Weir Coefficients: Dimensionless Parameters for Calculating Weir Discharge

The weir coefficient is a dimensionless parameter that incorporates the gravitational acceleration, discharge coefficient, and the geometric characteristics of the weir into a single value. It is used in the standard equation for weir discharge to simplify the calculation by consolidating multiple constants into one parameter.

Weir coefficient

In this article, we will discuss weir coefficients for different types of weirs as well as the factors affecting their value.

Weir Coefficients Explained

Weirs are structures built across open channels, such as rivers or streams, to control water flow and measure discharge. They typically consist of a low dam or wall with a series of notches or openings that regulate the passage of water.

Weir

Depending on the type and geometry of the weir, the equations used to calculate the discharge over the weir can have different forms. For example, for a rectangular sharp-crested weir, the discharge can be calculated using the following formula:

rectangular sharp-crested weir discharge formula

Where:

  • Q = flow rate [m3/s]
  • Cd = discharge coefficient [unitless]
  • B = crest width [m]
  • Ho = head over the weir measured from the weir crest to the upstream water surface [m]

On the other hand, for a triangular sharp-crested weir, the discharge is given by:

triangular sharp-crested weir discharge formula

Where:

  • Cc = contraction coefficient [unitless]
  • θ = notch angle [deg]

However, in practice, engineers often use a simple standard equation for weir discharge as follows:

weir discharge standard equation

Where:

  • C = weir coefficient [unitless]
  • L = length of the weir [m]
  • H = head over the weir [m]

Instead of having several constants in one equation, the above equation lumps these constants into one parameter called the weir coefficient. This coefficient incorporates the gravitational acceleration, the discharge coefficient, as well as the effects of the geometric characteristics of the weir.

It is important to note that weir coefficient is technically different from discharge coefficient, although they are often used interchangeably. The discharge coefficient is dimensionless, so it remains consistent in both English (U.S. Customary) Units and SI Units. However, the weir coefficient, being dependent on the gravitational constant, is not dimensionless and thus varies with the unit system employed.

Factors Affecting Weir Coefficient

Typically, the weir coefficient is determined experimentally. Weirs must be properly calibrated, and the value of the weir coefficient is specific to each weir setup. Several factors impact its value, including

Geometry of the Weir

The shape and dimensions of the weir influence the flow. For instance, sharp-crested weirs tend to have different coefficients compared to broad-crested weirs. A V-notch weir’s angle also plays a significant role; typically, a 90-degree V-notch may be used for flow measurement given its sensitivity to small changes in flow.

Surface Tension and Viscosity of Fluid

The fluid’s properties, such as viscosity and surface tension, can alter the flow pattern, affecting the coefficient. Viscous fluids tend to adhere to weir surfaces, causing a variation in the expected coefficient.

Flow Conditions

Submergence and approach velocities alter the flow characteristics. Submerged flow or a high-approach velocity can decrease the coefficient due to increased energy loss in the approach channel.

Weir Material and Surface Roughness

The material of the weir and its roughness affect the boundary layer development and the associated energy loss. Rough surfaces increase the friction and energy dissipation, altering the expected coefficient.

Temperature

Fluid temperature impacts its viscosity, which in turn can influence the weir coefficient.

Debris and Obstructions

Debris accumulating at the weir or any obstructions can restrict flow and affect the accuracy of the coefficient used for flow measurement.

Weir Coefficients of Different Types of Weirs

The weir coefficient value changes based on the weir’s geometry. This implies that a sharp-crested weir will have different weir coefficient values compared to a broad-crested weir. Additionally, rectangular, triangular, and trapezoidal sharp-crested weirs will each have distinct weir coefficients.

Broad-Crested Weirs

For broad-crested weirs, the typical weir coefficient values are listed in the table below, dependent on the measured head and breadth of crest of weir:

Sharp-Crested Weirs

For sharp-crested rectangular weirs without end contraction, the typical weir coefficient values are listed in the table below, also dependent on the measured head and height of crest above bed of channel of approach:

Weir Head vs. Weir Crest Height

On the other hand, for sharp-crested rectangular weirs with contractions suppressed at both ends:

Weir Head vs. Weir Length

It is important to note that these values are indicative and can be fine-tuned based on experimental data and specific site conditions. To determine the head, measure the height from the crest of the weir to the upstream water surface, ensuring that the flow is steady and the weir is fully functioning. Flow conditions must be free of upstream obstructions, with water flowing smoothly over the weir crest.

For accurate measurements, ensure that the downstream water does not submerge the weir, a condition known as “drowned flow,” which invalidates the standard coefficients provided. Adhering to standard procedures and correcting for site-specific conditions will lead to more accurate determinations of the weir coefficient and enable precise flow calculations.

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