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Bernoulli’s Principle Sample Essay & Outline

Abstract
The experiment to study Bernoulli’s theorem was conducted using an apparatus that consists of a classical venture with a horizontal test section consisting of various pressure tappings placed along its length to allow measurement of pressure, and a constant diameter for the inlet and the outlet. The complete combination of venturimeter with manometer tubes and a hand-held pump to control the required pressure above the manometers were used. During the experiment, water was fed through the inlet, flowing through a contraction, a throat and expansion before exiting through the outlet with a control valve that was used to control the rate of water flow in the experiment. The results of the experiment verify the Bernoulli’s principle.

Introduction

Bernoulli’s principle is one of the most widely applied physical principles in fluid mechanics since its formulation by a Swiss scientist named Daniel Bernoulli. The principle states that as the velocity of a fluid in motion increases, the pressure within the moving fluid decreases (Miller, Vandome and McBrewster). The principle is well demonstrated by the Bernoulli’s equation, which is an approximate equation that relates pressure, velocity and elevation for friction-less fluids with a constant density. The kinematic form of Bernoulli’s equation is:

……………………………………………….. (1)
Where:
: fluid velocity at the first point of measurement (m/s)
: gauge pressure at the first point of measurement (Pa)
: fluid density (kg/m3)
g : gravitational acceleration (m/s2)
h1 : the height of the fluid relative to a given reference (m)
V2 : the velocity of the fluid at the second point of measurement (m/s)

This equation can also expressed in terms of pressure head (m) by dividing each term in the equation with gravitational acceleration (g):

……………………………………………………... (2)

The Bernoulli’s principle can also be understood in terms of conservation of energy with some assumptions made on the field of fluid flow. As a fluid moves from a wider section to a narrower section, the corresponding fluid volume has to move a greater distance forward within the narrower section and thus at a greater velocity. At the same moment, the work done by the amount of work done by the corresponding fluid volumes in the wider and narrower pipe sections is expressed as the product of fluid pressure and volume (Pnueli and Gutfinger). We know that the pressure in the narrower section is greater, so the kinetic energy of the corresponding volume is greater. By the law of conservation of energy, the increase in kinetic energy in the narrower section must be balanced by a reduction in pressure in the wider section. From the equation above:

This experiment was performed with the aim of investigating the flow of water through a venture meter by the study of the interchange of pressure and kinetic energy, and conservation of energy as stated by the Bernoulli’s equation. In addition, the application of this equation to flow metering devices is also studied.

Methods

A.The experimental apparatus:

The experimental apparatus consisted of a horizontal test section with an inlet and outlet of constant diameters. The figure below shows the horizontal test section:

B.Procedure:

The test apparatus was leveled as well as possible using the adjustable feet. The position of the outlet control valve was then adjusted to ensure that there is the largest possible difference between the manometer levels, taking care to maintain the measurement levels within 0-300 mm. After a sufficient time was allowed for the stabilization of the manometer levels, the readings in each manometer was recorded down. Next, the flow rate (Q) was measured by measuring the time taken for 5 Litres of water to be collected in a bucket. Finally, the outlet control valve was closed slowly enough to allow manometers to stabilize as the flow rate reduced, and the readings recorded down. The steps described above were repeated for five different flow rates.

Results

The table below shows the raw experimental results obtained:

Table:
Run 1 Run 2 Run 3 Run 4 Run 5
Measuring bucket volume (Litres) 5 5 5 2 2
Discharge Time (sec) 32.2 31.1 33.7 31.5 35.9 37.1 19.1 19.8 32.4 31.2
31.8 30.8 32.8 32.2 35.0 33.2 19.5 19.2 28.2 30.0
32.7 33.1 34.5 34.4 36.6 37.2 18.4 18.9 28.4 26.5
Avrge time 31.95 33.18 35.83 19.15 29.45
Tapping position Manometer water levels (mm)
Manometer A 254 264 277 285 274
Manometer B 200 212 233 258 263
Manometer C 107 125 156 213 242
Manometer D 28 56 96 178 227
Manometer E 0 32 85 165 220
Manometer F 97 118 146 204 235
Manometer G 95 115 145 203 234
Manometer H 257 267 272 289 273

Calculations

As observed from the hydraulic flow lines, water pressure and energy varies in different sections. At high velocity, there is pressure drop, hence, low water head, and at low velocity, the pressure is higher, hence, high water head. The pressure is lowest at the constriction and increases as the cross section increases.

Question 2

Run 1:
Flow rate =
Area of inlet section (A1) = = ( = 0.00393
Velocity at the inlet (v1) = (Q/A) = =
Throat velocity:
Using the Bernoulli’s equation:

Since the pipe section is level in a horizontal position, , and therefore, the equation reduces to

(
= 5.143
Therefore, = 2.23m/s
Outlet velocity:

(

= 0.10114
Therefore, = 0.32m/s

Run 2:
Flow rate =
Area of inlet section (A1) = = ( = 0.00393
Velocity at the inlet (v1) = (Q/A) = =
Throat velocity:
(
= 4.6123
Therefore, = 1.62m/s
Outlet velocity:

(

= 0.01499m/s
Therefore, = 0.173m/s
Run 4:
Flow rate
Area of inlet section (A1) = = ( = 0.00393
Velocity at the inlet (v1) = (Q/A) = =
Velocity at the inlet (v1) =
Throat velocity:
(
= 2.355
Therefore, = 1.53 m/s
Outlet velocity:

(

= 0.0793
Therefore, = 0.28m/s
Question 3
Run 1:
= =

Run 4:
=

The venture discharge coefficient is not constant, but varies with Reynolds number.

Question 4

The flow rate in a pipe is influenced by frictional resistance due to pipe roughness and minor losses (fittings, entries, exits and valves). This leads to head loss in real flow rate. This losses are not taken into account in ideal flow rate, thus, the difference between the two.

Question 5

The effect of fluid viscosity and the internal frictional resistance of the pipe material. There is an assumption that there is no energy loss.

Question 6

Applications of Bernoulli’s equation

i. Airfoil: The Bernoulli’s equation is applied in the architecture of airfoils. The curving on top of a conventional airfoil is such that air traveling on the top surface moves at a higher speed than concurrent air moving on the bottom, somewhat flat surface. This relative motion of air results in a difference in pressure, with a higher pressure at the bottom and a low pressure zone on the top surface according to Bernoulli’s equation, providing a lift force.

ii. Drawing fluids upwards: Atomizers, carburetors and Chimneys use Bernoulli’s principle when drawing fluids from fluid reserves. Inside a perfume bottle, air moves relatively slowly, and therefore, according to Bernoulli’s equation, the air in the bottle has a relatively high pressure. When the squeeze pump is operated, air in the horizontal tube flows at a higher speed, thereby reducing the pressure towards the top part of the bottle. This pressure difference draws the perfume from the bottom of the bottle to the upper tube and squeezed out through the spray nozzle. A carburetor employs a similar principle to draw air.

iii. Baseball spin: As the ball travels through the air in a spinning manner, the velocity with which it moves with creates an air stream. The motion of the airstream is against the ball’s trajectory. This causes high air pressure on the top surface of the ball, and low air pressure on the lower bottom. According to Bernoulli’s equation the ball moves at a lower speed on top, and a higher speed at the bottom, causing the ball to accelerate downwards faster than at the top part, resulting in curving of the ball. With this phenomenon, an opponent can easily miscalculate the position of the ball.

Read also on Bernoulli application in surface milling experiment

Conclusion

The results obtained in this experiment are in agreement with Bernoulli’s principle. As the velocity of the fluid increases in a constriction, the pressure reduces, and when the velocity reduces in the wider section, the pressure increases. The discharge coefficient varies since the flow section has varied cross section areas, and depends on the Reynolds number.

Acknowledgement

I wish to acknowledge the lab instructor for a clear guide and assistance in performing this experiment. I also acknowledge every individual in my group for helping in carrying out the experiment and getting the required results.

References

Miller, Frederic P., Agnes F. Vandome and John McBrewster. Bernoulli's Principle. New York: Alphascript Publishing, 2010.
Pnueli, David and Chaim Gutfinger. Fluid Mechanics. U.K: Cambridge University Press, 1997.