Understanding Rectangular Notch Discharge

The discharge ($Q$) over a sharp-edged rectangular notch quantifies the volume of fluid flowing over it per unit time. This calculation depends on the notch's geometry and the fluid's properties.

Discharge Formula Basics

The flow rate over a sharp-edged rectangular notch is theoretically derived by integrating the flow velocity across the notch area. The velocity varies with the depth of the fluid (head, $h$).

The theoretical discharge ($Q_{th}$) is proportional to the notch width ($w$) and the head ($h$) raised to the power of $3/2$. The standard formula includes the coefficient of discharge ($C_d$) to account for real-world energy losses and flow patterns.

The widely accepted formula for discharge over a rectangular notch is:

$ Q = \frac{2}{3} C_d w \sqrt{2g} h^{3/2} $

Key variables:

• $C_d$: Coefficient of discharge (accounts for losses)
• $w$: Width of the rectangular notch
• $g$: Acceleration due to gravity
• $h$: Head (depth) of the fluid above the notch sill

Analyzing Notch Discharge Options

We need to find the expression that correctly represents the discharge over a sharp-edged rectangular notch.

The crucial factor in the formula is the term $h^{3/2}$, representing the head's influence on discharge. The options provided are:

• Option 1: $C_d a w \sqrt{2g} h^{5/2}$ - Incorrect power ($h^{5/2}$)
• Option 2: $C_d a w \sqrt{2g} h$ - Incorrect power ($h^1$)
• Option 3: $C_d a w \sqrt{2g} h^{3/2}$ - Correct power ($h^{3/2}$)
• Option 4: $C_d a w \sqrt{2g} h^{3/2}$ - Correct power ($h^{3/2}$)

Options 3 and 4 are identical and match the expected power dependency ($h^{3/2}$). The constant 'a' in the options likely represents the factor $\frac{2}{3}$ present in the standard formula, combined with other minor adjustments or simply serving as a placeholder constant within the context of the question's options.

Therefore, the discharge over a sharp-edged rectangular notch of width $w$ and depth $h$ is given by the form $C_d a w \sqrt{2g} h^{3/2}$.