Bernoulli’s Theorem Demonstration

CE 336

Department of CECEM

Date: 10/19/15

Fall 2015

Table of Contents

Purpose

Intro

Theory

Equipment

Set up and Procedure

Data Analysis

Discussion

Conclusion

References

Purpose

The purpose of this experiment is to demonstrate and analyze Bernoulli’s theorem using a

venturi meter. Static head for distinct flow rates through the venturi meter is recorded and kinetic

head is then calculated from independent measurement of volumetric flow rates. Static and

kinetic head is used to calculate total energy through the venturi pipe.

Introduction

A venturi meter is a flow meter utilized to measure flow through a pipe. The venturi meter makes

for a perfect candidate to demonstrate Bernoulli’s theorem, when the equation is applied at two

points along a streamline, one can compare the pressure heads, elevation heads, and velocity

head. It operates on the foundation that a decrease in flow area in any pipe results an increase in

velocity that is also accompanied by a decrease in pressure. The cross-sectional area of the

venturi meter is identified as the “throat”. Conservation of energy and conservation of mass rules

are used to make a connection between pressure differences with the velocity at pipe and throat

section in order to compute flow rates in the pipe. In order to obtain the values to demonstrate

Bernoulli’s theorem, we will analyze the flow through the venturi meter at six different areas,

where a manometer is attached. While performing this experiment it is important to maintain air

bubbles at a minimum within the venturi meter, and essentially out of the manometer tubes.

Theory

The law of conservation of energy goes hand in hand with Bernoulli’s theorem which is applied

to a fluid flow system. The theorem describes that for an ideal fluid with steady flow, the total

energy should be the same along the streamline. Furthermore, an ideal fluid is considered

inviscid, incompressible and irrotational. As these assumptions are made Bernoulli’s theorem

can be expressed by equation 1, where subscript 1 and 2 refer to points 1 and 2 in figure 1. From

this equation P and V are the pressure and velocity at a point in the streamline of the fluid flow

while z is the elevation of the point relative to the datum. Pγ

represents the “pressure energy per

unit weight” or pressure head. V2

2 g represents the “kinetic energy per unit weight or kinetic head.

Now, the sum of the pressure head and elevation head represents the static head, and sum of

static head and kinetic head represents Total head (H). Moreover, the (HGL) Hydraulic grade

line is the location of static along the flow direction, the location of the total energy is known as

Energy grade line (EGL). For an ideal fluid, total energy at every point should be equal and

constant, however as fluid passes through the narrow section of the throat, fluid loses energy due

to friction losses along the walls. Therefore total energy downstream in the venturi pipe is not the

same to the total energy upstream. The venturi meter is designed to reduce head losses to

minimum by creating a relatively streamlined contraction and a gradual expansion downstream

of the throat. Ultimately, the majority of the head loss in a venture meter is a result of friction

losses adjacent to the pipe walls rather than losses due to separated flows and inefficient mixing

motion.

Figure 1. Graphical representation of Bernoulli’s equation for ideal fluid flowing through a

venturi meter.

Equation

(1)P1

γ+z1+

V 12

2 g=

P2

γ+z2+

V 22

2 g

Equipment

The equipment being utilized in this experiment to demonstrate Bernoulli’s theorem consists of

several components. Figure 2 simply illustrates main parts of the equipment, figure 3 shows

outlet pipe with flow control valve, figure 4 shows bench flow control valve and scale. The

equipment being used consists of a inlet control valve, air bleed screw, and a flow control valve.

A hydraulic bench used to regulate flow through the venturi meter, a stopwatch or smart phone

to record time to collect specified volume of water in reservoir within hydraulic bench.

Figure 2. Hydraulic bench with venturi meter and manometers.

Figure 3. Outlet pipe with control flow valve. Figure 4. Bench Control Flow valve

Set up and Procedure

1. Make sure apparatus on the flat top of the bench is leveled and secured

2. Attempt to fully open the outlet flow control valve at the right hand end of the apparatus.

3. Close the bench flow control valve then start service pump.

4. Next, gradually open the bench flow control valve and allow the pipes to fill with water until

all air has been pushed out from the pipes.

5. If air bubbles remain in the pipe system, close both the bench flow control valve and the outle

flow control valve and open the air bleed screw. Remove cap from adjacent air inlet/outlet

connection. Open the bench flow control valve and allow flow through the manometers to purge

all air from them. Now, tighten the air bleed screw and slightly open both the bench valve and

outlet flow control valve.

6. Gradually increase the volume flowrate by opening the outlet flow control valve or the bench

flow control valve as required until maximum flowrate is obtained.

7. At this flow rate, measure the piezometric head.

8. Calculate the flow rate throught the venture pipe by recording the time to collect a known

volume of water in the tank.

9. Repeat steps 6-8 for at least 3 values of the inlet head.

Manometer Readings

(mm)

Tube Diameter

(mm)

C.S

Area

(mm2)

Low Flow Medium

Flow

High

Flow

Ave.

Velocity

(m/s)

Velocity

Head

V❑2

2 g

a 25 490.63 86 262 296 .27 .0037

b 13.9 151.67 72 190 208 .87 .0386

c 11.8 109.30 58 125 119 1.2 .0734

d 10.7 89.87 41 45 13 1.46 .1086

e 10.0 78.5 39 33 9 1.67 .1421

f 25.0 490.63 63 190 212 .27 .0037

Volume (L) 4 4 4

Time to collect (sec)

50.87 32.76 20.84

Discharge(L/s) .0786 .1221 .1920

Table 1. Flow rate, Kinetic head, and Cross sectional areas of tappings a-f.

Figure 2. Venturi meter with HGL and EGL drawn.

Discussion

Bernoulli’s theorem states that for an ideal fluid with steady flow, the total energy remains the

same along the streamline. From Figure 2, we can observe how the (EGL) energy grade line does

in fact remain constant throughout the streamline and essentially throughout the venturi meter.

The energy grade line is also the total head, which is the pressure head, plus the elevation head,

plus the kinetic head. In this situation the elevations are the same so they can be neglected in

both sides of the equation when comparing two points along the stream line. The (HGL)

Hydraulic grade line on the other hand does fluctuate with a significant pattern. The hydraulic

grade line is the sum of pressure head and elevation or the static head. For a better illustration of

the hydraulic grade line, it was also depicted when measuring the manometers for the different

flow rates. The HGL decreases along the throat section of the venturi meter, indicating the loss

of pressure and recuperates on the other side of the meter with the same cross sectional area. This

again demonstrates not only Bernoulli’s theorem but the laws of conservation of mass and

energy. From Table 1, we can better see the effects of the venturi meter on the flow of the water.

Most importantly the effect it has on the velocity of the water. The fastest average velocity was

at 1.67m/s at point e right before the cross sectional area is recovered. The slowest points where

at a and f which consisted of the same velocity. This can be considered as a demonstration of

convergent flow where the streamlines are smooth and in order for the most part. In the case of a

divergent flow we encounter stagnant forces that result in backpressure. This backpressure

created on the outlet side of the venturi meter will make a greater pressure on the right side of the

meter. While performing this experiment our group encountered a run with backpressure, where

we witnessed negative pressure in the throat section of the venturi meter, of course we

disregarded this run, but it gave us an example of a divergent flow. We believe we obtained this

flow due to major air bubbles in our system, we completely bled the system once more after that

run.

Conclusion

This experiment was conducted in order to demonstrate and understand Bernoullis theorem.

After calculating the different velocities throughout the venturi meter, it was easy to visualize the

effects of the meter on the flow of the water. Using Bernoulli’s equation we were able to

demonstrate the difference between the hydraulic grade line and energy grade line throughout the

venturi meter. When fluid flow is convergent it is easier to demonstrate Bernoulli’s theorem, as

opposed to when it is divergent, we must consider effects of stagnation. Another factor we did

not consider in this experiment that can also impact how flow acts in a pipe is the roughness of

the walls along the pipe. Furthermore, as long as the cross-sectional area recuperates after the

throat section of the venturi meter, Bernoulli’s theorem applies and can be demonstrated.

References

Sultana, 2015, “Fluid Mechanics Laboratory”, Instruction Manual.

Munson, B.R., T. H. Okiishi, W. W. Huebsch, A.P. Rothmayer, 2012, “Fundamentals of Fluid

Mechanics”, 7th edition, John Wiley.