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CET 240

Fundamental Hydrostatics

Be able to

• Compute pressure in water at various depths

• Compute the pressure on a submerged vertical surface

• Compute the pressure on a submerged inclined surface

• Compute the buoyant force on a submerged object

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Hydrostatic Pressure

• Pressure is defined as: p = F/A

where p = pressure, lbs/ft2 (N/m2)

F = force, pounds (newtons)

A = area, ft2 (m2)

Pascal’s Laws

• Water exerts pressure at a right angle to container walls or a submerged surface

• Pressure at any point in water at rest is equal in all directions

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Figure 3-1 For each of the infinitesimal elements of water at rest, the pressure acting on each surface in any direction has the same magnitude.

Pascal’s Principle

• The weight of a vertical column of water above the surface equals the force exerted there

• To compute the pressure of water at a depth z

– Rewrite the pressure equation using the definition of the weight of water (γ=W/V so W = γV) with (W=F) and V=Az

p = F/A = W/A = γV/A = γ(Az/A) = γz

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Figure 3-2 Pressure at a depth z in a volume of water.

Pascal’s Paradox

• Water pressure is not dependent on volume or shape of container

Figure 3-3 The pressure at point P is the same in each container provided the vertical depth z is the same.

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Pascal’s Paradox

The horizontal thrust on the dam is the SAME.

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Absolute and Gauge Pressure

• Pabs = pgauge + patm (as psia)

• Pabs (as psia) = pgauge (as psig) + patm (as psia, 14.7 for

example)

• Atmospheric Pressure is 14.7 psia (101 kPA) unless given a more exact figure

• A perfect vacuum is the lower possible pressure

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Hydrostatic Pressure (con’t.)

• Gauge pressure

– Pressure in excess of atmospheric pressure

– By convention, zero magnitude is assumed at the free surface

– Increases with depth below the surface

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Pressure on Plane Surfaces

• Determine the resultant force, FR, for the following plane surfaces in water at depth z:

– Horizontal submerged plane surface

– Vertical submerged plane surface intersecting the free surface

– Vertical plane surface completely submerged

– Inclined surface

• Location of FR called center of pressure, yR

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Figure 3-4 Pressure distribution on a submerged horizontal plane surface.

Pressure on Plane Surfaces (con’t.)

• Horizontal submerged plane surface

FR = pA or FR = γzlw

where z = depth below surface

l = length of surface

w = width of surface

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Figure 3-5 Pressure distribution on a submerged vertical plane surface.

Figure 3-6 Resultant force and center of pressure for a vertical plane surface, (a) intersecting the surface and (b) completely submerged.

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Pressure on Plane Surfaces (con’t.)

• Vertical submerged plane surface – If surface extends to free surface, pressure

distribution is triangular

FR = ½(z)(γz)w = (γz2/2)w

yR = ⅓(z)

– If surface is completely submerged, pressure distribution is trapezoidal

FR = [(γz1 + γz2)/2]lyw = (γly/2)w(z1 + z2)

yR = (ly/3) [2z1+z2 / z1+z2]

Figure 3-7 Pressure distribution on a submerged inclined plane surface.

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Measuring Pressure

• Piezometer

– Water rises until it reaches height proportional to pressure

– Length must be sufficient to prevent overflow

– Diameter should be large enough to prevent distortion due to adhesion

– Place tube perpendicular to pipe and flush with interface

Figure 3-12 Piezometer is used to measure water pressure in a pipe. In both (a) and (b), if the pressure are equal, the water levels in the piezometers are equal.

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Figure 3-13 Piezometers used to measure pressure in (a) a static and (b) a dynamic hydraulic system.

Measuring Pressure (con’t.)

• Manometer

– Used instead of piezometer when water pressure is relatively high

– Heavy liquid, such as mercury, measures force caused by an imbalance in tube

– Valve relieves trapped air

– Can be used to measure negative pressure

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Figure 3-14 Manometer used to measure water pressure in a pipe.

PROCEDURE FOR WRITING THE EQUATION FOR A

MANOMETER

1. Start from one end of the manometer and express the

pressure there in symbol form (e.g., PA refers to the pressure at point A). If one end is open the pressure is atmospheric pressure, taken to be zero gage pressure.

2. Add terms representing changes in pressure using Δp = γh; proceeding from the starting point and including each column of each fluid separately.

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PROCEDURE FOR WRITING THE EQUATION FOR A

MANOMETER Continued

3. When the movement from one point to another is downward, the pressure increases and the value of Δp is added. Conversely, when the movement from one point to the next is upward, the pressure decreases and Δp is subtracted. 4. Continue this process until the other end point is reached. The result is an expression for the pressure at that end point. Equate this expression to the symbol for the pressure at the final point, giving a complete equation for the manometer. 5. Solve the equation algebraically for the desired pressure at a given point or the difference in pressure between two points of interest.

6. Enter known data and solve for the desired pressure.

Figure 3-15 Schematic gauge of a bourdon gauge.

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0 + γmh = patm

γmh = patm

Barometer

Archimedes' Principle

• is a law of physics stating that the upward buoyant force exerted on a body immersed in a fluid is equal to the weight of the fluid the body displaces.

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Buoyancy

• Uplifting force exerted by water on a submerged solid object

• Difference between force pushing up on an object and force pushing down on an object

• Equal to the weight of water that would occupy the space occupied by the object – If buoyancy is greater than object weight, object

will rise to surface