Weirs for Flow Measurement

Lec r

Weirs for Flow Measurement

tu e 8

I. C o

The trapezoidal weir that is most often used is the so-called Cipoletti

• acted weir in which

the notch ends (sides) are not

vertical, as they are for a rectangular

tical corrections for end contraction are n is simpler

in a rectangular weir), splayed out at angle of 14° with the vertical, or nearly 1 horizontal to 4 vertical (tan 14°≈ 0.2493, not 0.25 exactly)

r than 1:4 in order to eliminate the effects of end contraction

• The sloping sides provides the advantage of having a stable discharge coefficient and true relationship of:

(1) 9) is:

ip letti Weirs

•

weir, which was reported in ASCE Transactions in 1894

This is a fully contr weir

• The effects of end contraction are compensated for by this trapezoidal notch shape, meaning that mathema unnecessary, and the equatio

• The side slopes of the notch are designed to correct for end contraction (as manifested

• Some researchers have claimed than the side slopes should be greate

3 / 2

Q CLh

=

The discharge equation by Addison (194

3 / 2 3 / 2

2

⎛

⎞

cip

Q

0.63

2g Lh

C

Lh

3

=

⎜

⎜ ⎟

⎛ ⎞

⎟

=

⎝ ⎠

(2) n above); and h is the upstream head, measured from the bottom (horizontal part)

cip = 3.37

⎝

⎠

where L is the weir length (equal to the width of the bottom of the crest, as show

of the weir crest

L & h are feet for Q

• The units for in cfs, with C

• The units for L & h are m for Q in m3/s, with Ccip = 1.86

• Eq. 2 is of the same form as a rectangular sharp-crested weir Eq. 2 (right-most side) is simpler than that for unsuppressed

• rectangular and

triangular sharp-crested weirs because the coefficient is a simple constant (i.e. no calibration curves are needed)

X. V-Notch Weirs

• Triangular, or V-notch, weirs are among the most accurate open channel constrictions for measuring discharge

• For relatively small flows, the notch of a rectangular weir must be very narrow so that H is not too small (otherwise the nappe clings to the downstream side of the plate)

ut high upstream heads

weir (rectangular or trapezoidal), so for the same maximum capacity, it can measure much smaller discharges, compared to a rectangular weir

is:

• Recall that the minimum hu value for a rectangular weir is about 2 inches (50 mm)

• But with a narrow rectangular notch, the weir cannot measure large flows witho correspondingly

• The discharge of a V-notch weir increases more rapidly with head than in the case of a horizontal crested

• A simplified V-notch equation

5/2

Q Ch

=

(3)

• Differentiating Eq. 3 with respect to h, 3 / 2

dQ

5

Ch

dh

=

2

(4)

• Dividing Eq. 4 by Eq. 3 and rearranging,

dQ

5 dh

Q

=

2 h

(5)

•

• discharge, even for relatively small flows

with a small head: h is not too small for small Q values, but you still must be able to measure the head, h, accurately

• A rectangular weir can accurately measure small flow rates only if the length, L, is sufficiently small, because there is a minimum depth value relative to the crest; but small values of L also restrict the maximum measurable flow rate

• The general equation for triangular weirs is:

It is seen that the variation of discharge is around 2.5 times the change in head for a V-notch weir

Thus, it can accurately measure the

u h d u x 0

Q C 2 2g tan

(h

h ) h dh

2

θ

⎛ ⎞

=

⎜ ⎟

−

⎝ ⎠

∫

x (6) because,

dA 2x dh

=

(7) u x

tan( / 2)

h

−

h

=

θ

x

(8) d

dQ C

=

2gh dA

(9)

• Integrating Eq. 6: 2.5 u

tan

h

2

⎜ ⎟

⎝ ⎠

(10) d

8

Q C

2g

15

θ

⎛ ⎞

=

• For a given angle, θ, and assuming a constant value of Cd, Eq. 10 can be reduced to Eq. 3 by clumping constant terms into a single coefficient

• A modified form of the above equation was proposed by Shen (1981): 5 / 2 e e

Q

2g C tan

h

15

2

=

⎜ ⎟

⎝ ⎠

(11)

8

h

θ

⎛ ⎞

where, e u

h

=

h

+

K

(12) • Q is in cfs for hu in ft, or Q is in m3/s for hu in m

• The Kh and Ce values can be obtained from the two figures below 3 -1

ngular (V-notch weirs):

ing

• Note that Ce is dimensionless and that the units of Eq. 11 are L T (e.g. cfs, m3/s, etc.)

Sharp-crested tria

• Shen (ibid) produced the following calibration curves based on hydraulic laboratory measurements with sharp-crested V-notch weirs

• The curves in the two figures below can be closely approximated by the follow equations:

(

)

⎡

⎤

≅

⎣

θ

θ −

+

⎦

13) h

K

0.001

1.395

4.296

4.135

(

for Kh in meters; and,

(

)

≅ θ

(14)

for in radians

• Of course, you multiply a value in degrees by π/180 to obtain radians

Some installations have an insertable metallic V-notch weir that can be placed in

θ −

+

e

C

0.02286

0.05734

0.6115

θ •

slots at the entrance to a Parshall flume to measure low flow rates during some months of the year

3

20

40

20

Notch Angle, (degrees)

θ

Notch Angle, (degrees)

40

θ

0.57

60

80

100

0

60

80

100

0.59

0.58

0.050

0.100

0.075

2

0.60

1

0.025

C

K (

m

m

)

h e

K

(in

hes)

c

h I. utro Weir

• ying cross-sectional shape with depth

• free flow)

A generalized weir equation can be X S

Sutro weirs have a var

• This weir design is intended to provide high flow measurement accuracy for both small and large flow rates

A Sutro weir has a flow rate that is linearly proportional to h (for

• written as:

f

Q

= + α

k

h

β (15)

where k = 0 for the V-notch and rectangular weirs, but not

• The Sutro weir functions like a rectangular weir for h ≤ d

• This type of weir is designed for flow measurement under free-flow conditions

• It is not commonly found in practice XII. Submerged Flow over Weirs

Single Curve

• Villamonte (1947) presented the following from his laboratory results: f 0.385 n d s f s u

h

Q

Q 1

K Q

h

⎛

⎛

⎞

⎞

⎜

⎟

=

−

⎜

⎟

=

⎜

⎝

⎠

⎟

⎝

⎠

f (16) lly, nf • The figure below shows that in applying Eq. 16, hu & hd are measured

• For hd≤ 0, Ks = 1.0 and the flow is free

• For hd > hu, there will be backflow across the weir

• For hu = hd, the value of Qs becomes zero (this is logical) • The value of Qf is calculated from a free-flow weir equation

• The exponent, nf, is that which corresponds to the free-flow equation (usua = 1.5, or nf = 2.5)

from the sill elevation

• Eq. 16 is approximately correct, but may give errors of more than 10% in the ty

Multiple Curves

• resby (1997) expanded on this approach, making laboratory measurements which could be used to generate a family of curves to define the submerged-flow coefficient, Ks

The following is based on an analysis of the laboratory data collected by Scoresby (ibid). The flow rate through a weir is defined as:

e-flow d low, as (unity) for free flow

0

Sco • f n s f u

Q K C LH

=

(17)

where Q is the flow rate; Cf and nf are calibration parameters for fre conditions; L is the “length” of the crest; Hu is the total upstream hydraulic hea with respect to the crest elevation; and Ks is a coefficient for submerged f defined above. As before, the coefficient Ks is equal to 1.0

and is less than 1.0 for submerged flow. Thus,

s

K

≤

1.

(18)

• Below is a figure defining some of the terms:

EL

h

_u

h

_d

2

V

2

g

P

HGL

H

_u

5h

_u

w

ei

r

flow

• The coefficient Ks can be defined by a family of curves based on the value of Hu/P and hd/Hu

Each curve can be approximated by a combination of an exponential function the

•

and a parabola

• The straight line that separates the exponential and parabolic functions in graph is defined herein as:

d s

h

⎛

⎞

u

H

K

=

A

⎜

⎟

+

B

(19)

• The exponential function is:

⎝

⎠

d s u

h

K

1

H

β

⎛

⎞

= α −

⎜

⎟

⎝

⎠

(20)

• The parabola is:

2

h

h

⎛

_d

⎞

⎛

_d

⎞

u u

a

b

c

H

H

=

⎜

⎟

+

⎜

⎟

+

⎝

⎠

⎝

⎠

(21)

• Below the straight line (Eq. 19) the function from Eq. 20 is applied

• And, Eq. 21 is applied above the straight line

• In Eq. 19, let A = 0.2 y B = 0.8 (other values could be used, according to judgment and data analysis)

• In any case, A+B should be equal to 1.0 so that the line passes through the point (1.0, 1,0) in the graph (see below).

s

K

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 -0.20 0.00 0.20 0.40 0.60 0.80 1.00 hd/Hu 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 0.2 < Hu/P < 2.0 K

• This curve is defined by Eq. 20, but the values of α and β depend on the value of Hu/P

• The functions are based on a separate analysis of the laboratory results from re the following:

Scoresby (ibid) and a

α = 0 24.

F

HG

H

I

KJ

+0 76. P

t ₍₂₂₎

β =0 014.

HG KJ

F I

Ht +0 23.

P (23)

• The point at which the two parts of the curves join is calculated in the following:

A

FG

h B h

H_t H_t

H

I

K

J

+ =

H

FG I

−

K

)

• Defining a function F, equal to zero,

J

α β 1 (24 d d u u

h

h

F A

B

1

0

H

H

β

⎛

⎞

⎛

⎞

=

⎜

⎟

+ − α −

⎜

⎟

⎝

⎠

⎝

⎠

=

(25) 1 d u d u

h

F

A

1

H

h

H

β−

⎛

⎞

∂

= + αβ −

⎜

⎟

⎛

⎞

⎝

⎠

∂ ⎜ ⎟

⎝

⎠

(26)

• With Eqs. 25 and 26, a numerical method can be applied to determine the value of hd/Hu

• Then, the value of Ks can be determined as follows:

d s u

h

K

A

H

⎛

⎞

B

=

⎜

⎟

+

⎝

⎠

(27) • the XIII. O e

• At large values of the angle setting the gate behaves like a weir, and at lower The resulting values of hd/Hu and Ks define the point at which the two parts of curves join together on the graph

vershot Gates

• So-called “overshot gates” (also known as “leaf gates,” “Obermeyers,”

“Langeman,” and other names) are weirs with a hinged base and an adjustabl angle setting (see the side-view figure below)

Steel cable

• s on either side of the gate leaf are attached to a shaft above and upstream of the gate, and the shaft rotates by electric motor to change the setting

angles it approximates a free overfall (but this distinction is blurred when it is recognized that these two conditions can be calibrated using the same basic

• These gates are manufactured by the Armtec company (Canada), Rubicon

• The figure below shows an overshot gate operating under free-flow conditions equation form)

(Australia), and others, and are easily automated

h

u

P

hinge

θ

L

• ons presented below for overshot gates are based on the

•

ssumed

• d submerged flow is:

The calibration equati

data and analysis reported by Wahlin & Replogle (1996)

The representation of overshot gates herein is limited to rectangular gate leafs in rectangular channel cross sections, whereby the specified leaf width is a

to be the width of the cross section, at least in the immediate vicinity of the gate; this means that weir end contractions are suppressed

The equation for both free an

=

1.5 s a e w e

Q K C C

G h

3

2 2g

(28) where Q is the discharge; θ is the angle of the opening (10°≤θ≤ 65°), measured

• , or

•

• viously-given equations for rectangular weirs Q is in m3/s; for he in ft, Q is in cfs

on may have significant error for opening angles outside of the from the horizontal on the downstream side; Gw is the width of the gate leaf; and he is the effective head

The effective head is defined as: he = hu + KH, where KH is equal to 0.001 m 0.0033 ft

KH is insignificant in most cases For θ = 90°, use the pre

• For he in m, • The calibrati

specified range

u

h

⎛

⎞

e

C

=

0.075

⎜

⎟

+

0.602

P

⎝

⎠

(29) g ) • but

n parameters for the overshot gate type.

where P is the height of the gate sill with respect to the gate hinge elevation (m or ft)

• The value of P can be calculated directly based on the angle of the gate openin and the length of the gate leaf (P = L sinθ, where L is the length of the gate)

• The coefficient Ca is a function of the angle setting, θ, and can be adequately described by a parabola:

2 a

C

=

1.0333 0.003848

+

θ −

0.000045

θ

(30 where θ is in degrees

The submerged-flow coefficient, Ks, is taken as defined by Villamonte (1947), with custom calibratio

2 C 1.5 d

h

K

_s

C 1

₁ u

h

⎡

⎛

⎞

⎤

⎢

⎥

=

−

⎜

⎟

⎢

⎝

⎠

⎥

⎣

⎦

(31) 2) (33) where, 1 1

C

1.0666 0.00111

for

60

C

1.0

for

60

=

−

θ

θ <

=

θ ≥

(3 and, 2 2

C

=

0.1525 0.006077

+

θ −

0.000045

θ

in which θ is in degrees The su

• bmerged-flow coefficient, Ks, is set equal to 1.0 when hd≤ 0

h

u

d

h

P

θ

IV •

X . Oblique and Duckbill Weirs

• What about using oblique or duckbill weirs for flow measurement?

The problem is that with large L values, the hu measurement is difficult because small ∆h values translate into large ∆Q

• Thus, the hu measurement must be extremely accurate to obtain accurate discharge estimations

oblique weir

duckbill weir

inverse duckbill weir

flow

XV. Approach Velocity

• The issue of approach velocity was raised above, but there is another standard way to compensate for this

• The reason this is important is that all of the above calibrations are based on zero (or negligible) approach velocity, but in practice the approach velocity may

• be significant To approximately compensate for approach velocity, one approach (ha ha!) method is to add the upstream velocity head to the head term in the weir equation

For example, instead of this…

•

(34) …use this (where V is the mean approach velocity, Q/A):

( )

nf

f f u

f n 2 f f u

Q

C h

2g

V

⎛

⎞

=

⎜

⎜

+

⎟

⎟

)

⎝

⎠

(35 or, f n 2

⎛

_f

⎞

f f u

Q

₂

Q

C h

2gA

=

+

6)

ns it is an iterative solution for Qf, which tends to complicate matters a

• . 36 may have

er, and both positive)

• usion: it is a logical way to account for approach velocity, but it can be XVI. E

• of

iment load

• Some weirs have underflow gates which can be manually opened from time to time, flushing out the sediment upstream of the weir

• The effect is that the discharge flowing over the weir can be increased due to a higher upstream “apron”, thus producing less flow contraction

• The approximate percent increase in discharge caused by silting in front of a rectangular weir is given below:

Percent Increase in Discharge

⎜

⎟

⎜

⎟

⎝

⎠

(3

which mea

lot, because the function is not always well-behaved

For known hu and A, and known Cf and nf, the solution to Eq

multiple roots; that is, multiple values of Qf may satisfy the equation (e.g. there may be two values of Qf that are very near each oth

• There may also be no solution (!*%&!#@^*) to the equation Concl

difficult to apply ffects of Siltation

One of the possible flow measurement errors is the effect of siltation upstream the weir

• This often occurs in a canal that carries a medium to high sed

X/W P/W 0 0.5 1.0 1.5 2.0 2.5 0.00 10% 13% 15% 16% 16% 0.25 5% 8% 10% 10% 10% 0.50 3% 4% 5% 6% 6% 0.75 1% 2% 2% 3% 3% 1.00 zero zero

P

h

_u

X

W

• W is the value of P when there is no sediment deposition upstream of the weir

• X is the horizontal distance over which the sediment has been deposited upstream of the weir – if X is very large, use the top of the sediment for

determining P, and do not make the discharge correction from the previous table

• The reason for the increase in discharge is that there is a change in flow lines upstream of the weir

• When the channel upstream of the weir becomes silted, the flow lines tend to straighten out and the discharge is higher for any given value of hu

References & Bibliography

Addison. 1949.

Kindsvater and Carter. 1957.

W.D. Dyer. 1894. The Cipoletti trapezoidal weir. Trans. ASCE, Vol. 32. tah State Univ., Logan, UT.

Wa

Flinn, A.D., and C.

Scoresby, P. 1997. Unpublished M.S. thesis, U hlin T., and J. Replogle. 1996.