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Pelton Wheel Performance
Experiment Findings · April 2017
DOI: 10.13140/RG.2.2.30344.93442
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Pelton Wheel Performance
MECN3007 - Mechanical Engineering Laboratory
Tumisang Kalagobe
Student number: 800363
A project report submitted to the Faculty of Engineering and the Built Environment,
University of the Witwatersrand, Johannesburg, in partial fulfilment of the
requirements for the degree of Bachelor of Science in Engineering.
Johannesburg, April 2017
University of the Witwatersrand, Johannesburg
School of Mechanical, Industrial & Aeronautical Engineering
INDIVIDUAL DECLARATION WITH TASK SUBMITTED FOR ASSESSMENT
I, the undersigned, am registered for the course MECN3007 - Mechanical Engineering Laboratory in the year 2017. I herewith submit the following task ”Pelton
Wheel Performance” in partial fulfilment of the requirements of the above course.
I hereby declare the following:
ˆ I am aware that plagiarism (the use of someone else’s work without their permis-
sion and / or without acknowledging the original source) is wrong;
ˆ I confirm that the work submitted herewith for assessment in the above course
is my own unaided work except where I have explicitly stated otherwise;
ˆ This task has not been submitted before. either individually or jointly, for any
course requirement, examination or degree at this or any other tertiary educational institution;
ˆ I have followed the required conventions in referencing the thoughts and ideas of
others;
ˆ I understand that the University of the Witwatersrand may take disciplinary
action against me if it can be shown that this task is not my own unaided work
or that I failed to acknowledge the sources of the ideas or words in my writing in
this task.
Signed this 3rd day of April 2017
Tumisang Kalagobe
800363
i
Abstract
A performance test was conducted for a pelton wheel turbine with the objective of
establishing the relationship between the mechanical efficiency and the velocity ratio. A
literature review found a number of useful mathematical models and some key physical
phenomena.
The performance test was conducted for a constant hydraulic head and flow rate while
varying the braking force. The raking force was gradually increased until the pelton
wheel stopped. Readings of the angular velocity and pressures were taken an processed
using the equations found in the literature review.
A parabolic relationship between the the efficiency and the velocity ratio was established and the peak efficiency was found. Some physical phenomena relating to the
flow field in the bucket were identified and analysed.
ii
Contents
Declaration
i
Abstract
iii
Contents
iv
List of Figures
vi
List of Tables
vi
1 Introduction
1
1.1
Background, motivation and literature review . . . . . . . . . . . . . .
1
1.2
Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2 Apparatus
3
2.1
Pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.2
Pressure transducer . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.3
Pelton wheel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.4
Spear valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2.5
Prony brake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
3 Methodology
4
3.1
Precautions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
3.2
Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
iii
4 Observations and data processing
6
4.1
Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
4.2
Data processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
4.3
Uncertainty analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
5 Results
8
6 Discussion
8
7 Conclusions
10
A Images
11
A.1 Bucket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B Methodology
B.1 Pump start up
11
11
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
B.2 Calibration of the digital pressure readings . . . . . . . . . . . . . . . .
12
B.3 Pump shut down . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
C Sample calculations
13
D Uncertainty analysis
15
E Health and Safety Risk Assessment
16
iv
List of Figures
1
Schematic of the pelton wheel experimental set up . . . . . . . . . . . .
3
2
Running plot of pelton wheel RPM vs 100N proney brake gauge reading
6
3
Running plot of pelton wheel RPM vs 250N proney brake gauge reading
7
4
Efficiency vs velocity factor . . . . . . . . . . . . . . . . . . . . . . . .
9
5
Pelton wheel bucket control volume [4] . . . . . . . . . . . . . . . . . .
11
6
Calibration graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
7
Health and safety assessment
. . . . . . . . . . . . . . . . . . . . . . .
16
Uncertainties of measurement equipment . . . . . . . . . . . . . . . . .
15
List of Tables
1
v
1
Introduction
1.1
Background, motivation and literature review
The pelton wheel is an impulse turbine that can reach initial pressure heads of more
than 50m [1]. Pelton wheels are generally used in locations where high water heads
are constantly available due to the solar energy driven water cycle [2].
The extractable power in a pelton wheel turbine is limited by the power of the input
jet of water. The power of the input jet can be found by Equation 1 [2].
Pin = ρgHQ
(1)
Pin is the input power in Watts, ρ is the density of the liquid in kg/m3 , g is the
acceleration due to gravity in m/s2 , H is the total head of the system in m and Q is
the flow rate of the fluid in m3 /s.
Aagr [2] found that a pelton wheel of diameter 70mm and a flow rate of 0.17l/s yielded
an efficiency of 0.47 ± 0.02. Inefficiencies in modern turbines are largely due to the
friction between moving components and leakages [3], and as such the output power is
always expected to be below the input power due to various factors.
The power delivered to the turbine by the jet can be quantified by using the relationship
between power, torque and angular velocity, shown in Equation 2.
Pwheel = Tjet ω
(2)
Pwheel is the power of the pelton wheel in Watts, Tjet is the torque delivered to the
pelton wheel by the input jet of water in N · m and ω is the angular velocity of the
pelton wheel in rad/s.
Potter [potter] showed that Equation 2 can be specified to a pelton wheel turbine
by performing a control volume analysis on a single bucket, as shown in Figure 5,
Appendix ??. This leads to Equation 3 [4].
Pwheel = ρQu(V1 − u)(1 − cos β2 )
(3)
u is the tangential velocity of the pelton wheel in m/s, V1 is the jet velocity in m/s
and β2 is the angle at which the water deflects off the bucket in radians.
1
A prony brake is used to measure the power that that is available at the output shaft
of the turbine [2]. This power can be found through the same power relationship as in
Equation 2. The only difference is that the output torque is that which is measured
by use of the prony brake.
Pout = Tbrake ω
(4)
Power losses due to friction, leakages, etc. make it necessary to evaluate the amount
of useful power out against the input power. The efficiency of a mechanical system is
defined as the ratio between the output and input powers of the system, as shown in
Equation 5.
Pin
(5)
Pout
In the specific case of the pelton wheel Equation 5, Equation 3 and Equation 1 can be
η=
combined and simplified to achieve Equation 6 [4].
η = 2φ(Cv − φ)(1 − cos β2 )
(6)
φ (Equation 7) is the speed factor and is defined as the ratio between the tangential
velocity (u) and the jet velocity (V1 ) while Cv is the nozzle efficiency.
φ=
rω
√
Cv 2gH
(7)
For a constant pump pressure the jet velocity remains constant, thus the speed factor
becomes only a function of the tangential velocity.
The maximum efficiency of the turbine can be found by taking the first derivative
of Equation 6 and setting it equal to zero. By this method it can be seen that the
maximum mechanical efficiency is expected to occur at the point where φ ≈ Cv /2.
1.2
Objectives
The objectives of this experiment are as follows:
1. To determine the variation of mechanical efficiency with flow rate and angular
velocity.
2. To compare the performance of the pelton wheel against the theoretical background.
3. To determine the maximum efficiency of the pelton wheel.
2
2
Apparatus
A basic schematic of the experimental set up is shown in Figure 1.
Figure 1: Schematic of the pelton wheel experimental set up
2.1
Pump
In normal pelton wheel applications there are usually naturally occurring high heads
from mountainous areas. As this experiment is conducted in a city where there are no
naturally occurring high heads, a pump is used to draw water from a reservoir that is
located within the laboratory.
2.2
Pressure transducer
A pressure transducer is located near the pump and is used to measure the input
pressure of the system. This transducer’s readings need to be calibrated as it does
not give pressure readings. The readings are simply arbitrary values with no unit of
measurement.
2.3
Pelton wheel
An Armfield pelton wheel of diameter 244mm is used in this experiment. This pelton
wheel is rated to output 1.56kW of power when there is a head of 30.5m being provided
3
at the inlet and the wheel is rotating at 800rpm. The housing of the pelton wheel has
a two analogue dials where one measures the angular velocity of the wheel in rpm and
the other measures the pressure at the nozzle inlet in kPa. It must be noted that the
buckets have an angle of 150o .
2.4
Spear valve
The spear valve is used to regulate the input water jet’s flow rate. The spear valve
accomplishes this through manually tightening and loosening it as necessary.
2.5
Prony brake
The prony brake is used to measure the output load of the pelton wheel. This load
would be the useful load in the case of an actual pelton wheel power plant. The prony
brake makes use of a belt that is tightened around a drum that is attached to the
pelton wheel’s output shaft. The drum has a diameter of 304.5mm and is hollow so
that cooling water can be added to avoid heat damage due to friction.
3
Methodology
The procedure for this experiment is quite extensive and as such Appendix B contains
some of the steps that are required to carry out the experiment.
3.1
Precautions
There are some steps that must be taken before and during the experimentation process
in order to ensure the safety of everyone that is present in the laboratory.
ˆ Ensure that the drum coolant region has a constant supply of water as the friction
between the belt and the drum is a fire hazard.
4
ˆ Do not add too much liquid to the drum as it could splash onto electrical com-
ponents.
ˆ Ensure that any persons near moving machinery are not wearing any loose items
of clothing and/or making direct physical contact with the aforementioned moving machinery.
ˆ There must be a constant water supply to the pump in order to avoid cavitation.
ˆ Ensure that there are no exposed electrical wires near the water.
3.2
Procedure
The following procedure is used to take the necessary readings for the experiment:
1. Calibrate the pressure transducer (please refer to Appendix B for this procedure).
2. Open the two valves just below the transducer and above the pump.
3. Open the spear valve to the desired flow rate and wait for the pelton wheel to
reach equilibrium.
4. Record the motor input pressure, nozzle analogue pressure and the angular speed
of the pelton wheel.
5. Tighten the proney brake until the 250N gauge moves by 2N.
6. Record the loads and the new angular velocity of the pelton wheel.
7. Repeat steps 5 and 6 until the brake is exerting enough force to stop the pelton
wheel.
8. Change the input flow rate by adjusting the spear valve and repeat steps 4, 5
and 6.
9. Repeat step 8 for a new flow rate.
10. Once the wheel stops then perform an unloading run and record the measurements as before.
11. Shut down the pelton wheel (refer to Appendix B.3).
5
4
Observations and data processing
4.1
Observations
The running plots in Figure 2 and Figure 3 show that the angular velocity of the pelton
wheel reduces as the braking force is increased. The last point on each graph indicates
the final reading before the pelton wheel stopped due to the high frictional forces
between the drum and the belt.
900
Q = 50.1 l/min
Q = 68 l/min
Q = 90.8 l/min (loading)
Q = 90.8 l/min (unloading)
800
Angular speed RPM
700
600
500
400
300
0
2
4
Force (N)
6
8
Figure 2: Running plot of pelton wheel RPM vs 100N proney brake gauge reading
The flow field generated by the deflection of the water off the bucket is particularly
interesting. At the highest speed there seems to be very little interaction between the
input jet and the buckets as the jet seems to just pass through the buckets without
touching them. As the speed reduces, the angle between the jet and the “reflected”
water gets smaller until it eventually goes to approximately zero when the pelton wheel
stops completely.
4.2
Data processing
The data processing in this experiment is carried out through the use of Equations 1
to ??. Sample calculations can be found in Appendix C.
6
900
Q = 50.1 l/min
Q = 68 l/min
Q = 90.8 l/min (loading)
Q = 90.8 l/min (unloading)
800
Angular speed RPM
700
600
500
400
300
0
5
10
15
Force (N)
20
25
30
Figure 3: Running plot of pelton wheel RPM vs 250N proney brake gauge reading
In order to find the hydraulic head supplied by the pump Bernoulli’s equation must be
used. Bernoulli can be simplified to Equation 8 since it is assumed that the velocity
and height components are insignificant at the pump exit.
H=
P
ρg
(8)
P is the pressure that is given by Equation 13 (Appendix B.2) in pascals.
The velocity ratio can now be determined through the use of Equation 7. Once the
speed ratio is known Equation 6 can be applied to solve for the efficiency of the turbine.
The braking torque of the prony brake is determined through a simple force balance,
leading to Equation 9. Refer to Appendix C for the full formulation.
T =
4.3
ddrum
(F1 − F2 )
2
(9)
Uncertainty analysis
The uncertainty in the three quantities of interest can be found through the error
propagation equation shown in Equation 10. For an equation where the terms are
being multiplied or divided this equation can be used for any number of variables.
s
2 2
2
∂f
∂f
∂f
δf =
δa +
δb + ... +
δz
(10)
∂a
∂b
∂z
7
In this experiment uncertainty arises due to the measurement equipment. The error
in any piece of equipment can be quantified by dividing the smallest increment by
two. This is done for all measurement equipment before the uncertainty is propagated
through Equation 10.
The final uncertainties in the efficiency and the velocity factor are given by Equation 11
and Equation 12.
5
δφ = 0.058δu
(11)
δη = (3.43 − 7.46φ)δφ
(12)
Results
Figure 4 (page 9) shows the relationship between the efficiency and velocity factor
of the turbine. Both are dimensionless values that are found through Equation 6 and
Equation 7. Error bars in accordance with subsection 4.3 and Appendix D are included
in order to show the variation of uncertainty for the two quantities.
6
Discussion
A performance evaluation was conducted on a pelton wheel turbine. Flow rates were
varied with a spear valve while the pump input head was kept constant. The primary
objective was to establish a way of describing the relationship between the mechanical
efficiency and the velocity factor.
The data collected and displayed in Figure 2 and Figure 3 shows the relationship
between the angular velocity of the turbine and the loads induced on the 100N and
200N prony brake gauges respectively for various flow rates. There appears to be a
“kick” about halfway down the curves where the gradient suddenly increases. The
location of this kick appears to be in a region where the maximum efficiency could be
expected.
As briefly discussed in section 4, the flow field of the water deflecting off of the bucket
varies according to the jet velocity. As the brakes are tightened and the pelton wheel is
forcefully slowed down, the flow field of the reflected water decreases in incident angle.
8
0.8
Efficiency
0.75
0.7
0.65
Q = 50.1 l/min
Q = 68 l/min
Q = 90.8 l/min (loading)
Q = 90.8 l/min (unloading)
0.6
0.2
0.3
0.4
0.5
0.6
0.7
Velocity factor
Figure 4: Efficiency vs velocity factor
The contact between the jet and bucket needs to be brief yet ensure maximal contact
in order to maximise the performance. If the wheel moves too fast there is limited
contact but if the wheel moves too slowly then the energy of the water is not fully
imparted on the wheel.
Figure 4 shows the parabolic relationship between the efficiency and the velocity factor.
The dome shape clearly shows that the point of maximal efficiency is approximately
half of the nozzle correction factor. The experimental data does not greatly deviate
from the theoretical line at any point, suggesting reasonable experimental accuracy.
The uncertainty of the experiment stems primarily from the measurement equipment
and is shown graphically in Figure 4. It can be seen that the uncertainty in the
efficiency tends to increase as the efficiency decreases.
The loading and unloading phases are done in order to test for hysteresis. The system
behaves in a similar manner in both phases. There is however, a slight looping as
can be seen in Figure 2 and Figure 3. This looping could be due to alterations in the
9
system during the loading. Steady state may not have been reached after each stage
of loosening the prony brake.
7
Conclusions
The following conclusions are made, based on the objectives set out in subsection 1.2:
1. The mechanical efficiency of a pelton wheel was found for a given flow rate and
hydraulic head. The efficiency was then plotted against the velocity factor.
2. The performance of the pelton wheel was evaluated against the theoretical line
that relates the mechanical efficiency and the velocity factor.
3. The maximum efficiency of the turbine was established by taking the first derivative of the efficiency equation and setting it to zero. It was found that the
maximum efficiency is where φ = Cv /2. This result corresponds to the graph of
efficiency vs velocity factor.
References
[1] GreenBug Energy - micro hydro. (n.d.). Types of Turbines - GreenBug Energy
- micro hydro. [online] Available at: http://greenbugenergy.com/get-educatedknowledge/types-of-turbines [Accessed 31 Mar. 2017].
[2] Agar, D. and Rasi, M. (2008). On the use of a laboratory-scale Pelton wheel water
turbine in renewable energy education. Renewable Energy, 33(7), pp.1517-1522.
[3] Liu, X., Luo, Y., Karney, B. and Wang, W. (2015). A selected literature review of
efficiency improvements in hydraulic turbines. Renewable and Sustainable Energy
Reviews, 51, pp.18-28.
[4] Potter, M. (2012). Mechanics of fluids. 4th ed. Stamford: Cengage Learning.
10
A
Images
The following section shows images that are useful during the course of the experiment.
A.1
Bucket
The control volume of the pelton wheel bucket is shown in Figure 5.
Figure 5: Pelton wheel bucket control volume [4]
B
Methodology
The following section contains the more tedious portions of the procedure outlined in
section 3.
B.1
Pump start up
1. Set motor speed controller to zero.
2. Close all valves around the pressure transducer.
3. Close the spear valve until finger tight.
4. Loosen the prony brake until the belt hangs loosely around the drum.
11
5. Turn on the flow meter and digital displays, noting that the digital display will
show a value of -40.7 if the flowmeter is off.
6. Turn on the motor power.
7. Slowly increase the pump speed at approximately 1 increment per second to the
desired motor speed, however the speed must not exceed 9.
B.2
Calibration of the digital pressure readings
The following procedure is used to calibrate the pressure transducer readings to actual
pressure values.
1. Start up the pump as described in Appendix B.1. Water should fill the small
hose next to the transducer with water.
2. Once the hose is filled shut off all the valves.
3. Shut off the pump, as described in Appendix B.3.
4. Turn the digital displays on.
5. Open the valve nearest the pressure transducer while holding the hose up. Raise
and lower the hose, noting the distances and their corresponding transducer readings.
6. Lower the hose back into the hole in the floor.
A linear regression is then applied to the data in conjunction with the use of ∆P = ρgh,
noting that the transducer readings are the independent variable and the pressure
change is the dependent variable. This yields Equation 13, which is plotted in Figure 6.
P = −10100C + 28135
(13)
Where P is the gauge pressure and C is the transducer reading. The linear regression
is extrapolated to include a wider range of pressures.
12
50000
0
Gauge pressure (Pa)
-50000
-100000
-150000
-200000
-250000
0
5
10
15
Pressure transducer counts
20
25
Figure 6: Calibration graph
B.3
Pump shut down
1. Close all valves around pressure transducer.
2. Close the spear valve.
3. Turn the motor speed down to zero at about 1 increment per second.
4. Wait for the pelton wheel to stop.
5. Switch off the pump, digital displays and flow meter at the wall.
6. Open the spear valve and clear out water from the supply line.
C
Sample calculations
For the purposes of this section, the point under consideration is where the angular
velocity of the pelton wheel is 850rpm and the flow rate is 50.1l/min. The input
13
pressure is found by using the calibration chart in Appendix B.2 since the digital
readout of the pump pressure is known.
P = −10100(20.1) + 28135 = 174875P a
The next step is to determine the hydraulic head of the system. This is done through
the use of Equation 8. The gravitational acceleration is assumed to be 9.61m/s2 since
the experiment was conducted in Johannesburg.
H=
174875P a
= 18.2m
(1000kg/m3 )(9.61m/s2 )
The tangential velocity of the pelton wheel is determined by first converting 850rpm
to radians.
850π
= 89.0rad/s
30
The tangential velocity can now be determined through the rigid body relationship.
0.244m
u = (89rad/s)
= 10.9m/s
2
ω=
The velocity factor is determined through the use of Equation 7.
φ=
10.9m/s
p
= 0.581
0.92 2(9.61m/s2 )(18.2m)
Finally, the efficiency can be calculated through Equation 6.
η = 2(0.581)(0.92 − 0.581)(1 − cos(2.62)) = 0.735
14
D
Uncertainty analysis
The uncertainty analysis outlined in subsection 4.3 is shown in greater detail in the
following section. Table 1 shows the known uncertainties of the system. It must be
noted that the uncertainty in the pressure is the standard variance obtained from the
linear regression when calibrating.
Table 1: Uncertainties of measurement equipment
Quantity
Uncertainty
δN
±5rpm
δdwheel
±0.5mm
δddrum
±0.25mm
δP
±331.5P a
Equation 10 is used to propagate the errors through to the efficiency and the velocity
ratio. Simplification of this equation leads to Equation 11 and Equation 12.
15
E
Health and Safety Risk Assessment
Risk Assessment
Activity:
Pelton wheel performance test
Venue:
North West Engineering Laboratory Basement
Risk type
Mechanical
Risk description
Severity
Fire due to friction between belt and Minor
pelton wheel drum
Fast moving machinery; loose
Minor
clothing may get caught and result in
injury
Electrical equipment is near water;
Insignificant
electrocution risk
Mechanical
Electrical
Signed:
Date:
Likelihood Risk score Action type
Unlikely
4 Elimination
Unlikely
4 Elimination
Rare
2 Elimination
Action
Use fire extinguisher to
quench flames
Switch off machinery
immediately and rectify
situation
Switch off all machinery
at the plug point
2017-04-03
Figure 7: Health and safety assessment
16
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Version 1.0
Revision date: 2017/02/17