Chapter 13 Open-Channel Flow Flow Control and Measurement in Channels Notes | EduRev

: Chapter 13 Open-Channel Flow Flow Control and Measurement in Channels Notes | EduRev

 Page 1


Chapter 13 Open-Channel Flow 
 
Flow Control and Measurement in Channels 
 
13-89C  
 
On the figure, diagram 1-2a is for frictionless 
 gate, 1-2b is for sluice gate with free outflow,  
and 1-2b-2c is for sluice gate with drown outflow, 
 including the hydraulic jump back to subcritical flow.   
 
 
 
 
 
 
 
 
 
. 
 
13-90C For sluice gates, the discharge coefficient C
d
 is defined as the ratio of the actual velocity through 
the gate to the maximum velocity as determined by the Bernoulli equation for the idealized frictionless 
flow case, for which  C
d
  = 1. Typical values of C
d
 for sluice gates with free outflow are in the range of 
0.55 to 0.60. 
 
13-91C The operation of broad crested weir is based on blocking the flow in the channel with a large 
rectangular block, and establishing critical flow over the block. Then the flow rate can be determined by 
measuring flow depths. 
 
13-92C In the case of subcritical flow, the flow depth y will decrease during flow over the bump. 
 
13-93C When the specific energy reaches its minimum value, the flow is critical, and the flow at this point 
is said to be choked. If the bumper height is increased even further, the flow remains critical and thus 
choked. The flow will not become supercritical. 
 
13-94C A sharp-crested weir is a vertical plate placed in a channel that forces the fluid to flow through an 
opening to measure the flow rate. They are characterized by the shape of the opening. For example, a weir 
with a triangular opening is referred to as a triangular weir. 
 
y
E
s1
 = E
s2a
 
(c) Drown 
outflow 
(a) Frictionless 
gate 
 
• 
2c
2b 
• 
1 
• 
2a 
• 
Supercritical
flow 
Subcritical 
flow 
E
s
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, 
you are using it without permission.   
13-45
Page 2


Chapter 13 Open-Channel Flow 
 
Flow Control and Measurement in Channels 
 
13-89C  
 
On the figure, diagram 1-2a is for frictionless 
 gate, 1-2b is for sluice gate with free outflow,  
and 1-2b-2c is for sluice gate with drown outflow, 
 including the hydraulic jump back to subcritical flow.   
 
 
 
 
 
 
 
 
 
. 
 
13-90C For sluice gates, the discharge coefficient C
d
 is defined as the ratio of the actual velocity through 
the gate to the maximum velocity as determined by the Bernoulli equation for the idealized frictionless 
flow case, for which  C
d
  = 1. Typical values of C
d
 for sluice gates with free outflow are in the range of 
0.55 to 0.60. 
 
13-91C The operation of broad crested weir is based on blocking the flow in the channel with a large 
rectangular block, and establishing critical flow over the block. Then the flow rate can be determined by 
measuring flow depths. 
 
13-92C In the case of subcritical flow, the flow depth y will decrease during flow over the bump. 
 
13-93C When the specific energy reaches its minimum value, the flow is critical, and the flow at this point 
is said to be choked. If the bumper height is increased even further, the flow remains critical and thus 
choked. The flow will not become supercritical. 
 
13-94C A sharp-crested weir is a vertical plate placed in a channel that forces the fluid to flow through an 
opening to measure the flow rate. They are characterized by the shape of the opening. For example, a weir 
with a triangular opening is referred to as a triangular weir. 
 
y
E
s1
 = E
s2a
 
(c) Drown 
outflow 
(a) Frictionless 
gate 
 
• 
2c
2b 
• 
1 
• 
2a 
• 
Supercritical
flow 
Subcritical 
flow 
E
s
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, 
you are using it without permission.   
13-45
Chapter 13 Open-Channel Flow 
13-95 Water is released from a reservoir through a sluice gate into an open channel. For specified flow 
depths, the rate of discharge is to be determined. vEES 
Assumptions 1 The flow is steady or quasi-steady. 2 The channel 
is sufficiently wide so that the end effects are negligible.    
Analysis The depth ratio y
1
/a and the contraction coefficient y
2
/a are 
           14
m 1
m 14
1
= =
a
y
    and      3
m 1
m 3
2
= =
a
y
 
The corresponding discharge coefficient is determined from Fig. 
13-38 to be C
d 
= 0.59. Then the discharge rate becomes 
/s m  48.9
3
= = = m) 14 )( m/s (9.81 2 m) m)(1 5 ( 59 . 0 2
2
1
gy ba C
d
V
&
 
a = 1 m
y
2
 = 3 m 
Sluice gate 
y
1
 = 14 m 
Discussion Discharge coefficient is the same as free flow because of small depth ratio after the gate. So, 
the flow rate would not change if it were not drowned. 
 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, 
you are using it without permission.   
13-46
Page 3


Chapter 13 Open-Channel Flow 
 
Flow Control and Measurement in Channels 
 
13-89C  
 
On the figure, diagram 1-2a is for frictionless 
 gate, 1-2b is for sluice gate with free outflow,  
and 1-2b-2c is for sluice gate with drown outflow, 
 including the hydraulic jump back to subcritical flow.   
 
 
 
 
 
 
 
 
 
. 
 
13-90C For sluice gates, the discharge coefficient C
d
 is defined as the ratio of the actual velocity through 
the gate to the maximum velocity as determined by the Bernoulli equation for the idealized frictionless 
flow case, for which  C
d
  = 1. Typical values of C
d
 for sluice gates with free outflow are in the range of 
0.55 to 0.60. 
 
13-91C The operation of broad crested weir is based on blocking the flow in the channel with a large 
rectangular block, and establishing critical flow over the block. Then the flow rate can be determined by 
measuring flow depths. 
 
13-92C In the case of subcritical flow, the flow depth y will decrease during flow over the bump. 
 
13-93C When the specific energy reaches its minimum value, the flow is critical, and the flow at this point 
is said to be choked. If the bumper height is increased even further, the flow remains critical and thus 
choked. The flow will not become supercritical. 
 
13-94C A sharp-crested weir is a vertical plate placed in a channel that forces the fluid to flow through an 
opening to measure the flow rate. They are characterized by the shape of the opening. For example, a weir 
with a triangular opening is referred to as a triangular weir. 
 
y
E
s1
 = E
s2a
 
(c) Drown 
outflow 
(a) Frictionless 
gate 
 
• 
2c
2b 
• 
1 
• 
2a 
• 
Supercritical
flow 
Subcritical 
flow 
E
s
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, 
you are using it without permission.   
13-45
Chapter 13 Open-Channel Flow 
13-95 Water is released from a reservoir through a sluice gate into an open channel. For specified flow 
depths, the rate of discharge is to be determined. vEES 
Assumptions 1 The flow is steady or quasi-steady. 2 The channel 
is sufficiently wide so that the end effects are negligible.    
Analysis The depth ratio y
1
/a and the contraction coefficient y
2
/a are 
           14
m 1
m 14
1
= =
a
y
    and      3
m 1
m 3
2
= =
a
y
 
The corresponding discharge coefficient is determined from Fig. 
13-38 to be C
d 
= 0.59. Then the discharge rate becomes 
/s m  48.9
3
= = = m) 14 )( m/s (9.81 2 m) m)(1 5 ( 59 . 0 2
2
1
gy ba C
d
V
&
 
a = 1 m
y
2
 = 3 m 
Sluice gate 
y
1
 = 14 m 
Discussion Discharge coefficient is the same as free flow because of small depth ratio after the gate. So, 
the flow rate would not change if it were not drowned. 
 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, 
you are using it without permission.   
13-46
Chapter 13 Open-Channel Flow 
13-96 Water flowing in a horizontal open channel encounters a bump. It will be determined if the flow 
over the bump is choked.  vEES 
  
Depression over the bump 
 
 
 
 
 
 
 
 
 
Assumptions 1 The flow is steady. 2 Frictional effects are negligible so that there is no dissipation of 
mechanical energy. 3 The channel is sufficiently wide so that the end effects are negligible.    
y
2
 y
2
 
y
1
=0.80 m y
1
=1.2 m  
V 
1 
=1.2 m/s V
1 
=2.5 m/s 
Bump Bump 
?z = 0.15 m ?z
b
=0.22 m
Analysis The upstream Froude number and the critical depth are  
         729 . 0
m) /s)(1.2 m (9.81
m/s 5 . 2
Fr
2
1
1
1
= = =
gy
V
 
        m 972 . 0
m/s 9.81
s) / m 5 . 2 ( m) 2 . 1 ( ) (
3 / 1
2
2 2
3 / 1
2
1
2
1
3 / 1
2
2
1 1
3 / 1
2
2
=
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
=
g
V y
gb
V by
gb
y
c
V
&
 
The flow is subcritical since Fr < 1, and the flow depth decreases over the bump. The upstream, over the 
bump, and critical specific energy is  
             m 52 . 1
) m/s 2(9.81
m/s) 5 . 2 (
m) 2 . 1 (
2
2
2 2
1
1 1
= + = + =
g
V
y E
s
     
 m 1.30 0.22 1.52
1 2
= - = ? - =
b s s
z E E 
 m 1.46
2
3
= =
c c
y E 
We have an interesting situation: The calculations show that E
s2
 < E
c
. That is, the specific energy of the 
fluid decreases below the level of energy at the critical point, which is the minimum energy, and this is 
impossible. Therefore, the flow at specified conditions cannot exist. The flow is choked when the specific 
energy drops to the minimum value of 1.46 m, which occurs at a bump-height of 
.  m 06 . 0 46 . 1 1.52
1 max ,
= - = - = ?
c s b
E E z
 
Discussion A bump-height over 6 cm results in a reduction in the flow rate of water, or a rise of upstream 
water level. Therefore, a 22-cm high bump alters the upstream flow. On the other hand, a bump less than 6 
cm high will not affect the upstream flow. 
 
 
 
 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, 
you are using it without permission.   
13-47
Page 4


Chapter 13 Open-Channel Flow 
 
Flow Control and Measurement in Channels 
 
13-89C  
 
On the figure, diagram 1-2a is for frictionless 
 gate, 1-2b is for sluice gate with free outflow,  
and 1-2b-2c is for sluice gate with drown outflow, 
 including the hydraulic jump back to subcritical flow.   
 
 
 
 
 
 
 
 
 
. 
 
13-90C For sluice gates, the discharge coefficient C
d
 is defined as the ratio of the actual velocity through 
the gate to the maximum velocity as determined by the Bernoulli equation for the idealized frictionless 
flow case, for which  C
d
  = 1. Typical values of C
d
 for sluice gates with free outflow are in the range of 
0.55 to 0.60. 
 
13-91C The operation of broad crested weir is based on blocking the flow in the channel with a large 
rectangular block, and establishing critical flow over the block. Then the flow rate can be determined by 
measuring flow depths. 
 
13-92C In the case of subcritical flow, the flow depth y will decrease during flow over the bump. 
 
13-93C When the specific energy reaches its minimum value, the flow is critical, and the flow at this point 
is said to be choked. If the bumper height is increased even further, the flow remains critical and thus 
choked. The flow will not become supercritical. 
 
13-94C A sharp-crested weir is a vertical plate placed in a channel that forces the fluid to flow through an 
opening to measure the flow rate. They are characterized by the shape of the opening. For example, a weir 
with a triangular opening is referred to as a triangular weir. 
 
y
E
s1
 = E
s2a
 
(c) Drown 
outflow 
(a) Frictionless 
gate 
 
• 
2c
2b 
• 
1 
• 
2a 
• 
Supercritical
flow 
Subcritical 
flow 
E
s
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, 
you are using it without permission.   
13-45
Chapter 13 Open-Channel Flow 
13-95 Water is released from a reservoir through a sluice gate into an open channel. For specified flow 
depths, the rate of discharge is to be determined. vEES 
Assumptions 1 The flow is steady or quasi-steady. 2 The channel 
is sufficiently wide so that the end effects are negligible.    
Analysis The depth ratio y
1
/a and the contraction coefficient y
2
/a are 
           14
m 1
m 14
1
= =
a
y
    and      3
m 1
m 3
2
= =
a
y
 
The corresponding discharge coefficient is determined from Fig. 
13-38 to be C
d 
= 0.59. Then the discharge rate becomes 
/s m  48.9
3
= = = m) 14 )( m/s (9.81 2 m) m)(1 5 ( 59 . 0 2
2
1
gy ba C
d
V
&
 
a = 1 m
y
2
 = 3 m 
Sluice gate 
y
1
 = 14 m 
Discussion Discharge coefficient is the same as free flow because of small depth ratio after the gate. So, 
the flow rate would not change if it were not drowned. 
 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, 
you are using it without permission.   
13-46
Chapter 13 Open-Channel Flow 
13-96 Water flowing in a horizontal open channel encounters a bump. It will be determined if the flow 
over the bump is choked.  vEES 
  
Depression over the bump 
 
 
 
 
 
 
 
 
 
Assumptions 1 The flow is steady. 2 Frictional effects are negligible so that there is no dissipation of 
mechanical energy. 3 The channel is sufficiently wide so that the end effects are negligible.    
y
2
 y
2
 
y
1
=0.80 m y
1
=1.2 m  
V 
1 
=1.2 m/s V
1 
=2.5 m/s 
Bump Bump 
?z = 0.15 m ?z
b
=0.22 m
Analysis The upstream Froude number and the critical depth are  
         729 . 0
m) /s)(1.2 m (9.81
m/s 5 . 2
Fr
2
1
1
1
= = =
gy
V
 
        m 972 . 0
m/s 9.81
s) / m 5 . 2 ( m) 2 . 1 ( ) (
3 / 1
2
2 2
3 / 1
2
1
2
1
3 / 1
2
2
1 1
3 / 1
2
2
=
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
=
g
V y
gb
V by
gb
y
c
V
&
 
The flow is subcritical since Fr < 1, and the flow depth decreases over the bump. The upstream, over the 
bump, and critical specific energy is  
             m 52 . 1
) m/s 2(9.81
m/s) 5 . 2 (
m) 2 . 1 (
2
2
2 2
1
1 1
= + = + =
g
V
y E
s
     
 m 1.30 0.22 1.52
1 2
= - = ? - =
b s s
z E E 
 m 1.46
2
3
= =
c c
y E 
We have an interesting situation: The calculations show that E
s2
 < E
c
. That is, the specific energy of the 
fluid decreases below the level of energy at the critical point, which is the minimum energy, and this is 
impossible. Therefore, the flow at specified conditions cannot exist. The flow is choked when the specific 
energy drops to the minimum value of 1.46 m, which occurs at a bump-height of 
.  m 06 . 0 46 . 1 1.52
1 max ,
= - = - = ?
c s b
E E z
 
Discussion A bump-height over 6 cm results in a reduction in the flow rate of water, or a rise of upstream 
water level. Therefore, a 22-cm high bump alters the upstream flow. On the other hand, a bump less than 6 
cm high will not affect the upstream flow. 
 
 
 
 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, 
you are using it without permission.   
13-47
Chapter 13 Open-Channel Flow 
13-97 Water flowing in a horizontal open channel encounters a bump. The change in the surface level over 
the bump and the type of flow (sub- or supercritical) over the bump are to be determined.  vEES 
  
Rise over the bump 
 
 
 
 
 
 
 
 
 
Assumptions 1 The flow is steady. 2 Frictional effects are negligible so that there is no dissipation of 
mechanical energy. 3 The channel is sufficiently wide so that the end effects are negligible.    
y
2
 y
2
 
y
1
=0.80 m y
1
= 0.8 m 
V  
1 
=1.2 m/s V
1 
= 8 m/s 
Bump Bump 
?z = 0.15 m ?z
b
=0.30 m
Analysis The upstream Froude number and the critical depth are  
         856 . 2
m) /s)(0.8 m (9.81
m/s 8
Fr
2
1
1
1
= = =
gy
V
 
        m 61 . 1
m/s 9.81
s) / m 8 ( m) 8 . 0 ( ) (
3 / 1
2
2 2
3 / 1
2
1
2
1
3 / 1
2
2
1 1
3 / 1
2
2
=
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
=
g
V y
gb
V by
gb
y
c
V
&
 
The upstream flow is supercritical since Fr > 1, and the flow depth increases over the bump. The upstream, 
over the bump, and critical specific energy are 
             m 06 . 4
) m/s 2(9.81
m/s) 8 (
m) 8 . 0 (
2
2
2 2
1
1 1
= + = + =
g
V
y E
s
     
 m 3.76 0.30 4.06
1 2
= - = ? - =
b s s
z E E 
 m 42 . 2
2
3
= =
c c
y E 
The flow depth over the bump can be determined from 
0
2
) (
2
1
2
1 2
2 1
3
2
= + ? - - y
g
V
y z E y
b s
   ?   0 m) 80 . 0 (
) m/s 2(9.81
m/s) 8 (
) m 30 . 0 06 . 4 (
2
2
2
2
2
3
2
= + - - y y 
Using an equation solver, the physically meaningful root of this equation is determined to be 0.846 m. 
Therefore, there is a rise of  
m 0.346 = + - = ? + - = 30 . 0 80 . 0 846 . 0 over bump Rise
1 2 b
z y y 
over the surface relative to the upstream water surface. The specific energy decreases over the bump from, 
4.06 to 3.76 m, but it is still over the minimum value of 2.42 m. Therefore, the flow over the bump is still 
supercritical. 
Discussion The actual value of surface rise may be different than the 4.6 cm because of the frictional 
effects that are neglected in the analysis. 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, 
you are using it without permission.   
13-48
Page 5


Chapter 13 Open-Channel Flow 
 
Flow Control and Measurement in Channels 
 
13-89C  
 
On the figure, diagram 1-2a is for frictionless 
 gate, 1-2b is for sluice gate with free outflow,  
and 1-2b-2c is for sluice gate with drown outflow, 
 including the hydraulic jump back to subcritical flow.   
 
 
 
 
 
 
 
 
 
. 
 
13-90C For sluice gates, the discharge coefficient C
d
 is defined as the ratio of the actual velocity through 
the gate to the maximum velocity as determined by the Bernoulli equation for the idealized frictionless 
flow case, for which  C
d
  = 1. Typical values of C
d
 for sluice gates with free outflow are in the range of 
0.55 to 0.60. 
 
13-91C The operation of broad crested weir is based on blocking the flow in the channel with a large 
rectangular block, and establishing critical flow over the block. Then the flow rate can be determined by 
measuring flow depths. 
 
13-92C In the case of subcritical flow, the flow depth y will decrease during flow over the bump. 
 
13-93C When the specific energy reaches its minimum value, the flow is critical, and the flow at this point 
is said to be choked. If the bumper height is increased even further, the flow remains critical and thus 
choked. The flow will not become supercritical. 
 
13-94C A sharp-crested weir is a vertical plate placed in a channel that forces the fluid to flow through an 
opening to measure the flow rate. They are characterized by the shape of the opening. For example, a weir 
with a triangular opening is referred to as a triangular weir. 
 
y
E
s1
 = E
s2a
 
(c) Drown 
outflow 
(a) Frictionless 
gate 
 
• 
2c
2b 
• 
1 
• 
2a 
• 
Supercritical
flow 
Subcritical 
flow 
E
s
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, 
you are using it without permission.   
13-45
Chapter 13 Open-Channel Flow 
13-95 Water is released from a reservoir through a sluice gate into an open channel. For specified flow 
depths, the rate of discharge is to be determined. vEES 
Assumptions 1 The flow is steady or quasi-steady. 2 The channel 
is sufficiently wide so that the end effects are negligible.    
Analysis The depth ratio y
1
/a and the contraction coefficient y
2
/a are 
           14
m 1
m 14
1
= =
a
y
    and      3
m 1
m 3
2
= =
a
y
 
The corresponding discharge coefficient is determined from Fig. 
13-38 to be C
d 
= 0.59. Then the discharge rate becomes 
/s m  48.9
3
= = = m) 14 )( m/s (9.81 2 m) m)(1 5 ( 59 . 0 2
2
1
gy ba C
d
V
&
 
a = 1 m
y
2
 = 3 m 
Sluice gate 
y
1
 = 14 m 
Discussion Discharge coefficient is the same as free flow because of small depth ratio after the gate. So, 
the flow rate would not change if it were not drowned. 
 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, 
you are using it without permission.   
13-46
Chapter 13 Open-Channel Flow 
13-96 Water flowing in a horizontal open channel encounters a bump. It will be determined if the flow 
over the bump is choked.  vEES 
  
Depression over the bump 
 
 
 
 
 
 
 
 
 
Assumptions 1 The flow is steady. 2 Frictional effects are negligible so that there is no dissipation of 
mechanical energy. 3 The channel is sufficiently wide so that the end effects are negligible.    
y
2
 y
2
 
y
1
=0.80 m y
1
=1.2 m  
V 
1 
=1.2 m/s V
1 
=2.5 m/s 
Bump Bump 
?z = 0.15 m ?z
b
=0.22 m
Analysis The upstream Froude number and the critical depth are  
         729 . 0
m) /s)(1.2 m (9.81
m/s 5 . 2
Fr
2
1
1
1
= = =
gy
V
 
        m 972 . 0
m/s 9.81
s) / m 5 . 2 ( m) 2 . 1 ( ) (
3 / 1
2
2 2
3 / 1
2
1
2
1
3 / 1
2
2
1 1
3 / 1
2
2
=
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
=
g
V y
gb
V by
gb
y
c
V
&
 
The flow is subcritical since Fr < 1, and the flow depth decreases over the bump. The upstream, over the 
bump, and critical specific energy is  
             m 52 . 1
) m/s 2(9.81
m/s) 5 . 2 (
m) 2 . 1 (
2
2
2 2
1
1 1
= + = + =
g
V
y E
s
     
 m 1.30 0.22 1.52
1 2
= - = ? - =
b s s
z E E 
 m 1.46
2
3
= =
c c
y E 
We have an interesting situation: The calculations show that E
s2
 < E
c
. That is, the specific energy of the 
fluid decreases below the level of energy at the critical point, which is the minimum energy, and this is 
impossible. Therefore, the flow at specified conditions cannot exist. The flow is choked when the specific 
energy drops to the minimum value of 1.46 m, which occurs at a bump-height of 
.  m 06 . 0 46 . 1 1.52
1 max ,
= - = - = ?
c s b
E E z
 
Discussion A bump-height over 6 cm results in a reduction in the flow rate of water, or a rise of upstream 
water level. Therefore, a 22-cm high bump alters the upstream flow. On the other hand, a bump less than 6 
cm high will not affect the upstream flow. 
 
 
 
 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, 
you are using it without permission.   
13-47
Chapter 13 Open-Channel Flow 
13-97 Water flowing in a horizontal open channel encounters a bump. The change in the surface level over 
the bump and the type of flow (sub- or supercritical) over the bump are to be determined.  vEES 
  
Rise over the bump 
 
 
 
 
 
 
 
 
 
Assumptions 1 The flow is steady. 2 Frictional effects are negligible so that there is no dissipation of 
mechanical energy. 3 The channel is sufficiently wide so that the end effects are negligible.    
y
2
 y
2
 
y
1
=0.80 m y
1
= 0.8 m 
V  
1 
=1.2 m/s V
1 
= 8 m/s 
Bump Bump 
?z = 0.15 m ?z
b
=0.30 m
Analysis The upstream Froude number and the critical depth are  
         856 . 2
m) /s)(0.8 m (9.81
m/s 8
Fr
2
1
1
1
= = =
gy
V
 
        m 61 . 1
m/s 9.81
s) / m 8 ( m) 8 . 0 ( ) (
3 / 1
2
2 2
3 / 1
2
1
2
1
3 / 1
2
2
1 1
3 / 1
2
2
=
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
=
g
V y
gb
V by
gb
y
c
V
&
 
The upstream flow is supercritical since Fr > 1, and the flow depth increases over the bump. The upstream, 
over the bump, and critical specific energy are 
             m 06 . 4
) m/s 2(9.81
m/s) 8 (
m) 8 . 0 (
2
2
2 2
1
1 1
= + = + =
g
V
y E
s
     
 m 3.76 0.30 4.06
1 2
= - = ? - =
b s s
z E E 
 m 42 . 2
2
3
= =
c c
y E 
The flow depth over the bump can be determined from 
0
2
) (
2
1
2
1 2
2 1
3
2
= + ? - - y
g
V
y z E y
b s
   ?   0 m) 80 . 0 (
) m/s 2(9.81
m/s) 8 (
) m 30 . 0 06 . 4 (
2
2
2
2
2
3
2
= + - - y y 
Using an equation solver, the physically meaningful root of this equation is determined to be 0.846 m. 
Therefore, there is a rise of  
m 0.346 = + - = ? + - = 30 . 0 80 . 0 846 . 0 over bump Rise
1 2 b
z y y 
over the surface relative to the upstream water surface. The specific energy decreases over the bump from, 
4.06 to 3.76 m, but it is still over the minimum value of 2.42 m. Therefore, the flow over the bump is still 
supercritical. 
Discussion The actual value of surface rise may be different than the 4.6 cm because of the frictional 
effects that are neglected in the analysis. 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, 
you are using it without permission.   
13-48
Chapter 13 Open-Channel Flow 
13-98 The flow rate in an open channel is to be measured using a sharp-crested rectangular weir. For a 
measured value of flow depth upstream, the flow rate is to be determined.  vEES 
 
 
V
1 
 
y
1
 = 2.2 m 
P
w
 = 0.75 m 
 Sharp-crested 
rectangular weir
 
 
 
 
 
 
 
 
 
Assumptions 1 The flow is steady. 2 The upstream velocity head is negligible. 3 The channel is sufficiently 
wide so that the end effects are negligible.    
Analysis The weir head is 
                m 45 . 1 75 . 0 2 . 2
1
= - = - =
w
P y H
The discharge coefficient of the weir is 
         771 . 0
m 75 . 0
m 45 . 1
0897 . 0 598 . 0 0897 . 0 598 . 0
rec ,
= + = + =
w
wd
P
H
C 
The condition H/P
w
 < 2 is satisfied since 1.45/0.75 = 1.93. Then the water flow rate through the channel 
becomes  
/s m 15.9
3
=
=
=
2 / 3 2
2 / 3
rec , rec
) m 45 . 1 ( ) m/s 2(9.81 m) 4 (
3
2
) 7714 . 0 (
2
3
2
H g b C
wd
V
&
 
Discussion The upstream velocity and the upstream velocity head are 
m/s 81 . 1
m) m)(2.2 (4
/s m 9 . 15
3
1
1
= = =
by
V
V
&
   and     m 167 . 0
) m/s 2(9.81
m/s) 81 . 1 (
2
2
2 2
1
= =
g
V
 
This is 11.5% of the weir head, which is significant. When the upstream velocity head is considered, the 
flow rate becomes 18.1 m
3
/s, which is about 14 percent higher than the value determined above. Therefore, 
it is good practice to consider the upstream velocity head unless the weir height P
w
 is very large relative to 
the weir head H.  
 
 
 
 
 
 
 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, 
you are using it without permission.   
13-49
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