Centre of Pressure on a Plane Vertical / Plane Inclined Surface

EN1702: Thermofluids 1 – Fluid mechanics – Laboratory

Date of Lab Report Submission: 18th March 2013

Date of Lab Exercise: 8th April 2013

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Centre of Pressure on a Plane Vertical / Plane Inclined Surface

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Table of Contents

Page

Aim 3 Objectives 3 Theory 3 Hydrostatic force

Experimental Centre of Pressure

Theoretical Centre of Pressure

Method 4 Equipment Used

Dimensions of the Apparatus

Experimental Procedure

Results 4 Table 1: Data Obtained from Experiment

Sample Experimental Equations

Graphs

Discussion 8 Conclusion 8 Appendices 9 Experimental Determination of Thermal Conductivity of Metal Results and Calculations

Aim

The Aim of this Laboratory was to determine the hydrostatic force and the position of the centre of pressure on a rectangular, plane vertical surface when partially and fully immersed in water.

Objectives

The objectives of this experiment were to first determine the magnitude of hydrostatic force exerted on the submerged area of the plane vertical surface, and to distinguish the variation in the centre of pressure of the submerged surface area with increasing depth of immersion. The data obtained was additionally used to calculate both the Experimental centre of pressure and the Theoretical centre of pressure, and compare the two.

Theory

Hydrostatic force:

The resultant force ‘F’ on one side of any plane submerged surface whether inclined or vertical, is equal to the area of the surface ‘A’ times the pressure ‘P’ at its centroid. Hydrostatic Force = Units (N)

Where ρ is the density of the liquid in which the surface is submerged and ȳ is the vertical depth of the centroid of the submerged surface area, below the free surface of the liquid. It can be shown that the resultant force, F acts normal to the surface and through a point, known as the centre of pressure, which lies below the centroid of the submerged surface. A = B(R2 – R1 or Y) Units (m2) – (R2 – R1):Fully Immersed (Y):Partially Immersed Density of water – 1,000 Kg/m3

Experimental Centre of Pressure:

ex = Units (m)

Theoretical Centre of Pressure:

For a plane vertical surface the depth of the centre of pressure, ‘’ below the free surface is given by:

th = Units (m)

Where Icc is the 2nd moment of inertia of the submerged surface area about a horizontal axis in the plane of the surface and passing through its centroid. Icc is given by: Icc = Units (Kg/m2)

When a plane submerged surface is inclined at an angle θ to the horizontal

axis, the depth of the centre of pressure is given by: th = Units (m)

Method

Equipment used:

TQ H11 Centre of Pressure Apparatus (SN: A0390/10)

X13 50g weights

Water – Jug

Dimensions of the Apparatus:

Lever arm = 200mm Inner radius of quadrant, R1 = 100mm Outer radius of quadrant, R2 = 200mm Breadth of quadrant, B = 75mm Experimental Procedure:

After assuring that the experimental apparatus is placed on a level surface, water should then be added to the tank situated on the same side of the assembly as the weight hanger until exact balance between the two tanks is obtained. A 50g weight should then be placed on the weight hanger and water added to the opposite tank until exact balance is once again obtained. Once level, both the Distance from the datum to the free surface and the Distance from the free surface to the centroid should me measured and noted. This procedure should be repeated, 50g at a time, until the 650g mark, each time taking not of the results shown by the apparatus. The equations shown in ‘Theory’ should then be used to calculate the Hydrostatic Force, Experimental centre of pressure and the Theoretical centre of pressure for each reading remembering to convert Grams – Newton’s.

Results

Table 1: Data obtained from experiment

Test No.

Weight

W

Distance from datum to water free surface Y

Distance from free surface to centroid ȳ

Hydrostatic Force

F

Experimental centre of pressure

ex

Theoretical centre of pressure

th

N

m

m

N

m

m

1

0.49

0.162

0.019

2.26

-0.118

0.134

2

0.98

0.144

0.028

2.96

-0.077

0.089

3

1.47

0.130

0.035

3.34

-0.041

0.075

4

1.96

0.120

0.040

3.53

-0.008

0.070

5

2.45

0.110

0.045

3.64

0.024

0.067

6

2.94

0.101

0.051

3.75

0.056

0.067

7

3.43

0.090

0.060

4.41

0.065

0.073

8

3.92

0.082

0.068

5.00

0.074

0.080

9

4.41

0.072

0.078

5.73

0.082

0.088

10

4.90

0.063

0.087

6.40

0.090

0.096

11

5.39

0.054

0.096

7.06

0.098

0.104

12

5.88

0.044

0.106

7.79

0.106

0.113

13

6.37

0.035

0.115

8.46

0.115

0.122

Sample Experimental Equations:

Partially Immersed: – Test No. 1

F =

F = 1000 9.81 0.075(0.162)

F = 2.26N

_______________________________________________________________________________________ ex =

ex =

ex = -0.118m

_______________________________________________________________________________________ Icc =

Icc =

Icc = 2.657×10-5 Kg/m2

_______________________________________________________________________________________ th =

th =

th = 0.134m

_______________________________________________________________________________________ Fully Immersed: – Test No. 13

F =

F = 1000 9.81 0.075(0.1)

F = 8.46N

_______________________________________________________________________________________ ex =

ex =

ex = 0.115m

_______________________________________________________________________________________ Icc =

Icc =

Icc = 6.25×10-6 Kg/m2

_______________________________________________________________________________________ th =

th =

th = 0.112m

_______________________________________________________________________________________

Graphs:

Distance from Datum to Free Surface vs. Distance From free Surface to Centroid:

Distance from Datum to Water Free Surface vs. Hrydrostatic Force:

Distance from datum to Free Surface vs. Experimental centre of Pressure:

Distance from Datum to Free Surface vs. Theoretical Centre of Pressure:

Experimental Centre of Pressure vs. Theoretical Centre of Pressure:

Discussion

The results obtained from this particular experiment seem to match those of previous experiments within reason. Showing an obvious change in Centre of Pressure on the submerged plane surface with increasing depth of immersion. Whilst the Distance from the free surface to the centroid (ȳ) increased, both the Experimental and Theoretical Centre of Pressure Increased. The Experimental procedure went ahead without any abnormal occurrences that would induce error in the results obtained. However the results gained by this experiment could potentially be inaccurate due to a number of reasons. It should be noted that a large discrepancy between the theoretical and experimental values of centre of pressure occurred. This is most likely due to errors in measurement of the height of the fluid inside of the tank. Another possible cause could have been that the apparatus was not sitting level on the counter where the experiment was performed. If the apparatus was not sitting level, the moment calculations will yield inaccurate results. A levelling device on or near the testing apparatus would aid in ensuring the moment balance is accurate. Another source of error would be the use of the accepted density of water, 1000Kg/m3, for the theoretical calculation of the Theoretical Hydrostatic Force. This accepted value is the density of sea water at 40C. The water used in this experiment was tap water at approximately 230C. However, if the actual density of the

tap water was used, the theoretical calculations would not differ greatly enough to compensate for the magnitude of the error. Conclusion

This report examined the changing centre of pressure on a submerged plane surface with increasing depth of immersion. The experiment took place in fair conditions to obtain results including The Hydrostatic Force, Experimental Centre of Pressure and the Theoretical Centre of Pressure of the submerged plane surface. In conclusion the results obtained by this experiment prove that with increased depth of immersion of the submerged plane surface there is an increased Centre of pressure, both experimentally and theoretically. As-well as an increased Hydrostatic Force acting on the plane surface.

Appendices

Experimental Determination of Thermal Conductivity of Metal: Results and Calculations

Table 1 – Data from Experiment

V (Volts)

I (Amps)

T1 0C

T2 0C

T3 0C

T4 0C

T5 0C

T6 0C

T7 0C

T8 0C

6

0.58

29.5

28.4

27.1

23.7

22.3

1809

17.3

16.7

12

1.26

48.4

44.8

41.2

34.2

30.7

24.8

21.4

17.9

Calculations: – 6V

Heat Flow Rate, Q = VI (W) = 6 x 0.58 = 3.48W

Cross Sectional Area, A = πr2 = (m2) = 4.908×10-4

Temperature at Hot face of the Specimen, Thotface = T4 0C = 23.70C Temperature at Cold face of the Specimen, Tcoldface = T5 0C = 22.30C Thermal Conductivity of the Specimen, (W/m 0C) = 151.9 W/m 0C Calculations: – 12V

Heat Flow Rate, Q = VI (W) = 12 x 1.26 = 15.12W

Cross Sectional Area, A = πr2 = (m2) = 4.908×10-4

Temperature at Hot face of the Specimen, Thotface = T4 0C = 34.2 0C Temperature at Cold face of the Specimen, Tcoldface = T5 0C = 30.7 0C Thermal Conductivity of the Specimen, (W/m 0C) = 264.1 W/m 0C