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INTRODUCTION On or about the 29th June 2013, we were embarked on an endeavour of completing our practical exercise in relation with the theoretical part of Fluid Mechanics II Module; the said session took place at Room D014A of the Vaal University of Technology. After a brief introductory excerpt by Mr. PITA, our instructor, data on two experiments; namely ―Hydrostatic Pressure‖ and ―Bernoulli Theorem Demonstration‖, had to be recorded, analysed and interpreted to constitute the backbone of this report. 1. HYDROSTATIC PRESSURE PRACTICAL Before any attempt to get into the core of this subject, we unassumingly believe that it is worth to underline that the effect of hydrostatic pressure is of major significance in many areas of engineering, such as shipbuilding, the construction of dykes, weirs and locks, and in sanitary and building services engineering. Furthermore, hydrostatic pressure is, the pressure exerted by a fluid at equilibrium due to the force of gravity. A fluid in this condition is known as a hydrostatic fluid. So our Hydrostatic pressure practical was to determine the hydrostatic pressure of water on a flat surface. Adding weight and then filling the tank with water to the point where the apparatus was in equilibrium so that we can calculate the force on the flat surface using the given equations. 1.1 AIM OF THE EXPERIMENT The aim of this experiment is to experimentally locate the centre of pressure of a vertical submerged surface, determine the position of the line of action of the thrust and compare the measurements to theoretical predictions. 1.2 SKETCH AND DESCRIPTION OF THE APPARATUS The equipment required for the accomplishment of our task is: •
The F1-12 Hydrostatic Pressure Apparatus
•
F1-10 Hydraulics Bench (water source)
A fabricated quadrant is mounted on a balance arm which pivots on knife edges. The knife edges coincide with the centre of arc of the quadrant. Thus, of the hydrostatic forces acting on the quadrant when immersed in water, only the force on the rectangular end face gives rise to a moment about the knife edges (forces on the curved surfaces resolve through the pivot and have no effect on the moment). This moment is counteracted by variable weights at a fixed distance from the pivot allowing the magnitude and position of the hydrostatic force to be determined for different water depths. The quadrant can be operated with the vertical end face partially or fully submerged, allowing the difference in theory to be investigated.
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The balance arm incorporates a weight hanger for the weights supplied and an adjustable counterbalance weight to ensure that the balance arm is horizontal before immersing the quadrant in water. The assembled balance arm is mounted on top of a clear acrylic tank which may be levelled by adjusting three screwed feet. Correct alignment is indicated on a circular spirit level mounted on the base of the tank. A level indicator attached to the side of the tank shows when the balance arm is horizontal. Water is admitted to the top of the tank by a flexible tube and may be drained through a cock in the side of the tank. The water level is indicated on a scale on the side of the quadrant. The key geometrical parameters of the device are: d: the submersion depth (distance from the bottom of the surface to the free surface) h’: the depth of the centre of pressure from the free surface h‖: the distance of the centre of pressure below the pivot position B: the width of the surface D: the height of the surface W: the weight (mg) of the hangar L: is the distance from the pivot point to the hangar weight H: is the distance from the pivot point to the bottom of the vertical surface h: either ½ the distance from the bottom of the face to the free surface if the surface is partially submerged or equal to D/2 if the surface is fully submerged.
MABENGO N.D. — N.D. — STUDENT STUDENT NO. 48591238 — 48591238 — July July 2013
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Figure 1. Diagram of the Armfield hydrostatic pressure apparatus with the key geometrical parameters defined with the surface partially submerged. F is the hydrostatic thrust and mg is the hangar weight
Figure 2 Photo of the Armfield hydrostatic pressure apparatus
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1.3 THEORY When the quadrant is immersed in water, it is possible to analyze the forces acting on the surfaces of the quadrant as: 1) The hydrostatic force at any point on the curved surfaces is normal to the surface and therefore resolves through the pivot point because the pivot point is located at the origin of the radii. Hydrostatic forces on the upper and lower curved surfaces therefore have no net torque effect; 2) The forces on the sides of the quadrant are equal and opposite horizontal forces; 3) The hydrostatic force on the vertical submerged face is counteracted by the balance weight. At equilibrium, the sum of the moments about the pivot point is zero.
[1]
Thus,
where m is the hanging mass, g is the acceleration due to gravity, L is the distance from the pivot point to the hanging mass, and
h”
is the distance from the pivot point to the
centre of pressure. With the mass, balance length, and hydrostatic force determined, the location of the centre of pressure on the end face may be determined for either a partially submerged vertical face or a fully submerged vertical face.
For the case of a partially submerged face the hydrostatic thrust is defined as
[2] Where A is the wetted surface area of the vertical face ( B∙d),
is the density of water,
and h is the mean depth of immersion (h = d/2). Therefore the hydrostatic thrust is
[3]
Substituting Eq. 3 into Eqn. 1 and solving for h” yields the experimentally determined distance between the pivot point and centre of pressure
[4]
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The theoretical depth of pressure below the free surface is
[5]
where I x is the 2nd moment of area of immersed section about an axis in t he free surface. Applying the parallel axes theorem yields
[6]
Substituting Eqn. 6 into Eqn. 5 yields
[7]
From geometry, the theoretical depth of centre of pressure below the pivot point is
[8] Combining Eqn. 7 and Eqn. 8 yields the theoretical depth of centre of pressure
[9] For the case where the vertical face of the quadrant is fully submerged, the hydrostatic thrust is
[10]
Substituting Eqn. 10 into Eqn. 1 the experimental distance between the pivot point and the centre of pressure is
[11]
The theoretical depth of pressure below the free surface is given by Eqn. 5. Applying the parallel axes theorem for the fully submerged surface yields
[12]
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Inserting Eqn. 12 into Eqn. 5 yields
[13]
Substituting Eqn. 13 into Eqn. 8 the theoretical depth of centre of pressure below the pivot point is
[14]
1.4 METHOD Setup
The empty tank had to be positioned on the hydraulic bench
The screwed feet was adjusted until the base was horizontal
The drain valve were to be closed to avoid losing water
Readings
A small mass of 50g had to be added to the balancing arm
Water had to be added to the tank until the balancing arm was horizontal
The depth of immersion readings from the scale on the face of the quadrant had to be recorded.
The same thing was to be repeated by adding more weight to the increment of 50g each until we had 5 readings and the quadrant was totally submerged.
1.5 RESULTS A priori technical data of the hydrostatic apparatus were given in the table bellow: Description
Symbol
Size
Length of balance
L
275 mm
Quadrant to pivot
H
200 mm
Height
D
100 mm
Width of quadrant
B
75 mm
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At the end of the 5 readings, the results obtained were recorded in a table as follows: Length
Quadrant
Width of
Quadrant
Quadrant
D (m)
B (m)
1
0.1
0.075
0.275
0.2
0.05
0.044
0.712206
0.189394
0.185333
2
0.1
0.075
0.275
0.2
0.1
0.064
1.506816
0.179036
0.178667
3
0.1
0.075
0.275
0.2
0.15
0.08
2.3544
0.171875
0.173333
4
0.1
0.075
0.275
0.2
0.2
0.094
3.250544
0.165988
0.168667
5
0.1
0.075
0.275
0.2
0.25
0.107
4.193775
0.160819
0.164620
of
to pivot H
Balance
(m)
L (m)
Depth of
Mass
Thrust
immersion
(kg)
(Fr) (N)
d (m)
2nd Moment
2nd
Height of
Experimental
moment theory
h’’(m)
h’’(m)
Calculations For reason of space, we have opted to only represent on this report sample calculations performed as they appear on the above table. Sample calculations of the Thrust (Fr) for a partitially submerged quadrant :
Fr
1 2
2
gBd
1 2
1000 9.81 0.075 0.044 0.712206N 2
Sample calculations of the Thrust (Fr) for a fully submerged quadrant :
Fr gBD d
D
0.1 1000 9.81 0.075 0.1 0.107 4.193775 N 2 2
Sample calculation of the Theoretical 2nd Moment h’’ for a par tially submerged quadrant :
h" H
d 3
0.2
0.044 3
0.185333m
Sample calculation of the Experimental 2nd Moment h” for a partially submerged quadrant :
h"
2mL 2
Bd
2 0.05 0.275 1000 0.075 0.044
2
0.189394m
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Sample calculation of the Theoretical 2nd Moment h’’ for a fully submerged quadrant :
D h" H
2
12
2
D d 2 0.2 D d
2
0.1
12
2
0.1 0.107 2 0.107 0.16462m 0.1 0.107
2
2
Sample calculation of the Experimental 2nd Moment h’’ for a fully submerged quadrant :
h"
mL
D BD d 2
0.25 0.275 0.1 1000 0.075 0.1 0.107 2
0.160819 m
Graphs THRUST vs DEPTH OF IMMERSION
) N ( e c r o F c i t a t s o r d y H
Depth of immersion (m)
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) m ( l a t n e m i r e p x E t n e m o M n
2
Depth of immersion (m)
) m ( l a c i t e r o e h T t n e m o M n
2
Depth of immersion (m)
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Discussion of results From the above graphs we can inconspicuously conclude that as the depth of the water increased, the hydrostatic thrust increased and the distance to the centre of pressure decreased. The centre of pressure moved closer to the centre of the vertical face as the depth increased. The experimental values for the distance to the centre of pressure were smaller than the theoretical distances at nearly all submersion depths. Since the same hydrostatic force is used to calculate the turning moments, the experimental turning moment was also smaller than the theoretical turning moment at nearly all submersion depths, this is the reason why the relationship between the 2 nd moment theoretical and the depth of immersion is purely linear and that of the 2 nd moment experimental and the depth of immersion is more or less linear . The reader should also bear in mind that as the depth of immersion increased, the second moment turning moment decreased . 1.6 CONCLUSION In this practical the turning moment and the distance to centre of pressure in relation to depth were determined. The objectives of this practical were to determine the hydrostatic thrust acting on a plane surface immersed in water when the surface is partially submerged or fully submerged, to determine the position of the line of action of the thrust and to compare the position determined by experiment with the theoretical position. The objectives of this endeavour were accomplished using a F1-10 Hydraulics Bench and F1-12 Hydrostatic Pressure Apparatus. As the depth of the water increased, the hydrostatic thrust increased and the centre of pressure moved closer to the centre of the vertical face. Causes of inaccurate readings The experimental distances to the centre of pressure were lower than the theoretical distances to the centre of pressure. Many factors may have contributed to this discrepancy. Water splashing onto the balance arm or quadrant would cause overestimation of the water depth for equilibrium, changes in water temperature would cause variations in the water density, and excess weight on the masses would cause underestimation of the experimental distance to the centre of pressure. What was learnt from this experiment? Besides the theoretical part about hydrostatic forces acting on plane surfaces, I have gained a lot in terms of familiarising myself with equipment that can be used in a Fluid Mechanics Lab: I was unaware of the fact that there is a countless of manufacturers of such equipment to advance the cause of science and technology in the country. Furthermore, I have gained more insight in the use of software such as Excel in the generation of graphs for results comparison.
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2. BERNOULLI’S THEOREM DEMONSTRATION 2.1 AIM OF EXPERIMENT The main objective of this experiment is to investigate the validity of the Bernoulli’s equation when applied to steady flow of water in a tapered duct. 2.2 DESCRIPTION OF THE APPARATUS The equipment used to for the test is: •
The F1-15 Bernoulli’s theorem demonstration Apparatus
•
F1-10 Hydraulics Bench (water source)
The test section is an accurately machined clear acrylic duct of varying circular cross section. It is provided with a number of side-hole pressure tappings which are connected to the manometers housed on the rig. These tappings allow the measurement of static pressure head simultaneously at each of 6 sections.
Fig. 3 accurately machined clear acrylic duct of varying circular cross section
The apparatus has the following elements:
Venturi meter Pad of manometer tubes Pump Water tank equipped with valves water controller Water hosts and tubes.
In addition to the above students should also use:
A stop watch Water
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Fig. 4 Photo of the Bernoulli’s Theorem Apparatus
2.3 SHORT SUMMARY AND THEORY In fluid dynamics, Bernoulli’s principle is best explained in the application that involves inviscid flow, whereby the speed of the moving fluid is increased simultaneously whether with the depleting pressure or the potential energy relevant to the fluid itself. In various types of fluid flow, Bernoulli’s principle usually relates to Bernoulli’s equation. Technically, different types of fluid involve different forms of Bernoulli’s equation. Bernoulli’s principle complies with the principle of conservation of energy. In a steady flow, at all points of the streamline of a flowing fluid is the same as the sum of all forms of mechanical energy along the streamline. It can be simplified as a constant practice of the sum of potential energy as well as kinetic energy. Fluid particles’ core properties are their pressure and weight. As a matter of fact, if a fluid is moving horizontally along a streamline, the increase in speed can be explained due to the fluid that moves from a region of high pressure to a lower pressure region and so with the inverse condition with the decrease in speed. In the case of a fluid that moves horizontally, the highest speed is the one at the lowest pressure, whereas the lowest speed is present at the highest pressure. Bernoulli’s principle relates much with incompressible fluids flow. Below is a common form of such an equation, where it is valid at any arbitrary point along a streamline when gravity is constant.
v2 2
gz
p
constant
[1]
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Where v: the fluid flow speed at a point on a streamline; g: the acceleration due to gravity z: the elevation of the point above a reference plane, with the positive zdirection pointing upward – so in the direction opposite to the gravitational acceleration. p: the pressure at the point, and : the density of the fluid at all points in the fluid. If equation [1] is multiplied with fluid density, , it can be rewritten as follows: 1 2
1 2
v
h z
2
gz p constant
[2]
q gh po gz constant
or
Where: q
v
2
p g
[3]
is the dynamic pressure,
is the piezometric head or hydraulic head (the sum of the elevation z
and the pressure head)
po = p + q is the total pressure (the sum of the static pressure p and dynamic pressure q). The above equations suggest there is a flow speed at which pressure is zero, and at even higher speeds the pressure is negative. Most often, gases and liquids are not capable of negative absolute pressure, or even zero pressure, so clearly Bernoulli’s equation ceases to be valid before zero pressure is reached. In liquids, when the pressure becomes too low, cavitations occur. The above equations use a linear relationship between flow speed squared and pressure. Generally in many applications of Bernoulli’s equations, it is common to neglect the values of g z term, since the change is so small compared to other values. Thus, the previous expression can be simplified as follows: P + q = p0
[4]
Where p0 is the total pressure and q is the dynamic pressure, whereas p usually refers as static pressure. Therefore, Total pressure = static pressure + dynamic pressure
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However, a few assumptions are taken into account in order to achieve the objectives of experiment, which are:
The fluid involved is incompressible (water); The flow is steady; The flow is frictionless.
2.4 METHOD (EXPERIMENTAL PROCEDURE) 1. The discharge valve and the inlet valve are opened. 2. The pump switch is opened. The flow control valve is then opened and the bench valve is adjusted to allow the flow through the manometer. 3. The air bleed screw is opened and the cap is removed from the adjacent air valve until the same level of water in manometers is reached. The bench valve is adjusted until a certain head difference of water is obtained. 4. The ball valve is closed and the time taken to accumulate a known volume of water in a measuring tube is taken to determine the volume of flow rate. 5. The whole process is repeated 3 times so as to obtain 3 different set of readings. 2.5 RESULTS Tables To allow the calculation of the dimensions of the test section, the tapping positions and the test section diameters are shown on the following table:
Tapping position
Manometer legend
Diameter (mm)
A
H1
25
B
H2
13.9
C
H3
11.8
D
H4
10.7
E
H5
10
F
H6
25
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Results obtained for the 3 set of data readings are recorded in th e tables bellow: Pressure difference = 29mm water Volume (m3) = 25 x 10-5 Time = 5s Flow rate (m3/s) = 5 x 10-5
Area of duct (m2)
Velocity (m/s)
Static head h, (m)
Setting
Pressure head
Dynamic head, (m)
Total head ho (m)
1
H1
490.9 x10-6
0.1019
0.274
0.0005
0.2745
1
H2
151.7 x10-6
0.3296
0.270
0.0055
0.2755
1
H3
109.4 x10-6
0.457
0.264
0.0106
0.2746
1
H4
89.9 x10-6
0.5561
0.258
0.0158
0.2738
1
H5
78.5 x10-6
0.6369
0.245
0.0207
0.2657
Pressure difference = 76mm water Volume (m3) = 455 x 10-6 Time = 5s Flow rate (m3/s) = 91 x 10 -6
Area of duct (m2)
Velocity (m/s)
Static head h, (m)
Setting
Pressure head
Dynamic head, (m)
Total head ho (m)
2
H1
490.9 x10-6
0.1854
0.295
0.0018
0.2968
2
H2
151.7 x10-6
0.5999
0.280
0.0183
0.2983
2
H3
109.4 x10-6
0.8318
0.265
0.0353
0.3003
2
H4
89.9 x10-6
1.0122
0.249
0.0522
0.3012
2
H5
78.5 x10-6
1.1592
0.219
0.0685
0.2875
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Pressure difference = 31 mm water Volume (m3) = 266 x 10-6 Time = 5s Flow rate (m3/s) = 53.2 x 10 -6
Area of duct (m2)
Static head h, (m)
Setting
Pressure head
Velocity (m/s)
Dynamic head, (m)
Total head ho (m)
2
H1
490.9 x10-6
0.108
0.277
0.0006
0.277
2
H2
151.7 x10-6
0.351
0.270
0.0063
0.276
2
H3
109.4 x10-6
0.486
0.264
0.0121
0.276
2
H4
89.9 x10-6
0.599
0.258
0.0178
0.276
2
H5
78.5 x10-6
0.678
0.246
0.0234
0.269
Difference (%) margin between using Bernoulli’s equation and continuity equation for the 3 set of readings Using Continuity Equation
Using Bernoulli’s equation
Pressure head
Total head, ho (m)
Static head, hi (m)
V a
2 g (h o hi )
Duct Area (m2)
V b
Difference
Q
V a V b
A
V b
H1
0.2745
0.274
0.0990
490.9 x10-6
0.1019
-2.8%
H2
0.2755
0.270
0.3285
151.7 x10-6
0.3296
-0.3%
H3
0.2746
0.264
0.4560
109.4 x10-6
0.4570
-0.2%
H4
0.2738
0.258
0.5568
89.9 x10-6
0.5561
0.1%
H5
0.2657
0.245
0.6373
78.5 x10-6
0.6369
0.1%
Pressure difference = 29mm water
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Using Continuity Equation
Using Bernoulli’s equation
Pressure head
Total head, ho (m)
Static head, hi (m)
V a
2 g (h o hi )
Duct Area (m2)
V b
Difference
Q
V a V b
A
V b
H1
0.297
0.295
0.187925517
490.9 x10-6
0.1854
1.36%
H2
0.298
0.280
0.599204473
151.7 x10-6
0.5999
-0.12%
H3
0.300
0.265
0.83221752
109.4 x10-6
0.8318
0.05%
H4
0.301
0.249
1.012009881
89.9 x10-6
1.0122
-0.02%
H5
0.288
0.219
1.159297201
78.5 x10-6
1.1592
0.01%
Pressure difference = 76mm water
Using Continuity Equation
Using Bernoulli’s equation
Pressure head
Total head, ho (m)
Static head, hi (m)
V a
2 g (h o hi )
Duct Area (m2)
V b
Difference
Q
V a V b
A
V b
H1
0.2776
0.277
0.1085
490.9 x10-6
0.1084
0.09%
H2
0.2763
0.27
0.3516
151.7 x10-6
0.3507
0.25%
H3
0.2761
0.264
0.4872
109.4 x10-6
0.4863
0.19%
H4
0.2758
0.258
0.5910
89.9 x10-6
0.5918
-0.14%
H5
0.2694
0.246
0.6776
78.5 x10-6
0.6777
-0.02%
Pressure difference = 31mm water
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Calculations The reader should bear in mind that these are only sample calculations as the full calculated data appear on the tables above.
Flow rate (Q) =
Velocity (V) =
25 10
5 105 m 3 / s
5
Q A
Dynamic head =
Total head
5
5 10
5
490.9 10
v2 2 g
6
0.1019 19.62
0.1019m / s
2
0.0005m
= Static head + Dynamic head = 0.274 + 0.0005 = 0.2745m
Graphs
) m ( t c u d f o a e r A
Velocity (m/s)
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) m ( d a e h l a t o T
Static head (m)
2.6 DISCUSSION From the above graphs we can easily deduce that as fluid flows from a wider pipe to a narrower one, the velocity of the flowing fluid increases. This is also shown in all the tables, where the velocity of water that flows in the tapered duct increases when the duct area decreases, regardless of the pressure difference and type of flow of each result taken. For instance, the velocities at pressure head H 5 at pressure difference of 29mm, 31mm and 76mm are 0.637, 0.678 and 1.1592 respectively. 2.7 CONCLUSION From the experiment conducted, the total head pressure increases and the velocity is increasing along the same channel. This is following exactly the Bernoulli’s principle for a steady flow of water. Cause of inaccurate readings There must be some error or weaknesses when taking the measurement of each data. One of them is, the observer must have not read the level of static head properly, where the eyes are not perpendicular to the water level on the manometer, this may cause some minor defects on the calculations. 2.8 RECOMMENDATION
Repeat the experiment several times to get the average value; Make sure the bubbles are fully removed and not left in the manometer; The eye of the observer should be parallel to the water level on the manometer; The values should be controlled slowly to maintain the pressure difference unchanged; The valve and bleed screw should regulate smoothly to reduce the errors; Make sure there is no leakage along the tube to avoid the water flowing out.
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REFERENCES 1. B.R. Munson, D.F. Young, and T.H. Okiishi, Fundamentals of Fluids Mechanics, 3 rd ed., 1998, Wiley. 2. C.F. Meyer, Principles of Fluid Mechanics, 2nd ed, 1995, CM TEK Lecture materials cc. 3. Armfield Limited, 2002, Instruction Manual F1-16, Ringwood, Hampshire. BH24 1DY England
MABENGO N.D. — STUDENT NO. 48591238 — July 2013