Design Project: A System of Fountains

Haotian Qiu, Omar Darwish, Tim Daniel ABJ, Monday 10-11:50 am

Jimmy Kim November 28, 2016

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I. Introduction

Whether they are used as an aesthetic display to impress and awe people or whether they are used

in your local school to drink water out of, fountains are found throughout society. Their widespread

use makes the study and understanding of their function an important area of study.

Our challenge is to be able to construct two water fountains that are constructed by splitting up

one source of water flow into two separate sources. The first source/fountain is required to hit a

target 3 feet away and be running at a constant flow rate of 3.6 liters per minute. From basic

kinematics, we concluded that the optimal angle need to achieve a distance of 3 feet was about

26.15°. The second fountain needs to be able to balance a Ping-Pong ball up in the air at a distance

of 15 inches above the pipe outlet. The ball is able to balance on the stream of water because the

pressure is lower on the film of water around the ball than the pressure of the stream. We

conducted several experiments and concluded that for our specific design, the optimal launch angle

needed to achieve this was 84° for a 1/4-inch inner diameter pipe. Both fountains are required to

exit the pipes at the same elevation.

Both major and minor head losses of the fountain system components are determined to create the

proper head losses that will allow the fountain system to meet the design requirements. These

losses depend on fluid inertia, surface roughness, pipe length, and viscous stresses. Major head

losses are those due to viscous interaction of the fluid with the pipe walls through straight portions

of pipe. It is determined by equation (1)

ℎ",$%&'( = 𝑓 ,-∙ /

0

12 (1)

where v is velocity of the flow, L is the pipe length, D is the pipe diameter, g is the gravitational

constant, and f is called the friction factor, which is defined by equation (2)

𝑓 = 3453040∗7/

0-,

(2)

2

where p1 - p2 is the pressure drop across the pipe length, and 𝜌 is the density of the fluid, in this

case water. Since the pipes provided are made out of acrylic, we assume that they are smooth and

there are no losses due to roughness. Our experimental data will ideally confirm this assumption.

Major losses can be estimated using the Moody diagram, which is a plot that corresponds a friction

factor to a certain Reynold’s number for a given pipe roughness. Additionally, the theoretical

friction factors can be calculated by the Haaland equation shown in equation (3)

9:= −1.8 ∗ 𝑙𝑜𝑔 B/-

D.E

9.99+ G.H

IJ (3)

where ε is the absolute roughness of the pipe and Re is the Reynold’s number.

Minor head losses are those due to inertial forces on the fluid that arise from bends are connectors.

and they are a function of geometry. They are calculated by equation (4)

ℎ",$KL'( =MN/0

12 (4)

where 𝐾, is a constant called the loss coefficient which is defined by equation (5)

𝐾, =1 34530

7/0 (5)

We assume the water flow inside the pipes obeys the Bernoulli Equation. Using the rearranged

Bernoulli equation, equation (6), with the addition of a head loss term, we are able to solve for the

total head loss needed between the two fountain paths to create a fountain that meets the design

requirements. In equation (6), va and vb are the exit velocities of the horizontal arcing fountain and

the sphere supporting fountain, respectively.

𝐻Q'Q%" =91𝑔 𝑣S1 − 𝑣%1 (6)

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A key assumption of the Bernoulli equation is that the fluid is incompressible. With low velocities

used in this lab, the assumption holds true. The fluid velocity and flow rate can be solved using

equation (7), where A is the cross sectional area of the pipe

𝐹𝑙𝑜𝑤𝑅𝑎𝑡𝑒 = 𝜌𝑣𝐴 (7)

Flows will be analyzed at a range of Reynold’s numbers to observe how the head losses are

affected. Reynold’s number is calculated by equation (7), where µ is the viscosity of the fluid.

𝑅𝑒 = 7/-\

(8)

II. Initial Design

a. Arcing Fountain

The assigned flow rate for the arcing fountain is 3.6 L/min and it is supposed to hit the target 3ft

(0.914 m) away. The initial velocity can be calculated using the following equation:

𝑉KLKQK%" =^_ (9)

Q = 3.6L/min for all pipes. 𝐴 is the cross sectional area of each pipe. The initial velocity can be

decomposed into velocity in x direction and velocity in y direction:

𝑉g = 𝑉KLKQK%" ∙ cos 𝛼 (10)

𝑉l = 𝑉KLKQK%" ∙ sin 𝛼 (11)

Where 𝛼 is the launching angle. After each pipe was tested, the best launching angle was

determined to be 26 degrees. At this angle the jet does not exceed the maximum height and can hit

the target with ease. The total time for the fountain to hit a target at the same height that it is

launched from can be calculated using the below formula:

𝑡 = 1mn2

(12)

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With total traveling time given, we can calculate the horizontal distance traveled by the water jet:

𝑥 = 𝑉g ∙ 𝑡 (13)

It is assumed that there is no other force except for gravity that is acting on the water jet.

Table 1 shows 𝑉g, 𝑉l, 𝑥 for each pipe oriented at launching angle of 26 degrees and flow rate of 3.6

L/min. Table 1: The initial velocity and horizontal distance

each pipe can hit at assigned flow rate.

Pipe inner diameter(in)

Velocity(m/s) 𝑽𝒙(m/s) 𝑽𝒚(m/s) t (s) Horizontal Distance (m)

0.17 4.10 3.68 1.81 0.37 1.35 0.19 3.37 3.02 1.48 0.30 0.91 0.25 1.89 1.70 0.83 0.17 0.29 0.38 0.84 0.76 0.37 0.08 0.06

From the table, the horizontal distance traveled by the jet from the 0.19-inch diameter pipe lies

in the most reasonable range (Close to 3ft). Consequently, the 0.19-inch diameter pipe is chosen

for the design for the arcing fountain. The 0.25-inch and 0.38-inch diameter pipes could not hit

the target no matter what launching angle is oriented. The 0.17 diameter could hit the target at

larger launching angles. We did not choose this pipe because it is easier to control the pipe at

low launching angles, and 0.19-inch pipe is more suitable for the supporting fountain. With the

exit pipes of both fountains chosen to be 0.19-inch inner diameter, we can reduce the

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uncertainties to the lowest level by having symmetry in the fountain design. A schematic is

shown in Figure 1 to exhibit the arcing fountain:

Figure 1: The arcing fountain design

From the schematic, the height of the exit from the base elbow, yarcing can be calculated using the

length of 2 connecting pipes and their oriented angles.

𝑦%(tKL2 = 4 ∙ (sin 63.85° + sin 26.15°) = 6𝑖𝑛𝑐ℎ𝑒𝑠 (14)

To”T”connector

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b. Sphere-supporting fountain

The supporting fountain is designed to have the same exit height as the arcing fountain, per design

requirements. The following schematic in Figure 2 shows the supporting fountain design.

Figure 2: The supporting fountain design

The launching angle is designed to be 𝛼 = 84° and the flow rate is Q =3.51 L/min. This is the state

at which the ping pong ball is most stable. The exit height of the supporting fountain from the

base can be calculated:

𝑦��33'(QKL2 = 6 ∙ sin 84°=5.96 inches (15)

The exit height of two fountains are approximately the same:

𝑦��33'(QKL2 ≈ 𝑦%(tKL2 (16)

To“T“connector

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Besides two fountains, a system of pipes are used to connect the two fountains and the pump,

according to the schematic in Figure 3:

Figure 3: The connecting tubes for the fountain system

From Figure 3, the connecting pipes and reducing elbows on both sides of the system are identical,

thus the head losses introduced are identical, too. This significantly reduced the possible error

introduced in the experiment. The ½’’ pipe was chosen because it has almost zero head loss at the

operating flow rate. Both horizontal pipes have length of 4 inches.

The fountain system is designed using the Bernoulli equation:

��72+ m�0

12+ 𝑍_ + ℎ_ =

��72+ m𝑩0

12+ 𝑍� + ℎ� (17)

The subscript A and B represents the arcing and supporting fountain, respectively. Z represents the

exit height of each fountain, and h represents the head loss for each fountain. P is the static pressure

at exit, which equals to atmospheric pressure and can be canceled out.

Since 𝑦��33'(QKL2 ≈ 𝑦%(tKL2, 𝑍_ and 𝑍� cancel out. Thus, the Bernoulli equation simplifies to:

ℎ_ − ℎ� =912(𝑉�1 − 𝑉_1) (18)

where 𝑉� = 3.37$� and 𝑉_ = 3.28$

�. Both velocities were calculated from the respective flow

rate and pipe cross sectional area.

Topump

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Thus the difference in head loss is:

ℎ_ − ℎ� = 0.0286𝑚 (19)

The PVC tubing is assumed to have zero roughness and the head loss introduced by the horizontal

connecting pipes are assumed to be identical. Any irregularities in the inner wall of the pipe is

assumed to be negligible. From the calculations above, the head loss for each fountain system is

almost the same. To provide the necessary head loss of 0.0286 m, an additional 0.03 in of 3/16’’

inner diameter pipe is added to the supporting fountain. Since the diameter of the pipe used for

each fountain is the same, the only variable is the total length of the pipe for each fountain and the

number of elbows used. The elbows used are identical on both sides of the fountain, and the total

pipe length for the supporting fountain is 6𝑖𝑛 + 2.03𝑖𝑛 = 8.03𝑖𝑛, while the total pipe length for

the arcing fountain is 4𝑖𝑛 + 4𝑖𝑛 = 8𝑖𝑛. If all assumptions hold, the exit jets should have the

desired velocities to meet the design requirements.

Table 2: A bill of materials for the components used in the fountain design

Component Quantity Length (inch) Loss (m) Published 𝐾,* 90° Tee 1 N/A N/A N/A 1/2'' OD Pipe 2 4 0 N/A 1/2'' to 5/16'' Elbow 2 N/A 1.02 0.37 5/16'' to 5/16'' Elbow 2 N/A 0 0.54 5/16'' OD Pipe 2 4 0.227 N/A 5/16'' OD Pipe 1 2.03 0.115 N/A 5/16'' OD Pipe 1 6 0.341 N/A

Note: All head losses are calculated under 3.5 L/min, which is accurate enough for both fountain systems.

*Sources: http://mimoza.marmara.edu.tr/~neslihan.semerci/ENVE204/L2.pdf https://www.plumbingsupply.com/ed-frictionlosses.html

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Table 3 includes the Reynolds number and according friction factor for each section of the pipe

under the designed flow rates (3.51 L/min for supporting fountain and 3.6 L/min for arcing

fountain). The methods and equations used to determine these values can be found in the

introduction of this report.

Table 3: Reynolds number and friction factor for each straight section.

Component Length (inch) 𝑅J f Supporting Fountain Q=3.51 L/min

1/2’’ Pipe 4 6780 0 5/16’’ Pipe 2.03 13600 0.026 5/16’’ Pipe 6 13600 0.026

Arcing Fountain Q=3.6 L/min

1/2’’ Pipe 4 6974 0 5/16’’ Pipe 4 13950 0.0265 5/16’’ Pipe 4 13950 0.0265

Multiple pumps were available of varying sizes and speeds. A Dayton submersible pump of model

number 1P811A was chosen because it is small enough to not take up much space in the testing

area, and it is able to provide the necessary flow rate of 3.51L/min + 3.6L/min = 7.11L/min for the

fountain design. Another benefit of this pump is that is does not vibrate like some of the other

pumps available. Vibrations may affect the fountain performance.

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III. Experimental Methods

The experiment is split into two parts. The first part is dedicated to find out the flow rate Q of each

tube at which the water jet is strong enough to suspend the ping pong ball 15-inch high. The second

part is to test the real head loss of different tubes and elbows. Then the experimental data is

compared with the moody chart, Haaland equation, and published 𝐾, values.

A schematic of the first experiment is shown in Figure 4:

Figure 4: The schematic for the supporting fountain testing

Four tubes with different inner diameters are used in this experiment: 0.17 inch, 0.188 inch, 0.25

inch, and 0.375 inch. All tubes tested are 4 inches long. First install the tube with appropriate

elbow so that it can rotate freely in one surface. To reduce leaking, only one tube and elbow were

used to connect the testing piece with the pump. The desired 𝑦�3�J(J is 15 inches, thus the flow

rates are adjusted for each pipe so that the difference between the exit of the tube and the ping

pong ball is 15 inches. From the scale reading, the exit of the tube is at 12-inches height.

Accordingly, the ping pong ball needs to reach 27-inches height on the scale. Turn on the pump

and adjust the flow rate so that the water jet passes 27-inch height. Then slowly place the ping

pong ball on the tip of the jet to see if it will stand. If not, adjust the flow rate so that the ping pong

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ball reaches 27-inch height. When the appropriate flow rate is reached, record the flow rate Q on

the flow meter. Then adjust the angle carefully to so that the ping pong ball is at the most stable

state. Record the project angle α from the calibrator. Repeat the same procedure for 4 pipes.

The schematics in Figures 5 and 6 show the setup for determining head loss of the tubes and elbows.

Figure 5: The schematic for the tube testing

Figure 6: The schematic for the elbow testing

a. Tube testing

All tubes tested have length L=1.74 m but with different inner diameters. The tubes are made from

PVC material and the surface roughness is assumed to be zero. The testing tubes are glued with

connectors on both ends and are ready to connect to the transducer and pump. First step is

connecting the pressure gage on both ends of the tube. Then connect one end of the pipe with the

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pump and turn it on to check if there is any leaking. Adjust the flow rate until pressure drop

readings on the pressure gage is positive. Then increase the flow rate by 0.5 L/min and record the

pressure drop again. Keep doing until at least 6 data points are collected. Repeat the same

procedure for each pipe. The flow rate at which each pipe produces a recordable pressure drop is

different. Table 4 shows the starting flow rate for each pipe tested:

Table 4: The starting flow rates for each pipe

Inner Diameter (in) 0.17 0.188 0.25 0.375 Starting Flow Rate (L/min) 1 1.5 3 5

The 0.375 in pipe is considered no loss because the minimum flow rate of the system is 3.51 L/min.

b. Elbow testing

3 different elbows were tested: 3/8’’ to 3/16’’ elbow, 3/16’’ to 3/16’’ elbow and 3/8’’ to 3/8’’

elbow. The procedure for elbow testing is same as pipe testing. The below table shows the starting

flow rate at which the pressure gage has a positive reading:

Table 5: The starting flow rates for each elbow

Elbow 3/8'' to 3/16'' 3/16'' to 3/16'' 3/8'' to 3/8'' Starting Flow Rate (L/min) 3 3.5 11

Since the flow rate for supporting fountain is 3.51 L/min and arcing fountain is 3.6 L/min, the

3/8’’ to 3/8’’ elbow is considered as no loss. The losses introduced by pipe and elbow are termed

as major loss and minor loss, respectively. As demonstrated in the introduction, they can be

expressed as:

ℎ$%&'( = 𝑓 ,-m0

12= �45�0

72 (20)

ℎ$KL'( =�45�072

(21)

where f is friction factor, which can be read from the Moody diagram. L =1.74 m for all pipes. 𝑉

represents the average flow speed, which could be calculated by flow rate and cross section area.

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The quantity 𝑃9 − 𝑃1 is directly read off from the pressure gage, which is the pressure drop along

the pipe or elbow at a certain flow rate. On the other hand, both head loss could be written as:

ℎ$%&'( = ℎ$KL'( = 𝐾,m0

12 (22)

Where 𝐾, is the loss coefficient, a quantity could be found in many publications for each pipe and

elbow used in the experiment. With the above equations the major loss, minor loss, and loss

coefficient 𝐾, can be calculated accordingly. Therefore, the experiment results can be compared

with published values.

IV. Experimental Results and Discussion

All pipe dimensions in this section are referenced by the inner diameter. The elbow connectors

are referenced by the inner diameter of the pipes that they connect to. This is done to avoid

confusion between inner and outer diameter references.

Table 6: Experimental data for determining the parameters necessary to suspend a plastic sphere in a water jet

Pipe inner diameter (in)

Flowrate (L/min) Angle Range (°) Most Stable Angle (°)

3/8 9.24 81-84 83

1/4 5.2 79-85 83

3/16 3.51 81-84 84

0.17 BALL WOULD NOT SUSPEND AT REQUIRED HEIGHT OF 15 IN

The data obtained for the purpose of determining the parameters necessary to support the plastic

sphere in a water jet are shown in Table 6. Since the requirement for the distance from the exit of

the tube to the suspended ball is 15 inches, data was only recorded for the parameters that satisfied

this requirement. All four diameter pipes were able to support the 1-inch diameter sphere except

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for the 0.17-inch diameter pipe as the small cross sectional area of the jet relative to the ball size

was insufficient to provide stability to the ball. Regardless of launch angle, the ball would suspend

momentarily in the jet before falling out of the stream. For this reason, no data was recorded for

the 0.17-inch diameter pipe as there were no conditions capable of meeting the design

requirement.

While the other three pipe diameters proved capable of suspending the sphere at the required height

of 15 inches, the ¼ inch pipe was the most stable and had the greatest range of angles that worked

to suspend the ball. For this reason, the initial design used the quarter inch pipe for the ball side

of the fountain system. It was soon realized, however, that using the quarter inch pipe forced us

to introduce significant uncertainty into the design due to the necessary head losses needed to

produce the correct flow rates. The pipe selection was therefore reconsidered. Using the ¼” pipe

required a flow rate of 5.2 L/min in order to suspend the ball at a height of 15 inches from the pipe

exit. Since the required flowrate for the other half of the fountain system is 3.6 L/min, significant

head loss between the two needed to be introduced. Due to the uncertainty of the head loss data,

it was desirable to make the exit flow rates as close to one another as possible so that the two sides

of the fountain could be nearly identical. The more symmetry there is between the two sides of

the fountain system, the more reliable the relative loss will be as identical components should

produce identical losses.

For this reason, the 3/16” pipe was chosen as only a 3.51 L/min flow rate is required to suspend

the ball at a height of 15”. Using the ⅜” would have been a poor choice as it required a 9.24 L/min

flow rate. 3.51 L/min flow rate is so close to the horizontal fountain flow rate of 3.6 L/min that

the required head loss between the two pipe systems is minimal. While the stability of the sphere

and the range of acceptable angles is less with the 3/16” pipe than the ¼” pipe, the improved

reliability of the loss calculations outweighs the risks of the ball being less stably supported.

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For the four diameter tubes tested the major losses were measured and studied. Figure 7 provides

the experimental friction factors compared to the Reynold’s number for the four pipe sizes. The

data is graphed on a log-log scale like the Moody diagram because plotting on equivalent scales

allows for easy visual comparison.

Figure 7: Friction factor vs Reynolds number for the four sizes of pipes tested

If there were no errors in the experimental data, the plot in Figure 7 would mirror the Moody

diagram at the corresponding Reynolds numbers. However, this is not the case as the data deviates

significantly from the Moody diagram. In the Moody diagram, as Reynold’s number increases,

the friction factor decreases slightly until it reaches fully turbulent flow at which point the friction

factors remain relatively constant. It is important to remember that the pipes used in this lab are

made of drawn plastic, which have a small roughness that can be treated as zero. Thus, the data

should correspond to points of zero roughness on the Moody diagram.

Theoretical friction factors for pipes with zero roughness can be obtained from the Haaland

equation (see introduction). These theoretical values are plotted versus the experimentally

determined values in Figure 8 to show the degree of deviance of the experimental data from the

theoretical data for zero roughness pipes.

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Figure 8: Experimental friction factors vs those predicted by the Haaland equation for zero roughness pipes

In order to examine the accuracy and reliability of the data, each pipe will be analyzed individually

followed by a discussion of the overall trends among all four pipe sizes. First, consider the ⅜”

inner diameter pipe. The flows tested through this pipe ranged from Reynold’s numbers of 11,200

to 22,400 which correspond on the Moody diagram to friction factors of 0.025 to 0.03 for zero

roughness pipes. The experimental friction factors were between 0.002 and 0.012, well below the

target values. The experimental values indicate roughness below zero on the Moody diagram,

which is clearly not possible. Since zero roughness pipes provide the least possible resistance to

flow, it is impossible to have friction factors below the zero roughness values. Thus, there is

undoubtedly some experimental error at play. See the error discussion to follow.

The ¼” pipe was tested at Reynold’s numbers between 8,700 and 17,400 which should produce

friction factors between 0.026 and 0.032 for zero roughness pipes, according to the Moody

diagram. The experimental friction factors are between 0.01 and 0.02. Looking at Figure 7, we

see that the higher Reynold’s numbers tested produced similar friction factors to one another,

whereas the lower Reynold’s numbers were inconsistent. We can therefore have greater faith in

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the accuracy of the points that are in closer agreement with one another. Similar to the ⅜” pipe,

the ¼” pipe also produced friction factors in the region below zero roughness on the Moody

diagram. However, there is less deviation in the ¼” pipe data than the ⅜” pipe.

Of the four pipes tested, the 3/16” pipe data most closely matches the Moody diagram. It was

tested at Reynold’s numbers ranging from 5,800 to 15,500 which correspond on the Moody

diagram to friction factors between 0.026 and 0.033 for zero roughness pipes. The experimental

values were between 0.024 and 0.028, closely resembling the theoretical values. The relative and

absolute roughness seem to be zero for this pipe. The agreement between theoretical and

experimental data lends confidence in the use of this data for the fountain design.

Looking at the 0.17” diameter pipe data gives a different conclusion than the previous three

pipes. Flows of Reynold’s numbers between 4,300 and 14,500 were tested to produce friction

factors between 0.045 and 0.085. These friction factors correspond to relative roughnesses

between 0.01 to 0.05 and absolute roughnesses ranging from 0.00188in to 0.00938in. This pipe is

the only one of the four tested for which the experimental data corresponded to a physically

achievable area of the Moody diagram.

A curious trend in the data among all four pipes is that the experimental friction factors increased

as pipe diameter decreased. This trend is clearly seen in Figure 2 as the cluster of points make a

vertical jump with each increase in pipe diameter. The 3/16” pipe data matches up the most

accurately with the theoretical line. Whether this is simply coincidence or not is difficult to

determine, but it is certainly beneficial for the fountain design as both paths of our pipe system

contain identical components until the end where the suspended sphere path has an additional

length of 3/16” pipe to provide the necessary head loss. The accuracy of the data for the identical

components is less important because the loss should be identical between the symmetrical

portions of the paths. Even if the data is inaccurate the losses will cancel each other out. The only

asymmetrical portion is composed of the 3/16” pipe, for which we have the most reliable data.

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The experimental data should match the Moody diagram within reason. The fact that significant

deviation occurred indicates that there were experimental errors affecting the results. There are

several potential sources of error, the most probable and significant of which will be considered in

the following discussion.

When testing major losses, it is important to maintain perfectly straight pipes as the major losses

are only those due to the viscous interaction between the pipe walls and the fluid, and not losses

due to inertial forces acting on the fluid that arise when the pipe has curves or bends. Care was

taken to keep the pipe as straight as possible, but it is difficult to avoid all curves so there was

likely some additional loss measured due to the unintended curvature. The smaller diameter pipes

were less rigid due to their thinness, which led to greater curvatures than the thicker

pipes. Additional curvature leads to additional losses, which is in agreement with the experimental

data that as pipe diameter decreases, the friction factor, and hence the losses, increase. This is a

likely source of error.

Another likely source of error is the uncertainty of the digital pressure transducers. The

transducers used seemed to be fairly insensitive to small pressure differences. For example, the

⅜” pipe produced a pressure drop reading of 0 psi for flow rates of less than 5 L/min. For any

amount of flow, no matter how small, there must exist some losses due to the interaction of the

fluid and pipe walls. Since the transducers were unable to detect pressure drops below certain

flow rates, perhaps across the board the measured pressure drops were lower than their actual

values. This would provide explanation for the experimental friction factors of the three largest

diameter pipes being lower than those for pipes of zero roughness according to the Moody

diagram.

There were two pressure transducers available to use for this lab, but the readings between the two

were in disagreement. For example, on week three of the design project a ⅜” to 3/16” reducing

elbow was tested to obtain the minor loss coefficient. After testing the elbow on one of the

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transducers and getting a pressure drop of 1.25 psi, the same exact setup, pump included, was

retested on the other transducer at the same flow rate and produced a pressure drop of 0.8 psi. The

discrepancy between the two measurements is so great that it cannot be attributed to normal

differences. The majority of the data was obtained from one of the transducers, but both were used

throughout the three weeks of testing. This uncertainty is unfortunate as it undermines conclusions

that are drawn from the data.

The friction factor versus Reynold’s number data in Figure 7 can be used to approximate the

critical Reynold’s number for each pipe. The critical Reynold’s number is the point at which the

pipe flow begins deviate from laminar flow as it begins its transition to turbulence. In the Moody

diagram, this point is a thin region near the start of the relative roughness curves. Laminar flow

occurs at the line defined by equation (23). Using this equation, we can determine the Reynold’s

numbers for which the flow is laminar in each pipe.

𝑓 = G�I�

(23)

According to the Moody diagram, the critical Reynold’s numbers should occur anywhere between

2,000 and 4,000, depending on the pipe roughness. The experimental values will differ due to the

experimental errors, but the trends between different sized pipes can still be usefully analyzed. The

best estimation of the critical Reynold’s numbers for each pipe tested are shown in Table 7.

Table 7: Critical Reynold’s number estimations for different pipe diameters

Pipe diameter (in) 0.17 3/16 1/4 3/8

Recr estimation 1000+ 2500 3400 5000

The 0.17” pipe data most closely resembles the general shape of the Moody diagram curves. Using

equation (23), we see that laminar flow for the 0.17” pipe occurs at Reynold’s numbers around

1,000. The lowest Reynold’s number at which data was taken was 4,273, and by this point we see

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that the graph, Figure 7, is fairly linear, indicating that it has reached turbulent flow. We can

therefore, conclude that the critical point occurs somewhere between 1,000 and 4,273. Since the

flow was turbulent by 4,273 the critical point likely occurred closer to 1,000, as is indicated by the

1000+ notation in Table 7. In reproductions of this lab, more points should be tested to obtain a

more precise critical point.

The 3/16” pipe once again produced the most reliable data as Reynold’s number for laminar flows

calculated from every data point tested produced consistent values around 2,500. The lowest

Reynold’s number tested for this pipe was 5,812, indicating that the critical point is somewhere

between 2,500 and 5,812, likely closer to 2,500 for the same reasons mentioned for the 0.17” pipe.

For the ¼” pipe the Reynold’s numbers for laminar flow obtained from the experimental data are

wide ranging. It is noticed, however that the half of the data produced similar values and the other

half produced inconsistent values. We will therefore concentrate on the similar data as it is likely

more reliable due to its repetition. The data indicates a Reynold’s number of 3,400 for laminar

flow. This is in the 2,000 to 4,000 range, so the critical value is likely near 3,400.

The critical Reynold’s number for the ⅜” pipe is the most difficult to analyze as the data has the

least consistency. However, we see once again that the higher Reynold’s numbers are in

agreement as to the Reynold’s numbers for laminar flow, whereas the lower ones vary

significantly. Based on the more consistent data, the best approximation of the critical Reynold’s

number is about 5,000. This is outside the 2,000 to 4,000 range given by the Moody diagram, but

is close enough that it can be used for comparison between the other pipe sizes.

Comparing the estimations of the critical Reynold’s numbers for all four pipe sizes, it is clear that

as pipe diameter increases, the critical Reynold’s number also increases. This trend makes

physical sense because flow is more likely to be turbulent through small pipes due to the high

velocities that arise from small cross sectional areas. Flow through a small diameter pipe will have

a much higher velocity than the same flow rate through a larger diameter pipe. The Reynold’s

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number is proportional to both the diameter as well as the velocity. However, a reduction in pipe

diameter by one half will produce a velocity more than double the initial value at the same flow

rate. Therefore, the smaller pipes will transition to turbulent flow at lower Reynold numbers.

Fully rough behavior is defined on the Moody diagram as the area in which the roughness curves

become linear. In this region the friction factor is essentially independent of Reynold’s

number. None of the tested points reached fully rough behavior. However, the 0.17” pipe was

close. Since the data indicated a roughness between 0.01 and 0.05, the f vs Re points on the Moody

diagram were in the regions where the transition zone from laminar to fully turbulent flow is thin

compared to smoother pipes. The thin transition region means that any non-laminar flow is close

to the fully turbulent, or fully rough region. High roughness pipes reach fully rough flow at

relatively low Reynold’s numbers. As pipe roughness decreases, the Reynold’s number at which

the fully rough zone begins increases until there is no fully rough region for perfectly smooth pipes.

Minor losses for each component used in the fountain design were also calculated and

studied. Published loss coefficient values were used in the initial stages of the design to get a

general idea of how different components would behave in the fountain system. The published

loss coefficients used as well as the experimental values obtained are listed in Table 8. All

experimental values listed are taken at 3.5 L/min flow rate as that is approximately the flow rate

used in the fountain design. The one exception to this is the ⅜” to ⅜” elbow for which the loss

coefficient listed is taken at 12 L/min as there was no measured loss below that flow rate.

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Table 8: Experimental and published loss coefficients

Connector Type Published KL value*

Experimental KL value

Percent Difference (%)

⅜” - ⅜” ID elbow 0.32 0.2 38

⅜” - 3/16” ID reducing elbow 0.37 3.0 711

3/16” to 3/16” ID elbow 0.54 0.9 67

*Sources: <https://www.plumbingsupply.com/ed-frictionlosses.html> for ⅜” and ¼” elbows <http://mimoza.marmara.edu.tr/~neslihan.semerci/ENVE204/L2.pdf >for ⅜” to 3/16” elbow

None of the experimental loss coefficients match closely with the published values used. The ⅜”

elbow is the closest, but even that one has over 30% difference from the published value. The

difference in the reducing elbow is so great that it is important to investigate the reasons for this

discrepancy. Upon further research, the source used for the reducing elbow coefficient was

realized to list the value that represented only the losses due to the contraction of the pipe, rather

than the entire elbow. This explanation clarifies the drastic difference in the two values. The 90-

degree bend will contribute significantly to the total losses, increasing the theoretical loss

coefficient to a value nearer the experimental value.

All three pipes were tested by measuring the pressure drop from the inlet to outlet of the

connector. There was, however, a short length of drawn plastic tubing on either end of the

connector that undoubtedly added some additional head loss to the measurements. Using the major

loss data, the excess losses could be approximated by subtracting the major loss of the additional

pipe length from the minor loss. Since the pipes were no more than an inch or two, the most the

additional loss could be is about 0.15m of head loss. While this value is not very large, it is enough

to affect the results.

The ⅜” elbow measured slightly lower than the published value. Of the three connectors measured,

this data is the least reliable as below flow rates of 12L/min we were unable to detect any pressure

loss from one end of the pipe to the other. Certainly head losses were present, but the transducers

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were unable to detect them. Because of this, it is likely that there were in fact greater losses through

this connector than were measured, so it makes sense why the experimental value was lower than

the published value.

As previously mentioned the loss coefficients were not consistent at different flow rates, or

different Reynold’s numbers. The loss coefficients are plotted with respect to Reynold’s number

in Figure 9 to show the relationship between the two.

Figure 9: Loss coefficient vs Reynold’s number for connectors considered for the fountain design

For the 3/8” to 3/16” elbow and 3/16” to 3/16” elbow, the loss coefficients tend to stabilize at

higher Reynold’s numbers. At low Reynold’s numbers, they either increase or decrease fairly

rapidly. For this reason, it is better to create a fountain design in which the flow through the elbows

is in the more stable region so that an accurate value of the head loss can be obtained. For flows

in the region where the loss coefficient is unstable, a slight change in flow rate may significantly

alter the loss and the fountain may not function properly. For example, our fountain design uses a

3/8” to 3/16” reducing elbow at a Reynold’s number of 15,000 which is right in the middle of the

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unstable region in Figure 9. Knowing that this presents significant uncertainty in our design, we

sought to avoid this by maintaining symmetry in our fountain circuitry. The design has identical

connectors in both fountain paths so that even if the measured losses are not right, the losses

through both paths would be equal and cancel each other out.

For the ⅜” to ⅜” elbow, the flow rates that will be used in the fountain design (about 3.5 L/min)

produce zero loss according to the experimental data. In fact, all flows below 12 L/min read zero

loss, which cannot be accurate because any time the fluid is forced to change direction in a pipe

there will be losses associated with that change. This uncertainty provides further incentive to

maintain symmetry to whatever extent possible so that the losses will be identical in both fountain

paths, and will thus be irrelevant to the relative head loss between the two fountain outlets.

The overall accuracy of the experimental results both for the major and minor losses is less than

ideal. While several potential sources of error have been mentioned that would reduce the errors

if they were addressed, it is of interest to quantify the degree of inaccuracy of the data. The percent

differences for the loss coefficients are listed in Table 8. These percentages are not necessarily an

accurate representation of the degree of error in our measurements as the published values used

are not the exact loss coefficients through the connectors used in this lab. The reducing elbow, for

example, has a greater loss than the published value as was previously discussed. The closest

match between published and experimental loss coefficient values was for the ⅜” elbow, and even

that had 38% difference.

For the major losses, the accuracy is more difficult to quantify. Ignoring the different pipe sizes

and just taking the collection of data points in Figure 7, it is clear that as Reynold’s number

increases, the friction factors decrease as a general trend. This trend matches the Moody diagram

and lends confidence to the data. As with the minor losses, the data stabilizes at higher Reynold’s

numbers. In Figure 8, the data points become increasingly concentrated as they near the ideal

line. To obtain a quantitative value for the accuracy of our data, we will take the closest point to

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the ideal line in Figure 8 for each pipe diameter and calculate the percent difference between

experimental and theoretical values. These values, listed in Table 9, confirm that the 3/16” pipe

produced the most accurate data. The other data varies increasingly, but as mentioned the

symmetry of the fountain system is such that only the 3/16” pipe data is needed for a reliable

fountain design.

Table 9: Deviation of experimental friction factors from theoretical friction factors for zero roughness pipes.

Pipe diameter (in) 0.17 3/16 1/4 3/8

Percent deviation from theoretical friction factors (%) 61 9 30 50

The uncertainties listed in Tables 8 and 9 are significant. Much of this uncertainty can be credited

to the lack of accuracy of the transducers as well as the lack of precision between measurements

by different transducers, as was previously discussed. There is, however, also the presence of

human error that contributes to the uncertainty of the data. While care was taken to obtain careful

and accurate measurements, there is always the possibility of improper setup of the experiments

that would lead to erroneous data. For example, the connectors that lead to the transducer pressure

sensors were difficult to snap on to the static pressure taps. There is a possibility that they were

not fully secured, which could lead to inaccurately low pressure measurements. Additionally,

there were water bubbles throughout the tubes leading to the transducer which may have caused

inaccurate pressure readings as well.

Another source of uncertainty is the randomness associated with turbulent flows. However, the

randomness is on a very small scale so it is highly unlikely that the turbulent nature of some of the

flows lead to errors of the degree we experienced.

The friction factors and loss coefficients are the quantities most affected by the uncertainties as

they are directly dependent on the pressure drop measurements. The pressure drops are the

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measurements with the greatest degree of uncertainty because there is both equipment

uncertainties as well as those due to human error at play. Other quantities such as dynamic

viscosity and Reynold’s number are less affected by the uncertainties. Dynamic viscosity for

example is solely a function of temperature, so its determination can be found in published tables

based on the temperature of the water used. The only variable that could introduce uncertainty to

the viscosity value is the purity of the water. Published values are based on pure water, so impure

water could have a slightly different viscosity. However, the impurities in the water used are likely

small enough that they can be neglected. Thus, the viscosity values are reliable.

Reynold’s numbers are a function of viscosity, density, pipe diameter, and fluid

velocity. Considering the uncertainty of each of these parameters will give an idea of the

uncertainty of the Reynold’s numbers. The velocity is simply a mathematical calculation of flow

rate over cross sectional area, for which the only associated uncertainty is from the flow rate

measurements of the Coriolis force mass flowmeter. These types of flowmeters are accurate to

within 0.1% uncertainty, so barring some failure of the device, the velocity calculations can be

trusted with little to no uncertainty. Pipe diameter is a measured quantity for which the only

uncertainty would be due to an improperly calibrated calipers. The density of water is a published

value that is essentially independent of temperature, so the uncertainty of that value is also very

small. Since all four variables that the Reynold’s number is dependent upon have very small

uncertainties, the uncertainty of the Reynold’s numbers used in the lab are very small as well.

V. Conclusions and Recommendations

In this experiment, it was determined that the most effective pipe for suspending a sphere at a

height of 15 inches is the 1/4-inch inner diameter pipe. It was able to stably support the sphere at

angles between 79-85 degrees, with 83 degrees being the most stable. The 3/16-inch diameter pipe

supports the sphere less stably, but was chosen for the fountain design because it allowed for

symmetry to be maintained.

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In testing the major losses through the four pipes, the data suggests that as pipe diameter decrease,

the friction factor increases. This data agrees with the Moody diagram as well as the theoretical

friction factors from the Haaland equation. The most reliable data is from the 3/16-inch pipe, upon

which the fountain design is dependent for the necessary head losses. None of the flows reached

fully rough behavior.

The minor losses and corresponding loss coefficients were measured. The reducing elbow was

found to have a loss coefficient of 3.0, much greater than the non-reducing elbows which had loss

coefficients of 0.2 for the 0.375-inch diameter pipe and 0.9 for the 0.25-inch diameter. There was

significant disagreement between the experimental loss coefficients and published

values. However, due to the reasons discussed in section VI of this report, the uncertainty

associated with the data is less critical due to the symmetry of the fountain design.

After testing in weeks 1-3, data was obtained that was used to formulate a final design. Initially,

1/4-inch inner diameter pipe was used as the outlet for the jet that suspends the sphere. However,

based on the experimental data, a change was made to use the 3/16-inch pipe instead in order to

maintain symmetry in the design and, thus, reduce uncertainty.

In order to improve future experiments of this type, some changes could be made to the data

gathering process. A track that would hold the pipes level and straight could be used to improve

the major loss data as it would remove unintended curvature from the pipes that affects the

measured pressure drop. Also, to improve the accuracy and reliability of the pressure drop

measurements, one should make sure the tubes connecting the transducer to the pipes being studied

are clear of water bubbles. The presence of bubbles in this lab may have negatively affected the

data.

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VI. Appendix

Tim: I was assigned team manager to oversee the writing of the report. I made sure the report was

progressing over break. The data reduction and analysis was my responsibility as well as the

discussion and results section of the report. I also worked with Haotian on writing the conclusion.

Haotian: I was mainly in charge of the experiment part. I was responsible for setting up the time

schedule for the team, and I took the data from the transducer and made sure they are in the correct

trend. In addition, I wrote the initial design and experimental methods in the report.

Omar: I was in charge of collecting data for the optimization of the angles needed to balance the

Ping-Pong Ball on top of the stream of Fountain B as well as helping with the design of the

fountains. In addition, I wrote the introduction for the report.