Fluid 03

1

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 University0928200909282009

2

MAIN TOPICSMAIN TOPICS

NewtonNewtonrsquorsquos 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 Equation

Static 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推導出推導出

解釋解釋Bernoulli equationBernoulli equation

3

NewtonNewtonrsquorsquos Second Law s Second Law 1616

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 decelerationexperiences an acceleration or decelerationAccording to According to NewtonNewtonrsquorsquos 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 accelerationits acceleration

F=maF=maIn 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 effectspossible to ignore viscous effectsThe forces acting on the particle Coordinates used The forces acting on the particle Coordinates used

作用在質點的合力等於質點質量乘以加速度

本章仍以非黏性流體為對象即假設流體沒有黏性忽略其黏性效應本章仍以非黏性流體為對象即假設流體沒有黏性忽略其黏性效應討論作用在質點上的力使用的座標討論作用在質點上的力使用的座標

4


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 Newtonrsquorsquos 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 itare needed to describe it

作用在質點上的作用在質點上的ForceForce

5


The most often used coordinate The most often used coordinate systems are systems are rectangular (xyz) rectangular (xyz) and cylindrical (rand cylindrical (rθθz) systemz) system選用合適的座標系統使用哪一種座標系統選用合適的座標系統使用哪一種座標系統

6


In this chapter the flow is confined to be In this chapter the flow is confined to be twotwo--dimensional motiondimensional motionAs 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 VVAs the particle moves As the particle moves it follows a particular pathit follows a particular pathThe 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 velocityof its initial position and velocity

以兩維流動為對象以兩維流動為對象

質點運動有它特質點運動有它特定的軌跡在軌跡定的軌跡在軌跡上上的的LocationLocation是是

初始位置與速度的初始位置與速度的函數函數

7


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 streamlinesstreamlinesFor 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)

流場內與質點速度相切的流場內與質點速度相切的線稱為線稱為streamlinestreamline

在穩態流場下在穩態流場下質點運動質點運動的速度向量與軌跡相切的速度向量與軌跡相切

質點的運動可以由它沿著質點的運動可以由它沿著StreamlineStreamline相對於某個參考原點的距離相對於某個參考原點的距離s=s=s(ts(t))來描述該來描述該streamlinestreamline的曲率半徑可以寫成的曲率半徑可以寫成R=R=R(sR(s) )

8


The The distance along the streamline is related to the distance along the streamline is related to the particleparticlersquorsquos speeds speed by by VV==dsdtdsdt and the radius of curvature is and the radius of curvature is related to shape of the streamlinerelated to shape of the streamlineThe 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

nsrrrr

rr

+=+==

RVa

dsdVVa

2

ns == CHAPTER 04 CHAPTER 04 再討論再討論

由由s=s=s(ts(t))來定義質點的速度來定義質點的速度 VV==dsdtdsdt

由速度由速度 VV==dsdtdsdt來定義加速度來定義加速度helliphellip

加速度的兩個分量加速度的兩個分量

先應用先應用

9

StreamlinesStreamlines

Streamlines past an airfoilStreamlines past an airfoil

Flow past a bikerFlow past a biker

10

F=ma along a Streamline F=ma along a Streamline 1414

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

11


Consider the small fluid particle of fluid particle of size of size of δδs by s by δδnn in the plane of the figure and δy normal to the figure For steady flow the component of Newtonrsquos second law along the streamline direction s

sVVV

sVmVmaF SS part

partρδ=

partpart

δ=δ=δsumWhere represents the sum of the s components of all the force acting on the particle

sumδ SF

沿著沿著ss方向的合力方向的合力

12


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

θγδminus=θδminus=δ sinVsinWWs

( ) ( ) Vspynp2ynppynppF SSSps δpartpart

minus=δδδminus=δδδ+minusδδδminus=δ

VspsinFWF psss δ⎟⎠⎞

⎜⎝⎛

partpart

minusθγminus=δ+δ=δ

sasVV

spsin ρ=

partpart

ρ=partpart

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

2s

sppSδ

partpart

=δ

單位體積

13


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 streamlineweight along the streamlineFor 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 producedis produced

sasVV

spsin ρ=

partpart

ρ=partpart

minusθγminus

0spsin =partpart

minusθγminus

IntegrationIntegrationhelliphelliphelliphellipParticle weightParticle weight

pressure gradientpressure gradient

造成質點速度改變的兩個因素造成質點速度改變的兩個因素

質點速度沒有改變則質點速度沒有改變則particle weightparticle weight與與pressure gradientpressure gradient相互抵銷相互抵銷

14

IntegrationIntegrationhelliphellip

sasVV

spsin ρ=

partpart

ρ=partpart

minusθγminus Rearranged and IntegratedRearranged and Integratedhelliphelliphelliphellip

( )

CgzV21dp

0dzVd21dp

dsdV

21

dsdp

dsdz

2

22

=++ρ

gtgtgtgt

=γ+ρ+gtgtρ=minusγminus

int 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 streamlinethe 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 signcannot be removed from under the integral sign

除非壓力與密度的關除非壓力與密度的關係很清楚係很清楚否則積分否則積分不能隨便拿開不能隨便拿開

15

Bernoulli Equation Along a StreamlineBernoulli Equation Along a Streamline

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

Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flowFrictionless flowFlow along a streamlineFlow along a streamline

ttanconsz2

Vp2

=γ+ρ+BERNOULLI EQUATIONBERNOULLI EQUATION

CgzV21dp 2 =++

ρint不可壓縮流體不可壓縮流體

一再提醒每一個結論（推導出來的方程式）都有它背後假設一再提醒每一個結論（推導出來的方程式）都有它背後假設條件即一路走來是基於這些假設才有如此結果條件即一路走來是基於這些假設才有如此結果

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 1738NONO 剪力剪力

16

Example 31 Pressure Variation along A Example 31 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 E31a From a more advanced theory of flow past shown in Figure E31a 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==--infininfin 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 已知流體沿著已知流體沿著streamlinestreamline的速度的速度

求沿著求沿著streamlinestreamline的的壓力變化壓力變化

sinsinθθ=0=0 s s xx

17

Example 31 Example 31 SolutionSolution1212

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=⎟⎟

⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

partpart

=partpart

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 (sinsinθθ=0)=0)

The acceleration term

sVV

sp

partpart

ρminus=partpart

(1)(1) sasVV

spsin ρ=

partpart

ρ=partpart

minusθγminus

The pressure gradient along the streamline is

( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)

18


The pressure gradient along the streamline

( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)

The pressure distribution along the streamline

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛ρminus=

2)xa(

xaVp

632

0

19

Example 32 The Bernoulli EquationExample 32 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 E32 Determine the as is shown in Figure E32 Determine the difference in the pressure between points (1) and (2)difference in the pressure between points (1) and (2)

求求point (1)point (1)與與point (2)point (2)的壓力差的壓力差

20

Example 32 Example 32 SolutionSolution

2V

2Vpp

20

21

12 ρ=ρ=minus

The Bernoullirsquos 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 z2

Vpz2

Vp γ+ρ+=γ+ρ+

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

將將Bernoulli equationBernoulli equation應用到沿著連接應用到沿著連接point (1)point (1)與與point (2)point (2)的的streamlinestreamline

21

F=ma Normal to a StreamlineF=ma Normal to a Streamline1212

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

RVV

RmVF

22

nρδ

=δ

=δsumWhere represents the Where represents the sum of the n components of all sum of the n components of all the force acting on the particlethe force acting on the particle

sumδ nF

法線方向的合力法線方向的合力

HydrocycloneHydrocyclone separatorseparator水力水力旋旋流分璃器流分璃器

22


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

θγδminus=θδminus=δ cosVcosWWn

( ) Vnpysp2ys)pp(ysppF nnnpn δpartpart


VRVV

npcosFWF

2

pnnn δρ

=δ⎟⎠⎞

⎜⎝⎛

partpart


RV

npcos

2ρ=

partpart

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

2n

nppnδ

partpart

=δ

23


RV

dndp

dndz 2ρ

=minusγminus

RV

npcos

2ρ=

partpart

minusθγminus 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 signunder the integral sign

IntegratedIntegratedhelliphellip

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

=++ρint int

Without knowing the n dependent Without knowing the n dependent in V=in V=V(snV(sn) and R=) and R=R(snR(sn) this ) this integration cannot be completedintegration cannot be completed

ParticleParticle weightweight

pressure gradientpressure gradient 造成質點速度改變的兩個因素造成質點速度改變的兩個因素

除非壓力與密度的關係很清除非壓力與密度的關係很清楚楚否則積分不能隨便拿開否則積分不能隨便拿開

除非除非V=V=V(snV(sn) ) 與與 R=R=R(snR(sn))很清很清楚否則積分不能隨便拿開楚否則積分不能隨便拿開

24

Bernoulli Equation Normal to a StreamlineBernoulli Equation Normal to a Streamline


Restrictions Steady flowRestrictions Steady flowIncompressible flowIncompressible flowFrictionless flow Frictionless flow NO shear forceNO shear forceFlow normal to a streamlineFlow normal to a streamline

CzdnRVp

2

=γ+ρ+ intBERNOULLI EQUATIONBERNOULLI EQUATION

CgzdnRVdp 2

=++ρint int

不可壓縮流體不可壓縮流體


25

If gravity is neglectedhellipFree 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 motionmotionIf 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 2ρ

=minusγminus

RV

dndp 2ρ

=minus 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 tornadolarger than it is near the center of the tornado

26

Aircraft wing tip vortexAircraft wing tip vortex

27

Example 33 Pressure Variation Normal to Example 33 Pressure Variation Normal to a Streamlinea Streamline

Shown in Figure E33a and E33b are two flow fields with circulaShown in Figure E33a and E33b 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 (dzdn=0)

已知速度分佈已知速度分佈求壓力場求壓力場

RV

npcos

2ρ=

partpart

minusθγminus

28


( ) 02

022

1 prrC21p +minusρ=

rV

rp 2ρ=

partpart

For flow in the horizontal plane (dzdn=0) The streamlines are circles partpartn=-partpartrThe radius of curvature R=r

For case (a) this gives

rCrp 2

1ρ=partpart

3

22

rC

rp ρ=

partpart

For case (b) this gives

0220

22 p

r1

r1C

21p +⎟⎟

⎠

⎞⎜⎜⎝

⎛minusρ=

RV

dndp

dndz 2ργ =minusminus

29

Physical InterpreterPhysical Interpreter1212

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 incompressibleand incompressibleApplication 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 ndashndash mass times acceleration (the mass times acceleration (the ρρVV222 term) 2 term) pressure (the p term) and weight (the pressure (the p term) and weight (the γγz term)z term)

CzdnRVp

2

=γ+ρ+ intCz2

Vp2

=γ+ρ+

基本假設基本假設

過程中有「質量過程中有「質量timestimes加速度」「壓力加速度」「壓力pp」與「重量」與「重量γγz z 」」

作用在質點上的兩個力作用在質點上的兩個力壓壓力與重力力與重力當當合力不平衡合力不平衡就會有加速度產生就會有加速度產生

RVa

dsdVVa

2

ns ==

如如何何解解讀讀

30


The Bernoulli equation is a mathematical statement of The Bernoulli equation is a mathematical statement of ldquoldquoThe 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 particlerdquordquo

Work done by force Work done by force FFtimestimesddWork done by weight Work done by weight γγz z Work done by pressure force pWork done by pressure force p

Kinetic energy Kinetic energy ρρVV2222

CdnRVzp

2

=ρ+γ+ intC2

Vzp2

=ρ+γ+

能量平衡外力作的功等於質點的動能改變能量平衡外力作的功等於質點的動能改變

外力有二外力有二一為壓力一為壓力一為重力一為重力

你可能會對其中的單位感到困惑沒錯單位是怪你可能會對其中的單位感到困惑沒錯單位是怪怪的但別忘了以上都是怪的但別忘了以上都是Based onBased on單位體積單位體積

31

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 γγ

czg2

VP 2

=++γ

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

沿著沿著streamlinestreamline 讓每一項都是長度單位讓每一項都是長度單位

32

Example 34 Kinetic Potential and Example 34 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 E34 A force applied to the shown in Figure E34 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

求求point (1) (2) (3)point (1) (2) (3)的三種型態的能量關的三種型態的能量關係係

停停止止移移動動

33

Example 34 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 flighta momentary stop at the top of its flight

streamlinethealongttanconszV21p 2 =γ+ρ+

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

34

Example 35 Pressure Variation in a Example 35 Pressure Variation in a Flowing StreamFlowing Stream

Consider the Consider the inviscidinviscid incompressible steady flow shown in Figure incompressible steady flow shown in Figure E35 From section A to B the streamlines are straight while frE35 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)

求求point (1)(2)point (1)(2)與與 point(3)(4)point(3)(4)的壓力變化的壓力變化

35


1221221 rhp)zz(rpp minus+=minus+=

ttanconsrzp =+

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

Using p2=0z1=0and 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)

36

intρminus= minus4

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(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)

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

37

Static Stagnation Dynamic and Static Stagnation Dynamic and Total PressureTotal Pressure1515

Each term in the Bernoulli equation can be interpreted as a Each term in the Bernoulli equation can be interpreted as a form of pressureform 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 ldquoldquostaticstaticrdquordquo relative to the moving relative to the moving fluid Hence it is termed the fluid Hence it is termed the static pressurestatic pressure helliphellip seen by the seen by the fluid particle as it movesfluid particle as it moves

C2

Vzp2

=ρ+γ+ Each term can be interpreted as a form of pressure

能量平衡的概念下每一項的單位壓力能量平衡的概念下每一項的單位壓力

pp是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體是流體流動下所量測的壓力量測時要緊貼著流體相對流動的流體而言它是而言它是「「staticstatic」故稱為「」故稱為「static pressurestatic pressure」」

38


The static pressureThe static pressure is measured in a flowing fluid using a is measured in a flowing fluid using a wall pressure wall pressure ldquoldquotaptaprdquordquo or a static pressure probe or a static pressure probe

hhhphp 34133131 γ=γ+γ=+γ= minusminusminusThe static pressureThe static pressureγγzz 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 changeschanges

利用利用wall wall pressurepressureldquoldquotaptaprdquordquo量測「量測「static static pressurepressure」」

γγzz不是真的壓力它代表著位能（高度）改變所產生的壓力變化不是真的壓力它代表著位能（高度）改變所產生的壓力變化

39


ρρVV2222 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 Hof HThe 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 V

21pp ρ+=Stagnation pressure

Static pressure Dynamic pressureDynamic pressure

ρρVV2222是將管插到流場中把流體擋下來在是將管插到流場中把流體擋下來在

管端管端（（22））衍生出來的壓力這個壓力等於衍生出來的壓力這個壓力等於

液柱高液柱高HH所產生的壓力所產生的壓力

假設（假設（11）與）與（（22））高度相同高度相同

兩者和兩者和稱為稱為sstagnationtagnation pressurepressure

40

Stagnation pointStagnation point

Stagnation point flowStagnation point flow

41


There 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 ldquoldquooveroverrdquordquoand some and some ldquoldquounderunderrdquordquo the objectthe objectThe dividing line is termed the The dividing line is termed the stagnation streamlinestagnation streamline and and terminates at the stagnation point on the bodyterminates at the stagnation point on the bodyNeglecting 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

將物體固定在流場中會在物體上出現將物體固定在流場中會在物體上出現stagnation pointstagnation point

忽略高度效應下忽略高度效應下 stagnation pointstagnation point處的壓力是流場中最大的壓力處的壓力是流場中最大的壓力

42


The 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 pressureThe Bernoulli equation is a statement that the total The Bernoulli equation is a statement that the total pressure remains constant along a streamlinepressure remains constant along a streamline

ttanconspz2

Vp T

2

==γ+ρ+

Total pressureTotal pressure沿著沿著streamlinestreamline是一個常數是一個常數

三者和三者和稱為稱為TTootaltal pressurepressure

43

The The PitotPitot--static Tube static Tube 1515

ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

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 calculatedfluid speed can be calculatedThis is This is the principle on which the the principle on which the PitotPitot--static tube is basedstatic 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 pressurevelocity into pressure

利用『沿著沿著streamlinestreamline的的total pressuretotal pressure是一個常數是一個常數』概念來量測流速的量具

量具可同時量測量具可同時量測staticstatic與與stagnation stagnation pressurepressure再據以計算流速再據以計算流速

44

Airplane Airplane PitotPitot--static probestatic probe

Airspeed indicatorAirspeed indicator

45


46


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 pressuremeasure the static and stagnation pressureAn accurate measurement of static pressure requires that An accurate measurement of static pressure requires that none of the fluidnone of the fluidrsquorsquos kinetic energy be converted into a s kinetic energy be converted into a pressure rise at the point of measurementpressure rise at the point of measurementThis requires a smooth hole with no burrs or imperfectionsThis requires a smooth hole with no burrs or imperfections

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

TapsTaps不可出現毛不可出現毛邊邊或瑕疵或瑕疵以免以免

造成不必要的壓造成不必要的壓升或壓降升或壓降

47


Typical pressure distribution along a Typical pressure distribution along a PitotPitot--static tubestatic 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 pressuremay be less than free stream static pressureIt 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 pressurepressure由壓力分布或變化可以了解由壓力分布或變化可以了解tapstaps要放在哪一個位置是要放在哪一個位置是很重要的以免高估或低估很重要的以免高估或低估static pressurestatic pressure

48


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 upstreamthat the center hole points directly upstream

( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=

Directional-finding Pitot-static tube

If θ=0圓柱會自行調整直到圓柱會自行調整直到PP11=P=P33

可以自動對位的可以自動對位的PitotPitot--static tubestatic tube

49

Example 36 Example 36 PitotPitot--Static TubeStatic Tube

An airplane flies 100mihr at an elevation of 10000 ft in a An airplane flies 100mihr at an elevation of 10000 ft in a standard atmosphere as shown in Figure E36 Determine standard atmosphere as shown in Figure E36 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 求第一點第二點的壓力求第一點第二點的壓力

機身下貼附機身下貼附PitotPitot--static tubestatic tube

50

Example 36 Example 36 Solution Solution 1212

2Vpp

21

12ρ

+=

psia1110)abs(ftlb1456p 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

3ftslug0017560=ρ

With V1=100mihr=1466fts and V2=0

)abs(ftlb)9181456(

2)sft7146)(ftslugs0017560(ftlb1456p2

222322

+=

+=

高度高度10000ft10000ft處的壓力與密度處的壓力與密度

51

psi13130ftlb918p 22 ==

In terms of gage pressure

psi131302Vpp

21

12 =ρ

=minus

The pressure difference indicated by the Pitot-static tube


52

Application of Bernoulli Equation Application of Bernoulli Equation 1212

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 γ+

ρ+=γ+

ρ+


Bernoulli equationBernoulli equation的應用的應用

沿著沿著streamlinestreamline任意兩點任意兩點

要應用要應用Bernoulli equationBernoulli equation時要注意問題本身有沒有符合下列限制條件時要注意問題本身有沒有符合下列限制條件

53


Free jetFree jetConfined flowConfined flowFlowrateFlowrate measurementmeasurement


54

Free Jets Free Jets 1313

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

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets

Flow from a tankFlow from a tank

56


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 V11In general dltlth and use the VIn general dltlth and use the V22

as average velocityas average velocity

For a For a sharpsharp--edged orificeedged orifice a a vena vena contractacontracta effecteffect occursoccursThe effect is the result of the The effect is the result of the inability of the fluid to turn the inability of the fluid to turn the sharp 90sharp 90degdeg cornercorner

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若

dltlthdltlth則以則以VV22代表平均速度代表平均速度

出現束縮效應出現束縮效應流體流體不可能不可能9090degdeg轉彎轉彎

57


Typical Typical flow patterns and flow patterns and contraction coefficientscontraction coefficients for for various round exit various round exit configurationconfigurationThe 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 flowshole 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 γ+

ρ+=γ+

ρ+

不同出口型態下的不同出口型態下的flow patternflow pattern與縮流係數與縮流係數

定義縮流係數定義縮流係數

58

Example 37 Flow From a TankExample 37 Flow From a Tank－－ GravityGravity

A stream of water of diameter d = 01m flows steadily from a tanA stream of water of diameter d = 01m flows steadily from a tank k of Diameter D = 10m as shown in Figure E37a of Diameter D = 10m as shown in Figure E37a 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 = 20m h = 20m

求求flowrateflowrate QQ

59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+

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

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)

Bernoulli equationBernoulli equation應用到應用到point(1)point(1)與與(2)(2)間間

Point(1)Point(1)與與(2)(2)的已知條件的已知條件

加上質量守恆條件加上質量守恆條件

60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=

Combining Equation 2 and 3

Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===

VV11nene0 (Q) vs V0 (Q) vs V11≒≒0 (Q0 (Q00))

4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61

Example 38 Flow from a TankExample 38 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=003m and exits to the atmosphere from a nozzle of D=003m and exits to the atmosphere from a nozzle of diameter d=001m as shown in Figure E38 diameter d=001m as shown in Figure E38 The pressure The pressure in the tank remains constant at 30kPa (gage)in the tank remains constant at 30kPa (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 hosethe hose

求求flowrateflowrate與與 Point(2)Point(2)的壓力的壓力理想氣體理想氣體

62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=

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

mkg261K)27315)(KkgmN9286(

kNN10]mkN)10103[(RTp

=+sdotsdot

times+==ρ

Bernoulli equationBernoulli equation應用到應用到point(1)(2)point(1)(2)與與(3)(3)間間

Point(1)Point(1)(2)(2)與與((33))的已知條件的已知條件

理想氣體的密度理想氣體的密度

63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

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

sm677AVAVHenceVAVA 23323322 ===

22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64

Example 39 Flow in a Variable Area PipeExample 39 Flow in a Variable Area Pipe

Water flows through a pipe reducer as is shown in Figure E39 TWater flows through a pipe reducer as is shown in Figure E39 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 onemanometer containing oil of specific gravity SG less than oneDetermine the manometer reading hDetermine the manometer reading h

reading hreading h

65


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 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122

21221 minus+minusγ=minus

66


h)SG1()zz(pp 1221 γminus+minusγ=minus

2121 phSGh)zz(p =γ+γ+γminusγminusminusγminus ll

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

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)AA(1AQh

2122

2 minusminus

=

Since VSince V22=QA=QA22

uarr- darr+

be independent of be independent of θθ

Point(1)(2)Point(1)(2)的壓力差的壓力差

67

Confined Flows Confined Flows 1414

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 jetthe free jetSuch 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 anotheris different from one section to anotherFor 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 equationthe Bernoulli equation

Tools Bernoulli equation + Continuity equation

受限流被限制在受限流被限制在devicedevice內部如內部如nozzlenozzlepipepipe

解解「「受限流受限流」」的工具的工具

68


Consider a fluid flowing through a fixed volume that has Consider a fluid flowing through a fixed volume that has one inlet and one outletone inlet and one outletConservation of mass requiresConservation of mass requiresFor For incompressible flowincompressible flow the continuity equation is the continuity equation is

222111 VAVA ρ=ρ

212211 QQVAVA ==加上「不可壓縮」條件連續方程式簡化加上「不可壓縮」條件連續方程式簡化

69


If the fluid velocity is increased If the fluid velocity is increased the pressure will decreasethe pressure will decreaseThis 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 pressurevapor pressure

Pressure variation and cavitationin a variable area pipe

受限流流經變化管徑的管當速度增加壓力就降低受限流流經變化管徑的管當速度增加壓力就降低

當速度持續增加壓力可能降當速度持續增加壓力可能降低到低於低到低於vapor pressurevapor pressure導導致初始空蝕現象出現致初始空蝕現象出現

VenturiVenturi channelchannel

70

Confined Flows Confined Flows 44 example of 44 example of cavitationcavitation

A example of A example of cavitationcavitation can be demonstrated with a can be demonstrated with a garden hosegarden hose If the hose is If the hose is ldquoldquokinkedkinkedrdquordquo a restriction in the a restriction in the flow area will resultflow area will resultThe water The water velocityvelocity through the restriction will be through the restriction will be relatively largerelatively largeWith 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 ndashndash a definite a definite ldquoldquohissinghissingrsquorsquo sound will sound will be producedbe producedThe sound is a result of The sound is a result of cavitationcavitation

水管扭結導致管的截面積縮小速度增加水管扭結導致管的截面積縮小速度增加

出現嘶嘶聲即表示空蝕現象出現出現嘶嘶聲即表示空蝕現象出現

71

Damage from Damage from CavitationCavitation

Cavitation from propeller

前進到高壓區前進到高壓區bubblebubble破裂破壞器具破裂破壞器具

72

Example 310 Siphon and Example 310 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 E310 Determine the maximum diameter hose as shown in Figure E310 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 147 of the tank Atmospheric pressure is 147 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 γγ and its and its vapor pressure vapor pressure ppvv

Max HMax H才能避免才能避免cavitationcavitation

73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

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 Vsft935ft)]5(15)[sft232(2)zz(g2V ==minusminus=minus=(1)(3)(1)(3)

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

74


22212

221

2112 V

21)zz(zV

21zV

21pp ρminusminusγ=γminusρminusγ+ρ+=

233222 )sft935)(ftslugs941(21ft)H15)(ftlb462()ftin144)(inlb414( minusminus=minus

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 0256 is 0256 psiapsia Hence for incipient Hence for incipient cavitationcavitation the lowest pressure in the system will be p = 0256 the lowest pressure in the system will be p = 0256 psiapsiaUUsing gage pressure sing gage pressure pp11 = 0 p= 0 p22=0256 =0256 ndashndash 147 = 147 = --144 144 psipsi

(2)(2)

ftH 228=

75

FlowrateFlowrate Measurement Measurement inin pipes 15pipes 15

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

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=

The theoretical The theoretical flowrateflowrate

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

Flow metersFlow meters的理論基礎的理論基礎

管流中量測流率的裝置管流中量測流率的裝置

76

Example 311 Example 311 VenturiVenturi MeterMeter

Kerosene (SG = 085) flows through the Kerosene (SG = 085) flows through the VenturiVenturi meter shown in meter shown in Figure E311 with Figure E311 with flowratesflowrates between 0005 and 0050 mbetween 0005 and 0050 m33s s Determine the range in pressure difference Determine the range in pressure difference pp11 ndashndash pp22 needed to needed to measure these measure these flowratesflowrates

Known Q Determine pKnown Q Determine p11--pp22

求維持流率所需求維持流率所需point(1)(2)point(1)(2)壓力差壓力差

77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus

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

360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78


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 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus

kPa116Nm10161 25 =times=

kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus

kPa116-ppkPa161 21 lele

79

FlowrateFlowrate Measurement Measurement sluice gate 25sluice gate 25

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 channelan open channelThe 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 awidth of the gate b and the gate opening a

212

212 )zz(1

)zz(g2bzQminus

minus=

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

22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+

水閘門調節與量測流率水閘門調節與量測流率

理論值理論值

80


In the limit of In the limit of zz11gtgtzgtgtz22 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 z11zz22 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 是求是求QQ的關鍵的關鍵

12 gz2bzQ =

12 gz2V =

特殊情況特殊情況zz11遠大於遠大於zz22

z2lta（束縮效應） Z2 = Cc a縮流係數 Cc

81


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 90degdeg corner A vena corner A vena contractacontracta results results with a contraction coefficient with a contraction coefficient CCcc=z=z22a less than 1a less than 1Typically CTypically Ccc~061 over the depth ratio range of 0ltaz~061 over the depth ratio range of 0ltaz11lt02lt02For For large value of azlarge value of az11 the value of C the value of Ccc increase rapidly increase rapidly

縮流係數縮流係數 CCc c 低於低於 11典型值約典型值約061061azaz11 越大越大 CCcc越高越高

82

Example 312 Sluice Gate

Water flows under the sluice gate in Figure E312a Water flows under the sluice gate in Figure E312a DertermineDertermine the the approximate approximate flowrateflowrate per unit width of the channelper unit width of the channel

83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=

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

With zWith z11=50m and a=080m so the ratio az=50m and a=080m so the ratio az11=016lt020=016lt020Assuming contraction coefficient is approximately CAssuming contraction coefficient is approximately Ccc=061 =061 zz22=C=Ccca=061(080m)=0488ma=061(080m)=0488mThe The flowrateflowrate

212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84


If we consider zIf we consider z11gtgtzgtgtz22 and neglect the kinetic energy of the upstream and neglect the kinetic energy of the upstream fluidfluid we would have we would have

( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85

FlowrateFlowrate Measurement Measurement weirweir 5555

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 Hchannel b and the head H of the water above the top of the weirof the water above the top of the weir

23111 Hg2bCgH2HbCAVCQ ===The The flowrateflowrate

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

堰待由實驗來決定待由實驗來決定

86

Example 313 WeirExample 313 Weir

Water flows over a triangular weir as is shown in Figure E313Water flows over a triangular weir as is shown in Figure E313Based 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 HH提高提高33倍流率增加倍流率增加

87


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(θθ 2)]2)]The The flowrateflowrate

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

251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==

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

( ) ( )( ) ( )

615Hg22tanCH3g22tanC

QQ

2501

2501

H

H3

0

0 =θθ

=

88

EL amp HGL EL amp HGL 1414

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

gp ρ

g2V 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==++

γ每一項化成長度單位－每一項化成長度單位－HEADHEAD

沿著沿著streamlinestreamline的的total headtotal head維持常數維持常數

89


Energy Line (EL) represents the total head heightEnergy 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 headselevation 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++

γ

zP+

γ

g2V2

沿著沿著streamlinestreamline的的total headtotal head連結成一條線連結成一條線

沿著沿著streamlinestreamline的的「「壓力頭壓力頭」＋「」＋「高度頭高度頭」」連結成一條線連結成一條線

ELEL與與HGLHGL的差的差代表沿代表沿sstreamlinetreamline各處的各處的「速度「速度頭頭」」

90


Httanconszg2

VP 2==++

γ

「「壓力頭壓力頭」＋「」＋「高度頭高度頭」」

91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92

Example 314 Example 314 Energy Line and Hydraulic Grade LineEnergy Line and Hydraulic Grade Line

Water is siphoned from the tank shown in Figure E314 through a Water is siphoned from the tank shown in Figure E314 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 malfunctionthe siphon to malfunction

空氣跑進去或水溢出來空氣跑進去或水溢出來

93

Example 314Example 314 SolutionSolution1212

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 horizontalthus 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 V222g 2g below the energy linebelow the energy line as shown in Figure E314 as shown in Figure E314

94


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 pressureline 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)

95

Restrictions on Use of the Bernoulli Restrictions on Use of the Bernoulli Equation Equation compressibility effects 14compressibility effects 14

The assumption of incompressibility is reasonable for The assumption of incompressibility is reasonable for most liquid flowsmost liquid flowsIn certain instances the assumption introduce considerable In certain instances the assumption introduce considerable errors for gaseserrors for gasesTo account for compressibility effectsTo account for compressibility effects

CgzV21dp 2 =++

ρint考慮壓縮效應考慮壓縮效應

96


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 ρρkk =Ct where k = Specific heat ratio=Ct where k = Specific heat ratio

22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++

ttanconsgzV21dPPC 2k

1k1

=++intminus

ttanconsgzV21dpRT 2 =++

ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus

理想氣體等溫流理想氣體等溫流

理想氣體等熵流理想氣體等熵流

97


To find the pressure ratio as a function of Mach number

1111a1 kRTVcVM ==The upstream Mach number

Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow

Incompressible flow21a

1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99

Example 315 Compressible Flow Example 315 Compressible Flow ndashndash Mach Mach NumberNumber

A Boeing 777 flies at Mach 082 at an altitude of 10 km in a A Boeing 777 flies at Mach 082 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 isentropicflow is incompressible isentropic

kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus

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

kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100

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 part

part+

partpart

=

( ) 0dzVd21dpds

tV 2 =γ+ρ++partpart

ρ Along a streamline

+ Incompressible condition+ Incompressible condition

2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int

考慮考慮非穩定效應非穩定效應

Oscillations Oscillations in a Uin a U--tubetube

101

Example 316 Unsteady Flow Example 316 Unsteady Flow ndashndash 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 E316 When released from Figure E316 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 frequencyfrequency

102

Example 316Example 316 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 interfaceHence z = 0 p1 =p2 = 0 z1 = 0 z2 = - z V1 = V2 = V rarrz = z ( t )

dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart

intintThe total length of the liquid colum

( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ

Liquid oscillationLiquid oscillation

2

MAIN TOPICSMAIN TOPICS

NewtonNewtonrsquorsquos 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 Equation

Static 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推導出推導出

解釋解釋Bernoulli equationBernoulli equation

3






4





5



6






7






8




nRVs

dsdVVnasa

dtVda

2

nsrrrr

rr

+=+==

RVa

dsdVVa

2





先應用先應用

9




10



11



sVVV

sVmVmaF SS part

partρδ=

partpart


sumδ SF


12








⎜⎝⎛

partpart


sasVV

spsin ρ=

partpart

ρ=partpart


2s

sppSδ

partpart

=δ

單位體積

13



sasVV

spsin ρ=

partpart

ρ=partpart

minusθγminus

0spsin =partpart

minusθγminus





14


sasVV

spsin ρ=

partpart

ρ=partpart


( )

CgzV21dp

0dzVd21dp

dsdV

21

dsdp

dsdz

2

22

=++ρ

gtgtgtgt





15




ttanconsz2

Vp2


CgzV21dp 2 =++




16




⎟⎟⎠

⎞⎜⎜⎝

⎛+= 3

3




17


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=⎟⎟

⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

partpart

=partpart

4

3

3

32

04

30

3

3

0 xa

xa1V3

xaV3

xa1V

xVV

sVV



sVV

sp

partpart

ρminus=partpart

(1)(1) sasVV

spsin ρ=

partpart

ρ=partpart

minusθγminus


( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)

18



( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛ρminus=

2)xa(

xaVp

632

0

19




20


2V

2Vpp

20

21

12 ρ=ρ=minus


z1=z2

2

22

21

21

1 z2

Vpz2

Vp γ+ρ+=γ+ρ+



21



RVV

RmVF

22

nρδ

=δ


sumδ nF



22







VRVV

npcosFWF

2

pnnn δρ

=δ⎟⎠⎞

⎜⎝⎛

partpart


RV

npcos

2ρ=

partpart


2n

nppnδ

partpart

=δ

23


RV

dndp

dndz 2ρ

=minusγminus

RV

npcos

2ρ=

partpart






CgzdnRVdp 2

=++ρint int






24




CzdnRVp

2


CgzdnRVdp 2

=++ρint int



25



RV

dndp

dndz 2ρ

=minusγminus

RV

dndp 2ρ


26


27



)b(r




RV

npcos

2ρ=

partpart

minusθγminus

28


( ) 02

022

1 prrC21p +minusρ=

rV

rp 2ρ=

partpart



rCrp 2

1ρ=partpart

3

22

rC

rp ρ=

partpart


0220

22 p

r1

r1C

21p +⎟⎟

⎠

⎞⎜⎜⎝

⎛minusρ=

RV

dndp


29




CzdnRVp

2

=γ+ρ+ intCz2

Vp2

=γ+ρ+




RVa

dsdVVa

2

ns ==


30





CdnRVzp

2

=ρ+γ+ intC2

Vzp2

=ρ+γ+




31

HeadHead


czg2

VP 2

=++γ






32





33

Example 34 Solution





34




35



ttanconsrzp =+




Point (1)~(2)

36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


3






4





5



6






7






8




nRVs

dsdVVnasa

dtVda

2

nsrrrr

rr

+=+==

RVa

dsdVVa

2





先應用先應用

9




10



11



sVVV

sVmVmaF SS part

partρδ=

partpart


sumδ SF


12








⎜⎝⎛

partpart


sasVV

spsin ρ=

partpart

ρ=partpart


2s

sppSδ

partpart

=δ

單位體積

13



sasVV

spsin ρ=

partpart

ρ=partpart

minusθγminus

0spsin =partpart

minusθγminus





14


sasVV

spsin ρ=

partpart

ρ=partpart


( )

CgzV21dp

0dzVd21dp

dsdV

21

dsdp

dsdz

2

22

=++ρ

gtgtgtgt





15




ttanconsz2

Vp2


CgzV21dp 2 =++




16




⎟⎟⎠

⎞⎜⎜⎝

⎛+= 3

3




17


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=⎟⎟

⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

partpart

=partpart

4

3

3

32

04

30

3

3

0 xa

xa1V3

xaV3

xa1V

xVV

sVV



sVV

sp

partpart

ρminus=partpart

(1)(1) sasVV

spsin ρ=

partpart

ρ=partpart

minusθγminus


( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)

18



( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛ρminus=

2)xa(

xaVp

632

0

19




20


2V

2Vpp

20

21

12 ρ=ρ=minus


z1=z2

2

22

21

21

1 z2

Vpz2

Vp γ+ρ+=γ+ρ+



21



RVV

RmVF

22

nρδ

=δ


sumδ nF



22







VRVV

npcosFWF

2

pnnn δρ

=δ⎟⎠⎞

⎜⎝⎛

partpart


RV

npcos

2ρ=

partpart


2n

nppnδ

partpart

=δ

23


RV

dndp

dndz 2ρ

=minusγminus

RV

npcos

2ρ=

partpart






CgzdnRVdp 2

=++ρint int






24




CzdnRVp

2


CgzdnRVdp 2

=++ρint int



25



RV

dndp

dndz 2ρ

=minusγminus

RV

dndp 2ρ


26


27



)b(r




RV

npcos

2ρ=

partpart

minusθγminus

28


( ) 02

022

1 prrC21p +minusρ=

rV

rp 2ρ=

partpart



rCrp 2

1ρ=partpart

3

22

rC

rp ρ=

partpart


0220

22 p

r1

r1C

21p +⎟⎟

⎠

⎞⎜⎜⎝

⎛minusρ=

RV

dndp


29




CzdnRVp

2

=γ+ρ+ intCz2

Vp2

=γ+ρ+




RVa

dsdVVa

2

ns ==


30





CdnRVzp

2

=ρ+γ+ intC2

Vzp2

=ρ+γ+




31

HeadHead


czg2

VP 2

=++γ






32





33

Example 34 Solution





34




35



ttanconsrzp =+




Point (1)~(2)

36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


4





5



6






7






8




nRVs

dsdVVnasa

dtVda

2

nsrrrr

rr

+=+==

RVa

dsdVVa

2





先應用先應用

9




10



11



sVVV

sVmVmaF SS part

partρδ=

partpart


sumδ SF


12








⎜⎝⎛

partpart


sasVV

spsin ρ=

partpart

ρ=partpart


2s

sppSδ

partpart

=δ

單位體積

13



sasVV

spsin ρ=

partpart

ρ=partpart

minusθγminus

0spsin =partpart

minusθγminus





14


sasVV

spsin ρ=

partpart

ρ=partpart


( )

CgzV21dp

0dzVd21dp

dsdV

21

dsdp

dsdz

2

22

=++ρ

gtgtgtgt





15




ttanconsz2

Vp2


CgzV21dp 2 =++




16




⎟⎟⎠

⎞⎜⎜⎝

⎛+= 3

3




17


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=⎟⎟

⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

partpart

=partpart

4

3

3

32

04

30

3

3

0 xa

xa1V3

xaV3

xa1V

xVV

sVV



sVV

sp

partpart

ρminus=partpart

(1)(1) sasVV

spsin ρ=

partpart

ρ=partpart

minusθγminus


( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)

18



( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛ρminus=

2)xa(

xaVp

632

0

19




20


2V

2Vpp

20

21

12 ρ=ρ=minus


z1=z2

2

22

21

21

1 z2

Vpz2

Vp γ+ρ+=γ+ρ+



21



RVV

RmVF

22

nρδ

=δ


sumδ nF



22







VRVV

npcosFWF

2

pnnn δρ

=δ⎟⎠⎞

⎜⎝⎛

partpart


RV

npcos

2ρ=

partpart


2n

nppnδ

partpart

=δ

23


RV

dndp

dndz 2ρ

=minusγminus

RV

npcos

2ρ=

partpart






CgzdnRVdp 2

=++ρint int






24




CzdnRVp

2


CgzdnRVdp 2

=++ρint int



25



RV

dndp

dndz 2ρ

=minusγminus

RV

dndp 2ρ


26


27



)b(r




RV

npcos

2ρ=

partpart

minusθγminus

28


( ) 02

022

1 prrC21p +minusρ=

rV

rp 2ρ=

partpart



rCrp 2

1ρ=partpart

3

22

rC

rp ρ=

partpart


0220

22 p

r1

r1C

21p +⎟⎟

⎠

⎞⎜⎜⎝

⎛minusρ=

RV

dndp


29




CzdnRVp

2

=γ+ρ+ intCz2

Vp2

=γ+ρ+




RVa

dsdVVa

2

ns ==


30





CdnRVzp

2

=ρ+γ+ intC2

Vzp2

=ρ+γ+




31

HeadHead


czg2

VP 2

=++γ






32





33

Example 34 Solution





34




35



ttanconsrzp =+




Point (1)~(2)

36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


5



6






7






8




nRVs

dsdVVnasa

dtVda

2

nsrrrr

rr

+=+==

RVa

dsdVVa

2





先應用先應用

9




10



11



sVVV

sVmVmaF SS part

partρδ=

partpart


sumδ SF


12








⎜⎝⎛

partpart


sasVV

spsin ρ=

partpart

ρ=partpart


2s

sppSδ

partpart

=δ

單位體積

13



sasVV

spsin ρ=

partpart

ρ=partpart

minusθγminus

0spsin =partpart

minusθγminus





14


sasVV

spsin ρ=

partpart

ρ=partpart


( )

CgzV21dp

0dzVd21dp

dsdV

21

dsdp

dsdz

2

22

=++ρ

gtgtgtgt





15




ttanconsz2

Vp2


CgzV21dp 2 =++




16




⎟⎟⎠

⎞⎜⎜⎝

⎛+= 3

3




17


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=⎟⎟

⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

partpart

=partpart

4

3

3

32

04

30

3

3

0 xa

xa1V3

xaV3

xa1V

xVV

sVV



sVV

sp

partpart

ρminus=partpart

(1)(1) sasVV

spsin ρ=

partpart

ρ=partpart

minusθγminus


( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)

18



( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛ρminus=

2)xa(

xaVp

632

0

19




20


2V

2Vpp

20

21

12 ρ=ρ=minus


z1=z2

2

22

21

21

1 z2

Vpz2

Vp γ+ρ+=γ+ρ+



21



RVV

RmVF

22

nρδ

=δ


sumδ nF



22







VRVV

npcosFWF

2

pnnn δρ

=δ⎟⎠⎞

⎜⎝⎛

partpart


RV

npcos

2ρ=

partpart


2n

nppnδ

partpart

=δ

23


RV

dndp

dndz 2ρ

=minusγminus

RV

npcos

2ρ=

partpart






CgzdnRVdp 2

=++ρint int






24




CzdnRVp

2


CgzdnRVdp 2

=++ρint int



25



RV

dndp

dndz 2ρ

=minusγminus

RV

dndp 2ρ


26


27



)b(r




RV

npcos

2ρ=

partpart

minusθγminus

28


( ) 02

022

1 prrC21p +minusρ=

rV

rp 2ρ=

partpart



rCrp 2

1ρ=partpart

3

22

rC

rp ρ=

partpart


0220

22 p

r1

r1C

21p +⎟⎟

⎠

⎞⎜⎜⎝

⎛minusρ=

RV

dndp


29




CzdnRVp

2

=γ+ρ+ intCz2

Vp2

=γ+ρ+




RVa

dsdVVa

2

ns ==


30





CdnRVzp

2

=ρ+γ+ intC2

Vzp2

=ρ+γ+




31

HeadHead


czg2

VP 2

=++γ






32





33

Example 34 Solution





34




35



ttanconsrzp =+




Point (1)~(2)

36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


6






7






8




nRVs

dsdVVnasa

dtVda

2

nsrrrr

rr

+=+==

RVa

dsdVVa

2





先應用先應用

9




10



11



sVVV

sVmVmaF SS part

partρδ=

partpart


sumδ SF


12








⎜⎝⎛

partpart


sasVV

spsin ρ=

partpart

ρ=partpart


2s

sppSδ

partpart

=δ

單位體積

13



sasVV

spsin ρ=

partpart

ρ=partpart

minusθγminus

0spsin =partpart

minusθγminus





14


sasVV

spsin ρ=

partpart

ρ=partpart


( )

CgzV21dp

0dzVd21dp

dsdV

21

dsdp

dsdz

2

22

=++ρ

gtgtgtgt





15




ttanconsz2

Vp2


CgzV21dp 2 =++




16




⎟⎟⎠

⎞⎜⎜⎝

⎛+= 3

3




17


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=⎟⎟

⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

partpart

=partpart

4

3

3

32

04

30

3

3

0 xa

xa1V3

xaV3

xa1V

xVV

sVV



sVV

sp

partpart

ρminus=partpart

(1)(1) sasVV

spsin ρ=

partpart

ρ=partpart

minusθγminus


( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)

18



( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛ρminus=

2)xa(

xaVp

632

0

19




20


2V

2Vpp

20

21

12 ρ=ρ=minus


z1=z2

2

22

21

21

1 z2

Vpz2

Vp γ+ρ+=γ+ρ+



21



RVV

RmVF

22

nρδ

=δ


sumδ nF



22







VRVV

npcosFWF

2

pnnn δρ

=δ⎟⎠⎞

⎜⎝⎛

partpart


RV

npcos

2ρ=

partpart


2n

nppnδ

partpart

=δ

23


RV

dndp

dndz 2ρ

=minusγminus

RV

npcos

2ρ=

partpart






CgzdnRVdp 2

=++ρint int






24




CzdnRVp

2


CgzdnRVdp 2

=++ρint int



25



RV

dndp

dndz 2ρ

=minusγminus

RV

dndp 2ρ


26


27



)b(r




RV

npcos

2ρ=

partpart

minusθγminus

28


( ) 02

022

1 prrC21p +minusρ=

rV

rp 2ρ=

partpart



rCrp 2

1ρ=partpart

3

22

rC

rp ρ=

partpart


0220

22 p

r1

r1C

21p +⎟⎟

⎠

⎞⎜⎜⎝

⎛minusρ=

RV

dndp


29




CzdnRVp

2

=γ+ρ+ intCz2

Vp2

=γ+ρ+




RVa

dsdVVa

2

ns ==


30





CdnRVzp

2

=ρ+γ+ intC2

Vzp2

=ρ+γ+




31

HeadHead


czg2

VP 2

=++γ






32





33

Example 34 Solution





34




35



ttanconsrzp =+




Point (1)~(2)

36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


7






8




nRVs

dsdVVnasa

dtVda

2

nsrrrr

rr

+=+==

RVa

dsdVVa

2





先應用先應用

9




10



11



sVVV

sVmVmaF SS part

partρδ=

partpart


sumδ SF


12








⎜⎝⎛

partpart


sasVV

spsin ρ=

partpart

ρ=partpart


2s

sppSδ

partpart

=δ

單位體積

13



sasVV

spsin ρ=

partpart

ρ=partpart

minusθγminus

0spsin =partpart

minusθγminus





14


sasVV

spsin ρ=

partpart

ρ=partpart


( )

CgzV21dp

0dzVd21dp

dsdV

21

dsdp

dsdz

2

22

=++ρ

gtgtgtgt





15




ttanconsz2

Vp2


CgzV21dp 2 =++




16




⎟⎟⎠

⎞⎜⎜⎝

⎛+= 3

3




17


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=⎟⎟

⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

partpart

=partpart

4

3

3

32

04

30

3

3

0 xa

xa1V3

xaV3

xa1V

xVV

sVV



sVV

sp

partpart

ρminus=partpart

(1)(1) sasVV

spsin ρ=

partpart

ρ=partpart

minusθγminus


( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)

18



( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛ρminus=

2)xa(

xaVp

632

0

19




20


2V

2Vpp

20

21

12 ρ=ρ=minus


z1=z2

2

22

21

21

1 z2

Vpz2

Vp γ+ρ+=γ+ρ+



21



RVV

RmVF

22

nρδ

=δ


sumδ nF



22







VRVV

npcosFWF

2

pnnn δρ

=δ⎟⎠⎞

⎜⎝⎛

partpart


RV

npcos

2ρ=

partpart


2n

nppnδ

partpart

=δ

23


RV

dndp

dndz 2ρ

=minusγminus

RV

npcos

2ρ=

partpart






CgzdnRVdp 2

=++ρint int






24




CzdnRVp

2


CgzdnRVdp 2

=++ρint int



25



RV

dndp

dndz 2ρ

=minusγminus

RV

dndp 2ρ


26


27



)b(r




RV

npcos

2ρ=

partpart

minusθγminus

28


( ) 02

022

1 prrC21p +minusρ=

rV

rp 2ρ=

partpart



rCrp 2

1ρ=partpart

3

22

rC

rp ρ=

partpart


0220

22 p

r1

r1C

21p +⎟⎟

⎠

⎞⎜⎜⎝

⎛minusρ=

RV

dndp


29




CzdnRVp

2

=γ+ρ+ intCz2

Vp2

=γ+ρ+




RVa

dsdVVa

2

ns ==


30





CdnRVzp

2

=ρ+γ+ intC2

Vzp2

=ρ+γ+




31

HeadHead


czg2

VP 2

=++γ






32





33

Example 34 Solution





34




35



ttanconsrzp =+




Point (1)~(2)

36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


8




nRVs

dsdVVnasa

dtVda

2

nsrrrr

rr

+=+==

RVa

dsdVVa

2





先應用先應用

9




10



11



sVVV

sVmVmaF SS part

partρδ=

partpart


sumδ SF


12








⎜⎝⎛

partpart


sasVV

spsin ρ=

partpart

ρ=partpart


2s

sppSδ

partpart

=δ

單位體積

13



sasVV

spsin ρ=

partpart

ρ=partpart

minusθγminus

0spsin =partpart

minusθγminus





14


sasVV

spsin ρ=

partpart

ρ=partpart


( )

CgzV21dp

0dzVd21dp

dsdV

21

dsdp

dsdz

2

22

=++ρ

gtgtgtgt





15




ttanconsz2

Vp2


CgzV21dp 2 =++




16




⎟⎟⎠

⎞⎜⎜⎝

⎛+= 3

3




17


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=⎟⎟

⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

partpart

=partpart

4

3

3

32

04

30

3

3

0 xa

xa1V3

xaV3

xa1V

xVV

sVV



sVV

sp

partpart

ρminus=partpart

(1)(1) sasVV

spsin ρ=

partpart

ρ=partpart

minusθγminus


( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)

18



( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛ρminus=

2)xa(

xaVp

632

0

19




20


2V

2Vpp

20

21

12 ρ=ρ=minus


z1=z2

2

22

21

21

1 z2

Vpz2

Vp γ+ρ+=γ+ρ+



21



RVV

RmVF

22

nρδ

=δ


sumδ nF



22







VRVV

npcosFWF

2

pnnn δρ

=δ⎟⎠⎞

⎜⎝⎛

partpart


RV

npcos

2ρ=

partpart


2n

nppnδ

partpart

=δ

23


RV

dndp

dndz 2ρ

=minusγminus

RV

npcos

2ρ=

partpart






CgzdnRVdp 2

=++ρint int






24




CzdnRVp

2


CgzdnRVdp 2

=++ρint int



25



RV

dndp

dndz 2ρ

=minusγminus

RV

dndp 2ρ


26


27



)b(r




RV

npcos

2ρ=

partpart

minusθγminus

28


( ) 02

022

1 prrC21p +minusρ=

rV

rp 2ρ=

partpart



rCrp 2

1ρ=partpart

3

22

rC

rp ρ=

partpart


0220

22 p

r1

r1C

21p +⎟⎟

⎠

⎞⎜⎜⎝

⎛minusρ=

RV

dndp


29




CzdnRVp

2

=γ+ρ+ intCz2

Vp2

=γ+ρ+




RVa

dsdVVa

2

ns ==


30





CdnRVzp

2

=ρ+γ+ intC2

Vzp2

=ρ+γ+




31

HeadHead


czg2

VP 2

=++γ






32





33

Example 34 Solution





34




35



ttanconsrzp =+




Point (1)~(2)

36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


9




10



11



sVVV

sVmVmaF SS part

partρδ=

partpart


sumδ SF


12








⎜⎝⎛

partpart


sasVV

spsin ρ=

partpart

ρ=partpart


2s

sppSδ

partpart

=δ

單位體積

13



sasVV

spsin ρ=

partpart

ρ=partpart

minusθγminus

0spsin =partpart

minusθγminus





14


sasVV

spsin ρ=

partpart

ρ=partpart


( )

CgzV21dp

0dzVd21dp

dsdV

21

dsdp

dsdz

2

22

=++ρ

gtgtgtgt





15




ttanconsz2

Vp2


CgzV21dp 2 =++




16




⎟⎟⎠

⎞⎜⎜⎝

⎛+= 3

3




17


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=⎟⎟

⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

partpart

=partpart

4

3

3

32

04

30

3

3

0 xa

xa1V3

xaV3

xa1V

xVV

sVV



sVV

sp

partpart

ρminus=partpart

(1)(1) sasVV

spsin ρ=

partpart

ρ=partpart

minusθγminus


( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)

18



( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛ρminus=

2)xa(

xaVp

632

0

19




20


2V

2Vpp

20

21

12 ρ=ρ=minus


z1=z2

2

22

21

21

1 z2

Vpz2

Vp γ+ρ+=γ+ρ+



21



RVV

RmVF

22

nρδ

=δ


sumδ nF



22







VRVV

npcosFWF

2

pnnn δρ

=δ⎟⎠⎞

⎜⎝⎛

partpart


RV

npcos

2ρ=

partpart


2n

nppnδ

partpart

=δ

23


RV

dndp

dndz 2ρ

=minusγminus

RV

npcos

2ρ=

partpart






CgzdnRVdp 2

=++ρint int






24




CzdnRVp

2


CgzdnRVdp 2

=++ρint int



25



RV

dndp

dndz 2ρ

=minusγminus

RV

dndp 2ρ


26


27



)b(r




RV

npcos

2ρ=

partpart

minusθγminus

28


( ) 02

022

1 prrC21p +minusρ=

rV

rp 2ρ=

partpart



rCrp 2

1ρ=partpart

3

22

rC

rp ρ=

partpart


0220

22 p

r1

r1C

21p +⎟⎟

⎠

⎞⎜⎜⎝

⎛minusρ=

RV

dndp


29




CzdnRVp

2

=γ+ρ+ intCz2

Vp2

=γ+ρ+




RVa

dsdVVa

2

ns ==


30





CdnRVzp

2

=ρ+γ+ intC2

Vzp2

=ρ+γ+




31

HeadHead


czg2

VP 2

=++γ






32





33

Example 34 Solution





34




35



ttanconsrzp =+




Point (1)~(2)

36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


10



11



sVVV

sVmVmaF SS part

partρδ=

partpart


sumδ SF


12








⎜⎝⎛

partpart


sasVV

spsin ρ=

partpart

ρ=partpart


2s

sppSδ

partpart

=δ

單位體積

13



sasVV

spsin ρ=

partpart

ρ=partpart

minusθγminus

0spsin =partpart

minusθγminus





14


sasVV

spsin ρ=

partpart

ρ=partpart


( )

CgzV21dp

0dzVd21dp

dsdV

21

dsdp

dsdz

2

22

=++ρ

gtgtgtgt





15




ttanconsz2

Vp2


CgzV21dp 2 =++




16




⎟⎟⎠

⎞⎜⎜⎝

⎛+= 3

3




17


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=⎟⎟

⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

partpart

=partpart

4

3

3

32

04

30

3

3

0 xa

xa1V3

xaV3

xa1V

xVV

sVV



sVV

sp

partpart

ρminus=partpart

(1)(1) sasVV

spsin ρ=

partpart

ρ=partpart

minusθγminus


( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)

18



( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛ρminus=

2)xa(

xaVp

632

0

19




20


2V

2Vpp

20

21

12 ρ=ρ=minus


z1=z2

2

22

21

21

1 z2

Vpz2

Vp γ+ρ+=γ+ρ+



21



RVV

RmVF

22

nρδ

=δ


sumδ nF



22







VRVV

npcosFWF

2

pnnn δρ

=δ⎟⎠⎞

⎜⎝⎛

partpart


RV

npcos

2ρ=

partpart


2n

nppnδ

partpart

=δ

23


RV

dndp

dndz 2ρ

=minusγminus

RV

npcos

2ρ=

partpart






CgzdnRVdp 2

=++ρint int






24




CzdnRVp

2


CgzdnRVdp 2

=++ρint int



25



RV

dndp

dndz 2ρ

=minusγminus

RV

dndp 2ρ


26


27



)b(r




RV

npcos

2ρ=

partpart

minusθγminus

28


( ) 02

022

1 prrC21p +minusρ=

rV

rp 2ρ=

partpart



rCrp 2

1ρ=partpart

3

22

rC

rp ρ=

partpart


0220

22 p

r1

r1C

21p +⎟⎟

⎠

⎞⎜⎜⎝

⎛minusρ=

RV

dndp


29




CzdnRVp

2

=γ+ρ+ intCz2

Vp2

=γ+ρ+




RVa

dsdVVa

2

ns ==


30





CdnRVzp

2

=ρ+γ+ intC2

Vzp2

=ρ+γ+




31

HeadHead


czg2

VP 2

=++γ






32





33

Example 34 Solution





34




35



ttanconsrzp =+




Point (1)~(2)

36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


11



sVVV

sVmVmaF SS part

partρδ=

partpart


sumδ SF


12








⎜⎝⎛

partpart


sasVV

spsin ρ=

partpart

ρ=partpart


2s

sppSδ

partpart

=δ

單位體積

13



sasVV

spsin ρ=

partpart

ρ=partpart

minusθγminus

0spsin =partpart

minusθγminus





14


sasVV

spsin ρ=

partpart

ρ=partpart


( )

CgzV21dp

0dzVd21dp

dsdV

21

dsdp

dsdz

2

22

=++ρ

gtgtgtgt





15




ttanconsz2

Vp2


CgzV21dp 2 =++




16




⎟⎟⎠

⎞⎜⎜⎝

⎛+= 3

3




17


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=⎟⎟

⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

partpart

=partpart

4

3

3

32

04

30

3

3

0 xa

xa1V3

xaV3

xa1V

xVV

sVV



sVV

sp

partpart

ρminus=partpart

(1)(1) sasVV

spsin ρ=

partpart

ρ=partpart

minusθγminus


( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)

18



( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛ρminus=

2)xa(

xaVp

632

0

19




20


2V

2Vpp

20

21

12 ρ=ρ=minus


z1=z2

2

22

21

21

1 z2

Vpz2

Vp γ+ρ+=γ+ρ+



21



RVV

RmVF

22

nρδ

=δ


sumδ nF



22







VRVV

npcosFWF

2

pnnn δρ

=δ⎟⎠⎞

⎜⎝⎛

partpart


RV

npcos

2ρ=

partpart


2n

nppnδ

partpart

=δ

23


RV

dndp

dndz 2ρ

=minusγminus

RV

npcos

2ρ=

partpart






CgzdnRVdp 2

=++ρint int






24




CzdnRVp

2


CgzdnRVdp 2

=++ρint int



25



RV

dndp

dndz 2ρ

=minusγminus

RV

dndp 2ρ


26


27



)b(r




RV

npcos

2ρ=

partpart

minusθγminus

28


( ) 02

022

1 prrC21p +minusρ=

rV

rp 2ρ=

partpart



rCrp 2

1ρ=partpart

3

22

rC

rp ρ=

partpart


0220

22 p

r1

r1C

21p +⎟⎟

⎠

⎞⎜⎜⎝

⎛minusρ=

RV

dndp


29




CzdnRVp

2

=γ+ρ+ intCz2

Vp2

=γ+ρ+




RVa

dsdVVa

2

ns ==


30





CdnRVzp

2

=ρ+γ+ intC2

Vzp2

=ρ+γ+




31

HeadHead


czg2

VP 2

=++γ






32





33

Example 34 Solution





34




35



ttanconsrzp =+




Point (1)~(2)

36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


12








⎜⎝⎛

partpart


sasVV

spsin ρ=

partpart

ρ=partpart


2s

sppSδ

partpart

=δ

單位體積

13



sasVV

spsin ρ=

partpart

ρ=partpart

minusθγminus

0spsin =partpart

minusθγminus





14


sasVV

spsin ρ=

partpart

ρ=partpart


( )

CgzV21dp

0dzVd21dp

dsdV

21

dsdp

dsdz

2

22

=++ρ

gtgtgtgt





15




ttanconsz2

Vp2


CgzV21dp 2 =++




16




⎟⎟⎠

⎞⎜⎜⎝

⎛+= 3

3




17


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=⎟⎟

⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

partpart

=partpart

4

3

3

32

04

30

3

3

0 xa

xa1V3

xaV3

xa1V

xVV

sVV



sVV

sp

partpart

ρminus=partpart

(1)(1) sasVV

spsin ρ=

partpart

ρ=partpart

minusθγminus


( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)

18



( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛ρminus=

2)xa(

xaVp

632

0

19




20


2V

2Vpp

20

21

12 ρ=ρ=minus


z1=z2

2

22

21

21

1 z2

Vpz2

Vp γ+ρ+=γ+ρ+



21



RVV

RmVF

22

nρδ

=δ


sumδ nF



22







VRVV

npcosFWF

2

pnnn δρ

=δ⎟⎠⎞

⎜⎝⎛

partpart


RV

npcos

2ρ=

partpart


2n

nppnδ

partpart

=δ

23


RV

dndp

dndz 2ρ

=minusγminus

RV

npcos

2ρ=

partpart






CgzdnRVdp 2

=++ρint int






24




CzdnRVp

2


CgzdnRVdp 2

=++ρint int



25



RV

dndp

dndz 2ρ

=minusγminus

RV

dndp 2ρ


26


27



)b(r




RV

npcos

2ρ=

partpart

minusθγminus

28


( ) 02

022

1 prrC21p +minusρ=

rV

rp 2ρ=

partpart



rCrp 2

1ρ=partpart

3

22

rC

rp ρ=

partpart


0220

22 p

r1

r1C

21p +⎟⎟

⎠

⎞⎜⎜⎝

⎛minusρ=

RV

dndp


29




CzdnRVp

2

=γ+ρ+ intCz2

Vp2

=γ+ρ+




RVa

dsdVVa

2

ns ==


30





CdnRVzp

2

=ρ+γ+ intC2

Vzp2

=ρ+γ+




31

HeadHead


czg2

VP 2

=++γ






32





33

Example 34 Solution





34




35



ttanconsrzp =+




Point (1)~(2)

36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


13



sasVV

spsin ρ=

partpart

ρ=partpart

minusθγminus

0spsin =partpart

minusθγminus





14


sasVV

spsin ρ=

partpart

ρ=partpart


( )

CgzV21dp

0dzVd21dp

dsdV

21

dsdp

dsdz

2

22

=++ρ

gtgtgtgt





15




ttanconsz2

Vp2


CgzV21dp 2 =++




16




⎟⎟⎠

⎞⎜⎜⎝

⎛+= 3

3




17


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=⎟⎟

⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

partpart

=partpart

4

3

3

32

04

30

3

3

0 xa

xa1V3

xaV3

xa1V

xVV

sVV



sVV

sp

partpart

ρminus=partpart

(1)(1) sasVV

spsin ρ=

partpart

ρ=partpart

minusθγminus


( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)

18



( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛ρminus=

2)xa(

xaVp

632

0

19




20


2V

2Vpp

20

21

12 ρ=ρ=minus


z1=z2

2

22

21

21

1 z2

Vpz2

Vp γ+ρ+=γ+ρ+



21



RVV

RmVF

22

nρδ

=δ


sumδ nF



22







VRVV

npcosFWF

2

pnnn δρ

=δ⎟⎠⎞

⎜⎝⎛

partpart


RV

npcos

2ρ=

partpart


2n

nppnδ

partpart

=δ

23


RV

dndp

dndz 2ρ

=minusγminus

RV

npcos

2ρ=

partpart






CgzdnRVdp 2

=++ρint int






24




CzdnRVp

2


CgzdnRVdp 2

=++ρint int



25



RV

dndp

dndz 2ρ

=minusγminus

RV

dndp 2ρ


26


27



)b(r




RV

npcos

2ρ=

partpart

minusθγminus

28


( ) 02

022

1 prrC21p +minusρ=

rV

rp 2ρ=

partpart



rCrp 2

1ρ=partpart

3

22

rC

rp ρ=

partpart


0220

22 p

r1

r1C

21p +⎟⎟

⎠

⎞⎜⎜⎝

⎛minusρ=

RV

dndp


29




CzdnRVp

2

=γ+ρ+ intCz2

Vp2

=γ+ρ+




RVa

dsdVVa

2

ns ==


30





CdnRVzp

2

=ρ+γ+ intC2

Vzp2

=ρ+γ+




31

HeadHead


czg2

VP 2

=++γ






32





33

Example 34 Solution





34




35



ttanconsrzp =+




Point (1)~(2)

36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


14


sasVV

spsin ρ=

partpart

ρ=partpart


( )

CgzV21dp

0dzVd21dp

dsdV

21

dsdp

dsdz

2

22

=++ρ

gtgtgtgt





15




ttanconsz2

Vp2


CgzV21dp 2 =++




16




⎟⎟⎠

⎞⎜⎜⎝

⎛+= 3

3




17


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=⎟⎟

⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

partpart

=partpart

4

3

3

32

04

30

3

3

0 xa

xa1V3

xaV3

xa1V

xVV

sVV



sVV

sp

partpart

ρminus=partpart

(1)(1) sasVV

spsin ρ=

partpart

ρ=partpart

minusθγminus


( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)

18



( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛ρminus=

2)xa(

xaVp

632

0

19




20


2V

2Vpp

20

21

12 ρ=ρ=minus


z1=z2

2

22

21

21

1 z2

Vpz2

Vp γ+ρ+=γ+ρ+



21



RVV

RmVF

22

nρδ

=δ


sumδ nF



22







VRVV

npcosFWF

2

pnnn δρ

=δ⎟⎠⎞

⎜⎝⎛

partpart


RV

npcos

2ρ=

partpart


2n

nppnδ

partpart

=δ

23


RV

dndp

dndz 2ρ

=minusγminus

RV

npcos

2ρ=

partpart






CgzdnRVdp 2

=++ρint int






24




CzdnRVp

2


CgzdnRVdp 2

=++ρint int



25



RV

dndp

dndz 2ρ

=minusγminus

RV

dndp 2ρ


26


27



)b(r




RV

npcos

2ρ=

partpart

minusθγminus

28


( ) 02

022

1 prrC21p +minusρ=

rV

rp 2ρ=

partpart



rCrp 2

1ρ=partpart

3

22

rC

rp ρ=

partpart


0220

22 p

r1

r1C

21p +⎟⎟

⎠

⎞⎜⎜⎝

⎛minusρ=

RV

dndp


29




CzdnRVp

2

=γ+ρ+ intCz2

Vp2

=γ+ρ+




RVa

dsdVVa

2

ns ==


30





CdnRVzp

2

=ρ+γ+ intC2

Vzp2

=ρ+γ+




31

HeadHead


czg2

VP 2

=++γ






32





33

Example 34 Solution





34




35



ttanconsrzp =+




Point (1)~(2)

36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


15




ttanconsz2

Vp2


CgzV21dp 2 =++




16




⎟⎟⎠

⎞⎜⎜⎝

⎛+= 3

3




17


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=⎟⎟

⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

partpart

=partpart

4

3

3

32

04

30

3

3

0 xa

xa1V3

xaV3

xa1V

xVV

sVV



sVV

sp

partpart

ρminus=partpart

(1)(1) sasVV

spsin ρ=

partpart

ρ=partpart

minusθγminus


( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)

18



( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛ρminus=

2)xa(

xaVp

632

0

19




20


2V

2Vpp

20

21

12 ρ=ρ=minus


z1=z2

2

22

21

21

1 z2

Vpz2

Vp γ+ρ+=γ+ρ+



21



RVV

RmVF

22

nρδ

=δ


sumδ nF



22







VRVV

npcosFWF

2

pnnn δρ

=δ⎟⎠⎞

⎜⎝⎛

partpart


RV

npcos

2ρ=

partpart


2n

nppnδ

partpart

=δ

23


RV

dndp

dndz 2ρ

=minusγminus

RV

npcos

2ρ=

partpart






CgzdnRVdp 2

=++ρint int






24




CzdnRVp

2


CgzdnRVdp 2

=++ρint int



25



RV

dndp

dndz 2ρ

=minusγminus

RV

dndp 2ρ


26


27



)b(r




RV

npcos

2ρ=

partpart

minusθγminus

28


( ) 02

022

1 prrC21p +minusρ=

rV

rp 2ρ=

partpart



rCrp 2

1ρ=partpart

3

22

rC

rp ρ=

partpart


0220

22 p

r1

r1C

21p +⎟⎟

⎠

⎞⎜⎜⎝

⎛minusρ=

RV

dndp


29




CzdnRVp

2

=γ+ρ+ intCz2

Vp2

=γ+ρ+




RVa

dsdVVa

2

ns ==


30





CdnRVzp

2

=ρ+γ+ intC2

Vzp2

=ρ+γ+




31

HeadHead


czg2

VP 2

=++γ






32





33

Example 34 Solution





34




35



ttanconsrzp =+




Point (1)~(2)

36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


16




⎟⎟⎠

⎞⎜⎜⎝

⎛+= 3

3




17


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=⎟⎟

⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

partpart

=partpart

4

3

3

32

04

30

3

3

0 xa

xa1V3

xaV3

xa1V

xVV

sVV



sVV

sp

partpart

ρminus=partpart

(1)(1) sasVV

spsin ρ=

partpart

ρ=partpart

minusθγminus


( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)

18



( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛ρminus=

2)xa(

xaVp

632

0

19




20


2V

2Vpp

20

21

12 ρ=ρ=minus


z1=z2

2

22

21

21

1 z2

Vpz2

Vp γ+ρ+=γ+ρ+



21



RVV

RmVF

22

nρδ

=δ


sumδ nF



22







VRVV

npcosFWF

2

pnnn δρ

=δ⎟⎠⎞

⎜⎝⎛

partpart


RV

npcos

2ρ=

partpart


2n

nppnδ

partpart

=δ

23


RV

dndp

dndz 2ρ

=minusγminus

RV

npcos

2ρ=

partpart






CgzdnRVdp 2

=++ρint int






24




CzdnRVp

2


CgzdnRVdp 2

=++ρint int



25



RV

dndp

dndz 2ρ

=minusγminus

RV

dndp 2ρ


26


27



)b(r




RV

npcos

2ρ=

partpart

minusθγminus

28


( ) 02

022

1 prrC21p +minusρ=

rV

rp 2ρ=

partpart



rCrp 2

1ρ=partpart

3

22

rC

rp ρ=

partpart


0220

22 p

r1

r1C

21p +⎟⎟

⎠

⎞⎜⎜⎝

⎛minusρ=

RV

dndp


29




CzdnRVp

2

=γ+ρ+ intCz2

Vp2

=γ+ρ+




RVa

dsdVVa

2

ns ==


30





CdnRVzp

2

=ρ+γ+ intC2

Vzp2

=ρ+γ+




31

HeadHead


czg2

VP 2

=++γ






32





33

Example 34 Solution





34




35



ttanconsrzp =+




Point (1)~(2)

36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


17


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus=⎟⎟

⎠

⎞⎜⎜⎝

⎛minus⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

partpart

=partpart

4

3

3

32

04

30

3

3

0 xa

xa1V3

xaV3

xa1V

xVV

sVV



sVV

sp

partpart

ρminus=partpart

(1)(1) sasVV

spsin ρ=

partpart

ρ=partpart

minusθγminus


( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)

18



( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛ρminus=

2)xa(

xaVp

632

0

19




20


2V

2Vpp

20

21

12 ρ=ρ=minus


z1=z2

2

22

21

21

1 z2

Vpz2

Vp γ+ρ+=γ+ρ+



21



RVV

RmVF

22

nρδ

=δ


sumδ nF



22







VRVV

npcosFWF

2

pnnn δρ

=δ⎟⎠⎞

⎜⎝⎛

partpart


RV

npcos

2ρ=

partpart


2n

nppnδ

partpart

=δ

23


RV

dndp

dndz 2ρ

=minusγminus

RV

npcos

2ρ=

partpart






CgzdnRVdp 2

=++ρint int






24




CzdnRVp

2


CgzdnRVdp 2

=++ρint int



25



RV

dndp

dndz 2ρ

=minusγminus

RV

dndp 2ρ


26


27



)b(r




RV

npcos

2ρ=

partpart

minusθγminus

28


( ) 02

022

1 prrC21p +minusρ=

rV

rp 2ρ=

partpart



rCrp 2

1ρ=partpart

3

22

rC

rp ρ=

partpart


0220

22 p

r1

r1C

21p +⎟⎟

⎠

⎞⎜⎜⎝

⎛minusρ=

RV

dndp


29




CzdnRVp

2

=γ+ρ+ intCz2

Vp2

=γ+ρ+




RVa

dsdVVa

2

ns ==


30





CdnRVzp

2

=ρ+γ+ intC2

Vzp2

=ρ+γ+




31

HeadHead


czg2

VP 2

=++γ






32





33

Example 34 Solution





34




35



ttanconsrzp =+




Point (1)~(2)

36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


18



( )4

3320

3

xxa1Va3

xp

sp +ρ

=partpart

=partpart

(2)(2)


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛ρminus=

2)xa(

xaVp

632

0

19




20


2V

2Vpp

20

21

12 ρ=ρ=minus


z1=z2

2

22

21

21

1 z2

Vpz2

Vp γ+ρ+=γ+ρ+



21



RVV

RmVF

22

nρδ

=δ


sumδ nF



22







VRVV

npcosFWF

2

pnnn δρ

=δ⎟⎠⎞

⎜⎝⎛

partpart


RV

npcos

2ρ=

partpart


2n

nppnδ

partpart

=δ

23


RV

dndp

dndz 2ρ

=minusγminus

RV

npcos

2ρ=

partpart






CgzdnRVdp 2

=++ρint int






24




CzdnRVp

2


CgzdnRVdp 2

=++ρint int



25



RV

dndp

dndz 2ρ

=minusγminus

RV

dndp 2ρ


26


27



)b(r




RV

npcos

2ρ=

partpart

minusθγminus

28


( ) 02

022

1 prrC21p +minusρ=

rV

rp 2ρ=

partpart



rCrp 2

1ρ=partpart

3

22

rC

rp ρ=

partpart


0220

22 p

r1

r1C

21p +⎟⎟

⎠

⎞⎜⎜⎝

⎛minusρ=

RV

dndp


29




CzdnRVp

2

=γ+ρ+ intCz2

Vp2

=γ+ρ+




RVa

dsdVVa

2

ns ==


30





CdnRVzp

2

=ρ+γ+ intC2

Vzp2

=ρ+γ+




31

HeadHead


czg2

VP 2

=++γ






32





33

Example 34 Solution





34




35



ttanconsrzp =+




Point (1)~(2)

36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


19




20


2V

2Vpp

20

21

12 ρ=ρ=minus


z1=z2

2

22

21

21

1 z2

Vpz2

Vp γ+ρ+=γ+ρ+



21



RVV

RmVF

22

nρδ

=δ


sumδ nF



22







VRVV

npcosFWF

2

pnnn δρ

=δ⎟⎠⎞

⎜⎝⎛

partpart


RV

npcos

2ρ=

partpart


2n

nppnδ

partpart

=δ

23


RV

dndp

dndz 2ρ

=minusγminus

RV

npcos

2ρ=

partpart






CgzdnRVdp 2

=++ρint int






24




CzdnRVp

2


CgzdnRVdp 2

=++ρint int



25



RV

dndp

dndz 2ρ

=minusγminus

RV

dndp 2ρ


26


27



)b(r




RV

npcos

2ρ=

partpart

minusθγminus

28


( ) 02

022

1 prrC21p +minusρ=

rV

rp 2ρ=

partpart



rCrp 2

1ρ=partpart

3

22

rC

rp ρ=

partpart


0220

22 p

r1

r1C

21p +⎟⎟

⎠

⎞⎜⎜⎝

⎛minusρ=

RV

dndp


29




CzdnRVp

2

=γ+ρ+ intCz2

Vp2

=γ+ρ+




RVa

dsdVVa

2

ns ==


30





CdnRVzp

2

=ρ+γ+ intC2

Vzp2

=ρ+γ+




31

HeadHead


czg2

VP 2

=++γ






32





33

Example 34 Solution





34




35



ttanconsrzp =+




Point (1)~(2)

36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


20


2V

2Vpp

20

21

12 ρ=ρ=minus


z1=z2

2

22

21

21

1 z2

Vpz2

Vp γ+ρ+=γ+ρ+



21



RVV

RmVF

22

nρδ

=δ


sumδ nF



22







VRVV

npcosFWF

2

pnnn δρ

=δ⎟⎠⎞

⎜⎝⎛

partpart


RV

npcos

2ρ=

partpart


2n

nppnδ

partpart

=δ

23


RV

dndp

dndz 2ρ

=minusγminus

RV

npcos

2ρ=

partpart






CgzdnRVdp 2

=++ρint int






24




CzdnRVp

2


CgzdnRVdp 2

=++ρint int



25



RV

dndp

dndz 2ρ

=minusγminus

RV

dndp 2ρ


26


27



)b(r




RV

npcos

2ρ=

partpart

minusθγminus

28


( ) 02

022

1 prrC21p +minusρ=

rV

rp 2ρ=

partpart



rCrp 2

1ρ=partpart

3

22

rC

rp ρ=

partpart


0220

22 p

r1

r1C

21p +⎟⎟

⎠

⎞⎜⎜⎝

⎛minusρ=

RV

dndp


29




CzdnRVp

2

=γ+ρ+ intCz2

Vp2

=γ+ρ+




RVa

dsdVVa

2

ns ==


30





CdnRVzp

2

=ρ+γ+ intC2

Vzp2

=ρ+γ+




31

HeadHead


czg2

VP 2

=++γ






32





33

Example 34 Solution





34




35



ttanconsrzp =+




Point (1)~(2)

36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


21



RVV

RmVF

22

nρδ

=δ


sumδ nF



22







VRVV

npcosFWF

2

pnnn δρ

=δ⎟⎠⎞

⎜⎝⎛

partpart


RV

npcos

2ρ=

partpart


2n

nppnδ

partpart

=δ

23


RV

dndp

dndz 2ρ

=minusγminus

RV

npcos

2ρ=

partpart






CgzdnRVdp 2

=++ρint int






24




CzdnRVp

2


CgzdnRVdp 2

=++ρint int



25



RV

dndp

dndz 2ρ

=minusγminus

RV

dndp 2ρ


26


27



)b(r




RV

npcos

2ρ=

partpart

minusθγminus

28


( ) 02

022

1 prrC21p +minusρ=

rV

rp 2ρ=

partpart



rCrp 2

1ρ=partpart

3

22

rC

rp ρ=

partpart


0220

22 p

r1

r1C

21p +⎟⎟

⎠

⎞⎜⎜⎝

⎛minusρ=

RV

dndp


29




CzdnRVp

2

=γ+ρ+ intCz2

Vp2

=γ+ρ+




RVa

dsdVVa

2

ns ==


30





CdnRVzp

2

=ρ+γ+ intC2

Vzp2

=ρ+γ+




31

HeadHead


czg2

VP 2

=++γ






32





33

Example 34 Solution





34




35



ttanconsrzp =+




Point (1)~(2)

36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


22







VRVV

npcosFWF

2

pnnn δρ

=δ⎟⎠⎞

⎜⎝⎛

partpart


RV

npcos

2ρ=

partpart


2n

nppnδ

partpart

=δ

23


RV

dndp

dndz 2ρ

=minusγminus

RV

npcos

2ρ=

partpart






CgzdnRVdp 2

=++ρint int






24




CzdnRVp

2


CgzdnRVdp 2

=++ρint int



25



RV

dndp

dndz 2ρ

=minusγminus

RV

dndp 2ρ


26


27



)b(r




RV

npcos

2ρ=

partpart

minusθγminus

28


( ) 02

022

1 prrC21p +minusρ=

rV

rp 2ρ=

partpart



rCrp 2

1ρ=partpart

3

22

rC

rp ρ=

partpart


0220

22 p

r1

r1C

21p +⎟⎟

⎠

⎞⎜⎜⎝

⎛minusρ=

RV

dndp


29




CzdnRVp

2

=γ+ρ+ intCz2

Vp2

=γ+ρ+




RVa

dsdVVa

2

ns ==


30





CdnRVzp

2

=ρ+γ+ intC2

Vzp2

=ρ+γ+




31

HeadHead


czg2

VP 2

=++γ






32





33

Example 34 Solution





34




35



ttanconsrzp =+




Point (1)~(2)

36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


23


RV

dndp

dndz 2ρ

=minusγminus

RV

npcos

2ρ=

partpart






CgzdnRVdp 2

=++ρint int






24




CzdnRVp

2


CgzdnRVdp 2

=++ρint int



25



RV

dndp

dndz 2ρ

=minusγminus

RV

dndp 2ρ


26


27



)b(r




RV

npcos

2ρ=

partpart

minusθγminus

28


( ) 02

022

1 prrC21p +minusρ=

rV

rp 2ρ=

partpart



rCrp 2

1ρ=partpart

3

22

rC

rp ρ=

partpart


0220

22 p

r1

r1C

21p +⎟⎟

⎠

⎞⎜⎜⎝

⎛minusρ=

RV

dndp


29




CzdnRVp

2

=γ+ρ+ intCz2

Vp2

=γ+ρ+




RVa

dsdVVa

2

ns ==


30





CdnRVzp

2

=ρ+γ+ intC2

Vzp2

=ρ+γ+




31

HeadHead


czg2

VP 2

=++γ






32





33

Example 34 Solution





34




35



ttanconsrzp =+




Point (1)~(2)

36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


24




CzdnRVp

2


CgzdnRVdp 2

=++ρint int



25



RV

dndp

dndz 2ρ

=minusγminus

RV

dndp 2ρ


26


27



)b(r




RV

npcos

2ρ=

partpart

minusθγminus

28


( ) 02

022

1 prrC21p +minusρ=

rV

rp 2ρ=

partpart



rCrp 2

1ρ=partpart

3

22

rC

rp ρ=

partpart


0220

22 p

r1

r1C

21p +⎟⎟

⎠

⎞⎜⎜⎝

⎛minusρ=

RV

dndp


29




CzdnRVp

2

=γ+ρ+ intCz2

Vp2

=γ+ρ+




RVa

dsdVVa

2

ns ==


30





CdnRVzp

2

=ρ+γ+ intC2

Vzp2

=ρ+γ+




31

HeadHead


czg2

VP 2

=++γ






32





33

Example 34 Solution





34




35



ttanconsrzp =+




Point (1)~(2)

36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


25



RV

dndp

dndz 2ρ

=minusγminus

RV

dndp 2ρ


26


27



)b(r




RV

npcos

2ρ=

partpart

minusθγminus

28


( ) 02

022

1 prrC21p +minusρ=

rV

rp 2ρ=

partpart



rCrp 2

1ρ=partpart

3

22

rC

rp ρ=

partpart


0220

22 p

r1

r1C

21p +⎟⎟

⎠

⎞⎜⎜⎝

⎛minusρ=

RV

dndp


29




CzdnRVp

2

=γ+ρ+ intCz2

Vp2

=γ+ρ+




RVa

dsdVVa

2

ns ==


30





CdnRVzp

2

=ρ+γ+ intC2

Vzp2

=ρ+γ+




31

HeadHead


czg2

VP 2

=++γ






32





33

Example 34 Solution





34




35



ttanconsrzp =+




Point (1)~(2)

36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


26


27



)b(r




RV

npcos

2ρ=

partpart

minusθγminus

28


( ) 02

022

1 prrC21p +minusρ=

rV

rp 2ρ=

partpart



rCrp 2

1ρ=partpart

3

22

rC

rp ρ=

partpart


0220

22 p

r1

r1C

21p +⎟⎟

⎠

⎞⎜⎜⎝

⎛minusρ=

RV

dndp


29




CzdnRVp

2

=γ+ρ+ intCz2

Vp2

=γ+ρ+




RVa

dsdVVa

2

ns ==


30





CdnRVzp

2

=ρ+γ+ intC2

Vzp2

=ρ+γ+




31

HeadHead


czg2

VP 2

=++γ






32





33

Example 34 Solution





34




35



ttanconsrzp =+




Point (1)~(2)

36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


27



)b(r




RV

npcos

2ρ=

partpart

minusθγminus

28


( ) 02

022

1 prrC21p +minusρ=

rV

rp 2ρ=

partpart



rCrp 2

1ρ=partpart

3

22

rC

rp ρ=

partpart


0220

22 p

r1

r1C

21p +⎟⎟

⎠

⎞⎜⎜⎝

⎛minusρ=

RV

dndp


29




CzdnRVp

2

=γ+ρ+ intCz2

Vp2

=γ+ρ+




RVa

dsdVVa

2

ns ==


30





CdnRVzp

2

=ρ+γ+ intC2

Vzp2

=ρ+γ+




31

HeadHead


czg2

VP 2

=++γ






32





33

Example 34 Solution





34




35



ttanconsrzp =+




Point (1)~(2)

36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


28


( ) 02

022

1 prrC21p +minusρ=

rV

rp 2ρ=

partpart



rCrp 2

1ρ=partpart

3

22

rC

rp ρ=

partpart


0220

22 p

r1

r1C

21p +⎟⎟

⎠

⎞⎜⎜⎝

⎛minusρ=

RV

dndp


29




CzdnRVp

2

=γ+ρ+ intCz2

Vp2

=γ+ρ+




RVa

dsdVVa

2

ns ==


30





CdnRVzp

2

=ρ+γ+ intC2

Vzp2

=ρ+γ+




31

HeadHead


czg2

VP 2

=++γ






32





33

Example 34 Solution





34




35



ttanconsrzp =+




Point (1)~(2)

36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


29




CzdnRVp

2

=γ+ρ+ intCz2

Vp2

=γ+ρ+




RVa

dsdVVa

2

ns ==


30





CdnRVzp

2

=ρ+γ+ intC2

Vzp2

=ρ+γ+




31

HeadHead


czg2

VP 2

=++γ






32





33

Example 34 Solution





34




35



ttanconsrzp =+




Point (1)~(2)

36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


30





CdnRVzp

2

=ρ+γ+ intC2

Vzp2

=ρ+γ+




31

HeadHead


czg2

VP 2

=++γ






32





33

Example 34 Solution





34




35



ttanconsrzp =+




Point (1)~(2)

36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


31

HeadHead


czg2

VP 2

=++γ






32





33

Example 34 Solution





34




35



ttanconsrzp =+




Point (1)~(2)

36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


32





33

Example 34 Solution





34




35



ttanconsrzp =+




Point (1)~(2)

36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


33

Example 34 Solution





34




35



ttanconsrzp =+




Point (1)~(2)

36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


34




35



ttanconsrzp =+




Point (1)~(2)

36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


35



ttanconsrzp =+




Point (1)~(2)

36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


36

intρminus= minus4

3

z

z

2

343 dzRVrhp


334z

z

2

4 rzprz)dz(R

Vp4

3

+=+minusρ+ int


Point (3)~(4)


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


37




C2

Vzp2




38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


38






39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


39



2112 V








40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


40



41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


41






42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


42



ttanconspz2

Vp T

2

==γ+ρ+



43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


43


ρminus=gtgt

ρ=minus

==asymp

ρ+==

)pp(2V

2Vpp

pppzz

2Vppp

43

243

14

41

232







44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


44



45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


45


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


46






47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


47




48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


48



( ) 21

12

31

PP2V

PP

⎥⎦

⎤⎢⎣

⎡ρminus

=

=




49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


49




50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


50


2Vpp

21

12ρ

+=




3ftslug0017560=ρ


)abs(ftlb)9181456(


222322

+=

+=


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


51

psi13130ftlb918p 22 ==


psi131302Vpp

21

12 =ρ

=minus



52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


52



2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+





53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


53




54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


54



gh2h2V

2Vh

2

=ργ

=

ρ=γ

At point (5)

)Hh(g2V +=

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

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

55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


55

Free JetsFree Jets


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


56





2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

VV22gtVgtV11若若



57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


57




h

jc A

AC =


2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+



58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


58




59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


59


22

221

2

11 zV21pzV

21p γ+ρ+=γ+ρ+

22

21 V

21ghV

21

=+


(1)(1)


(2)(2)


22

12 Vd

4VD

4π

=π

2

22

21

21

1 z2Vpz

2Vp γ+

ρ+=γ+

ρ+

22

1 V)Dd(V = (3)




60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


60


sm266)m1m10(1

)m02)(sm819(2)Dd(1

gh2V4

2

42 =minus

=minus

=


Thus

sm04920)sm266()m10(4

VAVAQ 322211 =

π===


4

4

2

2

0 )(11

2])(1[2

DdghDdgh

VV

QQ

D minus=

minus==

infin=

61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


61




62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


62


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+

2212

13 V

21ppandp2V ρminus=

ρ=



(1)(1)


33

2

1

mkg261K)27315)(KkgmN9286(


=+sdotsdot

times+==ρ




63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


63


sm005420Vd4

VAQ 33

233 =

π==sm069

mkg261)mN1003(2p2V 3

231

3 =times

=ρ

=

Thus Thus

oror



22

23232212

mN2963mN)1373000(

)sm677)(mkg261(21mN1003V

21pp

=minus=

minustimes=ρminus=

64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


64



reading hreading h

65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


65


2211 VAVAQ ==




(1)(1)

22221

211 zpV

21pzpV

21p γ++=γ++

])AA(1[pV21)zz(pp 2

122


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


66




⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=γminus

2

1

222 A

A1pV21h)SG1(


oror(2)(2)


2122

2 minusminus

=


uarr- darr+



67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


67






68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


68



222111 VAVA ρ=ρ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


69







70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


70





71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


71




72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


72





73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


73


32332

2221

211 zV

21pzV

21pzV

21p γ+ρ+=γ+ρ+=γ+ρ+



(1)(1)

22


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

74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


74


22212

221

2112 V

21)zz(zV

21zV





(2)(2)

ftH 228=

75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


75



2211

222

211

VAVAQ

V21pV

21p

==

ρ+=ρ+

])AA(1[)pp(2AQ 212

212 minusρ

minus=





76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


76





77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


77


33O2H kgm850)kgm1000(850SG ==ρ=ρ

22

2

A2])AA(1[Q

pp2

12

21

minusρ=minus




360)m100m0060()DD(AA 221212 ===

])AA(1[)pp(2AQ 212

212 minusρ

minus= EqEq 320 320

78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


78




22

22

21 ])m060)(4[(2)3601()850)(050(pp

πminus

=minus


kPa161Nm1160])m060)(4[(2

)3601()kgm850()sm0050(pp2

22

2323

21

==π

minus=minus


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


79



212

212 )zz(1

)zz(g2bzQminus

minus=


22221111

22221

211

zbVVAzbVVAQ

zV21pzV

21p

====

γ+ρ+=γ+ρ+


理論值理論值

80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


80





12 gz2bzQ =

12 gz2V =



81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


81




82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


82



83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


83


sm614)m05m4880(1

)m4880m05)(sm819(2)m4880(bQ 2

2

2

=minus

minus=

212

212 )zz(1

)zz(g2zbQ

minusminus

=



212

212 )zz(1

)zz(g2bzQminus

minus= Eq321Eq321

84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


84



( )( ) sm834m05sm8192m4880gz2zbQ 22

12 ===

85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


85






86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


86



87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


87


gH2



251

211 Hg2

2tanC)gH2(

2tanHCVACQ θ

=θ

==


( ) ( )( ) ( )


QQ

2501

2501

H

H3

0

0 =θθ

=

88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


88



gp ρ

g2V 2

zH





Httanconszg2

VP 2==++



89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


89





zg2

VP 2++

γ

zP+

γ

g2V2




90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


90


Httanconszg2

VP 2==++

γ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


91


Httanconszg2

VP 2==++

γ

Httanconszg2

VP 2==++

γ

92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


92




93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


93




94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


94




95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


95



CgzV21dp 2 =++


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


96




22

2

11

21 z

g2V

PPln

gRTz

g2V

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++


1k1

=++intminus


ρint

2

22

2

21

21

1

1 gz2

VP1k

kgz2

VP1k

k++

ρ⎟⎠⎞

⎜⎝⎛

minus=++

ρ⎟⎠⎞

⎜⎝⎛

minus



97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


97




Speed of sound

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12Compressible flow


1

12 M2k

ppp

=minus

98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


98


21a

1

12 M2k

ppp

=minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus1M

21k1

ppp 1k

k

21a

1

12

99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


99



kPa512pp

4710M2k

ppp

12

21a

1

12

=minus

===minus


kPa714pp

5501M2

1k1p

pp

12

1kk

21a

1

12

==minus

==⎥⎥

⎦

⎤

⎢⎢

⎣

⎡minus⎟

⎠⎞

⎜⎝⎛ ++=

minus minus

100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


100




sVV

tVaS part

part+

partpart

=

( ) 0dzVd21dpds




2222

S

S12

11 zV21pds

tVzV

21p 2

1

γ+ρ++partpart

ρ=γ+ρ+ int



101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


101



102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


102


dtdVds

dtdVds

tV 2

1

2

1

S

S

S

Sl==

partpart


( )

ll

l

g20zg2dt

zd

gdtdzV

zdtdVz

2

2

=ωrArr=+rArr

ρ=γ=

γ+ρ=minusγ


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Fluid 03