FUNDAMENTALS OF FLUID MECHANICS Chapter 8 Pipe Flowtaiwan921.lib.ntu.edu.tw/mypdf/fluid08.pdf · Osborne Reynolds, a British scientist and mathematician, was the first to distinguish

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FUNDAMENTALS OFFUNDAMENTALS OFFLUID MECHANICSFLUID MECHANICS

Chapter 8 Pipe Flow Chapter 8 Pipe Flow

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

National Taiwan UniversityNational Taiwan University12/21/200912/21/2009

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MAIN TOPICSMAIN TOPICS

General Characteristics of Pipe FlowGeneral Characteristics of Pipe FlowFully Developed Laminar FlowFully Developed Laminar FlowFully Developed Turbulent FlowFully Developed Turbulent FlowDimensional Analysis of Pipe FlowDimensional Analysis of Pipe FlowPipe Flow ExamplesPipe Flow ExamplesPipe Pipe FlowrateFlowrate MeasurementMeasurement

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IntroductionIntroduction

Flows completely bounded by solid surfaces are called Flows completely bounded by solid surfaces are called INTERNAL INTERNAL FLOWSFLOWS which include flows through which include flows through pipespipes ((Round cross sectionRound cross section), ), ductsducts (NOT (NOT Round cross sectionRound cross section)), nozzles, diffusers, sudden , nozzles, diffusers, sudden contractions and expansions, valves, and fittings. contractions and expansions, valves, and fittings. The basic principles involved are independent of the crossThe basic principles involved are independent of the cross--sectional sectional shape, although the details of the flow may be dependent on it.shape, although the details of the flow may be dependent on it.The flow regime (laminar or turbulent) of internal flows is primThe flow regime (laminar or turbulent) of internal flows is primarily arily a function of the Reynolds number.a function of the Reynolds number.

Laminar flow: Can be solved analytically.Laminar flow: Can be solved analytically.Turbulent flow: Rely heavily on semiTurbulent flow: Rely heavily on semi--empirical theories and empirical theories and experimental data.experimental data.

定義Internal flows：流體完全被solid surface包圍的流動

Internal flow的基本原理與截面形狀無關，但流場的細節分佈則與截面形狀有關

分類

依solid surface形狀可分成 pipes、 ducts（非圓）、nozzle、…等等

如此分類的意義？務實，便於分析。分類的依據，指標？如何界分如此分類的意義？務實，便於分析。分類的依據，指標？如何界分？？

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General Characteristics of General Characteristics of Pipe FlowPipe Flow

先談談管流的一般特徵…

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Pipe SystemPipe System

A pipe system include the pipes themselves A pipe system include the pipes themselves (perhaps of more than one diameter), the various (perhaps of more than one diameter), the various fittings, the fittings, the flowrateflowrate control devices valves , and control devices valves , and the pumps or turbines.the pumps or turbines.

管系：管本身與管配件

號稱管系統，含括？

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Pipe Flow vs. Open Channel FlowPipe Flow vs. Open Channel Flow

Pipe flow: Flows completely filling the pipe. (a)Pipe flow: Flows completely filling the pipe. (a)The pressure gradient along the pipe is main driving force.The pressure gradient along the pipe is main driving force.Open channel flow: Flows without completely filling the Open channel flow: Flows without completely filling the pipe. (b)pipe. (b)The gravity alone is the driving force.The gravity alone is the driving force.

Pipe flow

Open channel flow

流體一部分與大氣接觸

相對pipe flow，有所謂open-channel flow

沒有填滿管子的『管流』就非管流

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Laminar or Turbulent Flow Laminar or Turbulent Flow 1/21/2

The flow of a fluid in a pipe may be The flow of a fluid in a pipe may be Laminar ? Laminar ? Turbulent ?Turbulent ?Osborne ReynoldsOsborne Reynolds, a British scientist and mathematician, , a British scientist and mathematician, was the was the first to distinguishfirst to distinguish the difference between these the difference between these classification of flow by using a classification of flow by using a simple apparatussimple apparatus as as shown.shown.

管流依？標準如此分類

Reynolds先進行觀察

調節流率大小，觀察染料的崩裂情形

會有後續的分類，來自Reynolds

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Laminar or Turbulent Flow Laminar or Turbulent Flow 2/22/2

For For ““small enough small enough flowrateflowrate”” the dye streak will remain as a the dye streak will remain as a wellwell--defined line as it flows along, with only slight blurring due defined line as it flows along, with only slight blurring due to molecular diffusion of the dye into the surrounding water.to molecular diffusion of the dye into the surrounding water.

For a somewhat larger For a somewhat larger ““intermediate intermediate flowrateflowrate”” the dye the dye fluctuates in time and space, and intermittent bursts of irregulfluctuates in time and space, and intermittent bursts of irregular ar behavior appear along the streak.behavior appear along the streak.

For For ““large enough large enough flowrateflowrate”” the dye streak almost the dye streak almost immediately become blurred and spreads across the entire pipe inimmediately become blurred and spreads across the entire pipe ina random fashion.a random fashion.

染料分子擴散到附近，並出現輕微的模糊現象染料隨時空波動，沿著streak出現間歇性的、不規則的崩裂染料開始Random的崩

裂，並逐漸擴散充滿整個管子

當flowrate很低、中度到很大…，染料的崩裂情況不同！

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Time Dependence of Time Dependence of Fluid Velocity at a PointFluid Velocity at a Point

在某一位置記錄速度變動

流率

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Indication of Indication of Laminar or Turbulent FlowLaminar or Turbulent Flow

The term The term flowrateflowrate should be replaced by Reynolds should be replaced by Reynolds number, ,where V is the average velocitynumber, ,where V is the average velocity in in the pipe.the pipe.It is It is not only the fluid velocitynot only the fluid velocity that determines the that determines the character of the flow character of the flow –– its density, viscosity, and the pipe its density, viscosity, and the pipe size are of equal importance.size are of equal importance.For general For general engineering purposeengineering purpose, the flow in a round pipe , the flow in a round pipe

LaminarLaminarTransitionalTransitionalTurbulentTurbulent

μρ= /VDR e

2100R e <

4000＞R e

剛剛提到分類的標準？指標？

有共識的指標

該指標內涵不只是速度而已，還包括流體的黏度…

一般工程應用的角度

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Reynolds Number Reynolds Number 1/21/2

In honor of Osborne Reynolds (1842~1912), the British engineer In honor of Osborne Reynolds (1842~1912), the British engineer who first demonstrated that this combination of variables could who first demonstrated that this combination of variables could be be used as a criterion to distinguish between laminar and turbulentused as a criterion to distinguish between laminar and turbulent flow.flow.The Reynolds number is a measure of the ration of the inertia foThe Reynolds number is a measure of the ration of the inertia forces rces to viscous forces.to viscous forces.If the Reynolds number is small (Re

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Reynolds Number Reynolds Number 2/22/2

Flows with very small Reynolds numbers are commonly referred to Flows with very small Reynolds numbers are commonly referred to as as ““creeping flowscreeping flows””..For large Reynolds number flow, the viscous effects are small For large Reynolds number flow, the viscous effects are small relative to inertial effects and for these cases it may be possirelative to inertial effects and for these cases it may be possible to ble to neglect the effect of viscosity and consider the problem as one neglect the effect of viscosity and consider the problem as one involving a involving a ““nonviscousnonviscous”” fluid.fluid.Flows with Flows with ““largelarge”” Reynolds number generally are turbulent. Flows Reynolds number generally are turbulent. Flows in which the inertia forces are in which the inertia forces are ““smallsmall”” compared with the viscous compared with the viscous forces are characteristically laminar flowsforces are characteristically laminar flows..

Reynolds number很低的flow稱為creeping flows

Re>>1，表示inertial force為主，viscous force重要性很低，viscous effect可以忽略，此種流體可考慮被歸類為無黏性流體

是一個衡量inertial force / viscous force的指標

Reynolds number大的流體一般為turbulent flow，inertial force

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Example 8.1 Laminar or Turbulent FlowExample 8.1 Laminar or Turbulent Flow

Water at a temperature of 50Water at a temperature of 50℉℉ flows through a pipe of diameter D flows through a pipe of diameter D = 0.73 in. (a) Determine the minimum time taken to fill a 10= 0.73 in. (a) Determine the minimum time taken to fill a 10--oz oz glass (volume= 0.125ft3) with water if the flow in the pipe is tglass (volume= 0.125ft3) with water if the flow in the pipe is to be o be laminar. (b) Determine the maximum time taken to fill the glass laminar. (b) Determine the maximum time taken to fill the glass if if the flow is to be turbulent. Repeat the calculation if the waterthe flow is to be turbulent. Repeat the calculation if the watertemperature is 140temperature is 140℉℉..

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Example 8.1 Example 8.1 SolutionSolution

If the flow in the pipe is to maintain laminar, the minimum timeIf the flow in the pipe is to maintain laminar, the minimum time to to fill the glass will occur if the Reynolds number is the maximum fill the glass will occur if the Reynolds number is the maximum allowed for laminar flow, typically Re=2100. Thus allowed for laminar flow, typically Re=2100. Thus

s/ft486.0D/2100V =ρμ=

s85.8....VD)4/(

VQVt 2 ==ρπ

==

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流體在管內的發展歷程？從進口處開始…

How flowing fluid developed within pipe

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Entrance Region and Entrance Region and Fully Developed Flow Fully Developed Flow 1/51/5

Any fluidAny fluid flowing in a pipeflowing in a pipe had to enter the pipe at some had to enter the pipe at some location.location.The region of flow near where the fluid enters the pipe is The region of flow near where the fluid enters the pipe is termed the termed the entrance regionentrance region..

由進口端開始說明管流的發展

稱為進口區

總要有..開始

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The fluid typically enters the pipe with a nearly uniform The fluid typically enters the pipe with a nearly uniform velocity profile at section (1).velocity profile at section (1).The region of flow near where the fluid enters the pipe is The region of flow near where the fluid enters the pipe is termed the entrance region.termed the entrance region.As the fluid moves through the pipe, viscous effects cause As the fluid moves through the pipe, viscous effects cause it to stick to the pipe wallit to stick to the pipe wall ((the no slip boundary the no slip boundary conditioncondition))..

一開始幾乎是uniform flow，不受管的影響，但由

於No slip boundary condition的存在，管壁開始影響，即viscous effect開始浮現

在進口處…幾乎可以看成uniform flows，但稍微前進之後…

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A A boundary layerboundary layer in which viscous effects are important is in which viscous effects are important is produced along the pipe wall such that the initial velocity produced along the pipe wall such that the initial velocity profile changes with distance along the pipe,x , until the profile changes with distance along the pipe,x , until the fluid reaches the end of the fluid reaches the end of the entrance length, section (2), entrance length, section (2), beyond which the velocity profile does not vary with x.beyond which the velocity profile does not vary with x.The boundary layer has grown in thickness to completely The boundary layer has grown in thickness to completely fill the pipe. ??? fill the pipe. ??? 受到viscous effect影響的範圍稱為boundary layer，在層內，流體速度由管壁的ZERO（no slip condition）向管中心增加

NOTE: velocity profile順著管流方向改變，到某一位置（進口區結束）就維持穩定不再改變：換言之，Boundary layer厚度一直有變化，到某一位置（進口區結束）就不再改變

有時候邊界層不見得厚到填滿整個pipe！

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Viscous effects are of considerable importance within the Viscous effects are of considerable importance within the boundary layer. Outside the boundary layer, the viscous boundary layer. Outside the boundary layer, the viscous effects are negligible.effects are negligible.The shape of the velocity profile in the pipe depends on The shape of the velocity profile in the pipe depends on whether the flow is laminar or turbulent, as does the length whether the flow is laminar or turbulent, as does the length of the entrance region, of the entrance region, llll ..

eR06.0D=l

l 6/1eR4.4D

=ll

For laminar flowFor laminar flow For turbulent flowFor turbulent flow

Dimensionless entrance length

進口區長度？

BLBL內外，內外，viscous effectviscous effect重重要程度不同！要程度不同！

前一頁提到BL不見得填滿整個PIPE..依流況而定

回頭說到進口區有多長？

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Once the fluid reaches the end of the entrance region, Once the fluid reaches the end of the entrance region, section (2), the flow is simpler to describe because section (2), the flow is simpler to describe because the the velocity is a function of only the distance from the pipe velocity is a function of only the distance from the pipe centerline, r, and independent of x.centerline, r, and independent of x.The flow between (2) and (3) is termedThe flow between (2) and (3) is termed fully developed.fully developed.

進口區之後稱為「完全發展區」，在「完全發展區」內velocity profile已經定型

發展過程結束，velocity profiles不再改變！

此階段的velocity profiles只是function of r

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從pressure distribution來看管流的發展過程

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Pressure Distribution along PipePressure Distribution along PipeIn the entrance region of a pipe, the fluid In the entrance region of a pipe, the fluid accelerates or decelerates as it flows. There is accelerates or decelerates as it flows. There is a balance between pressure, viscous, and a balance between pressure, viscous, and inertia (acceleration) force.inertia (acceleration) force.

The magnitude of the The magnitude of the pressure gradient is pressure gradient is constant.constant.

The magnitude of the The magnitude of the pressure gradient is larger pressure gradient is larger than that in the fully than that in the fully developed region.developed region.

0pxp

<Δ

=∂∂

l

從壓力降的觀點來看管流變化

此區壓力降梯度大於完全發展區

xp∂∂

還在變動中..已經固定

參與角力者眾多…

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Fully Developed Laminar FlowFully Developed Laminar FlowThere are numerous ways to derive important There are numerous ways to derive important results pertaining to fully developed laminar flow:results pertaining to fully developed laminar flow:

From F = ma applied directly to a fluid element.From F = ma applied directly to a fluid element.From the From the NavierNavier--Stokes equations of motionStokes equations of motionFrom dimensional analysis methodsFrom dimensional analysis methods

局限於完全發展區且區內為Laminar flow

不同的切入方法不同的切入方法求解管流的求解管流的velocity distributionvelocity distribution

先討論流況被歸類為Laminar flow者……

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F = ma F = ma 切入切入

先討論先討論Force balanceForce balance（無關流況）－剪力、壓力（無關流況）－剪力、壓力與重力與重力 mama。。

再面對不能迴避的問題再面對不能迴避的問題：：剪應力與速度的關係剪應力與速度的關係？？Laminar flowLaminar flow或或Turbulent flowTurbulent flow，其剪應力與速，其剪應力與速度關係不同！度關係不同！

Laminar flowLaminar flow者，剪應力與速度關係比較簡單，者，剪應力與速度關係比較簡單，這也就是何以先集中火力討論這也就是何以先集中火力討論Laminar flowLaminar flow。。

可以想像者，可以想像者，Turbulent flowTurbulent flow比較複雜！後頭再比較複雜！後頭再討論。討論。

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From F=ma From F=ma 1/81/8

Considering a fully developed axisymmetric laminar flow in a long, straight, constant diameter section of a pipe. The Fluid element The Fluid element is a circular cylinder of fluid of length l and radius r centered on the axis of a horizontal pipe of diameter D. 在管流中選一個fluid element

先忽略重力項先忽略重力項

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Because the velocity is not uniform across the pipe, the initially flat end of the cylinder of fluid at time t become distorted at time t+δt when the fluid element has moved to its new location along the pipe.If the flow is fully developed and steady, the distortion on each end of the fluid element is the same, and no part of the fluid experiences any acceleration as it flows.

0tV=

∂∂r

0ixuuVV =∂∂

=∇⋅rr

SteadySteady Fully developedFully developed

因為velocity非uniform，因此從t發展到t+δt，fluid element移動新位置時形狀也有改變

因為在完全發展區，fluid element兩端的形狀改變

是相同的，且在區內的加速度為零且在區內的加速度為零

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Cr? =τ⇒τr

2p τ=

Δl

( ) ( )r

2p0r2rpprp 221τ

=Δ

⇒=πτ−πΔ−−πl

l

Apply the NewtonApply the Newton’’s second Law to the cylinder of fluids second Law to the cylinder of fluid

xx maF =

The force balance The force balance

Basic balance in forces needed to drive each fluid particle Basic balance in forces needed to drive each fluid particle along the pipe with constant velocityalong the pipe with constant velocity

Not function of rNot function of r

Not function of rNot function of r

Independent of rIndependent of r，要如何才能做到？，要如何才能做到？

B.C. r=0 B.C. r=0 ττ=0=0r=D/2 r=D/2 ττ= = ττww

Dr2 wτ=τ

力已平衡且加速度為0

力平衡維持等速度移動

左邊與r無關

由B.C.求常數

先忽略重力項先忽略重力項

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D4p wτ=Δ l

The pressure drop and wall shear stress are related byThe pressure drop and wall shear stress are related by

Valid for both laminar and turbulent flow.Valid for both laminar and turbulent flow.

LaminarLaminar

drdu

μ−=τ

Dr2 wτ=τ

r2p τ

=Δl

到目前為止，無關流體為Laminar或Turbulent flow

物理意義：在完全發展區內壓力降與管壁剪應力平衡

開始假設是Laminar flow

Turbulent flow？剪應力關係不單純！

注意沒有放入注意沒有放入『『重力重力』』因為水平擺放因為水平擺放

後頭再討論

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Since Since

With the boundary conditions: u=0 at r=D/2 With the boundary conditions: u=0 at r=D/2

drdu

μ−=τ

12 Cr

4purdr

2pdu

r2

pdrdu

+⎟⎟⎠

⎞⎜⎜⎝

⎛μΔ

−=⇒μΔ

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛μΔ

−=

∫∫ ll

l

lμΔ

−=16

pDC2

1

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−

μτ

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−

μΔ

=

2w

2

C

22

Rr1

4D)r(u

Dr21V

Dr21

16pD)r(ul

Velocity distributionVelocity distribution

D4p wτ=Δ l

切記切記：因為有假設：因為有假設Laminar flowLaminar flow

因為存在這個關係讓後續變得可為，若是Turbulent flow，就沒這麼簡單

目標達成

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The shear stress distributionThe shear stress distribution

Volume Volume flowrateflowratel2pr

drdu Δ

=μ=τ

l

r

μΔπ

=

π==π=⋅= ∫∫

128pDQ

2VR.....rdr2)r(uAduQ

4

C4R

0A

PoiseuillePoiseuille’’ss LawLawValid for Laminar flow only

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Average velocityAverage velocity

Point of maximum velocityPoint of maximum velocity

lμΔ

=π

==32

pDRQ

AQV

2

2average

0drdu

= at r=0at r=0

average

2

max V24pRUuu =

μΔ

−===l

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Making adjustment to account for Making adjustment to account for nonhorizontalnonhorizontal pipespipes

θγ−Δ→Δ sinpp l θθ>0 if the flow is uphill>0 if the flow is uphillθθ

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Example 8.2 Laminar Pipe FlowExample 8.2 Laminar Pipe Flow

An oil with a viscosity of An oil with a viscosity of μμ= 0.40 N= 0.40 N··s/ms/m22 and density and density ρρ= 900 = 900 kg/mkg/m33 flows in a pipe of diameter D= 0.20m . (a) What pressure flows in a pipe of diameter D= 0.20m . (a) What pressure drop, pdrop, p11--pp22, is needed to produce a , is needed to produce a flowrateflowrate of Q=2.0of Q=2.0××1010--55 mm33/s if /s if the pipe is horizontal with xthe pipe is horizontal with x11=0 and x=0 and x22=10 m? (b) How steep a hill, =10 m? (b) How steep a hill, θθ,must the pipe be on if the oil is to flow through the pipe at t,must the pipe be on if the oil is to flow through the pipe at the he same rate as in part (a), but with psame rate as in part (a), but with p11=p=p22? (c) For the conditions of ? (c) For the conditions of part (b), if ppart (b), if p11=200 =200 kPakPa, what is the pressure at section, x, what is the pressure at section, x33=5 m, =5 m, where x is measured along the pipe?where x is measured along the pipe?

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Example 8.2 Example 8.2 SolutionSolution1/21/2

210087.2/VDR e

35


With pWith p11=p=p22 the length of the pipe, the length of the pipe, ll, does not appear in the , does not appear in the flowrateflowrateequationequation

kPa200ppp 321 ===

ΔΔp=0 for all p=0 for all ll

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從從NavierNavier Stokes equationStokes equation切入切入透過合理假設，簡化透過合理假設，簡化…………

37

From the From the NavierNavier--Stokes EquationsStokes Equations 1/31/3

General motion of an incompressible Newtonian fluid is General motion of an incompressible Newtonian fluid is governed by the continuity equation and the momentum governed by the continuity equation and the momentum equationequation

0V =⋅∇r

VgpVVtV 2 rrrrr

∇ν++ρ∇−

=∇⋅+∂∂

Steady flowSteady flow

kggrr

−=

For steady, fully developed flow in a pipe, the velocity For steady, fully developed flow in a pipe, the velocity contains only an axial component, which is a function of contains only an axial component, which is a function of only the radial coordinateonly the radial coordinate i)r(uV

rr=

不可壓縮的牛頓流體簡化Navier-Stokes equation

在完全發展區內

由簡化的momentum equation與連續方程式來描述

加速度為0

速度僅與r有關

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Equation of Motion Equation of Motion chapter 6chapter 6

These are the differential equations of motion for anyThese are the differential equations of motion for anyfluid fluid satisfying the continuum assumptionsatisfying the continuum assumption..How to solve u,v,w ?How to solve u,v,w ?

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

+∂∂

ρ=∂σ∂

+∂τ∂

+∂τ∂

+ρ

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

+∂∂

ρ=∂τ∂

+∂σ∂

+∂τ∂

+ρ

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

+∂∂

ρ=∂τ∂

+∂τ∂

+∂σ∂

+ρ

zww

ywv

xwu

tw

zyxg

zvw

yvv

xvu

tv

zyxg

zuw

yuv

xuu

tu

zyxg

zzyzxzz

zyyyxyy

zxyxxxx

zzyyxx maFmaFmaF δ=δδ=δδ=δ 微分型式的運動方程式

非線性方程式

39

StressStress--Deformation Deformation chapter 6chapter 6

The stresses must be The stresses must be expressed in terms of the expressed in terms of the velocity and pressure velocity and pressure field.field.

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

μ=τ=τ

⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

μ=τ=τ

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

μ=τ=τ

∂∂

μ+⋅∇μ−−=σ

∂∂

μ+⋅∇μ−−=σ

∂∂

μ+⋅∇μ−−=σ

zv

yw

zu

xw

yu

xv

zw2V

32p

yv2V

32p

xu2V

32p

zyyz

zxxz

yxxy

zz

yy

xx

r

r

r

Cartesian coordinates

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The The NavierNavier--Stokes Equations Stokes Equations chapter 6chapter 6

UnderUnder incompressible flow with constant viscosity incompressible flow with constant viscosity conditionsconditions, , the the NavierNavier--Stokes equations are reduced to:Stokes equations are reduced to:

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂

∂+

∂

∂μ+ρ+

∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

+∂∂

ρ

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂

∂+

∂

∂μ+ρ+

∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

+∂∂

ρ

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂

∂+

∂

∂μ+ρ+

∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

+∂∂

ρ

2

2

2

2

2

2

z

2

2

2

2

2

2

y

2

2

2

2

2

2

x

zw

yw

xwg

zp

zww

ywv

xwu

tw

zv

yv

xvg

yp

zvw

yvv

xvu

tv

zu

yu

xug

xp

zuw

yuv

xuu

tu

再來一步步透過假設，簡化Navier-Stokes equations

假設不可壓縮且黏度是constant

41


The flow is governed by a balance of pressure, weight, and The flow is governed by a balance of pressure, weight, and viscous forces in the flow direction.viscous forces in the flow direction.

Vkgp 2rr

∇μ=ρ+∇

0V =⋅∇r

Simplify the Simplify the NavierNavier--Stokes equationStokes equation

簡化後的Navier-Stokes equation

顯示在完全發展區內pressure、weight與viscous force形成平衡

連續方程式

42


⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

μ=θρ+∂∂

rur

rr1sing

xp

l

pxp.const

xp Δ−

=∂∂

→=∂∂

i)r(uVrr

=

Function of, at most, only xFunction of, at most, only x Function of ,at most, only rFunction of ,at most, only r

IntegratingIntegrating Velocity profile u(r)=Velocity profile u(r)=

B.C. (1) r = R , u = 0 ;B.C. (1) r = R , u = 0 ;(2) r = 0 , u < (2) r = 0 , u < ∞∞ or r = 0 or r = 0 ∂∂u/u/∂∂r=0r=0

積分＋邊界條件

43

From Dimensional Analysis From Dimensional Analysis 1/31/3

Assume that the pressure drop in the horizontal pie, Assume that the pressure drop in the horizontal pie, ΔΔp, is p, is a function of the average velocity of the fluid in the pipe, a function of the average velocity of the fluid in the pipe, V, the length of the pipe, V, the length of the pipe, ll, the pipe diameter, D, and the , the pipe diameter, D, and the viscosity of the fluid, viscosity of the fluid, μμ. .

),D,,V(Fp μ=Δ l Dimensional analysisDimensional analysis

⎟⎠⎞

⎜⎝⎛φ=

μΔ

DVpD l an unknown function of the length to an unknown function of the length to

diameter ratio of the pipe.diameter ratio of the pipe.

Chapter 7

函數關係？UNKNOWN

44


DC

VpD l=

μΔ

where C is a constant.where C is a constant.

2DVCp μ

=Δl lμ

Δπ==

4pD)C4/(AVQ

The value of C must be determined by theory or experiment. The value of C must be determined by theory or experiment. For a round pipe, C=32. For duct of other crossFor a round pipe, C=32. For duct of other cross--sectional sectional shapes, the value of C is different.shapes, the value of C is different.

2DV32p lμ=ΔFor a round pipeFor a round pipe

假設是一個線性關係

常數C可由理論或實驗來推算

對圓型管而言，C = 32

lμΔ

=π

==32

pDRQ

AQV

2

2average

45


f is termed the friction factor, or f is termed the friction factor, or sometimes the Darcy friction factor.sometimes the Darcy friction factor.

For a round pipeFor a round pipe DRe64

DVD64

VD/V32

Vp

221

2

221

lll=

ρμ

=ρ

μ=

ρΔ

2V

Dfp

2ρ=Δ

l

2V

Dpf 2ρ

Δ= l

D4p wτ=Δ l

2w

V8

Re64f

ρτ

==For laminar flowFor laminar flow

2DV32p lμ=Δ

物理意義：在完全發展區內壓力降與管壁剪應力平衡

46

Example 8.3 Laminar Pipe Flow Properties Example 8.3 Laminar Pipe Flow Properties 1/21/2

The The flowrateflowrate, Q, of corn syrup through the horizontal pipe shown in , Q, of corn syrup through the horizontal pipe shown in Figure E8.3 is to be monitored by measuring the pressure differeFigure E8.3 is to be monitored by measuring the pressure difference nce between sections (1) and (2). It is proposed that Q=Kbetween sections (1) and (2). It is proposed that Q=KΔΔp, where the p, where the calibration constant, K, is a function of temperature, T, becauscalibration constant, K, is a function of temperature, T, because of e of the variation of the syrupthe variation of the syrup’’s viscosity and density with temperature. s viscosity and density with temperature. These variations are given in Table E8.3. (a) Plot K(T) versus TThese variations are given in Table E8.3. (a) Plot K(T) versus T for for 6060°°FF≤≤T T ≤≤ 160160°°F. (b) Determine the wall shear stress and the pressure F. (b) Determine the wall shear stress and the pressure drop, drop, ΔΔp=pp=p11--pp22, for Q=0.5 ft, for Q=0.5 ft33/s and T=100/s and T=100°°F. (c) For the conditions F. (c) For the conditions of part (b), determine the nest pressure force.(of part (b), determine the nest pressure force.(ππDD22/4)/4)ΔΔp, and the p, and the nest shear force, nest shear force, ππDDllττww, on the fluid within the pipe between the , on the fluid within the pipe between the sections (1) and (2).sections (1) and (2).

47

Example 8.3 Laminar Pipe Flow Properties Example 8.3 Laminar Pipe Flow Properties 1/21/2

48


21001380.../VDR e

49


The new pressure force and viscous force on the fluid within theThe new pressure force and viscous force on the fluid within the pipe pipe between sections (1) and (2) isbetween sections (1) and (2) is

lb84.5...2D2F

lb84.5...p4DF

wv

2

p

==τπ=

==Δπ

=

l

The values of these two forces are the same. The net The values of these two forces are the same. The net force is zero; there is no acceleration.force is zero; there is no acceleration.

50

當管流是當管流是Turbulent flowTurbulent flow…………

51

Fully Developed Turbulent FlowFully Developed Turbulent Flow

Turbulent pipe flow is actually more likely to occur than Turbulent pipe flow is actually more likely to occur than laminar flow in practical situations.laminar flow in practical situations.Turbulent flow is a very complex process.Turbulent flow is a very complex process.Numerous persons have devoted considerable effort in an Numerous persons have devoted considerable effort in an attempting to understand the variety of baffling aspects of attempting to understand the variety of baffling aspects of turbulence. Although a considerable amount if knowledge turbulence. Although a considerable amount if knowledge about the topics has been developed, about the topics has been developed, the field of turbulent the field of turbulent flow still remains the least understood area of fluid flow still remains the least understood area of fluid mechanics.mechanics.Much remains to be learned about the nature of turbulent flow.Much remains to be learned about the nature of turbulent flow.

實務上，管流為turbulent pipe flow的機率遠大於Laminar pipe flow

即便有很多投入，但對於turbulent pipe flow的了解還是有限

Turbulent flow的特性還是最吸引人去關注的議題

紊流，一個複雜的過程…

52

Transition from Laminar to Turbulent Transition from Laminar to Turbulent Flow in a Pipe Flow in a Pipe 1/21/2

For any flow geometry, there is one (or more) For any flow geometry, there is one (or more) dimensionless parameters such as with this parameter dimensionless parameters such as with this parameter value below a particular value the flow is laminar, whereas value below a particular value the flow is laminar, whereas with the parameter value larger than a certain value the with the parameter value larger than a certain value the flow is turbulent.flow is turbulent.The important parameters involved and their critical The important parameters involved and their critical values depend on the specific flow situation involved.values depend on the specific flow situation involved.

Consider a long section of pipe that is Consider a long section of pipe that is initially filled with a fluid at rest.initially filled with a fluid at rest.

For flow in pipe : 21004000 For flow along a plate Rex~5000

之前所使用的指標，小於某一值歸屬Laminar flow，大於某一值歸屬Turbulent flow

管內先塞滿流體

因flow situation不同

為了了解transition

似乎很難一刀畫下，存在一個過渡的階段…

討論討論LaminarLaminar turbulentturbulent flowflow的過程的過程

53

Transition from Laminar to Turbulent Transition from Laminar to Turbulent Flow in a Pipe Flow in a Pipe 2/22/2

As the valve is opened to start the flow, the flow velocity and,As the valve is opened to start the flow, the flow velocity and, hence, hence, the Reynolds number increase from zero (no flow) to their the Reynolds number increase from zero (no flow) to their maximum steady flow values.maximum steady flow values.For the initial time period the Reynolds number is small enough For the initial time period the Reynolds number is small enough for for laminar flow to occur.laminar flow to occur.At some time the Reynolds At some time the Reynolds number reaches 2100, and the number reaches 2100, and the flow begins its transition to flow begins its transition to turbulent conditions.turbulent conditions.Intermittent spots or burst Intermittent spots or burst appearappear……....

閥打開，管內流體開始流動，Re由0 增加

觀察不同時間的速度變動，並計算Re

Re > 2100崩裂

速度出現隨機波動

Re>2100開始由transition

速度

54

Description for Turbulent Flow Description for Turbulent Flow 1/41/4

Turbulent flows involve Turbulent flows involve randomly fluctuating randomly fluctuating parameters.parameters.The character of many of the The character of many of the important properties of the important properties of the flow (pressure drop, heat flow (pressure drop, heat transfer, etc.) depends strongly transfer, etc.) depends strongly on the existence and nature of on the existence and nature of the turbulent fluctuations or the turbulent fluctuations or randomness.randomness.

The timeThe time--averaged, averaged, ūū, and , and fluctuating, fluctuating, úú description of a description of a parameter for tubular flow.parameter for tubular flow.

A typical trace of the axial component of A typical trace of the axial component of velocity measured at a given location in velocity measured at a given location in the flow, u=u(t).the flow, u=u(t).

流體的參數與紊流的隨機波動特性有很強烈的關聯

紊流下，流體參數也都出現隨機波動的特性在特定點追蹤速度的軸向分量

注意其中的速度定義

進入turbulent status

55


Turbulent flows are characterized by random, threeTurbulent flows are characterized by random, three--dimensional dimensional vorticityvorticity..Turbulent flows can be described in terms of their mean Turbulent flows can be described in terms of their mean values on which are superimposed the fluctuations. values on which are superimposed the fluctuations.

( )∫+

=Tt

t

O

O

dtt,z,y,xuT1u

'uuu +=uu'u −=

以隨機的，三維的漩渦來描述紊流

描述turbulent flow

放大

其他參數也有這種現象……也可以如此描述

56


The time average of the fluctuations is zero.The time average of the fluctuations is zero.

The square of a fluctuation quantity is positive.The square of a fluctuation quantity is positive.

Turbulence intensity or the level of the turbulence Turbulence intensity or the level of the turbulence

( ) ( ) 0uTuTT1dtuu

T1'u

Tt

t

O

O

=−=−= ∫+

( ) 0dt'uT1)'u(

Tt

t

22 O

O

>= ∫+

( )

u

dt'uT1

u)'u(

2Tt

t

22

O

O⎥⎦

⎤⎢⎣

⎡

==ℑ∫

+ The larger the turbulence intensity, the larger The larger the turbulence intensity, the larger the fluctuations of the velocity. Wellthe fluctuations of the velocity. Well--designed wind tunnels have typical value of designed wind tunnels have typical value of ℑℑ=0.01, although with extreme care, values =0.01, although with extreme care, values as low as as low as ℑℑ=0.0002 have been obtained.=0.0002 have been obtained.

紊流強度

Turbulence intensity越大，速度變動越大

57


In some situations, turbulent flow characteristics are In some situations, turbulent flow characteristics are advantages. In other situations, laminar flow is desirable.advantages. In other situations, laminar flow is desirable.►►Turbulence: mixing of fluids.Turbulence: mixing of fluids.►►Laminar: pressure drop in pipe, aerodynamic drag on Laminar: pressure drop in pipe, aerodynamic drag on

airplane.airplane.

Turbulent flow有其優點，當然也有其缺點；Laminar flow亦是有優點，也有缺點

58

Shear Stress for Laminar Flow Shear Stress for Laminar Flow 1/21/2

Laminar flow is modeled as fluid particles that flow smoothly along in layers, gliding past the slightly slower or faster ones on either side.The fluid actually consists of numerous molecules darting about in an almost random fashion. The motion is not entirely random – a slight bias in one direction.As the molecules dart across a given plane (plane A-A, for example), the ones moving upward have come from an area of smaller average x component of velocity than the ones moving downward, which have come from an area of large velocity.

Laminar flow的剪應力

不能迴避的問題，剪力與速度的關係

59

Shear Stress for Laminar Flow Shear Stress for Laminar Flow 2/22/2

The momentum flux in the x direction across plane A-A give rise to a drag of the lower fluid on the upper fluid and an equal but opposite effect of the upper fluid on the lower fluid. The sluggish molecules moving upward across plane A-A must accelerated by the fluid above this plane. The rate of change of momentum in this processproduces a shear force. Similarly, the more energetic molecules moving down across plane A-A must be slowed down by the fluid below that plane.BY combining these effects, we obtain the well-known Newton viscosity law

dydu

yx μ=τShear stress is present only if there is a Shear stress is present only if there is a gradient in u = gradient in u = u(yu(y).).

如果流況歸類為Laminar flow，則

60

Shear Stress for Turbulent Flow Shear Stress for Turbulent Flow 1/21/2

The turbulent flow is thought as a The turbulent flow is thought as a series of random, threeseries of random, three--dimensional eddy type motions.dimensional eddy type motions.These eddies range in size from These eddies range in size from very small diameter to fairly large very small diameter to fairly large diameter.diameter.This eddy structure greatly This eddy structure greatly promotes mixing within the fluid.promotes mixing within the fluid.

Turbulent flow的剪應力

簡言之，紊流的剪力與速度關係無法像Laminar flow一樣

一連串random、3D、eddy型態的運動

61

Shear Stress for Turbulent Flow Shear Stress for Turbulent Flow 2/22/2

The flow is represented by (timeThe flow is represented by (time--mean velocity ) plus umean velocity ) plus u’’ and vand v’’(time randomly fluctuating velocity components in the x and y (time randomly fluctuating velocity components in the x and y direction).direction).The shear stress on the plane AThe shear stress on the plane A--AA

u

turbulentarminla'v'udyud

τ+τ=ρ−μ=τ

'v'uρ is called Reynolds stress introduced by is called Reynolds stress introduced by Osborne Reynolds.Osborne Reynolds.

0'v'u →ρ As we approach wall, and is zero at the wall As we approach wall, and is zero at the wall (the wall tends to suppress the fluctuations.)(the wall tends to suppress the fluctuations.)

The shear stress is not merely proportional to the gradient of the time-averaged velocity, .)y(u

不像Laminar flow的剪應力與速度梯度存在簡單的關係

剪力非與速度梯度成正比關係

在wall附近

稱為Reynolds stress

62

Structure of Turbulent Flow in a Pipe Structure of Turbulent Flow in a Pipe 1/21/2

Near the wallNear the wall (the viscous (the viscous sublayersublayer), the), the laminar shear laminar shear stress stress ττlamlam is dominant.is dominant.Away fromAway from the wall (in the outer layer) ,the wall (in the outer layer) , the turbulent the turbulent shear stress shear stress ττturbturb is is dominantdominant. . The transition between these two regions occurs in the The transition between these two regions occurs in the overlap layer.overlap layer.

管內紊流結構

近管壁處Laminar flow主導；近管

心處Turbulent flow主導

transition

越靠近管中心τturb > τlam

剪力比

速度曲線

管壁附近laminar shear stress為主

怎麼辦？分靠近牆壁與遠離牆壁者兩區……

63

Structure of Turbulent Flow in a Pipe Structure of Turbulent Flow in a Pipe 2/22/2

The relative magnitude of The relative magnitude of ττlamlam compared to compared to ττturbturb is a is a complex function dependent on the specific flow involved.complex function dependent on the specific flow involved.Typically the value of Typically the value of ττturbturb is 100 to 1000 times greater is 100 to 1000 times greater than than ττlam lam inin the outer region.the outer region.

兩者間的相對大小，依流況而定，一般是100~1000倍

64

Alternative Form of Shear Stress Alternative Form of Shear Stress 1/21/2

ττturbturb: requiring an accurate knowledge of the fluctuations : requiring an accurate knowledge of the fluctuations uu’’ and vand v’’, or , or The shear stress for turbulent flow is given in terms of the The shear stress for turbulent flow is given in terms of the eddy viscosity eddy viscosity ηη..

dyud

turb η=τThis extension of of laminar flow terminology This extension of of laminar flow terminology was introduced by J. was introduced by J. BoussubesqBoussubesq, a French , a French scientist, in 1877.scientist, in 1877.

ηη?? A semiempirical theory was proposed by L. L. PrandtlPrandtl to determine the value of ηη

'v'uρ

另一種表達shear stress的方式強調：只是另一種表達方式而已

65

Alternative Form of Shear Stress Alternative Form of Shear Stress 2/22/2

dyud2

mlρ=η2

2mturb dy

ud⎟⎟⎠

⎞⎜⎜⎝

⎛ρ=τ l

mixing length, is not constant throughout the flow field.

There is no general, allThere is no general, all--encompassing, encompassing, useful model that can accurately predict useful model that can accurately predict the shear stress throughout a general the shear stress throughout a general incompressible, viscous turbulent flow.incompressible, viscous turbulent flow.

不管那一種方式，都一樣『不簡單』

沒通用，可全面涵蓋的沒通用，可全面涵蓋的modelmodel，可以精確預估，可以精確預估viscous viscous turbulent flowturbulent flow的的shear stressshear stress

66

Turbulent Velocity Profile Turbulent Velocity Profile 1/51/5

Fully developed turbulent flow in a pipe can be broken into threFully developed turbulent flow in a pipe can be broken into three e region: the viscous region: the viscous sublayersublayer, the overlap region, and the outer , the overlap region, and the outer turbulent turbulent sublayersublayer..Within the Within the viscous viscous sublayersublayer the the shear stress is dominantshear stress is dominant compared compared with the turbulent stress, and the random, eddying nature of thewith the turbulent stress, and the random, eddying nature of the flow flow is essentially absent.is essentially absent.In the outer In the outer turbulent layerturbulent layer the the Reynolds stress is dominantReynolds stress is dominant, and , and there is considerable mixing and randomness to the flow.there is considerable mixing and randomness to the flow.Within the viscous Within the viscous sublayersublayer the fluid viscosity is an important the fluid viscosity is an important parameter; the density is unimportant. In the outer layer the opparameter; the density is unimportant. In the outer layer the opposite posite is true.is true.因為沒有一個簡單的shear stress vs. velocity gradient，所以…

說明不同sublayer，主導之stress不同

在viscous sublayer中黏度重要、密度不重要

Shear stress與velocity關係不簡單之下，還是得面對…如何寫出velocity profiles

前面已講過完全發展管流內分成三區

67


Considerable information concerning turbulent velocity profiles Considerable information concerning turbulent velocity profiles has has been obtained through the use ofbeen obtained through the use of dimensional analysis, and semidimensional analysis, and semi--empirical theoretical effortsempirical theoretical efforts..In the viscous In the viscous sublayersublayer the velocity profile can be written in the velocity profile can be written in dimensionless form asdimensionless form as

++ =ν

== yyuuuu

*

*

( ) 2/1w* /u ρτ=Where y is the distance measured from the wall y=RWhere y is the distance measured from the wall y=R--r.r.

is called the friction velocity.is called the friction velocity.

Law of the wallLaw of the wall

Is valid very near the smooth wall, for 5yu0*

≤ν

≤

探討velocity profiles的方法：因次分析與半經驗公式

不同layer有不同的公式

管壁附近

Kinematic viscosity

取自半經驗公式

取自半經驗公式

68


In the In the outer regionouter region the velocity should vary as the the velocity should vary as the logarithm of ylogarithm of y

In In transition regiontransition region or buffer layer or buffer layer

for 30yu*

>ν

0.5y

yuln5.2uu

+⎟⎠

⎞⎜⎝

⎛=

∗

∗

30yu7-5*

≤ν

≤⎟⎠

⎞⎜⎝

⎛=

−∗ y

Rln5.2u

uU forfor

Determined experimentally

管中心附近

69


0.5y

yuln5.2uu

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

∗

∗

ν=

*

*yu

uu

靠近管壁

管中心

70


The velocity profile for turbulent flow through a smooth pipe may also be approximated by the empirical powerpower--law equationlaw equation

The powerThe power--law profile is not law profile is not applicable close to the wall.applicable close to the wall.

n/1n/1

Rr1

Ry

Uu

⎟⎠⎞

⎜⎝⎛ −=⎟

⎠⎞

⎜⎝⎛=

Where the exponent, n, varies Where the exponent, n, varies with the Reynolds number.with the Reynolds number.

另一種公式

可其限制－近管壁處不管用

71

Example 8.4 Turbulent Pipe Flow Example 8.4 Turbulent Pipe Flow PropertiesProperties

Water at 20Water at 20℃℃ ((ρρ=998kg/m=998kg/m33 and and νν=1.004=1.004××1010--66mm22/s) flows through /s) flows through a horizontal pipe of 0.1a horizontal pipe of 0.1--m diameter with a m diameter with a flowrateflowrate of Q=4of Q=4××1010--22mm33/s /s and a pressure gradient of 2.59 and a pressure gradient of 2.59 kPa/mkPa/m. (a) Determine the . (a) Determine the approximate thickness of the viscous approximate thickness of the viscous sublayersublayer. (b) Determine the . (b) Determine the approximate centerline velocity, approximate centerline velocity, VVcc. (c) Determine the ration of the . (c) Determine the ration of the turbulent to laminar shear stress, turbulent to laminar shear stress, ττturbturb//ττlamlam at a point midway at a point midway between the centerline and the pipe wall (i.e., at r=0.025m)between the centerline and the pipe wall (i.e., at r=0.025m)

72


The thickness of viscous The thickness of viscous sublayersublayer, , δδss , is approximately, is approximately

5u*

s =ν

δ*s u

5 ν=δ( ) s/m255.0.../u 2/1w* ==ρτ=

2w m/N8.64...4

pD==

Δ=τ

l

mm02.0m1097.1...u

5 5*s =×==ν

=δ −

The centerline velocity can be obtained from the average velocitThe centerline velocity can be obtained from the average velocity and y and the assumption of a powerthe assumption of a power--law velocity profile law velocity profile

s/m09.54/)m1.0(

s/m04.0AQV 2

3

=π

== 5e 1007.5.../VDR ×==ν=

73

Example 8.4 Example 8.4 SolutionSolution2/32/3n/1n/1

Rr1

Ry

Uu

⎟⎠⎞

⎜⎝⎛ −=⎟

⎠⎞

⎜⎝⎛=

VR)1n2)(1n(

nVR2...dAuAVQ 22

c2 π=

++π==== ∫

n=8.4n=8.45

e 1007.5.../VDR ×==υ=

s/m04.6...V)1n2)(1n(

n2VV

c

2

c

==++

=

Dr2 wτ=τ Valid for laminar or turbulent flow

2turblam

2w

m/N4.32)m1.0(

)m025.0)(m/N8.64(2D

r2

=τ+τ=

=τ

=τ

74


2turblam

2w

m/N4.32)m1.0(

)m025.0)(m/N8.64(2D

r2

=τ+τ=

=τ

=τ

2n/)n1(

clam m/N0266.0R

r1nRV

drud

=⎟⎠⎞

⎜⎝⎛ −μ−=μ−=τ

−

12200266.0

0266.04.32

lam

lam

lam

turb =−

=ττ−τ

=ττ

75

Dimensional Analysis of Dimensional Analysis of Pipe FlowPipe Flow

76

Energy Considerations Energy Considerations 1/81/8

Considering the steady flow through the piping system, includingConsidering the steady flow through the piping system, including a a reducing elbow. The basic equation for conservation of energy reducing elbow. The basic equation for conservation of energy –– the the first law of thermodynamicsfirst law of thermodynamics

inShaftinnetCS

2

CVWQAdnV)gz

2Vpû(Vde

t&&

rrr+=⋅ρ++

ρ++ρ

∂∂

∫∫

∫∫∫

∫∫∫

⋅σ−⋅ρ+ρ∂∂

=+⇒

⋅ρ+ρ∂∂

=⋅σ++

CS nnCSCVinShaftinnet

CSCVCS nninShaftinnet

dAnVdAnVeVdet

WQ

dAnVeVdet

dAnVWQ

rrrr&&

rrrr&&

Energy equationEnergy equation

gz2

Vue2

++=

Page 1/8~7/8Page 1/8~7/8出現在出現在Chapter5Chapter5

Work done by normal Work done by normal stresses at the CSstresses at the CS

77

Rate of Work done by CVRate of Work done by CV

Shaft work : the rate of work transferred into throShaft work : the rate of work transferred into through ugh the CS by the shaft work ( negative for work transferred out, the CS by the shaft work ( negative for work transferred out, positive for work input required) positive for work input required) Work done by normal stresses at the CS:Work done by normal stresses at the CS:

Work done by shear stresses at the CS:Work done by shear stresses at the CS:

Other work Other work

othershearnormalShaft WWWWW &&&&& +++=

ShaftW&

∫∫ ⋅−=⋅σ=⋅δ= CSCS nnnormalnormal dAnVpdAnVVFWrrrrvr&

dAnVWCSshear

rr& ⋅τ+= ∫

∫∫∫ ⋅−+=⋅ρ+ρ∂∂

CSinnetshaftinnetCScvdAnVpWQdAnVeVde

trr&&r

r

Negligibly smallNegligibly small

藉由shaft傳遞的功

+輸入系統者，-輸出系統者

78


∫ =ρ∂∂ 0Vdet CV

mgz2

Vpûmgz2

VpûdAnVgz2

Vpûin

2

out

22

CS &&rr ∑∑∫ ⎟⎟

⎠

⎞⎜⎜⎝

⎛++

ρ+−⎟⎟

⎠

⎞⎜⎜⎝

⎛++

ρ+=⋅ρ⎥

⎦

⎤⎢⎣

⎡++

ρ+

inin

2

outout

2

2

CS

mgz2

Vpûmgz2

Vpû

dAnVgz2

Vpû

&&

rr

⎟⎟⎠

⎞⎜⎜⎝

⎛++

ρ+−⎟⎟

⎠

⎞⎜⎜⎝

⎛++

ρ+=

⋅ρ⎥⎦

⎤⎢⎣

⎡++

ρ+∫

When the flow is steadyWhen the flow is steadyThe integral of

dAnVgz2

Vpû2

CSrr⋅ρ⎥

⎦

⎤⎢⎣

⎡++

ρ+∫

??????

Uniformly distribution

Only one stream entering and leavingOnly one stream entering and leaving

Special & simple case

進一步假設

單進單出

79


( )

innetshaftinnet

inout

2in

2out

inoutinout

WQ

zzg2

VVppûûm

&&

&

+=

⎥⎦

⎤⎢⎣

⎡−+

−+⎟⎟

⎠

⎞⎜⎜⎝

⎛ρ

−⎟⎟⎠

⎞⎜⎜⎝

⎛ρ

+−

ρ+=

pûĥ

( ) in/netshaftin/netinout2in

2out

inout WQzzg2VVĥĥm &&& +=⎥

⎦

⎤⎢⎣

⎡−+

−+−

If shaft work is involvedIf shaft work is involved……..

OneOne--dimensional energy equation dimensional energy equation for steadyfor steady--inin--thethe--mean flowmean flow

EnthalpyEnthalpy The energy equation is written in terms The energy equation is written in terms of enthalpy.of enthalpy.

當 shaft work 包括進來

單維能量方程式

80


( )innetinoutin2inin

out

2outout qûûgz

2Vpgz

2Vp

−−−++ρ

=++ρ

( ) innetinout2in

2outinout

inout Qzzg2VVppûûm && =⎥

⎦

⎤⎢⎣

⎡−+

−+⎟⎟

⎠

⎞⎜⎝

⎛ρ

−⎟⎟⎠

⎞⎜⎝

⎛ρ

+−

m&÷

For steady, incompressible flowFor steady, incompressible flow……OneOne--dimensional energy equationdimensional energy equation

m/Qq innetinnet &&=

in

2in

inout

2out

out z2Vpz

2Vp γ+ρ+=γ+ρ+

0qûû innetinout =−−

wherewhere

For steady, incompressible, For steady, incompressible, frictionless flowfrictionless flow……

Bernoulli equationBernoulli equation

Frictionless flowFrictionless flow……

( ) innetshaftinnetinout2in

2out

inoutinout WQzzg2

VVppûûm &&& +=⎥⎦

⎤⎢⎣

⎡−+

−+⎟⎟

⎠

⎞⎜⎜⎝

⎛ρ

−⎟⎟⎠

⎞⎜⎜⎝

⎛ρ

+− gz2Vûe

2

++=

沒有 shaft work有normal stress做的功

沒有摩擦損失

之前得到的Bernoulli eq.

81


For steady, incompressible, For steady, incompressible, frictional flowfrictional flow……

0qûû innetinout >−−

lossqûû innetinout =−−

lossgz2

Vpgz2

Vpin

2inin

out

2outout −++

ρ=++

ρ

Defining “useful or available energy”… gz2

Vp 2++

ρ

Defining “loss of useful or available energy”…

Frictional flowFrictional flow……Loss發生在in out過程中

上游下游

因為1 2有摩擦損失， 2

222 gz2

Vp++

ρ自然而然就低於 1

211 gz2

Vp++

ρ

考量能量損失

82


( ) innetshatfinnetinout2in

2outinout

inout WQzzg2VVppûûm &&& +=⎥

⎦

⎤⎢⎣

⎡−+

−+⎟⎟

⎠

⎞⎜⎝

⎛ρ

−⎟⎟⎠

⎞⎜⎝

⎛ρ

+−

m&÷ )qûû(wgz2Vpgz

2Vp

innetinoutinnetshaftin

2inin

out

2outout −−−+++

ρ=++

ρ

For steady, incompressible flow with friction and shaft workFor steady, incompressible flow with friction and shaft work……

losswgz2

Vpgz2

Vpinnetshaftin

2inin

out

2outout −+++

ρ=++

ρ

g÷ Lsin2inin

out

2outout hhz

g2Vpz

g2Vp

−+++γ

=++γ

Q

W

gm

W

g

wh in/net

shaftin/netshaftin/netshaftS γ

=≡=&

&

&

glosshL =Head lossHead lossShaft headShaft head

有摩擦損失有軸功進來

在 in out 注入

在 in out 注入

83


For turbineFor turbineFor pumpFor pumpThe actual head drop across the turbineThe actual head drop across the turbine

The actual head drop across the pumpThe actual head drop across the pump

)0h(hh TTs >−=

Ps hh = hhpp is pump headis pump headhhTT is turbine headis turbine head

TLsT )hh(h +−=

pLsp )hh(h −=

Lsin

2inin

out

2outout hhz

g2Vpz

g2Vp

−+++γ

=++γ

in out 輸入

in out 輸出

想像：讓loss擴大

想像：讓loss減緩

84

總結1/8~7/8

Ls1

211

2

222 hhz

g2Vpz

g2Vp

−+++γ

=++γPipe system 內 LOCATION 1 LOCATION 2

orminmajor LLLhhh +=

85


Total head loss , Total head loss , hhLL, is regarded as the sum of major losses, , is regarded as the sum of major losses, hhLL majormajor, due to frictional effects in fully developed flow , due to frictional effects in fully developed flow in constant area tubes, and minor losses, in constant area tubes, and minor losses, hhLL minorminor, resulting , resulting from entrance, fitting, area changes, and so on.from entrance, fitting, area changes, and so on.

orminmajor LLLhhh +=

Head loss可以分成major loss與minor loss

86

Major Losses: Friction FactorMajor Losses: Friction Factor

The energy equation for steady and incompressible flow The energy equation for steady and incompressible flow with zero shaft work with zero shaft work

L2

2222

1

2111 hz

g2V

gpz

g2V

gp

=⎟⎟⎠

⎞⎜⎜⎝

⎛+

α+

ρ−⎟⎟

⎠

⎞⎜⎜⎝

⎛+

α+

ρ

L1221 h)zz(

gpp

+−=ρ−

>>>

For fully developed flow through a For fully developed flow through a constant area pipeconstant area pipe

For horizontal pipe, zFor horizontal pipe, z2 2 = z= z11L

21 hgp

gpp

=ρΔ

=ρ−

>>>

g2V

g2V 222

211 α=

α

簡化一

簡化二

87


lμΔπ

=128

pDQ4

PoiseuillePoiseuille’’ss LawLaw

Valid for Laminar flow only

lμΔ

=π

==32

pDRQ

AQV

2

2average

88

Major Losses: Major Losses: Laminar FlowLaminar Flow

In fully developed laminar flow in a horizontal pipe, the In fully developed laminar flow in a horizontal pipe, the pressure drop pressure drop

( )

2V

DR64

VD64

2V

DDV

D32h

2V

Dfp

DRe64

DVD64

V21

pDV

D32

D4/DV128

DQ128p

2

e

2

L

2

2

4

2

4

llll

ll

lll

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎝

⎛ρμ

=ρμ

=>>ρ

=Δ

=ρμ

=ρ

Δ

μ=

ππμ

=πμ

=Δ

Re64f arminla =

Friction FactorFriction Factor )2/V/()/D(pf 2ρΔ= l

先探討Laminar flow的major loss

水平、完全發展的laminar flow

Lhgp=

ρΔ

2V

Dfp

2ρ=Δ

l

g2V

Dfh

2

Ll

≡Q=VQ=V××A=VA=V××ππDD22/4/4

壓力降可以解出，然壓力降可以解出，然turbulent flowturbulent flow可就不是那麼容易。可就不是那麼容易。

89

Major Losses: Major Losses: Turbulent FlowTurbulent Flow 1/31/3

In turbulent flow we cannot evaluate the pressure drop analyticaIn turbulent flow we cannot evaluate the pressure drop analytically; lly; we must we must resort to experimental resultsresort to experimental results and use dimensional and use dimensional analysis to correlate the experimental data.analysis to correlate the experimental data.

( )ρμε=Δ ,,,,D,VFp l

In fully developed turbulent flow theIn fully developed turbulent flow thepressure drop, pressure drop, △△pp , caused by friction , caused by friction in a horizontal constantin a horizontal constant--area pipe is area pipe is known to known to depend on pipe diameter,D, depend on pipe diameter,D, pipe length, pipe length, ll, pipe roughness,e, , pipe roughness,e, average flow velocity,average flow velocity, V, fluid V, fluid densitydensityρρ, and fluid viscosity,, and fluid viscosity,μμ..

再探討Turbulent flow的major loss 困難的關鍵在無法解析解出壓力降

只得訴諸實驗，訴諸dimensional analysis

管壁粗糙度列出影響壓力降的因子

90

Major Losses: Turbulent Flow Major Losses: Turbulent Flow 2/32/3

Applying dimensional analysis, the result were a correlation of the form

Experiments showExperiments show that the nondimensional head loss is directly proportional to l/D. Hence we can write

⎟⎠⎞

⎜⎝⎛ εφ=

ρΔ

DRe,

DVp

221

l

⎟⎠⎞

⎜⎝⎛ εφ≡

DRe,f

g2V

Dfh

2

L major

l≡

2V

Dfp

2ρ=Δ

l

DarcyDarcy--WeisbachWeisbach equationequation

⎟⎟⎠

⎞⎜⎜⎝

⎛ εμ

ρφ=

ρΔ

D,

D,VD

Vp

221

l 諸多驗實驗顯示： head loss 與l/D成正比

把l/D拿出來是沒有問題的

Lhgp=

ρΔ

Frictional factor ?

透過dimensional analysis

來自累積的實驗結果

91

Roughness for PipesRoughness for Pipes影響frictional factor的因子之一

92

Friction Factor by L. F. MoodyFriction Factor by L. F. Moody

Depending on the specific Depending on the specific circumstances involved.circumstances involved.

93

About Moody ChartAbout Moody Chart

For laminar flow, f=64/Re, which is independent of the For laminar flow, f=64/Re, which is independent of the relative roughness.relative roughness.For very large Reynolds numbers, f=For very large Reynolds numbers, f=ΦΦ((εε/D), which is /D), which is independent of the Reynolds numbers. independent of the Reynolds numbers. For flows with For flows with very large value of Revery large value of Re, commonly termed , commonly termed completely turbulent flow (or wholly turbulent flow), the completely turbulent flow (or wholly turbulent flow), the laminar laminar sublayersublayer is so thin (its thickness decrease with is so thin (its thickness decrease with increasing Re) that the surface roughness completely increasing Re) that the surface roughness completely dominates the character of the flow near the wall.dominates the character of the flow near the wall.For flows with moderate value of Re, the friction factor For flows with moderate value of Re, the friction factor f=f=ΦΦ(Re,(Re,εε/D). /D).

從從Moody chartMoody chart看出什麼看出什麼

在Reynolds number很大時，f僅與粗糙度有關，原因…邊界層很薄很薄

在Laminar flow中，f與粗糙度無關

中度的Reynolds number…

94

Major Losses: Turbulent Flow Major Losses: Turbulent Flow 3/33/3

Colebrook – To avoid having to use a graphical method for obtaining f for turbulent flows.

Miler suggests that a single iteration will produce a result within 1 percent if the initial estimate is calculated from

⎥⎦⎤

⎢⎣⎡ +ε

−=fRe

51.27.3D/log0.2

f1

2

9.00 Re74.5

7.3D/log25.0f

−

⎥⎦⎤

⎢⎣⎡ +ε

=

Valid for the entire Valid for the entire nonlaminarnonlaminarrange of the Moody chart.range of the Moody chart.

Colebrook formulaColebrook formula

有沒有可以不用Moody chart的管道

僅適用於非Laminar flow範圍

猜f的初始值

95

Example 8.5 Comparison of Laminar or Example 8.5 Comparison of Laminar or Turbulent pressure DropTurbulent pressure Drop

Air under standard conditions flows through a 4.0Air under standard conditions flows through a 4.0--mmmm--diameter diameter drawn tubing with an average velocity of V = 50 m/s. For such drawn tubing with an average velocity of V = 50 m/s. For such conditions the flow would normally be turbulent. However, if conditions the flow would normally be turbulent. However, if precautions are taken to eliminate disturbances to the flow (theprecautions are taken to eliminate disturbances to the flow (theentrance to the tube is very smooth, the air is dust free, the tentrance to the tube is very smooth, the air is dust free, the tube does ube does not vibrate, etc.), it may be possible to maintain laminar flow.not vibrate, etc.), it may be possible to maintain laminar flow. (a) (a) Determine the pressure drop in a 0.1Determine the pressure drop in a 0.1--m section of the tube if the m section of the tube if the flow is laminar. (b) Repeat the calculations if the flow is turbflow is laminar. (b) Repeat the calculations if the flow is turbulent.ulent.

96


flowTurbulent700,13.../VDR e →==μρ=

Under standard temperature and pressure conditionsUnder standard temperature and pressure conditionsΡΡ=1.23kg/m=1.23kg/m33, , μμ=1.79=1.79××1010--55NN⋅⋅s/ms/mThe Reynolds numberThe Reynolds number

kPa179.0...V21

Dfp 2 ==ρ=Δ l

If the flow were laminarIf the flow were laminar

f=64/Re=f=64/Re=……=0.0467=0.0467

97


kPa076.1...V21

Dfp 2 ==ρ=Δ l

If the flow were turbulentIf the flow were turbulent

From Moody chart From Moody chart f=f=ΦΦ(Re,(Re,εε/D) =/D) =……0.0280.028

98

Minor Losses Minor Losses 1/51/5

Most pipe systems consist of Most pipe systems consist of considerably more than straight considerably more than straight pipes. These pipes. These additional additional componentscomponents (valves, bends, tees, (valves, bends, tees, and the like) add to the overall and the like) add to the overall head loss of the system.head loss of the system.Such losses are termed MINOR Such losses are termed MINOR LOSS.LOSS.

The flow pattern through a valveThe flow pattern through a valve

Pipe system不是只有直管而已，其他管元件所導致的損失

稱為稱為MINOR LOSSMINOR LOSS

Additional componentsAdditional components所導致的所導致的lossloss

99


The theoretical analysis to predict the details of flow The theoretical analysis to predict the details of flow pattern (through these additional components) is not, as pattern (through these additional components) is not, as yet, possible.yet, possible.The head loss information for essentially all components is The head loss information for essentially all components is given in dimensionless form and based on experimental given in dimensionless form and based on experimental data. The most common method used to determine these data. The most common method used to determine these head losses or pressure drops is to specify the head losses or pressure drops is to specify the loss loss coefficient, Kcoefficient, KLL

理論預測流體在元件內的流況根本做不到

以無因次化型式呈現head loss information 資料來自實驗

最常用的方法連結 Loss coefficient與head loss或壓力降

100


2L

22L

L V21Kp

V21

pg2/V

hK ormin ρ=Δ⇒

ρ

Δ==

Re),geometry(KL φ=

fDK

g2V

Df

g2VKh

Leq

2eq

2

LL ormin

=

==

l

lMinor losses are sometimes given in terms of an equivalent length leq

The actual value of KL is strongly dependent on the geometry of the component considered. It may also dependent on the fluid properties. That is

如何連結？

把minor loss視同另類major loss

相當於多長的管損失

101


For many practical applications the Reynolds number is For many practical applications the Reynolds number is large enough so that the flow through the component is large enough so that the flow through the component is dominated by inertial effects, with viscous effects being of dominated by inertial effects, with viscous effects being of secondary importance. secondary importance. In a flow that is dominated by inertia effects rather than In a flow that is dominated by inertia effects rather than viscous effects, it is usually found that pressure drops and viscous effects, it is usually found that pressure drops and head losses correlate directly with the dynamic pressure.head losses correlate directly with the dynamic pressure.This is the reason why the friction factor for very large This is the reason why the friction factor for very large Reynolds number, fully developed pipe flow is Reynolds number, fully developed pipe flow is independent of the Reynolds number.independent of the Reynolds number.

流體在元件內的Reynolds number很大，主導力來自inertial effect，viscous effect可略過

壓力降及壓力降及head losshead loss與與dynamic pressuredynamic pressure有關有關

Minor Minor lossloss’’FrictionFriction factorfactor與與ReRe無關無關

Viscous effect相對不重要

102


This is true for flow through pipe components.This is true for flow through pipe components.Thus, in most cases of practical interest the loss Thus, in most cases of practical interest the loss coefficients for components are a function of coefficients for components are a function of geometry geometry only,only,

)geometry(KL φ=

就像就像major lossmajor loss在在ReRe很大時，很大時，frictional frictional factorfactor僅與僅與roughnessroughness有關。有關。

103

Minor Losses Coefficient Minor Losses Coefficient Entrance flow 1/3Entrance flow 1/3

Entrance flow condition Entrance flow condition and loss coefficientand loss coefficient((aa) Reentrant, ) Reentrant, KKLL = 0.8= 0.8((bb) sharp) sharp--edged, edged, KKLL = 0.5 = 0.5 ((cc) slightly rounded, ) slightly rounded, KKLL = 0.2= 0.2((dd) well) well--rounded, rounded, KKLL = 0.04= 0.04

KKLL = function of rounding of = function of rounding of the inlet edge.the inlet edge.

依進口條件不同而異依進口條件不同而異

104


A vena A vena contractacontracta region may result because the fluid region may result because the fluid cannot turn a sharp rightcannot turn a sharp right--angle corner. The flow is said to angle corner. The flow is said to separate from the sharp corner.separate from the sharp corner.The maximum velocity velocity at section (2) is greater The maximum velocity velocity at section (2) is greater than that in the pipe section (3), and the pressure there is than that in the pipe section (3), and the pressure there is lower.lower.If this high speed fluid could If this high speed fluid could slow down efficiently, the slow down efficiently, the kinetic energy could be kinetic energy could be converted into pressure.converted into pressure.

轉角處出現vena contract

無法直角轉彎，自然由轉角處分離

(2)處速度高於(3)處速度，壓力回升

如果高速可以有效率緩下來，kinetic energy當然可以轉回壓力

壓力回不來壓力回不來

105


Such is not the case. Although Such is not the case. Although the fluid may be accelerated the fluid may be accelerated very efficiently, it is very very efficiently, it is very difficult to slow down difficult to slow down (decelerate) the fluid (decelerate) the fluid efficiently.(1)efficiently.(1) (2)(2)(2)(2) (3) The extra kinetic (3) The extra kinetic energy of the fluid is partially energy of the fluid is partially lost because of viscous lost because of viscous dissipation, so that the pressure dissipation, so that the pressure does not return to the ideal does not return to the ideal value.value.

Flow pattern and pressure distribution Flow pattern and pressure distribution for a sharpfor a sharp--edged entranceedged entrance

情況並非如此。加速度過程是OK的，但減速過程卻沒有效率

Kinetic energy在(2) (3)部分損

失，導致壓力回不到理想位置。

減速過程出現來自viscous dissipation的loss

106

Entrance HEAD LOSSEntrance HEAD LOSS

流體的流體的inertial inertial effectseffects主要是被主要是被流體內部的流體內部的shear shear stressstress給損失給損失掉掉，，少部分是肇少部分是肇

因於因於wall shear wall shear stressstress。。

107

Minor Losses Coefficient Minor Losses Coefficient Exit flowExit flow

Exit flow condition and Exit flow condition and loss coefficientloss coefficient((aa) Reentrant, ) Reentrant, KKLL = 1.0= 1.0((bb) sharp) sharp--edged, edged, KKLL = 1.0= 1.0((cc) slightly rounded, ) slightly rounded, KKLL = 1.0= 1.0((dd) well) well--rounded, rounded, KKLL = 1.0= 1.0

108

Minor Losses Coefficient Minor Losses Coefficient varied diametervaried diameter

Loss coefficient for sudden Loss coefficient for sudden contraction, expansion,typical contraction, expansion,typical conical diffuser.conical diffuser.

2

2

1L A

A1K ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

查表找KL

109

Minor Losses Coefficient Minor Losses Coefficient BendBend

Carefully designed guide vanes Carefully designed guide vanes help direct the flow with less help direct the flow with less unwanted swirl and disturbances.unwanted swirl and disturbances.

Character of the flow in bend Character of the flow in bend and the associated loss and the associated loss coefficient.coefficient.

彎管的KL

110

Internal Structure of ValvesInternal Structure of Valves

((aa) globe valve) globe valve((bb) gate valve) gate valve((cc) swing check valve ) swing check valve ((dd) stop check valve) stop check valve

Valves內部構造

111

Loss Coefficients for Pipe Loss Coefficients for Pipe ComponentsComponents

常見pipe components的KL

112

Example 8.6 Minor Loss Example 8.6 Minor Loss 1/21/2

Air at standard conditions is to flow through the test section Air at standard conditions is to flow through the test section [between sections (5) and (6)] of the closed[between sections (5) and (6)] of the closed--circuit wind tunnel circuit wind tunnel shown if Figure E8.6 with a velocity of 200 ft/s. The flow is drshown if Figure E8.6 with a velocity of 200 ft/s. The flow is driven iven by a fan that essentially increase the static pressure by the amby a fan that essentially increase the static pressure by the amount ount pp11--pp99 that is needed to overcome the head losses experienced by the that is needed to overcome the head losses experienced by the fluid as it flows around the circuit. Estimate the value of pfluid as it flows around the circuit. Estimate the value of p11--pp99 and and the horsepower supplied to the fluid by the fan.the horsepower supplied to the fluid by the fan.

113

Example 8.6 Minor Loss Example 8.6 Minor Loss 2/22/2

114


The maximum velocity within the wind tunnel occurs in the The maximum velocity within the wind tunnel occurs in the test section (smallest area). Thus, the maximum Mach number test section (smallest area). Thus, the maximum Mach number of the flow is Maof the flow is Ma55=V=V55/c/c55

91L9

299

1

211 hz

g2Vpz

g2Vp

−+++γ=++

γ

s/ft1117)KRT(cs/ft200V 2/1555 ===

The energy equation between points (1) and (9)The energy equation between points (1) and (9)

γ−

γ=−

9191L

pph The total head loss from (1) to (9).The total head loss from (1) to (9).

115


The energy across the fan, from (9) to (1)The energy across the fan, from (9) to (1)

91L55p55pa hVAhVAQhP −γ=γ=γ=

91L91

p hpph −=γ

−γ

=

1

211

p9

299 z

g2Vphz

g2Vp

++γ

=+++γ

HHpp is the actual head rise supplied is the actual head rise supplied by the pump (fan) to the air.by the pump (fan) to the air.

The actual power supplied to the air (horsepower, PThe actual power supplied to the air (horsepower, Paa) is obtained ) is obtained from the fan head byfrom the fan head by

116


The total head lossThe total head loss

hp3.62s/lbft34300...Ppsi298.0...)ft560)(ft/lb765.0(hpp

a

291L91

=⋅=====γ=− −

scrnozdif3corner2corner8corner7corner LLLLLLL91L hhhhhhhh ++++++=−

0.4K2.0Kg2

V6.0g2

VKhg2

V2.0g2

VKh

scrnoz

difdifcorner

LL

22

LL

22

LL

==

====

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Noncircular Ducts Noncircular Ducts 1/41/4

The empirical correlations for pipe flow may be used for The empirical correlations for pipe flow may be used for computations involving noncircular ducts, provided their computations involving noncircular ducts, provided their cross sections are not too exaggerated.cross sections are not too exaggerated.The correlation for turbulent pipe flow are extended for The correlation for turbulent pipe flow are extended for use with noncircular geometries by introducing the use with noncircular geometries by introducing the hydraulic diameterhydraulic diameter, defined as, defined as

PA4D h ≡

Where A is crossWhere A is cross--sectional area, and P sectional area, and P is wetted perimeter.is wetted perimeter.

非圓管，視同圓管D Dh

要要””像像””圓才說得通，圓才說得通，不可以將太誇張的截面不可以將太誇張的截面硬納入！硬納入！

P：與流體接觸的周長度

A為截面積

被流體潤濕的長度

118


For a circular ductFor a circular duct

For a rectangular duct of width b and height hFor a rectangular duct of width b and height h

The hydraulic diameter concept can be applied in the The hydraulic diameter concept can be applied in the approximate range approximate range ¼¼

119


The friction factor can be written as The friction factor can be written as f = C/f = C/ReRehh, where the , where the constant C depends on the particular shape of the duct, and constant C depends on the particular shape of the duct, and ReRehh is the Reynolds number based on the hydraulic is the Reynolds number based on the hydraulic diameter.diameter.The hydraulic diameter is also used in the definition of the The hydraulic diameter is also used in the definition of the friction factor, friction factor, , and the relative , and the relative roughness roughness εε/D/Dhh..

)g2/V)(D/(fh 2hL l=

利用Hydraulic diameter計算Reynolds number

利用hydraulic diameter來定義friction factor

g2V

Dfh

2

L major

l≡

120


For Laminar flow, the value of C = f·Reh have been obtained from theory and/or experiment for various shapes.For turbulent flow in ducts of noncircular cross section, calculations are carried out by using the Moody chart data for round pipes with the diameter replaced by the hydraulic diameter and the Reynolds number based on the hydraulic diameter.

The Moody chart, developed for round pipes, can also The Moody chart, developed for round pipes, can also be used for noncircular ducts. be used for noncircular ducts.

利用Moody chart處理非圓管

121

Friction Factor for Laminar Flow in Friction Factor for Laminar Flow in Noncircular DuctsNoncircular Ducts

f = C/f = C/ReRehhC？

122

Example 8.7 Noncircular DuctExample 8.7 Noncircular Duct

Air at temperature of 120Air at temperature of 120°°F and standard pressure flows from a F and standard pressure flows from a furnace through an 8furnace through an 8--in.in.--diameter pipe with an average velocity of diameter pipe with an average velocity of 10ft/s. It then passes through a transition section and into a s10ft/s. It then passes through a transition section and into a square quare duct whose side is of length a. The pipe and duct surfaces are duct whose side is of length a. The pipe and duct surfaces are smooth (smooth (εε=0). Determine the duct size, a, if the head loss per foot =0). Determine the duct size, a, if the head loss per foot is to be the same for the pipe and the duct.is to be th

FUNDAMENTALS OF FLUID MECHANICS Chapter 8 Pipe Flowtaiwan921.lib.ntu.edu.tw/mypdf/fluid08.pdf · Osborne Reynolds, a British scientist and mathematician, was the first to distinguish

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