Chapter 6: Viscous Flow in Ducts - University of...
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058:0160 Chapter 6-part4 Professor Fred Stern Fall 2018 1
Chapter 6: Viscous Flow in Ducts 6.4 Turbulent Flow in Pipes and Channels using mean-velocity correlations
1. Smooth circular pipe Recall laminar flow exact solution
dave
wu
f Re/6482 ==
ρτ
2000Re ≤=υ
duaved
A turbulent-flow “approximate” solution can be obtained simply by computing uave based on log law.
Bv
yuuu
+=*
* ln1κ
Where
rRyuByuu w −===== ;/;5;41.0);( * ρτκ
−+=
+=== ∫
κκ
πκπ
32ln221
2ln11
**
0
**
2
Bv
Ruu
drrBv
yuuRA
QuVR
ave
Or:
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34.1ln44.2*
* +=v
RuuV
8.0]log[Re2
02.1]log[Re99.12/1
2/12/1
−=
−=−
fff
d
d
Since f equation is implicit, it is not easy to see dependency on ρ, μ, V, and D 4/1Re316.0)( −= Dpipef
g
vDLfphf 2
2=
∆=
γ
Turbulent Flow:
4/74/54/14/3158.0 VDLp −=∆ µρ 75.175.44/14/3241.0 QDL −= µρ Laminar flow: 4/8 RLQp πµ=∆
EFD Adjusted constants f only drops by a factor of 5 over 104 < Re < 108
4000 < ReD < 105 Blasius (1911) power law curve fit to data
Nearly linear Only slightly with μ
Drops weakly with pipe size
Near quadratic (as expected)
V2
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p∆ (turbulent) decreases more sharply with D than p∆ (laminar) for same Q; therefore, increase D for smaller p∆ . 2D decreases p∆ by 27 for same Q.
BRuuru
uu
+==
=υκ
*
**max ln1)0(
Combine with
κυκ 23ln1 *
* −+= BRuuV
Vu
VuuuV
uu
uV
κκκ 231
23
23 *
max*
max*max
* +=⇒−=⇒−=⇒ Also
8/8/18/1
*
2
2*
2
2* fVu
Vuf
Vfandu w
w =⇒=⇒==ρ
ρρτρτ
ffV
uV
u 3.118/231
231
*max +=+=+=⇒
κκ Or:
For Turbulent Flow: ( ) 1
max
3.11−
+= fuV
Recall laminar flow:
5.0/ max =uV
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2. Turbulent Flow in Rough circular pipe
),( +++ = kyfU )/,(Re dkff d=
)(ln1 +++ ∆−+= kBByUκ
which leads to three roughness regimes: 1. k+ < 4 hydraulically smooth 2. 4 < k+ < 60 transitional roughness (Re dependence) 3. k+ > 60 full rough (no Re dependence)
+−
+−= −
−
11.1
2/12/1
7.3/
Re9.6log8.1~
Re51.2
7.3/log2
dk
fdkf
d
d
There are basically four types of problems involved with uniform flow in a single pipe:
1. Determine the head loss, given the kind and size of pipe along with the flow rate, Q = A*V
2. Determine the flow rate, given the head, kind, and size of pipe
3. Determine the pipe diameter, given the type of pipe, head, and flow rate
4. Determine the pipe length, given Q, d, hf, ks, µ, g
Log law shifts downward
Moody diagram
Approximate explicit formula
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058:0160 Chapter 6-part4 Professor Fred Stern Fall 2018 6 1. Determine the head loss
The first problem of head loss is solved readily by obtaining f from the Moody diagram, using values of Re and ks/D computed from the given data. The head loss hf is then computed from the Darcy-Weisbach equation. f = f(ReD, ks/D)
2
2fL Vh f hD g
= = ∆ ( )
−+−=∆γγ
2121
ppzzh
=
+∆ zp
γ
ReD = ReD(V, D)
2. Determine the flow rate The second problem of flow rate is solved by trial, using a successive approximation procedure. This is because both Re and f(Re) depend on the unknown velocity, V. The solution is as follows:
1) solve for V using an assumed value for f and the Darcy-Weisbach equation
2/12/1
f fD/L
gh2V −⋅
=
known from note sign given data
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2) using V compute Re 3) obtain a new value for f = f(Re, ks/D) and reapeat as
above until convergence
Or can use Re
2/12/32/1 2
=
LghDf f
ν
scale on Moody Diagram
1) compute 2/1Re f and ks/D 2) read f
3) solve V from g2
VDLfh
2
f =
4) Q = VA 3. Determine the size of the pipe
The third problem of pipe size is solved by trial, using a successive approximation procedure. This is because hf, f, and Q all depend on the unknown diameter D. The solution procedure is as follows:
1) solve for D using an assumed value for f and the Darcy-Weisbach equation along with the definition of Q
5/15/1
f2
2f
ghLQ8D ⋅
π=
known from given data
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2) using D compute Re and ks/D 3) obtain a new value of f = f(Re, ks/D) and repeat as above
until convergence
4. Determine the pipe length The four problem of pipe length is solved by obtaining f from the Moody diagram, using values of Re and ks/D computed from the given data. Then using given hf ,V, D, and
calculated f to solve L from fDh
VgL f2
2= .
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3. Concept of hydraulic diameter for noncircular ducts
For noncircular ducts, τw= f(perimeter); thus, new
definitions of 28V
f wρτ
= and 22V
C wf ρ
τ= are required.
Define average wall shear stress
dsP
Pww ∫=
0
1 ττ ds = arc length, P = perimeter
Momentum:
0=
∆+−∆
Lz
WALPLpA w γτ
( )PALzph w
//
γτγ =+∆=∆
A/P =Rh= Hydraulic radius (=R/2 for circular pipe and
2/RLh w
γτ
=∆ )
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Energy:
PALhh w
L /γτ
==∆
( )
−=
+−=
−=
∆=
dxpd
PA
dxzpd
PA
dxdh
PA
Lh
PA
w
^γγγτ non-circular duct
Recall for circular pipe:
dxpdD
dxpdR
wˆ
4ˆ
2−=−=τ
In analogy to circular pipe:
PADD
PA
dxpdD
dxpd
PA
hhh
w4
4
^
4
^=⇒=⇒
−=
−=τ
For multiple surfaces such as concentric annulus P and A based on wetted perimeter and area
υε
ρτ h
DhDw VD
DfV
fhh=== Re)/,(Re8
2
gV
DLf
RLfV
RLhh
hhh
wL 28
22
====∆γ
ργτ
However, accuracy not good for laminar flow (40%) and marginal turbulent flow (15%).
Hydraulic diameter
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a. Accuracy for laminar flow (smooth non-circular
pipe) Recall for pipe flow:
==
==
64Re
16Re)(#
0
0
0 fP
CPPPoiseuille
f
fc f
Recall for channel flow:
hD
hhVhf
Re42 Re
96Re4824
===ρµ
hD
hhf
f
VhC
fC
Re42 Re
24Re126
4/
===
⇒=
ρµ
==
==
96Re
24Re)(#
0
0
0
h
hf
Df
Dfc
fP
CPPPoiseuille
Therefore:
32
9664
2416
0
0
0
0====
hf
f
hf
f
Donbasedchannel
pipe
Donbasedchannelc
pipec
P
P
P
P
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Thus, if we could not work out the laminar theory and chose to use the approximation 64Re ≈
hDf or 16Re ≈hDfC ,
we would be 33 percent low for channel flow.
For laminar flow, 0P varies greatly, therefore it is better to use the exact solution
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b. Accuracy for turbulent flow(smooth non-circular pipe)
For turbulent flow, Dh works much better especially if combined with “effective diameter” concept based on ratio of exact laminar circular and noncircular duct P0 numbers, i.e. fCP 0/16 or fP0/64 . First recall turbulent circular pipe solution and compare with turbulent channel flow solution using log-law in both cases Channel Flow
dYBuyhuh
Vh
+
−= ∫ υκ
*
0
* )(ln11
−+=κυκ1ln1 *
* Bhuu
hhB
hBPAD
Bh 442
)2(4lim4=
+==
∞→ h= half width
Define υυhVVDh
Dh
4Re ==
Y=h-y wall coordinate
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( ) 19.1Relog2 2/12/1 −=− ffhD
Therefore error in Dh concept relatively smaller for turbulent flow. Note ( ) 8.0Re64.0log2)( 2/12/1 −=− fchannelf
hD
Define Deffective hf
fh D
channelPcircleP
D24)(
16)(~64.0
0
0
==
=
(therefore, improvement on Dh is)
υ
effeffD
VD=Re
hC
Ch
f
feff D
circularnonP
circlePD
circularnonPcircleP
Df
f
)(
)(
)()(
0
0
0
0
−=
−=
Or
hC
hf
eff DcircularnonP
DcircularnonP
Df
)(16
)(64
00 −=
−=
Very nearly the same as circular pipe 7% to large at Re = 105 4% to large at Re = 108
From exact laminar solution
Laminar solution
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