DNT 233 Thermo Fluid
Laboratory Module
EXPERIMENT 1
HYDROSTATIC PRESSURE
1.0 OBJECTIVE
1.1
To determine the centre of pressure on both submerged and partially submerged plane
surface.
To compare the centre of pressure between experimental and theoretical.
1.2
2.0 INTRODUCTION AND THEORY
2.1
Hydrostatic Forces on Plane Surface
Consider a plane surface of arbitrary shape and orientation submerged in a static fluid as shown
in Figure 1. If P represents the local pressure at any point on the surface and h represents the
depth of fluid above any point on the surface, from the basic physics, we can easily show that
the net hydrostatic force on a plane surface is given by:

F  PdA
A
(1)
Free Surface
p = pa
q
h
hcg
Resultant Force
F = PcgA
dA = dxdy
CG
Side
View
CP
Plan view of arbitrary
plane surface
Figure 1: Hydrostatic Pressure on Plane Surface
The hydrostatic force on one side of a plane surface submerged in a static fluid equals to the
product of the fluid pressure at the centroid of the surface times the surface area in contact with
the fluid. Thus, basic physics says that the hydrostatic force is a distributed load equal to the
integral of the local pressure force over the area. Pressure acts normal to a surface, therefore
the direction of the resultant force will always be normal to the surface.
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DNT 233 Thermo Fluid
Laboratory Module
In most cases, since it is the net hydrostatic force that is desired and the contribution of
atmospheric pressure Pa will act on both sides of a surface, the result of atmospheric pressure
Pa will cancel and the net force is obtained by:
F   g h cg A
(2)
F  Pcg A
(3)
Pcg is now the gauge pressure at the centroid of the area in contact with the fluid. Therefore, to
obtain the net hydrostatic force, F on a plane surface,
Determine depth of centroid, hcg for the area in contact with the fluid
Determine the gauge pressure at the centroid Pcg
Calculate F = PcgA
Table 1 shows the centroid and other geometric properties of several areas.
Table 1
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DNT 233 Thermo Fluid
Laboratory Module
The effective point of application of force which is normally called the “Center of
Pressure, CP” of the hydrostatic force and this is not necessary the same as the centroid.
The location of the resultant force is determined by integrating the moment of the distributed
fluid load on the surface about each axis and equating this to the moment of the resultant force
about that axis. Therefore, for the moment about the y axis:

F y cp  y P dA
A
(4)
Applying a procedure similar to that used previously to determine the resultant force, we obtain:
Ycp  
 g sinq I xx
Pcg A
0
(5)
Where, Ixx is defined as the Moment of Inertia, or

I xx  2 nd moment of the area
(6)
Therefore, the resultant force will always act at a distance ycp below the centroid of the surface
(except for the special case of sin θ = 0). Proceeding in a similar manner for the x location, and
defining I xy = product of inertia, we obtain:
X cp  
 g sinq I xy
Pcg A
(7)
Where Xcp can be either positive or negative since I xy can be either positive or negative. For
areas with a vertical plane of symmetry through the centroid, the y-axis (e.g. squares, circles,
isosceles triangles, etc.), the center of pressure is located directly below the centroid along the
plane of symmetry, Xcp = 0.
For most problems where we have a single, homogeneous fluid and the surface pressure is at
atmospheric, the fluid specific weight  cancels in the equation for Ycp and Xcp and we have the
following simplified expressions:
Ycp  
X cp  
I xx sinq
h cg A
I xy g sinq
hcg A
(8)
(9)
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DNT 233 Thermo Fluid
2.2
Laboratory Module
Hydrostatic Pressure on a Vertical Plane Surface
Figure 2 , illustrates a hydrostatic pressure demonstration setup where the center of pressure of
an immersed and partial immersed vertical plane surface can be determined.
L
H
W
C
P
D
Figure 2: Hydrostatic Pressure Demonstration
Referring to Figure 2,
L
D
B
H
C
P
= Distance between Suspended Mass and Fulcrum
= Height of End Surface
= Width of End Surface
= Total Depth of Quadrant
= Centroid of End Surface
= Center of Pressure of Plane Surface
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DNT 233 Thermo Fluid
Laboratory Module
Partial Immerged Vertical Plane Surface
Fulcrum
H
Quadrant
h h’ h”
d
C
P
D
F
B
Figure 3: Partial Immersed Vertical Plane Surface
d
= Depth of Immersion in Water
F
= Hydrostatic Force exerted on Quadrant
h
= Depth of Centroid from the Water Surface
h’
= Depth of Center of Pressure from the Water Surface
h”
= Distance between Fulcrum and the Center of Pressure
For a partially immersed plane surface as shown in Figure 3,
A  Bd
(10)
d
h
2
(11)
Therefore, the hydrostatic force, F (Equation 2) is simplified as:
Bd 2
F g
2
(12)
If the system as shown in Figure 3 is equilibrium with the moment of horizontal arm, then,
M
fulcrum
 Fh "WL
Fh "  WL  mg L
(13)
Substituting Equation 12 into Equation 13, we obtain,
h"
mg L 2mL

F
Bd 2
Experiment
(14)
Theoretically, the center of pressure from the water surface, h’ is expressed in Equation 8:
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DNT 233 Thermo Fluid
 I sinq 

h '  h  Ycp  h    xx
 hcg A 


Laboratory Module
(15)
The orientation of the surface from the horizontal = 90° gives 1 for sin θ, then
h' h
I xx
hA
(16)
and
I xx 
Bd 3
12 (For a Rectangular Surface)
(17)
where I xx is the moment of inertia of the submerged surface (please refer to Table 1)
The center of pressure bellow the fulcrum is finally determined by,
h "  h ' H  d
(18)
Thus, substituting Equation 16 and 17 into Equation 18 will finally give,
h"  H 
d
3
Theory
(19)
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DNT 233 Thermo Fluid
Laboratory Module
Fully Immerged Vertical Plane Surface
Fulcrum
H
Quadrant
h
D/2
d
h’ h”
C
P
D
F
B
Figure 4: Fully Immersed Vertical Plane Surface
Referring to Figure 4,
D
F
h
h’
h”
= Depth of Immersion in Water
= Hydrostatic Force exerted on Quadrant
= Depth of Centroid from the Water Surface
= Depth of Center of Pressure from the Water Surface
= Distance between Fulcrum and the Center of Pressure
The hydrostatic force of a fully immersed vertical plane surface, F is determined as:
D

F   g Ah   gBD d  
2

(20)
If the system as shown in Figure 4 is equilibrium with the moment of horizontal arm, then,
M
fulcrum
 Fh "WL
Fh "  WL  mg L
Thus,
h '' 
mgL

F
mL
Experiment (21)
D

BD  d  
2

Theoretically, the center of pressure from the water surface, h’ is expressed in Equation 8:
 I sinq 

h '  h  Ycp  h    xx
 hcg A 


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DNT 233 Thermo Fluid
Laboratory Module
The orientation of the surface from the horizontal = 90° gives 1 for sinθ, then
h' h
I xx
hA


 3

D   BD
1
1 

h '  d    


D
2
12 BD 


 d   
2


2
D   D2 

 d     
2  2 


D

d  
2

(22)
The center of pressure bellow the fulcrum is finally determined by,
h "  h ' H  d
D   D2 


d


  
2
2



  H  d Theory (23)
h '' 
D

d  
2

2
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DNT 233 Thermo Fluid
Laboratory Module
3.0 EQUIPMENT/APPARATUS
Figure 5: Assembly Diagram of Hydrostatic Pressure
DESCRIPTION
The Hydrostatic Pressure apparatus consists of a specially constructed quadrant mounted on a
balance arm which pinots on knife edges and it also correspond to the centre of arc of the
quadrant. This means that only the hydrostatic force acting on the rectangular end face will
provide a moment about the knife edge. The balance arm also has an adjustable
counterbalance, and a balance pan for the weights supplied.
The quadrant assembly is mounted on top of a clear reservoir tank provided with levelling
screws on its feet. A spirit level attached to the base of the tank assures accuracy of the
alignment. The tank is filled with water from the top and can be drained through a valve at the
side. A scale is provided on the side of the quadrant to indicate the water level in the tank.
The torque exerted by the hydraulic thrust is measured by direct weighing. Tests may be carried
out over a range of water level.
Specification
Quadrant
Material: PVC
Total Depth of Quadrant, H:175mm
Height of Fulcrum above Quadrant: 100
mm
Height of End Surface, D: 75 mm
Width, B: 73 mm
Reservoir Tank
Material
:
Clear Acrylic
Volume: 5.5 L
Balance Arm
Distance between
Suspended Mass and
Fulcrum, L:
260 mm
Set of Weights:
50g/each
Overall
Dimension
Length :
340 mm
Width :
130 mm
Height :
290 mm
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DNT 233 Thermo Fluid
Laboratory Module
4.0 EXPERIMENTAL PROCEDURE
No.
1.
Procedure
Place the Hydrostatic Pressure Apparatus on top of a Table.
2.
Make sure that the apparatus is installed properly as in Figure 5.
3.
Level the apparatus using the adjustable feet facilitate by the spirit level
attached.
4.
Adjust the counter-weight to level the balance arm to horizontal position.
5.
Make sure that the drain valve is closed and slowly add water into the tank until
the water surface just touches the quadrant base, thus establishing a datum
level corresponding to zero at the level scale.
6.
Place a 50 g weight on the balance pan and slowly add water into the tank until
the balance arm is again horizontal.
7.
Record the mass of weight, m and depth of immersion, d in Table 1.
8.
Repeat steps 6 and 7 with increasing number of weights until the tank is full of
water.
9.
Drain off the water after the experiment is completed.
10.
End of experiment.
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DNT 233 Thermo Fluid
Name :
Laboratory Module
______________________________
Date : ______________
Matrix No : ______________________________
5.0 DATA & RESULTS
Observation Table:
Height of Quadrant , D
=______________________ m
Water Density
=___________________kg/m3
Width of Quadrant , B
=______________________m
Gravity
=___________________m/s2
Length of Balance, L
=______________________m
Quadrant to pivot, H
=______________________m
Table 1
Mass,
m
Depth of
Immersion ,
d
kg
m
Immersion Sketch
(refer figure 2)
Show water level and all
dimensions
Immersion
Type
Thrust, F
2nd Moment
Experimental
2nd Moment
Theory
Error
N
h”
h”
%
0.05
0.10
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DNT 233 Thermo Fluid
Mass,
m
Depth of
Immersion ,
d
kg
m
Laboratory Module
Immersion Sketch
(refer figure 2)
Show water level and all
dimensions
Immersion
Type
Thrust,F
2nd Moment
Experimental
2nd Moment
Theory
Error
N
h”
h”
%
0.15
0.20
0.25
0.30
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DNT 233 Thermo Fluid
Name :
______________________________
Matrix No :
_____________________________
Laboratory Module
Date : ______________
7.0
DISCUSSION / EVALUATION & QUESTION
7.1.
Briefly summarize the key results of experiment
- Discuss about the trends and the nature of your results
7.2.
Explain the significance of your findings
- First tell about the experiment objective
- Explain the significance about your results and why is it of interest to you
7.3.
Explain any unusual difficulties or problems which may have led to poor results
- Gives 3 problems in your experiment
7.4.
Offer suggestions for how the experimental procedure or design could be improved.
- Give 3 suggestion how to improves your results
7.5.
Compare your experimental results with theoretical results.
7.6
What have you understand about Hydrostatic Pressure?. Explain in your own words.
7.7
Give three examples the application of Hydrostatic Pressure.
7.8
What will happen if Quadrant weight is increased?.
8.0
REFERENCE (3%)
- Gives at least 3 reference
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