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CU06997 Fluid Dynamics
Flow in pipes and closed conduits
4.1 Introduction (page 91)
4.2 The historical context (page 91-93)
4.3 Fundamental concepts of pipe flow (page 94-97)
4.4 Laminar flow (page 97-100)
4.5 Turbulent flow (page 100 β 111)
1
Pipe with head loss
41
2
44
2
11
22 H
g
uh
g
uh
4411 AuAuQ
Pressure
Head
Total
Head
Head loss
1
Reynolds number:
p 93 (pipe), p 127 (open channel)
π = Absolute viscosity [m2/s]
π = Kinematic viscosity [kg/ms]
water, 20Β°C= 1,00 β 10β6
π = Density of liquid [kg/m3]
π = Velocity [m/s]
D = Hydraulic diameter [m]
R = Hydraulic Radius = D/4 [m]
π π = Reynolds Number [1]
πΉπ > ππππ Turbulent flow
πΉπ < ππππ Laminar flow
π π =π. 4π
π
π π =π β π β π·
π=
π β π·
π
1
Laminar flow, frictional head loss
[Energieverlies tgv wrijving]
βπ =32 β π β πΏ β π
π β π β π·2
βπ = frictional head loss βH [m]
π = Absolute viscosity [kg/ms]
πΏ = Length between the Head Loss [m]
π = mean velocity [m/s]
D = Hydraulic Diameter [m]
π = Density of liquid [kg/m3]
π = earths gravity [m/s2] 2
Total Head
Pressure Head
Laminar flow, wall shear stress
[Schuifspanning]
π0=
4 β π β π
π
Ο0 = shear stress at solid boundary [N/m2]
π = Absolute viscosity [kg/ms]
π = mean velocity [m/s]
R = Hydraulic Radius [m]
2
Head loss /Energy loss [m]
β’ Turbulent flow
β’ Friction loss (wrijvingsverlies)
β’ Local loss (lokaal verlies)
[m] 2g
uΞΞ
2
β’ ΞH = Head loss or Energy loss [m]
β’ u2/2g = Velocity head [m]
β’ ΞΎ (ksie) = Loss coΓ«fficiΓ«nt [1]
3
Darcy-Weisbach
2g
u
2g
u
4ΞΞ
22
f R
L
β’ ΞH = Head loss by friction [m]
β’ u2/2g = Velocity head [m]
β’ L = Length [m]
β’ Ξ» = (lamda) = Friction coΓ«fficiΓ«nt[1]
β’ ΞΎ (ksie) = Loss coΓ«fficiΓ«nt [1]
β’ R = hydraulic radius [m]
R
Lf
4
Total Head
Pressure Head
3
Remarks friction loss Darcy-Weisbach
β’ Ξ» (boundary roughness) depends on material and
construction. Ξ» often between 0,01 and 0,10
β’ Ξ» is not a constant, depends on βboundary layerβ.
βSmoothβ or βRoughβ, Most of the time βSmoothβ
How to calculate Ξ» !!!
3
β’ During exams Fluid Dynamics, the Ξ» will be given
Colebrook-White transition formula
1
π= β2 β πππ
ππ
3,70 β π·+
2,51
Reβ π
π = Friction coefficient [1]
D = Hydraulic Diameter 4R [m]
kS = surface roughness [m]
(k-waarde)
Difficult to solve
Could use figure 4.5 page 105
Nowadays computers?
3
Moody diagram
3
Colebrook-White and Darcy Weisbach
π = β2 2π β π· β ππβ πππππ
3,70π·+
2,51Ο
D 2πβπ·βππ
with ππ =βπ
πΏ
π = Average velocity [m/s]
D = Hydraulic Diameter (4R) [m]
kS = surface roughness [m]
π = Kinematic viscosity [kg/ms]
Sf = slope of hydraulic gradient [-]
hf = frictional head loss (βHf) [m]
πΏ = Length between the Head Loss [m]
3
Turbulent flow ,
Mean boundary shear stress
π0 = π β π β π β π0
Ο0 = shear stress at solid boundary [N/m2]
R = Hydraulic Radius [m]
π0 = Slope of channel bed [1]
In sewer minimum shear stress value
(0.5 β 1.5 N/m2)
3
Local head losses
[m] 2g
uΞΞ
2
l
4
Head loss Sudden Pipe Enlargement
2g
VVΞΞ
2
21
l
βπ»π= (1 β
π΄1
π΄2)2β
π12
2π
4
Head loss Sudden Pipe Enlargement
βπ»π =(π1 β π2)2
2π βπ»π= (1 β
π΄1
π΄2)2β
π12
2π ππ = (1 β
π΄1
π΄2)2
βπ»π = Head Loss due to sudden pipe enlargement [m]
ππ = Loss coefficient due to sudden pipe enlargement [1]
π΄ = Wetted Area [m2]
π = Mean Fluid Velocity [m/s]
π = earths gravity [m/s2]
1= Before enlargement
2= After enlargement
4
Head loss Sudden Pipe Contraction
βπ»π= (π΄1
π΄3β 1)2β
π22
2π and π΄3 β 0,6 β π΄2 βπ»π= 0,44 β
π22
2π
βπ»π = Head Loss due to sudden pipe contraction [m]
π2 = Mean Fluid Velocity after sudden pipe contraction [m/s]
π = earths gravity [m/s2]
4
Local head loss coefficients
βπ»π = ππ βπ’2
2π
ππ = ππ
4