FUNDAMENTALS OF FLUID MECHANICS Chapter 3 Fluids in ...

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1 FUNDAMENTALS OF FUNDAMENTALS OF FLUID MECHANICS FLUID MECHANICS Chapter 3 Fluids in Motion Chapter 3 Fluids in Motion - - The Bernoulli Equation The Bernoulli Equation Jyh Jyh - - Cherng Cherng Shieh Shieh Department of Bio Department of Bio - - Industrial Industrial Mechatronics Mechatronics Engineering Engineering National Taiwan University National Taiwan University 09/28/2009 09/28/2009

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    FUNDAMENTALS OFFUNDAMENTALS OFFLUID MECHANICSFLUID MECHANICS

    Chapter 3 Fluids in Motion Chapter 3 Fluids in Motion -- The Bernoulli EquationThe Bernoulli Equation

    JyhJyh--CherngCherng ShiehShiehDepartment of BioDepartment of Bio--Industrial Industrial MechatronicsMechatronics Engineering Engineering

    National Taiwan UniversityNational Taiwan University09/28/200909/28/2009

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    MAIN TOPICSMAIN TOPICS

    NewtonNewtons Second Laws Second LawF=ma Along a StreamlineF=ma Along a StreamlineF=ma Normal to a StreamlineF=ma Normal to a StreamlinePhysical Interpretation Physical Interpretation of of Bernoulli EquationStatic, Stagnation, Dynamic, and Total PressureStatic, Stagnation, Dynamic, and Total PressureApplication of the Bernoulli EquationApplication of the Bernoulli EquationThe Energy Line and the Hydraulic Grade LineThe Energy Line and the Hydraulic Grade LineRestrictions on Use of the Bernoulli EquationRestrictions on Use of the Bernoulli Equation

    Bernoulli Equation

    Bernoulli equationBernoulli equation

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    NewtonNewtons Second Law s Second Law 1/61/6

    As a fluid particle moves from one location to another, it As a fluid particle moves from one location to another, it experiences an acceleration or deceleration.experiences an acceleration or deceleration.

    According to According to NewtonNewtons second law of motions second law of motion, the net force acting , the net force acting on the fluid particle under consideration must equal its mass tion the fluid particle under consideration must equal its mass times mes its acceleration.its acceleration.

    F=maF=ma In this chapter, we consider the motion of In this chapter, we consider the motion of inviscidinviscid fluidsfluids. That is, . That is,

    the fluid is assumed to have the fluid is assumed to have zero viscosityzero viscosity. For such case, . For such case, it is it is possible to ignore viscous effects.possible to ignore viscous effects.

    The forces acting on the particle ? Coordinates used ?The forces acting on the particle ? Coordinates used ?

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    NewtonNewtons Second Law s Second Law 2/62/6

    The fluid motion is governed byThe fluid motion is governed byF= Net pressure force + Net gravity forceF= Net pressure force + Net gravity force

    To apply NewtonTo apply Newtons second law to a fluid, s second law to a fluid, an appropriate an appropriate coordinate system must be chosen to describe the coordinate system must be chosen to describe the motionmotion. In general, the motion will be three. In general, the motion will be three--dimensional dimensional and unsteady so that and unsteady so that three space coordinates and timethree space coordinates and timeare needed to describe it.are needed to describe it.

    ForceForce

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    NewtonNewtons Second Law s Second Law 3/63/6

    The most often used coordinate The most often used coordinate systems are systems are rectangular (x,y,z) rectangular (x,y,z) and cylindrical (r,and cylindrical (r,,z) system.,z) system.

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    Streamline?Streamline? Streamline coordinate!Streamline coordinate!

    As is done in the study of dynamics, the motion of each As is done in the study of dynamics, the motion of each fluid particle is described in terms of its velocity, V, which fluid particle is described in terms of its velocity, V, which is defined as the time rate of change of the position of the is defined as the time rate of change of the position of the particle.particle.

    The particleThe particles velocity is a vector quantity with a s velocity is a vector quantity with a magnitude and direction.magnitude and direction.

    As the particle moves about, it follows a particular path, As the particle moves about, it follows a particular path, the shape of which is governed by the velocity of the the shape of which is governed by the velocity of the particle.particle.

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    Streamline?Streamline? Streamline coordinate!Streamline coordinate!

    The location of the particle along the path is a function of The location of the particle along the path is a function of where the particle started at the initial time and its velocity where the particle started at the initial time and its velocity along the path.along the path.

    If it is If it is steady flowsteady flow, each successive particle that passes , each successive particle that passes through a given point will follow the same path.through a given point will follow the same path.

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    Streamline?Streamline? Streamline coordinate!Streamline coordinate!

    For such cases the path is a fixed line in the For such cases the path is a fixed line in the xx--zz plane.plane.Neighboring particles that pass on either side of point (1 Neighboring particles that pass on either side of point (1

    following their own paths, which may be of a different following their own paths, which may be of a different shape than the one passing through (1).shape than the one passing through (1).

    The entire The entire xx--zz plane is filled with such paths.plane is filled with such paths.

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    Streamline?Streamline? Streamline coordinate!Streamline coordinate!

    For steady flows each particle slides along its path, and its For steady flows each particle slides along its path, and its velocity is everywhere tangent to the path.velocity is everywhere tangent to the path.

    The lines that are tangent to the velocity vectors The lines that are tangent to the velocity vectors throughout the flow field are called throughout the flow field are called streamlines.streamlines.

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    Streamline?Streamline? Streamline coordinate!Streamline coordinate!

    For many situations it is easiest to describe the flow in For many situations it is easiest to describe the flow in terms of the terms of the streamlinestreamline coordinate based on the coordinate based on the streamlines. The particle motion is described in terms of streamlines. The particle motion is described in terms of its distance, its distance, s = s = s(ts(t),), along the streamline from some along the streamline from some convenient origin and the local radius of curvature of the convenient origin and the local radius of curvature of the streamline, streamline, R = R = R(sR(s).).

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    NewtonNewtons Second Law s Second Law 4/64/6

    In this chapter, the flow is confined to be In this chapter, the flow is confined to be twotwo--dimensional motion.dimensional motion.

    As is done in the study of dynamics, the motion of each As is done in the study of dynamics, the motion of each fluid particle is described in terms of its velocity vector fluid particle is described in terms of its velocity vector VV..

    As the particle moves, As the particle moves, it follows a particular path.it follows a particular path.The location of the particle along the path The location of the particle along the path is a function is a function of its initial position and velocity.of its initial position and velocity.

    LocationLocation

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    NewtonNewtons Second Law s Second Law 5/65/6

    For steady flowsFor steady flows, each particle slides along its path, and , each particle slides along its path, and its velocity vector is everywhere tangent to the path. The its velocity vector is everywhere tangent to the path. The lines that are tangent to the velocity vectors throughout the lines that are tangent to the velocity vectors throughout the flow field are called flow field are called streamlinesstreamlines..

    For such situation, the particle motion is described in For such situation, the particle motion is described in terms of its distance, s=s(t), along the streamline from terms of its distance, s=s(t), along the streamline from some convenient origin and the local radius of curvature some convenient origin and the local radius of curvature of the streamline, R=R(s).of the streamline, R=R(s).

    streamlinestreamline

    StreamlineStreamlines=s=s(ts(t))streamlinestreamlineR=R=R(sR(s) )

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    NewtonNewtons Second Law s Second Law 6/66/6

    The The distance along the streamline is related to the distance along the streamline is related to the particleparticles speeds speed by by VV==ds/dtds/dt, and the radius of curvature is , and the radius of curvature is related to shape of the streamline.related to shape of the streamline.

    The acceleration is the time rate of change of the velocity The acceleration is the time rate of change of the velocity of the particleof the particle

    The The components of accelerationcomponents of acceleration in the in the s and ns and n directiondirection

    nRVs

    dsdVVnasa

    dtVda

    2

    ns

    RVa

    dsdVVa

    2

    ns CHAPTER 04 CHAPTER 04

    s=s=s(ts(t)) VV==ds/dtds/dt

    VV==ds/dtds/dt

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    StreamlinesStreamlines

    Streamlines past an airfoilStreamlines past an airfoil

    Flow past a bikerFlow past a biker

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    F=ma along a Streamline F=ma along a Streamline 1/41/4

    Isolation of a small fluid particle in a flow field.fluid particle in a flow field.

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    F=ma along a Streamline F=ma along a Streamline 2/42/4

    Consider the small fluid particle of fluid particle of size of size of s by s by nn in the plane of the figure and y normal to the figure.

    For steady flow, the component of Newtons second law along the streamline direction s

    sVVV

    sVmVmaF SS

    Where represents the sum of the s components of all the force acting on the particle.

    SF

    ss

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    F=ma along a Streamline F=ma along a Streamline 3/43/4

    The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the streamline directionstreamline direction

    The The net pressure forcenet pressure force on the particle in the streamline on the particle in the streamline directiondirection

    sinVsinWWs

    Vspynp2ynppynppF SSSps

    VspsinFWF psss

    sasVV

    spsin

    Equation of motion Equation of motion along the streamline along the streamline directiondirection

    2s

    sppS

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    F=ma along a Streamline F=ma along a Streamline 4/44/4

    A change in fluid particle speed is accomplished by the A change in fluid particle speed is accomplished by the appropriate combination of pressure gradient and particle appropriate combination of pressure gradient and particle weight along the streamline.weight along the streamline.

    For fluid For fluid static situationstatic situation, the balance between pressure , the balance between pressure and gravity force is such that and gravity force is such that no change in particle speedno change in particle speedis produced.is produced.

    sasVV

    spsin

    0spsin

    IntegrationIntegrationParticle weightParticle weight

    pressure gradientpressure gradient

    particle weightparticle weightpressure gradientpressure gradient

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    IntegrationIntegration....

    sasVV

    spsin

    Rearranged and IntegratedRearranged and Integrated

    CgzV21dp

    0dzVd21dp

    dsdV

    21

    dsdp

    dsdz

    2

    22

    along a streamlinealong a streamlineWhere Where C is a constant of integration C is a constant of integration to be to be determined by the conditions at some point on determined by the conditions at some point on the streamline.the streamline.

    In general it is not possible to integrate the pressure term In general it is not possible to integrate the pressure term because the density may not be constant and, therefore, because the density may not be constant and, therefore, cannot be removed from under the integral sign.cannot be removed from under the integral sign.

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    Bernoulli Equation Along a StreamlineBernoulli Equation Along a Streamline

    For the special case of For the special case of incompressible flowincompressible flow

    Restrictions : Steady flow.Restrictions : Steady flow.Incompressible flow.Incompressible flow.Frictionless flow.Frictionless flow.Flow along a streamline.Flow along a streamline.

    ttanconsz2

    Vp2

    BERNOULLI EQUATIONBERNOULLI EQUATION

    CgzV21dp 2

    The Bernoulli equation is a very The Bernoulli equation is a very powerful tool in fluid mechanics, powerful tool in fluid mechanics, published by Daniel Bernoulli published by Daniel Bernoulli (1700~1782) in 1738.(1700~1782) in 1738.NONO

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    Example 3.1 Pressure Variation along A Example 3.1 Pressure Variation along A StreamlineStreamline Consider the Consider the inviscidinviscid, incompressible, steady flow, incompressible, steady flow along the along the

    horizontal streamline Ahorizontal streamline A--B in front of the sphere of radius a, as B in front of the sphere of radius a, as shown in Figure E3.1a. From a more advanced theory of flow past shown in Figure E3.1a. From a more advanced theory of flow past a a sphere, the fluid velocity along this streamline issphere, the fluid velocity along this streamline is

    Determine the pressure variation along the streamline from pDetermine the pressure variation along the streamline from point A oint A far in front of the sphere (far in front of the sphere (xxAA==-- and Vand VAA= V= V00) to point B on the ) to point B on the sphere (sphere (xxBB==--aa and Vand VBB=0)=0)

    3

    3

    0 xa1VV streamlinestreamline

    streamlinestreamline

    sinsin=0=0 s s xx

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    Example 3.1 Example 3.1 SolutionSolution1/21/2

    4

    3

    3

    32

    04

    30

    3

    3

    0 xa

    xa1V3

    xaV3

    xa1V

    xVV

    sVV

    The equation of motion along the streamline (The equation of motion along the streamline (sinsin=0)=0)

    The acceleration term

    sVV

    sp

    (1)(1) sasVV

    spsin

    The pressure gradient along the streamline is

    4

    3320

    3

    xx/a1Va3

    xp

    sp

    (2)(2)

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    Example 3.1 Example 3.1 SolutionSolution2/22/2

    The pressure gradient along the streamline

    4

    3320

    3

    xx/a1Va3

    xp

    sp

    (2)(2)

    The pressure distribution along the streamline

    2)x/a(

    xaVp

    632

    0

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    Example 3.2 The Bernoulli EquationExample 3.2 The Bernoulli Equation

    Consider the flow of air around a bicyclist moving through stillConsider the flow of air around a bicyclist moving through still air air with with velocity Vvelocity V00, as is shown in Figure E3.2. Determine the , as is shown in Figure E3.2. Determine the difference in the pressure between points (1) and (2).difference in the pressure between points (1) and (2).

    point (1)point (1)point (2)point (2)

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    Example 3.2 Example 3.2 SolutionSolution

    2V

    2Vpp

    20

    21

    12

    The Bernoullis equation applied along the streamline that passes along the streamline that passes through (1) and (2)through (1) and (2)

    z1=z2

    2

    22

    21

    21

    1 z2Vpz

    2Vp

    (1) is in the free stream VV11=V=V00(2) is at the tip of the bicyclists nose V2=0

    Bernoulli equationBernoulli equationpoint (1)point (1)point (2)point (2)streamlinestreamline

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    F=ma Normal to a StreamlineF=ma Normal to a Streamline1/21/2

    For steady flow, the For steady flow, the component of Newtoncomponent of Newtons s second law in the second law in the normal normal direction ndirection n

    RVV

    RmVF

    22

    n

    Where represents the Where represents the sum of the n components of all sum of the n components of all the force acting on the particle.the force acting on the particle.

    nF

    HydrocycloneHydrocyclone separatorseparator

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    F=ma Normal to a StreamlineF=ma Normal to a Streamline2/22/2

    The The gravity force (weight)gravity force (weight) on the particle in the on the particle in the normal normal directiondirection

    The The net pressure forcenet pressure force on the particle in the on the particle in the normal normal directiondirection

    cosVcosWWn

    Vnpysp2ys)pp(ysppF nnnpn

    VRVV

    npcosFWF

    2

    pnnn

    RV

    npcos

    2

    Equation of motion Equation of motion normal to the streamlinenormal to the streamlineNormal directionNormal direction

    2n

    nppn

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    IntegrationIntegration....

    RV

    dndp

    dndz 2

    RV

    npcos

    2

    RearrangedRearranged

    Normal to Normal to the streamlinethe streamline

    In general it is not possible to In general it is not possible to integrate the pressure term because integrate the pressure term because the density may not be constant and, the density may not be constant and, therefore, cannot be removed from therefore, cannot be removed from under the integral sign.under the integral sign.

    IntegratedIntegrated

    A change in the direction of flow of a fluid particle is A change in the direction of flow of a fluid particle is accomplished by the appropriate combination of pressure accomplished by the appropriate combination of pressure gradient and particle weight normal to the streamlinegradient and particle weight normal to the streamline

    CgzdnRVdp 2

    Without knowing the n dependent Without knowing the n dependent in V=in V=V(s,nV(s,n) and R=) and R=R(s,nR(s,n) this ) this integration cannot be completed.integration cannot be completed.

    ParticleParticle weightweight

    pressure gradientpressure gradient

    V=V=V(s,nV(s,n) ) R=R=R(s,nR(s,n))

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    Bernoulli Equation Normal to a StreamlineBernoulli Equation Normal to a Streamline

    For the special case of For the special case of incompressible flowincompressible flow

    Restrictions : Steady flow.Restrictions : Steady flow.Incompressible flow.Incompressible flow.Frictionless flow. Frictionless flow. NO shear forceNO shear forceFlow normal to a streamline.Flow normal to a streamline.

    CzdnRVp

    2

    BERNOULLI EQUATIONBERNOULLI EQUATION

    CgzdnRVdp 2

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    If gravity is neglectedFree vortex

    A larger speed or density or smaller radius of curvature of A larger speed or density or smaller radius of curvature of the motion required a larger force unbalance to produce the the motion required a larger force unbalance to produce the motion.motion.

    If gravity is neglected or if the flow is in a horizontalIf gravity is neglected or if the flow is in a horizontal

    RV

    dndp

    dndz 2

    RV

    dndp 2

    Pressure increases with distance away from the center Pressure increases with distance away from the center of curvature. Thus, the pressure outside a tornado is of curvature. Thus, the pressure outside a tornado is larger than it is near the center of the tornado.larger than it is near the center of the tornado.

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    Aircraft wing tip vortexAircraft wing tip vortex

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    Example 3.3 Pressure Variation Normal to Example 3.3 Pressure Variation Normal to a Streamlinea Streamline Shown in Figure E3.3a and E3.3b are two flow fields with circulaShown in Figure E3.3a and E3.3b are two flow fields with circular r

    streamlines. The streamlines. The velocity distributionsvelocity distributions are are

    )b(r

    C)r(V)a(rC)r(V 21

    Assuming the flows are steady, inviscid, and incompressible with streamlines in the horizontal plane (dz/dn=0).

    RV

    npcos

    2

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    Example 3.3 Example 3.3 SolutionSolution

    020221 prrC21p

    rV

    rp 2

    For flow in the horizontal plane (dz/dn=0). The streamlines are circles /n=-/rThe radius of curvature R=r

    For case (a) this gives

    rCrp 2

    1

    3

    22

    rC

    rp

    For case (b) this gives

    0220

    22 pr

    1r1C

    21p

    RV

    dndp

    dndz 2

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    Physical InterpreterPhysical Interpreter1/21/2

    Under the basic assumptions: Under the basic assumptions: the flow is steady and the fluid is the flow is steady and the fluid is inviscidinviscid and incompressible.and incompressible.

    Application of F=ma and integration of equation of motion along Application of F=ma and integration of equation of motion along and normal to the streamline result inand normal to the streamline result in

    To To produce an acceleration, there must be an unbalance of produce an acceleration, there must be an unbalance of the resultant forcethe resultant force, , of which only pressure and gravity were of which only pressure and gravity were considered to be importantconsidered to be important. Thus, there are three process . Thus, there are three process involved in the flow involved in the flow mass times acceleration (the mass times acceleration (the VV22/2 term), /2 term), pressure (the p term), and weight (the pressure (the p term), and weight (the z term).z term).

    CzdnRVp

    2

    Cz2Vp

    2

    ppz z

    RVa

    dsdVVa

    2

    ns

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    Physical InterpreterPhysical Interpreter2/22/2

    The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of The The work done on a particle of all force acting on the particle is ework done on a particle of all force acting on the particle is equal qual to the change of the kinetic energy of the particleto the change of the kinetic energy of the particle..Work done by force : Work done by force : FFdd..Work done by weight: Work done by weight: z z Work done by pressure force: pWork done by pressure force: p

    Kinetic energy: Kinetic energy: VV22/2/2

    CdnRVzp

    2

    C2Vzp

    2

    Based onBased on

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    HeadHead

    An alternative but equivalent form of the Bernoulli An alternative but equivalent form of the Bernoulli equation is obtained by dividing each term by equation is obtained by dividing each term by

    czg2

    VP 2

    Pressure HeadPressure Head

    Velocity HeadVelocity Head

    Elevation HeadElevation Head

    The Bernoulli Equation can be written in The Bernoulli Equation can be written in terms of heights called headsterms of heights called heads

    streamlinestreamline

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    Example 3.4 Kinetic, Potential, and Example 3.4 Kinetic, Potential, and Pressure EnergyPressure Energy Consider the flow of water from the syringe Consider the flow of water from the syringe

    shown in Figure E3.4. A force applied to the shown in Figure E3.4. A force applied to the plunger will produce a pressure greater than plunger will produce a pressure greater than atmospheric at point (1) within the syringe. atmospheric at point (1) within the syringe. The water flows from the needle, point (2), The water flows from the needle, point (2), with relatively high velocity and coasts up to with relatively high velocity and coasts up to point (3) at the top of its trajectory. Discuss point (3) at the top of its trajectory. Discuss the energy of the fluid at point (1), (2), and (3) the energy of the fluid at point (1), (2), and (3) by using the Bernoulli equation. by using the Bernoulli equation.

    point (1) (2) (3)point (1) (2) (3)

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    Example 3.4 Solution

    The sum of the three types of energy (kinetic, potential, and prThe sum of the three types of energy (kinetic, potential, and pressure) essure) or heads (velocity, elevation, and pressure) must remain constanor heads (velocity, elevation, and pressure) must remain constant. t.

    The pressure gradient between (1) and (2) The pressure gradient between (1) and (2) produces an acceleration to eject the water produces an acceleration to eject the water form the needle. Gravity acting on the form the needle. Gravity acting on the particle between (2) and (3) produces a particle between (2) and (3) produces a deceleration to cause the water to come to deceleration to cause the water to come to a momentary stop at the top of its flight.a momentary stop at the top of its flight.

    streamlinethealongttanconszV21p 2

    The motion results in a change in the magnitude of each type of energy as the fluid flows from one location to another.

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    Example 3.5 Pressure Variation in a Example 3.5 Pressure Variation in a Flowing StreamFlowing Stream Consider the Consider the inviscidinviscid, incompressible, steady flow shown in Figure , incompressible, steady flow shown in Figure

    E3.5. From section A to B the streamlines are straight, while frE3.5. From section A to B the streamlines are straight, while from C om C to D they follow circular paths. to D they follow circular paths. Describe the pressure variation Describe the pressure variation between points (1) and (2)and points(3) and (4)between points (1) and (2)and points(3) and (4)

    point (1)(2)point (1)(2) point(3)(4)point(3)(4)

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    Example 3.5 Example 3.5 SolutionSolution1/21/2

    1221221 rhp)zz(rpp

    ttanconsrzp

    R= , , for the portion from for the portion from A to BA to B

    Using p2=0,z1=0,and z2=h2-1

    Since the radius of curvature of the streamline is infinite, the pressure variation in the vertical direction is the same as if the fluids were stationary.

    Point (1)~(2)

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    4

    3

    z

    z

    2

    343 dzRVrhp

    With pp44=0 and z=0 and z44--zz33=h=h44--33 ,this becomes

    334z

    z

    2

    4 rzprz)dz(RVp

    4

    3

    Example 3.5 Example 3.5 SolutionSolution2/22/2

    Point (3)~(4)

    For the portion from For the portion from C to DC to D

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    Static, Stagnation, Dynamic, and Static, Stagnation, Dynamic, and Total PressureTotal Pressure1/51/5Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a

    form of pressure.form of pressure.

    pp is the actual thermodynamic pressure of the fluid as it is the actual thermodynamic pressure of the fluid as it flows. To measure this pressure, one must move along flows. To measure this pressure, one must move along with the fluid, thus being with the fluid, thus being staticstatic relative to the moving relative to the moving fluid. Hence, it is termed the fluid. Hence, it is termed the static pressurestatic pressure seen by the seen by the fluid particle as it movesfluid particle as it moves. .

    C2

    Vzp2

    Each term can be interpreted as a form of pressure

    ppstaticstaticstatic pressurestatic pressure

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    Static, Stagnation, Dynamic, and Static, Stagnation, Dynamic, and Total PressureTotal Pressure2/52/5The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a

    wall pressure wall pressure taptap, or a static pressure probe., or a static pressure probe.hhhphp 34133131 The static pressureThe static pressure

    zz is termed the is termed the hydrostatic hydrostatic pressurepressure. It is not actually a . It is not actually a pressure but does represent the pressure but does represent the change in pressure possible due change in pressure possible due to potential energy variations of to potential energy variations of the fluid as a result of elevation the fluid as a result of elevation changes.changes.

    wall wall pressurepressuretaptapstatic static pressurepressure

    zz

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    Static, Stagnation, Dynamic, and Static, Stagnation, Dynamic, and Total PressureTotal Pressure3/53/5VV22/2/2 is termed the is termed the dynamic pressuredynamic pressure. It can be interpreted . It can be interpreted

    as the pressure at the end of a small tube inserted into the as the pressure at the end of a small tube inserted into the flow and pointing upstream. flow and pointing upstream. After the initial transient After the initial transient motion has died out, the liquid will fill the tube to a height motion has died out, the liquid will fill the tube to a height of H.of H.

    The fluid in the tube, including that at its tip (2), will be The fluid in the tube, including that at its tip (2), will be stationary. That is, Vstationary. That is, V22=0, or point (2) is a stagnation point.=0, or point (2) is a stagnation point.

    2112 V2

    1pp Stagnation pressure

    Static pressure Dynamic pressureDynamic pressure

    VV22/2/222HH

    1122

    sstagnationtagnation pressurepressure

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    Stagnation pointStagnation point

    Stagnation point flowStagnation point flow

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    Static, Stagnation, Dynamic, and Static, Stagnation, Dynamic, and Total PressureTotal Pressure4/54/5There is a stagnation point on any stationary body that is There is a stagnation point on any stationary body that is

    placed into a flowing fluid. Some of the fluid flows placed into a flowing fluid. Some of the fluid flows overoverand some and some underunder the object.the object.

    The dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the body.terminates at the stagnation point on the body.

    Neglecting the elevation Neglecting the elevation effects, effects, the stagnation the stagnation pressure is the largest pressure is the largest pressure obtainable along a pressure obtainable along a given streamlinegiven streamline..

    stagnation pointstagnation point

    stagnation pointstagnation point

    stagnation pointstagnation point

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    Static, Stagnation, Dynamic, and Static, Stagnation, Dynamic, and Total PressureTotal Pressure5/55/5The The sum of the static pressure, dynamic pressure, and sum of the static pressure, dynamic pressure, and

    hydrostatic pressure is termed the total pressurehydrostatic pressure is termed the total pressure..The Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total

    pressure remains constant along a streamline.pressure remains constant along a streamline.

    ttanconspz2

    Vp T2

    Total pressureTotal pressurestreamlinestreamline

    TTootaltal pressurepressure

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    The The PitotPitot--static Tube static Tube 1/51/5

    /)pp(2V

    2/Vpp

    pppzz

    2/Vppp

    43

    243

    14

    41

    232

    Knowledge of the values of the static and Knowledge of the values of the static and stagnation pressure in a fluid implies that the stagnation pressure in a fluid implies that the fluid speed can be calculated.fluid speed can be calculated.

    This is This is the principle on which the the principle on which the PitotPitot--static tube is based.static tube is based.

    Static pressureStatic pressure

    Stagnation pressureStagnation pressure

    PitotPitot--static static stubesstubes measure measure fluid velocity by converting fluid velocity by converting velocity into pressure.velocity into pressure.

    streamlinestreamlinetotal pressuretotal pressure

    staticstaticstagnation stagnation pressurepressure

  • 49

    Airplane Airplane PitotPitot--static probestatic probe

    Airspeed indicatorAirspeed indicator

  • 50

    The The PitotPitot--static Tube static Tube 2/52/5

  • 51

    The The PitotPitot--static Tube static Tube 3/53/5

    The use of The use of pitotpitot--static tube depends on the ability to static tube depends on the ability to measure the static and stagnation pressure.measure the static and stagnation pressure.

    An accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluids kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurement.pressure rise at the point of measurement.

    This requires a smooth hole with no burrs or imperfections.This requires a smooth hole with no burrs or imperfections.

    Incorrect and correct design of static pressure taps.Incorrect and correct design of static pressure taps.

    TapsTaps

  • 52

    The The PitotPitot--static Tube static Tube 4/54/5

    Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tube.static tube.

    The pressure along the surface of an object varies from the The pressure along the surface of an object varies from the stagnation pressure at its stagnation point to value that stagnation pressure at its stagnation point to value that may be less than free stream static pressure.may be less than free stream static pressure.

    It is important that the pressure taps be properly located to It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static ensure that the pressure measured is actually the static pressure.pressure.tapstapsstatic pressurestatic pressure

  • 53

    The The PitotPitot--static Tube static Tube 5/55/5

    Three pressure taps are drilled into a small circular Three pressure taps are drilled into a small circular cylinder, fitted with small tubes, and connected to three cylinder, fitted with small tubes, and connected to three pressure transducers. pressure transducers. The cylinder is rotated until the The cylinder is rotated until the pressures in the two side holes are equalpressures in the two side holes are equal, thus indicating , thus indicating that the center hole points directly upstream.that the center hole points directly upstream.

    21

    12

    31

    PP2V

    PP

    Directional-finding Pitot-static tube.

    If =0PP11=P=P33

    PitotPitot--static tubestatic tube

  • 54

    Example 3.6 Example 3.6 PitotPitot--Static TubeStatic Tube

    An airplane flies 100mi/hr at an elevation of 10,000 ft in a An airplane flies 100mi/hr at an elevation of 10,000 ft in a standard atmosphere as shown in Figure E3.6. Determine standard atmosphere as shown in Figure E3.6. Determine the pressure at point (1) far ahead of the airplane, point (2), the pressure at point (1) far ahead of the airplane, point (2), and the pressure difference indicated by a and the pressure difference indicated by a PitotPitot--static static probe attached to the fuselage. probe attached to the fuselage.

    PitotPitot--static tubestatic tube

  • 55

    Example 3.6 Example 3.6 Solution Solution 1/21/2

    2Vpp

    21

    12

    psia11.10)abs(ft/lb1456p 21

    The static pressure and density at the altitude

    If the flow is steady, inviscid, and incompressible and elevation changes are neglected. The Bernoulli equation

    3ft/slug001756.0

    With V1=100mi/hr=146.6ft/s and V2=0

    )abs(ft/lb)9.181456(

    2/)s/ft7.146)(ft/slugs001756.0(ft/lb1456p2

    222322

    10,000ft10,000ft

  • 56

    psi1313.0ft/lb9.18p 22

    In terms of gage pressure

    psi1313.02Vpp

    21

    12

    The pressure difference indicated by the Pitot-static tube

    Example 3.6 Example 3.6 Solution Solution 2/22/2

  • 57

    Application of Bernoulli Equation Application of Bernoulli Equation 1/21/2

    The Bernoulli equation can be appliedThe Bernoulli equation can be applied between any two between any two points on a streamline providedpoints on a streamline provided that the other that the other three three restrictionsrestrictions are satisfied. The result isare satisfied. The result is

    2

    22

    21

    21

    1 z2Vpz

    2Vp

    Restrictions : Steady flow.Restrictions : Steady flow.Incompressible flow.Incompressible flow.Frictionless flow.Frictionless flow.Flow along a streamline.Flow along a streamline.

    Bernoulli equationBernoulli equation

    streamlinestreamline

    Bernoulli equationBernoulli equation

  • 58

    Application of Bernoulli Equation Application of Bernoulli Equation 2/22/2

    Free jet.Free jet.Confined flow.Confined flow.FlowrateFlowrate measurementmeasurement

    Bernoulli equationBernoulli equation

  • 59

    Free Jets Free Jets 1/31/3

    Application of the Bernoulli equation between points (1) Application of the Bernoulli equation between points (1) and (2) on the streamlineand (2) on the streamline

    gh2h2V

    2Vh

    2

    At point (5)

    )Hh(g2V

    2

    22

    21

    21

    1 z2Vpz

    2Vp

    pp11=p=p22=0=0 zz11=h=hzz22=0=0VV11=0=0

    pp11=p=p55=0=0 zz11==h+Hh+Hzz22=0=0VV11=0=0

  • 60

    Free JetsFree Jets

    Flow from a tankFlow from a tank

  • 61

    Free Jets Free Jets 2/32/3

    For the For the horizontal nozzlehorizontal nozzle, the , the velocity at the centerline, Vvelocity at the centerline, V22, , will be greater than that at the will be greater than that at the top Vtop V11..

    In general, d

  • 62

    Free Jets Free Jets 3/33/3

    Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configuration.configuration.

    The diameter of a fluid jet is The diameter of a fluid jet is often smaller than that of the often smaller than that of the hole from which it flows.hole from which it flows.

    Define Cc = contraction coefficient Define Cc = contraction coefficient

    h

    jc A

    AC

    AjAj=area of the jet at the vena =area of the jet at the vena contractacontractaAh=area of the holeAh=area of the hole

    2

    22

    21

    21

    1 z2Vpz

    2Vp

    flow patternflow pattern

  • 63

    Example 3.7 Flow From a TankExample 3.7 Flow From a Tank GravityGravity

    A stream of water of diameter d = 0.1m flows steadily from a tanA stream of water of diameter d = 0.1m flows steadily from a tank k of Diameter D = 1.0m as shown in Figure E3.7a. of Diameter D = 1.0m as shown in Figure E3.7a. Determine the Determine the flowrateflowrate, Q,, Q, needed from the inflow pipe if needed from the inflow pipe if the water depth remains the water depth remains constantconstant, h = 2.0m., h = 2.0m.

    flowrateflowrate QQ

  • 64

    Example 3.7 Example 3.7 SolutionSolution1/21/2

    22

    221

    2

    11 zV21pzV

    21p

    22

    21 V2

    1ghV21

    The Bernoulli equation applied between points The Bernoulli equation applied between points (1) and (2)(1) and (2) isis

    (1)(1)

    With With pp11 = p= p22 = 0, z= 0, z11 = h, and z= h, and z22 = 0= 0

    (2)(2)

    For steady and incompressible flow, For steady and incompressible flow, conservation of mass requiresconservation of mass requiresQQ11 = Q= Q22, where Q = AV. Thus, A, where Q = AV. Thus, A11VV11 =A=A22VV2 2 , or , or

    22

    12 Vd

    4VD

    4

    2

    22

    21

    21

    1 z2Vpz

    2Vp

    22

    1 V)Dd(V (3)

    Bernoulli equationBernoulli equationpoint(1)point(1)(2)(2)

    Point(1)Point(1)(2)(2)

  • 65

    Example 3.7 Example 3.7 SolutionSolution2/22/2

    s/m26.6)m1/m1.0(1

    )m0.2)(s/m81.9(2)D/d(1

    gh2V4

    2

    42

    Combining Equation 2 and 3

    Thus,

    s/m0492.0)s/m26.6()m1.0(4

    VAVAQ 322211

    VV110 (Q) vs. V0 (Q) vs. V110 (Q0 (Q00))

    4

    4

    2

    2

    0 )/(11

    2])/(1/[2

    DdghDdgh

    VV

    QQ

    D

  • 66

    Example 3.8 Flow from a TankExample 3.8 Flow from a Tank--PressurePressure

    Air flows steadily from a tank, through a hose of diameter Air flows steadily from a tank, through a hose of diameter D=0.03m and exits to the atmosphere from a nozzle of D=0.03m and exits to the atmosphere from a nozzle of diameter d=0.01m as shown in Figure E3.8. diameter d=0.01m as shown in Figure E3.8. The pressure The pressure in the tank remains constant at 3.0kPa (gage)in the tank remains constant at 3.0kPa (gage) and the and the atmospheric conditions are standard temperature and atmospheric conditions are standard temperature and pressure. pressure. Determine the Determine the flowrateflowrate and the pressure in and the pressure in the hose.the hose.

    flowrateflowrate Point(2)Point(2)

  • 67

    Example 3.8 Example 3.8 SolutionSolution1/21/2

    32332

    2221

    211 zV2

    1pzV21pzV

    21p

    2212

    13 V2

    1ppandp2V

    For steady, For steady, inviscidinviscid, and incompressible flow, the Bernoulli equation , and incompressible flow, the Bernoulli equation along the streamlinealong the streamline

    With zWith z11 =z=z22 = z= z33 , V, V11 = 0, and p= 0, and p33=0=0

    (1)(1)

    The density of the air in the tank is obtained from the perfect The density of the air in the tank is obtained from the perfect gas law gas law

    33

    2

    1

    m/kg26.1K)27315)(Kkg/mN9.286(

    kN/N10]m/kN)1010.3[(RTp

    Bernoulli equationBernoulli equationpoint(1)(2)point(1)(2)(3)(3)

    Point(1)Point(1)(2)(2)((33))

  • 68

    Example 3.8 Example 3.8 SolutionSolution2/22/2

    s/m00542.0Vd4

    VAQ 332

    33

    s/m0.69m/kg26.1

    )m/N100.3(2p2V 323

    13

    Thus, Thus,

    oror

    The pressure within the hose can be obtained from The pressure within the hose can be obtained from EqEq. 1. 1and the continuity equationand the continuity equation

    s/m67.7A/VAV,HenceVAVA 23323322

    22

    23232212

    m/N2963m/N)1.373000(

    )s/m67.7)(m/kg26.1(21m/N100.3V

    21pp

  • 69

    Example 3.9 Flow in a Variable Area PipeExample 3.9 Flow in a Variable Area Pipe

    Water flows through a pipe reducer as is shown in Figure E3.9. TWater flows through a pipe reducer as is shown in Figure E3.9. The he static pressures at (1) and (2) are measured by the inverted Ustatic pressures at (1) and (2) are measured by the inverted U--tube tube manometer containing oil of specific gravity, SG, less than one.manometer containing oil of specific gravity, SG, less than one.Determine the manometer reading, h.Determine the manometer reading, h.

    reading hreading h

  • 70

    Example 3.9 Example 3.9 SolutionSolution1/21/2

    2211 VAVAQ

    For steady, For steady, inviscidinviscid, incompressible flow, the Bernoulli equation along , incompressible flow, the Bernoulli equation along the streamlinethe streamline

    The continuity equationThe continuity equation

    Combining these two equations Combining these two equations

    (1)(1)

    22221

    211 zpV2

    1pzpV21p

    ])A/A(1[pV21)zz(pp 212

    221221

  • 71

    Example 3.9 Example 3.9 SolutionSolution2/22/2

    h)SG1()zz(pp 1221

    2121 phSGh)zz(p

    2

    1

    222 A

    A1pV21h)SG1(

    This pressure difference is measured by the manometer and determThis pressure difference is measured by the manometer and determine ine by using the pressureby using the pressure--depth ideas developed in depth ideas developed in Chapter 2Chapter 2..

    oror(2)(2)

    SG1g2)A/A(1A/Qh

    2122

    2

    Since VSince V22=Q/A=Q/A22

    - +

    be independent of be independent of

    Point(1)(2)Point(1)(2)

  • 72

    Confined Flows Confined Flows 1/41/4

    When the fluid is physically constrained within a device, When the fluid is physically constrained within a device, its pressure cannot be prescribed a priori as was done for its pressure cannot be prescribed a priori as was done for the free jet.the free jet.

    Such casesSuch cases include include nozzle and pipesnozzle and pipes of of various diametervarious diameterfor which the fluid velocity changes because the flow area for which the fluid velocity changes because the flow area is different from one section to another.is different from one section to another.

    For such situations, it is necessary to use the concept of For such situations, it is necessary to use the concept of conservation of mass (the continuity equation) along with conservation of mass (the continuity equation) along with the Bernoulli equation.the Bernoulli equation.

    Tools: Bernoulli equation + Continuity equation

    devicedevicenozzlenozzlepipepipe

  • 73

    Confined Flows Confined Flows 2/42/4

    Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outlet.one inlet and one outlet.

    Conservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow, the continuity equation is, the continuity equation is

    222111 VAVA

    212211 QQVAVA

  • 74

    Confined Flows Confined Flows 3/43/4

    If the fluid velocity is increased, If the fluid velocity is increased, the pressure will decrease.the pressure will decrease.

    This pressure decrease can be This pressure decrease can be large enough so that the large enough so that the pressure in the liquid is reduced pressure in the liquid is reduced to its to its vapor pressure.vapor pressure.

    Pressure variation and cavitationin a variable area pipe.

    vapor pressurevapor pressure

    VenturiVenturi channelchannel

  • 75

    Confined Flows Confined Flows 4/4 example of 4/4 example of cavitationcavitation

    A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hose.garden hose. If the hose is If the hose is kinked,kinked, a restriction in the a restriction in the flow area will result.flow area will result.

    The water The water velocityvelocity through the restriction will be through the restriction will be relatively large.relatively large.

    With a sufficient amount of restriction the sound of the With a sufficient amount of restriction the sound of the flowing water will change flowing water will change a definite a definite hissinghissing sound will sound will be produced.be produced.

    The sound is a result of The sound is a result of cavitationcavitation..

  • 76

    Damage from Damage from CavitationCavitation

    Cavitation from propeller

    bubblebubble

  • 77

    Example 3.10 Siphon and Example 3.10 Siphon and CavitationCavitation

    Water at 60Water at 60 is siphoned from a large tank through a constant is siphoned from a large tank through a constant diameter hose as shown in Figure E3.10. Determine the maximum diameter hose as shown in Figure E3.10. Determine the maximum height of the hill, H, over which the water can be siphoned withheight of the hill, H, over which the water can be siphoned without out cavitationcavitation occurring. The end of the siphon is 5 ft below the bottom occurring. The end of the siphon is 5 ft below the bottom of the tank. Atmospheric pressure is 14.7 of the tank. Atmospheric pressure is 14.7 psiapsia..

    The value of H is a function of both the The value of H is a function of both the specific weight of the fluid, specific weight of the fluid, , and its , and its vapor pressure, vapor pressure, ppvv..

    Max. HMax. Hcavitationcavitation

  • 78

    Example 3.10 Example 3.10 SolutionSolution1/21/2

    32332

    2221

    211 zV2

    1pzV21pzV

    21p

    For ready, For ready, inviscidinviscid, and incompressible flow, the Bernoulli equation , and incompressible flow, the Bernoulli equation along the streamline from along the streamline from (1) to (2) to (3)(1) to (2) to (3)

    With With zz1 1 = 15 ft, z= 15 ft, z22 = H, and z= H, and z33 = = --5 ft. Also, V5 ft. Also, V11 = 0 (large tank), p= 0 (large tank), p11 = 0 = 0 (open tank), p(open tank), p33 = 0 (free jet), and from the continuity equation A= 0 (free jet), and from the continuity equation A22VV22 = = AA33VV33, or because the hose is constant diameter V, or because the hose is constant diameter V22 = V= V33.. The speed of The speed of the fluid in the hose is determined from the fluid in the hose is determined from EqEq. 1 to be. 1 to be

    (1)(1)

    22

    313 Vs/ft9.35ft)]5(15)[s/ft2.32(2)zz(g2V (1)(3)(1)(3)

    (1)(2)(1)(2)

  • 79

    Example 3.10 Example 3.10 SolutionSolution2/22/2

    22212

    221

    2112 V2

    1)zz(zV21zV

    21pp

    233222 )s/ft9.35)(ft/slugs94.1(21ft)H15)(ft/lb4.62()ft/.in144)(.in/lb4.14(

    Use of Use of EqEq. 1 . 1 between point (1) and (2) then gives the pressure pbetween point (1) and (2) then gives the pressure p22 at the at the toptop of the hill asof the hill as

    The The vapor pressure of water at 60vapor pressure of water at 60 is 0.256 is 0.256 psiapsia. . Hence, for incipient Hence, for incipient cavitationcavitation the lowest pressure in the system will be p = 0.256 the lowest pressure in the system will be p = 0.256 psiapsia..UUsing gage pressure: sing gage pressure: pp11 = 0, p= 0, p22=0.256 =0.256 14.7 = 14.7 = --14.4 14.4 psipsi

    (2)(2)

    ftH 2.28

  • 80

    FlowrateFlowrate Measurement Measurement inin pipes 1/5pipes 1/5

    Various flow meters are Various flow meters are governed by the governed by the Bernoulli Bernoulli and continuity equationsand continuity equations..

    2211

    222

    211

    VAVAQ

    V21pV

    21p

    ])A/A(1[)pp(2AQ 212

    212

    The theoretical The theoretical flowrateflowrate

    Typical devices for measuring Typical devices for measuring flowrateflowrate in pipesin pipes

    Flow metersFlow meters

  • 81

    Example 3.11 Example 3.11 VenturiVenturi MeterMeter

    Kerosene (SG = 0.85) flows through the Kerosene (SG = 0.85) flows through the VenturiVenturi meter shown in meter shown in Figure E3.11 with Figure E3.11 with flowratesflowrates between 0.005 and 0.050 mbetween 0.005 and 0.050 m33/s. /s. Determine the range in pressure difference, Determine the range in pressure difference, pp11 pp22, needed to , needed to measure these measure these flowratesflowrates..

    Known Q, Determine pKnown Q, Determine p11--pp22

    point(1)(2)point(1)(2)

  • 82

    Example 3.11 Example 3.11 SolutionSolution1/21/2

    33O2H kg/m850)kg/m1000(85.0SG

    22

    2

    A2])A/A(1[Q

    pp2

    12

    21

    For steady, For steady, inviscidinviscid, and incompressible flow, the relationship between , and incompressible flow, the relationship between flowrateflowrate and pressure and pressure

    The density of the flowing fluidThe density of the flowing fluid

    The area ratioThe area ratio

    36.0)m10.0/m006.0()D/D(/AA 221212

    ])A/A(1[)pp(2AQ 212

    212

    EqEq. 3.20. 3.20

  • 83

    Example 3.11 Example 3.11 SolutionSolution2/22/2

    The pressure difference for the The pressure difference for the smallest smallest flowrateflowrate isis

    The pressure difference for the The pressure difference for the largest largest flowrateflowrate isis

    22

    22

    21 ])m06.0)(4/[(2)36.01()850)(05.0(pp

    kPa116N/m1016.1 25

    kPa16.1N/m1160])m06.0)(4/[(2

    )36.01()kg/m850()/sm005.0(pp2

    22

    2323

    21

    kPa116-ppkPa16.1 21

  • 84

    FlowrateFlowrate Measurement Measurement sluice gate 2/5sluice gate 2/5

    The sluice gate is often used to regulate and measure the The sluice gate is often used to regulate and measure the flowrateflowrate in in an open channel.an open channel.

    The The flowrateflowrate, Q, , Q, is function of the water depth upstream, zis function of the water depth upstream, z11, the , the width of the gate, b, and the gate opening, a.width of the gate, b, and the gate opening, a.

    212

    212 )z/z(1

    )zz(g2bzQ

    With pWith p11=p=p22=0, the =0, the flowrateflowrate

    22221111

    22221

    211

    zbVVAzbVVAQ

    zV21pzV

    21p

  • 85

    FlowrateFlowrate Measurement Measurement sluice gate 3/5sluice gate 3/5

    In the limit of In the limit of zz11>>z>>z22, this result simply becomes , this result simply becomes

    This limiting result represents the fact that if the depth This limiting result represents the fact that if the depth ratio, zratio, z11/z/z22, is large, , is large, the kinetic energy of the fluid the kinetic energy of the fluid upstream of the gate is negligibleupstream of the gate is negligible and the fluid velocity and the fluid velocity after it has fallen a distance (zafter it has fallen a distance (z11--zz22)~z)~z11 is approximately is approximately

    ZZ2 2 ?? ?? QQ

    12 gz2bzQ

    12 gz2V

    zz11zz22

    z2

  • 86

    FlowrateFlowrate Measurement Measurement sluice gate 4/5sluice gate 4/5

    As we discussed relative to flow from an orifice, As we discussed relative to flow from an orifice, the fluid the fluid cannot turn a sharp 90cannot turn a sharp 90 corner. A vena corner. A vena contractacontracta results results with a contraction coefficient, with a contraction coefficient, CCcc=z=z22/a, less than 1/a, less than 1..

    Typically CTypically Ccc~0.61 over the depth ratio range of 0

  • 87

    Example 3.12 Sluice Gate

    Water flows under the sluice gate in Figure E3.12a. Water flows under the sluice gate in Figure E3.12a. DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channel.per unit width of the channel.

  • 88

    Example 3.12 Example 3.12 SolutionSolution1/21/2

    s/m61.4)m0.5/m488.0(1

    )m488.0m0.5)(s/m81.9(2)m488.0(bQ 2

    2

    2

    212

    212 )z/z(1

    )zz(g2zbQ

    For steady, inviscid, incompreesible flow, the flowerate per uniFor steady, inviscid, incompreesible flow, the flowerate per unit widtht width

    With zWith z11=5.0m and a=0.80m, so the ratio a/z=5.0m and a=0.80m, so the ratio a/z11=0.16

  • 89

    Example 3.12 Example 3.12 SolutionSolution1/21/2

    If we consider zIf we consider z11>>z>>z22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid, we would have, we would have

    s/m83.4m0.5s/m81.92m488.0gz2zbQ 22

    12

  • 90

    FlowrateFlowrate Measurement Measurement weirweir 5/55/5

    For a typical rectangular, sharpFor a typical rectangular, sharp--crested, the crested, the flowrateflowrate over the top of over the top of the weir plate is dependent on the weir plate is dependent on the weir height, Pthe weir height, Pww, the width of the , the width of the channel, b, and the head, H,channel, b, and the head, H, of the water above the top of the weir.of the water above the top of the weir.

    2/3111 Hg2bCgH2HbCAVCQ The The flowrateflowrate

    Where CWhere C11 is a constant to be determined.is a constant to be determined.

  • 91

    Example 3.13 WeirExample 3.13 Weir

    Water flows over a triangular weir, as is shown in Figure E3.13.Water flows over a triangular weir, as is shown in Figure E3.13.Based on a simple analysis using the Bernoulli equation, determiBased on a simple analysis using the Bernoulli equation, determine ne the dependence of the dependence of flowrateflowrate on the depth H. on the depth H. If the If the flowrateflowrate is Qis Q00when H=Hwhen H=H00, estimate the , estimate the flowrateflowrate when the depth is increased to when the depth is increased to H=3HH=3H00.. HH33

  • 92

    Example 3.13 Example 3.13 SolutionSolution

    gH2

    For steady , For steady , inviscidinviscid , and incompressible flow, the average speed , and incompressible flow, the average speed of the fluid over the triangular notch in the weir plate is of the fluid over the triangular notch in the weir plate is proportional toproportional toThe flow area for a depth of H is H[H tan(The flow area for a depth of H is H[H tan( /2)]/2)]The The flowrateflowrate

    where Cwhere C11 is an unknown constant to be determined experimentally.is an unknown constant to be determined experimentally.

    2/51

    211 Hg22

    tanC)gH2(2

    tanHCVACQ

    An increase in the depth by a factor of the three ( from HAn increase in the depth by a factor of the three ( from H00 to 3Hto 3H00 ) ) results in an increase of the results in an increase of the flowrateflowrate by a factor ofby a factor of

    6.15Hg22/tanCH3g22/tanC

    QQ

    2/501

    2/501

    H

    H3

    0

    0

  • 93

    EL & HGL EL & HGL 1/41/4

    For For steady, steady, inviscidinviscid, incompressible flow, incompressible flow , the total energy , the total energy remains constant along a streamline.remains constant along a streamline.

    g/p

    g2/V 2

    zH

    The head due to local static pressure (pressure energy)The head due to local static pressure (pressure energy)

    The head due to local dynamic pressure (kinetic energy)The head due to local dynamic pressure (kinetic energy)

    The elevation head ( potential energy )The elevation head ( potential energy )

    The total head for the flowThe total head for the flow

    Httanconszg2

    VP 2

    HEADHEAD

    streamlinestreamlinetotal headtotal head

  • 94

    EL & HGL EL & HGL 2/42/4

    Energy Line (EL) : represents the total head height.Energy Line (EL) : represents the total head height.

    Hydraulic Grade Line (HGL) height: Hydraulic Grade Line (HGL) height: represents the sum of the represents the sum of the elevation and static pressure heads.elevation and static pressure heads.

    The difference in heights between the The difference in heights between the EL and the HGLEL and the HGL represents represents the dynamic ( velocity ) headthe dynamic ( velocity ) head

    zg2

    VP 2

    zP

    g2/V2

    streamlinestreamlinetotal headtotal head

    streamlinestreamline

    ELELHGLHGLsstreamlinetreamline

  • 95

    EL & HGL EL & HGL 3/43/4

    Httanconszg2

    VP 2

  • 96

    EL & HGL EL & HGL 4/44/4

    Httanconszg2

    VP 2

    Httanconszg2

    VP 2

  • 97

    Example 3.14 Example 3.14 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line

    Water is siphoned from the tank shown in Figure E3.14 through a Water is siphoned from the tank shown in Figure E3.14 through a hose of constant diameter. A small hole is found in the hose at hose of constant diameter. A small hole is found in the hose at location (1) as indicate. When the siphon is used, will water lelocation (1) as indicate. When the siphon is used, will water leak out ak out of the hose, or will air leak into the hose, thereby possibly caof the hose, or will air leak into the hose, thereby possibly causing using the siphon to malfunction?the siphon to malfunction?

  • 98

    Example 3.14Example 3.14 SolutionSolution1/21/2

    Whether air will leak into or water will leak out of the hose dWhether air will leak into or water will leak out of the hose depends epends on on whether the pressure within the hose at (1) is less than or whether the pressure within the hose at (1) is less than or greater than atmosphericgreater than atmospheric. Which happens can be easily determined . Which happens can be easily determined by using the energy line and hydraulic grade line concepts. Withby using the energy line and hydraulic grade line concepts. With the the assumption of steady, incompressible, assumption of steady, incompressible, inviscidinviscid flow it follows that the flow it follows that the total head is constanttotal head is constant--thus, the energy line is horizontal.thus, the energy line is horizontal.

    Since the hose diameter is constant, it follows from the continuSince the hose diameter is constant, it follows from the continuity ity equation (AV=constant) that the water velocity in the hose is coequation (AV=constant) that the water velocity in the hose is constant nstant throughout. Thus throughout. Thus the hydraulic grade line is constant distance, Vthe hydraulic grade line is constant distance, V22/2g, /2g, below the energy linebelow the energy line as shown in Figure E3.14. as shown in Figure E3.14.

  • 99

    Example 3.14Example 3.14 SolutionSolution2/22/2

    Since the pressure at the end of the hose is atmospheric, it folSince the pressure at the end of the hose is atmospheric, it follows that lows that the hydraulic grade line is at the same elevation as the end of the hydraulic grade line is at the same elevation as the end of the hose the hose outlet. outlet. The fluid within the hose at any point above the hydraulic gradeThe fluid within the hose at any point above the hydraulic gradeline will be at less than atmospheric pressure.line will be at less than atmospheric pressure.

    Thus, Thus, air will leak into the hose through the hole at point (1). air will leak into the hose through the hole at point (1).

  • 100

    Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 1/4compressibility effects 1/4

    The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flows.most liquid flows.

    In certain instances, the assumption introduce considerable In certain instances, the assumption introduce considerable errors for gases.errors for gases.

    To account for compressibility effectsTo account for compressibility effects

    CgzV21dp 2

  • 101

    Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 2/4compressibility effects 2/4

    For isothermal flow of perfect gas

    For isentropic flow of perfect gas the density and pressure For isentropic flow of perfect gas the density and pressure are related by are related by P / P / kk =Ct, where k = Specific heat ratio=Ct, where k = Specific heat ratio

    22

    2

    11

    21 z

    g2V

    PPln

    gRTz

    g2V

    ttanconsgzV21dPPC 2k

    1k1

    ttanconsgzV21dpRT 2

    2

    22

    2

    21

    21

    1

    1 gz2

    VP1k

    kgz2

    VP1k

    k

  • 102

    Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 3/4compressibility effects 3/4

    To find the pressure ratio as a function of Mach number

    1111a1 kRT/Vc/VM The upstream Mach number

    Speed of sound

    1M2

    1k1p

    pp 1kk

    21a

    1

    12Compressible flow

    Incompressible flow21a

    1

    12 M2k

    ppp

  • 103

    Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 4/4compressibility effects 4/4

    21a

    1

    12 M2k

    ppp

    1M2

    1k1p

    pp 1kk

    21a

    1

    12

  • 104

    Example 3.15 Compressible Flow Example 3.15 Compressible Flow Mach Mach NumberNumber A Boeing 777 flies at Mach 0.82 at an altitude of 10 km in a A Boeing 777 flies at Mach 0.82 at an altitude of 10 km in a

    standard atmosphere. Determine the stagnation pressure on the standard atmosphere. Determine the stagnation pressure on the leading edge of its wing if the flow is incompressible; and if tleading edge of its wing if the flow is incompressible; and if the he flow is incompressible isentropic.flow is incompressible isentropic.

    kPa5.12...pp

    471.0...M2k

    ppp

    12

    21a

    1

    12

    For incompressible flowFor incompressible flow For compressible isentropic flowFor compressible isentropic flow

    kPa7.14....pp

    55.0...1M2

    1k1p

    pp

    12

    1kk

    21a

    1

    12

  • 105

    Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation unsteady effectsunsteady effects

    For unsteady flow V = V ( s , t )For unsteady flow V = V ( s , t )

    To account for unsteady effects To account for unsteady effects

    sVV

    tVaS

    0dzVd21dpds

    tV 2

    Along a streamline

    + Incompressible condition+ Incompressible condition

    2222

    S

    S12

    11 zV21pds

    tVzV

    21p 2

    1

    Oscillations Oscillations in a Uin a U--tubetube

  • 106

    Example 3.16 Unsteady Flow Example 3.16 Unsteady Flow UU--TubeTube

    An incompressible, An incompressible, inviscidinviscid liquid liquid is placed in a vertical, constant is placed in a vertical, constant diameter Udiameter U--tube as indicated in tube as indicated in Figure E3.16. When released from Figure E3.16. When released from the the nonequilibriumnonequilibrium position shown, position shown, the liquid column will oscillate at a the liquid column will oscillate at a specific frequency. Determine this specific frequency. Determine this frequency.frequency.

  • 107

    Example 3.16Example 3.16 SolutionSolutionLet points (1) and (2) be at the air-water interface of the two columns of the tube and z=0 correspond to the equilibrium position of the interface.Hence z = 0 , p1 =p2 = 0, z1 = 0, z2 = - z , V1 = V2 = V z = z ( t )

    dtdVds

    dtdVds

    tV 2

    1

    2

    1

    S

    S

    S

    S

    The total length of the liquid colum

    /g20zg2dt

    zd

    gdtdzV

    zdtdVz

    2

    2

    Liquid oscillationLiquid oscillation