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.