Statics CVEN 311 

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

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

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???

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?

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?

Pressure Variation with Direction (Pascal’s law) y x psspss pxypxy pyxpyx  yy xx ss Body forces Surface forces Equation of Motion F = ma p x  y - p s  s sin  Independent of direction!

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?

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!

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

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

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

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!!!!

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 kPa 14.7 psi ______ m H mm Hg Suction vacuum (gage pressure) Local barometer reading in Hg

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

1000 kg/m 3 1 x10 -3 N·s/m N/m 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

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

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)

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 = p 3 11 22 For small h 1 use fluid with high density. Mercury! + h 1  2 - h 2  1 p1p1 p1p1 U-tube manometer

- 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

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

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

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

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