# Weirs - Flow Rate Measure

## Weirs can be used for measuring flow rates in open channels and rivers - common in connection to water supply and sewage plants

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**Weirs** are structures consisting of an obstruction such as a dam or bulkhead placed across the open channel with a specially shaped opening or notch. The weir results an increase in the water level, or head, which is measured upstream of the structure. The flow rate over a weir is a function of the head on the weir.

Common weir constructions are the rectangular weir, the triangular or v-notch weir, and the broad-crested weir. Weirs are called sharp-crested if their crests are constructed of thin metal plates, and broad-crested if they are made of wide timber or concrete.

Water level-discharge relationships can be applied and meet accuracy requirements for sharp-crested weirs if the installation is designed and installed consistent with established ASTM and ISO standards.

Rectangular weirs and triangular or v-notch weirs are often used in water supply, wastewater and sewage systems. They consist of a sharp edged plate with a rectangular, triangular or v-notch profile for the water flow.

Broad-crested weirs can be observed in dam spillways where the broad edge is beneath the water surface across the entire stream. Flow measurement installations with broad-crested weirs will meet accuracy requirements only if they are calibrated.

Other available weirs are the trapezoidal (Cipolletti) weir, Sutro (proportional) weir and compound weirs (combination of the previously mentioned weir shapes).

### Rectangular Weir

The flow rate measurement in a rectangular weir is based on the Bernoulli Equation principles and can be expressed as:

q= 2/3c_{d}b (2 g)^{1/2}h^{3/2}(1)

where

q= flow rate (m^{3}/s)

h= head on the weir (m)

b= width of the weir (m)

g= 9.81 (m/s^{2}) - gravity

c_{d}= discharge constant for the weir - must be determined

*c _{d}* must be determined by analysis and calibration tests. For standard weirs -

*c*- is well defined or constant for measuring within specified head ranges.

_{d}#### The Francis Formula - Imperial Units

Flow through a rectangular weir can be expressed in imperial units with the Francis formula

q = 3.33 (b - 0.2 h) h^{3/2}(1b)

where

q = flow rate (ft^{3}/s)

h = head on the weir (ft)

b = width of the weir (ft)

Alternative with heigth in inches and flow in gpm:

### Triangular or V-Notch Weir

For a triangular or v-notch weir the flow rate can be expressed as:

q= 8/15c_{d }(2 g)^{1/2}tan(θ/2) h^{5/2}(2)

where

θ= v-notch angle

### Broad-Crested Weir

For the broad-crested weir the flow rate can be expressed as:

q=c_{d}h_{2}b ( 2 g (h_{1}- h_{2}) )^{1/2 }(3)

### Measuring the Levels

For measuring the flow rate it's obviously necessary to measure the flow levels, then use the equations above for calculating. It's common to measure the levels with:

- ultrasonic level transmitters, or
- pressure transmitters

Ultrasonic level transmitters are positioned above the flow without any direct contact with the flow. Ultrasonic level transmitters can be used for all measurements. Some of the transmitters can even calculate a linear flow signal - like a pulse signal or *4 - 20 mA* signal - before transmitting it to the control system.

Pressure transmitters can be used for the sharp-crested weirs and for the first measure point in broad-crested weir. The pressure transmitter outputs a linear level signal - *4 - 20 mA* - and the flow must in general be calculated in the control system.

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