## INTRODUCTION the water in the tank at any time,

INTRODUCTION

Tanks
are major components for many engineering processes, and it is very important
to be able to understand and predict their impact on a system. Understanding
and predicting the behavior of tanks comes from comprehending and being able to
apply mass and energy balances, such as Bernoulli’s Principle for a flowing
fluid. In order to accurately predict the behavior of the fluid in the tank, a
model must first be developed. The model is used to predict the height of the
water in the tank at any time, or it can be used to predict the time elapsed
after any change in height of the water in the tank.

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The
model developed in this experiment is based on Bernoulli’s Principle, and it
accounts for the friction of the tube connecting to the tank while disregarding
the friction of the tank itself. The model is split up into two zones: the cylindrical
section and the conical section. The data obtained from the cylindrical section
was relatively accurate, but the data obtained from the conical section was not
accurate. This indicates that Bernoulli’s Principle is difficult to use as a
model for tanks with complex shapes and non-ideal fluid flow.

EXPERIMENTAL EQUIPMENT
AND PROCEDURES

Tube Sizes

Tube No.

Tube ID (in)

Length (in)

Length (ft)

1

1/8

24

2

2

3/16

3

0.25

3

3/16

12

1

4

3/16

24

2

5

5/16

12

1

6

5/16

24

2

Tank Components

Specifications

Cylinder

16″ ID x 12″
Height

Cone

16″ ID top x
12″ Height x 1.38″ ID bottom

Pipe

Schedule 40, 1.25″

Tee

Schedule 40, 1.25″

Bell Reducer and Bushing

for the tubes to connect
to

Barbed Fitting

connects the plastic tube
to the tee

Plastic Tube

used to determine height
of tank water

Ruler

measured height of tank
water in plastic tube

Support Stand

holds entire system

Bucket

used for tank to drain
into

1.25″ Sch. 40 pipe section

7.25″ of water above tee

1.25″ Sch. 40 tee

Plastic tube

Bell Reducer and Bushing

Tubes connect here

For this
experiment, the tank was filled with water to a height of 30.5 inches above the
tube. A tube was attached to the bushing, and a bucket was placed directly
under the tube to catch the draining water. The valve was opened. As the height
of the water decreased, the time elapsed and the water height were recorded
until the height decreased to roughly 12 inches above the tube. The valve was
then closed, and the water in the bucket was poured back into the tank. This
process was repeated for each tube. Once each tube was used, the water was
completely drained from the tank and was poured out. The tubes were returned to
their rack, and the lab area was cleaned and dried to ensure no falls. After
the experiment was completed, the data for each tube was plotted as time vs
water height above the tube.

In order
to predict the behavior of this tank, a model was developed. The tank was
separated into two different zones: the cylindrical portion and the conical
portion. For simplicity’s sake, and because the water stopped draining before
it completely emptied the conical section, the portion of the system containing
the 1.25 inch schedule 40 pipe section to the bushing where the tubes connected
were ignored as if it did not exist. The model assumed that the system was in
steady state, had inviscid flow, followed a streamline, and incompressible
flow. For the cylindrical portion, the head form of the Bernoulli Equation was
used to create the model, where the top of the water inside of the cylinder
portion was point one and the exit of the tube was point two.

+  + z1 – Ws
=  +  + z2 + hD

For this
system, the pressure at point one and two were both atmospheric. Therefore,
both pressure terms were equal to one another and were removed from the
equation. Since the exit of the tube was the end of this experiment, z2
was considered to have a value of zero. There was not a pump in the system, so
Ws also had a value of zero. This made the Bernoulli Equation
simplify to

+ z1 =  + hD,

where z1 was
the height of the water at any time (z1 = h). At this point, the
velocity at the top of the water inside the cylinder was calculated by
determining the volumetric flow rate of the cylinder portion (the first eight
inches drained) and dividing that by the cross-sectional area of the cylinder
(V1 =  ). The contunity
equation (?1A1V1 = ?2A2V2)
was then used to determine the velocity exiting the tube. Since the fluid is
assumed to have laminar flow, the Reynolds number and fanning friction factor
were used to determine the friction head of the tube.

NRE =   (dimensionless)     ffanning =   (dimensionless)          hD = 4ffanning()()  (ft)

When the
velocities were calculated, it was assumed that the velocities were constant,
and it was determined that V2 >>> V1. Therefore,
the V1 term was considered to have a value of zero and V2
= – . These simplifications resulted in an equation that could be
integrated to determine h at any t.

h = ( + hD

The c value was calculated by making h = 8 inches when t
= 0 seconds and by using the previously determined hD value.

For the conical portion, the head form of the Bernoulli
Equation was used to create the model, where the top of the water inside of the
conical portion was point two and the exit of the tube was point one.

+  + z1 – Ws
=  +  + z2 + hD.

For this
system, the pressure at point one and two were both atmospheric. Therefore,
both pressure terms were equal to one another and were removed from the
equation. There was not a pump in the system, so Ws also had a value
of zero. This made the Bernoulli Equation simplify to

+ z1 + hD
=  + z2,

where V2 = –  <<< V1. The hD value that was previously calculated for the cylindrical portion was assumed to be the same value for the conical portion. V2 was treated as being equal to zero in order to simplify V1, so that V1 = (.             The continuity equation was then used to V1A1 = V2A2. This relation simplified to ((? = - A2. A2 was determined through the use of the geometry of the cone.                   r       h                                      r2 = h(tan?)              This was used in determining the cross-sectional area A2 = ? h2(tan?)2, so that (? = -  ? h2(tan?)2. This equation was integrated with respect to h and t to yield t = .           EXPERIMENTAL RESULTS AND DISCUSSION Cylinder Model: h = ( + hD  Cone Model: t = The cylinder model was used to calculate the height of the water as time elapsed until the time where the cylinder portion was supposed to be empty (water height 22.5 inches above tube). From that time, the cone model was used to determine the time elapsed as the height of water above the tube decreased by 1 inch or .5 inch increments until the water height above the tube was roughly 12 inches. For the cylindrical model, the data mostly fits. This is because the friction of the cylinder is negligible, so the model is able to predict the flow. The conical data does not fit the experimental data. This is because the fluid dynamics changed as the water level decreased, causing the model to become even more ineffective. For the cylindrical model, the data mostly fits. This is because the friction of the cylinder is negligible, so the model is able to predict the flow. The conical data does not fit the experimental data. This is because the fluid dynamics changed as the water level decreased, causing the model to become even more ineffective. For the cylindrical model, the data mostly fits. This is because the friction of the cylinder is negligible, so the model is able to predict the flow. The conical data does not fit the experimental data. This is because the fluid dynamics changed as the water level decreased, causing the model to become even more ineffective. For the cylindrical model, the data mostly fits. This is because the friction of the cylinder is negligible, so the model is able to predict the flow. The conical data does not fit the experimental data. This is because the fluid dynamics changed as the water level decreased, causing the model to become even more ineffective. For the cylindrical model, the data mostly fits. This is because the friction of the cylinder is negligible, so the model is able to predict the flow. The conical data does not fit the experimental data. This is because the fluid dynamics changed as the water level decreased, causing the model to become even more ineffective. For the cylindrical model, the data mostly fits. This is because the friction of the cylinder is negligible, so the model is able to predict the flow. The conical data does not fit the experimental data. This is because the fluid dynamics changed as the water level decreased, causing the model to become even more ineffective. RELATIONSHIP BETWEEN THEORY AND EXPERIMENTAL RESULTS Bernoulli's principle is based on the inverse relationship between a fluid's potential energy and kinetic energy. The Bernoulli equation used to develop the cylindrical model and the conical model also takes into account the friction acting against the flow of the fluid caused by the tube. For the cylindrical portion, the model is able to also take advantage of Torricelli's law, V2 = , because the initial velocity is practically zero, P1 and P2 are both atmospheric pressure, and there is minimal friction with no external work. However, there is not an equal amount of fluid flowing into the tank as the fluid leaving the tube.1 This causes the model to be slightly off for the cylindrical portion.  For the conical portion, the model also utilizes both the Bernoulli equation and Torricelli's law. However, this is not an accurate model because the cone shape of the tank induces the Coriolis effect vortex. The Coriolis effect vortex occurs at the end of the cone when the water level is lower and the exiting flow rate is too high.2 CONCLUSIONS AND RECOMMENDATIONS             The cylindrical model was appropriately accurate. It was not exact, but the results for Tubes 1-4 are not far from the experimental results. For Tubes 1, 3, and 4, the model ends roughly 2 inches above the experimental water height. For Tubes 2, 5, and 6, the model ends roughly 3 inches above the experimental water height. The conical model was not accurate. The only reason that it is touching the experimental data on the graphs is because the first data point is set to where the experiment started draining through the cone. For Tubes 1-4, the conical model added roughly 15 minutes to the draining time. For Tubes 5-6, the conical model added roughly 6 minutes to the draining time. This shows how the increase in ID of the tube increases the flow rate from the tube. In order to better the model, there needs to be a mathematical way to factor in the Coriolis effect for the conical portion of the model.                   REFERENCES 1. Nevers, N. d. (2004). Fluid Mechanics for Chemical Engineers, Third Edition. New York, NY: McGraw-Hill. 2. Peck, D. A. (1998). United States of America Patent No. US5790619.

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