Water Flow Through A Slot

Concept Question 5.01.22 Incorrect Water flows into a rectangular slot that is at an angle β with respect to the wall as shown below. Wall Width-W Rectanget slot V elocity-V The correct relation between the velocities V and V2 is Vi sin Click if you would like to Show Work for this question: Open Show Work. A slot is a weir regardless of whether it fits into any standard model of a weir. The important characteristic of a weir that it is free surface flow and at some point the flow is at minimum energy (at transition from sub critical to super critical critical depth if you like). You can calculate the flow through a weir/slot from this principle.

Water Flow Through A Horizontal Slot
Water Flow Through A Slot Machines
Water Flow Through A Slot Jackpots

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

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 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. If the notch plate is mounted on the supporting bulkhead such that the water does not contact or cling to the downstream weir plate or supporting bulkhead, but springs clear, the weir is a sharp-crested or thin-plate weir.

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, the Sutro (proportional) weir and the compound weirs (combination of the previously mentioned weir shapes). By combining V-notch weirs with broad chested weirs - larger range of flow can be measured with accuracy.

Rectangular Weir

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

q = 2/3 c_d b (2 g)^1/2 h^3/2 (1)

where

q = flow rate (m³/s)

h = elevation head on the weir (m)

b = width of the weir (m)

g = 9.81 (m/s²) - 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_d - is well defined or constant for measuring within specified head ranges.

The lowest elevation (h = 0) of the overflow opening of the sharp-crested weirs or the control channel of broad-crested weirs is the head measurement zero reference elevation.

Rectangular Weir Flow Rate Measurement Calculator

c_d - discharge constant

b - width of weir (m)

h - height of weir (m)

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 (ft³/s)

h = head on the weir (ft)

b = width of the weir (ft)

Alternative with height in inches and flow in gpm:

Triangular or V-Notch Weir

The triangular or V-notch, thin-plate weir is an accurate flow measuring device particularly suited for small flows.

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

q = 8/15 c_d(2 g)^1/2 tan(θ/2) h^5/2 (2)

where

Water Flow Through A Horizontal Slot

θ = v-notch angle

Broad-Crested Weir

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

q = c_d h₂ b ( 2 g (h₁ - h₂) )^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 digital pulse signal or an analog 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 - typical 4-20 mA - and the flow must be calculated in the transmitter or the control system.