Weir: Problem Solving

Water flow in open channels is often measured using hydraulic structures such as weirs, which allow precise calculation of discharge. In a rectangular channel, flow rates are measured using three types of weirs: rectangular sharp-crested, triangular sharp-crested, and broad-crested. The weir head is set at a fixed height above the channel bottom, simplifying calculations and enabling the relationship between depth and flow rate to be analyzed.

For the rectangular sharp-crested weir, the flow rate depends on the weir coefficient, the channel width, gravitational acceleration, and the fluid depth. When the weir head is fixed, the equation simplifies by reducing the variables to channel width and depth. This configuration allows the flow rate to be plotted against depth for visualizing the relationship between flow depth and discharge.

The triangular sharp-crested weir, or V-notch weir, incorporates an additional parameter, the notch angle, which directly affects the flow rate equation. This weir is particularly suited for low-flow conditions and is governed by the relationship where the discharge is proportional to the product of the tangent of half the notch angle, the square root of gravitational acceleration, and the fluid depth raised to the power of five over two. The discharge coefficient is typically obtained from standard references or experimental data. By plotting the discharge coefficient against depth, the weir's performance can be analyzed over varying flow conditions. This weir is highly sensitive to changes in the notch angle and is especially effective in measuring small flow rates accurately.

For the broad-crested weir, the flow rate is influenced by the discharge coefficient, channel width, gravitational acceleration, and fluid depth. When the weir head is fixed, the equation simplifies to a function of depth alone. This type of weir is robust for higher flow rates but requires careful design to satisfy constraints on block length and head-to-depth ratios.

Comparing the flow rate versus depth across these weir types allows for the evaluation of their efficiency and accuracy under different conditions, aiding in the selection of the optimal design for the specified flow range.