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 velocity 2 = total pressure = constant
Fig 1: Bernoulli Equation
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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 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.
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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.
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x ) = v ( x x (t )), In steady flow the velocity is constant with respect to time, v = v ( x )), 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
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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 cross -section the fluid moves away from crosssection 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
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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— 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:
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(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: 2
The term v / (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:
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(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.
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Experimental Setup:
Fig 3: 3: Bernoulli’s apparatus
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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.
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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 the Tapping Positions of the Venturi 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 Weight of
pisometer and pitot tube reading at different tapping(mm) Obs.
water
no
with
a
b
c
d
e
f
Time(s)
bucket
M
P
M
P
M
P
M
P
M
P
M
P
(Kg)
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
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Calculated data: Density of water at 26.8ºC = 996.513 kg/m
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Table-3: Table of Flow Flow Rate Measurement Measurement
Obs. no
Mass of
Mass of empty bucket (kg)
(water + bucket) (kg)
1 2
0.6237
3
Mass of
Time
water (kg)
(s)
Mass flow
Volumetric
rate
flow
(kg/s)
rate Q (m /s)
3
2.9
2.2763
40.4
0.056344
5.65×10
-05
5.0
4.3763
40.6
0.107791
1.08×10
-04
7.1
6.4763
40.1
0.161504
1.62×10
-04
Table-4: Table of calculated data
Volumetri c flow rate
5.65E-05
1.08E-04
1.62E-04
Tub e No
Pressure head (pizometer ) P/ρg mm
Theoretical head 2 V /2g+P/ρg mm
Pitot tube pressure head
Diamete r(m)
Cross sectional area (m2)
Velocity (m/s)
Velocity head V2/2g *1000 mm
a
2.50E-02
4.91E-04
1.15E-01
0.675581643
200
200.6755816
200
b
1.39E-02
1.52E-04
3.72E-01
7.049424254
193
200.0494243
198
c d
1.18E-02 1.07E-02
1.09E-04 8.99E-05
5.18E-01 6.28E-01
13.70843346 20.15215249
185 180
198.7084335 200.1521525
198 198
e
1.00E-02
7.85E-05
7.20E-01
26.43026459
164
190.4302646
194
f
2.50E-02
4.91E-04
1.15E-01
0.675581643
177
177.6755816
178
a
2.50E-02
4.91E-04
2.20E-01
2.468473421
217
219.4684734
221
b
1.39E-02
1.52E-04
7.11E-01
25.75753293
194
219.7575329
198
c d
1.18E-02 1.07E-02
1.09E-04 8.99E-05
9.91E-01 1.20E+00
50.08854817 73.63292557
185 147
235.0885482 220.6329256
240 238
e
1.00E-02
7.85E-05
1.38E+00
96.57220022
94
190.5722002
237
f
2.50E-02
4.91E-04
2.20E-01
2.468473421
160
162.4684734
173
a
2.50E-02
4.91E-04
3.30E-01
5.554065197
258
263.5540652
267
b c
1.39E-02 1.18E-02
1.52E-04 1.09E-04
1.07E+00 1.49E+00
57.95444909 112.6992334
221 182
278.9544491 294.6992334
262 267
d
1.07E-02
8.99E-05
1.80E+00
165.6740825
120
285.6740825
267
e f
1.00E-02 2.50E-02
7.85E-05 4.91E-04
2.06E+00 3.30E-01
217.2874505 5.554065197
73 140
290.2874505 145.5540652
263 173
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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 m
Volumetric flow rate, Q =
-04
= 1.08×10
3
m /s
Velocity Calculation: -2
Diameter of cross section a = 25mm = 2.50×10 m Area, A =
d
2
=
4
( 2.50 10
2
)
2
2
m
4 -4
2
= 4.91× 10 m
Q
Velocity, v =
A
-4
-4
= 1.08×10 / 4.91× 10 m/s =0.21996
Similarly, For diameter b = m,
A=
(13 .9 10
3
-4
-4
v = 1.08×10 /1.52×10
For diameter c = m, A=
2
2
m
4 2
=1.52×10 -4
)
m,
= 0.7105 m/s (11 .8 10
3
)
2
2
m
4 -4
2
=1.09×10 m , -4
-4
-1
v =1.08×10 /1.09×10 =0.9908ms -3
For diameter d = 10.7×10 m, A=
(10 .7 10
4
3
)
2
2
-4
2
m =0.899×10 m ,
13
-4
-4
v = 1.089×10 /0.899×10
-1
=1.2013ms
-3
For diameter e = 10×10 m,A= -4
(10 10
-4
For diameter f = 25×10 m,
-4
)
2
2
A=
-4
-4
2
m =0.785×10 m ,
4
v = 1.08×10 / 0.785×10 = 1.3757 ms
-3
3
-1
( 25 10
4
3
)2
2
-4
2
m =4.91×10 m ,
-1
v = 1.08×10 / 4.91×10 =0.21996 ms
Total Head Calculation: For cross section at a,
Velocity head =
v
2
2g
=
(0.21996) 2 9.8 9.8
2
m = 2.4684 mm
Pressure head (observed), P/ρg = 217 mm Total head, H= (
v2 2g
P g
) = 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
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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 experim ent 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 B ernoulli’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 re lated to pressure measurements by using Bernoulli’ Bernoulli’ss 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 withou t 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.
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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
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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 a nd 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.
.
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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.
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Literature Cited
Franzini, Joseph B. ; Finnemore, E. John, Fluid Mechanics with Engineering th
Application, 9 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#Derivatio http://en.wikipedia.org/wik i/Bernoulli's_principle#Derivations_of_Bernoulli_equa ns_of_Bernoulli_equatio tio n Date Cited: 04/07/2012
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Nomenclature: Table 05: List of symbols used throughout the report Symbol
Name
Unit
Dimension
A
Cross Cross section sectional al area
m
L
d
Diameter
m
L
g
Acceleration Acceleration due to gravity
m/s
L
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 T
Q
Volum Volumetr etric ic Flow Flow rate rate
m /s
L /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
Densi Density ty
kg/m kg/m
M/L
µ
Visco Viscosit sity y
Ns/m Ns/m
ML T
γ
Specifi Specificc weight weight
N/m
ML T
-
-
-
-
-
-
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Appendix
Application and Modification of Bernoulli’s 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.
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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 p 2 ,
We now have an expression for velocity obtained from two pressure measurements and the application of the Bernoulli equation.
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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 static tube 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
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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.
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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
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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
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