4tf3 lecture 01 fall10

8/8/2019 4TF3 Lecture 01 Fall10

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Thursday September 9Thursday September 9, , 20102010

MAN TECH 4TF3 MAN TECH 4TF3

IntroductionIntroduction

PressurePressure - - ManometryManometry

Fluid MechanicsFluid MechanicsFluid MechanicsFluid MechanicsFluid MechanicsFluid MechanicsFluid MechanicsFluid Mechanics

BehaviourBehaviour of of fluidsfluids Fluid: Liquid or gasFluid: Liquid or gas Wide subject with variety of applicationsWide subject with variety of applications

Extreme range of physical parameters Extreme range of physical parameters Length Length -- Pipe flows: Pipe flows: FromFrom nanonano--scale tubes to 4scale tubes to 4- -ftft

diameter oil pipelines.diameter oil pipelines.

Speed:Speed: From hypersonic flow of air around aircraftsFrom hypersonic flow of air around aircraftsand meteorites (10and meteorites (10 44 m/s)m/s)

to creeping flow of magma (10to creeping flow of magma (10 --88 m/s)m/s)

Pressure Pressure :: From pressure inside hydraulic ramsFrom pressure inside hydraulic rams(10,000 psi) to sound waves (10(10,000 psi) to sound waves (10 --6 6 psi) psi) Vid 1 & 2


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Definition of a FluidDefinition of a FluidDefinition of a FluidDefinition of a Fluid

Solids: Strong intermolecular forces Closely spaced molecules Hard

Liquids: Weaker intermolecular forces Molecules spaced further apart Soft, but hardly compressible

Gases: Almost no intermolecular forces Molecules are free to move Expands to occupy all available space

FLUI

D

S

Fluid DefinitionFluid DefinitionFluid DefinitionFluid Definition

A substance that deforms continuously(flows) when acted on by a shearing stressof any magnitude

Continuum Approach Do not consider motion of individual molecules Consider small volumes and average over its molecules Properties change continuously within the fluid

F

Surface


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DensityDensityDensityDensityMass per unit volumeMass per unit volume Units:Units: Kg/mKg/m 33, slug/ft, slug/ft 33,, lblbmm /ft /ft 33 Small dependence on temperature for liquidsSmall dependence on temperature for liquids Liquids are practically incompressible (density does notLiquids are practically incompressible (density does not

depend on pressure)depend on pressure)

Strong dependence on temperature and pressure for gassesStrong dependence on temperature and pressure for gasses

Specific weight & gravitySpecific weight & gravitySpecific weight & gravitySpecific weight & gravity

Specific weight ( ): Weight per unit volume = g

Units: N/m 3 or lb f /ft 3

Water: 9800 N/m 3

Specific gravity (SG): ratio of fluid density todensity of water at a specific temperature

Reference density: water at 4 C ( = 1000 Kg/m 3)

Units: dimensionless

C O H o

SG4@2

=


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ViscosityViscosityViscosityViscosityVideo 3 and 5

Shear Thinninghttp://www.youtube.com/watch?v=TT8RwyrHA1Y

Shear Thickeninghttp://www.youtube.com/watch?v=vNzTYzjLgKE&feature=related

ViscosityViscosityViscosityViscosity

Shearing experimentPlace a material between two parallel plates and apply a

force on the top plate

Solid material: Plate will move slightly and then stop!Liquid material?


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Liquid material: Plate will move continuously at a velocity U Fluid sticks on boundaries ( NO-SLIP CONDITION)

Fluid is stationary at bottom plate Fluid moves with velocity U at top plate

A linear fluid velocity profile is developed


Forces on upper plate

P: Force applied by you

A: Shear stress from fluid

It is

rate of shearing strain

VISCOSITY (Absolute or dynamic)

dydu

=

dydu NEWTONIAN

FLUIDS


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Newtonian fluidsNewtonian fluidsNewtonian fluidsNewtonian fluids

Viscosity is constant with shearing strain

Simplest case Many common liquids are Newtonian

NonNonNonNon----Newtonian FluidsNewtonian FluidsNewtonian FluidsNewtonian Fluids

Viscosity depends on shearing strainShear Thinning: Viscosity

drops with shearing strain(liquid polymers, paint)

Shear Thickening: Viscosityrises with shearing strain(quicksand)

Bingham Plastic: Yield stressmust be exceeded for flow(mayonnaise, toothpaste)


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ViscosityViscosityViscosityViscosity - --- TemperatureTemperatureTemperatureTemperature

Viscosity dependsgreatly on temperatureLiquids: Viscosity drops with

temperatureGasses : Viscosity increases

with temperature

Pr 1.65

Compressibility of FluidsCompressibility of FluidsCompressibility of FluidsCompressibility of Fluids

Bulk Modulus (E v)

Change in pressure required to compress thevolume V by dV

Liquids: Ev 3 105

psia pressure of 3000 psi is required tocompress a liquid by 1%

practically incompressible

V dV dp

E v / =

/ d dp

E v =or

Vid 7


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Compressibility of GassesCompressibility of GassesCompressibility of GassesCompressibility of GassesGasses can be compressed (or expanded)Isothermally: P/ = constant

Then, E v = pIsentropically: P/ k = constant

Then, E v = kp

Bulk modulus depends on pressure (the higher thepressure, the higher the modulus, the harder tocompress a gas)

Air at 1 atm (14.7 psi) . It is k = 1.4 and E v = 20.6 psicompare to water

Speed of SoundSpeed of SoundSpeed of SoundSpeed of SoundDisturbances in a fluid propagate at the acoustic

velocity , or speed of sound ( c)c depends on changes of

pressure and density

Isentropic process (negligible heat transfer)

Air at 15.5C: c = 340 m/sWater at 20C: c = 1481 m/s

v E c

d dp

c ==

kpc =For gasses

For ideal gasses kRT c =

NOTE: Ev = and c= For incompressible materials


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VaporVaporVaporVaporVaporVaporVaporVapor PressurePressurePressurePressurePressurePressurePressurePressureObservation: Liquids (e.g. gasoline) evaporate when left in an open

containerExplanation: Liquid molecules escape the liquid phase and enter the gas

Experiment: Completely fill a container with a liquidand lift one end of it (without letting any air in).Result: The space between the liquid and the containerwill be filled with vapor. The pressure of the vapor iscalled vapor pressure

Vapor pressure is a property of the liquid.It depends heavily on temperature

Surface TensionSurface TensionSurface TensionSurface TensionSurface TensionSurface TensionSurface TensionSurface TensionThe surface of a liquid (gasThe surface of a liquid (gas- -liquid interface) behaves as aliquid interface) behaves as amembrane or skin (e.g. it can support a razor blade)membrane or skin (e.g. it can support a razor blade)

Results from unbalance of molecular forcesResults from unbalance of molecular forcesalong the surfacealong the surface

AA tensile forcetensile force acts in the plane of the surfaceacts in the plane of the surfaceat any line along the surfaceat any line along the surface

This force is calledThis force is called Surface Tension (Surface Tension ( ))

Units of Units of :: N/m,N/m, lblb f f /ft ( /ft (Force/LengthForce/Length ))

Videos


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Surface TensionSurface TensionSurface TensionSurface TensionSurface TensionSurface TensionSurface TensionSurface Tension

Pressure inside a dropPressure inside a drop

Capillary riseCapillary riseWater rises inside a narrow tubeWater rises inside a narrow tube

Force Balance: 2 R = p R2

Hence

Pressure is higher inside droplet

R p p p ei

2==

Pr 1.95

Capillary riseCapillary riseCapillary riseCapillary riseCapillary riseCapillary riseCapillary riseCapillary riseFree Body Diagram

Force Balance

R2 h = 2 R cos

and

Rh

cos2=

Liquid may drop too!Depends on angle


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STATIC FLUIDS

Pressure at a PointPressure at a PointPressure at a PointPressure at a Point How does pressure at a point vary with direction?Consider the triangular wedge of fluid:

No shear stress Gravity in negative z direction Neglect forces on x-axis


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Pressure at a pointPressure at a pointPressure at a pointPressure at a point - --- 2222

Triangular wedge Assume acceleration in y ( a y) and z (a z) directionsForce balances in y and z directions

It is:

and the force balance equations yield

zs z z

ys y y

as y xs y x

s x p y x pF

as y x

s x p z x pF

22cos

2sin

==

==

Note: Multiplying pressure by area gives force

sin and cos s zs y ==

Pascals lawPascals lawPascals lawPascals law

And as x, y, and z tend to zero, we get p y = p z = p s

( )2

and 2

za p p

ya p p zs z ys y

+==

Pressure at a point in a fluid at rest or inmotion is independent of direction as long asthere are no shearing stresses present

Pascals Law


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Pressure FieldPressure FieldPressure FieldPressure Field How does pressure vary from point to point?Consider the rectangular element of fluid: Fluid volume is x y z Pressure at center is p Pressure at sides is expressed

using pressure derivatives No shearing

Pressure FieldPressure FieldPressure FieldPressure Field - --- 2222Pressure at sides are expressed using derivativesFor example, the pressure at the top surface is:

Two kinds of forces act on the fluid element:

Surface forces (due to pressure on the exposed surfaces Body forces (the weight of the liquid)

2 z

z p

p

+

Pressure at centerPressure derivative

Distance from center


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Pressure FieldPressure FieldPressure FieldPressure Field - --- 3333

Forces in y-direction (only surface forces)

Forces in x-direction (only surface forces)

Forces in y-direction (surface and body forces)

j j jF)))

2

2

z y x y p

z x y

y p

p z x y

y p

p y

=

+

=

iF)

z y x x p

x

=

kkF )) z y x z y x z p

z = weight

Pressure FieldPressure FieldPressure FieldPressure Field - --- 4444

Assuming the fluid elements acceleration is a , theforce balance yields:

Then:

General equation of motion for a fluid without shearing stresses

akk ji

akk ji

z y x z y x z y x z p

y p

x p

z y x z y x z y x z p

z y x y p

z y x x p

=

+

+

=

))))

))))

p Pressure gradient

ak =)

p


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Fluid at RestFluid at RestFluid at RestFluid at Rest

Fluid at rest (or in solid body motion) a = 0

And in component form

Pressure does not vary in x and y directions.

Pressure varies with elevation Pressure decreases as we move upward in a fluid

0k =)

p

0=

x p 0=

y p =

z p

Incompressible FluidIncompressible FluidIncompressible FluidIncompressible Fluid Incompressible fluid: A fluid with constant densityIf also g is assumed constant

= g is constant (: specific weight)And integration is easy

( )12122

1

2

1

z z p pdzdp z p p

p

z

z===

h p p +=

21

Hydrostatic pressure distribution


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Pressure HeadPressure HeadPressure HeadPressure HeadUse hydrostatic pressure distribution to express

pressure differences in terms of height of fluid

Pressure in fluids is often measured from free surface

Pressure depends only on height of liquid (not shape) Pressure on same level on continuous liquid is same

21 p ph

= h: pressure head

h p p p B A +== 0

Pr. 2.5

Hydraulic equipmentHydraulic equipmentHydraulic equipmentHydraulic equipment Pressure on same level on continuous liquid is same

Base of all hydraulic devices (jack, brakes, lifts, etc.) Apply a small force F 1 (on a small piston) to achieve

a high force F 2 (on a big piston)


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Compressible FluidsCompressible FluidsCompressible FluidsCompressible Fluids

Compressible fluid: Density changes with pressure(any gas is compressible)

For high elevation changes:

And

If T=T 0 (constant with elevation)

=dzdp Density (and specific weight) of gasses is small

Neglect pressure changes in gasses when elevationchanges are small (order of few hundred feet)

RT p= Ideal gas

==

1

1

2

1

1 p p

z

z dzT R

g

p

dp

RT

gp

dz

dp

( )=0

1212 exp RT

z zg p p

Pressure in AtmospherePressure in AtmospherePressure in AtmospherePressure in AtmosphereFor the atmosphere, this correction does not make

such a huge difference


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Standard AtmosphereStandard AtmosphereStandard AtmosphereStandard Atmosphere

Idealized representation of mean condition of earthsatmosphere

Temperature changes with elevationsee text

Pressure MeasurementPressure MeasurementPressure MeasurementPressure MeasurementAbsolute pressure: Pressure value relative to perfect vacuumGage pressure: Pressure value relative to the local atmospheric

pressureVacuum pressure: Used for pressure values lower that the local

atmospheric pressure. Difference from atmospheric value isreported as a positive number.

Pr 2.21


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BarometerBarometerBarometerBarometer

Used for measuring atmospheric pressure

patm = p vapor + h

But for mercury, p vapor 0 and

patm = h

h 760 mm

ManometryManometryManometryManometryMeasure pressure using liquid rise in a tubePiezometer Tube

11 h p p p atm A +==

Gage pressure at A

1, h p gage A =

Absolute pressure at A

Disadvantages:p A must be higher thanatmosphericp A must be low (so that h 1 issmall)Fluid must be liquid V2


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2

UUUU----tube Manometertube Manometertube Manometertube Manometer

1 p p A =1112 h p p +=

223 h p p atm +=32 p p =

1122 hh p p atm A +=Then

or

1122, hh p gage A =

If A contains a gas 1 0 and

22, h p gage A =

Differential UDifferential UDifferential UDifferential U- ---tube Manometertube Manometertube Manometertube ManometerMeasure pressure difference between A and B

112233

112233

hhh p p

phhh p

B A

A B

+=

=++

Pr 2.24, 2.36


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2

InclinedInclinedInclinedInclined Tube ManometerTube ManometerTube ManometerTube Manometer

Used for measuring small pressure differencesVertical height difference results to pressure difference

A B phlh p =++ 112233 sin

Changing the angle you can convert small pressuredifferences to large l2 values

Pressure Measurement DevicesPressure Measurement DevicesPressure Measurement DevicesPressure Measurement DevicesPressure Measurement DevicesPressure Measurement DevicesPressure Measurement DevicesPressure Measurement DevicesBourdon pressure gageBourdon pressure gage

Measures gage pressureMeasures gage pressure Only static pressure (does not respond quickly)Only static pressure (does not respond quickly)Pressure acts on elastic structure, deforms it,Pressure acts on elastic structure, deforms it,

deformation is read (as pressure) on a dialdeformation is read (as pressure) on a dial

V3


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Pressure Measurement DevicesPressure Measurement DevicesPressure Measurement DevicesPressure Measurement DevicesPressure Measurement DevicesPressure Measurement DevicesPressure Measurement DevicesPressure Measurement DevicesPressure transducerPressure transducerProvide electrical outputProvide electrical outputBourdon gage withBourdon gage with LVDTLVDT Diaphragm transducerDiaphragm transducer

Fast responseFast response

4tf3 lecture 01 fall10

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