Discharge Over a Rectangular Weir

A weir is a barrier constructed across the width of a river or stream to alter the flow characteristics of water, often resulting in a change in the water level. Weirs are fundamental hydraulic structures commonly used for measuring the flow rate or discharge of water in open channels, rivers, and canals.

Rectangular Weir Characteristics

A rectangular weir is a specific type of weir characterized by a rectangular opening or crest over which water flows. The crest of a rectangular weir is typically horizontal, and its side walls are vertical. This straightforward design makes it a popular choice for accurate flow measurement, especially in controlled environments, due to its predictable hydraulic behavior.

Discharge Formula for Rectangular Weirs

The theoretical discharge over a rectangular weir is determined by considering the flow through small horizontal strips across the weir's opening and integrating the velocity over the total head of water. However, in practical applications, factors such as energy losses and the contraction of the water stream (known as the nappe) as it passes over the weir's crest affect the actual flow. To account for these real-world effects, a coefficient of discharge (\(C_d\)) is introduced into the theoretical formula to yield the actual discharge.

The universally accepted formula for the discharge (\(Q\)) over a rectangular weir is:

$$Q = \frac{2}{3}C_d L\sqrt{2g} H^{3/2}$$

Rectangular Weir Formula Terms Explained

Let's understand each term within the discharge formula for a rectangular weir:

• Q: This represents the discharge or the volumetric flow rate of water over the weir. It is typically expressed in units of cubic meters per second \((m^3/s)\) or cubic feet per second \((ft^3/s)\).
• \(\frac{2}{3}\): This is a constant factor that emerges from the integration process during the theoretical derivation of the flow equation for a rectangular weir.
• \(C_d\): This is the coefficient of discharge, a dimensionless value. It accounts for the difference between the theoretical and actual discharge, incorporating effects like frictional losses and nappe contraction. For sharp-crested rectangular weirs, \(C_d\) usually falls within the range of 0.6 to 0.62.
• L: This denotes the effective length of the crest of the rectangular weir. It is the horizontal dimension over which the water flows and is measured in meters (m) or feet (ft).
• \(\sqrt{2g}\): This part of the formula relates to the velocity of the water. Here, g represents the acceleration due to gravity, which is approximately \(9.81 \, m/s^2\) in metric units or \(32.2 \, ft/s^2\) in imperial units.
• \(H^{\frac{3}{2}}\): This term incorporates the head of water over the weir. H is the measured height of the water surface above the crest of the weir, typically in meters (m) or feet (ft). The exponent \(\frac{3}{2}\) is a direct result of the integration performed during the derivation of the flow equation.

Discharge Formula Options Analysis

The question requires identifying the correct formula for the discharge over a rectangular weir. We will compare the standard, established formula with the given options:

Based on this detailed analysis, the formula \(\frac{2}{3}Cd\sqrt {2g} L{H^{\frac{3}{2}}}\) is the accurate representation of the discharge over a rectangular weir.