Tài liệu Open channel hydraulics for engineers. Chapter 5 spillways pptx

Classified in accordance with the shape of the notch, there are rectangular weirs, triangular or V-notch weirs, trapezoidal weirs, and parabolic weirs.. Now, we consider a rectangular we

Chapter SPILLWAYS

_

5.1 Introduction

5.2 General formula

5.3 Sharp-crested weir

5.4 The overflow spillway

5.5 Broad-crested weir

_

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

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 – each with its separate problems

Fig 5.1 General view of a spillway

A weir is a notch of regular form through which water flows The term is also applied to

the structure containing such a notch Thus a weir may be a depression in the side of a tank, a reservoir, or a channel, or it may be an overflow dam or other similar structure Classified in accordance with the shape of the notch, there are rectangular weirs, triangular

or V-notch weirs, trapezoidal weirs, and parabolic weirs Weirs are usually designed to control water levels in rice fields or wetlands They are commonly used as a means of flow measurement

Of primary importance for hydraulic structures considered in this chapter is the magnitude

of backwater level they cause upstream of the structure for the given discharge; that is, the head-discharge relationship for the structure Both gradually varied and rapidly varied flows are possible through these structures, but one-dimensional methods of analysis usually are sufficient and well-developed in this branch of hydraulics Essential to the

crest

face

toe

“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

Fig 5.2 Rectangular weir

Let, H = height of the water above the crest of the weir,

L = length of the weir, and

Cd = coefficient of discharge (see Chapter 1)

Let us consider a horizontal strip of water of thickness dh at a depth h from the water surface as shown in Fig 5.2

Assuming that the flow does not contract when passing over the weir (i.e neglecting streamline curvature), and that the pressure is atmospheric across the vertical section above the weir and the upstream velocities are small, the velocity of the water through the strip can theoretically be derived to be:

H

dh

h water surface

L

theoretical velocity = 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

The total discharge over the weir may be found by integrating the above equation within the limits 0 and H:



H 3

3

o

0

2



or the discharge per unit width, q, is:

3 d

2

3

where H is to be conceived of as the total upstream specific-energy head on the weir crest supposing that the upstream velocities are negligibly small; Cd is a discharge coefficient, which can be approximated by Rehbock’s experimental formula (1929):

Cd = 0.611 + 0.08 H

P

 

 

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, Cd becomes equal to 0.611 In this case, Eq (5-1) can be written as:

3

q1.80 H [1.80] = m½s-1

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 HD, the head over the spillway crest We obtain:

3 D

where HD may be termed the design head

Example 5.1: A rectangular weir, 4.5 m long, has a head of water 30 cm Determine the discharge over the weir, if the coefficient of discharge is 0.6

Solution:

Given: length of weir: L = 4.5 m

head of water: H = 30 cm = 0.3 m

coefficient of discharge: Cd = 0.6

Discharge over the weir Q?

Using the relation:

3 d

2

3

Example 5.2: The daily record of rainfall over a catchment area is 0.2 million m3 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 106 m3 per day

discharge into the reservoir: Q = 80% of rainfall

Q = 0.8 x 0.2 x 106 m3/day = 1.85 m3/s head of water: H = 1 m

Let, L = length of the weir

Take: coefficient of discharge: Cd = 0.6

Using the relation:

3 d

2

3



5.3.1 Experiments on sharp-crested rectangular weirs

All tests on weirs of this type were made with the nappe fully aerated When the

crest length L of a horizontal weir (see Fig 5.3) is shorter than the width of the channel b,

as well as in the case of V-notch weirs, aeration is automatic However, for horizontal weirs extending over the full width of the channel, i.e L/b = 1, air at atmospheric pressure must be provided by vents Otherwise the air beneath the nappe will be exhausted, causing

a reduction of pressure beneath the nappe, with a corresponding increase in the discharge for a given head

Fig 5.3 Weirs, definition sketch

L

b

crest

V

2

V

2g



H

2

V 2g



P

H

crest

section through

sharp-crested weir

horizontal- crested weir L/b < 1

V-notch weir round-crested

weir

The sharp-crested weir is of fundamental interest, because its associated theory forms a basis for the design of spillways Because the edge is sharp, opportunities for boundary-layer development are limited to the vertical face of the weir, where velocities are low; we may therefore expect the flow to be substantially free from viscous effects and the resultant energy dissipation

Fig 5.4 The sharp-crested weir

Fig 5.4 shows a longitudinal section of flow over such a weir An elementary analysis can

be made by assuming that the flow does not contract as it passes over the weir (i.e 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

2

H

V

2 g

2

 



 

 

 

 

 

 

 

the depth h being measured downwards form the total energy-head line, and not from the upstream water surface Vo is the approach velocity to the weir

The effect of the flow contraction may be expressed by a contraction coefficient Cc leading finally to the result:

(5-6)

where the discharge coefficient:

We should expect both Cc and the ratio (Vo2/2gH) to be dependent on the boundary geometry alone, in particular on the ratio H/P; it follows that Cd should be a function of H/P alone, which was indeed found by Rehbock (1929); see Eq (5-2)

H

h

Vo

45

A

B

2 o

V 2g



2

V 2g

p



total energy-head line

P

C

2

V 2g



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 Cd

 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:

3

e e

In these equations kL and kH are factors representing the effects of viscosity and surface tension, and the subscript e indicates effective values, that is, He is the effective energy-head By treating the variables in this manner the authors were able to obtain a single linear relationship between Ce and H/P for all values of H The values of kL and kH were obtained

by trying successive values of kL and kH until the values of Ce 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 Ce, with correcponding values of kH and kL are given hereafter

 The Kindsvater and Carter tests yielded

e

H

C 1.78 0.22

P

kH = 0.001 m; kL = - 0.001 m

 The Bazin tests yielded

e

H

P

kH = 0.004 m; kL = 0 m

 The Schoder and Turner tests yielded

e

H

P

kH = 0.001 m; kL = 0 m

 The US Bureau of Reclamation (USBR) tests yielded

e

H

C 1.78 0.24

P

kH = 0.001 m; kL = 0 m

5.3.2. Other types of sharp-crested weirs used for flow measurement

Of the many types of weirs developed in the last 100 years, only a few survived and find practical use today Excluding the suppressed (i.e without lateral contraction effects) rectangular weir, only the contracted rectangular weir and the triangular weir (see Fig 5.5) are employed with any frequency Other types, such as the parabolic, the circular and the compound-form weirs have been used from time to time for special applications

 The contracted rectangular weir

This type of weir was subject to considerable experimentation in the USA in the past century Most notable were the large scale tests by Francis, 1835, and Hamilton Smith,

1884 The Francis experiments were conducted with weirs between 2.44 m to 3.05 m in width, with a crest made of a cast iron plate, 6.3 mm in width and carefully planed and machined in the upstream corner Francis suggested that the total discharge was diminished with respect to the suppressed weir due to the contractions occurring at the sides An empirical correction was devised, that decreased the width of the weir by 5% of the head h (see Fig 5.5) for each lateral contraction The Francis discharge equation was:

d

with Cd = 0.623

The more modern formulae of Hegly, 1921, and the Swiss Society of Engineers (SIA),

1924, for rectangular contracted weirs reproduce basically the same idea These formulae are quoted below:

Hegly:

2



SIA:

2

b 3.615 3

(5-17)

In these relations, b is the width of the weir and B the width of the rectangular channel

Fig.5.5 The contracted rectangular thin-plate weir (top) and triangular thin-plate weir (bottom) and detail of crest and sides of notch (right)

1 cm to 2 cm

 /4 radians

upstream face of weir plate

B

H

P

b

4 to 5 H max

head

measuring

section

B

H

P

4 to 5 H max

head

measuring

 Triangular or V-Notch weir

Because the relative error in the measurement of the head becomes important for the smaller heads (say less than 0.05 m) in a rectangular weir, it has been found advantageous

to replace this type of weir by triangular weirs or notches, when the discharges to be measured are smaller than about 0.020 m3/s This type of weir was devised by James Thomson in England, 1858 Thomson experimented with 90 notches, but it was found later that other angles were needed for particular applications, in the range from 15 to 120

In this type of weir the crest should be finished as in the rectangular weir, with a sharp upstream corner and a crest width of the order of 1 mm to 3 mm As the weir nappe does not span the whole width of the channel, it is not necessary to provide aeration ducts beyond the weir plate

For historical reasons (derived from Weisbach’s theory) it has become customary to define

a flow equation for a triangular weir with a notch angle  as:

2

8



In many instances the product (8/15)Cd is replaced by a more convenient non-dimensional coefficient m This coefficient is expected to vary with the fluid properties and the weir geometry in much the same way as in the case of the rectangular weir The velocity-of-approach effect is, however, much smaller than for rectangular weirs, as the cross-sectional area of the notch is small in comparison with the channel cross-section This condition would ordinarily apply to weirs designed for accurate flow measurement On the other hand the surface tension effects are of the same order as in the case of the rectangular weirs

5.4.1 The spillway crest

Normally the crest is shaped so as to conform to the lower surface of the nappe from a sharp-crested weir, as shown in Fig 5.6 The pressure on the crest will then be atmospheric, provided that the resistance of the solid surface to flow does not induce a material change in the pressure distribution This could be happen only if the boundary layer over the crest were very thick; and it is known (see Henderson, 1966) that the boundary layer, which will grow effectively only from the neighbourhood of the point A, is

in fact a very small fraction of the head over the crest Therefore we may expect the pressure over the crest to be atmospheric, and in this fact lies the virtue of this crest shape; pressures above atmospheric will reduce the discharge, and pressures below atmospheric will increase the discharge, but at the risk of introducing instability and cavitation

Fig 5.6 The crest and the equivalent weir of an overflow spillway

We consider the case of the high spillway For the equivalent weir, H/P = 0 and the substitution of the Eq (5-2) into Eq (5-1) then leads to the result of Eq (5-3):

3

q1.80 H [1.80] = m½s-1

Experiments show that the rise from the weir crest to the high point of the nappe (the spillway crest) is 0.11 H, as in Fig 5.6 Using this fact we can express Eq (5-19) in terms

of HD, the head over the spillway crest We obtain Eq (5-4):

3 D

q2.14 H [2.14] = m½s-1

where HD may be termed the design head; as we have seen, operation at this head will make the pressure over the crest atmospheric However, the spillway will also have to operate at lower heads, and possibly higher heads as well

The former will evidently result in above-atmospheric pressures on the crest

As to the details of the crest shape, extensive experiments by the U.S Bureau of Reclamation (USBR) have resulted in the development by the U.S Army Corps of Engineers of curves which can be described by simple equations, yet approximately close

to the nappe profiles measured in the USBR experiments The profile for a vertical upstream face is shown in Fig 5.7; others were also developed for various angles of the upstream face to the vertical

H HD

0.11H A

C

P

Vo