INTRODUCTION

Bernoulli’s theorem which is known as Bernoulli’s principle, states that an increase in the

speed of moving air or a flowing fluid is accompanied by a decrease in the air or fluid’s

pressure. Swiss scientist, Daniel Bernoulli (1700-1782), demonstrated that, in most cases the

pressure in a liquid or gas decreases as the liquid or gas move faster. This is an important

principle involving the movement of a fluid through the pressure difference. Suppose a fluid

is moving in a horizontal direction and encounters a pressure difference. This pressure

difference will result in a net force, which is by Newton’s Second Law will cause an

acceleration of the fluid.

Bernoulli’s theorem states that the total energy (pressure energy, potential energy and

kinetic energy) of an incompressible and non-viscous fluid in steady flow through a pipe

remains constant throughout the flow, provided there is no source or sink of the fluid along

the length of the pipe. This statement is due to the assumption that there is no loss energy

due to friction.

This theorem deals with the facts that when there is slow flow in a fluid, there will be

increase in pressure and when there is increased flow in a fluid, there will be decrease in

pressure. If the elevation remains constant, velocity and pressure, energy to or from the

system can be calculated by this equation-

Static pressure + dynamic pressure = total pressure = constant

Static pressure + 1/2 x density x velocity2 = total pressure = constant

Fig 1: Bernoulli Equation

So the main objective of this experiment is to justify the validity of Bernoulli’s theorem for

water flow through a circular conduit.

The converging-diverging nozzle apparatus is used to show the validity of Bernoulli’s

Equation. It is also used to show the validity of the continuity equation where the fluid flows

is relatively incompressible. The data taken will show the presence of fluid energy losses,

often attributed to friction and the turbulence and eddy currents associated with a

separation of flow from the conduit walls.

Theory

Clearly stated that the assumptions made in deriving the Bernoulli’s equation is:

The liquid is incompressible.

The liquid is non-viscous.

The flow is steady and the velocity of the liquid is less than the critical velocity for

the liquid.

There is no loss of energy due to friction.

Derivation of Bernoulli equation from Newton’s second law:

The Bernoulli equation for incompressible fluids can be derived by integrating the Euler

equations, or applying the law of conservation of energy in two sections along a streamline,

ignoring viscosity, compressibility, and thermal effects.

The simplest derivation is to first ignore gravity and consider constrictions and expansions in

pipes that are otherwise straight, as seen in Venturi effect. Let the x axis be directed down

the axis of the pipe.

Define a parcel of fluid moving through a pipe with cross-sectional area "A", the length of

the parcel is "dx", and the volume of the parcel A dx. If mass density is ρ, the mass of the

parcel is density multiplied by its volume m = ρ A dx. The change in pressure over distance

dx is "dp" and flow velocity v = dx / dt.

Apply Newton's Second Law of Motion Force F =mass . acceleration and recognizing that the

effective force on the parcel of fluid is -A dp. If the pressure decreases along the length of

the pipe, dp is negative but the force resulting in flow is positive along the x axis.

In steady flow the velocity is constant with respect to time, v = v(x) = v(x(t)), so v itself is not

directly a function of time t. It is only when the parcel moves through x that the cross

sectional area changes: v depends on t only through the cross-sectional position x(t).

With density ρ constant, the equation of motion can be written as

by integrating with respect to x

where C is a constant, sometimes referred to as the Bernoulli constant. It is not a universal

constant, but rather a constant of a particular fluid system. The deduction is: where the

speed is large, pressure is low and vice versa.

In the above derivation, no external work-energy principle is invoked. Rather, Bernoulli's

principle was inherently derived by a simple manipulation of the momentum equation.

Fig 2: Flow of water through Bernoulli Apparatus

A streamtube of fluid moving to the right. Indicated are pressure, elevation, flow speed,

distance (s), and cross-sectional area. Note that in this figure elevation is denoted as h,

contrary to the text where it is given by z. Another way to derive Bernoulli's principle for an

incompressible flow is by applying conservation of energy. In the form of the work-energy

theorem, stating that the change in the kinetic energy Ekin of the system equals the net

work W done on the system;

Therefore, the work done by the forces in the fluid = increase in kinetic energy.

The system consists of the volume of fluid, initially between the cross-sections A1 and A2. In

the time interval Δt fluid elements initially at the inflow cross-section A1 move over a

distance s1 = v1 Δt, while at the outflow cross-section the fluid moves away from cross-

section A2 over a distance s2 = v2 Δt. The displaced fluid volumes at the inflow and outflow

are respectively A1 s1 and A2 s2. The associated displaced fluid masses are – when ρ is the

fluid's mass density – equal to density times volume, so ρ A1 s1 and ρ A2 s2. By mass

conservation, these two masses displaced in the time interval Δt have to be equal, and this

displaced mass is denoted by Δm:

The work done by the forces consists of two parts:

The work done by the pressure acting on the areas A1 and A2

The work done by gravity: the gravitational potential energy in the volume A1 s1 is lost, and

at the outflow in the volume A2 s2 is gained. So, the change in gravitational potential energy

ΔEpot,gravity in the time interval Δt is

Now, the work by the force of gravity is opposite to the change in potential energy,

Wgravity = −ΔEpot,gravity: while the force of gravity is in the negative z-direction, the work—

gravity force times change in elevation—will be negative for a positive elevation change

Δz = z2 − z1, while the corresponding potential energy change is positive. So:

And the total work done in this time interval is

The increase in kinetic energy is

Putting these together, the work-kinetic energy theorem W = ΔEkin gives:

or

After dividing by the mass Δm = ρ A1 v1 Δt = ρ A2 v2 Δt the result is:

or, as stated in the first paragraph:

(Eqn. 1), which is also Equation (A)

Further division by g produces the following equation. Note that each term can be described

in the length dimension (such as meters). This is the head equation derived from Bernoulli's

principle:

(Eqn. 2a)

The middle term, z, represents the potential energy of the fluid due to its elevation with

respect to a reference plane. Now, z is called the elevation head and given the designation

elevation.

A free falling mass from an elevation z > 0 (in a vacuum) will reach a speed

when arriving at elevation z = 0.

Or when we rearrange it as a head:

The term v2 / (2 g) is called the velocity head, expressed as a length measurement. It

represents the internal energy of the fluid due to its motion.

The hydrostatic pressure p is defined as

, with p0 some reference pressure, or when we rearrange it as a head:

The term p / (ρg) is also called the pressure head, expressed as a length measurement. It

represents the internal energy of the fluid due to the pressure exerted on the container.

When we combine the head due to the flow speed and the head due to static pressure with

the elevation above a reference plane, we obtain a simple relationship useful for

incompressible fluids using the velocity head, elevation head, and pressure head.

(Eqn. 2b)

If we were to multiply Eqn. 1 by the density of the fluid, we would get an equation with

three pressure terms:

(Eqn. 3)

We note that the pressure of the system is constant in this form of the Bernoulli Equation. If

the static pressure of the system (the far right term) increases, and if the pressure due to

elevation (the middle term) is constant, then we know that the dynamic pressure (the left

term) must have decreased. In other words, if the speed of a fluid decreases and it is not

due to an elevation difference, we know it must be due to an increase in the static pressure

that is resisting the flow.

All three equations are merely simplified versions of an energy balance on a system.

Experimental Setup:

Fig 3: Bernoulli’s apparatus

Fig 4: Diagram Showing the Diameter of venture meter at various point

Procedure:

The room temperature was measured.

A piezometer is attached to the pipe through which water flows. Then for any flow of

water manometer readings were taken.

The mass flow rate of water was measured by collecting water in a bucket for a

period of time.

Then the bucket was weighted in a manual balance. The empty bucket was weighted

separately.

By varying the flow rate with the valve attached to the pipe, several data were taken.

Observed data:

Temperature of empty bucket= 0.6 kg

Room temperature = 26.8º C

Temperature of water = 27.5º C

Table-1: Diameters of the Tapping Positions of the Venturi Tube

Tapping Position Diameter (mm)

a 25.0

b 13.9

c 11.8

d 10.7

e 10.0

f 25.0

Table 2: Reading of manometer and pitot tube at different tapping

Obs. no

pisometer and pitot tube reading at different tapping(mm)

Weight of

water with

bucket (Kg)

Time(s)a b c d e f

M P M P M P M P M P M P

1 200 200 193 198 185 198 180 198 164 184 177 178 2.9 40.4

2 217 221 194 198 185 240 147 238 94 237 160 173 5.0 40.6

3 258 267 221 262 182 267 120 267 73 263 140 173 7.1 40.1

Calculated data:

Density of water at 26.8ºC = 996.513 kg/m3

Table-3: Table of Flow Rate Measurement

Obs. no

Mass of empty

bucket (kg)

Mass of

(water + bucket) (kg)

Mass of water (kg)

Time (s)

Mass flow

rate

(kg/s)

Volumetric

flow

rate Q (m3/s)

10.6237

2.9 2.2763 40.4 0.056344 5.65×10-05

2 5.0 4.3763 40.6 0.107791 1.08×10-04

3 7.1 6.4763 40.1 0.161504 1.62×10-04

Table-4: Table of calculated data

Volumetric flow rate

Tube

NoDiamete

r(m)

Cross sectional area (m2)

Velocity (m/s)

Velocity head V2/2g *1000 mm

Pressure head

(pizometer) P/ρg mm

Theoretical head

V2/2g+P/ρg mm

Pitot tube

pressure head

5.65E-05

a 2.50E-02 4.91E-04 1.15E-01 0.675581643 200 200.6755816 200b 1.39E-02 1.52E-04 3.72E-01 7.049424254 193 200.0494243 198c 1.18E-02 1.09E-04 5.18E-01 13.70843346 185 198.7084335 198d 1.07E-02 8.99E-05 6.28E-01 20.15215249 180 200.1521525 198e 1.00E-02 7.85E-05 7.20E-01 26.43026459 164 190.4302646 194f 2.50E-02 4.91E-04 1.15E-01 0.675581643 177 177.6755816 178

1.08E-04

a 2.50E-02 4.91E-04 2.20E-01 2.468473421 217 219.4684734 221b 1.39E-02 1.52E-04 7.11E-01 25.75753293 194 219.7575329 198c 1.18E-02 1.09E-04 9.91E-01 50.08854817 185 235.0885482 240d 1.07E-02 8.99E-05 1.20E+00 73.63292557 147 220.6329256 238e 1.00E-02 7.85E-05 1.38E+00 96.57220022 94 190.5722002 237f 2.50E-02 4.91E-04 2.20E-01 2.468473421 160 162.4684734 173

1.62E-04

a 2.50E-02 4.91E-04 3.30E-01 5.554065197 258 263.5540652 267b 1.39E-02 1.52E-04 1.07E+00 57.95444909 221 278.9544491 262c 1.18E-02 1.09E-04 1.49E+00 112.6992334 182 294.6992334 267d 1.07E-02 8.99E-05 1.80E+00 165.6740825 120 285.6740825 267e 1.00E-02 7.85E-05 2.06E+00 217.2874505 73 290.2874505 263f 2.50E-02 4.91E-04 3.30E-01 5.554065197 140 145.5540652 173

Sample calculation:

Sample calculation for observation 2

Volumetric flow rate calculation:

Mass of water,M = 4.3763 kg

Time interval, t = 40.6 s

Mass flow rate, m = 0.1078 kg/s

Volumetric flow rate, Q =

mρ

= 1.08×10-04 m3/s

Velocity Calculation:

Diameter of cross section a = 25mm = 2.50×10-2m

Area, A =

πd2

4=

π×(2 .50×10−2 )2

4 m2

= 4.91× 10-4 m2

Velocity, v =

QA

= 1.08×10-4 / 4.91× 10-4 m/s

=0.21996

Similarly,

For diameter b = m, A=

π×(13 .9×10−3 )2

4 m2

=1.52×10-4 m2,

v = 1.08×10-4 /1.52×10-4 = 0.7105 m/s

For diameter c = m, A=

π×(11.8×10−3 )2

4 m2

=1.09×10-4m2,

v =1.08×10-4 /1.09×10-4 =0.9908ms-1

For diameter d = 10.7×10-3m, A=

π×(10 . 7×10−3 )2

4 m2=0.899×10-4m2,

v = 1.089×10-4 /0.899×10-4 =1.2013ms-1

For diameter e = 10×10-3m,A=

π×(10×10−3 )2

4 m2=0.785×10-4m2,

v = 1.08×10-4 / 0.785×10-4 = 1.3757 ms-1

For diameter f = 25×10-3m, A=

π×(25×10−3 )2

4 m2=4.91×10-4m2,

v = 1.08×10-4 / 4.91×10-4 =0.21996 ms-1

Total Head Calculation:

For cross section at a,

Velocity head =

v2

2g =

(0 .21996)2

2× 9 .8 m = 2.4684 mm

Pressure head (observed), P/ρg = 217 mm

Total head, H= (

v2

2g+ Pρg ) = 217+2.4684 mm = 219.4684 mm

Similarly,

For cross section at b, H= 219.7575 mm

For cross section at c, H=235.0885 mm

For cross section at d, H=220.6329 mm

For cross section at e, H=190.5722 mm

For cross section at f, H=162.4684 mm

DISCUSSION

The objectives of this experiment is to investigate the validity of the Bernoulli equation

when applied to the steady flow of water in a tapered duct and to measure the flow rates

and both static and total pressure heads in a rigid convergent and divergent tube of known

geometry for a range of steady flow rates. This experiment is based on the Bernoulli’s

principle which relates between velocities with the pressure for an inviscid flow. To achieve

the objectives of this experiment, Bernoulli’s theorem demonstration apparatus was used.

This instrument was combined with a venturi meter and the pad of manometer tubes which

indicate the pressure of h1 until h8 but for this experiment only the pressure in manometer

h1 until h6 being measured. A venturi is basically a converging-diverging section (like an

hourglass), typically placed between tube or duct sections with fixed cross-sectional area.

The flow rates through the venturi meter can be related to pressure measurements by using

Bernoulli’s equation. From the result obtained through this experiment, it is been observed

that when the pressure difference increase, the flow rates of the water increase and thus

the velocities also increase for both convergent and divergent flow.

Though the experiment was done very carefully and properly, but there is some

discrepancy in the result. The total head was not same in different points. The possible

reasons are-

There was leak in pitot tube

In Bernoulli’s theorem the fluid is considered as inviscid and incompressible. But

practically the water, used in this experiment as the working fluid does not satisfy this

assumption.

Since the venturi tube cannot be thermally isolated from the surrounding completely,

there are some possible heat transfer between tube and surrounding which is not

account in the theorem. It introduces a permanent frictional resistance in the pipeline.

The use of mean velocity without kinetic energy correction factor (α) introduces some

error in the results. Here we assume that α = 1. But it varies with Reynolds number. The

variation of α with Reynolds number is given in appendix 2.

The piezometer readings were fluctuating continuously during the experiment due to

unsteady supply. Since capillarity makes water rise in piezometer tube, it introduce

some error in calculation of the static head.

Contraction loss: There is a marked drop in pressure due to the increase in velocity and

to the loss of energy in turbulence.

Fig 5: Flow at sudden contraction of cross section

Directional velocity fluctuation due to turbulence increase pitot tube readings and

hence we got large value of total head.

A large head loss occurs at the entrance of the pitot tube due to sudden contraction.

Expansion loss: In sudden expansion there is a state of excessive turbulence. The loss

due to sudden expansion is greater than the loss due to a corresponding contraction.

This is so because of the inherent instability of flow in expansion where the diverging

path of the flow tend to encourage the formation of eddies within the flow. In

converging flow there is a dampening effect on eddy formation and hence loss is less

than diverging flow. It is reflected by the drastic decrease of total head in figure 6.

Fig 6: Flow at sudden enlargement of cross section

Conclusion

From the result obtained, we can conclude that the Bernoulli’s equation is valid for

convergent and divergent flow as both of it does obey the equation. The results were quite

satisfactory and very near the authentic data at the two of the three observations

performed. But we also know the reasons and handicaps for which the discrepancies

appeared and discussed it thoroughly in the discussion section. Therefore considering all the

evens and odds we can come to a conclusion that the experiment was a very successful one

which not only introduced us with the properties of Bernoulli’s Theorem and fluid flow

system but also enabled us to understand the deviations and inconsistencies we have to

face in the real life situations comparing to the theoretical.

.

RECOMMENDATION

1. Repeat the experiment for several times to get the average values in order to get more

accurate results.

2. Make sure the trap bubbles must be removing first before start running the experiment.

3. The eye position of the observer must be parallel to the water meniscus when taking the

reading at the manometers to avoid parallax error.

4. The valve must be control carefully to maintain the constant values of the pressure

difference as it is quite difficult to control.

5. The time keeper must be alert with the rising of water volume to avoid error and must be

only a person who taking the time.

6. The leakage of water in the instrument must be avoided.

Literature Cited

Franzini, Joseph B. ; Finnemore, E. John, Fluid Mechanics with Engineering Application, 9th edition. McGraw -Hill ,New York,1997, Page (142,121)

Mott, Robert L, “ Applied Fluid Mechanics”, 5th ed., Prentice Hall Upper Saddle River, 2000, Page (272,276)

Derivations of Bernoulli equation, http://en.wikipedia.org/wiki/Bernoulli's_principle#Derivations_of_Bernoulli_equationDate Cited: 04/07/2012

Nomenclature:

Table 05: List of symbols used throughout the report

Symbol Name Unit Dimension

A Cross sectional area m2 L2

d Diameter m L

g Acceleration due to gravity m/s2 L2

hp Pressure head m L

ht Total head m L

hv Velocity head m L

m Mass of water kg M

p Pressure bars ML-1T-2

Q Volumetric Flow rate m3/s L3/T

Mass flow rate kg/s M/T

v Average Velocity m/s L/T

π Constant value 3.14 Dimensionless

z Vertical height m L

ρ Density kg/m3 M/L3

µ Viscosity Ns/m2 ML-1T-1

γ Specific weight N/m3 ML-2T-2

Appendix

Application and Modification of Bernoulli’s Equation

A) Stagnation pressure and dynamic pressure:

If a stream of uniform velocity flows into a blunt body, the stream lines take a

pattern similar to this:

Fig 7: Streamlines around a blunt body

Note how some move to the left and some to the right. But one, in the centre, goes

to the tip of the blunt body and stops. It stops because at this point the velocity is zero - the

fluid does not move at this one point. This point is known as the stagnation point.

From the Bernoulli equation we can calculate the pressure at this point. Apply

Bernoulli along the central streamline from a point upstream where the velocity is u1 and

the pressure p1 to the stagnation point of the blunt body where the velocity is zero, u2 = 0.

Also z1 = z2.

This increase in pressure which brings the fluid to rest is called the dynamic pressure.

The blunt body stopping the fluid does not have to be a solid. I could be a static

column of fluid. Two piezometers, one as normal and one as a Pitot tube within the pipe can

be used in an arrangement shown below to measure velocity of flow.

Using the above theory, we have the equation for p2 ,

We now have an expression for velocity obtained from two pressure measurements

and the application of the Bernoulli equation.

B) Pitot Static Tube:

The necessity of two piezometers and thus two readings make this arrangement a

little awkward. Connecting the piezometers to a manometer would simplify things but there

are still two tubes. The Pitot statictube combines the tubes and they can then be easily

connected to a manometer. A Pitot static tube is shown below. The holes on the side of the

tube connect to one side of a manometer and register the static head, (h1), while the central

hole is connected to the other side of the manometer to register, as before, the stagnation

head (h2).

Fig 8: A Pitot-static tube

Consider the pressures on the level of the centre line of the Pitot tube and using the

theory of the manometer,

We know that , substituting this in to the above gives

The Pitot/Pitot-static tubes give velocities at points in the flow. It does not give the

overall discharge of the stream, which is often what is wanted. It also has the drawback that

it is liable to block easily, particularly if there is significant debris in the flow.

C) Flow Through A Small Orifice:

We are to consider the flow from a tank through a hole in the side close to the base.

The general arrangement and a close up of the hole and streamlines are shown in the figure

below

Fig 9: Tank and streamlines of flow out of the sharp edged orifice

The shape of the holes edges are as they are (sharp) to minimize frictional losses by

minimizing the contact between the hole and the liquid - the only contact is the very edge.

Looking at the streamlines you can see how they contract after the orifice to a

minimum value when they all become parallel; at this point, the velocity and pressure are

uniform across the jet. This convergence is called the vena contracta. (From the Latin

'contracted vein'). It is necessary to know the amount of contraction to allow us to calculate

the flow.

We can predict the velocity at the orifice using the Bernoulli equation. Apply it along

the streamline joining point 1 on the surface to point 2 at the centre of the orifice.

At the surface velocity is negligible (u1 = 0) and the pressure atmospheric (p1 = 0).At

the orifice the jet is open to the air so again the pressure is atmospheric (p2 = 0). If we take

the datum line through the orifice then z1 = h and z2 =0, leaving

This is the theoretical value of velocity.

D) Energy losses due to friction:

In a real pipe line there are energy losses due to friction - these must be taken into

account as they can be very significant.

Fig 10: Hydraulic Grade line and Total head lines for a constant diameter pipe with friction

The total head - or total energy per unit weight – is constant. We are considering

energy conservation, so if we allow for an amount of energy to be lost due to friction the

total head will change