Lab1 Fluid Manual

UOIT

School of Energy Systems and Nuclear Science

Experiment # 1

Hydrostatic Pressure Unit

Fall 2007

LABORATORY MANUAL for ENGR2860U-Fluid Mechanics Experiment # 4: Hydrostatic Pressure Unit 1.1 Objective The object of this experiment is to locate the center of pressure of an immersed rectangular surface and compare this position with that predicted value by theory. 1.2 Introduction and Theoretical Background When a surface is submerged in a fluid, forces develop on the surface due to the fluid. The determination of these forces is of major significance in many areas of engineering, such as shipbuilding, the construction of dykes, weirs and locks, and in sanitary and building services engineering. Surfaces that are horizontal or are subjected to gas pressure have essentially uniform pressure over their entire surface. Therefore, the total force resulting from the pressure is equal to the product of the pressure and the area of the surface. For this case the resultant force acts as the centroid of the area, and its line of action is normal to the area. If a plane surface is not horizontal and if it is acted on by a hydrostatic force such as that produced by static liquids, then the pressure is linearly distributed over the surface, and a more general type of analysis must be made to evaluate the magnitude of the resultant force and the location of its line of action. For fluids at rest the force must be perpendicular to the surface since there are no shearing stresses present. The pressure will vary linearly with depth if the fluid is incompressible. For a horizontal surface, such as the bottom of a liquid-filled tank (Figure 1), the magnitude of the resultant force is simply as follows: (1) pAFR = with hghp ρ=γ= (2)

where p is the uniform pressure on the bottom and A is the area of the bottom.

Figure 1. Pressure and resultant hydrostatic force developed on the bottom of an open tank

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LABORATORY MANUAL for ENGR2860U-Fluid Mechanics

Figure 2. Notation for hydrostatic force on an inclined plane surface of arbitrary shape If atmospheric pressure acts on both sides of the bottom, as is illustrated, the resultant force on the bottom is simply due to the liquid in the tank. Since the pressure is constant and uniformly distributed over the bottom, the resultant force acts through the centroid of the area as shown in Figure 1. For the more general case in which a submerged plane surface is inclined, as is illustrated in Figure 2, the determination of the resultant force acting on the surface is more involved. The magnitude of the resultant force on an inclined plane surface of arbitrary shape can be written as θγ= SinAyF cR (3a)

or more simply as

AhF cR γ= (3b)

where hc is the vertical distance from the fluid surface to the centroid of the area. Note that the magnitude of the force is independent of the angle θ and depends only on the specific weight of the fluid, the total area, and the depth of the centroid of the area below the surface. 1.3 Equipment The schematic layout and photo of the hydrostatic pressure unit are shown in Figures 3 and 4, respectively. The unit’s water vessel is designed as a ring segment with constant cross-section (Figure 5). The force due to weight G of the water always produces the same moment of momentum referred to the centre of motion O as the resultant force FR of the active surface running through the centre of pressure D. Consequently, this apparatus can be used to determine the force of pressure Fp and the centre of pressure.

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1. 2. 3. 4. 5.

6. 7. 8.

Figure 3. A schematic layout of the hydrostatic pressure unit

Figure 4. Two photos of the hydrostatic pressure unit

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Water vessel Detent Slider Stop pin Water level scale Rider Weights Handle

LABORATORY MANUAL for ENGR2860U-Fluid Mechanics To illustrate the point, imagine the ring segment completely filled. The force due to weight W applied to the centre of volume of the water can be broken down into two components: • A radially applied component Wr running precisely through the centre of motion, and • A tangential component Wt with a lever arm r acting on the centre of motion O. The radial component Wr exerts no momentum on centre of motion O because its lever arm is zero. Regardless of the water level, the following relation may be written. (4) rWhF tcR = which means that the force due to weight W of the water volume always exerts the same moment of the momentum as the force FR at the centre of the pressure D.

Figure 5. Function mode of the hydrostatic pressure unit

1.4 Operating Instructions and Procedure The following procedure should be followed during the experiments by taking into account Figure 3.

1. Set the water vessel (1) to an angle of α using the detent (2) as shown in the figure. 2. Counterbalance the unit with a rotating slider (3): The stop pin (4) must be precisely in the middle of

the hole for this. 3. Mount the rider (6), set the lever arm on the scale (e.g. l=150 mm). 4. Top up with water until the unit is balanced (stop pin (4) at centre of hole). 5. Read off water level s and enter it in the prepared worksheet (see appendix). 6. Increase the appended weights (7) in increments of 0.5 - 1 N and repeat the measurement. 7. Set the water vessel (1) to a different angle of α and repeat steps 1-6.

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Figure 6. Experimental apparatus

1.5 Calculations 1.5.1 Center of Pressure for α = 0o

At a water level s, below the 100 mm mark, the height of the active surface changes with the water level. If the water level is above that mark, the height of the active surface is always 100 mm (Figure 7). For a water level s < 100 mm (Pressure has a triangular profile):

2sp γ

= (5)

bsA = (6)

s61e = (7)

s31mm200ID −= (8)

where s is the water level, e is the distance between the centre of pressure D and the planar centre of force C of the active surface, ID is the distance to centre of motion of the unit and b=75 mm is the width of liquid vessel. For a water level s ≥ 100mm (Pressure has a trapezoidal profile):

( )mm50sp −γ= (9)

b(mm)100A = (10)

( )( )mm50s12

mm100e2

−= (11)

emm150ID += (12)

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Figure 7. Pressure profile on the acting surface α=0o (a) s<100mm, (b) s≥100mm 1.5.2 Center of Pressure for α > 0ο

For water level s < sh (Figure 8):

( )2

ssp t−γ

= (13)

bhA = (14)

6he = (15)

cos

hα

tss −= (16)

h31mm200ID −= (17)

For water level s≥ sh:

( α−− )γ= cosmm50ssp t (18)

( )

⎟⎠

⎞⎜⎝

⎛α

−=

mm50cos

ss12

mm100et

2

(19)

emm150ID += (20)

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(a) (b)

Figure 8. Pressure on the acting surface α>0ο (a) s<100mm, (b) s>100mm 1.5.3. Resultant Force The hydrostatic pressure acting on the surface A (also called active surface) can be represented as resultant force FR, of which the line of application leads through the centre of pressure D. The size of this resultant force corresponds to the hydrostatic pressure at the planar centre of force C of the surface A. It may be calculated in a similar manner as given in Equation (1) where the hydrostatic pressure p is given by the Equation (2). cyCosp αγ= (21)

Figure 9. Hydrostatic force on an inclined plane surface of the hydrostatic pressure unit

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LABORATORY MANUAL for ENGR2860U-Fluid Mechanics 1.5.4. Balance of Moments For checking out the theory, a balance of moments around the centre of O can be established and compared to those predicted by hydrostatic theory.

(22) DRa)0( IFIW0M ===∑

where Wa is the appended weight and I is the lever arm of appended weight referred to centre of motion O.

Figure 10. Illustration of moment balance

When doing your calculations, make sure to include the forces due to pressure on the curved surfaces. Explain why the forces acting on the curved surfaces would not contribute in your calculations in finding the center of pressure. 1.6 Worksheet for Experimental Data

Angle (α) (o)

Lowest water level (st) (mmWC)

Highest water level (sh) (mmWC)

Lever arm

I (mm) Appended weight

Wa (N) Water level reading

s (mm) Calculated lever arm

ID (mm) Resultant force

FR (N)

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LABORATORY MANUAL for ENGR2860U-Fluid Mechanics Nomenclature A : Surface (active surface) (m2) b : Width of liquid container (m) e : Distance between centre of pressure and planar centre of gravity (m) FR : Resultant force (N) g : Gravitational acceleration (m/s2) h : Height of active surface (m) hc : Vertical distance from the fluid surface to the centroid of the area (m) I : Lever arm of force due weight (m) ID : Lever arm of resultant force FR (m) M : Moment (Nm) p : Hydrostatic pressure in planer centre of gravity of active surface (bar) s : Water level reading/scale (m) sh : Water level at highest point of active surface (m) st : Water level at lowest point of measuring vessel (m) W : Force due to weight of the appended weights (N) Wa : Appended weight (N) Wt : Radial component of the force due to weight (N) Wt : Tangential component of the force due to weight (N) yp : Distance between centre of pressure liquid level along the active surface (m) Greek letters γ : Specific weight (N/m3) θ,α : Angle (o) ρ : Density (kg/m3) References 1. Equipment for Engineering Education, Instruction and Operation Manuals, Gunt Hamburg Germany,

1998. 2. Munson, B. A., Young, D. F., and Okiishi, T. H. Fundamentals of Fluid Mechanics. 4th Edition. John

Wiley & Sons, Inc., NY, 2002. 3. Crowe, C. T., Elger, D. F., and Roberson, J. A. Engineering Fluid Mechanics. Seventh Edition. John

Wiley & Sons, Inc., NY, 2001.

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Lab1 Fluid Manual

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