Open channel flow is flow of a liquid in a conduit with a free surface
subjected to atmospheric pressure.
Examples: flow of water in rivers, canals, partially full sewers and drains and flow
of water over land.
Free surface
Datum

y
A
B
T
Figure. Sketch of open channel geometry
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Dr. Indrajeet Sahu

Application
• Practical applications are the
determination of:
• Flow depth in rivers, canals and other
conveyance conduits,
• Changes in flow depth due to channel
controls. Example: Weirs, spillways, and
gates,
• Changes in river stage during floods,
• Surface runoff from rainfall over land,
• Optimal channel design
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Types of
channels
Man made
• Channel designed and made by
human
• Examples: earth or concrete lined
drainage and irrigation
• Prismatic channel (no change in
geometry with distance)
Natural
• Examples: River and streams
• Changes with spatial and
temporal (non prismatic channel)
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Dr. Indrajeet Sahu

Types of Open Channel
 Prismatic and non-prismatic
channels
• Prismatic channel is the channel which cross-
sectional shape, size and bottom slope are constant.
Most of the man-made (artificial) channels are prismatic
channels over long stretches.
• Examples of man-made channels are irrigation canal,
flume, drainage ditches, roadside gutters, drop, chute,
culvert and tunnel.
• All natural channels generally have varying cross-
sections and therefore are non-prismatic.
• Examples of natural channels are tiny hillside rivulets,
through brooks, streams, rivers and tidal estuaries.
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Rigid and mobile boundary channels
• Rigid channels are channels with boundaries that
is not deformable. Channel geometry and
roughness are constant over time.
• Typical examples are lined canals, sewers and non-
erodible unlined canals.
• Mobile boundary channels are channels with
boundaries that undergo deformation due to the
continuous process of erosion and deposition due
to the flow.
• Examples are unlined man-made channels and natural
rivers.
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Canals - is usually a long and mild-sloped
channel built in the ground, which
may be unlined or lined with stoned
masonry, concrete, cement, wood
or bituminous material.
Griboyedov Canal, St. Petersburg, Russia
Terusan Wan Muhammad Saman, Kedah
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This flume diverts water from White River,
Washington to generate electricity Bull Run Hydroelectric Project diversion flume
Open-channel flume in laboratory
Flumes - is a channel of wood, metal, concrete, or masonry, usually supported
on or above the surface of the ground to carry water across a
depression.
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Drop - the change in elevation is within a short distance.
The spillway of Leasburg Diversion Dam is a vertical hard basin drop
structure designed to dissipate energy
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Stormwater sewer - is a drain or drain system designed to drain excess rain
from paved streets, parking lots, sidewalks and roofs.
Storm drain receiving urban runoff
Storm sewer
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FLOW IN OPEN CHANNEL
STEADY FLOW UNSTEADY FLOW
UNIFORM FLOW NON-UNIFORM FLOW
RAPIDLY VARIED FLOW
GRADUALLY VARIED FLOW
TEMPORAL (Time)
SPATIAL (Space)
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Dr. Indrajeet Sahu

Types of flow
Based on temporal
(Time, t) and
Spatial (Space,x)
Time Criteria
Steady flow (dy/dt = 0).
Water depth at one point
same all the time. (Flow
constant with time)
Unsteady flow (dy/dt ≠ 0).
Water depth changes all the
time. (Flow variation with
time)
Space criteria
Uniform flow (dy/dx = 0).
Water depth same along
the whole length of flow.
Non-uniform flow (dy/dx
≠ 0). Water depth
changes either rapidly or
gradually flow
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Types / Classification of Open Channel Flows
Open channel flow conditions can be characterised with respect to space (uniform or non-uniform flows)
and time (steady or unsteady flows).
Space - how do the flow conditions change along the reach of an open channel system.
a. Uniform flow - depth of flow is the same at every section of the flow dy/dx = 0
b. Non-uniform flow - depth of flow varies along the flow dy/dx  0
Time - how do the flow conditions change over time at a specific section in an open channel system.
c. Steady flow - depth of flow does not change/ constant during the time interval under
consideration dy/dt = 0
d. Unsteady flow - depth of flow changes with time dy/dt  0
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Types / Classification of Open Channel Flows
a. Uniform flow
b. Non-uniform flow
c. Steady flow
d. Unsteady flow
y
y
y1
y2
y
Time = t1
y
Time = t2
y1
Time = t1
y1
Time = t2
y
t3
t2
t1
y
t3
t2
t1
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Sluice
Hydraulic
jump
Flow over
weir
Hydraulic
drop
Contraction
below the sluice
RVF RVF
GVF RVF
GVF RVF
GVF
 The flow is gradually varied if the depth changes slowly over a comparatively long distance.
 Examples of gradually varied flow (GVF) are flow over a mild slope and the backing up of flow
(backwater).
 The flow is rapidly varied if the depth changes abruptly over a comparatively short distance.
 Examples of rapidly varied flow (RVF) are hydraulic jump, hydraulic drop, flow over weir and flow under a
sluice gate.
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States of flow
Flow vary with
following forces:
• Viscous
• Inertia
• Gravity
Defines by
Reynolds
number (Re)
and Froude
numbers (Fr)
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Re < 500 , the flow is laminar
500 < Re < 12500, the flow is transitional
Re > 12500 , the flow is turbulent
The flow is laminar if the viscous forces are dominant relative to inertia.
Viscosity will determine the flow behaviour. In laminar flow, water
particles move in definite smooth paths.

VR

Re
The flow is turbulent if the inertial forces are dominant than the viscous
force. In turbulent flow, water particles move in irregular paths which
are not smooth.
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State of Flow
Effect of gravity - depending on the effect of gravity forces relative to inertial
forces, the flow may be subcritical, critical and supercritical.
- Froude number represents the ratio of inertial forces to
gravity forces,
gD
V

Fr
where V is the velocity, D is the hydraulic depth of a conduit
and g is the gravity acceleration (g = 9.81 m/s2).
Fr < 1 , the flow is in subcritical state
Fr = 1 , the flow is in critical state
Fr > 1 , the flow is in supercritical state
gD
V 

gD
V 

gD
V 

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1.5 Regimes of Flow
A combined effect of viscosity and gravity may produce any one of the
following four regimes of flow in an open channel:
a. subcritical - laminar , when Fr < 1 and Re < 500
b. supercritical - laminar , when Fr > 1 and Re < 500
c. supercritical - turbulent , when Fr > 1 and Re > 12500
d. subcritical - turbulent , when Fr < 1 and Re > 12500
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Froude Number
A flow is called critical if the flow velocity is equal to
the velocity of a gravity wave having small amplitude.
The flow is called subcritical flow, if the flow velocity is
less than the critical velocity
The flow is called supercritical flow if the flow velocity
is greater than the critical velocity.
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Flow Parameters and Geometric Elements
a. Depth of flow y is the vertical measure of water depth.
Normal depth d is measured normal to the channel bottom.
d = y cos
For most applications, d  y when   10%. cos 0.1 = 0.995.
Free surface
Datum


So = bottom slope
Sw = water surface slope
b. Flow or discharge Q is the volume of fluid passing a cross-section
perpendicular to the direction of flow per unit time.
Mean velocity V is the discharge divided by the cross-sectional area
A
Q
V 
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1.1 Geometric Elements
c. Wetted perimeter P is the length of channel perimeter that is wetted or
covered by flowing water.
A = cross sectional area covered
by flowing water
B = bottom width
T = top width
A
P
y
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1.1 Geometric Elements
d. Hydraulic radius R is the ratio of the flow area A to wetted perimeter P.
B
T
A
P
y
P
A
R 
e. Hydraulic depth D is the average depth of irregular cross section.
T
A
D 

width
top
area
flow
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Channel shape
Area
A
Top width
T
Wetted perimeter
P
By B B + 2y
Table. Open channel geometries
y
B
T
Rectangular
y
z
T
Triangular
1 zy2 2zy 2
1
2 z
y 
By + zy2 B + 2zy 2
1
2 z
y
B 

y
z
T
Trapezoidal
1
B
y
T
Circle

D
 

 sin
8
2

D
 in radian
2
D

 in radian






2
sin
D
 in angle
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Parameters of Open Channels
Wetted Perimeter (P) :The Length of contact between Liquid
and sides and base of Channel
Hydraulic Mean Depth or Hydraulic Radius (R): If cross
sectional area is A, then R = A/P
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Depth of flow section (d) : depth of flow normal to the direction
of flow.
Flow depth (y)
Top width (T) : the width of channel section at the free surface.
Hydraulic depth (D) : D = A/T
Base slope (So) : So = tan θ
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 Freeboard: Vertical distance between the highest
water level anticipated in the design and the top of
the retaining banks. It is a safety factor to prevent
the overtopping of structures.
 Side Slope (Z): The ratio of the horizontal to vertical
distance of the sides of the channel. Z = e/d = e’/D
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Dr. Indrajeet Sahu

Table 1: Maximum Canal Side Slopes (Z)
Sand, Soft Clay 3: 1 (Horizontal: Vertical)
Sandy Clay, Silt Loam,
Sandy Loam
2:1
Fine Clay, Clay Loam 1.5:1
Heavy Clay 1:1
Stiff Clay with Concrete
Lining
0.5 to 1:1
Lined Canals 1.5:1
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M.Hanif Chaudry, Open Channel Flow 2nd Edition, Springer, 2008
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Continuity Equation
Inflow – Outflow = Change in Storage
Inflow
1 2
A
A
3
Section AA
Change in Storage
Outflow
3a
3b
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General Flow Equation
Q = vA
Flow rate
(m3/s)
Avg. velocity
of flow at a
cross-section
(m/s)
Area of the
cross-section
(m2)
Equation 1
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Uniform flow in Open Channel
i
Sw
So
yo
Flow
Energy lines
Water Surface
For uniform flow (in prismatic channel), i = Sw = So
yo= normal depth for uniform flow only
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Dr. Indrajeet Sahu

Resistance Equation
1. Chezy Equation
 By Antoine Chezy (France), 1768
2. Manning Equation
 By Robert Manning (Irish), 1889
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Chezy Equation
Introduced by the French engineer Antoine Chezy in 1768
while designing a canal for the water-supply system of
Paris
 Because i = So, than
Ri
C
v 
o
RS
C
v 
o
RS
AC
Q 
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Dr. Indrajeet Sahu

Chezy Equation
 Where C = Chezy coefficient
= L1/2/T (Unit m1/2/s)
150
<
C
<
60
s
m
s
m
where 60 is for rough and 150 is for smooth
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Dr. Indrajeet Sahu

Manning Equation
 Most popular for open channels around the world
(English system)
1/2
o
2/3
h S
R
1
n
V 
1/2
o
2/3
h S
R
49
.
1
n
V 
2
/
1
3
/
2
1
o
h S
AR
n
Q 
Dimensions of n?
(SI units!)
Bottom slope
n = Manning roughness
coefficient
= T/L1/3 (Unit s/m1/3)
VA
Q 
very sensitive to n
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Lined Canals n
Cement plaster 0.011
Untreated gunite 0.016
Wood, planed 0.012
Wood, unplaned 0.013
Concrete, trowled 0.012
Concrete, wood forms, unfinished 0.015
Rubble in cement 0.020
Asphalt, smooth 0.013
Asphalt, rough 0.016
Natural Channels
Gravel beds, straight 0.025
Gravel beds plus large boulders 0.040
Earth, straight, with some grass 0.026
Earth, winding, no vegetation 0.030
Earth , winding with vegetation 0.050
n = f (surface roughness,
channel irregularity, stage...)
d = median size of bed material
6
/
1
038
.
0 d
n 
6
/
1
031
.
0 d
n  d in ft
d in m
Manning roughness coefficient, n
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During the design stages of an open channel, the channel cross-section,
roughness and bottom slope are given.
The objective is to determine the flow velocity, depth and flow rate, given
any one of them. The design of channels involves selecting the channel
shape and bed slope to convey a given flow rate with a given flow depth. For
a given discharge, slope and roughness, the designer aims to minimize the
cross-sectional area A in order to reduce construction costs
Most Economical Section of Channels
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A section of a channel is said to be most economical when the cost of
construction of the channel is minimum.
But the cost of construction of a channel depends on excavation and
the lining. To keep the cost down or minimum, the wetted perimeter,
for a given discharge, should be minimum.
This condition is utilized for determining the dimensions of economical
sections of different forms of channels.
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Most economical section is also called the best section or most
efficient section as the discharge, passing through a most economical
section of channel for a given cross sectional area A, slope of the bed S0
and a resistance coefficient, is maximum.
Hence the discharge Q will be maximum when the wetted
perimeter P is minimum.
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Dr. Indrajeet Sahu

The most ‘efficient’ cross-sectional shape is determined for uniform flow
conditions. Considering a given discharge Q, the velocity V is maximum for
the minimum cross-section A. According to the Manning equation the
hydraulic diameter is then maximum.
It can be shown that:
1. the wetted perimeter is also minimum,
2. the semi-circle section (semi-circle having its centre in the surface) is
the best hydraulic section
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Most Economical Rectangular Channel
Because the hydraulic radius is equal to the water cross
section area divided by the wetted parameter, Channel
section with the least wetted parameter is the best
hydraulic section
Rectangular section
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Dr. Indrajeet Sahu

D
B
A 
 B
D
2
P 

D
A
2D
P 

0
dD
dP

2
2
2
2
0
2
D
D
B
D
A
D
A
dD
dP












D
B
2 

2
B
D 
Most Economical Rectangular Channel
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Dr. Indrajeet Sahu

D
)
D
n
(B
A 

2
1
2 n
D
B
P 


D
n
D
A
B 

2
1
2 n
D
)
nD
D
A
(
P 



0
dD
dP







 0
1
2 2
2
n
n
D
A
dD
dP
n
D
A
n
1
2 2
2



D
D
n
B
n
D
D
nD)
(B
n
2
1
2 2
2 





2
D
n
2
B
n
1
D 2 


or
Most Economical Trapezoidal Channel
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Trapezoidal open channel as shown Q=10m3/s, velocity =1.5m/s, for most
economic section. find wetted parameter, and the bed slope n=0.014.
 
m
D
D
D
D
A
D
kD
B
A
m
V
Q
A
B
D
D
B
D
kD
B
k
D
78
.
1
667
.
6
)
2
3
6055
.
0
(
667
.
6
5
.
1
10
6055
.
0
2
2
3
2
2
3
1
2
2
1
2
2
2


















Example 5
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Dr. Indrajeet Sahu

m
P
k
D
D
P
k
D
B
P
49
.
7
2
3
1
78
.
1
2
)
78
.
1
(
6055
.
0
1
2
6055
.
0
1
2
2
2
2


















cont.
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The best side slope for Trapezoidal section
3
1

k



 60

0

dk
dP
Other criteria for economic Trapezoidal section
D
OF
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Dr. Indrajeet Sahu

Circular section


2
sin
8
4
2
2
d
d
A 

d
r
P 

2 

d
D 95
.
0
154 

 

Maximum Flow using Manning
d
D 81
.
0
75
.
128 

 

Maximum Velocity using Chezy
Most Economical Circular Channel
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Two separate condition are obtained
1.) Condition for max velocity
2.) Condition for max discharge
1.) Condition for max velocity
 For maximum velocity of flow the depth of circular channel should be equal to 0.81
times the Diameter of Channel.
(d=0.81D)
 The hydraulic mean depth is equal to 0.3D
M=0.3D
2.) Condition for max discharge
 For maximum discharge of flow the depth of circular channel should be equal to
0.95 times the Diameter of Channel.
(d=0.95D)
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Dr. Indrajeet Sahu

Example 4
Circular open channel as shown d=1.68m, bed slope = 1:5000, find the Max. flow rate & the
Max. velocity using Chezy equation, C=70.
 
s
VA
Q
V
m
P
A
R
d
P
d
d
A
S
R
C
V
h
h
/
m
496
.
1
17
.
2
69
.
0
m/s
69
.
0
5000
1
485
.
0
70
485
.
0
5
.
4
17
.
2
m
4.5
68
.
1
180
154
m
17
.
2
154
2
sin
8
68
.
1
180
154
4
68
.
1
2
sin
8
4
3
2
2
2
2
2






























 154

Max. flow rate
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Dr. Indrajeet Sahu

 
m/s
748
.
0
5000
1
57
.
0
70
57
.
0
3775
.
3
93
.
1
m
378
.
3
68
.
1
180
75
.
128
m
93
.
1
75
.
128
2
sin
8
68
.
1
180
75
.
128
4
68
.
1
2
sin
8
4
2
2
2
2
2




















V
m
P
A
R
d
P
d
d
A
S
R
C
V
h
h






 75
.
128

Max. Velocity
Example 4 cont.
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g
V
y
Z
Energy
Total
2
2



Energy Principles in Open Channel Flow
D
Water Surface
T.E.L
channel bed
Referring to the figure shown, the total energy of a flowing liquid per unit
weight is given by
Where:
Z = height of the bottom of channel above
datum,
y = depth of liquid,
V = mean velocity of flow.
If the channel bed is taken as the datum (as shown), then the total energy
per unit weight will be.
This energy is known as specific energy, Es. Specific energy of a flowing
liquid in a channel is defined as energy per unit weight of the liquid
measured from the channel bed as datum
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D
Water Surface
T.E.L
channel bed
The specific energy of a flowing liquid can be re-written in the form:
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It is defined as the curve which shows the variation of specific energy (Es ) with depth of
flow y. It can be obtained as follows:
Let us consider a rectangular channel in which a constant discharge is taking place.
Specific Energy Curve (rectangular channel)
But
2
2
2 A
g
Q
y
Es 

Or
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Dr. Indrajeet Sahu

g
V
2
2
H
G
E
The graph between specific energy (x-axis) and depth (yaxis)
may plotted.
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Dr. Indrajeet Sahu

Referring to the diagram above, the following features can be
observed:
 The depth of flow at point C is referred to as critical
depth, yc. It is defined as that depth of flow of liquid
at which the specific energy is minimum, Emin, i.e.; Emin
@ yc . The flow that corresponds to this point is called
critical flow (Fr = 1.0).
 For values of Es greater than Emin , there are two corresponding depths. One depth is greater
than the critical depth and the other is smaller then the critical depth, for example ; Es1 @
y1 and y2 These two depths for a given specific energy are called the alternate depths.
 If the flow depth y > yc , the flow is said to be sub-critical (Fr < 1.0). In this case Es
increases as y increases.
 If the flow depth y < yc , the flow is said to be super-critical (Fr > 1.0). In this case Es
increases as y decreases. HHM VCE
Dr. Indrajeet Sahu

Froude Number (Fr)
h
r
D
g
V
F 
T
A
Width
Surface
Water
Area)
(Wetted
Flow
of
Area


h
D
T
T
Flow
Fr
Sub-critical
1 > Fr
Critical
1 = Fr
Supercritical
1 < Fr
g
A
T
Q
Fr 3
2
2

HHM VCE
Dr. Indrajeet Sahu

Rectangular Channel
3
1
2









g
q
yC
h
r
D
g
V
F 
1
At critical Flow
D
g
V
D
g
V
F
h
r 

For rectangular section
3
1
2
2









g
B
Q
yc
q=Q/B
a) Critical depth, yc , is defined as that depth of flow of liquid at which the specific
energy is minimum, Emin,
b) Critical velocity, Vc , is the velocity of flow at critical depth.
HHM VCE
Dr. Indrajeet Sahu

Rectangular Channel
c) Critical, Sub-critical, and Super-critical Flows:
Critical flow is defined as the flow at which the specific energy
is minimum or the flow that corresponds to critical depth. Refer
to point C in above figure, Emin @ yc .
and therefore for critical flow Fr = 1.0
If the depth flow y > yc , the flow is said to be sub-critical. In this case Es
increases as y increases. For this type of flow, Fr < 1.0 .
If the depth flow y < yc , the flow is said to be super-critical. In this case
Es increases as y decreases. For this type of flow, Fr > 1.0 .
HHM VCE
Dr. Indrajeet Sahu

Rectangular Channel
d) Minimum Specific Energy in terms of critical depth:
At (Emin , yc ) ,
HHM VCE
Dr. Indrajeet Sahu

Other Sections
)
2
(
2
)
5
3
(
c
c
c
c
D
n
B
D
D
n
B
E


 Trapezoidal section




sin
)
2
sin
2
(
16
)
cos
1
(
2




d
d
Ec Circular section
c
c D
E
4
5
 Triangle section
c
c D
E
2
3

Rectangular section
at critical flow Fr =1 where: 1
3
2
2


g
A
T
Q
Fr
HHM VCE
Dr. Indrajeet Sahu

Example 1
1
3
2

g
A
T
Q
 
 
 
1
81
.
9
3
1
4
.
2
3
1
2
4
.
2
33
.
1
2
3
2
3
2

































c
c
c
c
c
c
D
D
D
g
D
nD
B
nD
B
Q
m
31
.
0

c
D
Determine the critical depth if the flow is 1.33m3/s. the
channel width is 2.4m
HHM VCE
Dr. Indrajeet Sahu

Example 2
4
4
81
.
9
25
3
2
3
1
2
3
1
2
2
3
1
2
2
B
B
B
g
B
Q
yc 


























0.006
2
D
0.016
1
25
2
P
A
R
S
R
n
1
V
3
2
c
3
2














B
D
B
B
D
B
D
B
D
c
c
c
c
Rectangular channel , Q=25m3/s, bed slope =0.006, determine the
channel width with critical flow using manning n=0.016
HHM VCE
Dr. Indrajeet Sahu

m
B
B
B
B
B
B
B
B
B
B
3
0.006
8
4
0.016
1
4
25
0.006
4
2
4
0.016
1
4
25
3
2
3
5
3
1
3
2
3
2
3
2
3
2









































HHM VCE
Dr. Indrajeet Sahu

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