Multi-bladed Pumps. Does the number of propellor blades affect the efficiency of a water pump?
Pumps & Physics
Research and Rationale
What's new?
When I was thinking about which aspect of physics to investigate for my investigation, I knew it was a good idea to choose something that really interested me. At the time I was becoming more and more fascinated by subatomic particles. I liked the fact that much of it was new and not understood properly, unlike the classical physics that everyone associates the subject with. Unfortunately, high energy physics does not translate into good practical coursework. However, while reading Six Easy Pieces, a book adapted from Richard Feynman's famous textbook The Feynman Lectures on Physics, I noticed that a very common everyday phenomenon is still not properly understood by physicists. Encouraged by the prospect of discovering something new, I read on.
Chaotic ideas
Feynman wrote (on page 66) "There is a physical problem that is common to many fields, that is very old, and that has not been solved...It is the analysis of circulating or turbulent fluids...No-one can analyse it from first principles"
"Wow - something science can't explain" I thought.
I looked on the internet for further details and I found a poster from World Maths Year 2000 (http://www.newton.cam.ac.uk/wmy2kposters/march/), showing just the type of unpredictable fluid motion that Feynman was writing about. It's a new and exciting branch of maths called chaos theory and it is just beginning to be understood mathematically. The main idea is that simple systems can show very complicated behaviour that seems to have no repeating pattern. The sums that describe these systems are difficult to get your head round and appear to be way beyond my abilities as an A-Level maths student.
Despite this, I felt something chaotic was an excellent phenomenon to look into for this task - it's a chance to do some experimental work where there isn't a perfect formula or a flawless explanation in any textbook. I couldn't rely on distorting my results to fit a simple law, so my experimentation had to be rigorous.
Limitations
It was important to find a subject that was practical to investigate at school. While I was watching water swirl down the drain as I filled the kettle at home, I wondered how widely-used machines like ship's propellers cope with the unpredictable world of chaos. Propellers have an unusual and distinctive shape designed to reduce turbulence. I wanted to investigate why this particular shape works so well - and if it can tell us anything about turbulent flow. Conveniently, water and propellers are easy-to-use in school labs (or so I thought!).
Best of all, I thought, if I could model the situation but ignore the effect of turbulent water, I could look at the mechanics of the propeller, and then compare the theory with what happens in real life. It seemed like a good mix of fresh ideas and traditional physics problems.
I talked about my plans to some of my teachers and one of them mentioned that his son had done a PhD degree in the formation of bubbles by marine propellers - an effect called cavitation. This encouraged me to continue with this project, knowing that it relates to current areas of research and is an important and worthwhile topic.
Research
It turns out that one of the most interesting applications of pumps is in fire engines. As fire services are public organisations they make available plenty of high-quality, free information online. Engineering sites were also useful.
* The Physics Behind Firefighting
American high-school physics project
http://ffden-2.phys.uaf.edu/212_fall2003.web.dir/Matt_Taylor/Matt1.dwt
* How Fire Engines Work
General information
http://science.howstuffworks.com/fire-engine.htm
* Bedfordshire & Luton Fire and Rescue Service
My local fire brigade, who I actually went to visit to find out more
http://www.bedsfire.gov.uk/index.htm
* American Turbine: Pump Calculations
Web-based program for working out quantities in pumping
http://americanturbine.net/formulacalc/pump.htm
* Impeller Design
The engineering that goes into pumps
http://homepage.mac.com/mrbach/mixdesign.htm
* Firefighting.com
Useful data on pumps but uses frames so I can't give a full URL
http://www.firefighting.com
* How Things Work
A simple explanation of propellers and aerofoils
Lesley Firth, Kingfisher, 1983 p13
* The Physics of Firefighting
Some simple principles explained
Physics Teacher, vol 28, p 599
* Firefighting
Contains a bit of physics but interesting background information
Jack Gottschalk, Dorling Kindersley, 2002, ISBN: 0789489090, p128
* Go with the flow
Article about modelling granular and fluid motion
New Scientist, 2 August 2003, p38-39
Preliminary Experiments
I wanted to find the most efficient propeller design. From research I found out that propellers have different shapes for different tasks, so my first goal was to get a propeller up and working, and then look at what I could change to make it run more efficiently.
These are the variables I aimed to evaluate for their effect on power transfer efficiency in preliminary tests:
* The speed of rotation
* The size of the propeller
* Since speed of rotation is less time consuming to collect data for, I'll look at it first. I intend to plot a graph of speed of rotation vs. output flow rate.
Considering the shape of a ship's propeller, I expected to be looking at these variables later on:
* The number of blades on the impeller
* The shape of the blades
* The orientation of the blades (what angle they are in relation to the axis of rotation)
The physics principles that are important here are mechanical ones. The efficiency of the propeller depends on how much of its power goes into pushing water outwards and how much is wasted on heating the water up or causing it to form whirlpools.
New Scientist's article Go with the flow mentioned the Bernoulli Effect, which is observed on aircraft wings and on propeller blades.
Lower pressure
Higher pressure
A blade with a curved plane and a flat plane forces some air or water on a longer route over the curve, and the rest takes the shorter flat route. The longer journey over the curved plane causes a drop in pressure, which translated to lift in planes, and thrust in propellers.
According to all the textbooks, the optimum number of blades, the blade angle, the speed of rotation and the size of propeller all contribute to the efficiency. It seems like I've got my work cut out for me. I'm going to concentrate on rotation speed and its effect on water flow rate outwards. Let's see what the preliminary tests show.
Water flows in
Axle
Propeller
Watertight casing
Water flows out
Planning
Risk Assessment1,2
Apparatus or procedure
Hazard
Precautions
All apparatus
Accident or fire
Supervise the experiment at all times and clear away at the end of the session. Store all equipment safely and securely.
Boiling water for shaping polypropene propellers
Risk of scalding
Take care with boiling water, paying attention at all times. Stand well back from the saucepan and do not move it while the water is hot. Use a heat-insulating towel to manipulate the hot polypropene.
Electric circuit in general
Risk of fire from short circuiting etc.
Use insulated wires, keep connections clean and dry, and always supervise the apparatus while current is flowing.
Do not leave the set-up unattended without unplugging the mains supply.
Use wires of appropriate diameter to prevent overheating resulting in fire.
Rapidly rotating propeller
Possibility of injury from contact with rotating blades of propeller
Leave motor switched off until ready to record data. Take care to keep your distance from the propeller, especially fingers.
Heavy equipment (power pack, retort stands)
Falling equipment could injure
Ensure stands etc. are sturdily placed and avoid placing equipment near the very edge of the work bench.
Power pack
Output: 13V 5A DC
Input: 230V mains AC
Risk of electrocution from mains input
(risk of injury from output voltage is minimal)
Keep power pack away from the wet part of the apparatus (to prevent conduction through water). In my experiment, I will keep all the electrics on a shelf above the level of the water-containing apparatus.
Ensure all water-containing equipment is as waterproof as possible, and have towels to hand to soak up spills.
Do not leave the set-up unattended without unplugging the mains supply.
Preliminary findings
In the research and rationale section, I ...
This is a preview of the whole essay
Input: 230V mains AC
Risk of electrocution from mains input
(risk of injury from output voltage is minimal)
Keep power pack away from the wet part of the apparatus (to prevent conduction through water). In my experiment, I will keep all the electrics on a shelf above the level of the water-containing apparatus.
Ensure all water-containing equipment is as waterproof as possible, and have towels to hand to soak up spills.
Do not leave the set-up unattended without unplugging the mains supply.
Preliminary findings
In the research and rationale section, I identified variables I wanted to investigate. I conducted preliminary experiments to found out which variables were the most practical to focus on. The basic aim is to narrow my search down to one or possibly two variables and then find the most power-efficient value for each variable.
Size of propeller was very difficult to control since I found that the propeller will only stir the water unless it tightly fits the container. Small propellers did not displace any water. Only propellers with a diameter 1 or 2mm less than the diameter of the container were effective in pumping water. As such, I decided not to consider investigating this variable.
Angle of propeller blade inclination is possible to vary, but I found the range of angles possible with the materials I had chosen were too limited. I developed a method of cutting out rectangles of polypropene sheeting, boiling them in water and bending them to the right shape, but the blades often snapped and it was tricky to get the blades to remain at the chosen angle as they cooled and hardened. I decided to keep blade inclination constant.
45° might seem to be an appropriate angle of inclination to choose for all the propellers I will compare, but most propellers I found photographs of from my research showed shallower angles of blade inclination. I have decided that all my propellers will be inclined at 30° because it is easier to make the propellers this shape and I assume that this is a more efficient angle than 45° since many propellers are about this angle.
Speed of rotation turned out to be very simple to control with the use of the variable voltage power pack. I investigated the effect of power input on rotation speed (or angular velocity of the propeller as I call it from here on in).
Using a stroboscope, I determined the linear relationship between the voltage supplying the motor (V) and the angular velocity (?) of the propeller shaft in air. I adjusted the frequency of the strobe light until the propeller appeared not to rotate.
At this frequency, the time between flashes of the strobe and the time for one blade of the propeller to reach the former position of the blade before it is equal. If you find the angle in radians (?) between two adjacent blades and multiply it by the frequency (f) of the stroboscope (the time between flashes), you are left with the angular velocity (?) of the propeller, i.e. the rate of rotation.
? = ?f
In the table below, V and f were determined experimentally and ? was calculated by multiplying f by ?. Since the frequency is only known to two significant figures, the angular velocity can only be determined to 2 s.f.
Angle between blades, ?
degrees
72
Angle between blades, ?
radians
0.4?
V
V
0
2.25
4.25
6.25
8.75
0.00
3.00
±0.25
f
s-1
0
3
26
36
50
57
74
±0.5
?
rad s-1
0
6
32
45
63
72
93
±0.5
Once the propeller is immersed in water the relationship between ? and V changes. The relationship is non-linear and, unlike the graph above, is different for every propeller.
In light of the preliminary experiments I will modify this method to vary the power supplied to the drill that drives the propeller. It will not matter that the speed of rotation varies depending on how much the water resists the motion of the propeller. The only data that are needed to calculate the efficiency of the system are power input and useful power output.
Efficiency
At this point it is important to mention that I am concentrating on the efficiency of the propeller at displacing water. Percentage efficiency = useful power output / power input × 100%, or rewritten in symbols, ? = Puseful out / Pin. Also, power input is proportional to input voltage since current is constant at 5 A in my equipment.
P = VI and I = 5 › Power (Watts) = 5 x voltage (Volts).
Review of purpose of investigation
The focus of this investigation is to determine the optimum number of blades for a propeller to have to maximise energy-efficiency. Experiments will compare propellers with 2, 4 and 6 blades. The energy efficiency of the three propellers when displacing water will be determined and compared. Their efficiency may not be independent of the rate of rotation. This too will be investigated and analysed.
The analysed results will show which of the three propellers is most energy efficient in at each rate of rotation investigated.
Extract from Eric Weisstein's World of Physics
http://scienceworld.wolfram.com/physics/Screw.html
A screw is a simple machine that is actually a version of the inclined plane. The pitch of the screw corresponds to the inclination of the plane: a higher pitch (i.e., more threads per length) means less inclination, and thus easier turning, but also more turning that needs to be done to travel a given length. As with the other simple machines, the required force is reduced, but the amount of work done is the same.
Apparatus
3V max. variable voltage power pack
Retort stands and clamps
5 cm ruler
Silicone polymer window sealant
Garden hosepipe
Expanded polystyrene for supports
Multimeter (±0.25V, ±0.25A tolerance)
Polypropene sheet for making propellers
PET lemonade bottles (2 Litre capacity)
Plastic funnel for filling
Stopwatch
Collection bottle with 2 litre mark (± 0.002 L)
Cordless electric screwdriver/drill
Steel axle
Volumetric burette
PET pudding basins to contain propeller
Water
Colour-coded wires and crocodile clips
Saucepan, hotplate and tongs for heating and reshaping polypropene into propellers
Scissors and craft knife for cutting out propeller shapes from polypropene sheet
Apparatus set-up
These diagrams show how I designed the equipment. The circuit diagram connected to the drill represents the power pack, and its voltage selector is displayed as a variable resistor. The plastic volute is the container that houses the propeller.
To begin with, water fills the water tank and the plastic volute. Activating the power pack supplies an electric current to the drill, which rotates the propeller.
Variables to control
Variable
How I will control it
Viscosity of water
Constant at constant temperature and pressure
Power and speed of rotation of propeller
Use a power pack instead of a battery to supply the cordless drill. Use the same power pack, axle and drill throughout the experiment. Rotation speed does not vary linearly with power but carefully designing the experiment can avoid problems.
Room temperature and pressure
Constant at 20°C due to central heating. Atmospheric pressure changes are insignificant to this experiment.
Plan for laboratory sessions
Session and duration
Targets
Before lab work begins
Build the waterproof sections of the apparatus and seal them with silicone polymer. Buy a cordless drill.
First two hours
Set up all apparatus, construct the propellers and test the experiment to ensure it works as planned
Second two hours
Measure the time taken to raise 2 Litres of water through 50cm vertically by each of the three propellers, with 65W power input
Third two hours
Repeat the previous session's experiment, but with the power set at 35W.
Fourth two hours
By considering the results collected before this session, decide which range of power input to investigate in detail
Fifth two hours
Continue gathering results for chosen range of power inputs
Remaining time
Investigate turning points and anomalies as necessary
In between lab sessions
Complete results tables, draws graphs as appropriate and start to analyse findings. Use analysis to modify strategy and to make decisions on how to progress.
While I was designing which equipment to use and how to use it, I thought carefully about accuracy and sensitivity. The major difficulty with this experiment is the unpredictable nature of the propellers - unlike many other things physics, it is not easy to find a good estimate of what will happen in textbooks or online.
One way of ensuring good results is to measure the variables to a reasonable number of significant figures. The multimeter I chose to use is quick to respond to changes in current or potential difference and has fine graduations on its scale, providing high sensitivity. It also has very tight tolerances as it is designed for use in high performance electronics, which contributes to the accuracy of the results I will gather.
The multimeter is significantly more accurate and sensitive some of the digital alternatives at school. It responds to changes much quicker too.
I have had to design and build quite a large amount of equipment just to make this project possible. To measure the volume of water pumped out by the system, I will calibrate the water collection bottle with graduations. To make sure they are very sensitive and accurate, I will use the high quality, high accuracy laboratory glassware available at school for use in chemistry and biology. The percentage error on the volume graduations on these pieces of equipment is very small (around 0.0003%).
References for planning section
. Cambridge University Department of Physics
Physics risk assessment form
http://www.phy.cam.ac.uk/cavendish/hands/forms/RAform.pdf
2. CLEAPSS Secondary Schools website
http://www.cleapss.org.uk/secfr.htm
Implementing
Modifications to plan
Problem
Solution
How to water-seal the entire system
Careful application of silicone sealant and gaffer tape at all junctions. Apparatus tested underwater by pressurising with air using a bike pump. Leaks located by bubbles escaping where seals were incomplete.
How to get water to flow from the water reservoir into the propeller cavity, without providing any extra pressure that would reduce the workload of the propeller
Height of water reservoir bottle adjusted until water just reaches the top of the propeller cavity, without spilling out the output hole
How to accurately measure the volumes of water used in each experiment
Volumetric glassware borrowed from chemistry department
Calculation of power efficiency of pumping system
?E = mg?h
P = Et-1
Useful power output = power spent on raising water against the force of the Earth's gravitational field
Useful power output = (mass of water raised (mwater) × strength of gravity at sea level (g) × height through which the water is raised (?h)) / time taken (t)
Pout = mwaterg?ht-1
The mass of water is proportional to its volume at constant temperature and atmospheric pressure. In these experiments, the temperature and pressure have been constant at 293K (20°C) and 105 Pa respectively. Under these conditions, water has a density of 998.2 kgm-3 (according to the Nuffield Advanced Science Data Book, Nuffield-Chelsea Curriculum Trust, Longman, 1984). Therefore, the time taken to raise the water and the number of blades on each propeller are the only variables in my experiment.
Pout = mwaterg?ht-1
mwater = 998.2 kgm-3 × 2×10-3 m3 = 1.996 kg ˜ 2.00 kg (3 s.f.)
g = 9.81 Nkg-1 (3 s.f.)
?h = 0.500 m (3 s.f.)
mwaterg?h = 2.00 × 9.81 × 0.500 = 9.81 (3 s.f.)
Pout = 9.81 ÷ t
Efficiency = Puseful output / Pin
Efficiency = (9.81 / t) / Pin
Efficiency = 9.81 / tPin
At this stage it will be easier to plot graphs of the measured variables. Just remember that the efficiency of the system at a certain point on the graph is inversely proportional to the area under the graph between the origin and that point. I will plot graphs of power input vs. efficiency later on in the analysis section.
Results table 1: effect of number of blades at 65W input power
Number of blades on propeller
Voltage / V
Current / A
Power / W
Height water raised / m
Volume of water / L
Volume water / m3
Mass of water / kg
Time taken / s
2
3
5
65
0.5
2
2 ×10-3
2.00
25.2
2
3
5
65
0.5
2
2 ×10-3
2.00
24.9
2
3
5
65
0.5
2
2 ×10-3
2.00
25.3
Average
25.1
4
3
5
65
0.5
2
2 ×10-3
2.00
8.8
4
3
5
65
0.5
2
2 ×10-3
2.00
9.2
4
3
5
65
0.5
2
2 ×10-3
2.00
9.1
Average
9.0
6
3
5
65
0.5
2
2 ×10-3
2.00
8.6
6
3
5
65
0.5
2
2 ×10-3
2.00
8.3
6
3
5
65
0.5
2
2 ×10-3
2.00
8.4
Average
8.4
exact
± 0.05
± 0.01
± 0.38
± 0.0005
± 0.05
± 5 ×10-5
± 0.005
± 0.1
Results table 2: effect of number of blades at 35W input power
Number of blades on propeller
Voltage / V
Current / A
Power / W
Height water raised / m
Volume of water / L
Volume water / m3
Mass of water / kg
Time taken / s
2
7
5
35
0.5
2
2 ×10-3
2.00
233.9
2
7
5
35
0.5
2
2 ×10-3
2.00
232.3
2
7
5
35
0.5
2
2 ×10-3
2.00
234.5
Average
233.6
4
7
5
35
0.5
2
2 ×10-3
2.00
200.1
4
7
5
35
0.5
2
2 ×10-3
2.00
202.0
4
7
5
35
0.5
2
2 ×10-3
2.00
201.2
Average
201.1
6
7
5
35
0.5
2
2 ×10-3
2.00
63.2
6
7
5
35
0.5
2
2 ×10-3
2.00
62.5
6
7
5
35
0.5
2
2 ×10-3
2.00
60.7
Average
62.1
exact
± 0.05
± 0.01
± 0.32
± 0.005
± 0.05
± 5 ×10-5
± 0.005
± 0.1
Observations from the graph
Looking at the graph on the previous page, the most striking thing is how much the results for the 6-blade propeller differ from the 4- and 2-blade propellers.
I first wondered if the 6-blade results were anomalous - but the experiment was repeated three times and error bars are included on the graph. I would normally repeat the test to confirm the results are not anomalous, but in this case there is no good reason to believe they are. Just because the difference in performance between 2 and 4 blades is small, why does that mean the difference in performance between 4 and 6 blades must be small?
Revised strategy
I think the best way to progress is to collect more data over a greater range of input powers. The three repetitions used in each test so far have been very concordant - look how close together the plots are on the graph. I see no reason to change the number of repetitions at this point.
In the research and rationale section, I discovered how fluid systems are highly complex and small input changes lead to large output changes. Therefore, I expect that the relationship between power input and time taken to pump 2L is not linear. I may be proved wrong (in which case I will have to eat my hat) but I expect to find that the straight lines on the graph become curves as soon as more data is collected.
For one thing, according the current graph (which assumes a linear relationship in the absence of information to the contrary), at about 68W input power, it will take 0 seconds to pump the water. It will always take some small amount of time to pump the water, so this line of best fit is not quite right. The line should be an asymptote to the x-axis.
Initial analysis and conclusions
As for initial analysis, the graph shows that 'upgrading' a propeller from 2 to 4 blades only slightly reduces the amount of time it takes to pump 2L - at 35W input power, a 2-blade propeller takes 235 seconds and a 4-blade propeller takes 200 seconds. However, the move to 6 blades reduces the time massively to about 60 seconds, which is less than one-third of the 4-blade elution time.
Clearly, the effect of six blades is making much more difference than just four blades. I expect that this happens because with four blades, if you imagine the propeller as a disc with some portions cut out, most of the volume of the disc is in fact empty. The force exerted on the water by one blade in the propeller may push the water forward (in the useful direction) or it may force the water back in the opposite direction through the gaps in the propeller.
On the other hand, the distance between two blades on the six-blade propeller is much less and as a consequence, water pushed forward by a blade has much less time to be accelerated back toward the propeller by gravity before another blade exerts more propulsion force on the water.
At the moment it is not clear to me why the relationship is not linear, but I suspect that the propeller must first totally overcome the force exerted by gravity on the water (weight) during the time between two blades hitting the water. I think this may be where the 2 and 4 blade propellers are failing - they are using most of their power to stir the water. The power is uselessly transferred to providing the water with thermal energy. In the 6 blade system, I expect that the water is not stirred because force is applied more evenly.
Here is a simple diagram of how I imagine the force experienced by water in the time after it is pushed by a blade:
The first blade strikes at time A. The force is rapidly spread throughout the incompressible water.
Time
Force
A
B
C
At times B and C the second and third blades strike and the effect is similar. Over time, water does not experience equal force. This jerky distribution of force with time may lead to eddies forming, which consume power by heating the water rather than moving it all in one direction.
With six blades, the time between force-maxima in the water would be less since for a given rate of rotation; a blade will pass through any given point in the plane of the propeller more often than with fewer blades. The water experiences a narrower range of forces with six blades than it would with two or three blades.
Time
Force
B
C
A
More modifications
Problem
Solution
The vibrations from the drill rotation cause the whole apparatus to slide along the work surface.
I have made some solid supports from expanded polystyrene to hold everything in place.
Further gathering of results
Looking at the graph and considering my estimation that the relationships are non-linear, I feel it is important to collect data for a range of power input in between the two values investigated so far. It seems sensible to take readings at approximately even intervals of power input. To get good results within the time limitations of this project, I will repeat each test three times as this approach has proved successful already. I will consider the range of power inputs possible with the current equipment.
I have now performed a preliminary test to find the minimum input power possible. I found that with the six blade propeller, 10W (2V, 5A) was not enough power to pump any water at all. Only stirring took place. 15W (3V, 5A) was, however, enough power to make water to trickle through the apparatus. The same result was true for the 2 and 4-blade propellers. I will measure the time taken to displace 2L of water at input powers of 15, 20, 25, 30, (35 already measured), 40, 45, 50, 55, and 60 W.
Results table 3: effect of number of blades at various input powers
Blades = number of blades, V = voltage input, I = current input, P = power input, ?h = change in height of water, vol = volume of water displaced, m = mass of water displaced, t = time taken
blades
V
I
P
?h
vol
vol
m
t
power out
efficiency
#
V
A
W
m
L
m3
kg
s
W
%
2
3
5
5
0.5
2
0.002
2
298.8
0.033
0.22
2
3
5
5
0.5
2
0.002
2
298.8
0.033
0.22
2
3
5
5
0.5
2
0.002
2
299.8
0.033
0.22
2
4
5
20
0.5
2
0.002
2
283.1
0.035
0.17
2
4
5
20
0.5
2
0.002
2
283.1
0.035
0.17
2
4
5
20
0.5
2
0.002
2
282.2
0.035
0.17
2
5
5
25
0.5
2
0.002
2
266.2
0.037
0.15
2
5
5
25
0.5
2
0.002
2
268.2
0.037
0.15
2
5
5
25
0.5
2
0.002
2
265.6
0.037
0.15
2
6
5
30
0.5
2
0.002
2
247.8
0.040
0.13
2
6
5
30
0.5
2
0.002
2
249.1
0.039
0.13
2
6
5
30
0.5
2
0.002
2
249.4
0.039
0.13
2
7
5
35
0.5
2
0.002
2
233.9
0.042
0.12
2
7
5
35
0.5
2
0.002
2
232.3
0.042
0.12
2
7
5
35
0.5
2
0.002
2
234.5
0.042
0.12
2
8
5
40
0.5
2
0.002
2
212.8
0.047
0.12
2
8
5
40
0.5
2
0.002
2
212.6
0.046
0.12
2
8
5
40
0.5
2
0.002
2
212.1
0.047
0.12
2
9
5
45
0.5
2
0.002
2
88.2
0.052
0.12
2
9
5
45
0.5
2
0.002
2
87.5
0.052
0.12
2
9
5
45
0.5
2
0.002
2
88.4
0.052
0.12
2
0
5
50
0.5
2
0.002
2
49.1
0.066
0.13
2
0
5
50
0.5
2
0.002
2
50.5
0.066
0.13
2
0
5
50
0.5
2
0.002
2
50.8
0.066
0.13
2
1
5
55
0.5
2
0.002
2
92.9
0.106
0.19
2
1
5
55
0.5
2
0.002
2
92.7
0.107
0.19
2
1
5
55
0.5
2
0.002
2
92.1
0.107
0.19
2
2
5
60
0.5
2
0.002
2
44.6
0.216
0.37
2
2
5
60
0.5
2
0.002
2
44.7
0.220
0.37
2
2
5
60
0.5
2
0.002
2
44.7
0.221
0.37
2
3
5
65
0.5
2
0.002
2
25.2
0.389
0.60
2
3
5
65
0.5
2
0.002
2
24.9
0.394
0.61
2
3
5
65
0.5
2
0.002
2
25.3
0.388
0.60
4
3
5
5
0.5
2
0.002
2
287.3
0.034
0.23
4
3
5
5
0.5
2
0.002
2
286.5
0.034
0.23
4
3
5
5
0.5
2
0.002
2
289.4
0.034
0.23
4
4
5
20
0.5
2
0.002
2
268.0
0.037
0.18
4
4
5
20
0.5
2
0.002
2
267.8
0.036
0.18
4
4
5
20
0.5
2
0.002
2
269.5
0.037
0.18
4
5
5
25
0.5
2
0.002
2
245.6
0.040
0.16
4
5
5
25
0.5
2
0.002
2
246.2
0.039
0.16
4
5
5
25
0.5
2
0.002
2
248.0
0.040
0.16
4
6
5
30
0.5
2
0.002
2
227.0
0.044
0.14
4
6
5
30
0.5
2
0.002
2
225.2
0.043
0.15
4
6
5
30
0.5
2
0.002
2
226.8
0.043
0.14
4
7
5
35
0.5
2
0.002
2
200.1
0.049
0.14
4
7
5
35
0.5
2
0.002
2
202.0
0.049
0.14
4
7
5
35
0.5
2
0.002
2
201.2
0.049
0.14
4
8
5
40
0.5
2
0.002
2
69.6
0.058
0.14
4
8
5
40
0.5
2
0.002
2
68.5
0.058
0.15
4
8
5
40
0.5
2
0.002
2
69.9
0.057
0.14
4
9
5
45
0.5
2
0.002
2
16.8
0.084
0.19
4
9
5
45
0.5
2
0.002
2
17.6
0.084
0.19
4
9
5
45
0.5
2
0.002
2
17.6
0.084
0.19
4
0
5
50
0.5
2
0.002
2
59.6
0.163
0.33
4
0
5
50
0.5
2
0.002
2
59.7
0.164
0.33
4
0
5
50
0.5
2
0.002
2
60.2
0.163
0.33
4
1
5
55
0.5
2
0.002
2
34.2
0.292
0.52
4
1
5
55
0.5
2
0.002
2
34.5
0.287
0.52
4
1
5
55
0.5
2
0.002
2
33.6
0.292
0.53
4
2
5
60
0.5
2
0.002
2
25.3
0.400
0.65
4
2
5
60
0.5
2
0.002
2
24.7
0.400
0.66
4
2
5
60
0.5
2
0.002
2
25.2
0.395
0.65
4
3
5
65
0.5
2
0.002
2
8.8
0.522
0.80
4
3
5
65
0.5
2
0.002
2
9.2
0.511
0.79
4
3
5
65
0.5
2
0.002
2
9.1
0.514
0.79
6
3
5
5
0.5
2
0.002
2
242.7
0.040
0.27
6
3
5
5
0.5
2
0.002
2
244.5
0.040
0.27
6
3
5
5
0.5
2
0.002
2
241.3
0.040
0.27
6
4
5
20
0.5
2
0.002
2
74.4
0.057
0.28
6
4
5
20
0.5
2
0.002
2
72.7
0.057
0.28
6
4
5
20
0.5
2
0.002
2
72.7
0.057
0.28
6
5
5
25
0.5
2
0.002
2
20.2
0.081
0.33
6
5
5
25
0.5
2
0.002
2
21.0
0.083
0.32
6
5
5
25
0.5
2
0.002
2
18.9
0.082
0.33
6
6
5
30
0.5
2
0.002
2
84.7
0.117
0.39
6
6
5
30
0.5
2
0.002
2
84.6
0.116
0.39
6
6
5
30
0.5
2
0.002
2
83.9
0.117
0.39
6
7
5
35
0.5
2
0.002
2
63.2
0.155
0.44
6
7
5
35
0.5
2
0.002
2
62.5
0.157
0.45
6
7
5
35
0.5
2
0.002
2
60.7
0.162
0.46
6
8
5
40
0.5
2
0.002
2
47.7
0.205
0.51
6
8
5
40
0.5
2
0.002
2
48.0
0.204
0.51
6
8
5
40
0.5
2
0.002
2
47.9
0.204
0.51
6
9
5
45
0.5
2
0.002
2
38.2
0.262
0.57
6
9
5
45
0.5
2
0.002
2
37.7
0.260
0.58
6
9
5
45
0.5
2
0.002
2
37.7
0.259
0.58
6
0
5
50
0.5
2
0.002
2
29.5
0.338
0.67
6
0
5
50
0.5
2
0.002
2
29.5
0.339
0.67
6
0
5
50
0.5
2
0.002
2
28.9
0.337
0.68
6
1
5
55
0.5
2
0.002
2
9.5
0.480
0.91
6
1
5
55
0.5
2
0.002
2
9.7
0.496
0.91
6
1
5
55
0.5
2
0.002
2
9.7
0.500
0.91
6
2
5
60
0.5
2
0.002
2
2.8
0.775
.28
6
2
5
60
0.5
2
0.002
2
3.3
0.727
.23
6
2
5
60
0.5
2
0.002
2
2.9
0.768
.27
6
3
5
65
0.5
2
0.002
2
8.6
.141
.75
6
3
5
65
0.5
2
0.002
2
8.3
.182
.82
6
3
5
65
0.5
2
0.002
2
8.4
.168
.80
exact
± 0.05
± 0.01
± 0.5
± 0.005
± 0.05
± 5 ×10-5
± 0.005
± 0.1
± 0.0005
± 0.005
Analysis
Validity of results
The most surprising outcome of the experiment was the very low efficiency of the system. I know from my study of physics that no energy transfer is 100% efficient, but a typical value for the efficiency of a system is somewhere between 5% and 80%. The efficiency of the propeller-pumping system I designed for this investigation varies from around 0.2% to 2%, which is unusually low.
I rechecked my calculations twice and found no mistakes but it is often difficult to spot your own mistakes, so I devised a small experiment to test the efficiency of the drill at doing the work as the propeller system, but without the propeller.
I suspect that the propellers are the cause of most of the inefficiency in the system. Other sources of loss of power may include resistive forces against the flow of water provided by the inner surfaces of the hoses and bottles - but these are fairly smooth and I don't expect they will be the main culprit.
The diagram on the left shows how I set up the equipment. The drill's motor will raise 2 kg of water through a height of 0.5 m. The strength of the gravitational field that the drill is working against is approximately 9.81 Nkg-1.
The efficiency of the system is given by:
? = 100mg?ht-1 / pin
...where
? is efficiency in percent
m is the mass of water displaced
g is the gravitational field strength
?h is the change in height
t is the time taken and...
pin is the power delivered to the motor.
I used the maximum power available with the power pack, 65W (13V, 5A). At these settings, the efficiency of the 6-blade propeller was 1.8%, the 4-blade propeller was 0.8% efficient and the 2-blade propeller only 0.6% efficient.
Voltage / V
Current / A
Power
input / W
?height / m
Volume water / L
Volume water / m3
Mass water / kg
Time taken / s
Power output / W
Efficiency / %
3
5
65
0.5
2
2 ×10-3
2.00
3.3
3.0
4.6
3
5
65
0.5
2
2 ×10-3
2.00
3.1
3.2
4.9
3
5
65
0.5
2
2 ×10-3
2.00
3.1
3.2
4.9
Average
3.2
3.1
4.8
Are the main results valid?
From the results given above it is clear that the drill is only about 5% efficient in this situation, which is quite a low efficiency for an electric motor. Even so, depending on the choice of propeller, this system is between 3 and 8 times more efficient at elevating water than my propeller-based systems.
These results give a good indication that the calculated efficiencies of the propeller systems are in fact correct and I did not make a mistake. The low efficiency can be attributed bad design (my fault). Although the propeller system would not be of much practical use, it has been successful in showing which propeller design is most efficient and therefore it has served its purpose.
Which propeller is most efficient?
Is this true for any input power?
From the power vs. efficiency graph, it is clear that the 6-blade propeller is always the most efficient one over the range of power input I investigated. Especially above 55W, it is two to three times more efficient than the other propellers studied. Below 20W, there is much less difference in efficiency between the three different propellers.
Why is the graph the shape it is?
The shape of the curve for the 6-blade propeller suggests its efficiency increases exponentially with input power. I would attribute this to overcoming the resistive forces in the system. At lower power, most of the power supplied to the water is used to overcome friction. This explains the shallow gradient of the curve below 50W. Once the power to overcome these forces has been supplied, water can begin to use the remaining power to move against gravity.
Why is six-bladed propeller most efficient?
As for why the six-bladed propeller is most efficient, I still believe the explanation I gave in the preliminary analysis is correct. Essentially, with two or four blades, most of the power delivered by the blades is spent on overcoming the frictional force in the system, caused by eddies forming around the propeller. With six blades, the flow of water is less turbulent due to a more consistent and less intermittent transfer of power. Less turbulent flow is much more efficient.
Evaluation
Calculation of percentage error in results:
Efficiency of a propeller system, ? = 100mg?h / tpin
Maximum error of variables:
m ± 2.5%
g ± 0.05%
?h ± 1.0%
t ± 1.2%
pin ± 3.3%
Maximum error in efficiency calculated = 100 x (1.025m) x (1.0005g) x (1.01?h) / (1.012t x 1.033pin)
Suppose efficiency error is at maximum, i.e. m = 2, g = 9.81, ?h = 0.5, pin = 65 and t = 8.3
The maximum value of efficiency = 100 x (1.025 x 2) x (1.0005 x 9.81) x (1.01 x 0.5) / (0.988 x 8.3 x 0.967 x 65) = 1.9713 %
The efficiency normally calculated would be 100 x 2 x 9.81 x 0.5 / (8.3 x 65) = 1.8184 %
Percentage error in efficiency = 100 x (?max - ?normal) / ?normal
= 100 x (1.9713 - 1.8184) / 1.8184 = ± 8.4 % error
For a school-based experiment, this is an extremely low percentage error. Error bars are indicated on the graphs.
The reason the vertical error bars are much larger on the efficiency-power graph than the time-power graphs is because the efficiency value incorporates the error in mass, gravity, time, change in height and power input, whereas the value of time only includes the error in time.
Overall, the choice of equipment of appropriate accuracy and sensitivity has meant the data collected are reliable. The conclusions drawn about the relative efficiencies of the propellers are fully justified by the results, and the small percentage error in the results has lead to concrete conclusions. This is probably due to the large amount of time I spent developing the experiment, and the many modifications implemented.
The explanations I have given for the behaviour of the propellers could be more detailed. I have found it difficult to use theory from the A-Level course because it does not go into much detail on fluid dynamics (with good reason - it's very complicated!). I have tried to be imaginative and provide some possible explanations.
I researched fluid dynamics and the Bernoulli Effect quite thoroughly before starting the practical work, but I found it was not explained to any great depth in A-Level texts from other courses. Research on the internet led to much more quantitative theory on fluid dynamics, but the mathematics used to express it was beyond what I have learned.
Suggested improvements
This investigation was very difficult at times, but very interesting. I am sure it will crop up again at university in one form or another. The modifications I found most helpful were those that simplified the problem and allowed easy measuring.
With that in mind, if I were to repeat the investigation or study this area in the future, I would design a simpler set-up with more prefabricated components used. As ever, if greater accuracy or sensitivity were required in the data, different measuring equipment could be used. Also, a greater change in height could be used to exaggerate the differences between results and make them more sensitive.
The modifications made would almost certainly depend on how the results and conclusions were going to be used.