The Effect of Tesla Valves on the Flow Rate of Water
Red Encabo / Mikolaj Pal
Macomb Mathematics Science Technology Center
AP Physics
Section 12C
Mr. McMillan / Mr. Supal / Mr. Acre / Mrs. Dewey
6 December 2016
The Effect of Tesla Valves on the Flow Rate of Water
The purpose of this research was to determine whether Tesla valves had a
significant effect on the water’s flow rate. Diagnosing the Tesla valve’s impact on the
flow of a fluid would provide supplementary information on its optimization and
potential in real-world functions. The researchers predicted that a one-segment Tesla
valve would reduce its flow rate the least, and a three-segment valve would reduce its
flow rate the greatest. To verify whether the hypothesis was true, the researchers wanted
to acquire the flow rates before and after the utilization of the valve and use those values
to analyze the change in flow. To calculate the flow rate, the researchers measured the
amount of water that filled a container in the measured time interval. They subsequently
accounted the bin’s length and width and multiplied the dimensions by the measured
height. Descriptive statistics established the correlation between the number of valve
segments and the flow rate. A one-way ANOVA test was conducted to determine the
significance in the differences between the average flow rates in valves of one, two, and
three segments. It was discovered that the three average flow rates before connecting the
valves to the hose were 33.4 cm3/s, 32.0 cm3/s, and 33.1 cm3/s, respectively. After
applying the valves, the flow rates dropped to 29.9 cm3/s, 26.4 cm3/s, and 22.6 cm3/s,
respectively. The flow rate on average expelled the least in a one-segment valve and
expelled the greatest in a three-segment valve. There was significant evidence of a
negative correlation between the number of segments and the flow rate. Tesla valves
could reduce overall costs while maintaining or improving efficiency in many of a
valve’s general uses including pipelines for oil and natural gas production, sewage
systems for private households, and engines and hydraulic cylinders for motor vehicles.
Table of Contents
Introduction ......................................................................................................................... 1
Review of Literature ........................................................................................................... 6
Problem Statement ............................................................................................................ 16
Experimental Design ......................................................................................................... 17
Data and Observations ...................................................................................................... 20
Data Analysis and Interpretation ...................................................................................... 27
Conclusion ........................................................................................................................ 37
Acknowledgements ........................................................................................................... 42
Appendix A: Statistical Equations and Sample Calculations ........................................... 44
Appendix B: Professional Contact .................................................................................... 50
Works Cited ...................................................................................................................... 52
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Introduction
The Tesla valve bears a striking resemblance with the veins of the human leg: in
1916, Nikola Tesla, an inventor, engineer, and physicist invented and patented a valve
that, like one-way valves, allows fluid to flow in one direction and prevent fluid from
flowing in the opposite direction. A valve is a device that controls the passage of a fluid
through a pipe or duct. A fluid’s flow and direction is controlled in any valve, but with
diverse designs, types, and models—such as check valves and swing valves—valves are
pertinent in numerous industrial applications. Valves, for instance, control the flow of oil
under extreme temperatures and pressures to prevent oil erosion in offshore or onshore
drilling. In another example, valves control fluid flow in a ship’s pipe system to
efficiently generate power, manage wastewater, and control heating, ventilation, and air
conditioning.
Figure 1. Check Valve
As previously stated, there are different types of valves; most check valves have
moving parts that rely on the fluid to open or close the passage. Traditional check valves,
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as seen in Figure 1, rely on discs inside them to allow flow to pass forward. Opening the
disc allows fluid to flow forward, while closing it causes the fluid to flow in a reverse
direction.
Figure 2. Swing Valve
Swing valves, as seen in Figure 2, are check valves that contain attached movable
discs on the arm which block reverse flow and push back to enable forward flow.
Figure 3. Tesla Valve
Since Tesla valves, like check valves, allow fluid to flow in only one direction,
they are a type of one-way check valve. However, since Tesla valves do not use moving
parts, they are often advantageous in efficiency and reliability. Instead of relying on
moving parts, flow control is dependent on the geometry of the valve’s design: according
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to the diagram in Figure 3, the loops of the valve are notably shaped to control the fluid’s
flow. The loops make it in order that the fluid itself becomes the force restricting it to
flow at a rate nearly-equal or equal to the flow rate before the application of the valve.
Many conventional valves moreover use moving parts which are prone to depreciation;
given that Tesla valves do not use moving parts, they mostly do not experience this issue
(Gamboa).
Knowing the features of Tesla valves is essential: they have the potential to
reduce maintenance costs, improve efficiency, and boost productivity. Unfortunately,
only a few number of scientific literatures have discussed the Tesla valve’s optimization,
hindering the valve’s utilization. The purpose of this research was to experiment with
Tesla valves of different sizes and demonstrate the significant impact they had on the
fluid’s flow rate. The researchers intended to create a generalization about Tesla valves
for any real-world application. Discovering their impact on the fluid’s flow is
fundamental to shaping the scientific community’s beliefs in the efficiency and
performance of the Tesla valve. Take the hydraulic system as an example, which utilizes
fluid to power machines. The most vital part of the hydraulic system is the valve, which
directs the flow of a liquid through the hydraulic system. If the research’s data, in one
case, suggests that resistance in the blocking direction is ten times greater than the
unimpeded direction in a Tesla valve, then scientists can use that information and
implement the design of Tesla valves onto future hydraulic systems that could better
prevent ruptures and accidents while effectively controlling the fluid’s flow. In a more
general standpoint, they can improve the maintenance and reliability of nearly any system
("Hydraulic Valves”).
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To determine the significance the Tesla valve had on the fluid’s flow rate, the
researchers initially observed the growth of the fluid level in a bin. With the faucet turned
on halfway, the researchers allowed the fluid to exit through the hose (and valve) and into
the bin for sixty seconds. Using a meter stick, the height of the fluid was measured and
afterwards multiplied by the bin’s length and width to calculate the volume of fluid that
filled the container during a time interval; after calculating the volume, the flow rate was
computed. The trials were performed thirty times--ten times each for a Tesla valve of
one, two, and three segments. Each trial additionally required preparation before the
Tesla valve was applied. The researchers calculated the flow rates before and after the
utilization of the Tesla valve to perform a descriptive test which visually depicted and
compared the changes in the distribution of values.
This research was applicable to future research. For instance, doctors, nurses, and
those of the medical field can take account the features of the Tesla valve to design a
proficient tool that extracts blood and poison while preventing fluids from returning back.
Applying the Tesla valve’s features to a tool in this instance can save millions of lives
and improve the medical methods doctors use to transfer blood from one individual to
another individual or substance. Tesla valves are potentially capable of advancing the
medical field. Researchers can furthermore use the information discovered from this
experiment to develop auxiliary tests. For example, provided that the number of valve
segments and flow rate simultaneously change, scientists can take the research further by
conducting an experiment that takes account of the connections between two objects and
determines their impact on the fluid’s flow rate. Being knowledgeable of Tesla valves is
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not only imperative for present work, but also vital for practical and relevant applications
and future research.
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Review of Literature
Check valves are one-way valves that allow a fluid to flow in one direction
without creating backflow, a term that describes an undesired flow of water in the reverse
direction. The prevention of backflow ultimately eliminates any source of “water
hammer,” a knocking noise in water that occurs when a tap is turned off in check valves.
This is important because water hammers are signs of air exiting a pipe or connection.
(“Check Valve Installation and Benefits”). Allowing flow in one direction, they are selfactuated, meaning they do not require assistance to open and close.
Figure 4. Check Valve
Regarding the operation of check valves (Figure 4), they rely on the fluid to open
and close. The discs inside the valves allow flow to pass forward, opening the valve; then
the discs close the valve as the flow of the fluid decreases (or the reverse direction of the
flow of the fluid increases). How this applies is dependent on the design of the check
valve. Additionally, the disc and seat ring of the check valve relies on fluid back-
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pressure, allowing for a greater seat leakage rate. Properly sized check valves for specific
applications do the operation the best; this includes the condition in which the disc is
stable (Shorts).
The significance of check valves is shown industrially and domestically.
Industrially, check valves are commonly used in chemical and power plants for feed
water control, dump lines, nitrogen monitoring, and sampling. They are moreover applied
to fuels and oxidizers to keep gas cylinders from becoming prone to flammability
(“Nuclear Check Valves & Excess Flow Check Valves”). Domestically, check valves are
used agriculturally, as they are commonly found in irrigation sprinklers and drip
irrigation sprinklers to prevent drainage when the system is shut off. Some rainwater
harvesting systems use check valves to avoid contamination of rainwater supply. In short,
check valves are common in various fluid-centered situations such as pumping, industrial
processing, and domestic use. There are various types of check valves with such
applications, including swing, wafer, and lift valves (Hartmann-Bresgen, Petra, and
Kleophas).
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Figure 5. Swing Check Valve
A type of check valve, swing check valves contain attached movable disks on the
arm to block reverse flow and enable forward flow. The design of this valve (Figure 5) is
advantageous in minimizing pressure loss and simplifying fluid flow, while it creates a
sense of flexibility for horizontal or vertical layouts. The configuration of the open
position is required for the valve to close when necessary.
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Figure 6. Wafer Check Valve
An advantage with the wafer check valve design (Figure 6) is its portability:
wafer valves tend to be lighter and thinner than conventional check valves, including
swing check valves. At the same time, its design simplifies the ability to close and
increases its multi-functionality in high-performance water hammer absorption (“Valve
Types & Configurations”).
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Figure 7. Lift Check Valve
As a fluid flows through a lift check valve, pressure builds up against the disk,
causing it to rise off of the body (Figure 7). This allows the fluid to flow through the
entire valve. When the fluid flows reversely, the lift check valve design allows the fluid
to quickly close the disk. In addition, lift check valves offer a more effective seal
compared to other types of check valves. Similar to the swing valve, lift check valves can
be configured vertically and horizontally (“Lift Check Valves”).
Although check valves are self-actuated, it is difficult to determine their openclose status and assess the condition of internal parts, since the moving parts in check
valves are enclosed. Valve discs are prone to sticking in open position. When configuring
check valves, they have their own limitations, so an individual is required to understand
what is essential to installing any check valve (Chugh). (For instance, the designs of the
swing and lift valves unintentionally impede fluid flow.)
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Figure 8. Tesla Valve
The Tesla valve (Figure 8) is a special one-way check valve that allows fluid to
flow in one direction while blocking it in the other without using moving parts. This
gives it an advantage compared to other check valves, since moving parts are prone to
wear over time. In the forward direction, it allows fluid to flow with almost no resistance,
while fluid is directed into a segmented series of loops in the reverse direction. Since the
fluid goes through a series of loops, pressure is built up against the flow of the fluid and
ultimately blocks the flow (West). However, a disadvantage of the valve is its potential to
allow some liquid to flow in reverse: there is no physical blocking device that disallows
the reverse flow of fluid.
If the fluid flowed through the valve from the right to the left side, it would flow
through a nearly-straight path. The fluid experiences the least resistance from the right
side to the left side, so the fluid flows through the valve with little to no change. From the
left to the right, however, the segments restrict the airflow through the valve by forcing
the fluid to go left or right of each loop. As a result, the fluid moving right returns against
the flow, and the fluid moving left interacts with the fluid returning against the flow.
Generally, the fluid itself becomes the force that restricts it to flow at a rate nearly-equal
or equal to the flow rate.
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Figure 9. Flow Developed in Unimpeded (Left) and Blocking (Right) Direction
According to other research, researchers have discovered that a valve’s design
primarily determines the regulation of fluid flow (Figure 9). They demonstrate this in an
experiment where micro valves are simulated through a calculation liquid framework that
utilizes laminar stream (flow regime characterized by high momentum diffusion and low
momentum convection) and basic calculation. They took account of each valve’s
diameter, length, position, and density. Testing the larger valves, less pressure was
emitted in the water flow. The lack of pressure applied to the direction of the flow
deregulated the fluid flow and increased the flow rate of the fluid.
Figure 10. Reverse Flow Velocity Based on Diameter
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It was furthermore discovered that valves designed without inlet and outlet flow
channels had a higher overall diodicity (or higher flow rate when losses are linear or
quadratic in both directions). The top flow channel, seen in Figure 10, experiences lower
flow rate ranging between 0-2.65 m/s compared to the original channel, which ranges
between 2.98-6.61 m/s. In the end, the research moreover came to the conclusion that
wider, less dense valves experience less flow pressure, resulting in higher flow rate
(Nobakht, Shahsavan, and Paykami).
In determining what design best suited the regulation of fluid flow, Truong and
Nguyen’s study optimized the design of the tesla valve with different construction
methods and configuration according on different flow rates. Geometry construction
methods and design optimization parameters were proposed along with varied design
parameters used to form different valve configurations. Two-dimensional steady flow
models of these different valves was made using an “ANSYS FLOTRAN” numerical
software. The study determined what the three best valves designs were by taking
account of its diodicity.
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Figure 11. Diodicity vs. Radius (R/W)
The research found that diodicity was inversely proportional to the valve’s radius,
meaning the larger the radius, the smaller the diodicity (Figure 11). Large angles reduce
the amount of flow entering the curve section in forward flow direction, dropping the
pressure of the fluid flow. It also helps the flow in the curve section block the flow in the
straight section while in reverse. This is important since there are different potential
applications that require Tesla valves to use different flow rates, configurations, and
different sizes (Truong and Nguyen). Given that the diodicity decreases as the radius of a
valve increases, it can be assumed that valves of greater length and diameter decrease the
flow rate of a fluid more proficiently than smaller valves.
Overall, the purpose of our research is to determine if the tesla valve regulates
fluid flow more effectively than conventional one-way valves. Research has shown that
the designs of each valve affect the performance of regulating the fluid flow and
preventing backflow: valves with a greater radius and lower density experience fluid flow
at higher rates compared to valves with a lower radius and higher density. Tesla valves
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have many potential applications such as dishwashing and maintenance for irrigation, and
it can be less costly to maintain compared to traditional one-way valves (Bardell).
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Problem Statement
Problem:
To determine the effect that the Tesla valve at different lengths (one segment, two
segments, and three segments) has on the flow rate of a fluid (cm3/s).
Hypothesis:
The Tesla valve of three segments experiences the lowest flow rate; the Tesla
valve of one segment experiences the highest flow rate.
Data Measured:
The dependent variable--flow rate as it exits the valve (cm3/s)--is determined by
taking into account (the independent variable) the length of the Tesla valve (in segments).
Segments are combined to create three Tesla valves composing of one segment, two
segments, and three segments. The height of the fluid is multiplied by the bin’s length
(54.5 cm) and width (8.5 cm) to calculate the volume of the fluid (cm3) that filled the
container during a time interval. The flow rate is calculated by dividing the volume by the
time interval (sixty seconds). An ANOVA test and descriptive test are conducted to
determine whether the means of the flow rate are significantly different using the sample
size and sample means of the flow rate before and after the application of the Tesla valve.
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Experimental Design
Materials:
(3) Tesla Valve Segments
10.0 m Standard [Non-Pocket] Hose
(with 0.02 m Diameter)
54.5 x 8.5 cm Plastic Bin
Aven 40-Watt Soldering Iron (with
Fine Tip)
Dell Latitude 13 Education Series (3340)
TI-Nspire CX Handheld Calculator
Sharpie Permanent Marker
Gorilla Duct Tape
Meter Stick
Procedure:
Experimental Setup
1. Work in an isolated laboratory where a standard or non-pocket hose with a 2.0 cm
diameter (and a minimum length of 1000.0 cm) can be applied to a water spigot of the
same diameter. Screw one end of the hose onto the water spigot as tightly as possible.
2. Create a hole with a diameter equivalent to the diameter of the hose on one short side
of the bin using a 40-watt soldering iron. (A soldering iron with a fine tip is better at
creating precise holes.) After creating the hole, attach the opposite end of the hose in
the hole and secure it using gorilla duct tape. This will reduce the movement of the
hose.
3. Position the tank 50.0 cm away from the sink. Repeat this step daily to achieve
accurate results in fluid velocity.
Conducting the Experiment
4. Turn on the spigot halfway to its maximum level. Using an alarm clock, begin timing
(seconds), and stop timing when the timer reaches sixty seconds. Additionally, mark
the halfway position with a marker for future reference to ensure data accuracy.
5. Measure the height of the water (cm) using a meter stick.
6. Calculate the volume (cm3) by multiplying the bin’s length (cm), its width (cm), and
the measured height. The bin’s dimensions can be found in Appendix A.
7. Divide the volume found on step 6 by the time interval the water filled the container
(sixty seconds) to calculate the flow rate (cm3/s). Record the flow rate (without the
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Tesla valve) on table 1.
8. Apply one segment of the Tesla valve by tightening the open end of the valve to the
open end of the hose. After the Tesla valve is stabilized to the open end of the hose,
repeat steps 4-7.
9. Perform steps 4-8 until ten randomized trials, using one segment of the Tesla valve,
have been conducted. Record the flow rates on table 1.
10. Calculate the average flow rates (cm3/s) before and after the application of a onesegment valve by dividing the sum of the flow rates found on steps 4-8 by the number
of trials (ten trials).
11. Perform steps 4-8 until ten randomized trials, using two segments of the Tesla valve
have been conducted. Record the flow rates on table 2.
12. Calculate the average flow rates (cm3/s) before and after the application of a twosegment valve by dividing the sum of the flow rates found on steps 4-8 by the number
of trials (ten trials).
13. Perform steps 4-8 until ten randomized trials using three segments of the Tesla valve
have been conducted. Record the flow rates on table 3.
14. Calculate the average flow rates (cm3/s) before and after the application of a threesegment valve by dividing the sum of the flow rates found on steps 4-8 by the number
of trials (ten trials).
Performing the Analysis
15. Conduct the ANOVA test provided in Appendix A. The test determines whether the
means of the flow rate are significantly inequivalent using the sample size and sample
means of the flow rate found in steps 9-14.
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Diagram:
Figure 12. Experimental Setup
Figure 12 visually depicts the experimental setup, which shows that three
segments of the Tesla valve were applied on the open end of the hose; they were attached
to the hole of the rectangular bin and the water spigot located on the top of the sink. With
the faucet turned on halfway, the researchers allowed the fluid to exit through the hose
(and valve) and into the bin for sixty seconds. By measuring the height of the fluid in the
given time interval, the researchers multiplied the found height by the bin’s dimensions to
calculate the volume. The volume was then divided by the time interval, sixty seconds, to
find the flow rate. This setup determined the fluid velocity before and after the Tesla
valve is applied.
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Data and Observations
Q=
V lwh
=
t
60
Figure 13. Flow Rate Equation
To calculate the flow rate (Q) of the fluid, as shown in Figure 13, multiply the
length of the bin, and width of the bin, and the height of the water. Divide the product by
the time, which is sixty seconds for all trials.
Table 1
Trials with One Tesla Valve Segment
#
Volume
Before
(cm3)
Volume
After
(cm3)
Time
(s)
Flow Rate
Before
(cm3/s)
Flow Rate
After
(cm3/s)
1
1,853.0
1,667.7
30.9
27.8
2
2,084.6
1,714.0
34.7
28.6
3
2,038.3
1,945.7
34.0
32.4
4
1,806.7
1,853.0
30.1
30.9
5
1,992.0
1,575.1
33.2
26.3
6
2,038.3
1,992.0
34.0
33.2
7
2,038.3
1,945.7
34.0
32.4
8
1,992.0
1,575.1
33.2
26.3
9
2,223.6
1,621.4
37.1
27.0
60.0
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#
10
Volume
Before
(cm3)
1,992.0
Volume
After
(cm3)
Time
(s)
2,038.3
60.0
Flow Rate
Before
(cm3/s)
Average Flow Rate (cm3/s)
Flow Rate
After
(cm3/s)
33.2
34.0
33.4
29.9
Table 1 shows the recorded volumes (cm3) before and after one segment (10.0
cm) of the Tesla valve was applied to the hose. It also provides the fluid’s flow rate
(cm3/s) before and after one segment of the Tesla valve is applied. Before the Tesla valve
was applied, the average flow rate of the fluid is 33.4 cm3/s; the average flow rate of the
fluid is 29.9 cm3/s with the Tesla valve.
Table 2
Trials with Two Tesla Valve Segments
#
Volume
Before
(cm3)
Volume
After
(cm3)
Time
(s)
Flow Rate
Before
(cm3/s)
Flow Rate
After
(cm3/s)
1
1,806.7
1,667.7
30.1
27.8
2
1,853.0
1,436.1
30.8
23.9
3
2,177.3
1,621.4
36.3
27.0
4
1,992.0
1,806.7
33.2
30.1
5
1,806.7
1,621.4
30.1
27.0
6
1,945.7
1,482.4
32.4
24.7
7
1,899.3
1,436.1
31.7
23.9
60.0
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#
Volume
Before
(cm3)
Volume
After
(cm3)
8
2,084.6
1,760.4
9
1,899.3
1,528.7
10
1,760.3
1,482.4
Time
(s)
Flow Rate
Before
(cm3/s)
60.0
Average Flow Rate (cm3/s)
Flow Rate
After
(cm3/s)
34.7
29.3
31.7
25.5
29.3
24.7
32.0
26.4
Table 2 shows the recorded volumes (cm3) before and after two segments of the
Tesla valve was applied to the hose. It also provides the flow rate (cm3/s) before and after
two segments of the Tesla valve was applied. Before the Tesla valve was applied, the
average flow rate of the fluid is 32.0 cm3/s; the average flow rate of the fluid is 26.4
cm3/s with the Tesla valve. Based on the table 2’s findings, the mean fluid flow rate with
two Tesla valve segments is 3.5 cm3/s less than the mean fluid flow rate with one Tesla
valve segment. This information is important, as the mean fluid flow rates (regardless of
the number of segments) could be or not be significantly equal.
Table 3
Trials with Three Tesla Valve Segments
#
Volume
Before
(cm3)
Volume
After
(cm3)
1
2,177.3
1,436.1
2
1,945.7
1,343.4
3
1,945.7
1,343.4
Time
(s)
60.0
Flow Rate
Before
(cm3/s)
Flow Rate
After
(cm3/s)
36.3
23.9
32.4
22.4
32.4
22.4
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#
Volume
Before
(cm3)
Volume
After
(cm3)
Time
(s)
Flow Rate
Before
(cm3/s)
Flow Rate
After
(cm3/s)
4
1,806.7
1,482.4
30.1
24.7
5
1,899.3
1,297.1
31.7
21.6
6
2,038.3
1,389.8
34.0
23.2
7
1,806.7
1,250.8
30.1
20.8
8
2,084.6
1,343.4
34.7
22.4
9
2,038.3
1,389.8
34.0
23.2
10
2,084.3
1,297.1
34.8
21.6
33.1
22.6
60.0
Average Flow Rate (cm3/s)
Table 3 shows the recorded volumes (cm3) before and after three segments of the
Tesla valve was applied to the hose. It also provides the flow rate (cm3/s) before and after
three segments of the Tesla valve was applied. Before the Tesla valve was applied, the
average flow rate of the fluid was 33.1 cm3/s; the average flow rate of the fluid is 22.6
cm3/s with the Tesla valve. Based on the table 3’s findings, the mean fluid flow rate with
three Tesla valve segments is 3.8 cm3/s less than the mean fluid flow rate with two Tesla
valve segments. The mean fluid flow rate with three Tesla valve segments is 7.3 cm3/s
less than the mean fluid flow rate with one Tesla valve segment. This information is
important, as the mean fluid flow rates (regardless of the number of segments) could or
could not be significantly equal. Note: the fluid flow rate for each trial was calculated
using the flow rate equation and the bin’s given length (cm) and width (cm) in Appendix
A.
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Figure 14. Two Tesla Valve Segments
Figure 14 shows two three-dimensional printed segments of the Tesla valve. Each
segment of the valve contains two “loops” connected to a main tube, which reverse the
flow of the fluid and direct it back into the main tube; this process resultantly slows down
the fluid going through the main tube. Initially, it was found that the Tesla valve segment
on the left was printed incorrectly, because it partially missed a loop. Testing the Tesla
valve segments with missing elements showed that any fluid could leak out of the valve
and skew the fluid’s flow rate. This was solved by subsequently printing and gluing the
element onto the segment; this helped us not only do the experiment properly, but it also
reduced the time consumed in printing the parts.
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Figure 15. Positioning the Tesla Valve
One of the essential steps of setting up the experiment is making the Tesla valve
parallel to the ground, as shown in Figure 15. The purpose of positioning the Tesla valve
horizontally was to increase the accuracy and consistency of the data. Before beginning
the first few trials of the experiment, it was found that the valve’s position played an
important role in determining the water level: positioning the valve at an angle below the
imaginary horizontal axis would increase the fluid flow rate, while positioning the valve
at an angle above the imaginary horizontal axis would decrease the fluid flow rate. The
experiment had to be restarted because of this. Hence, the trials were re-conducted with
the valve positioned horizontally in order that none of the data collected in the process is
skewed.
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Figure 16. Measuring the Water Level
Before calculating the fluid’s flow rate, the water level was measured, as shown
in Figure 16. Appendix A provides the length and width of the bin in cm, which are 54.5
cm and 8.5 cm, respectively. Going into the trials, it was observed that as the number of
Tesla valve segments increased, the average water level decreased. Using the flow rate
equation provided in Appendix A, it was discovered that the lower average water level
resulted in a lower average fluid flow rate.
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Data Analysis and Interpretation
To collect reliable data on the fluid’s flow rate (cm3/s), the experiment took
account of the control, randomization, and replication of the trials. Considering the
lurking variables, it was assumed that [lurking variables] pressure (atm) and temperature
(°C) would have significant effects on the fluid’s flow rate.
During the conduction of the preliminary trials, a pocket hose was used, whose
elasticity allows it to compress or stretch. The elasticity of the pocket hose, given that it
compresses or stretches, would exert more or less pressure on the fluid going through the
hose, ultimately changing its flow rate. To prevent pressure from being a lurking variable,
a pocket hose was initially replaced with a regular 10.0 m hose for the final experimental
design. Fluid leakage from the connections between the hose and the Tesla valve
segments were blocked with gorilla duct tape, which was cut into lengths of 5.0 cm.
Last but not least, a simple random sample (SRS) on the order the trials were
conducted was done to randomize the experiment. Using a TI-Nspire calculator, the
numbers 1-10 were graphed added onto a spreadsheet; a pre-installed randomization
feature was used to order the trials without repeating the numbers. This was performed
three times to give the experiment three sets of randomized trials from 1-10. Ten trials
were conducted for one segment, two segments, and three segments of the Tesla valves
(thirty trials overall) to meet the sampling requirement for the analysis of variance
(ANOVA) test; this was to moreover provide information about the reliability of the
conclusions the data would draw.
Encabo – Pal 28
Figure 17. Box Plots of Flow Rates (cm3/s) for One Segment
Figure 18. Box Plots of Flow Rates (cm3/s) for Two Segments
Figure 19. Box Plots of Flow Rates (cm3/s) for Three Segments
Encabo – Pal 29
The box plots of the fluid’s flow rates before (blue; labeled as “flowbefore”) and
after (orange; labeled as “flowafter”) one Tesla valve segment is applied to the hose show
important differences in the minimum (min), maximum (max), first quartile (Q1), third
quartile (Q3), mean, and median values. Figure 17 shows that the first quartile and third
quartile of the orange dot plot is 27.0 cm3/s and 32.4 cm3/s, respectively. This means that
half of the flow rates after the application of the Tesla valve is between 27.0 cm3/s and
32.4 cm3/s. The range of the flow rates after the valve, given the minimum (26.3 cm3/s)
and the maximum (34.0 cm3/s), is 7.7 cm3/s. On average, the flow rate of the fluid after
the valve is 29.9 cm3/s; the median flow rate is 29.8 cm3/s, 0.1 cm3/s less than the mean.
The first quartile and third quartile of the blue dot plot is 33.2 cm3/s and 34.0 cm3/s
(respectively), meaning that half of the flow rates before the application of the Tesla
valve is between 33.2 cm3/s and 34.0 cm3/s. The range of the flow rates after the valve,
given the minimum (30.1 cm3/s) and the maximum (37.1 cm3/s), is 7.0 cm3/s. On
average, the flow rate of the fluid after the valve is 33.6 cm3/s; the median flow rate is
33.6 cm3/s, 0.2 cm3/s greater than the mean.
There are various elements in Figure 17 worth noting: for instance, over
approximately seventy-five percent of the data for the flow rate with the valve is less than
the minimum/first quartile of the flow rate (33.2 cm3/s) without the valve. Moreover,
three outliers of 30.1 cm3/s (minimum), 30.9 cm3/s, and 37.1 cm3/s (maximum) were
moreover found when conducting the experiment before the Tesla valve was applied;
these values deviate remarkably from the remainder of the fluid’s flow rates. Without
these outliers, the range of the flow rates before the valve would be significantly smaller,
Encabo – Pal 30
and the mean and median of the data would be more precise. Outliers probably occurred
E the experiment due to the various pressures the spigot exerted into the hose.
Figure 18 shows that the first quartile and third quartile of the orange dot plot is
24.7 cm3/s and 27.8 cm3/s, respectively. This means that half of the flow rates after the
application of the Tesla valve is between 24.7 cm3/s and 27.8 cm3/s. The range of the
flow rates after the valve, given the minimum (23.9 cm3/s) and the maximum (30.1
cm3/s), is 6.2 cm3/s; this is 1.5 cm3/s less than the range calculated with the information
(after the valve) from Figure 17. On average, the flow rate of the fluid after the valve is
26.4 cm3/s; the median flow rate is 26.3 cm3/s, 0.1 cm3/s less than the mean.
The first quartile and third quartile of the blue dot plot is 30.1 cm3/s and 33.2
cm3/s (respectively), meaning that half of the flow rates before the application of the
Tesla valve is between 30.1 cm3/s and 33.2 cm3/s. The range of the flow rates after the
valve, given the minimum (29.3 cm3/s) and the maximum (36.3 cm3/s), is 7.0 cm3/s; this
is equal to the range calculated with the information (before the valve) from Figure 17.
On average, the flow rate of the fluid after the valve is 32.0 cm3/s; the median flow rate is
31.7 cm3/s, 0.3 cm3/s less than the mean.
All of the data for the flow rate with the valve is less than the first quartile of the
flow rate (30.1 cm3/s) without the valve. Unlike the data provided in Figure 17, no
outliers were found. The average flow rate of the orange dot plot is 3.5 cm3/s less than the
average flow rate of the orange dot given in Figure 17. The number of Tesla valve
segments, or the length of the Tesla valve, could have played a significant role in
determining the fluid’s flow rate.
Encabo – Pal 31
Figure 19 shows that the first quartile and third quartile of the orange dot plot is
21.6 cm3/s and 23.2 cm3/s, respectively. This means that half of the flow rates after the
application of the Tesla valve is between 21.6 cm3/s and 23.2 cm3/s. The range of the
flow rates after the valve, given the minimum (20.8 cm3/s) and the maximum (24.7
cm3/s), is 3.9 cm3/s; this is 3.8 cm3/s less than the range calculated with the information
(after the valve) from Figure 17 and 2.3 cm3/s less than the range calculated with the
information (after the valve) from Figure 18. On average, the flow rate of the fluid after
the valve is 22.6 cm3/s; the median flow rate is 22.4 cm3/s, 0.2 cm3/s less than the mean.
The first quartile and third quartile of the blue dot plot is 31.7 cm3/s and 34.7 cm3/s
(respectively), meaning that half of the flow rates before the application of the Tesla
valve is between 31.7 cm3/s and 34.7 cm3/s. The range of the flow rates after the valve,
given the minimum (30.1 cm3/s) and the maximum (36.3 cm3/s), is 6.2 cm3/s; this is 0.8
cm3/s less than the ranges calculated with the information (before the valve) from Figure
17 and Figure 18. On average, the flow rate of the fluid after the valve is 33.1 cm3/s; the
median flow rate is 33.2 cm3/s, 0.1 cm3/s greater than the mean.
All of the data for the flow rate with the valve is less than the minimum flow rate
(30.1 cm3/s) without the valve. Furthermore, no outliers were found. The average flow
rate of the orange dot plot, 22.6 cm3/s, is 3.8 cm3/s less than the average flow rate of the
orange dot given in Figure 18 and 7.3 cm3/s less than the average flow rate of the orange
dot given in Figure 17. Again, the number of Tesla valve segments, or the length of the
Tesla valve, could have played a significant role in determining the fluid’s flow rate.
Encabo – Pal 32
Figure 20. F Distribution of Flow Rates (cm3/s) for One Segment
Figure 21. F Distribution of Flow Rates (cm3/s) for Two Segments
Figure 22. F Distribution of Flow Rates (cm3/s) for Three Segments
Encabo – Pal 33
F distribution histograms help show variance of the data; in this case, these
distributions help show the variance of the flow rates before (blue; labeled as
“flowbefore”) and after the valve (orange; labeled as “flowafter”), and whether they meet
the distribution requirement of the ANOVA test. Figure 20 shows that the distribution of
the flow rates, before and after the application of the Tesla valve, is normally distributed
[as shown by TI-Nspire’s probability density function (pdf) analysis]. The majority of the
found flow rates before the Tesla valve range between approximately 33.0 cm3/s and 34.0
cm3/s; the majority of the found flow rates after the Tesla valve are at approximately 26.0
cm3/s and 32.0 cm3/s. These distributions successfully meet the requirement of the
ANOVA test that they have to be normally distributed.
Figure 21 shows that the distribution of the flow rates, before and after the
application of the Tesla valve, is normally distributed [as shown by TI-Nspire’s
probability density function (pdf) analysis]. The majority of the flow rates before the
Tesla valve (blue histogram) are found at approximately 32.0 cm3/s; the majority of the
flow rates after the Tesla valve (orange histogram) are found at approximately 24.0
cm3/s, 25.0 cm3/s, and 27.0 cm3/s. These distributions successfully meet the requirement
of the ANOVA test that they have to be normally distributed.
Figure 22 shows that the distribution of the flow rates, before and after the
application of the Tesla valve, is normally distributed [as shown by TI-Nspire’s
probability density function (pdf) analysis]. The majority of the flow rates before the
Tesla valve (blue histogram) are found at approximately 32.0 cm3/s; the majority of the
flow rates after the Tesla valve (orange histogram) are found at approximately 22.0
Encabo – Pal 34
cm3/s. These distributions successfully meet the requirement of the ANOVA test that
they have to be normally distributed.
To determine whether the differences in the fluid’s flow rate before and after
applying the Tesla valve onto the hose are significant, the ANOVA test was conducted.
The ANOVA test determines whether there are any statistically significant differences
between the average flow rates one segment of the Tesla valve is applied, two segments
of the Tesla are applied, and three segments of the Tesla valve are applied.
Before conducting the ANOVA test, the following required assumptions have to
be met. As mentioned previously, a simple random sample (SRS) was conducted to
randomize the experiment and reduce bias. In addition, there needs to be at least three
independent samples. The experiment meets this need, since the three given independent
samples are: a sample given one Tesla valve segment, a sample given two Tesla valve
segments, and a sample given three Tesla valve segments. Each population furthermore
must have a normal distribution. As seen in Figures 20, 21, and 22, the distributions for
each data are fairly normally distributed; none of the collected data has resulted in
extreme skewness of the data’s distribution. It can be assumed that each data set was
pulled from a normal population. Since each population has a normal distribution, the
Central Limit Theorem verifies that the distribution of the populations makes conducting
the ANOVA test safer to do. Given that these values are only estimators of the true
population standard deviations, which remain unknown, the ANOVA test proceeded. All
of this criteria allows the validity of conducting an ANOVA test.
Encabo – Pal 35
H0: μone segment = μtwo segment = μthree segment
Ha: Not all μone segment, μtwo segment, and μthree segment are equal.
Figure 23. ANOVA Test Hypothesis
A one-way ANOVA test determines if the average flow rate when one Tesla
valve segment is applied, the average flow rate when two Tesla valve segments are
applied, and the average flow rate when three Tesla valve segments are applied are
significantly not equal. According to Figure 23, the null hypothesis states that regardless
of the number of Tesla valve segments, the means are equal (or the differences between
the means are insignificant). The alternate hypothesis implies that the number of Tesla
valve segments affect the means, meaning that not all means are equal.
"Title" "ANOVA"
"F" 25.925091157372
"PVal" 0.000000520684
"df" 2.
"SS" 264.386
"MS" 132.193
"dfError" 27.
"SSError" 137.674
"MSError" 5.09903703704
"sp" 2.25810474448
"CLowerList" "{28.424838281447,24.924838281447,21.154838281447}"
"CUpperList" "{31.355161718553,27.855161718553,24.085161718553}"
"List" "{29.89,26.39,22.62}"
Figure 24. ANOVA Test Statistics
Figure 24 shows the ANOVA test for the analysis of the significance of one, two,
and three segment tesla valves on the flow rate (cm3/s) from the TI-NSpire Student
Encabo – Pal 36
Software. A total of thirty trials, the data was consistent and fairly normally distributed in
its collection. When performing the ANOVA test on the calculator, the value for F is
approximately the same, with a p-value of approximately 5.21 × 10−7 , which would be
lower than the default alpha level of five percent (0.05). To verify the accuracy of the
calculator’s results, the ANOVA test was re-conducted using the procedures in Appendix
A. The p-value using the steps in Appendix A was very similar to the p-value using the
calculator. Hence, the conclusion from the ANOVA test can still be upheld. Knowing that
the p-value is approximately 5.21 × 10−7, which is lower than the alpha level of 0.05,
the test rejected the null hypothesis. There is significant evidence that the mean flow rates
of the one Tesla valve segment, two Tesla valve segments, and three Tesla valve
segments are not equal. There is approximately a zero percent chance of getting the
means again by chance alone if the null hypothesis was true.
Encabo – Pal 37
Conclusion
The purpose of this research was to determine the effect that the Tesla valve at
different lengths (one segment, two segments, and three segments) has on the flow rate of
a fluid (cm3/s). Prior to the experiment, the researchers predicted that the flow rate
(cm3/s) was the highest in a one-segment Tesla valve and the lowest in a three-segment
Tesla valve. The researchers accepted the hypothesis, given that the data agreed with the
prediction and that the average flow rates were significantly different.
The flow rate was found by measuring the fluid level rise (cm) in sixty seconds,
multiplying it by the bin’s length (cm) and width (cm) [to find the volume (cm3)], and
dividing the volume by the time (sixty seconds). The average flow rates of the fluid in
valves of one, two, and three segments were 29.9 cm3/s, 26.4 cm3/s, and 22.6 cm3/s,
respectively. (The three average flow rates before applying the Tesla valve to the hose
were 33.4 cm3/s, 32.0 cm3/s, and 33.1 cm3/s, in respective to their sets.)
In accepting the hypothesis, the researchers looked at two variables: the individual
flow rates (or the flow rate in a trial) and the average flow rates. The flow rates from
table 1, table 2, and table 3 showed that the average flow rate decreased as the number of
Tesla valve segments increased. This was initially interpreted according to the differences
between the flow rate before and after the valve was applied. With one segment, the
researchers found a 3.5 cm3/s difference between the flow rate before and after the
application of the valve; with two and three segments, they found a respective 5.6 cm3/s
and 10.5 cm3/s difference between the flow rate before and after the application of the
valve. The boxplots and histograms from the descriptive statistics further contributed to
the idea that the difference between the average flow rates before and after the valve
Encabo – Pal 38
increased as the number of segments increased. As the number of segments increased, the
researchers saw the minimum and maximum flow rates after the application of the valve
decrease. For instance, box plots showed that when two valve segments were applied to
the hose, the minimum flow rate dropped from 26.3 cm3/s to 23.9 cm3/s.
To determine whether the differences between the average flow rates were
significant, a one-way ANOVA test was conducted. The ANOVA test calculated an Fstatistic of 25.9, which the researchers used to find a p-value of 5.21 * 10-7 . Since the pvalue was lower than the alpha level (0.05), it can be said that there was evidence that the
average flow rates in one-segment, two-segment, and three-segment Tesla valves were
significantly different. Going back to the average flow rates, the average flow rate was
the highest in a one-segment Tesla valve and the lowest in a three-segment Tesla valve:
the average flow rate in one segment was 29.9 cm3/s; the average flow rate in two
segments was 26.4 cm3/s; the average flow rate in three segments was 22.6 cm3/s.
Figure 25. Tesla Valve Design
Looking back at the Tesla valve’s design (as shown in Figure 25), the Tesla
valve’s loops allowed fluid to flow through it preferentially in only one direction. If the
fluid flowed through the valve from the right to the left side, it would flow through a
Encabo – Pal 39
nearly-straight path. The fluid experienced the least resistance from the right side to the
left side, so the fluid flowed through the valve with little to no change. Nonetheless, the
researchers tested the flow going from the left side to the right side. From the left side to
the right side, the segments restricted the airflow through the valve by forcing the fluid to
go left or right of each loop. Ultimately, the fluid that moved right would return against
the flow, and the fluid that moved left would interact with the fluid that returned against
the flow. Generally, the fluid itself became the force that restricted it to flow at a rate
nearly-equal or equal to the flow rate before the application of the valve; thus it explains
why the flow rate decreased as the number of segments increased: the fluid experienced
greater force of flow as it approached the end of the valve, resultantly creating a greater
resistance towards the flow and decreasing the flow rate.
The results of this research agreed with the current scientific work with valves. As
previously mentioned, the research’s data showed that adding the number of Tesla valve
segments increased the number of loops the fluid flowed through, eventually increasing
the force of flow and resistance the decreasing the flow rate. When taking account what
the scientific community can gather from this experiment, the tests proved the Tesla
valves’ effectiveness to reduce the fluid’s flow rate, making the Tesla valve an efficient
alternative to the one-way check valve. Scientists and engineers looking see the Tesla
valve’s utilization and optimization in oil production systems, motor vehicles, and
machines in private households have discovered that the Tesla valve’s use of less moving
parts made the valve more efficient. Regardless of the material, moving parts in several
conventional valves tend to wear out in a couple of years. Given that tesla valves do not
use moving parts, they can be better depending on what the application is due to the fact
Encabo – Pal 40
that any moving part will wear over time no matter what the material used in it. The
scientific community can use the research’s information to improve efficiency in various
applications. These include, but are not limited to, regulating the flow of air into the
vehicle’s motors to keep the vehicle’s engine running efficiently at all speeds, shutting
off water going through pipes in dishwashers and sinks, and taking in air in fuel cells to
allow fuel in motor vehicles to be drawn out. Again, the average flow rates and
differences between the flow rates (before and after the valve’s utilization) show that the
Tesla valve significantly affected the flow rate (“Valve Applications”).
On the weaknesses, fluid leakage occurred in the connection of parts, such as the
connection between the valve and the hose. Because of this, the researchers observed a
lower fluid level. Fluid leakage reduced the pressure exerted against the fluid’s flow,
increasing the flow rate of the fluid within the valve itself. Efforts were made to reduce
the leakage by sealing all connections with hot glue and gorilla tape. This procedure
helped prevent the fluid from exiting the hose and the valve, maintaining the consistency
of the flow rates throughout the experiment. Additionally, one of the valve segments
missed a part of one loop; the researchers discovered this issue when they found that
water exited from the back of the Tesla valve. The missing part was subsequently glued
back onto the segment, and the valve performed as normal.
Future research could be conducted to further determine the Tesla valve’s
effectiveness on the flow rate of a fluid. For instance, future research could emphasize the
performance of the Tesla valve with fluids of greater viscosity, including motor oil,
molasses, and vegetable glycerin; it could additionally emphasize the performance of the
Tesla valve with fluids of different temperatures, such as a low temperature of 10° C, a
Encabo – Pal 41
standard temperature of 20° C, and a high temperature of 30° C. Fluid temperatures could
determine whether the Tesla valve significantly changes the flow of any liquid, regardless
of its condition. Determining the valve’s effectiveness on fluids of various temperatures
is imperative to the research, since engineers, for example, have found that motor vehicle
engines operate at very high temperatures. It results in the new question of how Tesla
valves fare in different environments. Last but not least, Tesla valves with more loops
could be tested, since the experiment proved that the flow rate decreased as the number of
segments increased. The researchers could test a Tesla valve with an excessive number of
loops to see whether the valve would almost completely (if not entirely) halt the fluid.
The conclusion made from the experiment’s results was essential for future research.
Encabo – Pal 42
Acknowledgements
In the course of conducting the preliminary and final work of our experiment,
Mikolaj and I underwent issues with our general experimental design and have often
questioned its significance to the scientific community given that the procedures to the
experiment are incredibly basic. We received some backlash from our peers, who claimed
that the research we were doing was “too elementary” or “something a child could do.”
The simplicity of the research kept it from being able to generate new questions about the
optimization of Tesla valves. This was important since the intention of this research is to
expand people’s beliefs on what a Tesla valve could be utilized and optimized to do.
Thankfully, our research has been improved, and parts of it have been justified by
our teachers. Without the assistance of our teachers from the Macomb Mathematics
Science and Technology Center, our research would not have reached a certain quality
we desired it to have.
We would like to initially thank Mr. McMillan for providing class time to conduct
our trials, allowing us to conduct our experiment using the clamps that were available in
his classroom, and simplifying our experimental design. Mikolaj and I initially had it in
order that we determined the flow rate of the fluid by finding the time it took for the
fluid’s height to increase from 10.0 cm to 50.0 cm. McMillan suggested that our
experiment could be made simpler by measuring the fluid’s height in a sixty-second time
interval. As a result, our trials were conducted more quickly, and we completed our
experiment faster than we expected. He additionally provided us an important lesson
about simplicity: research should not always emphasize its complexity. Too many
students had great ideas for their research but made their experimental designs more
Encabo – Pal 43
complicated than they should be. It made us believe over the thought that our experiment
was too simple. Generally, it was one idea about the experimental design that we were
doing correctly (and that most groups were doing incorrectly).
Speaking of being able to conduct our experiment, we are thankful for Mr. Supal
who allowed the students to use his computer lab during class periods. He permitted us to
use the lab sink, which was imperative to conducting the experiment. Without it, we
would have had to devise an alternative experimental design which could have been
incredibly expensive. (Fact: the cost of the bin alone was around thirty dollars.)
Furthermore, we greatly appreciate Mr. Acre’s assistance with the type of testing
that our research should conduct. We were stuck between doing a design of experiment, a
t-test, and an ANOVA test. At first, we believed a design of experiment would be the
most effective in determining the Tesla valve’s significance on the fluid’s flow rate.
However, Mr. Acre and several others have asserted that we were only complicating the
path to our goal by performing a design of experiment. He told us to conduct an ANOVA
test instead, because 1) our final experimental design used three populations and 2) we
were trying to determine whether the differences between the average flow rates were
significant.
Again, we would like to sincerely thank them for assisting us with our research.
Most importantly, however, we would like to give thanks to the entire MMSTC program.
The program in general has helped us to prepare for college by requiring us (since
freshmen year) to do the kind of research people would do there. More than ever, we feel
prepared and confident about doing the same type of work in college.
Encabo – Pal 44
Appendix A: Statistical Equations and Sample Calculations
A. Finding the Flow Rate (cm3/s):
Table 4
Bin Length and Width
Length
(cm)
Width
(cm)
54.5
8.5
Table 4 provides the given bin length (cm) and width (cm), which is respectively
at 54.5 cm and 8.5 cm. The length is multiplied by the width, and the product of these
values are multiplied by the water level (or height) after sixty seconds to calculate the
volume (cm3) before or after the application of the Tesla valve.
V lwh
Q= =
t
60
Figure 26. Flow Rate Equation
According to Figure 26, to calculate the flow rate (Q) of the fluid multiply the
length of the bin, and width of the bin, and the height of the water. Divide the product by
the time, which is sixty seconds for all trials.
B. Conducting the ANOVA Test:
An ANOVA test is a statistical analysis tool that separates the variance found in
the set of data into random factors and systematic factors. Random factors have no
significant influence on the data; systematic factors have significant influence on the
Encabo – Pal 45
data. An ANOVA test allows researchers to compare the means of three or more
populations and determine whether or not there are any significant differences.
Before conducting the ANOVA test, the following required assumptions have to
be met. As mentioned previously, a simple random sample (SRS) was conducted to
randomize the experiment and reduce bias. In addition, there needs to be at least three
independent samples. The experiment meets this need, since the three given independent
samples are: a sample given one Tesla valve segment, a sample given two Tesla valve
segments, and a sample given three Tesla valve segments. Each population furthermore
must have a normal distribution. As seen in Figures 20, 21, and 22, the distributions for
each data are fairly normally distributed; none of the collected data has resulted in
extreme skewness of the data’s distribution. It can be assumed that each data set was
pulled from a normal population. Since each population has a normal distribution, the
Central Limit Theorem verifies that the distribution of the populations makes conducting
the ANOVA test safer to do. Given that these values are only estimators of the true
population standard deviations, which remain unknown, the ANOVA test proceeded. All
of this criteria allows the validity of conducting an ANOVA test.
Table 5
Sample Sizes, Means, and Standard Deviations for Tesla Valve Segments
# of
Segments
ni
x̅i
si
1
10
29.9
3.01
2
10
26.4
2.21
3
10
22.6
1.16
Encabo – Pal 46
Table 5 provides the sample sizes ni, means x̅i, and standard deviations si for each
of the valves. The sample size is the number of trials completed; ten trials were
conducted for a Tesla valve with one segment, ten trials were conducted for a Tesla with
two segments, and ten trials were conducted for a Tesla valve with three segments.
According to the data from the experiment, the average fluid flow rates for a Tesla valve
with one, two, and three segments are 29.9 cm3/s, 26.4 cm3/s, and 22.6 cm3/s,
respectively. The standard deviation is a measure of how spread out the flow rates from
the trials are compared to the average flow rate. Given the data from the trials, the means
and standard deviations for each segment were calculated using one-variable statistics on
the TI-Nspire student software.
df = n − 1
df = 3 − 1
df = 2
Figure 27. Calculating the Degrees of Freedom
Given the number of categories n, the degrees of freedom df can be calculated by
subtracting the number of categories by one. The number of categories n is three, since
there were three different valves used in the experiment. Using the steps seen in Figure
27, the degrees of freedom are two.
Encabo – Pal 47
# 𝑜𝑓 𝑠𝑒𝑔𝑚𝑒𝑛𝑡𝑠
X̅ =
∑𝑖=1
𝑛𝑖 𝑥𝑖
=
𝑁
(𝑛1 ∗ 𝑥̅1 ) + (𝑛2 ∗ 𝑥̅2 ) + (𝑛3 ∗ 𝑥̅3 )
𝑁
(10 ∗ 29.9) + (10 ∗ 26.4) + (10 ∗ 22.6)
X̅ =
X̅ = 26.3
30
Figure 28. Calculating the Grand Mean
Given the sample sizes n, means x̅, number of trials N, and standard deviations s
for Tesla valves of one segment, two segments, and three segments, the grand mean X̅
can be calculated by finding the sum of the sample sizes multiplied by their respective
means and dividing it by the number of trials. The population N is 30, since ten trials of
each material were conducted. Using the steps seen in Figure 28, the grand mean is
approximately 26.3.
MSG = Mean Square Group
# of segments
MSG =
∑i=1
ni (xi −X)2
=
I−1
n1 (x̅1 − X̅)2 + n2 (x̅2 − X̅)2 + n3 (x̅3 − X̅)2
MSG =
I−1
10(29.9 − 26.3)2 + 10(26.4 − 26.3)2 + 10(22.4 − 26.3)2
3−1
MSG =
MSG = 140.9
129.6 + 0.1 + 152.1
2
Figure 29. Calculating Mean Square Group (MSG)
Encabo – Pal 48
Given the sample sizes n, means x̅, number of populations I, and standard
deviations s for Tesla valves of one segment, two segments, and three segments, the
mean square group (MSG) could be calculated by finding the sum of the sample sizes
multiplied by the squared difference between the means and the grand mean and dividing
by one less than the number of populations. Using the steps seen in Figure 29, the mean
square group is approximately 140.9.
MSE = Mean Square ErrorMSE =
# of segments
∑i=1
(ni −1)si 2
=
N
2
(n1 − 1)s1 + (n2 − 1)s22 + (n3 − 1)s32
MSE =
N−I
(10−1)3.012 + (10−1)2.212 + (10−1)1.162
30−3
81.5409 + 43.9569+12.1104
MSE =
MSE =5.1
27
Figure 30. Calculating Mean Square Error (MSE)
Given the sample sizes n, number of trials N, number of populations I, and
standard deviations s for Tesla valves of one segment, two segments, and three segments,
the mean square error (MSE) can be calculated by finding the sum of the squared standard
deviations multiplied by one less than the sample sizes and dividing it by the difference
between the number of trials and the number of populations. There are thirty trials and
three unique populations. Using the steps seen in Figure 30, the mean square error is
approximately 5.1.
Encabo – Pal 49
F=
𝑚𝑒𝑎𝑛 𝑠𝑞𝑢𝑎𝑟𝑒 𝑔𝑟𝑜𝑢𝑝
𝑚𝑒𝑎𝑛 𝑠𝑞𝑢𝑎𝑟𝑒 𝑒𝑟𝑟𝑜𝑟
=
𝑀𝑆𝑏𝑒𝑡𝑤𝑒𝑒𝑛
𝑀𝑆𝑤𝑖𝑡ℎ𝑖𝑛
=
140.9
5.1
= 27.6
Figure 31. Calculating the F-Statistic
Given the values for the mean square group (Figure 29) and the mean square error
(Figure 30), F can be calculated by dividing the mean square group by the mean square
error, resulting in 27.6 as shown in Figure 31.
Encabo – Pal 50
Appendix B: Professional Contact
Senior Research Professional Consultant Contact Form
Names: Red Encabo, Mikolaj Pal
Research Topic: The Effect of Tesla Valves on the Fluid Flow Rate
Professional Contact Information
Name: Ronald Louis Bardell
Title: Doctor of Philosophy
Organization: Microplumbers Microsciences LLC
Phone (Area Code and Extension): (612) 605-1574
Email: [email protected]
Mailing Address: 5530 Canfield Pl N, Seattle, Washington
Dialogue Information
1. Contact Goal:
We wanted to learn how to experiment with Tesla valves and analyze their effect on the
flow rate of a fluid.
2. At least three potential questions to help reach your goal:
A. Given that you have done research at the University of Washington on valves and are
currently operating a business that emphasizes fluid viscosity (MicroPlumbers
Microsciences, LLC.), you have the proficient experience in testing the valves. What are
the three factors you would say are the most important in determining fluid viscosity?
B. How can we test Tesla valves and compare them with other conventional one-way
valves?
C. Part of your work also emphasizes simulation, design, and modeling; how can we
simulate, model, and design our experiment to make our analysis of the regulation of
flow within valves more vivid?
D. How does your analysis work (please explain with depth)?
3. Additional Information
Encabo – Pal 51
Response:
“Hi Red and Mikolaj,
The Tesla valve differs from the other two valves. It doesn't stop flow. It could be
considered a "leaky" valve, because it simply slows the flow more in one direction
through the valve than flow in the other direction.
The Wikipedia page:
https://en.wikipedia.org/wiki/Viscosity
has a very good discussion of viscosity.
The most detailed description of our use and modeling of Tesla valves is in my
dissertation. Please find it attached (Section 1.5 on page 15 describes what is contained
in each chapter). We used them as a component of micropumps, applying oscillatory flow
to create a net flow in one direction.
I hope the success of your studies reflects all your hard work and you have great
success.”
Encabo – Pal 52
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Encabo – Pal 53
Mechanism in Different Tesla-Type Microvalves." Journal of Applied Research
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Sept. 2016. <http://www.nsti.org/procs/Nanotech2003v1/9/M72.05>.
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<http://www.globalspec.com/pfdetail/valves/applications>.
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<https://www.kitz.co.jp/english2/type_checkvalve.html>.
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<http://fluidpowerjournal.com/2013/10/teslas-conduit/>.