MODULE 5.7: WEIRS
a. Description
An overflow hydraulic structure extending across an open channel normal to the direction of
flow for the purpose of flow measurement and control. In general, weirs are classified in
accordance to their upstream characteristics and shapes:
1. Sharp-crested Type: This has a thin upstream edge so fashioned that the passing mass of
liquid simply touches a line. Shapes are normally rectangular (suppressed and contracted),
triangular and trapezoidal. The figure below shows an example of a sharp-crested weir.
In particular, a rectangular sharp-crested weir may either be “suppressed”, that is, where
its crest length L extends from wall to wall of the channel or where L is equal to the width
of the channel W, or L = W, or “contracted”, when L > W. The triangular and trapezoidal
sharp-crested weirs are both categorized to be of the latter type.
NOTE: In the figure above, the properties of flow and that of the weir are shown: The
nappe is the downward falling stream of liquid; the head H (measured above the crest to
the liquid surface at a distance of about 6H upstream of the weir); the height of the weir,
denoted as P. The effective upstream depth is d, which is the sum, (P + H). V A is the
channel’s velocity of approach.
2. Broad-crested Type: The figure below shows two sub-types of a broad-crested weir. One
has a rounded upstream corner (a), where the mass of liquid passes smoothly and slides
over the surface of the crest. The (b), the other sub-type, has a square upstream corner in
which the stream of liquid first “jumps” at the corner and slides over the downstream
surface of the crest.
As in orifice flow, there are two types of flow over the weir, that is, free flow if the liquid
surface downstream is sufficiently below the crest elevation, and submerged type if the
liquid surface downstream is above the crest, but below the upstream liquid surface.
b. Sharp-crested Rectangular Weirs
(i) Suppressed Type (L = W)
In the determination of the rate of flow over a suppressed sharp-crested rectangular weir
(in the figure below), take a rectangular element area dA = Ldh across the flow area, as
shown. Utilizing the principle of the orifice, the theoretical elemental flow through the
elemental rectangular opening is,
𝑉1 2
𝑑𝑄𝑡 = 𝐿𝑑ℎ(𝑉2 ) = 𝐿𝑑ℎ √2𝑔[ℎ +
]
2𝑔
where 𝑉1 is the same as the mean velocity of approach VA, h is the variable head, 0 < h <
H
𝑉2 is the velocity at point 2 in the mass of liquid and obtained by the energy equation
between 1 and 2 (head losses neglected)
Integrate above differential to obtain the total theoretical rate of flow, from h = 0 to h = H,
𝑉𝐴 2 1/2
𝑄𝑡 = √2𝑔 𝐿 ∫ [ℎ +
] 𝑑ℎ
2𝑔
0
𝐻
Assuming VA to be a constant velocity, the theoretical discharge is finally determined as,
(𝑉𝐴 )2 3
(𝑉𝐴 )2 3
2
𝑄𝑡 = ( ) √2𝑔 𝐿 {[𝐻 +
]2 − [
]2 }
3
2𝑔
2𝑔
Use a correction factor C’ to take care of the head losses initially neglected, the actual flow
rate is obtained as,
(𝑉𝐴 )2 3
(𝑉𝐴 )2 3
2
𝑄𝑎 = 𝐶′ ( ) √2𝑔 𝐿 {[𝐻 +
]2 − [
]2 }
3
2𝑔
2𝑔
𝑄𝑎 = 𝐶 𝐿 {[𝐻 +
(𝑉𝐴 )2
2𝑔
3
]2 − [
(𝑉𝐴 )2
2𝑔
3
]2 }
2
where: C = 𝐶′ (3) √2𝑔 (unit is in m1/2/s)
L is in meter (m)
VA is in meter per second (m/s)
NOTE: (1) if the ratio H/P is sufficiently small and the velocity head of approach is
negligible, or
(𝑉𝐴 )2
2𝑔
≈ 0, the equation above reduces to,
𝑄𝑎 = 𝐶 𝐿 𝐻
3/2
(2) from his more than 80 experiments, conducted between 1848 and 1852, using
weir lengths, 1.00 < L < 5.20 m, and heads, 18 < H < 50 cm, Francis obtained results
which suggested that the correction factor, C’ ≈ 0.622 (C = 1.84). Substituted to the
equation, the following relations, known as the Francis Formula for suppressed weir
rectangular weir is obtained:
Francis Formula
𝑄𝑎 = 1.84 𝐿 𝐻3/2
(3) Specifications of a Standard Suppressed Rectangular Weir
•
Weir plate is vertical with smooth upstream face;
•
Crest is horizontal and normal to the direction of flow/must be sharp to
cause the passing liquid to spring free from the edge;
•
The pressure on the upper and lower nappes is atmospheric;
•
The channel approach is uniform in section and its surface free from waves
(4) Other Standard Suppressed Rectangular Weir Formulas (English Units)
Bazin Formula
𝑄𝑎 = [3.248 + (
0.0789
𝐻 2
3/2
)] [1 + 0.55( ) ]𝐿𝐻
𝐻
𝑑
King Formula
𝑄𝑎 = 3.34 [1 + 0.56(𝐻/𝑑)2 ]𝐿𝐻1.47
(ii) Contracted Type (L < W)
The figure below shows a rectangular weir with an opening cut through its plate having
sharp upstream vertical edges. The effect of such an opening is to reduce the amount of
flow from that which would occur over a standard suppressed weir with the same length L
and under the same head H. This is known as a contracted rectangular weir. The effective
length of flow L’, as shown, is smaller than the actual length L by an amount EC, called as
the end-contraction correction, so that,
𝐿′ = 𝐿 − 𝑛𝐸𝑐
where n is the number of end contractions.
Contracted Rectangular Weir
NOTE:
(1) From the results of his investigation on contracted weir, Francis suggested the value
of the end-contraction correction as,
𝐸𝑐 = 0.10𝐻
which when applied to end contraction correction gives,
′
𝐿′ = 𝐿 − 0.10𝑛𝐻
𝐿 = 𝐿 − 0.10𝑛𝐻
Therefore, the Francis Formula for contracted rectangular weir becomes,
Francis Formula
𝑄𝑎 = 1.84 (𝐿 − 0.10𝑛𝐻)𝐻
3/2
(2) From his investigation on end contraction effect, V.M. Cone in 1916 developed a more
precise flow formula (English Units) for a contracted rectangular weir,
V.M. Cone Formula
𝑄𝑎 = 3.247𝐿𝐻
1.48
0.566𝐿1.8
−[
1.8
(1 + 2𝐿
]𝐻1.9
)
(3) To ensure complete contraction, the distance S in the figure above must have a
minimum normal distance of 2H both from the side and the bottom of the weir plate.
c. Triangular or V-notch Weirs
The triangular or V-notch weir is illustrated in the figure below with vertex angle 𝜃, liquid
surface length L and head over the vertex, H. The elemental opening shown has an area,
dA = xdh, under the variable head h (0 < h < H). Following the development of theoretical
flow for rectangular weirs, head losses and velocity head of approach neglected, the
discharge is obtained as,
𝐻
𝑄𝑡 = ∫ 𝑥𝑑ℎ√2𝑔ℎ
0
𝜃
Using basic mathematics principles, x = L(H – h)/H and L = 2h tan(𝜃/2), which when
substituted to the above equation and integrated from h = 0 to h = H results to,
𝑄𝑡 =
8
𝜃
√2𝑔 tan ( )𝐻 5/2
15
2
A correction factor C’ is considered to obtain the actual flow, or
𝑄𝑎 = 𝐶 𝐻5/2
8
𝜃
where: 𝐶 = 𝐶′(15)√2𝑔tan (2)
NOTE:
(1) For a vertex angle 𝜃 = 90°, and a value of C’ = 0.60, a relation known as the approximate
formula is obtained,
Approximate Formula
𝑄𝑎 = 1.42 𝐻5/2
(2) Formulas developed by different experimenters on the V-notch or triangular weirs, using
the following parameters:
(i) James Thomson (1861): C’ = 0.593; 𝜃 = 90°; Head H varying from 5 to 18 cm
𝑄𝑎 = 1.40 𝐻5/2
(ii)
James Barr (1908): Extension of J. Thomson works, using 8 < H < 25cm.
𝑄𝑎 = 2.48 𝐻
2.48
(𝑖𝑛 𝐸𝑛𝑔𝑙𝑖𝑠ℎ 𝑢𝑛𝑖𝑡𝑠)
(iii) V.M. Cone (1916): Expressed in the English units
𝑖𝑓 𝜃 = 30°; 𝑄𝑎 = 0.685 𝐻2.48
𝑖𝑓 𝜃 = 60°; 𝑄𝑎 = 1.45 𝐻2.47
(iv) A.T. Lenz undertook a comprehensive study of the flow over triangular weirs using
liquids other than water. In the English units, he obtained the formula
A.T. Lenz Formula
𝑄𝑎 = [2.395 +
𝑁
𝜃 5/2
𝑛 ] tan( ) 𝐻
2
𝐻
where N and n are given in the table below, with ratio
𝑁
𝐻𝑛
having a minimum value
of 0.090.
𝜽
𝟑𝟎°
𝟒𝟓°
𝟔𝟎°
𝟗𝟎°
N
0.131
0.102
0.087
0.068
n
0.576
0.579
0.582
0.588
d. Trapezoidal Weirs
The isosceles trapezoidal weir in the figure below has an opening which can be divided
into two parts: a suppressed rectangular weir and an isosceles triangular part (combination
of the two right triangular parts at the contracted ends). Applying the principles utilized in
the previous lesson, and assuming approach velocity head and head losses negligible,
theoretical flow is,
𝑄𝑡 =
2
8
𝜃
√2𝑔𝐿𝐻 3/2 +
√2𝑔 tan ( )𝐻 5/2
3
15
2
Introducing the correction factors C’ and C”, respectively, for the rectangular and
triangular parts, the reduces to the actual discharge,
𝑄𝑎 = 𝐶𝑅 𝐿𝐻 3/2 + 𝐶𝑇 𝐻 5/2
2
𝜃
8
where: 𝐶𝑅 = 𝐶 ′ 3 √2𝑔; 𝐶𝑇 = 𝐶" 15 √2𝑔 tan (2)
NOTE: A trapezoidal weir which automatically eliminates the end contraction effect on a
contracted rectangular weir, as investigated by Francis, was proposed by C. Cipoletti. He
𝜃
𝜃
proved than for tan (2) = 0.25 or for (2) = 14°, the amount of liquid that is supposed to
flow over the weir through the triangular end of the trapezoidal weir, is compensated by
the effect of end contraction. Under these special conditions, Cipoletti proposed the
following relation,
Cipoletti Formula
𝑄𝑎 = 1.86𝐿𝐻 3/2 (𝑆𝐼 𝑈𝑛𝑖𝑡𝑠)
𝑄𝑎 = 3.37𝐿𝐻 3/2 (𝐸𝑛𝑔𝑙𝑖𝑠ℎ 𝑈𝑛𝑖𝑡𝑠)
closely expressing Francis formula on suppressed rectangular weir, with crest length L and
head H.
e. Broad-Crested Weirs (Rectangular)
Two common types of broad-crested weirs are shown:
(a) Rounded upstream corner: The correction factor C’ in the rectangular weir formula for
this type ranges from 0.50 to 0.57.
(b) Ogee-shaped spillway: For this type, the correction factor C’ ranges from 0.60 to 0.75
with a fixed reservoir surface, the flow over these two structures will be approximately
equal, but the head on type (a) weir will be slightly greater than that of type (b). For the
same flow the smaller coefficient is compensated by the larger head and vice versa.
f. Submerged Sharp-Crested Weirs
Submerged weirs are categorized in accordance to the type of nappe developed as the liquid
mass passes over the structure:
(i) The plunging nappe type: the nappe is similar to the free-flow type;
(ii) The surface nappe type: the nappe remains on or near the surface, leaving the liquid
below undisturbed. Below is an example of a submerged sharp-crested weir.
The net discharge over a submerged sharp-crested weir was investigated by
Villemonte, among many other experimenters, and from the results of his tests (1947),
he came up with the empirical formula,
Villemonte Formula
𝑄𝑎 = 𝑄 ′ (1 − 𝑆 𝑛 )0.385
Where Q’ is the flow which would result when the head H1, and the weir is not submerged.
The exponent n is dependent on the type of weir used, that is, 1.5 for rectangular weirs and
2.5 for triangular weirs. The variable S = H2/H1, and is called as the submergence ratio.
NOTE: The equation above is accurate within 5% for a range 0 < S < 0.90. In particular,
for a submerged V-notch weir where 𝜃 = 90°, the accuracy is within 0.50% for S < 0.70.
Compared to a free flow type weir, the head loss in submerged weirs is much less.
g. Unsteady Flow in Weirs
The principles applied on the unsteady flow conditions in orifices are also applicable to
weirs under unsteady flow conditions. The specific unsteady flow problem that was
considered is the determination of the time required for the liquid surface to drop from one
elevation to another elevation. Following the same line of reasoning, for the liquid surface
in a weir to drop in the range H1 < H < H2, the time is,
𝐻1
𝑡= ∫
𝐻2
𝐴 𝑑ℎ
𝑄𝑊
Where A is in general function of the variable head H and A represents the horizontal area
of the reservoir controlled by the weir at elevation H, measured above the crest of the weir.
The flow QW is the flow over the weir, that is,
(i)
𝑄𝑎 = 1.84 𝐿 𝐻 3/2, for a suppressed rectangular weir (Francis Formula)
(ii)
𝑄𝑎 = 1.42 𝐻 5/2 , for a 90° V-notch weir (Approximate Formula)
(iii) 𝑄𝑎 = 1.86𝐿𝐻 3/2, for a trapezoidal weir (Cipoletti Weir)
And so on, dependent on the type of weir considered.
SAMPLE PROBLEMS:
1. A suppressed rectangular weir with L =3.66m (12ft); H = 47cm (1.54 ft); and P = 1.07m
(3.50ft) is used. Find the discharge using the following formulas:
a. Bazin Formula
b. King’s Formula
c. Francis Formula
2. There is a suppressed rectangular weir across a 6.10m wide channel. The depth of flow is
90cm with a mean velocity of 75 cm/s. What will be the height of the weir that will increase
the depth of flow to 1.50m, from the original depth of 90cm. Assume a correction factor of
0.622.
3. A contracted rectangular weir has the following properties: Length, L = 1.83m (6ft); Head,
H = 47cm (1.55ft). Find the flow rate by the Francis Formula (using the end-contraction
correction). Assume two-end contraction and check by the Cone Formula.
4. A V-notch weir with vertex angle of 90° and flow rate, Q = 32 liters/sec. Find the head, H,
over the weir using the approximate formula. Check by the Barr and AT Lenz relations.
5. A Cipoletti weir is 1.00m high and 3.60m long. Find the head, H, for the discharge to be
1.80 m3/s.
6. A Cipoletti weir L = 9.20m, P = 0.75m is built across a rectangular channel 10.50m wide,
and flows under a head, H = 37.50cm. Some distance upstream is a rectangular sharp
crested weir with P = 1.05m, the middle of which is along the centerline of the channel. If
the head on this second weir is 50cm, what is the length of its crest?
7. A V-notch weir is located at one end of a large tank having a horizontal square section 10m
x 10m. If the initial head on the weir is 1.20m and it takes 375 seconds to discharge 100m3
of water, what could have been the vertex angle 𝜃 of the weir. Use a correction factor C’ =
0.60.