1
)=
p
1
ρ
+
α
1
U
1
2
2
+
gy
1
–
p
2
ρ
+
α
2
U
2
2
2
+
gy
2
(
p
/
+
U
2
/2+
gy
)
represents the mechanical
energy per unit mass at a
flow cross section.
(
u
2

u
1

Q
/
m
) is equal to the difference in
mechanical energy per unit mass
between sections 1 & 2.
This term represents the conversion of
mechanical energy at section (1) to unwanted
thermal energy (
u
2

u
1
) and loss of energy via
heat transfer (
Q
/
m
). So it is the total head
loss
.
2
2
2
2
2
1
2
1
1
1
t
2
2
gy
U
p
gy
U
p
h
l
80
h
l
t
=
h
l
+
h
l
m
h
l
 head loss due to frictional effects in fully
developed flow in constant area conduits.
h
lm

minor losses due to entrances, fittings,
area changes, etc.
So, for a fully developed flow through a
constantarea pipe,
2
1
2
1
y
y
g
p
p
h
l
and if
y
1
=
y
2
,
h
l
=
p
1
–
p
2
ρ
=
∆
p
ρ
For laminar flow,
–
d
p
d
x
=
∆
p
L
=
128
μ
Q
π
D
4
Q
=
U
(
π
D
2
/4)
⇒
∆
p
= 32
L
D
μ
U
D
h
l
= 32
L
D
μ
U
ρ
D
=
L
D
U
2
2
64
μ
ρ
U
D
=
64
Re
L
D
U
2
2
f
=
64
Re
⇒
h
l
=
f
L
D
U
2
2
81
8.3. Turbulent flow
In turbulent flow we cannot evaluate the pressure drop
analytically. We must use experimental data and
dimensional analysis. In fully developed turbulent flow,
the pressure drop,
p
, due to friction in a horizontal
constantarea pipe is known to depend on the pipe
diameter,
D
, the pipe length,
L
, the pipe roughness,
e
, the
average flow velocity,
U
, the fluid density,
, and the
fluid viscosity,
.
Therefore,
p
=
p
(
D
,
L
,
e
,
U
,
,
)
Dimensional analysis,
D
e
,
D
L
,
D
U
ρ
μ
U
P
1
2
Δ
h
l
=
p
/
D
e
,
D
L
,
U
h
l
Re
1
2
Experiments show that the nondimensional
head loss is directly proportional to
L
/
D
, hence:
D
e
Re
D
L
U
h
l
,
2
2
or
D
e
Re
D
L
U
h
l
,
2
2
2
1
Define the friction factor,
f
=
3
(
Re
,
e
/
D
)
So,
h
l
=
f
(
L
/
D
) (
U
2
/
2)
f
is determined experimentally. Moody Diagram
.
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 Fall '08
 ZOHAR
 Fluid Dynamics, Incompressible Flow, Turbulent Flow