Understanding Discharge Over a Rectangular Notch

When water flows over a notch or weir, the rate at which the water volume passes is called the discharge. A rectangular notch is a type of notch with a rectangular opening. The amount of discharge over a rectangular notch depends on several factors, including the dimensions of the notch and the height of the water level above the bottom of the notch, known as the head.

Formula for Discharge over a Rectangular Notch

The theoretical formula for the discharge \(Q\) over a rectangular notch is derived based on basic fluid mechanics principles, considering the velocity of water flowing through a strip of the notch at a certain depth. The actual discharge is slightly less due to factors like viscosity and surface tension, accounted for by a coefficient of discharge.

The commonly used formula for discharge \(Q\) over a rectangular notch is given by:

\( Q = C_d \cdot \frac{2}{3} \cdot L \cdot \sqrt{2g} \cdot H^{3/2} \)

Where:

• \(Q\) is the discharge (volume per unit time).
• \(C_d\) is the coefficient of discharge, which is an experimental value (typically around 0.6 to 0.62 for rectangular notches).
• \(L\) is the length of the rectangular notch.
• \(g\) is the acceleration due to gravity (approximately \(9.81 \, m/s^2\)).
• \(H\) is the head of water over the crest of the notch (the height of the water surface above the bottom of the notch).

Analyzing the Relationship between Discharge and Head

In the formula \( Q = C_d \cdot \frac{2}{3} \cdot L \cdot \sqrt{2g} \cdot H^{3/2} \), let's examine the terms:

• \(C_d\) is a constant for a given notch and flow conditions.
• \(\frac{2}{3}\) is a constant.
• \(L\) is the length of the notch, which is constant for a specific notch.
• \(\sqrt{2g}\) is a constant, as \(g\) is constant.

All these terms \((C_d \cdot \frac{2}{3} \cdot L \cdot \sqrt{2g})\) can be grouped together as a single constant value for a given notch. Let's call this combined constant \(K\). So, the formula simplifies to:

\( Q = K \cdot H^{3/2} \)

This equation clearly shows that the discharge \(Q\) is directly proportional to the head \(H\) raised to the power of \(3/2\).

Therefore, the discharge over a rectangular notch is directly proportional to \(H^{3/2}\).

Revision Table: Key Factors for Rectangular Notch Discharge

Additional Information on Notches and Weirs

Notches and weirs are structures used to measure the flow rate of water in open channels or tanks. They work by creating a known relationship between the water level upstream of the structure (the head) and the discharge.

Different types of notches exist, including:

• Rectangular Notch: As discussed, \( Q \propto H^{3/2} \). Simple design.
• Triangular Notch (V-notch): Often used for measuring low discharges accurately. The formula for discharge is \( Q = C_d \cdot \frac{8}{15} \cdot \tan(\frac{\theta}{2}) \cdot \sqrt{2g} \cdot H^{5/2} \), where \(\theta\) is the angle of the notch. For a triangular notch, \( Q \propto H^{5/2} \).
• Trapezoidal Notch (Cippoletti Weir): A combination of a rectangular notch and two triangular notches at the sides. The side slopes are typically 1 horizontal to 4 vertical. Designed such that the discharge through the triangular sections compensates for the reduction in discharge due to end contractions of the rectangular section. The formula is typically \( Q = 1.859 L H^{3/2} \) (using metric units and empirical constant).

Understanding the relationship between discharge and head for each type of notch is crucial for flow measurement in hydraulics.