FACULTY OF Science, technology, engineering and mathematics
(fostem)
Bachelor of ENGineering (HONS) IN MECHANICAL ENGINEERING
(BMEGI)
MEE 3220FLUID MECHANICS 1
Experiment 1Measurements ofHydrostatics Force
NameID
EHSAN SAMOHDONYEGHA TUNEMIAlbert Law Lee Tai
I14005275I13003778I14004759
Date of conducted: 5th Feb 2015Date of submission: 12th Feb
2014
Experiment 1: Measurements of Hydrostatic Force
1.1Objectives
i.To determine the hydrostatic thrust acting on a plane surface
immersed in water.ii.To determine the position of the line of
action of the thrust and to compare the position determined by
experiment with theory.
1.2Introduction and Theory
Buoyancy, also known as upthrust, is a very important concept
dealing with objects which are wholly or partially immersed in a
fluid, especially in the aspect of marine engineering. For
instance, the draught of a pontoon or ferry can be determined by
knowing the buoyant force acting on it, which equals the weight of
the given pontoon or ferry. Practically, the size of vessels is
expressed in terms of displacement or displacement tonnage, which
is the weight of the sea water displaced by the vessel when it is
floating. Under safety consideration, the greatest allowable
displacement of the vessel when floating is called full load
displacement, and the line where the water level reaches on the
vessels rail is known as Plimsoll line, usually drawn on the outer
shell of the vessel.
For a given fluid with density , we know that the gauge pressure
at a point below free surface at a distance measured from the free
surface is given by , where is the constant value of gravitational
acceleration 9.81 N kg-1. For a given rectangular plane vertically
immersed in the given fluid, we can obtain the force due to fluid
pressure on each element of area . Since we know from the
definition of pressure that,
Rearranging the equation we will have . Substituting and , and
the equation becomes
Summing up all the forces on all elements of area over the
immersed rectangular plane, we have the resultant force,
The quantity is the first moment of area under the rectangular
plane about the free surface of the fluid, and stands for the
vertical depth from the free surface to the centroid of the
immersed surface.
Note that the resultant force R should be perpendicular to the
immersed surface. We know from the definition of fluid that, if a
fluid is at rest, there can be no shearing forces acting and,
therefore, all forces in the fluid must be perpendicular to the
planes upon which they act. Since the fluid particles are relative
at rest, we can assume that there are only perpendicular forces
acting upon the surface.
At the same time it is important to know about the position of
the center of pressure as well since the density of the fluid is
not uniform along the vertical direction, therefore the center of
pressure need not to coincide the centroid in vertical direction.
The depth of the center of pressure measured from the free surface
is determined by
where is the radius of gyration of the immersed surface about
the axis where the extension area of immersed surface meets the
free surface of the fluid, and is the angle between the immersed
surface and the free surface. For a partially or exact wholly,
vertically immersed rectangular plane, , ,
Therefore, the equation becomes,
This equation indicates that the distance of center of pressure
measured from the fluids free surface is two-third of the height of
immersed vertical rectangular surface. That is also where the
resultant force due to fluid pressure acting on the plane.
Since we now can simplify the pressure forces on a given surface
immersed in a fluid with known value of density, it would be much
easier to analyse the static equilibrium condition of the mechanism
shown in the experiment below.
Figure 1.1 The hydrostatic pressure apparatus graphical
representation
Figure 1.2 Hydrostatic pressure apparatus
The apparatus shown above is the hydrostatic pressure apparatus.
The main parts of the apparatus are the pivot, quadrant, weight
hanger and the counterbalance. The position of pivot relative to
the quadrant is very important to the accuracy of this experiment.
The pivot is mounted on the balance arm where coincides with the
center of arc of the quadrant. Since we know that a fluid at rest
will not be having any shearing forces and all the forces in the
fluid must be perpendicular to the surface in contact, which is the
surface of the quadrant, in this case. The line of action of the
forces will definitely pass through the pivot. In order to create a
moment on the balance arm, there must be forces acting on the
quadrant and moment arm which is the perpendicular distance from
the line of action of the force to the pivot. As a result, all the
elementary pressure forces acting on the curve surface of the
quadrant will not generate a significant moment upon the apparatus,
and therefore the apparatus will not rotate about pivot P.
If the fluid, say water, is poured into the tank and the
quadrant is partially immersed in the water, when taking the
moments about pivot P, only the effects of the pressure force on
the end face of the quadrant and the weights added need to be
considered as the pressure forces on the curve surface will not
create any moment or only create a moment that is sufficiently
small to be neglected. The moment arm of both fluid pressure force
and weights can be measured directly before carrying out the
experiment.
Since the whole system is in static equilibrium, the algebraic
summation of all the moments acting on the system should be equal
to zero, which can be expressed in mathematical expression . The
theoretical values of hydrostatic force can be determined by
equating the counter-clockwise moments and the clockwise moments,
while the experimental values of the hydrostatic force can be
determined using equation . The height of the immersed part of the
quadrant can be determined from the scale on the side of the
quadrant. Note that the counterbalance will also generate a moment
about pivot P, and since its weight and moment arm are constant,
its moment should be a constant value as well.
Figure 1.3 The scale can be used to measure the height of the
immersed part of the quadrant.
1.3Procedures
1. Before carrying out the experiment, one ensured that the
fluid tank of the hydrostatic pressure apparatus is on the
horizontal water level by checking the circular spirit level
mounted beside the fluid tank.2. The dimensions of the quadrant
end-face (both length and width), the distance between the pivot
and the weights and the distance between the pivot and the bottom
of the quadrant were measured directly on the apparatus.3. The
position of the counterbalance and the number of weights were
adjusted according to level indicator in order to balance the arm
to horizontal position. (The fluid tank was empty.)
Figure 1.4 The level indicator on the end of the balance arm4.
The drain valve was closed before pouring water into the tank to
avoid water from draining out of the tank.5. Extra weights were
added progressively, in our case the weight was added 20g every
time, initially 120g.6. A certain amount water was added (water
level should at least reach the minimum level of the height scale
on the side of quadrant) to balance the arm back to horizontal
position, and the height of water level was taken down.7. Step 5
and 6 was repeated until the water level reaches the top of the
water level scale.8. Then the procedures were repeated reversely,
the height of remaining water was recorded for every time 20g of
weight was removed.9. The experimental data were recorded and
tabulated in the Results and Calculations part.
1.4Results and Calculations
The distance from pivot to the weights, The distance from pivot
to the bottom of the quadrant, Dimensions of end face of quadrant:
(length d3 x width d4)First attempt: Adding weight progressively
and pouring in water to maintain balance.Mass of Weight, m (g)Water
Level, h (mm)Weight, W=mg (N)
200831.962
220892.158
240942.354
260992.551
2801042.747
3001092.943
3201133.139
3401183.335
3601233.532
3801283.728
4001333.924
4201384.120
We can find the resultant force due to water pressure by using
equation. This equation can transform into a more convenient
expression. When the water poured in was immersing the end face of
quadrant, and , which is
Where the density of water , gravitational acceleration and the
width of the end face of the quadrant are constant. From this
relationship we know that resultant force R is directly
proportional to , that is . For given values of h, we can determine
the resultant force R respectively.After the end face of quadrant
had wholly been immersed (which means that ), , therefore .
Water Level, h (mm)Resultant Force, R (N)(, )
832.636
893.030
943.381
993.749
1044.124
1094.499
1134.799
1185.174
1235.549
1285.924
1336.299
1386.674
When the water is immersing the end face of the quadrant, the
distance from the free surface to its center of pressure is based
on the derivation in the Introduction and Theory part, and
therefore the moment arm of resultant force is .
After the end face of quadrant is wholly immersed, the distance
of center of pressure from free surface of water can be derived as
below:
Since
Water Level, h (mm)D (m)Moment Arm of R(m)
830.0550.183
890.0590.181
940.0630.180
990.0660.178
1040.0700.177
1090.0730.175
1130.0770.175
1180.0810.174
1230.0850.173
1280.0890.172
1330.0940.172
1380.0980.171
The experimental result can be verified by comparing the moment
generated by water pressure force and weight. The magnitude of
these two moments should cancel each other to remain itself in
equilibrium. That is,
Moment Arm of R(m)Resultant Force, R (N)Weight, W(N)Moment of
R(Nm)Moment of W(Nm)
0.1832.6361.9620.4830.540
0.1813.0302.1580.5500.594
0.1803.3812.3540.6070.647
0.1783.7492.5510.6670.701
0.1774.1242.7470.7280.755
0.1754.4992.9430.7890.809
0.1754.7993.1390.8370.863
0.1745.1743.3350.8980.917
0.1735.5493.5320.9590.971
0.1725.9243.7281.0201.025
0.1726.2993.9241.0801.079
0.1716.6744.1201.1411.133
The value of moment of W can be assumed to be the theoretical
values of moment of water pressure resultant force R. Comparing
both column of results, we have a maximum discrepancy of around
0.056 Nm, and therefore the result of this experiment can be
treated as reliable and accurate.
1.5Discussions
Based on the water level versus resultant force table presented
above, we can plot a graph to represent their relationship.
From the diagram and the relationship equation derived in
previous part, the water pressure resultant force (acting similar
to upthrust) and depth of water level can be described
approximately linked by a linear relationship (even before h =
9.8cm the relationship between R and h is binomial , however the
slope of the binomial curve doesnt change much, and therefore can
be treated as approximately linear). Based on the graph, the slope
of the best-fit line is approximately the same based on the data
points collected. From the scatter diagram above, we can conclude
that as the depth of water level increases, the resultant water
pressure force increases proportionally to the increase in depth of
water level.
Except of the resultant force R, the depth of the center of
pressure is also affected by the depth of immersion. We know that
before h = 9.4cm, the center of pressure is always two-third of
depth of immersion measured from the free surface of water.
However, the relationship between these two becomes complicated,
which is . The relationship is graphically represented in the graph
shown below.
Based on the graph above, the relationship between D and h is to
be more likely a binomial or exponential trend. As the depth of
immersion increases, the depth of center of pressure will increase,
and its rate of increase is getting faster as well since the slope
of the curve is becoming greater.
And we also looked into the experiment to check the
discrepancies between theoretical and experimental results
obtained. Back to the table where we compare the moments generated
by resultant upthrust force and the weight.
Moment of R(Nm)Moment of W(Nm)Discrepancy(Nm)
0.4830.5400.056
0.5500.5940.044
0.6070.6470.040
0.6670.7010.034
0.7280.7550.027
0.7890.8090.020
0.8370.8630.026
0.8980.9170.019
0.9590.9710.012
1.0201.0250.005
1.0801.0790.001
1.1411.1330.008
The discrepancy can be considered relatively small and
sufficient to be neglected. However, it is important to identify
the reasons behind the discrepancies occurred in the experiment.
Firstly, we made an assumption on the density of water to be 1000
kg m-3 but this might not be exactly true, as 1000 kg m-3 is the
greatest density of water at 4 degree Celsius. During experiment,
impurities and the different in temperature may lead to minor
discrepancies to the experimental results.
And secondly we should be cautious on the location of pivot and
the smooth curve of the quadrant. The difference we obtained
between the moments of resultant force R and weights added W may be
caused due to this major reason. As mentioned earlier, if the
location of pivot P coincides the center of arc of the curve
surface of quadrant, the line of action of the water pressure force
acting on the curve surface will be perpendicular to the surface
and thus passing through the pivot, which results in no moment arm
and therefore no moment is generated. But this can only be achieved
by assuming that the pivot is exactly coincides the center of arc
of the curve surface of quadrant, which might not be experimentally
true. In most real cases, these elemental water pressure force will
cause a small amount of moment on the whole system rotating about
pivot P. Therefore, theoretically speaking the curve surface of the
quadrant has nothing to do with the experimental result, only if
the curve is smooth and having the pivot as its center of arc.
Thirdly, we considered that the discrepancies might be resulted
by the inaccuracy labelling of weight value on the weights. The
difference of the actual weights and the labelled weights may be
small, however if we multiply the weight with the moment arm to
obtain its rotational moment, the small difference can be enhanced
and result in a larger difference in the magnitude of moment.
This experiment can relate us to the design of a water dam.
Below is the graphical sectional view of Mullaperiyar Dam in the
Indian state of Kerala.
Figure 1.5 Mullaperiyaar Dam (Cross Sectional View)We know that
in the reservoir of water dam, as the head from the water surface
increases, the water pressure will increase according to the
formula where h is the head of a point in the water contained.
Therefore, the deeper it goes, the wall of the dam should be built
thicker in order to withstand the hydrostatic force caused by the
water pressure. We know that the pressure force on a flat surface
immersed in fluid will be parallel to each other, and add up to
become greater in magnitude if they are in the same direction. If
the surface is curved, every elemental pressure force are pointing
at different direction, and therefore they will cancel each other
partially (or even wholly cancel each other if the surface is
circle in shape). This concept is also applied in building a dam,
for example the El Atazar Dam near Madrid. This kind of design is
known as arch dam. An arch dam is designed so that the pressure
force of water will partially cancel each other and also press
against the arch to compress and strengthen the dam structure. It
is more suitable in narrow gorges or canyons, and since it requires
less construction materials, it is more economical and practical in
remote areas.
Figure 1.6 El Atazar Dam1.6Conclusion
This experiment gives a moderate accuracy to measure the
pressure force on the end face of quadrant. The key concept of this
experiment is to know that since the balance arm is in equilibrium
before pouring in water, the moment created by the water pressure
force on the end face of quadrant should be equal to the moment
created by weights. And since the pressure force on the curve
surface will not create any moment on the system about pivot, it
needs not to be considered. As the water level increases, the
resultant pressure force and its center of pressure will vary as
well. To improve the accuracy of this experiment, the weight should
be weighed in advance to confirm its actual weight value. Besides,
this experiment can also be done by replacing water with other
liquids, such as alcohol with known density to compare the
experimental results obtained.
1.7References
1. Displacement (ship) - Wikipedia, the free encyclopedia.
2014.Displacement (ship) - Wikipedia, the free encyclopedia.
[ONLINE] Available at:
http://en.wikipedia.org/wiki/Displacement_(ship).2. Arch dam -
Wikipedia, the free encyclopedia. 2014.Arch dam - Wikipedia, the
free encyclopedia. [ONLINE] Available at:
http://en.wikipedia.org/wiki/Arch_dam.