Statics

Definitions and ApplicationsStatics: no relative motion between adjacent fluid layers.Shear stress is zeroOnly _______ can be acting on fluid surfacesGravity force acts on the fluid (____ force)Applications:Pressure variation within a reservoir Forces on submerged surfacesTensile stress on pipe wallsBuoyant forcespressurebody

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

Upstream face of Hoover DamUpstream face of Hoover Dam in 1935Crest thickness: 13.7 m Base thickness: 201 mWHY???

What do you think?Lake Mead, the lake behind Hoover Dam, is the world's largest artificial body of water by volume (35 km3). Is the pressure at the base of Hoover Dam affected by the volume of water in Lake Mead?

What do we need to know?Pressure variation with directionPressure variation with locationHow can we calculate the total force on a submerged surface?

Pressure Variation with Direction(Pascals law)yxpsdspxdypydxqdydxdsBody forcesSurface forcesEquation of MotionF = mapxdy - psds sinqPressure is independent of direction!

Pressure Field (pressure variation with location)In the absence of shearing forces (no relative motion between fluid particles) what causes pressure variation within a fluid?Consider a soda can in spaceThrow the soda can to another astronautThrow the soda can toward the moonWhat causes pressure gradients?

Pressure FieldSmall element of fluid in pressure gradient with arbitrary __________.Forces acting on surfaces of elementPressure is p at center of elementaccelerationMassSame in x!Now lets sum the forces in the y direction

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

Apply Newtons Second LawMass of element of fluidSubstitute into Newtons 2nd LawObtain a general vector expression relating pressure gradient to acceleration and write the 3 component equations.We are effectively accelerating upward at g when we are at rest on earths surface!Text version of eq.At rest3 component equationsA surface of constant pressure?

Pressure Variation When the Specific Weight is ConstantWhat are the two things that could make specific weight (g) vary in a fluid?Changing densityChanging gravityg = rgPiezometric head is constant in a static incompressible fluidConstant specific weight!Generalize to any a!

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?hWhat is the pressure at the top of the tank?Suppose I define pressure and elevation as zero at the water surface. What is the piezometric head everywhere in the tank? Zero!No!Free surface

Units and Scales of Pressure Measurement6894.76 Pa/psiStandard atmospheric pressureLocal atmospheric pressureAbsolute zero (complete vacuum)Absolute pressureGage pressure1 atmosphere101.325 kPa14.7 psi______ m H20760 mm HgSuction vacuum(gage pressure)Local barometer reading10.34

Mercury Barometer (team work)What is the local atmospheric pressure (in kPa) when R is 750 mm Hg?RP2 = Hg vapor pressureAssume constant r

Pressure Variation in a Compressible FluidPerfect gas at constant temperature (Isothermal)Perfect gas with constant temperature gradient

Perfect Gas at Constant Temperature (Isothermal)Mgas is molecular massr is function of pIntegrate

Perfect Gas with Constant Temperature Gradient The atmosphere can be modeled as having a constant temperature gradientb = 0.00650 K/mLapse rateMt. Everest8,850 m

Chart1

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P (kPa)

Elevation (m)

Pressure (kPa)

Sheet1

M0.029for air

Beta-0.00651

g9.806

R8.314

P0100,000

T0300

y (m)P (kPa)

10099

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P (kPa)

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Sheet3

MBD00918011.unknown

Pressure MeasurementBarometersManometersStandardDifferentialPressure Transducers

Measure atmospheric pressurePressure relative to atm.Pressure difference between 2 pts.

Standard ManometersWhat is the pressure at A given 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?Ahp = ghh = p/gh = 500,000 Pa/(9800 N/m3)h = 51 mWhy is this a reasonable pressure?gage

Manometers for High PressuresFind the gage pressure in the center of the sphere. The sphere contains fluid with g1 and the manometer contains fluid with g2.What do you know? _____Use statics to find other pressures.P1 = 0h1h2=P3g1g2For small h1 use fluid with high density.Mercury!+ h1g2- h2g1P1

Differential Manometers- h2gHg- h3gwh1h3MercuryFind the drop in pressure between point 1 and point 2.p1p2Waterh2orifice= p2p1 - p2 = (h3-h1)gw + h2gHgp1 - p2 = h2(gHg - gw)p1+ h1gw

Procedure to keep track of pressuresStart at a known point or at one end of the system and write the pressure there using an appropriate symbolAdd to this the change in pressure to the next meniscus (plus if the next meniscus is lower, and minus if higher)Continue until the other end of the gage is reached and equate the expression to the pressure at that pointp1 + Dp = p2

Pressure TransducersExcitation: 10 Vdc regulated Output: 100 millivoltsAccuracy: 1% FS Proof Pressure: 140 kPa (20 psi) for 7 kPa modelNo Mercury!Can be monitored easily by computerMyriad of applications Volume of liquid in a tankFlow ratesProcess monitoring and controlFull Scale

Types of Diaphragms Used for Pressure MeasurementsStainless SteelStrain gages bonded to the stainless steelTypical full scale output of 3 mV/VPiezoresistiveStrain gage diffused into silicon wafersTypical full scale output of 10 mV/V

SiliconIdeal material for receiving the applied forcePerfect crystalReturns to its initial shape (no hysteresis)Good elasticityNo need for special bonding between material receiving force and strain gage

Pressure Sensor FailureHigh pressures rupture crystal (beware of resulting leak!)Water hammer High speed pressure waves (speed of sound)Result from flow transients such as rapidly shutting valvesInstall pressure snubber!Incompatible materialsElastic tubing or gas chamber

Absolute vs. Gage vs. DifferentialAbsolutePort 2 sealed with vacuum on bottom side of silicon crystalGagePort 2 open to atmosphereDifferentialBoth ports connected to systemPort 1Port 2

Summary for StaticsPressure is independent of Pressure increases withconstant densitygas at constant temperaturegas with constant temperature gradientPressure scalesunitsdatumPressure measurementdirectiondepthp = ghUse ideal gas law

ReviewPressure increases or decreases as we move in the direction of the acceleration vector?The free surface is _______ to the acceleration vector.What is an equation that describes the change in pressure with depth in a fluid?Suppose a tank of fuel is accelerating upward at 2g. What is the change in pressure with depth in the fuel?

normal

Statics exampleWhat is the air pressure in the cave air pocket?

Statics LabHow did the bubbler work?How does the pressure sensor read pressure at the bottom of the tank?Must the pump be running if the water depth is decreasing?

Somebody finally got smart and came up with an above-ground pool thats got a deep end and a shallow end.