This document discusses optimal and efficient channel cross-sections for open channel flow. It defines an optimal channel section as having minimum construction costs, considering excavation and lining costs. An efficient channel section maximizes discharge for a given cross-sectional area and roughness. The most hydraulically efficient shape allows the greatest flow for a given area. Formulas for uniform flow like Chezy and Manning equations are presented to calculate velocity and discharge as a function of roughness, slope, hydraulic radius and other variables. Examples are given to demonstrate computing the most efficient dimensions for a rectangular channel.

1. 1. The geometry of the trapezoidal channel is displayed in the figure below
with all the necessary information. Based on the given information calculate
the depth.
3. A given trapezoidal channel has the following dimensions: B=10m, m=1.5,
y=3m, and is designed to carry 40 m3/s. Using the same cross-sectional area
and side slopes, how much would the most efficient cross-section increase
the present hydraulic radius and flow capacity? What would be the
corresponding depth and bed width?
Additional information
n=0.030, S= 1° and
Q=25m3/s
Exercises
2. Derive the most economical section for
rectangular / trapezoidal channel?

2. Lecture 2
2

3. Optimal (efficient) channel cross section
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.
A channel section is said to be efficient if it gives the maximum discharge
for the given shape, area and roughness.
The most hydraulically-efficient shape of channel is the one which can
pass the greatest quantity of flow for any given area or, equivalently, the
smallest area for a given quantity of flow.
 most economical section
As the cost of construction(excavation, channel lining is directly related
with the area of the channel & perimeter.

Dp/dy=0

4. Example
What are the best dimensions y and B for a rectangular brick
channel designed to carry 5 m3/sec of water in uniform flow
with S0 = 0.001, and n = 0.015?
Ans:

5. Velocity distribution in open channels
The measured velocity in an open channel will always vary across the
channel section because of friction along the boundary.
Neither is this velocity distribution usually ax symmetric (as it is in pipe
flow) due to the existence of the free surface. It might be expected to
find the maximum velocity at the free surface where the sheer force is
zero but this is not the case.
The maximum Velocity is usually found just below the surface.
Non scouring and Non silting velocity should have to be designed.
Design consideration

6. *Accurate V is found b/n 0.2 D to 0.8D
*For deep channel Vmax occurs at D/3 form the free surface.

8. 1.4. Uniform Flow in open channel
Uniformflow(steady uniformflow)
The main feature of uniform flow in an open channel is
The discharge, area of flow, depth, and velocity remain constant throughout the reach
The energy line, the water surface line, and the bed line all are parallel in other words, the slopes of the energy line (Sf), the
hydraulic gradient line (Sw), and the bed line (So) are equal
uniform flow is an exact balance between gravity and frictional forces
This is so when there is an exact balance between the gravity and resistance forces
8

9. Uniform flow is considered to be steady only, since unsteady uniform flow is
practically nonexistent. In natural streams, even steady uniform flow is rare, for
rivers and streams in natural states scarcely ever experience a strict uniform flow
condition.
The results obtained from this assumption are understood to be approximate and
general, but they offer a relatively simple and satisfactory solution to many
practical problems.
When flow occurs in an open channel, the water encounters resistance as it flows
downstream. This resistance is generally counteracted by the components of
gravity forces acting on the body of the water in the direction of motion. A uniform
flow will be developed if the resistance is balanced by the gravity forces.

10. • Several equations are available to calculate the rate of flow in an open channel,
Chezy and manning equations are commonly used. They are used for estimating
velocity as well as discharge of channel flow.
• The Continuity equation and manning formula are the two basic equations which
are used for various problems of uniform flow computation.
• the computation consist of discharge, velocity of flow, normal depth, roughness
Coeffiecnt, channel slope, Size of cross section( A ,R, K).

11. Where S0- bed slope of channel , Sw- Water surface slope , S- Slope of EGL
W – Weight of water , 0 – Shear stress, L- Length of channel
11
Uniform flow in open channel
 uniform steady and one dimensional flow and
 Forces acting on the liquid in the reach b/n two sections
A. Chezy Formula

12. 12
Force due to Gravity in the direction of flow = Wsin
Frictional Force occurs at the bottom = PL
Uniform flow is the exact balance between the gravity and friction force
Wsin = PL
A L sin = P.L But sin  = hf/L = S, solving for , V2
From bed shear stress theory o= Kv2
Therefore, kv2=RS,
constant (b/c  & k-are constant)
Then V2=
o

o

S
R
S
P
A
.
. 
 
RS
k
 2
C
k


This is the Chezy –formula
o

P = perimeter

13. B. Manning Formula
The other the most widely used formula uniform flow in open
channels is that published by the Irish engineer Robert Manning.
V=
A relation between the Chezy’s C and Manning’s n may be
obtained by comparing equations
13
2
1
3
2
0
1
S
R
n
n
R
C
6
1

The Manning equation has the great benefits that it is simple, accurate and
due to it long extensive practical use; there exists a wealth of publicly
available values of n for a very wide range of channels.

15. Example

16. 16