# Weir Head: Analyzing Hydraulic Energy for Weir Flow

In open channel analysis, weir head refers to the height of the water surface above the crest of a weir. It represents the energy exerted by the flowing water, and is an important parameter in determining the flow rate over the weir.

In this article, we will discuss the significance of weir head in analyzing weir flow, including its components and its role in determining discharge for different types of weirs such as sharp-crested and broad-crested weirs.

## Understanding Weir Head

Weirs are barriers built across rivers or streams to control the flow of water and measure its discharge. Typically made of wood, concrete, stone, or other materials, these barriers are designed to create a change in water elevation, allowing to regulate water levels, divert water for various purposes, and monitor the quantity of flowing water.

Weirs play an important role in water management, helping prevent flooding, ensuring a stable water supply, and facilitating the measurement of streamflow for environmental and engineering purposes. They are commonly used in irrigation systems, hydropower plants, and environmental monitoring, providing an effective means of managing and utilizing water resources.

One of the essential parameters used to analyze weir flow is weir head. Weir head quantifies the energy of the water flowing over a weir in terms of the height of an equivalent static water column, expressed in units of distance such as meters or feet.

In general, weir head can be divided into two components: elevation head and velocity head.

The elevation head is the vertical measure from the weir crest—the point where water begins to overflow—to the water’s surface at an upstream position. This is represented by Ho in the diagram above.

On the other hand, the velocity head represents the kinetic energy of the fluid per unit weight and is associated with the velocity of the stream as it approaches the weir. This is represented by ho in the diagram above and it can be calculated using the following equation:

Where:

• ho = velocity head [m]
• Vo = mean flow velocity [m/s]
• g = gravitational acceleration [9.81 m/s2]

It would be more theoretically appropriate to include the energy coefficient, α, in the velocity head to accommodate the nonuniform velocity distribution. However, experimental results indicate that its value only ranges between 1.00 and 1.08 for flows approaching a weir, so it can be assumed to be equal to 1.

## Calculating Flow Rate from Weir Head

Weir head is typically used to calculate for the rate of discharge over a weir. For sharp-crested rectangular weirs, for example, the discharge per unit width can be calculated using the equation:

Where:

• q = discharge per unit width [m2/s]
• Ho = elevation head over the weir [m]

However, in practice, a discharge coefficient is typically introduced to refine the equation to account for contraction and other effects. Hence, the modified equation for flow per unit width of the rectangular weir can be expressed by:

Where:

• Cd = discharge coefficient [unitless]

The discharge coefficient is also affected by weir head. In fact, based on Rehbock’s experimental results for rectangular weirs, Cd may be approximated as:

For triangular weirs, the flow rate is computed differently. Incorporating the contraction coefficient and the internal angle of the weir, the discharge is described by the equation:

Where:

• Q = discharge [m3/s]
• Cc = contraction coefficient [unitless]
• α = internal angle of the weir [deg]

Lastly, for broad-crested weirs, the discharge can be calculated as follows:

Where:

• C = weir coefficient [unitless]
• B = channel width [m]
• H = elevation head over the weir [m]

## Weir Head Measurement

To accurately determine the flow rate over a weir, it is important to measure the weir head correctly. Notice that the formulas for calculating the discharge mentioned above only consider the elevation head, not the velocity head. Therefore, our focus should be on measuring the elevation head, which is the vertical distance between the weir crest and the water surface upstream.

To account for the drawdown effect in weir flow, where the water level decreases just before flowing over the crest, it is important to measure the head at a point located at a minimum distance of 3 to 4 times the elevation head from the crest. Additionally, the minimum height of the crest should be at least 2 to 3 times the elevation head relative to the channel bottom.

In the case of broad-crested weirs, it is essential to consider the relative magnitudes between the length of the weir and the weir head.

If the weir’s length in the direction of flow is short, the flow depth on the crest may vary with distance, resulting in potentially curvilinear or even separated flow. This can lead to errors in discharge computation if critical parallel flow is assumed. On the other hand, if the weir is too long, viscous effects become significant and necessitate a correction.

A weir can be considered long if its length-to-head ratio is greater than 3. In such cases, the assumed weir head should be reduced by the maximum displacement thickness of the boundary layer to account for viscous effects.

Lastly, it is important to note that in weir analysis, the pressure head is assumed to be zero. This assumption is based on the theoretical development of weirs, which assumes atmospheric pressure above and below the nappe.

Therefore, it is necessary to vent the underside of the nappe to ensure that the lower side of the nappe is at atmospheric pressure. Failure to vent the nappe may lead to non-atmospheric pressure under the jet, resulting in erroneous discharge computations.

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