Lecture 3 - Pressure Distribution in Fluid

1Lecture3

Mechanics of Fluids 1: Lecture 3: Pressure Distribution in FluidDepartment of Mechanical Engineering MEHB223

PRESSURE PRESSURE DISTRIBUTION DISTRIBUTION

IN FLUIDIN FLUIDIN FLUIDIN FLUID


2ChapterSummary


FluidPressureataPoint;Absolute,VacuumandGaugePressure

PressureVariationwithElevation PressureMeasuringDevices Barometer,Open

EndManometer,DifferentialManometer HydrostaticParadoxy HydrostaticForcesonSubmergedSurfaces

Planesurface,CurvedSurfaces Buoyancy StabilityofImmersedandFloatingBodies

3.1Introduction

Fluids in general exerts both normal and shearing


Fluids,ingeneral,exertsbothnormalandshearingforcesonsurfacesincontact

Shearingforcesareproducedonlywithrelativemotion

Withoutrelativemotion~onlynormalforce=>calledPressureForces

No Relative Motion Means : NoRelativeMotionMeans: Stationary Movingwiththesameconstantvelocity Movingwiththesamevaryingvelocity

33.2FluidPressureAtAPoint


By balancing the forces in the vertical and horizontal directions : ~

Px = Pz = Pn

=> Pressure at a point is the same in all directions

=> PASCALs Law

=> The conclusion is also applicable if there is relative motion

3.3PressureTransmissionIn a closed system, due to Pascals L th h d d


Law, the pressure change produced at one point in the system will be transmitted throughout the entire systemPrinciple of the Hydraulic Lift

2

2

1

1

AF

AF =

4Example3.1A hydraulic jack has dimensions as shown. If one exerts a f h h dl f h k h l d


force or 100N on the handle of the jack, what load F2, can the jack support? Neglect lifter weight.

3.4Absolute,VacuumandGagePressure

Pressure normally measured


Pressurenormallymeasuredaspressuredifference

Thepressureaboveabsolutevacuum:AbsolutePressure

RelativePressureaboveatmosphericPressure:GagePPressure

RelativePressurebelowatmosphericPressure:VacuumPressure

53.5PressureVariationWithElevation

Balancing the force in s direction : -


Balancing the force in s direction :

But sin = dz/dl, so : -g

dzdP =

singddP =l

dz

For incompressible fluid, is constant : -Constant gz P =+ = Piezometric Pressure (Pascal)Constant z gP =+ = Piezometric Head (meter)

3.5PressureVariationWithElevation... IfA >B whichdistributioniscorrect?


If If BB > > AA which distribution is correct ?which distribution is correct ?

XXXX

63.5PressureVariationWithElevation... Therefore,forincompressiblefluid,inanystaticfluid

h h d h h h fl d


thepiezometricheadisconstantthroughoutthefluid Thusatanypointwithinthefluid:

( )Datum2211 z gP z gP z gP +=+=+1

datum

2

3 zz

z

3

21

3.5PressureVariationWithElevation... Example:Pressureatapoint,hbelow

the surface of a stationary fluid :


thesurfaceofastationaryfluid:

gh P P ah +=

Example : Case Involving 2 fluids. Example : Case Involving 2 fluids. Determine the gage pressure at the Determine the gage pressure at the bottom of the tankbottom of the tank

The equation has to be applied to each The equation has to be applied to each fluid separatelyfluid separately

73.5PressureVariationWithElevation...

CompressibleFluid(e.g.atmosphericair)


Troposphere : Troposphere : RTPgg

dzdP == Function of Temperature Function of Temperature

( )( ) Rgoo

o

oo

TzzTPP

zzTT

=

=

oT Stratosphere : Stratosphere :

( )RT

gzz

o

cons

o

ePP

TT=

=

3.6PressureMeasuringDevices

Barometer usetomeasureatmosphericpressure


Applying the equation from 1 to 2 : -

2m21m1 gz P gz P +=+P1 = Patm ; z2-z1 = h ; P2 = Pvap

83.6PressureMeasuringDevices...

Piezometer Consistsofsingleverticalt b t th t i t d i t i


tubeopenatthetopinsertedintoapipeunderpressurewhichrisesinthetubedependingonthepressure.Ifthetopofthetubeisopentoatmospherethenitreadsgagepressureatthatlocationinthepipe

ghP fgage =

3.6PressureMeasuringDevices... OpenendUtubeManometer:


lghgPP fmatm += 4If f


Differentialmanometer:


( ) ( )abggh P-P f2fm21 =Static pressure change from

A to B

Static pressure change due to

the system

Static pressure change due to difference in

elevation

3.6PressureMeasuringDevices... Example:Obtaintheexpressionforpressureoftheairin

thetank.


10

3.6PressureMeasuringDevices... Example:Thepistonshownweigh100N.Initsoriginalposition,the

pistonisrestrainedfrommovingtothebottomofthecylinderbymeanofametalstop.Assumingthereisneitherfrictionnoleakage


betweenthepistonandcylinder,whatvolumeofoil(S=0.85)wouldhavetobeaddedtothe1cmtubetocausethepistontorise1cmfromitsinitialposition?

1cm

6 cm

4 cm

4 cm

4 cm



11

3.7HydrostaticParadox Pressureexertedbyafluidisonlydependentontheverticalheadanditsdensity Itisindependentoftheweightofthefluidpresent


paradox

The pressures are the same although the weight of the liquids are obviously different

3.8HydrostaticForceonSubmergedSurfaces

Importance of Hydrostatic Force Calculations


p f yExamples : -

12

3.8.1HydrostaticForceonPlaneSurfaces


hh

dF=PdA yArea, AAtmospheric pressure is

ignored since both sides are open to atmosphere

Consider the magnitude of the hydrostatic force on one side

of the plane

x

y

dA

edge view

normal view

elemental area

centroid

AsinygAhgF__

h ==Hydrostatic Force = Pressure at the Centroid x Area

open to atmosphere

3.8.1 Hydrostatic Force on Plane Surfaces3.8.1 Hydrostatic Force on Plane Surfaces

- The resultant hydrostatic force acts through the CENTRE OF PRESSURE (COP)


h

Fh

- The resultant hydrostatic force acts through the CENTRE OF PRESSURE (COP)- The slanted distance of COP from the centroid, ycp, is determined by : -

Ay

Iy _xx

cp =

- And xcp, is given by : -

centroid

Area, A

Fh

h

ycpCP

Ay

Ix _

xycp =

- If the shape is symmetrical about y axis xcp is zero

13

3.8.1 Hydrostatic Force on Plane Surfaces3.8.1 Hydrostatic Force on Plane Surfaces- Centroid and Second Moment of Area, Ixx of regular shapes : -


This is given in This is given in Appendix . No Appendix . No Need to RememberNeed to Remember


Example 1 : Find the magnitude of the hydrostatic force and its line


w

Example 1 : Find the magnitude of the hydrostatic force and its line of action from the hinge. Calculate the force F applied at the middle of the gate required to hold the gate in its vertical position

h

hinge

Fgh

14

3.8.1 Hydrostatic Force on Plane Surfaces3.8.1 Hydrostatic Force on Plane SurfacesExample 2 : Find the magnitude of the force, P required to just start opening the 2m wide gate Neglect the weight of the gate


opening the 2m wide gate. Neglect the weight of the gate.


Example 3 : Find the magnitude of the force G required to just start


h

h

hinge

Example 3 : Find the magnitude of the force, G required to just start opening the gate. Neglect the weight of the gate.

h

w

hinge

G

15


Example 4 : An elliptical gate covers the end of a pipe 4m in


diameter. If the gate is hinged at the top, what normal force, F is required to open the gate when water is 8m deep above the top of the pipe and the pipe is open to atmosphere on the other side? Neglect the weight of the gate.

3.8.2 Hydrostatic Force on Curve Surfaces3.8.2 Hydrostatic Force on Curve Surfaces


h

elemental area, dAdF=hdA

v

C v x

y

l

H

dF

dF

dFy

x ds

Vertical projection

Area, A v

16


- Horizontal Component of the Force, Fx : -


x

proj_vertproj_vert_CentroidVV

_

x AAhgF ==_P

- Line of action of Fx : -

vv

_v_xx

v_cp

Ah

Iy =

- Vertical Component of the Force, Fy : -

SurfaceAbove Fluid of Weight== surface_abovey gF- Line of action of Fy is through the centroid of the Fluid Above the

surface


- The Resultant Hydrostatic Force, F is : - 22 FFF


C v

Centroid of fluid above

surface

CP of vertical

2y

2x FFF +=

Vertical projection

C v

Fy

FxyCPv

CP of vertical projection

F

Note the Horizontal and Vertical Component of the Force Acts From Different Points

17


- Water underneath the surface ?


FyCentroid of

imaginary fluid above surface

CP f ti l

- The force will be exactly of the same magnitude but now acts in the opposite direction.

CP of vertical projectionFy

F

- Need to use imaginary surface in order to calculate vertical component

surface_above_imaginaryy gF =


Example 1 : Surface AB is a circular arc with radius of 2m and a depth of a


m into the paper. The distance EB is 4m. The fluid above surface AB is water and atmospheric pressure prevails on the free surface of water and on the bottom side of surface AB. Find the magnitude and line of action of the hydrostatic force acting on surface AB.

18


Example 2 : Determine the hydrostatic force acting on this gate ?


C

DCB

Example 2 : Determine the hydrostatic force acting on this gate ?

DB

A

3.8.2 Hydrostatic Force on Curve Surfaces3.8.2 Hydrostatic Force on Curve SurfacesExample 3 : What force must be exerted through the bolts to hold the dome in place? The metal dome and pipe weigh 6 kN The dome has


dome in place? The metal dome and pipe weigh 6 kN. The dome has

no bottom. Here = 80 cm.

19

3.9 Bouyancy3.9 Bouyancy- Definition : Vertical Force on a body immersed in a stationary fluid.

It arises because the pressure varies with depth


A B

CDCOG

Vol4Vol2

Vol3

f

It arises because the pressure varies with depth.- Consider a body partly immersed in a fluid : -

FMCD

Mg

Vol1 bVol displaced of Weight =

== bouyancy1f FgMg

- Act through the centroid of the displace volume => Centre of Bouyancy

Archimedes Principle

3.9 Bouyancy3.9 Bouyancy

Example 1 :A metal part (2) is hanged by a thin cord from a floating d bl k ( ) h d bl k h f f d


wood block (1). The wood block has a specific gravity of 0.3 and dimension 50 x 50 x 10 mm. The metal part has volume 6600 mm3. Find the mass m2 of the metal part and the tension in the cord.

20


Example 2 : The partially submerged wood pole is attached to the ll b h h h l l b d h


wall by a hinge as shown. The pole is in equilibrium under the action of weight and buoyant forces. Determine the density of the wood?


Hydrometry .. Hydrometer Hydrometry .. Hydrometer


is a device use to measure the is a device use to measure the specific gravity of a liquidspecific gravity of a liquid

Based on the buoyancy Based on the buoyancy principleprinciple

The depth of the hydrometer is The depth of the hydrometer is dependent on the specific dependent on the specific gravity of the liquidgravity of the liquid

21

3.10 Stability of Immersed Body3.10 Stability of Immersed Body- Stability depends on the relative position of Centre of Gravity (COG) and Centre of Bouyancy (COB)


Centre of Bouyancy (COB)

- If COB > COG = > Stable

- If COB < COG = > Unstable

3.11 Stability of Floating Body3.11 Stability of Floating Body- The previous rule is not applicable to floating body because the

COB of displaced volume will change as the object is displaced : -


COB of displaced volume will change as the object is displaced :

- Thus more involve d analysis is needed .

22


End of End of Lecture Lecture 33

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Lecture 3 - Pressure Distribution in Fluid