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Statics CVEN 311

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Definitions and Applications ä Statics: no relative motion between adjacent fluid layers. ä Shear stress is zero ä Only _______ can be acting on fluid surfaces ä Gravity force acts on the fluid (____ force) ä Applications: ä Pressure variation within a reservoir ä Forces on submerged surfaces ä Tensile stress on pipe walls ä Buoyant forces ä Statics: no relative motion between adjacent fluid layers. ä Shear stress is zero ä Only _______ can be acting on fluid surfaces ä Gravity force acts on the fluid (____ force) ä Applications: ä Pressure variation within a reservoir ä Forces on submerged surfaces ä Tensile stress on pipe walls ä Buoyant forces pressure body

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Motivation? ä What are the pressure forces behind the Hoover Dam?

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Upstream face of Hoover Dam Upstream face of Hoover Dam in 1935 Tall: 220 m (726 ft) Crest thickness: 13.7 m (50 ft) Base thickness: 201 m (660 ft - two footballs fields) WHY???

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What do you think? Lake Mead, the lake behind Hoover Dam, is the world's largest artificial body of water by volume (35 km 3 ). Is the pressure at the base of Hoover Dam affected by the volume of water in Lake Mead?

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What do we need to know? ä Pressure variation with direction ä Pressure variation with location ä How can we calculate the total force on a submerged surface? ä Pressure variation with direction ä Pressure variation with location ä How can we calculate the total force on a submerged surface?

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Pressure Variation with Direction (Pascal’s law) y x psspss pxypxy pyxpyx yy xx ss Body forces Surface forces Equation of Motion F = ma p x y - p s s sin Independent of direction!

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Pressure Field ä In the absence of shearing forces (no relative motion between fluid particles) what causes pressure variation within a fluid? p1p1 p2p2 p3p3 Which has the highest pressure?

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Pressure Field zy x Small element of fluid in pressure gradient with arbitrary __________. Forces acting on surfaces of element Pressure is p at center of element acceleration Mass… Same in x!

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Simplify the expression for the force acting on the element Same in xyz! This begs for vector notation! Forces acting on element of fluid due to pressure gradient

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Apply Newton’s Second Law Mass of element of fluid Substitute into Newton’s 2 nd Law Obtain a general vector expression relating pressure gradient to acceleration and write the 3 component equations. Text version of eq. At rest (independent of x and y) 3 component equations

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Pressure Variation When the Specific Weight is Constant What are the two things that could make specific weight ( ) vary in a fluid? = g Compressible fluid - changing density Changing gravity Piezometric head is constant

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Example: Pressure at the bottom of a Tank of Water? Does the pressure at the bottom of the tank increase if the diameter of the tank increases? h p = h NO!!!! 1 1 2 2

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6894.76 Pa = 1 psi Units and Scales of Pressure Measurement Standard atmospheric pressure Local atmospheric pressure Absolute zero (complete vacuum) Absolute pressure Gage pressure 1 atmosphere 101.325 kPa 14.7 psi ______ m H 2 0 760 mm Hg Suction vacuum (gage pressure) Local barometer reading 10.34 29.92 in Hg

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What is the local atmospheric pressure (in kPa) when R is 750 mm Hg? Mercury Barometer R 1 2 p 2 = Hg vapor pressure

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1000 kg/m 3 1 x10 -3 N·s/m 2 9800 N/m 3 101.3 kPa A few important constants! ä Properties of water Density: _______ Viscosity: ___________ Specific weight: _______ ä Properties of the atmosphere ä Atmospheric pressure ______ ä Height of a column of water that can be supported by atmospheric pressure _____ ä Properties of water Density: _______ Viscosity: ___________ Specific weight: _______ ä Properties of the atmosphere ä Atmospheric pressure ______ ä Height of a column of water that can be supported by atmospheric pressure _____ 10.3 m

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Pressure Measurement ä Barometers ä Manometers ä Standard ä Differential ä Pressure Transducers ä Barometers ä Manometers ä Standard ä Differential ä Pressure Transducers Weight or pressure

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A Standard Manometers What is the pressure at A in terms of h? Pressure in water distribution systems commonly varies between 25 and 100 psi (175 to 700 kPa). How high would the water rise in a manometer connected to a pipe containing water at 500 kPa? What is the pressure at A in terms of h? Pressure in water distribution systems commonly varies between 25 and 100 psi (175 to 700 kPa). How high would the water rise in a manometer connected to a pipe containing water at 500 kPa? h p = h h = p/ h = 500,000 Pa/9800 N/m 3 h = 51 m Not very practical! piezometer tube container ( 72 psi)

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Find the gage pressure in the center of the sphere. The sphere contains fluid with 1 and the manometer contains fluid with 2. What do you know? _____ Use statics to find other pressures. Find the gage pressure in the center of the sphere. The sphere contains fluid with 1 and the manometer contains fluid with 2. What do you know? _____ Use statics to find other pressures. p 1 = 0 h1h1h1h1 ? h2h2h2h2 Manometers for High Pressures 1 1 2 2 3 3 = p 3 11 22 For small h 1 use fluid with high density. Mercury! + h 1 2 - h 2 1 p1p1 p1p1 U-tube manometer

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- h 2 Hg - h 3 w Differential Manometers h1h1 h3h3 Mercury Find the drop in pressure between point 1 and point 2. p1p1 p2p2 Water h2h2 orifice = p 2 p 1 - p 2 = (h 3 -h 1 ) w + h 2 Hg p 1 - p 2 = h 2 ( Hg - w ) p1p1 p1p1 + h 1 w

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Procedure to keep track of pressures ä Start at a known point or at one end of the system and write the pressure there using an appropriate symbol ä Add to this the change in pressure to the next meniscus (plus if the next meniscus is lower, and minus if higher) ä Continue until the other end of the gage is reached and equate the expression to the pressure at that point ä Start at a known point or at one end of the system and write the pressure there using an appropriate symbol ä Add to this the change in pressure to the next meniscus (plus if the next meniscus is lower, and minus if higher) ä Continue until the other end of the gage is reached and equate the expression to the pressure at that point p 1 + p = p 2

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Pressure Transducers ä Excitation: 10 Vdc regulated ä Output: 100 millivolts ä Accuracy: ±1% FS ä Proof Pressure: 140 kPa (20 psi) for 7 kPa model ä No Mercury! ä Can be monitored easily by computer ä Myriad of applications ä Volume of liquid in a tank ä Flow rates ä Process monitoring and control ä Excitation: 10 Vdc regulated ä Output: 100 millivolts ä Accuracy: ±1% FS ä Proof Pressure: 140 kPa (20 psi) for 7 kPa model ä No Mercury! ä Can be monitored easily by computer ä Myriad of applications ä Volume of liquid in a tank ä Flow rates ä Process monitoring and control Full Scale

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Summary for Statics ä Pressure is independent of ä Pressure increases with ä constant density ä Pressure scales ä units ä datum ä Pressure measurement ä Pressure is independent of ä Pressure increases with ä constant density ä Pressure scales ä units ä datum ä Pressure measurement direction depth p = h

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Statics example What is the air pressure in the cave air pocket?

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