Journal of Civil Engineering and Architecture 13 (2019) 195-203
doi: 10.17265/1934-7359/2019.03.004
D
DAVID
PUBLISHING
Discharge Coefficient Measurements Using Heron’s
Fountain
Gevo Abcarian, Zainab Algharib, Omran Hussain, Ana Martin, Abraham Villa, Francisco Villalobos, Tadeh
Zirakian and David Boyajian
Department of Civil Engineering and Construction Management, California State University, Northridge, CA 91330, USA
Abstract: Civil Engineering design students at CSUN (California State University, Northridge), aimed to demonstrate the pneumatic
action of liquid water as it flows through an airtight one-way vessel system which is known as Heron’s Fountain. This project explores
hydraulic and pneumatics principles commonly found in environment control systems, such as the non-isothermal heating facilities
located on the CSUN campus. Since this was a simply constructed version of an ancient Greek fountain, its development required the
collaboration of the team to execute its simple function. The parameters involved were diameter, length, height, and density. This
analysis utilizes Pascal and Bernoulli’s equations to reinforce the principles of fluid mechanics. The fountain action is described based
on flow rate and head loss is described by Darcy’s equation. Friction loss with an angled fitting attached to the fountain head is
described by Reynold’s equation. The experiment observed the performances of two types of reentrant tube fittings for head loss:
straight and angled. The experiment enhanced the educational experience of the research team by bringing together creative ideas from
different educational and cultural backgrounds. The results of the experiment concluded with a 0.58% error for the straight fitting and
5.3% error for an angled fitting.
Key words: Heron’s Fountain, hydraulic principles, air pressure, pneumatics, friction factor, engineering education.
1. Introduction
Heron’s Fountain begins in antiquity with the Greek
mathematician and engineer, Heron of Alexandria. He
invented mechanical devices powered by air, water and
steam which were used for a variety of reasons. He
enjoyed using his inventions for educational purposes
and taught pneumatic principles by how his devices
worked. Heron’s Fountain is an apparatus which
responds when liquid water is added to the fountain
basin and generates a water jet from the fountain head.
This instrument operates using pneumatic principles
and the principles of non-isothermal flow [1].
Pneumatic systems are commonly found in dams,
transmission systems and power stations. Teaching the
principles of hydraulics to aspiring engineers can be
challenging but rewarding in the end [2]. Heron’s
Corresponding author: Tadeh Zirakian, Ph.D., P.E.,
assistant professor, research fields: civil engineering,
structural/earthquake engineering, engineering education.
Fountain is a device used to teach and demonstrate the
principles of hydraulics with water and air pressure.
Many scientists have dreams of perpetual motion
machines that do not dissipate energy. This fountain is
not continuous and therefore it is not perpetual since it
only works for a certain amount of time until the
velocity becomes zero [3]. The interesting device is a
demonstration of hydrostatic pressure. Heron’s
Fountain is the first model to incorporate both
automatic recharging energy and flow-triggering of the
fountain [4]. Unfortunately, the fountain has several
setbacks that can only be re-energized by some manual
transfer of liquid which can cause liquid spillage. The
team constructed the project by using plastic tanks and
vinyl tubing to obtain the flow rate of the water jet and
compared it to the theoretical values derived from
Pascal’s and Bernoulli’s equations [5, 6]. Comparisons
of the results were made and studied after various trials.
The objective of this project is to enhance the
knowledge about the principles of hydraulics in a
196
Discharge Coefficient Measurements Using Heron’s Fountain
simple manner and to provide a better understanding of
air pressure using this educational design.
The inventor, Heron of Alexandria sold his
inventions to wealthy patrons who enjoyed the
splendor of his mechanical inventions as centerpieces
for grand occasions. Fig. 1 illustrates the original
invention called “A Satyr Pouring Water from a
Wine-skin into a full Washing Basin, without making
the contents overflow” also known as Heron’s
Fountain.
2. Physical Model Details and Testing
The research team gathered reusable materials that
allowed better visibility of the liquid as it runs
throughout the system. The materials chosen to achieve
best results were three 1.3-gallon containers with
plastic lids, vinyl tubing and adhesive shown in Table 1.
It is best to construct Heron’s Fountain as air-tight as
possible in order to achieve best results.
2.1 Construction Procedures
Three containers were aligned in a stacked
configuration. Three pieces of vinyl tubing of different
diameters were cut into the appropriate lengths. Holes
were drilled through the lids to pass the tubing through.
With the holes aligned, the water-resistant adhesive
was applied to secure the assembly and create an
air-tight vessel. Silicone adhesive was also applied on
the lid threads and sealed with adhesive tape. Fig. 2
illustrates the placement of the parts. As indicated in
Fig. 1 Original drawing of Heron’s Fountain [3].
Table 1
Count
1
1
1
1
1
Orifice fittings and various vinyl tubing sizes.
Dimension
½”
½”
¾” - 24.1” (Tube #1)
¼” - 6.50” (Tube #2)
½” - 11.8” (Tube #3)
Material/Items
Straight fitting
90° fitting
Clear vinyl tube
Clear vinyl tube
Clear vinyl tube
Discharge Coefficient Measurements Using Heron’s Fountain
197
Fig. 2 Project drawing of a custom-made Heron’s Fountain based on original construction.
the drawing, Tank 1 is open to the atmosphere while
the other tanks are closed. Tank 3 carries the stream
into the basin. The fountain process was initiated by
pouring water into the basin. Fig. 7, provided in the
Appendix, shows the final constructed fountain.
2.2 Test Procedures
Tank 2 was filled with one gallon of warm water
funneled through Tube 3. Warm water will increase
internal pressure in Tank 2 as water drains into Tank 3.
The rising water level inside Tank 3 generates internal
air pressure, forcing air through Tube 2 into Tank 2. As
Tank 2 increases in internal air pressure, this pressure
forces the reserve water through Tube 3 up into the
basin of Tank 1. Tube 3 is the point of the experimental
control where the two fittings are attached. The
hydraulic pressure is created as water enters Tank 1 and
increased air pressure in Tank 3. Tank 2 creates a rising
jet of water through an orifice attached to Tube 3 in
Tank 1. The performance tests occur at this point with
the two fittings by testing the flow rate, , of each
orifice.
3. Theoretical Analysis
Simplicity is at the core design and construction. In
the development of the project, the use of imagination
and creative thinking were used to test the performance
of a variety of orifices attached to the fountain’s head
as shown in Fig. 3. The air pressure, , in Tank 2 was
determined using Eq. (1), the moment air was forced
into Tank 2. Before this movement occurs the velocity
of the water-level is zero. The velocity V of the water
jet was determined using Bernoulli’s Eq. (2) in order to
calculate the flow rate using Eq. (3).
(1)
where,
is pressure inside the middle container,
is gravitational acceleration,
is water density,
is height of water life,
is air density, and
is
height of air.
(2)
where,
is specific weight of fluid,
and
is the datum height.
where,
is flow rate and
is velocity
(3)
is cross sectional area.
4. Discussion of Results
To find the flow rate between Tank 1 and Tank 2,
Bernoulli’s equation was used.
(4)
198
Discharge Coefficient Measurements Using Heron’s Fountain
where, head losses =
= pipe loss + fittings losses
and Pipe loss is described as:
(5)
the expression
was used for the fittings.
Therefore,
is classified for each type of fitting.
The general equation for head loss can be described
as follows:
Eq. (4) was applied between Tank 3 and Tank 2
where
, therefore
and
cancel. This
yields the following equation to solve for :
(8)
Pressure at each point is determined by applying Eq.
(4) with the head losses to find the velocities in the
pipes.
Eqs. (4) and (5) were applied to Tank 1 and Tank 2 to
solve for
. The theoretical flow rate of the straight
fitting
∑
(6)
The
factors represent the friction coefficients of
each fitting and are shown in Table 2 [5].
These angled fittings are attached at the threated
side which is indicated as the entrance, as shown in
Fig. 3.
Eq. (4) is used between Tank 1 and Tank 3 to
determine pressure at Tank 3,
. Because the
cross-sectional area of Tank 1 is equal to the
cross-sectional area of Tank 3,
therefore
and
cancel and create the following expression
0.
(7)
Table 2
0.0067
is
experimental average flow rate concluded was
#1 Straight fitting
#2 90° fitting
K1
Entrance
0.78
0.78
Fig. 3 Fittings: ½” - 90° (elbow) and ½” - 180° (straight).
The
=
3
0.00705 [ft /sec]. The expression used to determine
this percentage is:
%
100
(9)
This yields the percent error to be 5.3%.
The theoretical flow rate of the angle fitting is
. The
0.0058
average flow rate concluded was
3
experimental
= 0.00566
[ft /sec]. yielding to a percent error of 0.58%. Sample
calculations of friction factor ( ) are provided in the
Appendix.
Friction constant values of each reentrant fitting.
Experiment
.
K2
Elbow
30
K3
Exit
1
1
Discharge Coefficient Measurements Using Heron’s Fountain
The experimental research indicates the flow rates of
angled and straight fittings at three separate trials. As
shown in Figs. 4 and 5, the correlation among these
flow rates per each trial for angled and straight fittings
is 99% based on the R2 values. Due to a lower friction
factor in the straight fitting attachment, the flow rate
through the system was more efficient.
The theoretical velocity for the straight fitting was
4.91 [ft/s] while the angled fitting was 4.23 [ft/s]. The
experimental velocity for the straight fitting was 5.2
[ft/s] while the angled fitting was 4.29 [ft/s]. The
velocity value in the angled fitting is less than the
velocity in the straight fitting for both the experimental
and the theoretical due to the head loss in the angled
fitting. The research team came to the conclusion that
the percentage error for the straight fitting and the
angle fitting were due to the size of the tubes, since
Tubes 2 and 3 are smaller in diameter compared to
Tube 1. Minor leakage of air pressure tends to give
such percentage of errors.
Straight Fitting Flowrate (ft3/s)
0.008
0.00705
0.00646
Flowrate, Q ( ft3/s )
0.007
R² = 0.9904
0.00563
0.006
0.005
0.004
0.003
0.002
0.001
0.000
1
2
3
Trial
Fig. 4 Straight fitting flow rate per trial.
Angled Fitting Flowrate (ft3/s)
0.007
R² = 0.9937
Flowrate, Q (ft3/s)
0.006
0.00566
0.00549
0.00537
1
2
3
0.005
0.004
0.003
0.002
0.001
0.000
Trial
Fig. 5 Angled fitting flow rate per trial.
199
200
Discharge Coefficient Measurements Using Heron’s Fountain
Fig. 6 Group member Francisco Villalobos demonstrates Hero’s fountain at student youth STEM program.
5. Educational Objectives
Each member in the research team was able to
understand basic hydraulic principles that this device
teaches. With students having different backgrounds
and different levels of knowledge about the subject, it
was necessary to discuss and conduct research about
the topic with one another throughout meetings.
Learning about hydraulics in this simple manner will
provide a better comprehension in future related
classes and professional practice. Furthermore, the
members were able to gain theoretical knowledge on
how to apply Bernoulli’s equation to a hydraulic
mechanism
in
which
enhances
members’
understanding about the analysis and properties of
fluids.
Every member in the research team comes from a
minority background, including women in engineering.
Through teamwork and creativity, the students were
able to bring this effort to light, despite different
cultural backgrounds. This model could be used to cast
attention on science and engineering to young high
school students, as it can be explored by them
physically. In addition, the members of the research
team can serve as role models to younger minority
students. This effort was demonstrated to the student
youth at a Los Angeles STEM program as seen in Fig.
6, which carries on the tradition of hands on learning
and teamwork.
During this demonstration, the children were able to
have a hands-on learning experience by capitalizing on
the fountains size.
This presentation included a brief presentation with
slides of pictures, graphs and results. This collaborative
effort utilized simple techniques and methods to
conduct the experiment and produce results. The
design and construction used a creative approach to test
the flow rates of two types of orifice reentrant
fittings
attached
to
the
fountain’s head.
The
collaborative nature of the project brought purpose and
meaning to the project while enriching the participation
in the research of this ancient invention that was
modeled for teaching purposes. Heron’s invention was
used as a basis to test the performance of the attached
fittings.
6. Conclusion
In this educational research, Heron’s Fountain was
designed,
constructed,
tested,
and
theoretically
analyzed. This project was conducted to find the flow
rate of two different orifice fittings, including a straight
orifice and an angled orifice. Many principles have
been used in this project such as, the flow rate equation
and the Bernoulli’s, Darcy’s, and Reynold’s equations.
Therefore, this research effort has enhanced the usage
of such equations and engineering applications. Some
challenges the research team experienced included,
where to begin the construction, finding the pressure of
the Tanks, and preventing leakage. The team was led by
Discharge Coefficient Measurements Using Heron’s Fountain
women in engineering and with their innovative ideas,
References
the goal was achieved after discussing different
[1]
approaches that could be applied to this project.
Through intense research, the team was able to find a
solution to finalize the Heron’s Fountain project.
[2]
Acknowledgments
The student research team would like to express their
sincere and profound appreciation for the support
provided by Dr. Tadeh Zirakian and Dr. David
Boyajian, Professors of Civil Engineering in the
Department of Civil Engineering and Construction
Management at California State University,
Northridge, and also Marcial Martin, Engineer Advisor
Process and Mechanical at STANTEC, Bakersfield,
CA, USA.
[3]
[4]
[5]
[6]
201
Kezerashvili, R., and Sapozhnikov, A. 2003. “Magic
Fountain.” Arxiv ID: physics/0310039. ArXiv.org,
Cornell University.
Georgescu, A.-M., Georgescu, S.-C., and Stroia, L. 2014.
“Heron’s Fontaine Demonstrator. Revista Română de
Inginerie Civilă.” Romanian Journal for Civil
Engineering 5: 87-94.
Greenwood, J., and Woodcroft, B. 1971. “The
Pneumatics of Hero of Alexandria” introduced by Marie
Boas Hall.
Ong, P. P. 1992. “Hero’s Fountain: Reversible Model.
(Apparatus for Demonstrating Hydrostatics).” The
Physics Teacher 30 (7): 436.
Lindeburg, M. 2018. PE Civil Reference Manual (16th
ed.). Belmont, California: Professional Publications, Inc.
Gerhart, P. M., Gerhart, A. L., and Hochstein, J. I. 2016.
Munson, Young, and Okiishi’s Fundamentals of Fluid
Mechanics (8th ed.) New Jersey: Wiley.
Discharge Coefficient Measurements Using Heron’s Fountain
202
Appendix
Fig. 7 Final construction of Heron’s Fountain. Based on the schematic diagram the fountain was constructed and tested
until there was no leakage.
In addition, sample calculations for determination of the friction factor are also provided in the following. Two methods were used to
solve for the friction factor for A. the straight fitting and B. the angle fitting.
A. Solving for the friction factor as a function of Reynold’s Number
Considering
0.0006
for a plastic pipe, we can get:
0.0001
An initial approximation of Reynold’s can be obtained by using the experimental velocity
0.0000092
Therefore, the flow is turbulent:
ft
@ 80
sec
23550
4000
5.2
Discharge Coefficient Measurements Using Heron’s Fountain
From the Moody Diagram with
0.0001 and
19300, therefore is f = 0.0278
B. Solving for the friction factor using the experimental velocity
As a first approximation
4.26
, so
23550
Therefore, the flow is turbulent:
From the Moody Diagram with
0.0001 and
4000
19300, therefore is
Note that only in laminar flows does the following apply (Re < 2100):
= 0.028.
.
203