4.
EGL
HGL
HGL & EGL
HGL & EGL
V2/2g
Gradual expansion of conduit allows kinetic
energy to be converted to pressure head with
much smaller hL at the outlet. Then the HGL
approaches the EGL
P/g
Z
HT
Turbine
Neglecting entrance
and exit losses
6.
V2/2g
EGL
HGL
hL due to partially
closed valve
hL due to entrance
hL due to outlet
Valve
Consider entrance
and outlet losses
P/g
Z
Reference datum
HGL & EGL
7.
V2/2g
EGL
HGL
hL due to partially
closed valve
hL due to outlet
Valve
Neglecting entrance
and outlet losses
P/g
Z
Reference datum
HGL & EGL
8.
V1
2/2g
EGL
HGL
hL due to entrance
hL due to outlet
Consider entrance
and outlet losses
Larger V2
2/2g
because smaller pipe diameter here
P1/g
Z
Reference datum
65. (125)
(150)
(25) (25)
hT
hL(1)
hL (2)
hL (2)
hp
hL(1)
Pump
Turbine
EGL
EGL
150
h
)
h
h
(
25 p
2
,
L
1
,
L
25
h
)
h
h
(
150 T
2
,
L
1
,
L
ft
17
h
h 2
,
L
1
,
L
ft
17
h
h 2
,
L
1
,
L
Case: “a” Case: “b”
80. g
az
h
Z
g
az
h
Z
1
p 1
p
1
2 p
p h
g
2
p
p 1
2
Density Density
Free Fall
of a Liquid az = - g
Upward Acceleration
of a Liquid az = + g
Free Fall of a Fluid Body
Fig. ( ) - The Effect of Acceleration on the Pressure of a Liquid
During Free Fall and Upward Acceleration
81. az 0 az 0
g h g h
h
Fig. ( ) - Variation of the Magnitude of Pressure with the Variation of az
Variation.
87. Not all flows with circle streamlines are rotational. To
illustrate this, we consider two:
Incompressible.
Steady,
Two-dimensional flows,
Both of which have circular streamlines in the r
- plane.
Comparison of Two Circular Flow
88. Free Vortex: A free vortex approximates to naturally occurring
circular flows (e.g. circumferential component of the flow
down a drain hole or round a river bend) in which there is no
source of energy.
Forced Vortex: A forced vortex is a circular motion
approximating to the pattern generated by the action of a
mechanical rotor on a fluid. The rotor forces the fluid to
rotate at uniform rotational speed rad/sec, so V = r. A
forced vortex is a rotational flow.
Comparison of Two Circular Flow
89. Flow “A”: Vr = 0 and V = c/r (ii)
We have to note that:
The “bathtub vortex” formed when water drains through a
bottom hole in a tank, is a good approximation to the free
vortex.
Comparison of Two Circular Flow
r and are constants
The streamlines are circles
(const. r), and the potential
Lines are radial Spokes
(const. )
90. Flow “B”: Vr = 0 and V = r (i)
We have to note that:
The motion of a liquid contained in a tank that is rotated
about its axis with angular velocity corresponds to a
forced vortex.
Comparison of Two Circular Flow
r and are constants
The streamlines are circles
(const. r), and the potential
Lines are radial Spokes
(const. )
91.
Solid lines are streamlines,
Dashed lines are potential lines
Streamlines
Potential lines
92.
Fig. ( ) – Velocity distribution in a Forced Vortex.
Forced Vortex.
V = r
93.
V
r
Fig. ( ) – Velocity distribution in a Free Vortex.
Free Vortex.
V = c / r
95. s
/
ft
028
.
0
449
1
60
745
Q
discharge
The 3
Applying the energy equation (Bernoulli’s eq. ) between the points identified
as “1” and “2” in the shown figure, we have
2
2
L
p
1
2
g
2
V
g
p
z
h
h
g
2
V
g
p
z
For the given condition, p1= pamt = 0, V1 = V2 0 and taking the water
surface at the well as a reference datum (Z1=0 & Z2 = 120 ft ), the above
equation gives
0
0
96.
0
2
.
32
94
.
1
12
40
120
50
.
10
h
0
0
0
2
p
Solving for hp gives hp = 222.7 ft.
We can now calculate the power delivered by the pump
hp
70
.
0
550
7
.
222
028
.
0
2
.
32
94
.
1
h
Q
g
power p
To calculate the mechanical efficiency of the pump,
70
.
0
0
.
1
70
.
0
power
pump
h
Q
g
Efficiency
p
Thus, at the stated condition, the pump is 70% efficient.
115. 0
dy/dx
Fr
1.0 1.5 2.0
0.5
SE < So
SE > So
SE = So
The shown figure is valid for
channels of any constant cross-
sectional shape, provided the
Froude number is interpreted.
116. 2
r
o
E
F
1
S
S
dx
dy
It is seen that the rate of change of water depth, dy/dx, depends on:
the local slope of the channel bottom, So,
the slope of the energy line, SE , and
the Froude number Fr.
By using the concepts of the specific energy and critical conditions (Fr
= 1.0), it was possible to determine how the depth of a flow in an open
channel changes with distance along the channel.
117. 0
dy/dx
Fr
1.0 1.5 2.0
0.5
SE < So
SE > So
SE = So
The shown figure is valid for
channels of any constant cross-
sectional shape, provided the
Froude number is interpreted.
As shown by the shown figure,
the value of dy/dx can be either
negative, zero, or positive,
depending on the values of
these three parameters.
That is, the channel flow depth may be constant or it may increase or
decrease in the flow direction, depending on the values of So, SE, and
Fr.
118. The behavior of subcritical flow may be the opposite of that for
supercritical flow, as shown by the denominator, 1-Fr
2 of the above
shown equation.
In this section we will consider only rectangular cross-sectional
channels when using this equation
