OPEN CHANNEL HYDRAULICS FOR ENGINEERS
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Chapter 5: SPILLWAYS

90
Chapter


SPILLWAYS
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5.1. Introduction
5.2. General formula
5.3. Sharp-crested weir
5.4. The overflow spillway
5.5. Broad-crested weir
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Summary
Spillways are familiar hydraulic structures built across a stream to control the water level.
This chapter emphasizes the classification of weirs and spillways as well as the application
of hydraulic formulas for designing their shape and dimensions.

Key words
Spillway; weir; crest; design head

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5.1. INTRODUCTION
Spillways are used at both large and small dams for letting flood flows pass,
thereby preventing overtopping and failure of the dam. A spillway as sketched in Fig. 5.1
is the most common type. Three zones can be distinguished: the crest, the face and the toe

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Chapter 5: SPILLWAYS

91
“hydraulic approach” is the specification of empirical discharge coefficients that have been
well established by laboratory experiments and verified in the field. The determination of
controls in hydraulic analysis also is important, and critical depth often is the control of
interest. The energy equation and the specific-energy head diagram are useful tools in the
hydraulic analyses of this chapter.

5.2. GENERAL FORMULA

The equation for discharge over a weir cannot be derived exactly, because not only
the flow pattern of one weir differs from that of another, but also the flow pattern for a
given weir varies with the discharge. Furthermore, the number of variables involved is too
large to warrant a rigorous analytical approach. Approximate derivations are presented in
most texts. These derivations show effects of gravitational forces in an approximate
manner, but do not include the effects of viscosity, surface tension, the ratios of the
dimensions of the weir to the dimensions of the approach channel, the nature of the weir
crest, and the velocity distribution in the approach channel. A simplified derivation will be
made here to show the general character of the relationship between the discharge and the
most important variables and to demonstrate the nature of the effect of some of the
variables. The derivation will be made for sharp-crested weirs, but as will be shown later, a
similar derivation would apply to weirs that are not sharp-crested. Now, we consider a
rectangular weir, over which the water is flowing as shown in Fig. 5.2.
2gh
(ii)
 The discharge per unit width, q, flowing over a weir is generally expressed as:

d
dq C area of strip theoretical velocity  d
dq C .L.dh. 2gh
(iii)
The total discharge over the weir may be found by integrating the above equation within
the limits 0 and H:

H
3
H
2
3
2
d d d
o
0
h 2
Q C .L.dh. 2gh C .L. 2g. C .L. 2g.H
3
3
2
 
 


where P is measured from top of the crest of the weir to the bottom of the reservoir; P is
called the weir height. Assuming P very large, C
d
becomes equal to 0.611. In this case, Eq.
(5-1) can be written as:

3
2
q 1.80 H
[1.80] = m
½
s
-1
(5-3)

Experiments show that the rise from the sharp weir crest to the highest point of the nappe
(i.e. the “spillway crest”) is 0.11H (see Figs. 5.4 and 5.6). Using this fact we can express
Eq. (5-3) in terms of H
D
, the head over the spillway crest. We obtain:

3
2
D
q 2.14 H
[2.14] = m
½
s
-1

= 1.31 m
3
/s Ans.

Example 5.2: The daily record of rainfall over a catchment area is 0.2 million m
3
. It has
been found that 80% of the rainfall reaches the storage reservoir and then passes over a
rectangular weir. What should be the length of the weir, if the water is not to rise more than
1 m above the crest? Assume a suitable value of the coefficient of discharge for the weir.

Solution:
Given: rainfall = 0.2 x 10
6
m
3
per day
discharge into the reservoir: Q = 80% of rainfall
Q = 0.8 x 0.2 x 10
6
m
3
/day = 1.85 m
3
/s
head of water: H = 1 m
Let, L = length of the weir
Take: coefficient of discharge: C
d
= 0.6
Fig. 5.3. Weirs, definition sketch
L
b
crest
V
2
V
2g


H
P
crest
2


2
V
2g

P
H
crest
section through
sharp-crested weir
horizontal-
crested weir
L/b < 1

neglecting streamline curvature), and that the pressure is atmospheric across the whole
section AB. Under these assumptions the velocity at any point such as C is equal to
2gh

(Henderson, 1966), and the discharge q per unit width accordingly equal to:

2
o
2
o
3 3
V
2 2
2 2
H
2g
o o
V
2g
V V2
2ghdh 2g H
3 2g 2g
 

 
 
 
 
 
 

q C 2g H C 2g.H
3 2g 2g 3
 
   
 
   
   
 
   
 
(5-6)
where the discharge coefficient:
3 3
2 2
2 2
o o
d c
V V
C C 1
2gH 2gH
 
   
 
  
   
 
   
 
(5-7)
We should expect both C

p


total energy-head line
P
C
2
V
2g


OPEN CHANNEL HYDRAULICS FOR ENGINEERS
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Chapter 5: SPILLWAYS

95
In early experiments on weirs only small quantities of water were available. In most cases
results are given in the form of Eq. (5-1), with a discharge coefficient C
d
.

 Tests on weirs of this type were conducted by Kindsvater and Carter (1959). Their
tests cover a range of values of H/P from approximately 0.1 to 2.5, a range of heads from 3
cm to 22 cm, and weir heights from 9 cm to 44 cm. They also varied the weir length and
the channel width from 3 cm to 82 cm. In presenting their data they adopted the method
used by Rehbock of including the effect of H in the main body of the equation. Kindsvater
and Carter also introduced a method that includes the effect of the weir length L in the
main body of the equation. Their method is shown in the following three equations:

and k
H
until the values of C
e
were obtained that were the
most independent of H and L. They did this not only for their own data, but for several
other groups of experiments as well.

Their equations for C
e
, with correcponding values of k
H
and k
L
are given hereafter.
 The Kindsvater and Carter tests yielded
e
H
C 1.78 0.22
P
 
(5-11)
k
H
= 0.001 m; k
L
= - 0.001 m

 The Bazin tests yielded
e