FUNDAMENTALS OF FLUID MECHANICS Chapter 8 …charnnarong.me.engr.tu.ac.th/charnnarong/My...
Transcript of 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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Entrance Region and Fully Developed Flow 2/4
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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Entrance Region and Fully Developed Flow 3/4
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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Entrance Region and Fully Developed Flow 4/4
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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Description for Turbulent Flow 2/4
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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Description for Turbulent Flow 3/4
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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Description for Turbulent Flow 4/4
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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Turbulent Velocity Profile2/4
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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Turbulent Velocity Profile3/4
0.5y
yuln5.2uu
*
*
yuuu
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Turbulent Velocity Profile4/4
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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Energy Considerations 2/8
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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Energy Considerations 3/8
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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Energy Considerations 4/8
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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Energy Considerations 5/8
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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Energy Considerations 6/8
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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Energy Considerations 7/8
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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Energy Considerations 8/8
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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Major Losses: Turbulent Flow 2/3
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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Major Losses: Turbulent Flow 3/3
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
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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Noncircular Ducts 2/4
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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Noncircular Ducts 3/4
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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Noncircular Ducts 4/4
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
謝志誠
73
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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74
Single-Path Systems 2/2
謝志誠
75
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
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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
謝志誠
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.