FUNDAMENTALS OF FLUID MECHANICS Chapter 8 …charnnarong.me.engr.tu.ac.th/charnnarong/My...

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FUNDAMENTALS OFFLUID MECHANICS

Chapter 8 Pipe Flow

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MAIN TOPICS

General Characteristics of Pipe FlowFully Developed Laminar FlowFully Developed Turbulent FlowDimensional Analysis of Pipe FlowPipe Flow ExamplesPipe Flowrate Measurement

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Introduction

Flows completely bounded by solid surfaces are called INTERNAL FLOWS which include flows through pipes (Round cross section), ducts (NOT Round cross section), nozzles, diffusers, sudden contractions and expansions, valves, and fittings.

The basic principles involved are independent of the cross-sectional shape, although the details of the flow may be dependent on it.

The flow regime ( laminar or turbulent ) of internal flows is primarily a function of the Reynolds number.Laminar flow: Can be solved analytically.Turbulent flow: Rely Heavily on semi-empirical theories and

experimental data.

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Pipe System

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

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Pipe Flow and Open Channel Flow

Pipe flow: Flows completely filling the pipe. (a)The main driving force is the pressure gradient along the pipe.

Open channel flow: Flows without completely filling the pipe. (b)The gravity alone is the driving force.

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Laminar or Turbulent Flow1/2

The flow of a fluid in a pipe may be Laminar ? Turbulent ? Osborne Reynolds, a British scientist and mathematician, was the

first to distinguish the difference between these classification of flow by using a simple apparatus as shown.

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Laminar or Turbulent Flow2/2

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

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

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

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Laminar or Turbulent Flow

當流率很小時，染劑的軌跡依然隨著水流保持明顯的細條，染劑分子會稍微擴散到周圍水流，並呈現輕微混濁現象。

當流率增加時，染劑軌跡會隨時間與位置振動，並呈現間歇性的不規則現象。

當流率足夠大時，染劑軌跡幾乎是迅速迸裂開來，並以隨機方式在管中擴散開來。

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Time Dependence of Fluid Velocity at a Point

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Indication of Laminar or Turbulent FlowThe term flowrate should be replaced by Reynolds

number, ,where V is the average velocity in the pipe.

It is not only the fluid velocity that determines the character of the flow – its density, viscosity, and the pipe size are of equal importance.

For general engineering purpose, the flow in a round pipe LaminarTurbulent

/VDRe

2100Re 4000Re＞

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Entrance Region and Fully Developed Flow 1/4

Any fluid flowing in a pipe had to enter the pipe at some location.

The region of flow near where the fluid enters the pipe is termed the entrance region.

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The fluid typically enters the pipe with a nearly uniform velocity profile at section (1).

The region of flow near where the fluid enters the pipe is termed the entrance region.

As the fluid moves through the pipe, viscous effects cause it to stick to the pipe wall (the no slip boundary condition).

The boundary layer in which viscous effects are important is produced along the pipe wall such that the initial velocity profile changes with distance along the pipe,x , until the fluid reaches the end of the entrance length, section (2), beyond which the velocity profile does not vary with x.

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The boundary layer has grown in thickness to completely fill the pipe.

Viscous effects are of considerable importance within the boundary layer. Outside the boundary layer, the viscous effects are negligible.

The length of entrance region depends on whether the flow is laminar or turbulent.

ee R06.0

D

6/1

ee R4.4

D

For laminar flow For turbulent flow

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

The flow between (2) and (3) is termed fully developed.

任何流體欲在圓管中流動，則必須在某一位置導入圓管；其中，流體在進入圓管的區域，稱之為入口區（entrance region）；若從流體沿著pipe前進的發展過程來看，在截面（1）處的速度接近均勻分佈，再往下游，則由於黏性效應使得boundary layer逐漸成長，並使速度曲線隨著X軸逐漸改變，直到流體達到『入口長度』的尾端，也就是說，在截面（2）處以後，速度分佈曲線就不再變化了！至於管流中的速度曲線？則依流動為laminar flow，或turbulent flow以及入口長度le 而定！

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

The magnitude of the pressure gradient is constant.

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

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Fully Developed Laminar Flow

There are numerous ways to derive important results pertaining to fully developed laminar flow:From F=ma applied directly to a fluid element.From the Navier-Stokes equations of motionFrom dimensional analysis methods

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From F=ma 1/8

Considering a fully developed axisymmetric laminar flow in a long, straight, constant diameter section of a pipe.

The Fluid element is a circular cylinder of fluid of length and radius r centered on the axis of a horizontal pipe of diameter D.

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From F=ma 2/8

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+t 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

0ixuuVV

Steady Fully developed

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From F=ma 3/8

Cr? r

2p

r

2p0r2rpprp 21

21

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

xx maF The force balance

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

Not function of r

Not function of r

Independent of r

B.C. r=0 =0r=D/2 = w

Dr2 w

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From F=ma 4/8

D4p w

The pressure drop and wall shear stress are related by

Valid for both laminar and turbulent flow.

Laminar

drdu

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From F=ma 5/8

Since

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

drdu

12 Cr

4purdr

2pdu

r2

pdrdu

16

pDC2

1

2

w

2

C

22

Rr1

4D)r(u

Dr21V

Dr21

16pD)r(u

Velocity distributionVelocity distribution

D4p w

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From F=ma 6/8

The shear stress distribution

Volume flowrate2pr

drdu

128pDQ

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

4

C4

R

0A

Poiseuille’s Law

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From F=ma 7/8

Average velocity

Point of maximum velocity

32pD

RQ

AQV

2

2average

0drdu

at r=0

average

2

max V24

pRUuu

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From F=ma 8/8

Making adjustment to account for nonhorizontal pipes

sinpp >0 if the flow is uphill<0 if the flow is downhill

r2sinrp

32DsinrpV

2

average

128DsinrpQ

4

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From the Navier-Stokes Equations 1/2

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

0V

Vkgp

VgpVVtV

2

2

Steady flow

kgg

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

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From the Navier-Stokes Equations 2/2

rur

rr1sing

xp

pxp.const

xp

i)r(uV

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

IntegratingIntegrating Velocity profile u(t)=

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

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Fully Developed Turbulent Flow

Turbulent pipe flow is actually more likely to occur than laminar flow in practical situations.

Turbulent flow is a very complex process.Numerous persons have devoted considerable effort in an

attempting to understand the variety of baffling aspects of turbulence.

The field of turbulent flow still remains the least understood area of fluid mechanics.

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Transition from Laminar to Turbulent Flow in a PipeA pipe is initially filled with a fluid at rest. As the valve is

opened to start the flow, the flow velocity and, hence, the Reynolds number increase from zero (no flow) to their maximum steady flow values.

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Description for Turbulent Flow 1/4

Irregular, random nature is the distinguishing feature of turbulent flow.

The character of many of the important properties of the flow (pressure drop, heat transfer, etc.) depends strongly on the existence and nature of the turbulent fluctuations or randomness indicated.

The time-averaged, ū, and fluctuating, ú description of a parameter for tubular flow.

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

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Calculation of the heat transfer, pressure drop, and many other parameters would not be possible without inclusion of the seemingly small, but very important, effects associated with the randomness of the flow.

Due to the macroscopic scale of the randomness in turbulent flow, mixing processes and heat transfer processes are considerably enhanced in turbulent flow compared to laminar flow.

In some situations, turbulent flow characteristics are advantages. In other situations, laminar flow is desirable.

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Turbulent flows are characterized by random, three-dimensional vorticity.

Turbulent flows can be described in terms of their mean values on which are superimposed the fluctuations.

Tt

t

O

Odtt,z,y,xu

T1u

'uuu uu'u

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0uTuTT1dtuu

T1'u Tt

t

O

O

0dt'uT1)'u( Tt

t

22 O

O

u

dt'uT1

u)'u(

2Tt

t

2

2O

O

The time average of the fluctuations is zero.

The square of a fluctuation quantity is positive.

Turbulence intensity or the level of the turbulenceThe larger the turbulence intensity, the larger the fluctuations of the velocity. Well-designed wind tunnels have typical value of =0.01, although with extreme care, values as low as =0.0002 have been obtained.

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Shear Stress for Laminar Flow

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 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 process produces a shear force.

The Shear stress distribution

dydu

yx

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Shear Stress for Laminar Flow

dydu

yx

當分子由較小u的平面，穿過A – A平面，向上運動，即發生動量通量，這

會使得A-A平面下側的流體作用在A – A平面上側的流體，並形成阻力，一旦分子通過A – A平面後，必然會被A – A平面上側的流體帶走，此一動

量改變過程，以巨觀尺寸來看，即是形成『剪應力』。

同理，A – A平面上側活躍的分子往A –A平面下側越過時，也會產生減速現象。因此，我們可以大膽的說：剪應力僅有

存在u = u （ y ）梯度時才有。除了上

述原因外，再加上分子間的吸引力，我

們可以獲得知名的『牛頓黏性定律』：

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Shear Stress for Turbulent Flow

The turbulent flow is thought as a series of random, three-dimensional eddy type motions.

The flow is represented by (time-mean velocity ) plus u’ and v’ (time randomly fluctuating velocity components in the x and y direction).

The shear stress

u

turbulentarminla'v'udyud

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

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

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Shear Stress for Turbulent Flow

除了laminar flow的分子隨機運動仍會出現外，另外一個重要的因素必須加

以考量，就是必須將紊流視為一連串隨機的三維漩渦運動型式，其中，漩渦

尺寸可以是極小，也可以是相當大。這些漩渦結構在流體中助長了混合作用

，也大幅提高了越過A – A平面的x方向動量傳輸，也就是說，有限量的流體

隨機傳輸穿越平面，並形成相當大的剪力，其中，u’ , v’（隨機速度分量）恰巧可以用來說明之。因此，A – A平面上出現的剪應力，可以表示成：

turbulentarminla'v'udyud

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

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

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Structure of Turbulent Flow in a Pipe

Near the wall (the viscous sublayer), the laminar shear stress lamis dominant.

Away from the wall (in the outer layer) , the turbulent shear stress turb is dominant. turb is 100 to 1000 times greater than lam in the outer region

The transition between these two regions occurs in the overlap layer.

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Alternative Form of Shear Stress

An alternate form for the shear stress for turbulent flow is given in terms of the eddy viscosity .

dyud

turb

2

m2

dyudl

2

2

mturb dyudl

Introduced by J. Boussubesq.

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

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Turbulent Velocity Profile1/4

Considerable information concerning turbulent velocity profiles has been obtained through the use of dimensional analysis, and semi-empirical theoretical efforts.

In the viscous sublayer the velocity profile can be written in dimensionless form as

yyuuuu

*

*

2/1

w* /u

Where y is the distance measured from the wall y=R-r.

is called the friction velocity.

Law of the wall

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

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In the overlap region the velocity should vary as the logarithm of y

In transition region or buffer layer

for 30yu*

0.5y

yuln5.2uu

30yu7-5*

y

Rln5.2u

uU for

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0.5y

yuln5.2uu

*

*

yuuu

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

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

n1

n1

Rr1

Ry

Uu

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

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Energy Considerations 1/8

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

in/Shaftin/netCS

2

CV WQAdnV)gz2

Vpu(Vdet

CS nnCSCVin/Shaftin/net

CSCVCS nnint/Shafin/net

dAnVdAnVeVdet

WQ

dAnVeVdet

dAnVWQ

Energy equation

gz2

Vue2

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0Vdet CV

mgz2

Vpumgz2

VpudAnVgz2

Vpuin

2

out

22

CS

in

in

2

out

out

2

2

CS

mgz2

Vpumgz2

Vpu

dAnVgz2

Vpu

When the flow is steadyThe integral of

dAnVgz2

Vpu2

CS

???

Uniformly distribution

Only one stream entering and leavingOnly one stream entering and leaving

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in/netshaftin/net

inout

2in

2out

inoutinout

WQ

zzg2

VVppuum

puh

in/netshaftin/netinout

2in

2out

inout WQzzg2

VVhhm

If shaft work is involved….

One-dimensional energy equation for steady-in-the-mean flow

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in/netinoutin

2inin

out

2outout quugz

2Vpgz

2Vp

in/netinout

2in

2outinout

inout Qzzg2

VVppuum

m

For steady, incompressible flow without shaft work…

m/Qq in/netin/net

in

2in

inout

2out

out z2Vpz

2Vp

0quuin/netinout

where

For steady, incompressible, frictionless flow without shaft work…

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For steady, incompressible, frictional flow…

0quuin/netinout

lossquuin/netinout

lossgz2

Vpgz2

Vpin

2inin

out

2outout

Defining “useful or available energy”… gz2

Vp 2

Defining “loss of useful or available energy”…

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in/netshaftin/netinout

2in

2outinout

inout WQzzg2

VVPPuum

m )quu(Wgz2

Vpgz2

Vpin/netinoutin/netshaftin

2inin

out

2outout

For steady, incompressible flow with friction and shaft work…

lossWgz2

Vpgz2

Vpin/netshaftin

2inin

out

2outout

gLsin

2inin

out

2outout hhz

g2Vpz

g2Vp

Q

W

gm

W

g

Wh in/netshaftin/netshaftin/netshaft

S

glosshL Head lossShaft head

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For turbineFor pumpThe actual head drop across the turbine

The actual head drop across the pump

)0h(hh TTs Ps hh hp is pump head

hT is turbine head

TLsT )hh(h

pLsp )hh(h

Lsininin

outoutout hhz

gVpz

gVp

22

22

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

orminmajor LLL hhh

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Energy equation for pipe flow

Lp hzg

Vphzg

Vp 2

22

22

1

21

11

22

where = Kinetic energy correction coefficient

for uniform flow

for Laminar or Turbulent flows

V1 and V2 are average velocities

1

1

Uniform Laminar Turbulent1 2 2.1

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Major Losses: Friction Factor

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

L2

2222

1

2111 hz

g2Vpz

g2Vp

L1221 h)zz(

gpp

For fully developed flow through a constant area pipe

For horizontal pipe, z2=z1 L21 h

gp

gpp

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Major Losses: Laminar Flow

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

g2V

DR64

VD64

g2V

DgDV

D32h

2V

Dfp

DRe64

DVD64

V21

pDV

D32

D4/DV128

DQ128p

2

e

2

L

2

2

4

2

4

Re64f arminla

Friction Factor

2/V//Dpf 2

g2V

Dfh

2

L

22.p...128

pDQ4

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Major Losses: Turbulent Flow 1/3

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

,,,,D,VFp

In fully developed turbulent flow thepressure drop, △p , caused by friction in a horizontal constant-area pipe is known to depend on pipe diameter,D, pipe length,L, pipe roughness,e, average flow velocity, V, fluid densityρ, and fluid viscosity,μ.

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

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

DRe,

DV21

p2

DRe,f

g2V

Dfh

2

Lmajor

2V

Dfp

2

Darcy-Weisbach equation

D,

D,VD

V21

p2

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Roughness for Pipes

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Friction Factor by L.F. Moody

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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

fRe51.2

7.3D/log0.2

f1

2

9.00 Re74.5

7.3D/log25.0f

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Minor Losses 1/7

The flow in a piping system may be required to pass through a variety of fittings, bends, or abrupt changes in area.

Additional head losses are encountered, primarily as a result of flow separation.

These losses will be minor if the piping system includes long lengths of constant area pipe.

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Minor Losses 2/7

The minor losses are computed based on experimental data. The most common method used to determine these head losses or pressure drops is to specify the loss coefficient, KL

fDK

g2V

Df

g2VKh

V21Kp

V21

pg2/V

hK

Leq

2eq

2

LL

2L

22

L

L

ormin

ormin

Re,geometryKL

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Minor Losses 3/7

Entrance flow condition and loss coefficient(a) Reentrant, KL = 0.8(b) sharp-edged, KL = 0.5 (c) slightly rounded, KL = 0.2(d) well-rounded, KL = 0.04

KL = function of rounding of the inlet edge.

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Minor Losses 4/7

A vena contracta region may result because the fluid cannot turn a sharp right-angle corner. The flow is said to separate from the sharp corner.

(2)(3) The extra kinetic energy of the fluid is partially lost because of viscous dissipation, so that the pressure does not return to the ideal value.

Flow pattern and pressure distribution for a sharp-edged entrance

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Minor Losses 5/7

Exit flow condition and loss coefficient(a) Reentrant, KL = 1.0(b) sharp-edged, KL = 1.0(c) slightly rounded, KL = 1.0(d) well-rounded, KL = 1.0

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Minor Losses 6/7

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

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Minor Losses 7/7

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

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

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Internal Structure of Valves

(a) globe valve(b) gate valve(c) swing check valve (d) stop check valve

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Loss Coefficients for Valves

Source: Robert W. Fox, Alan T. McDonald, and Philip J. Pritchard, INTRODUCTION TO FLUID MECHANICS, Sixth Edition, JOHN WILEY & SONS, INC.

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Noncircular Ducts 1/4

The empirical correlations for pipe flow may be used for computations involving noncircular ducts, provided their cross sections are not too exaggerated.

The correlation for turbulent pipe flow are extended for use with noncircular geometries by introducing the hydraulic diameter, defined as

For a circular ductPA4Dh

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

DPA4Dh

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For a rectangular duct of width b and height h

The hydraulic diameter concept can be applied in the approximate range ¼<ar<4. So the correlations for pipe flow give acceptably accurate results for rectangular ducts.

b/harar1h2

)hb(2bh4

PA4Dh

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The friction factor can be written as f=C/Reh, where the constant C depends on the particular shape of the duct, and Reh is the Reynolds number based on the hydraulic diameter.

The hydraulic diameter is also used in the definition of the friction factor, hL=f( /Dh)V2/2g, and the relative roughness /Dh.

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For Laminar flow, the value of C=fReh 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.

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Pipe Flow Examples

The energy equation, relating the conditions at any two points 1 and 2 for a single-path pipe system

by judicious choice of points 1 and 2 we can analyze not only the entire pipe system, but also just a certain section of it that we may be interested in.

g2V

Dfh

2

Lmajor

orminmajor LLL2

2222

1

2111 hhhz

g2V

gpz

g2V

gp

Major loss Minor lossg2

VKh2

LL ormin

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Single-Path Systems 1/2

Find pressure dropΔp, for a given pipe (L and D), and flow rate, and Q

Find L for a given Δp, D, and QFind Q for a given Δp, L, and DFind D for a given Δp, L, and Q

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Single-Path Systems 2/2

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Find Δp for a given L , D, and Q

The energy equation

The flow rate leads to the Reynolds number and hence the friction factor for the flow.

Tabulated data can be used for minor loss coefficients and equivalent lengths.

The energy equation can then be used to directly to obtain the pressure drop.

orminmajor LLL2

2222

1

2111 hhhz

g2V

gpz

g2V

gp

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76

Find L for a given Δp, D, and Q

The energy equation

The flow rate leads to the Reynolds number and hence the friction factor for the flow.

Tabulated data can be used for minor loss coefficients and equivalent lengths.

The energy equation can then be rearranged and solved directly for the pipe length.

orminmajor LLL2

2222

1

2111 hhhz

g2V

gpz

g2V

gp

謝志誠

77

Find Q for a given Δp, L, and D

These types of problems required either manual iteration or use of a computer application.

The unknown flow rate or velocity is needed before the Reynolds number and hence the friction factor can be found.

First, we make a guess for f and solve the energy equation for V in terms of known quantities and the guessed friction factor f.

Then we can compute a Reynolds number and hence obtain a new value for f.

Repeat the iteration process f→ V→ Re→ f until convergence

謝志誠

First guessf0

0

02f

PLDV

DV0

1Re

givenD

1Re

f1

1

12f

PLDV)Re,/( 11 Dff

DV1

2Re

Stop, if Ren+1 = Ren

……..

Example: Find Q for a given Δp, L, and D

謝志誠

79

Find D for a given Δp, L, and Q

These types of problems required either manual iteration or use of a computer application.

The unknown diameter is needed before the Reynolds number and relative roughness, and hence the friction factor can be found.

First, we make a guess for f and solve the energy equation for D in terms of known quantities and the guessed friction factor f.

Then we can compute a Reynolds number and hence obtain a new value for f.

Repeat the iteration process f→ D→ Re and e/D→ f until convergence

謝志誠

First guessf1

guessD 0

0DD

1Re

f2

PfQD

1

2

218

)Re,/( 001 Dff

Stop, if fn+1 = fn

……

11

4ReDQ

guessnextD

1

)Re,/( 112 Dff

00

4ReDQ

0Re

Second guess

1Re

Example: Find D for a given Δp, L, and Q

謝志誠

81

Multiple-Path Systems Series and Parallel Pipe System

321 LLL321 hhhQQQQ

321BA LLLL321 hhhhQQQ

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82

Multiple-Path Systems Multiple Pipe Loop System

3L2L

3L1LB

2BB

A

2AA

2L1LB

2BB

A

2AA

321

hh

31hhzg2

VPzg2

VP

21hhzg2

VPzg2

VP

QQQ

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83

Multiple-Path Systems Three-Reservoir System

If valve (1) was closed, reservoir B reservoir CIf valve (2) was closed, reservoir A reservoir CIf valve (3) was closed, reservoir A reservoir BWith all valves open….

CBhhzg2

VPzg2

VP

BAhhzg2

VPzg2

VP

QQQ

3L2LC

2CC

B

2BB

2L1LB

2BB

A

2AA

321

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84

Pipe Flowrate Meters1/2

The theoretical flow rate may be related to the pressure differential between section 1 and 2 by applying the continuity and Bernoulli equations.

Then empirical correction factors may be applied to obtain the actual flow rate.

2

2

221

2

11

CSCV

gz2

Vpgz2

Vp

0AdVVdt

Basic equation

謝志誠

85

Pipe Flowrate Meters2/2

2

1

2

2

221 A

A12Vpp

212

212 )A/A(1

)pp(2V

Theoretical mass flow rateTheoretical mass flow rate

412

21222ideal )/D(D1ρ

)p2(pAAVQ

pQideal ??Qactual

謝志誠

86

Pipe Flowrate MetersOrifice Meter

4

21OOidealO 1

pp2ACQCQ

4/dA 20 Area of the hole in the orifice plate

)/VDRe,D/d(CC OO Orifice meter discharge coefficient

謝志誠

87

Pipe Flowrate MetersNozzle Meter

4

21nnidealn 1

pp2ACQCQ

4/dA 2n Area of the hole

)/VDRe,D/d(CC nn

Nozzle meter discharge coefficient

謝志誠

88

Pipe Flowrate Meters Venturi Meter

4

21TVidealV 1

pp2ACQCQ

4/dA 2T Area of the throat

The Venturi discharge coefficient)/VDRe,D/d(CC VV

謝志誠

89

Linear Flow MeasurementFloat-type Variable-area Flow Meters

The ball or float is carried upward in the tapered clear tube by the flowing fluid until the drag force and float weight are in equilibrium.

Float meters often called rotameters.

Rotameter-type flow meter

謝志誠

90

Linear Flow MeasurementTurbine Flow Meter A small, freely rotating

propeller or turbine within the turbine meter rotates with an angular velocity that is function of the average fluid velocity in the pipe. This angular velocity is picked up magnetically and calibrated to provide a very accurate measure of the flowrate through the meter.

Turbine-type flow meter

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91

Volume Flow Meters 1/2

Measuring the amount (volume or mass) of fluid that has passed through a pipe during a given time period, rather than the instantaneous flowrate.

Nutating disk flow meters is widely used to measurement the net amount of water used in domestic and commercial water systems as well as the amount of gasoline delivered to your gas tank.

Nutating disk flow meter

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92

Volume Flow Meters 2/2

Bellow-type flow meter is a quantity-measuring device used for gas flow measurement.

It contains a set of bellows that alternately fill and empty as a result of the pressure of gas and the motion of a set of inlet and outlet valves.

Bellows-type flow meter.(a) Back case emptying, back diaphragm filling.(b) Front diaphragm filling, front case emptying.(c) Back case filling, back diaphragm emptying. (d) Front diaphragm emptying, front case filling.

FUNDAMENTALS OF FLUID MECHANICS Chapter 8 …charnnarong.me.engr.tu.ac.th/charnnarong/My...

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Transcript of FUNDAMENTALS OF FLUID MECHANICS Chapter 8 …charnnarong.me.engr.tu.ac.th/charnnarong/My...