FULLY DEVELOPED TURBULENT PIPE FLOW

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FULLY DEVELOPED TURBULENT PIPE FLOW CLASS 2 -
REVIEW
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FULLY DEVELOPED, STEADY, NO BODY FORCES, LAMINAR
PIPE FLOW
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0
0
FSx FBx ?/?t (?cvu?dVol ) ?csu?V?dA Eq.
(4.17)
? (r/2)(dp/dx) Eq 8.13a
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? (r/2)(dp/dx)
?yx ?(du/dy)?uv uavg UC/L(1-r/R)1/n Q
?0 uldyV Q/A V/UC/L 2n2/(2n1)(n1)
?yx ?(du/dy) u - (R2/4?)(dp/dx) x 1
(r/R)2 UC/L1-(r/R)2 Q ?0 uldyV
Q/A V/UC/L 1/2
(empirical)
a
a
turbulent
laminar
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u(r)/Uc/l (y/R)1/n (R-r/R)1/n (1-r/R)1/n
n6-10
Laminar Flow u/Uc/l 1-(r/R)2
u(r)/Uc/l (y/R)1/n
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Eq. 8.30 one of
the most important and useful equations in
fluid mechanics Fox et al.
ENTER ENERGY EQUATION
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?V12/ (2) p1/(?) gz1?V22/ (2) p2/(?) gz2
hlT hlT has units of enery per unit mass
V2
?V12/ (2g) p1/(?g) z1?V22/ (2g) p2/(?g)
z2 HlT HlT has units of
enery per unit weight L
from hydraulics during 1800s
one of the most important and useful equations
in fluid mechanics Fox et al.
Allows calculations of capacity of an oil pipe
line, what diameter water main to install or
pressure drop in an air duct,
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?V12/ (2g) p1/(?g) z1?V22/ (2g) p2/(?g)
z2 HlT
?A ½ V(r)2 ? V(r)dA ? (dm/dt) ½ V 2 ? ?A ?
V(r)3 dA / (dm/dt) V2

• Turbulent Flow V(r)/Uc/l (1-r/R)1/n
• Laminar Flow V(r)/Uc/l 1 (r/R)2

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?V12/ (2g) p1/(?g) z1?V22/ (2g) p2/(?g)
z2 HlT

• ?A ? V(r)3 dA / (dm/dt) V2
• 1 for potential flow
• 2 for laminar flow
• ? 1 for turbulent flow

V(r)/Uc/l (y/R)1/n

• (Uc/l/V)3 2n2 / (3 n)(3 2n)
• 1.08 for n 6 ? 1.03 for n 10

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?V1avg2/ (2g) p1/(?g) z1 ?V2avg2/ (2g)
p2/(?g) z2 Hl (Eq. 8.30)

Hl
(Eq. 8.34)
V Q/Area
Hl
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BREATH
(early 20th Century turbulent pipe flow
experiments)
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fF ?wall /(1/2) ?V2
Similarity of Motion in Relation to the Surface
Friction of Fluids Stanton Pannell Phil.
Trans. Royal Soc., (A) 1914
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fF ?wall /(1/2) ?V2
1914
fF ?wall /(1/2) ?V2 fD (?p/L)D/(1/2) ?V2
(?p/L)2R2/2 ½ ?V2 4?wall /(1/2)
?V2 4 fF
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BREATH
(rough pipe turbulent flow experiments)
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Original Data of Nikuradze
?p ? U?
Stromungsgesetze in Rauhen Rohren, V.D.I. Forsch.
H, 1933, Nikuradze
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aside
Sir Isaac Newton (1642 1727)
?p ? uavg2 Newton believed that drag ?
uavg2 arguing that each fluid particle would lose
all their momentum normal to the body. Drag
Mass Flow x Change in Momentum Drag dp/dt ?
(?U?A)?U ? ?U?2A Drag/Area ? ?U?2
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• Fully rough zone where have flow separation
• over roughness elements and ?p V2
• k ?u/? k lt 4 hydraulically smooth
• 4 lt k lt 60 transitional regime k gt 60 fully
  rough (no ? effect)
• White 1991 Viscous Fluid Flow

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Curves are from average values good to /- 10
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BREATH
(Moody Diagram)
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Hl f (L/D)V2/(2g)
laminar
t u r b u l e n t
f 64/Re and is proportional to ? in laminar
flow f is not a function of ?/D in laminar flow
f const. and is not a function of ? at high
enough Re turbulent flows in a rough pipe f is
usually a function of ?/D in turbulent flows
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Curves are from average values good to /- 10
fD (?p/L)D/(1/2) ?V2 Darcy friction
factor ReD UD/?
For new pipes, corrosion may cause e/D for old
pipes to be 5 to 10 times greater.
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fF -2.0log(e/D/3.7 2.51/(RefF0.5) If first
guess is fo 0.25log(e/D/3.7
5.74/Re0.9-2 should be within 1 after 1
iteration
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For turbulent flow in a smooth pipe and ReD lt
105, can use Blasius correlation f
0.316/ReD0.25 which can be rewritten as ?wall
0.0332 ? V2 (?/RV)1/4)
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For turbulent flow and Re lt 105 can use Blasius
correlation fD 0.316/Re0.25 Which can be
rewritten as ?wall 0.0332 ? V2 (?/RV) PROOF
fD 4 fF 0.316 ?1/4 / (V1/4 D1/4)
4?wall/(1/2 ?V2) ?wall (0.0395 ?V2) ?1/4 /
(V1/4 (2R)1/4) ?wall (0.0332 ?V2) ? /
(VR)1/4 QED
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Question? Looking at graph imagine that pipe
diameter, length, viscosity and density is
fixed. Is there any region where an increase in V
results in an increase in pressure drop?
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Question? Looking at graph imagine that pie
diameter and kinematic viscosity and density is
fixed. Is there any region where an increase in V
results in an increase in pressure drop?
Everywhere!!!!!!!
Turbulent flow
Instead of non-dimensionalizing ?p by ½ ?V2 use
D3 ? /( ?2L)
transition
?pD3 ?/(?2L)
Laminar flow
From Tritton
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Some history
Moody Diagram
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f function of V, D, roughness and viscosity f
is dimensionless
Hl
Antoine Chezy 1770 for channels V2P
AS extrapolate this for pipe Hl
(4/C2)(L/D)V2 Gaspard Riche de Prony (1800) Hl
(L/D)(aV bV2) C a and b are not
dimensionless C a and b are not a function of
roughness
Antoine Chezy
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Hl
f function of V, D and roughness f is
dimensionless
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Traditional to call f the Darcy friction factor
although Darcy never proposed it in that form
Hl
f is a function of ? and D better estimates of f
Could be dropped for rough pipes
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Hl
?
?
?w/( ½ ?Vavg2) prob 8.83
Combined Weisbachs equation with Darcy and
other data, compiled table for f but used
hydraulic radius.
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Hl
Eq. 8.34
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4000lt ReR lt 80000
Full range of turbulent Reynolds numbers
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Re
f
1/?f
Re/?f
These equations are obviously too complex to be
of practical use. On the other hand, if the
function which they embody is even approximately
valid for commercial surfaces in general, such
extremely important information could be made
readily available in diagrams or tables.
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Hl
f ?p/(?g)D2g/(LV2) f ?p/LD/1/2?V2
The author does not claim to offer anything
particularly new or original, his aim merely
being to embody the now accepted conclusion in
convenient form for engineering use.
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Hl
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T H E
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