Center of Pressure

KAS 1331: LABORATOR LABORATORY Y1 CENTRE OF PRESSURE

Introduction:

From this experiment experiment we are enable to measure measure the moment due to the total fluid thrust thrust on a wholly, or partially, submerged plane surface to be directly measured and compared with theoretical analysis. The plane area may be tilted relative to the vertical so that the general case may be studied. The water is contained in a clear Perspex quadrant, the cylindrical sides of which have their central axes coincident with the axis about which the turning moments are measured. The total fluid pressures on these curved surfaces therefore exert no moment about this pivot, the only moment being due to the fluid pressure on the plane test surface. This moment is simply measured by weights suspended from a level arm.

OBJECTIVE:

• • •

To determine the hydrostatic thrust acting on a plane surface immersed in water. 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 object of this experiment was to calculate the hydrostatic force a fluid exertson a submerged plane surface and then compare the experimental hydrostatic force to thetheoretical hydrostatic force.

THEORY:

Below is a diagrammatic representation of the apparatus defining the physical dimensions, this nomenclature will be used throughout this theory discussion. Whilst the theory for the partly submerged and fully submerged plane is the same, it will be clearer to consider the two separately.

Where: L is the horizontal distance between the pivot point and the balance pan. 1

UNIVERSITI INDUSTRI SELANGOR (UNISEL) DEPARTMENT OF CIVIL ENGINEERING

KAS 1331: LABORATORY 1 CENTRE OF PRESSURE

D is the height of the quadrant face. B is the width of the quadrant face. H is the vertical distance between the bottom of the quadrant face and the pivot arm. C is the centroid of the quadrant. P is the Centre of Pressure on the quadrant face. Theory-Partly Submerged Vertical Plane Surface

Below is a diagrammatic represented of the apparatus defining the physical dimensions, in addition to those shown earlier. This nomenclature will be used throughout this theory discussion.

Where: d is the depth of submersion. F is the hydrostatic thrust exerted on the quadrant. h is the depth of the centroid. h’ is the depth of the Pressure P. h” is the distance of the line of action of thrust below the pivot. This line of action passes through the Centre of Pressure, P.

Theory - Partly Submerged Vertical Plane Surface – Thrust on Surface

The hydrostatic thrust F can be defined as: F = ρgAh (Newtons) Where area A = Bd d And h = C = 2 2



Hence F = pg

Bd 2

……….. (1)

2

Partly Submerged Vertical Plane Surface – Experimental Depth of Centre of Pressure

The Moment, M, can be defined as: M = Fh” (Nm) A balancing moment is produced by the weight, applied to the hanger at the end of the balance arm. The moment is proportional to the length of the balance arm, L For the static equilibrium the two moments are equal. That is: Fh” = WL = mgL By substitution of the derived hydrostatic thrust, F from (1) we have: h" =

mgL F

=

2mL ρBd

2

(meters)

Partly Submerged Vertical Plane Surface – Theoretical Depth of Centre of Pressure

The theoretical result for depth of centre of pressure P, below the free- Surface is: h' =

Ix

……….. (2)

Ah

Where: Ix is the 2nd moment of the area in immersed section about an axis in the free surface. By use of the parallel axes theorem. Ix = Ic + Ah²

Ix

=

Bd 2 12

2

d  Bd 3  + Bd  = (m4 ) 3  2  3

……….. (3)



The depth of the centre of pressure below the pivot is h” = h’ + H – d (m)

……….. (4)

Substitution of (3) into (2) and thence to (4) yield the theoretical result of: h" = H −

d 3

Theory – Fully Submerged Vertical Plane surface

Where: d is the depth of submersion. F is the hydrostatic thrust exerted on the quadrant. h is the depth of the centroid. h’ is the depth of Centre of Pressure, P. h” is the distance of the line of action of thrust below the pivot. This line of action passes through the centre of pressure, P. Theory – Fully Submerged Vertical Plane Surface – Hydrostatic Thrust

The hydrostatic thrust F can be defined as: F = ρgAh

D  = ρgBD  d −  ( N )  2 

4

……….. (5)



Theory – Fully Submerged Vertical Plane Surface – Experimental Depth of Centre of Pressure

The moment, M, can be defined as: M = Fh” (Nm) A balancing moment is produced by the weight, W, applied to the hanger at the end of the balance arm. The moment is proportional to the length of the balance ann, L For static equilibrium the two moments are equal. That is: Fh” = WL = mgL By substitution of the derived hydrostatic thrust, F, from (5) we have: h" =

mL

( m)

 D  ρBD d −   2 

Theory – Fully Submerged Vertical Plane Surface – Theoretical Depth of Centre of Pressure

The theoretical result for depth of centre of pressure, F, below the free-surface is: h' =

Ix Ah

Where: Ix is the 2nd moment of the area of immersed section about an axis in the free-surface. By use of the parallel axes theorem: Ix = Ic + Ah²

 D 2  D  2  4 +  d −   ( m ) I" = BD  12  2    The depth of the centre of Pressure below the pivot is: h” = h’ + H – d (m) 5



substitution as before yield the theoretical result of: D2 h" =

12

2

D  +   d −   2  + H − d D d− 2

Apparatus/EQUIPMENT:

•

The Hydraulics Bench

•

The Hydrostatic Pressure Apparatus

•

A set of weights

•

A jug

•

Calipers or ruler, for measuring the dimension of the quadrant

•

A measuring cylinder, or other container of water of similar height to the hydrostatic pressure tank

•

A length of small bore flexible tubing

Technical Data The following dimensions from the equipment are used in the appropriate calculations. If required these values may be checked as part of the experimental procedure and replaced with your own measurements. Length of Balance L 275 mm Distance from weight hanger to pivot

Quadrant to Pivot

H

200 mm

Base of quadrant face to pivot height

Height of Quadrant

D

100 mm

Height of vertical quadrant face

Width of Quadrant

B

75 mm

Width of vertical quadrant face

Method/PROCEDURE:

6



The dimensions B, D of the quadrant end-face and the distance H and L was be measured. EQUIPMENT SET UP

The empty F1-12 tank on the hydraulic bench is positioned, and the screwed foot is adjusted until the built-in circular spirit level indicated that the based is horizontal. The balance arm is positioned on the knife edges. The weight hangar was located in the groove at the end of the balance aim is horizontal. TAKING A SET OF RESULTS

1) A small mass (50g) is added to the weight hangar. 2) Water is added to the tank. This can be done in one or two ways:

•

The jug is used, and water is poured into the tank the triangular aperture adjacent to the pivot point. Time is allowed for the water in the tank to settle.

•

Alternately, the measuring cylinder frill of water is set beside the tank. The small bore tubing is filled with water, and the end sealed (a thumb over each end is suitable) is hold. One end is placed below the water surface in the cylinder and the other end into the tank. This will set up a siphon system. Water can be added to the tank by pouring it into the measuring cylinder using the jug, then allowing time for the water levels to balance. Siphoning allows greater control over the water entering the tank and less disturbance of the water is produced within the tank.

3) Water is added until the hydrostatic thrust on the end-face of the quadrant causes the balance arm to rise. There is ensured that is no water spilled on the upper surfaces of the quadrant or the sides, above the water level.

7



4) Add of the water is continued until the balance arm is horizontal, this is measured by the base of the balance arm is aligned with the top or bottom of the central marking on the balance rest (either can be used but it must be kept consistent during the experiment). It is founded easier to slightly over-fill the tank, and the equilibrium position is obtained by the drain cock is opened to allow a small outflow. The depth of immersion from the scale on the face of the quadrant is read; more accurate results can be obtained from the reading with the line of sight below the surface, to avoid the effects of surface tension. 5) The above procedure is repeated for each load increment produced by adding a further weight to the weight hanger. The weights supplied allow increments of ten, twenty, and fifty grains to be used, depending on the number of samples required. Fifty gram – intervals are suggested for initial set of results, which will give a total of nineteen samples. 6) This procedure is continued until the water reaches the top of the upper scale on the quadrant face. The procedure is repeated in reverse, by progressively removing the weights. 7) Any factors that we think are likely to affect the accuracy of our results is noted.

Result/Data& CALCULATION: Result Height of quadrant D (m) 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100

Width of Length quadrant of B (m) Balance L (m) 0.075 0.275 0.075 0.275 0.075 0.275 0.075 0.275 0.075 0.275 0.075 0.275 0.075 0.275 0.075 0.275 0.075 0.275 0.075 0.275 0.075 0.275

Quadrant of Pivot H (m)

Mass m (kg)

Depth d (m)

0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200

0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.510

0.048 0.066 0.082 0.096 0.108 0.120 0.131 0.144 0.156 0.169 0.172

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2nd moment Expt. h” (m) 0.848 0.159 1.602 0.168 2.474 0.164 3.390 0.159 4.291 0.157 5.297 0.153 6.313 0.150 7.628 0.141 8.953 0.136 10.507 0.128 10.883 0.126

Thrust F (N)

2nd moment theory h” (m) 0.184 0.178 0.173 0.168 0.164 0.160 0.156 0.152 0.148 0.144 0.143



Height Width Length of of of quadrant quadrant Balance D (m) B (m) L (m) 0.100 0.075 0.275

Quadran t of Pivot H (m) 0.200

0.100

0.075

0.275

0.200

0.100

0.075

0.275

0.200

0.100

0.075

0.275

0.200

0.100

0.075

0.275

0.200

0.100

0.075

0.275

0.200

0.100

0.075

0.275

0.200

0.100

0.075

0.275

0.200

0.100

0.075

0.275

0.200

0.100

0.75

0.275

0.200

0.100

0.075

0.275

0.200

Mass m (kg)

Depth d (m)

0.05 0 0.10 0 0.15 0 0.20 0 0.25 0 0.30 0 0.35 0 0.40 0 0.45 0 0.50 0 0.51 0

0.048

Thrust 2nd 2nd F (N) moment moment Expt. theory h” (m) h” (m) -0.147 -0.917 -0.267

0.066

1.177

0.229

0.202

0.082

2.354

0.172

0.176

0.096

3.384

0.160

0.168

0.108

4.267

0.158

0.164

0.120

5.150

0.157

0.162

0.131

5.960

0.158

0.160

0.1.44

6.916

0.156

0.158

0.156

7.800

0.156

0.158

0.169

8.755

0.154

0.157

0.172

8.976

0.153

0157

Calculation 1) Calculation for Partly Submerged Vertical Plane Surface – Thrust on Surface

F = ρg Bd² 2 = 1000 x 9.81 x 0.075 x( 0.048)2 2 = 0.848 N 2) ,Calculation for Fully Submerged Vertical Plane Surface – Hydrostatic Thrust

F = ρgBD (d – D) 2 9



= 1000 x 9.81 x 0.075 x 0.1 (0.048 – 0.1) 2 = -0.147N

3) Calculation for Partly Submerged Vertical Plane Surface – Experimental Depth of Centre of Pressure

h” = mgL F = 0.05 x 9.81 x 0.275 0.848 = 0.159 m 4) Calculation for Partly Submerged Vertical Plane Surface – Theoretical Depth of Centre of Pressure h” = H – d 3 = 0.2 – 0.048 3 = 0.184 m 5) Calculation for Fully Submerged Vertical Plane Surface – Experimental Depth of Centre of Pressure h” = mL ρBD (d – D) 2

=

0.050 x 0.275 1000 x 0.075 x 0.1 x (0.048 – 0.1) 2 = -0.917 m 6) Calculation for Fully Submerged Vertical Plane Surface – Theoretical Depth of Centre of Pressure

D2 + (d – D)2 h” = 12 2 +H–d d–D 2 10



0.12 + (0.048 – 0.1)2 = 12 2 + 0.2 – 0.048 0.048 – 0.1 2 = -0.267m

DISCUSSION & Recommendation:

Discussion

For summing the moments about the pivot of the apparatus, the buoyant force isneglected. As seen in the apparatus setup, the fluid resides inside the torus.The presence of buoyancy comes from the air outside of the torus. Because the densityof air is a mere fraction of that of the material of the torus and the fluid it contains, it canbe neglected in the hydrostatic force calculations.The weight of the torus can also be neglected. Because the center of the curvatureof the torus is at the location of the pivot, it is negated. The weight of the torus was notincluded in the calculations because the device was calibrated with ballast water so as tobegin the experiment with a net moment of zero about the pivot

RECOMMENDATION

source of error would be the use of the accepted density of water, 1000kg/m^3, for the theoretical calculation of the hydrostatic force. This accepted value is thedensity of sea water at 4oC. The water used in this experiment was tap water atapproximately 23oC. However, if the actual density of the tap water was used, thetheoretical calculations would not differ greatly enough to compensate for the magnitudeof the error. CONCLUSION

IT

WAS NOTED THAT A LARGE DISCREPANCY BETWEEN THE THEORETICAL AND EXPERIMENTALVALUES

OCCURRED.

THIS

IS MOST LIKELY DUE TO ERRORS IN MEASUREMENT OF THE HEIGHT OF THEFLUID INSIDE

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OF THE TORUS.

A NOTHER POSSIBLE

CAUSE COULD BE THAT THE APPARATUS WAS NOTSITTING LEVEL ON

THE COUNTER WHERE THE EXPERIMENT WAS PERFORMED . IF THE APPARATUS IS NOTSITTING LEVEL, THE MOMENT CALCULATIONS WILL YIELD INACCURATE RESULTS .

A

LEVELING DEVICE ONOR NEAR THE

TESTING APPARATUS WOULD AID IN ENSURING THE MOMENT BALANCE IS ACCURATE .

References REFERENCE

Introduction to Fluid Mechanics, 3rd Edition William S. Janna (1993) A Manual for the Mechanics of Fluids Laboratory William S. Janna (2008) Fluid Mechanics with Engineering Application E. John Finnemore Theory and Problems of College Physics Frederick J. Bueche Fluid Mechanics (with Engineering Application) Joseph B. Franzini R.J Grade, Fluid Mechanics through Problem New Age International Publishers, 1997. Fluid and Hydraulic Practice, 2nd Edition, the Millennium. Lab sheet.

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Appendix

13


Center of Pressure

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