Triangular Notch Discharge Formula Explained

A triangular notch, also known as a V-notch, is a type of weir used in hydraulics to measure the flow rate (discharge) of a fluid, particularly in open channels. It features a V-shaped opening over which the liquid flows.

Understanding the Key Components

The formula for discharge through a triangular notch incorporates several parameters:

• Q: Represents the discharge or flow rate, typically measured in cubic meters per second (m³/s) or liters per second (L/s).
• \(C_d\): This is the coefficient of discharge, a dimensionless factor that accounts for energy losses due to friction and the contraction of the jet. It's determined experimentally and usually ranges between 0.60 and 0.65 for V-notches.
• \(\theta\): The angle of the V-notch, measured in degrees. The notch is often designed with specific angles like 90°.
• \(H\): The head of the fluid, which is the vertical distance from the bottom of the notch to the free surface of the liquid upstream. It's usually measured in meters (m).
• \(g\): The acceleration due to gravity, approximately 9.81 m/s² on Earth.

The Standard Discharge Formula

The widely accepted formula for calculating the discharge (Q) through a triangular notch is:

$$Q = \frac{8}{{15}}{C_d}\cdot \tan \frac{\theta }{2} \times \sqrt {2g} \cdot {H^{5/2}}$$

This formula is derived by integrating the flow rate over elementary horizontal strips of the V-notch, considering the velocity of flow across each strip.

Derivation Insight

The derivation involves considering a thin horizontal strip of the notch at a depth \(h\) from the free surface, with a width \(b\). The width \(b\) varies with depth \(h\) according to the notch angle \(\theta\). The velocity of water flowing over this strip is approximately \(\sqrt{2gh}\) (based on Torricelli's theorem). The discharge through this elemental strip (\(dQ\)) is the product of the area of the strip (\(b \cdot dh\)) and the velocity. Integrating \(dQ\) from the apex of the notch (\(h=0\)) to the free surface (\(h=H\)), and applying the coefficient of discharge (\(C_d\)), yields the final formula. The \(\frac{8}{15}\) coefficient and the \(H^{5/2}\) term arise specifically from the integration process over the triangular geometry.

It is important to use this specific formula for triangular notches, as formulas for rectangular or trapezoidal notches differ.