photo by Joe Turcio
TABLE OF CONTENTS
3.1 List Of Symbols
3.2 Velocity Analysis
3.3 Nozzle Flow Coefficient
3.4 Power Output
3.5 Variation of Power Output With Speed
3.6 Efficiency
4.1 Experimental Program
4.2 Experimental Procedure
4.3 Speed Measurement
The Pelton Wheel is the only form of impulse turbine in common industrial use. It is a robust and simple machine which is ideal for the production of power from low volume water flows at a high head with reasonable efficiency.
Cussons P6240 Pelton Wheel, although a model, reproduces all the characteristics of full size machines and allows an experimental program to determine both the performance of the turbine and to verify design theory.
Cussons P6240 Pelton Wheel consists of a Degener 4717 model Pelton Wheel mounted on a base plate and fitted with a friction dynamometer as illustrated in Figure 2 overleaf. The design of the Degener Pelton Wheel follows typical industrial practice with a horizontal shaft, single horizontal jets produced by a single nozzle fitted with a needle or spear regulator, and a wheel fitted with multiple (16) elliptical ridged buckets at a mean diameter of 100mm.
The nozzle is positioned in the same plane as the wheel and arranged so that the jet of water impinges tangentially on to the buckets. The nozzle and a single bucket are illustrated in Figure 1 below. Each "bucket" is divided by a "splitter" ridge which divides the jet into two equal parts. The buckets are shaped so as to prevent the jet to the preceding bucket being intercepted too soon. After flowing round the inner surface of the bucket, the fluid leaves with a relative velocity almost opposite in direction to the original jet. The desired maximum deflection of the jet (180°) cannot be achieved without the fluid leaving one bucket striking the following one, and so in practice the deflection is limited to approximately 155° (i.e. see Figure 1 below 18025). The bottom of the casing is open to allow the water leaving the buckets to drain away. The front face of the casing is transparent Perspex allowing easy observation of the behaviour of the water jet and assessment of exit angles.
The nozzle is controlled by movement of the spear regulator along the axis of the nozzle which alters the annular space between the spear and the housing, the spear being shaped so as to induce the fluid to coalesce into a circular jet of varying diameter according to the position of the spear. A static pressure tapping is provided to allow measurement of the inlet water pressure.
The friction dynamometer consists of a 60mm diameter brake wheel fitted with a fabric brake band. The brake band is tensioned by a weight hanger and masses with the fixed end being secured via a spring balance to the support frame.
The speed of the turbine can be measured using Cussons P4740 Optical Tachometer which is available as an optional accessory.
Figure 1. Details of Pelton Wheel Buckets.
Figure 2 General Arrangement of Cussons Pelton Wheel note that our setup does not have the friction brake.
3.1 list of symbols
A_{r} 
area of incident jet 
m^{2} 
C 
loss coefficient 

C_{v} 
coefficient of velocity for jet 

F 
force exerted on bucket 
N 
g 
acceleration due to gravity 
9.806 m/s^{2} 
H 
inlet head 
m 
k_{1} 
frictional resistance coefficient 

mass flow rate 
kg/s 

rate of flow of jet 
m/s 

R 
radius of turbine rotor 
m 
U 
tangential velocity of wheel 
m/s 
V_{1} 
incident jet velocity 
m/s 
V_{r1} 
incident jet velocity relative to bucket 

V_{r2} 
emergent jet velocity relative to bucket 

power output 
watts 

h 
wheel efficiency 
% 
q 
angle between incident and emergent jets 
rads 
r 
density of water torque 
kg/m^{3} 
t 
torque 
Nm 
w 
angular velocity 
rads/s 
3.2 Velocity Analysis
Consider a Pelton Wheel rotating in an anticlockwise direction with an angular velocity w due to the combined action of an incident water jet of velocity V_{1} and a clockwise resisting torque t as shown in Figure 3a. The velocity analysis will use a bucket on the Pelton Wheel as a reference as shown in Figure 3b.
a) Absolute Velocities 
b) Velocities Relative to Bucket 
Figure 3. Velocity Analysis 
The velocity of the incident jet relative to the bucket is given by
Since the incident and emergent jets are both exposed to atmospheric pressure the magnitude of the emergent jet velocity will be only slightly less than the incident jet velocity due to frictional resistance which can be allowed for by introducing a frictional resistance coefficient k_{1} so that
The jet will be deflected so that the emergent jet is at an acute angle q to the incident jet. The change in the component of velocity in the plane of the wheel (i.e. in the line of the incident jet) will be
which can be written as
3.3 Nozzle Flow Coefficient
The discharge through the nozzle form the inlet head H is given by
3.4 Power Output
The force exerted on the bucket by the water jet is obtained from Newton's second law as the rate of change of momentum between the incident and emergent jets in the plane of the wheel.
The torque acting on the shaft of the Pelton Wheel is then
and the power output is
substituting for and
3.5 Variation of Power Output With Speed
For a given head and nozzle area the power output will be a maximum if C_{v} and k_{1} have their highest obtainable values, if cos q = 1 and if U (V_{1} – U) is a maximum.
U (V_{1} – U) will be a maximum when V_{1} = 2U, that is, when the incident jet velocity is twice the Pelton Wheel bucket speed. For cos q =1 then q = 0 and relative to the buckets the incident jet is deflected through 180°, furthermore if k_{1} = 1 then V_{r1} = V_{r2}.
With V_{1} = 2U the V_{r1} = U (and ignoring friction across the buckets V_{r2} = U) the absolute velocity of the emergent jet in the plane of the wheel will then be zero, the whole of the velocity head of the incident jet will have been converted into useful work and the water will effectively fall off the trailing edge of the buckets. The maximum power output is given by
If resisting torque is reduced then wheel will accelerate, D V will reduce and the limiting case which would not be obtainable in practice would be when U = V_{1}, for this condition the torque and power output would both be zero. If the resisting torque is increased the wheel can become stalled so that U = 0 and the stall torque will be
3.6 Efficiency
The input hydraulic power to the Pelton Wheel is the product of the inlet pressure and flow rate.
and the efficiency of the Pelton Wheel is
Since the input hydraulic power depends only on the head and the nozzle area and is independent of the Pelton Wheel speed then the efficiency is directly proportional to the power output and thus maximum power and maximum efficiency occur at the same conditions
Substituting for D V
For maximum power V_{1} = 2U
Substituting for
In an ideal situation, C_{v}, k_{1} and cos q are all equal to 1.0 and hence the ideal maximum efficiency is unity.
4.1 Experimental Program
The fundamental experiment which can be carried out with Cussons Pelton Wheel is to investigate the performance of the machine for a range of flow rates and rotational speeds.
There are two controls which can be varied independently; the load applied to the brake and the position of the nozzle regulating spear. The position of the nozzle regulating spear controls the nozzle flow rate whilst varying the friction brake load will vary the speed of the turbine wheel.
4.2 Experimental Procedure
The easiest way to organise the experiment is to set the nozzle regulating spear and conduct a number of tests from zero load to maximum load. The speed is influenced by the torque applied by the electromagnetic brake.
4.3 Speed Measurement
The Pelton Wheel speed in RPM is displayed on the control panel and recorded in the data set.
Experiment Pelton wheel
Aim To investigate the performance of a Pelton Wheel.
Equipment Preparation Set up the quipment as follows.
Experimental Procedure
Caution: Be careful not to jam the spear at the end of its travel. Doing so could cause the spear to disconnect from the knob requiring disassembly of the turbine an considerable time delay.
Results and Analysis
Record the data using the LabVIEW VI. Set the number of runs to 5000 so the VI does not stop before all your data is recorded. Note that each time the VI is stopped the current data set is filed. Restarting the VI will cause a new data file to be generated. This is not a problem so long as you capture all your files before leaving.
a) Record the results on a copy of the result sheet.c) Plot graphs of torque, power and efficiency against Pelton Wheel speed for each setting of the spear regulator (approx. constant flow rate).