differential pressure in turbine spiral case. Guide vane and runner blade
angles. Generator current, voltage and power.
4. Note the following on sheets
a. Guide vane angle according to fixed scale (both in mm and degrees), see Figure 4.5
b. Runner blade opening according to fixed scale, see Figure 4.5
c. Time the measurements was started
d. Measurement time
e. Measurement number
f. Generator Power according to control room dial
g. Generator Current according to control room dial
h. Generator Voltage according to control room dial
i. The reactive power according to control room dial
j. Adjacent turbine generator power according to control room dial
5. Repeat from point 1.
Figure 4.5 – Fixed scales. Picture to the left: Runner blade angle [°]. Center picture: Guide vane angle [°]. Picture to the right: Guide vane angle [mm] (Sendelius, 2015) .
29
5 Results
5.1 Statistical analysis
5.1.1 Comparability of the results and measurement error
A statistical comparison of the results from the two days was performed for each blade angle in
order to investigate whether or not the resulting efficiencies from day 1 and day 2 could be
evaluated together or would have to be evaluated separately.
The results were evaluated separately for the different blade angles, to avoid any effect on the
results.
The measurement data was ordered according to intended guide vane setting and day for each of
the runner blade angles. Using the “twofactor factorial with reproduction” command in MS Excel, an
ANOVAtable was generated for each of the blade angles.
Table 5.1 – ANOVAtable for comparing differences between days for runner blade angle 1°.
Source of variation
Sum of squares
Degrees of freedom
Mean sum of squares
F Pvalue FCrit
Guide vane setting 0.000645777 4 0.000161444 13.91605 1.57E06 2.689628
Day 7.12228E07 1 7.12228E07 0.061392 0.805997 4.170877
Interaction 9.64059E06 4 2.41015E06 0.207748 0.932106 2.689628
Error 0.000348039 30 1.16013E05
Total 0.001004168 39
The null hypothesis concerning the 1° runner blade angle cannot be rejected as the pvalue in Table
5.1 is 0.806, which is above 0.05. The null hypothesis means no significant difference between the
efficiencies from the two days. The measurements from day 1 and day 2 can then be evaluated
together for a runner blade angle 1°.
Table 5.2 – ANOVAtable for comparing differences between days for runner blade angle 4°.
Source of variation
Sum of squares
Degrees of freedom
Mean sum of squares
F Pvalue FCrit
Guide vane setting 0.000416249 4 0.000104 9.186414 5.71E05 2.689628
Day 1.7153E07 1 1.72E07 0.015142 0.902885 4.170877
Interaction 4.71266E06 4 1.18E06 0.104006 0.980233 2.689628
Error 0.000339835 30 1.13E05
Total 0.000760969 39
For the 4° runner blade angle, the pvalue in Table 5.2 is found to be 0.903 > 0.05. We cannot reject
the null hypothesis for this runner blade angle either.
To check the probability for the β error for both runner blade angles, the Φ2 value is calculated for
both cases according to Equations 3.12 and 3.13 with the values from the operating characteristic
curve (Montgomery, Appendix V. Operating Characteristic Curves for the Fixed Effects Model
30
Analysis of Variance, 2005) This method assumes equal variance of the samples; this was not the case
during the experiment. The variance was different for each of the days for both runner blade angles.
The variance chosen for this analysis was calculated according to Equation 3.15 to avoid optimistic
estimation of β.
5.1
  5.2
√
5.3
Table 5.3  Probability of β error for both runner blade test statistics when deciding whether or not all measurements can be evaluated together.
Angle n a b µ1 µ2 σ D Φ2 Φ v1 v2 β 1° 4 5 2 0.9433 0.9435 3.80E05 2.7E04 39.47 6.3 4 30 <0.01
4° 4 5 2 0.9425 0.9423 2.90E05 1.3E04 14.41 4.1 4 30 <0.01
From Table 5.3 it is found that the β error for both runner blade angles is less than 0.01. Therefore,
the difference in the efficiency function of the day is insignificant.
The random measurement error for the efficiency can be found for each runner blade in Table 5.1
and Table 5.2 to be 1.16E05 for 1° runner blade angle and 1.13E05 for 4° runner blade angle.
5.1.2 Sensor comparison
The readings were summarized as a function of the angle change for both the runner blade and guide
vanes. The fixed scales readings were subtracted with the station sensor readings. In order to
compare the resulting differences statistically, an equal number of measurement values were
necessary. Therefore, a number of values were removed, depending on if the guide vanes or the
runner blades were under investigation. This removal was done by sorting the measurement values
after a column with random values generated through the command “=slump()” and then remove
the bottom values until an equal amount of measurements was achieved.
31
Figure 5.1 – The difference between the fixed scale and the station sensor plotted for each of the two runner blade angles, the measurements are ordered in the internal order that they were performed for each angle.
There is a difference between the magnitudes of the scales for the two runner blade angles
investigated, see Figure 5.1.
The pvalue for a onefactor factorial using the command in MS Excel for the difference between the
different angles is 1.9·10^52. It is thus highly probable that this difference in readings is not only due
to faulty readings of the fixed scale. There is a difference in the angle readings for the station
function of the angle.
This deviation between fixed scale and station sensor for the runner blade is a known problem for
some machines and the ambition is that this deviation is at its worst at the time of the index test.
In order to investigate the runner blade mechanism, the responses for the two sensor outputs
depending on direction of movement were investigated. Since only two runner blade angles were
used for the lower angle, only decrease can be investigated from the lower angle and only increase
from the higher angle. The lower blade angle difference was thus adjusted with the ratio of the
average difference at 4° increase and the average at 1° decrease.
5.4
The measurements were sorted into two columns depending on the runner blade angle and then
into two subcategories depending on whether the angle was changed or left unchanged.
Table 5.4  Average differences between fixed scales and station sensors function of the movement direction of the runner blade angle.
Diff 1° decrease Diff 1° neutral Diff 4° neutral Diff 4° increase
Unadjusted 0.270 0.259 0.704 0.735
1° differences Adjusted
0.735 0.704 0.704 0.735
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
Difference between fixed scales and station sensor
Diff 4°
Diff 1°
32
A statistical analysis was performed, a twofactor factorial anova for the adjusted differences, which
can be seen in Table 5.5.
Table 5.5 – Anovatable for comparing differences between days for runner blade angle 4°.
Source of variation
Sum of squares
Degrees of freedom
Mean sum of squares
F Pvalue FCrit
Neutral or changed position 0.02996 1 0.029959 3.959664 0.050626 3.981896
Runner blade angle 0.00167 1 0.001669 0.22063 0.640063 3.981896
Interaction 0.00097 1 0.000973 0.128602 0.720996 3.981896
Error 0.51449 68 0.007566
Total 0.5741 71
The pvalue for the difference depending on the runner blade angle is larger than the predecided
significance α = 0.05. Thus no significant difference between the adjusted difference for 1° and 4°
runner blade angle can be discerned. There is a difference in the readings between when the runner
blade was changed and when it was not which is on the verge of being significant, P=0.0506. This
may be due to the angle which drifted during the measurements and was closer to the read value on
the fixed scales after one or more measurements.
The probability for β error was investigated for the difference between the angles, once again using
Equations 3.12 and 3.13 and assuming equal variance. The variance was calculated through Equation
3.15.
5.5
    5.6
√
√ 5.7
Table 5.6  Probability of β error when evaluating differences in readings from fixed scales and station sensor for the runner blade.
Aspect under investigation
n a b µ1 µ2 Φ2 Φ v1 v2 β
Difference between 1° and 4°
18 2 2 0.7105 0.7202 0.0120 5.76 2.4 1 68 0.07
From Table 5.6 the β error probability derived from the experiment was 0.07 for 18 replicates. This is
slightly high to draw any conclusions about the adjusted difference. In summary, 18 replicates are
too few to investigate the adjusted difference, 22 replicates are necessary in this case.
33
The difference in the readings between the fixed scales and the station sensor for the guide vanes is
presented, see Figure 5.2.
Figure 5.2 – Fixed scale and station sensor readings for the guide vanes arranged after internal order of measurement for each angle each day.
For the guide vanes, the measurements with no change were not considered because only five
measurement points were available. The major interest was in the vanes response to an input.
The differences were sorted into two columns based upon the blade angle increase “+” or decrease
““. The resulting two columns were not equal in length. Four measurement values were selected
randomly and removed in the same way as for the runner blades. A onefactor ANOVA was executed,
and yielded the results in Table 5.7 and Table 5.8.
Table 5.7 – One factor ANOVA table for comparing the differences between the fixed scale and the station sensor depending on if the guide vane angle was increased or decreased.
Source of variation
Sum of squares
Degrees of freedom
Mean sum of squares
F Pvalue FCrit
Increase or decrease 0.377134 1 0.377134 9.959529 0.002411 3.986269
Error 2.499198 66 0.037867
Total 2.876332 67
Table 5.8 – Summary of teststatistics from the data used in the ANOVA in Table 5.7.
SUMMARY Groups Number Sum Mean value Variance
+ 34 11.37856 0.334664 0.042709
 34 6.31446 0.185719 0.033024
24
25
26
27
28
29
30
31
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77
Guide vane angle [°]
Order, arranged after day and position
Fixed scale
Sensor
34
By reading the pvalue from Table 5.7 to 0.0024 < 0.05 it is possible reject the nullhypothesis. The
probability for the β error was estimated by calculating the dvalue through Equation 3.14, using the
test statistics from Table 5.8. Since the variances differed a mean variance was calculated through
Equation 3.15.
 
5.8
√
√ 5.9
From the operating characteristic curve for a two sided ttest, with 34 replicates, the β error is below
0.05. The minimum number of replicates is according to the same operating characteristic curve 15.
Therefore, it is likely to be a difference in the response of the guide vane sensors depending on the
direction of movement. The different readings vary roughly 0.15° as a function of the guide vanes
movement direction.
This sensor difference when to the right of the propeller curve does not have a significant effect as
the efficiency drop is insignificant.
A difference in the sensor readings has been spotted. To determine if this difference is or not just
due to different placement of the sensors, the square root of the differential pressure has been
compared with the readings from both sensors. The square root of the differential pressure is used
since the flow rate is calculated by taking this multiplied with a constant, see Equation 2.19.
Figure 5.3 – Square root of differential pressure versus guide vane angle for fixed scale (black) and station sensor (grey) for 1° runner blade angle.
In Figure 5.3, the difference between the sensors can be spotted. Furthermore, the same guide vane
angle may result in different differential pressures and vice versa. For reference, a difference of 0.3 in
the square root of the differential pressure represents a difference of 2.1 m3/s in flow rate.
24,5
25
25,5
26
26,5
27
27,5
28
37 37,5 38 38,5 39 39,5
Gu
ide
van
e an
gle
[°]
Square root of differential pressure reading
diffp vs guide vanes fixed, alfa 1
diffp vs guide vanes station, alfa 1
35
Figure 5.4 – The square root of the differential pressure versus the fixed scale guide vane angle readings. Light grey squares and dark grey rhombs are readings when decreasing and increasing the guide vane angle, respectively.
In Figure 5.4 no clear trend between differential pressure and direction of movement is found.
Figure 5.5  The square root of the differential pressure versus the station sensor guide vane angle readings. Light grey rhombs and dark grey are readings when decreasing and increasing the guide vane angle, respectively.
In Figure 5.5 no clear trend of behavior due to the guide vanes movement direction can be spotted.
24
25
26
27
28
29
30
31
37 38 39 40 41 42 43 44 45 46
Gu
ide
van
e an
gle
[°]
Square root of differential pressure reading
diffp vs guide vanes fixed +
diffp vs guide vanes fixed 
24
25
26
27
28
29
30
31
37 38 39 40 41 42 43 44 45 46
Gu
ide
van
e an
gle
[°]
Square root of differential pressure reading
diffp vs guide vanes station +
diffp vs guide vanes station 
36
Both Figure 5.4 and Figure 5.5 display variation in differential pressure for the same guide vane
angle, especially for high guide vane angles. This variation can be seen for low guide vane angles too
(25° and 27.5°) for the station sensor but not in the same extent for the fixed scale readings. Worth
notice is the clear separation between the two runner blade angles and the resulting difference in
discharge between runner blade angles.
Figure 5.6 – Fixed scale readings on the vertical axis and station sensor readings on the horizontal axis. Light grey and dark grey are decreasing and increasing movement of the guide vane angle, respectively. The reference line marks the line at which the station sensor and the fixed scale display the same value.
In Figure 5.6 almost all readings follow a linear pattern. For a majority of the measurements where
the vane angle was decreased the two readings match. This is not true for the measurements where
the angle was increased.
5.1.3 Sensitivity to deviations in guide vane and runner blade angles
The intention was to repeat the same measurement point several times. However, the
measurements were performed with deviation in the settings. The results can however be used to
evaluate the distribution of the efficiency during repeated measurements.
The efficiency was calculated according to Equation 2.10
5.10
The density ρ was assumed to 998.98 kg/m3 and the gravity acceleration constant g to 9.81 m/s2. The
turbine power, head and flow rate were calculated as follows.
The flow rate for each measurement point was calculated according to Equation 2.19
24
25
26
27
28
29
30
31
24 25 26 27 28 29 30 31
Fixe
d s
cale
[°]
Station sensor [°]
+

Reference
37
5.11
The flow coefficient k and the exponential factor n were decided by the consultants from Sweco and
h was the measured differential pressure in the spiral case.
The head used for calculating the efficiency was the gross head, Equation 2.12
(( )
(
)
) 5.12
Where A0 and Atw were obtained from the blueprints, the flow rate calculated as in Equation 2.19, zhw
and ztw were measured at the headwater after trash rack and tailwater, respectively.
The turbine power was calculated according to Equation 2.13
5.13
Where Pgen was the measured generator power. Pgenf is the generator loss calculated by Equation
2.15
( ) (
)
5.14
Where Ptom, Ptommag, Pbel, Pbelmag and Im were known. The generator current I was measured.
The head was normalized to the rated head. The flow rate was normalized according to the similarity
laws, Equation 2.18. Since A turbine diameter ratio of 1 was assumed because the flow rate was
normalized.
(
)
√
√ √
√ 5.15
The resulting efficiency for the runner blade angle read on the fixed scale is presented in Figure 5.7.
All efficiencies have been normalized for confidentiality reasons.
38
Figure 5.7  Normalized efficiency function of the normalized flow rate (left vertical axis) for two blade angles: triangles for 1° and squares for 4° blade angle. The fixed scales readings of the runner blade angle (right vertical axis) have the rhomb symbol.
In Figure 5.7, the efficiencies vary more for low flow rates as the runner blade angle varies with
about 0.2 degrees. The variation of the efficiency around the peak of the propeller curve is barely
larger than the usual measurement error. No difference in the variation of the runner blade angle at
the peak of the propeller curve compared to the rest of the curve can be discerned. The variation in
the runner blade angle does not seem to be the major factor for the efficiency.
The distribution of the efficiency appears to be dependent to a large extent on the flow rate. The
guide vane angle, controlling the flow rate, is now investigated.
0,5
1
1,5
2
2,5
3
3,5
4
4,5
0,98
0,985
0,99
0,995
1
1,005
1,01
250 260 270 280 290 300 310 320 330
Fixe
d s
cale
ru
nn
er
bla
de
an
gle
[°]
Effi
cie
ncy
[%
]
Flow [m3/s]
Blade angle 1
Blade angle 2
Runner blade angle
39
Figure 5.8 – Normalized efficiency as a function of the normalized flow rate (left vertical axis) for each of the blade angles: triangles for 1° and squares for 4° blade angle. On the right vertical axis are the guide vane angle readings of the fixed scales marked with the rhomb symbol.
The major factor affecting the variation in the guide vane angle appears to be the position on the
propeller curve, see Figure 5.. For the lowest flow rates at both runner blade angles, the extreme
values differ more than 1 % in some cases. At the other side of the curves, at flow rates of 270 m3/s
and 310 m3/s the difference is instead on the scale of 0.3 %.
When comparing the variation of the guide vanes and the variation of the efficiency in Figure 5. with
the results regarding the variation of the response of the guide vane angle this variation is smaller
than the usual measurement error at index tests.
It is possible that there is an efficiency loss during the operation of the turbine due to the difference
in readings between fixed scales and station sensors for the runner blades. The difference and
subsequent control error results in a lower flow rate than the one intended to be used for 4° runner
blade angle, thus the efficiency may be significantly lowered. How much and if this assumption is
correct cannot be determined from the data of this experiment.
5.2 Comparison with index test
5.2.1 Index test curve
The efficiency and the flow rate were calculated in the same way as the efficiencies from the
experiment in the previous chapter. The difference is that the values used were measured in a
sequential order during the index test. Only the two angles also measured during the experiment are
analyzed.
In Figure 5.9 the index test propeller curves are presented.
24,7
25,7
26,7
27,7
28,7
29,7
30,7
0,98
0,985
0,99
0,995
1
1,005
1,01
250 260 270 280 290 300 310 320 330
Fixe
d s
cale
gu
ide
van
e a
ngl
e [
°]
Effi
cie
ncy
[%
]
Flow [m3/s]
Blade angle 1
Blade angle 2
Guide vane angle
40
Figure 5.9 – Propeller and guide vane versus flow curves from the index test. The propeller curve depicting the efficiency is marked by rhombs and the curve showing the guide vane angle corresponding to each flow is depicted with squares.
In Figure 5.9, a clear peak for the efficiency and approximately linear relationship between the flow
and the guide vane angle are observed.
5.2.2 Experiment average curve compared to index test curve
The propeller curves from the index test are compared to the average value obtained from the
experiment.
The average value curve was derived by taking the average values for each intended setting. An
average cam curve for the experiment measurements can be derived. The settings were however not
exact and some of the measurements ended up in another setting interval than the one intended.
The cam curve derived from the experimental values is only intended to be used for comparison with
the index curve.
60
65
70
75
80
85
0,986
0,988
0,99
0,992
0,994
0,996
0,998
1
1,002
1,004
1,006
250 260 270 280 290 300 310 320 330
Gu
ide
van
e a
ngl
e [
%]
Effi
cie
ncy
[%
]
Flow [m3/s]
Eff index
Guide vane index
41
Figure 5.10 – Cam curve comparison between index test and experiment. Rhombs for the efficiency and squares for the guide vane angles at the index test. The results from the experiment; the average efficiency is marked by crosses and the average guide vane angle by dots. The guide vane angle reading is from the station sensor for both index test and experiment.
In Figure 5.10, for a runner blade angle of 1°, the curves have almost the same shape, except the
peak efficiency appears for a slightly higher flow rate. For a runner blade angle of 4°, the peaks
coincide, although the experiment curve has two points with almost the exact same efficiency. A
sharper efficiency slope for low flow rates is observed; slope smaller for higher flow rates compared
to the index test. The flow rates and guide vane angles appears to coincide fairly well for both runner
blade angles but better for the lower runner blade angle. For the second runner blade angle, the
guide vane angles are lower than during the index test, except for the first one. The two methods
yielded no difference in efficiency. The measurement error for the efficiency is typically considered
to be around 0.2 % and the difference between the peak efficiencies is smaller than this.
The index test curves are now plotted with all resulting efficiencies and guide vane angles from the
experiment.
62
67
72
77
82
87
92
97
0,986
0,988
0,99
0,992
0,994
0,996
0,998
1
1,002
1,004
1,006
250 260 270 280 290 300 310 320 330
Gu
ide
van
e a
ngl
e [
%]
Effi
cie
ncy
[%
]
Flow [m3/s]
Eff index
average eff exp
Guide vane index
average guide vanes exp
42
Figure 5.11 – Index test curve compared to all measurements from experiment.
Comparing Figure 5.10 with Figure 5.11, the efficiency at the high flow rates for the second runner
blade angle are low, and the efficiency at the lowest flow rate were among the larger values. This is
confirmed further by looking at Figure 5.12, where black bars represent the standard deviation in
flow rate, efficiency and guide vane opening.
Figure 5.12 – Same curves as in Figure 5.10 but with black bars to represent the standard deviation. For the efficiency curve the vertical error bar is the standard deviation in efficiency and the horizontal error bar is the standard deviation for the flow rate. For the guide vanes the vertical error bar is the standard deviation in guide vane angle and the horizontal bar is the standard deviation for the different flows used at that point.
0
20
40
60
80
100
0,98
0,985
0,99
0,995
1
1,005
1,01
250 260 270 280 290 300 310 320 330
Gu
ide
van
e a
ngl
e [
%]
Effi
cie
ncy
[%
]
Flow [m3/s]
Eff index
eff exp
Guide vane index
guide vanes exp
63
68
73
78
83
88
93
98
103
108
0,985
0,987
0,989
0,991
0,993
0,995
0,997
0,999
1,001
1,003
1,005
260 270 280 290 300 310 320
Effi
cie
ncy
[%
]
Flow [m3/s]
Eff index
average eff exp
Guide vane index
average guide vanes exp
43
The standard deviation in guide vane opening is small since the black bars are difficult to discern in
the figure. The standard deviation in flow rate and efficiency is however larger.
In Figure 5.12, the shape of the experiment test propeller curve is closer to the theoretical shape
than the index test propeller curve because the slope is flatter after the best efficiency point.
The variation in head on the efficiency has not been investigated. Today the efficiency is assumed to
be constant when normalizing the flow with respect to the head variation. It is possible that some of
the variation in the distribution of the efficiency is due to this variation.
44
6 Recommendations Regarding the difference in the guide vane sensors more investigations are necessary for two
reasons; the station sensors calibration was not taken into account, and if this phenomenon is
present in other machines.
As for the difference between fixed scales and stations sensors for the runner blade, the station
sensors need to be recalibrated. Furthermore, a test of 22 replicates is necessary for the angles
variation to determine with certitude if the sensor response varies with the direction of movement.
Also of interest, is to investigate the relationship of the difference in the readings with respect to the
runner blade. The number of measurements necessary to assess this question is possible to find
using my measurement data. Index test data can be used for deriving this relationship if required
number of measurements is less than five.
More measurement points around the expected BEP during index tests may be of use to avoid
choosing a guide vane angle and flow rate at which the efficiency varies significantly.
A final recommendation is to investigate the relation between the distribution of the actual head and
the efficiency. This may allow to validate the constant efficiency assumption made when normalizing
the flow rate to one head using the measurement data collected during the experiment.
45
7 Conclusions The station sensor for the runner blade angle is in need of an adjustment. By looking at the sensor
outputs, no hysteresis phenomenon resulting in different responses function of the blade movement
(increased or decreased angle) could be seen; the results are inconclusive.
A sensor difference exists for the guide vane mechanisms. The difference between the fixed scale
and the station sensor was larger when the guide vane angle was increased, resulting in an average
of 0.1° difference between closure and opening.
The variation in the efficiency around the best efficiency point was negligible. The result from the
different measurement methods appears to give roughly the same result and therefore it shouldn’t
be necessary to change the method for the index test. An option may however be to make repetition
points if the initial values for a propeller curve are higher than expected since they seem to vary
more for low flows.
Furthermore, it is worse if the angle deviation for the guide vanes result in smaller than intended
flow rates since the peak is sharper for low flows.
46
8 References Alvarez, H. (2006). Energiteknik Del 1. Lund: Studentlitteratur.
Bard, A. (1993). Att utföra indexprov och utvärdera resultaten. Stockholm : STF ingenjörsutbildning
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Cervantes, M. (2015, March). Propellerkurvor. (H. Isaksson, Interviewer)
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Pumps (ss. 813). Boca Raton: Lewis Publishers.
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47
Krivchenko, G. (1993). 4.1 Guide vane operating gear. i G. Krivchenko, Hydraulic machines: Turbines
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Appendix A Measurement schedules for the experiment
Dag 1
Provpunkt Pådrag, position
Löphjul [°] Check
Provpunkt
Pådrag, position
Löphjul [°] Check
1 11 1
21 24 4
2 24 4
22 21 4
3 13 1
23 22 4
4 23 4
24 21 4
5 25 4
25 22 4
6 14 1
26 23 4
7 15 1
27 11 1
8 13 1
28 12 1
9 15 1
29 14 1
10 22 4
30 23 4
11 21 4
31 13 1
12 24 4
32 25 4
13 22 4
33 11 1
14 12 1
34 13 1
15 23 4
35 21 4
16 14 1
36 11 1
17 14 1
37 12 1
18 15 1
38 24 4
19 12 1
39 25 4
20 25 4
40 15 1
Dag 2
Provpunkt Pådrag, position
Löphjul [°] Check
Provpunkt
Pådrag, position
Löphjul [°] Check
1 22 4
21 14 1
2 14 1
22 13 1
3 15 1
23 12 1
4 11 1
24 22 4
5 23 4
25 22 4
6 21 4
26 25 4
7 15 1
27 14 1
8 15 1
28 23 4
9 13 1
29 13 1
10 11 1
30 21 4
11 22 4
31 25 4
12 23 4
32 25 4
13 12 1
33 11 1
14 11 1
34 14 1
15 12 1
35 24 4
16 21 4
36 13 1
17 15 1
37 24 4
18 25 4
38 23 4
19 21 4
39 24 4
20 12 1
40 24 4