Lecture 2 manning

Lecture

1. 9

Open Channel Flow Equations

The equations of open channel flow are based on uniform flow
conditions. Some of these equations have been derived using basic
conservation laws (e.g. conservation of energy) whereas others have been
derived using an empirical approach.
1- Continuity Equation.
One of the fundamental concepts which must be satisfied in all flow
problems is the continuity of flow. The continuity equation states that the
mass of fluid per unit time passing every section in a stream of fluid is
constant. The continuity equation may be expressed as follows:
Q = A1V1 = A2V2
Where
Q: is the discharge,
A: is the cross-sectional flow area, and
V: is the mean flow velocity.
2- Energy Equation.
The basic principle used most often in hydraulic analysis is conservation
of energy or the energy equation. For uniform flow conditions, the energy
equation states that the energy at one section of a channel is equal to the
energy at any downstream section plus the intervening energy losses. The
energy equation, expressed in terms of the Bernoulli equation, is:


Where:
z = Distance above some datum; y = Depth of flow;
v
2
/2g= velocity head; g = Acceleration of gravity; and
hL = head losses

2. 10

Uniform flow and the Development of velocity equations
Several equations have been empirically derived for computing the
average flow velocity within an open channel. Famous equations are the
Chezy and Manning Equations. Assuming steady uniform and turbulent
flow conditions, the mean flow velocity in an open channel can be computed
as:
When uniform flow occurs, gravitational forces exactly balance the
frictional resistance forces which apply as a shear force along the boundary
(channel bed and walls) as shown in fig below.

3. 11


Figure of forces on a channel length in uniform flow

F1- - F2 -P L??????=0
=
For small slope sin=tan= S
S=
= …………………..(1)
= (Shear stress at pipe boundary)……….(2)
=friction factor (Darcy coefficient)
Eq.(1)=Eq.(2)

4. 12

= , =g
V=
grouping the constants together as one equal to C
V=C (Chezy formula)
For steady uniform flow Se= S0=Sw
V=C
V = Mean velocity,
C = Chezy coefficient
S = Channel slope,
R = Hydraulic Radius,

Many studies have been made of the evaluation of C for different
natural and manmade channels. One of these studies was made by Manning
who suggested the following relation:

The equation of velocity becomes:
(SI units)

(British units)
And then:
Q =

5. 13

V = Mean velocity
n = Manning coefficient of roughness, depend on the type of material of the
channel
S = Channel slope, is governed by topography
R = Hydraulic Radius,

The Manning equation has the great benefits that it is simple, accurate
due to it long extensive practical use,

Below is a table of a few typical values of Manning's n

6. 14

Computations for steady uniform flow
We can use Manning's formula to calculate steady uniform flow. In
steady uniform flow the flow depth is known as normal depth(y). By using
the Manning equation, the section factor for uniform flow is and
equal to:


Additional relation is needed besides Mannings equation which is
freeboard (f). Freeboard is the vertical distance from the water surface to the
top of the channel. It should be sufficient to prevent the over topping of the
channel by waves or a fluctuating water surface. The following formula has
recommended
(English units)
f= freeboard, ft
c= 1.5 for Q> 3000 cfs
c=2.5 for
y= depth of flow

EX 1: A concrete lined trapezoidal channel with uniform flow has a normal
depth is (2m). The base width is 5m and the side slopes are equal to
(1V:2 H), Manning's (n) can be taken as (0.015), And the bed slope
(S0 = 0.001)

What are:
a) Discharge (Q)
b) Mean velocity (V)

7. 15

C) Depth of section
d) Section factor
e) Reynolds number (Re)

EX 2: Using the same channel as above, if the discharge is known to be
(30 m
3
/s) in uniform flow, what is the normal depth?
EX 3: Design a trapezoidal irrigation channel to carry a discharge of
(15 cumecs) with a permissible velocity of (0.75 m/sec). Assume bed slope
as (1/3600), and (Manning coefficient =0.03) and side slope (1V:1H).
Ex 4: water flows in a rectangular concrete open channel (n=0.013). The
average sheer stress along the wetted perimeter of the channel is (48.3 pa),
(S=0.0028, B=4.8 y). compute:
1- The normal depth of flow
2- Velocity using Chezy formula

Ex: the Fig. below shows a cross section of a canal designed to carry
(1590 cfs). The canal is lined with concrete (n=0.014). Find the grade of
the canal in ft/mile Z=1.5, Ans: s=0.000147 or 0.776 ft./mile










EX 5: A (24in) diameter cast iron pipe (n=0.012) placed on a slope (1/400)
carries water at a depth of (5.6in). What is the flow rate.

8. 16

Note: for circular channel flowing partly full, the following relations can be
used:






R=A/P




(EX 6) A 500 mm-diameter concrete pipe on a (1:500) slope is to
carry water at a velocity of (0.18 m/s). Find the depth of the flow.
(n=0.013)

9. 17

Weighted Manning’s Coefficient (Composite Roughness) in
Compound channels
In the previous discussion, we assumed that the flow surface at a channel
cross section has the same roughness (n) along the entire wetted perimeter.
However, this is not always true. For example, if the channel bottom and
sides are made from different materials, then the Manning n for the bottom
and sides may have different values. To simplify the computations, it
becomes necessary to determine a value of n, designated by (ne) that may be
used for the entire section. This value of ne is referred to as the equivalent n
for the entire cross section.

Let us consider a channel section that may be subdivided into N
subareas having wetted perimeter Pi and Manning constant, ni, (i = 1, 2, · · ·
,N). By assuming that the mean flow velocity in each of the subareas is equal
to the mean flow velocity in the entire section, the following equation may
be derived:
The Horton composite roughness equation shown below is applied for
open, regular, and irregular channels such as natural floodplains to compute
the weighted Manning’s coefficient (Equivalent Manning ne):

Where:
n: Roughness Coefficient
P: Wetted Perimeter
R: Hydraulic Radius