1.9 The broad-crested weir | Lecture notes Laboratory Practices and Management

If the specific energy is a minimum dH,/dy

we may write

-29)

Equation 1-29 is valid only for steady flow with parallel streamlines in a channel of

small slope. If the velocity distribution coefficient, U, is assumed to be unity, the crite-

rion for critical flow becomes

Vc--

(g A,/B,)O.SO (1 -30)

2g 2Bc

-2

Provided that the tailwater level of the measuring structure is low enough to enable

the depth of flow at the channel contraction to reach critical depth, Equations 1-2,

1-23, and

1-30

allow the development of a discharge equation for each measuring

device, in which the upstream total energy head

(HI)

is the only independent variable.

Equation

1-30

states that at critical flow the average flow velocity

(g A,/B,)n.5n

It can be proved that this flow velocity equals the velocity with which the smallest

disturbance moves in an open channel, as measured relative to the flow. Because of

this feature, a disturbance or change in a downstream level cannot influence an up-

stream water level if critical flow occurs in between the two cross-sections considered.

The 'control section' of a measuring structure is located where critical flow occurs

and subcritical, tranquil, or streaming flow passes into supercritical, rapid, or shooting

flow.

Thus, if critical flow occurs at the control section of a measuring structure, the

upstream water level is independent of the tailwater level; the flow over the structure

is then called 'modular'.

1.9

The

broad-crested

weir

A broad-crested weir is an overflow structure with

horizontal crest above which

the deviation from a hydrostatic pressure distribution because of centripetal acce-

leration may be neglected. In other words, the streamlines are practically straight

and parallel.

obtain this situation the length of the weir crest in the direction

of flow (L) should be related to the total energy head over the weir crest as

H,/L

0.50.

H,/L

0.07

because otherwise the energy losses above the

weir crest capbemay occur on the-creqt;

iI,/L_>

0.50,

tha; onlylight curvature-lGs occurTabove the crest and a hydrostatic

pressure distribution may be assumed.

If the measuring structure is

designed that there are no significant energy losses

in the zone of acceleration upstream of the control section, according to Bernoulli's

equation (1-23):

-=c

~V,'/2g

aV2/2g

(1-31)

where

equals the total upstream energy head over the weir crest as shown in Figure

1.9 The broad-crested weir, Lecture notes of Laboratory Practices and Management

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I

If the specific energy is a minimum dH,/dy = O , we may write

( 1 -29)

AC

Provided that the tailwater level of the measuring structure is low enough to enable

It can be proved that this flow velocity equals the velocity with which the smallest

A broad-crested weir is an overflow structure with a horizontal crest above which

the deviation from a hydrostatic pressure distribution because of centripetal acce-

and parallel. To obtain this situation the length of the weir crest in the direction

of flow (L) should be related to the total energy head over the weir crest as

weir crest c a p b e m a y occur on the-creqt; i I , / L _ > 0.50,

so tha; onlylight curvature-lGs occurTabove the crest and a hydrostatic

(1-31)

1.10. Substituting Q = VA and putting c1 = 1 .O gives

Q = A (2g(Hl -Y)}~.’~ (1-32)

Provided that critical flow occurs at the control section (y = y,), a head-discharge

equation for various throat geometries can now be derived from

For a rectangular control section in which the flow is critical, we may write A, =

Broad-crestedweir with rectangular control section

I

.I

Substitution of this relation and A, = b, into Equation 1-33 gives, after simplification

This formula is based on idealized assumptions such as: absence of centripetal forces

Figure 1.13 Dimensions of a parabolic control section

As explained in Section 1.9.1, correction coefficients have to be introduced to obtain

a practical head-discharge equation. Thus

Q = Cd C V J T fg hI2.O (1-43)

1.9.3 Broad-crested weir with triangular control section

For a triangular control section with A, = y: tan- and B, = 2yc tan- (see Figure

1.14), we may write Equation 1-30 as:

Hence

Substituting those relations and A, = y: tan- into Equation 1-33 gives

I

It also shows that, if both b, and z, are known the ratio y,/Hl is a function of H I.

Values of yc/HIas a function of z, and the ratio Hl/bcare shown in Table I. I.

coefficient gives as a head-discharge equation

Since for each combination of b,, zo and H,/b,, the ratio y,/H, is given in Table 1.1,

the discharge Q can be computed because the discharge coefficient has a predictable

value. In this way a Q-HI curve can be obtained. If the approach velocity head vI2/2g

is negligible, this curve may be used to measure the discharge. If the approach velocity

has a significant value, v12/2gshould be estimated and h, = H I -v12/2gmay be obtained

in one or two steps.

In the literature the trapezoïdal control section is sometimes described as the sum

of a rectangular and a triangular control section. Hence, along similar lines as will

be shown in Section I. 13 for sharp-crested weirs, a head-discharge equation is obtained

by superposing the head-discharge equations valid for a rectangular and a triangular

control section. For broad-crested weirs, however, this procedure results in a strongly

variable C,-value, since for a given H, the critical depth in the two superposed equa-

tions equals 2/3H, for a rectangular and 4/5Hcfor a triangular control section. This

difference of simultaneous y,-values is one of the reasons why superposition of various

head-discharge equations is not allowed. A second reason is the significant difference

in mean flow velocities through the rectangular and triangular portions of the control

section.

Figure I. 16 Dimensions of a trapezoidal control section

1.9.6 Broad-crested weir with circular control section

For a broad-crested weir with a circular control section we may write (see Figure

A, = -d,Z(O-sin€))

B, = d,sin l/,O and ( 1 -57)

d

y, = 2

(1 -cos ' / , O ) = d, sin2 1/

Substitution of values for A, and B, into Equation 1-30 gives

vc A, d, O-sin

2g 2B, 16 sin '/,O

weir crest as

For each value of yJd, = sin2 1/40 a matching value of the ratios A,/d,Z and HJd,

can now be calculated with the above equations. These values, and the additional

values on the dimensionless ratios v,2/2gdCand yJH, are presented in Table 1.2.

For a circular control section we may use the general head-discharge relation 'given

earlier (Equation 1-33)

where the discharge coefficient Cd has been introduced for similar reasons to those

explained in Section 1.9.1. The latter equation may also be written in terms of dimen-

sionless ratios as

Figure 1.17 Dimensions of a circular control section

Table 1.2 Ratios for determining the discharge Q of a broad-crested weir and long-throated flume with

circular control section (Bos 1985)