Understanding the Francis Formula and End Contractions

The Francis formula is a fundamental equation used in fluid mechanics and hydraulics to calculate the discharge ($Q$) over a weir. A critical aspect of this formula is its handling of 'end contractions', which are adjustments made when the weir crest does not extend fully across the channel width.

The formula is often expressed as:

$$ Q = C_d \times (L - n \times C_e) \times H^{3/2} $$

For the specific context of the Francis formula, the coefficient $C_d$ is typically taken as $1.84$, and $C_e$ (the end contraction allowance) is usually $0.1 \times H$. Thus, the formula becomes:

$$ Q = 1.84 \times (L - n \times 0.1 H) \times H^{3/2} $$

Where the key variables are:

• $Q$: Represents the discharge rate (volume per unit time).
• $L$: Denotes the total crest length of the weir.
• $H$: Represents the hydraulic head, which is the vertical distance from the weir crest to the upstream water surface.
• $n$: Signifies the number of end contractions.
• $C_e$: The contraction allowance for each end, typically dependent on $H$.

An end contraction effectively reduces the weir's length available for flow. It occurs at each end of the weir where it does not meet the channel wall.

Analyzing the Specific Weir Scenario: Crest Length Equals Channel Width

The question poses a specific condition: the crest length of the weir ($L$) is exactly equal to the width of the channel ($W$). Mathematically, this is represented as:

$$ L = W $$

This condition implies that the weir spans the entire width of the channel, extending from one side wall to the other without any gaps. Such a weir setup is termed a suppressed weir.

Determining the Number of End Contractions ('n') for a Suppressed Weir

In a suppressed weir, where $L = W$, the weir crest runs flush against both channel walls. Consequently, there are no exposed ends of the weir crest that are separated from the channel boundaries. This means there are no end contractions.

According to the definition used in the Francis formula and related hydraulic principles, when a weir is suppressed (i.e., $L = W$), the number of end contractions ($n$) is zero.

Substituting $n=0$ into the Francis formula for a contracted weir yields the formula for a suppressed weir:

$$ Q = 1.84 \times (L - 0 \times 0.1 H) \times H^{3/2} $$

Which simplifies to:

$$ Q = 1.84 \times L \times H^{3/2} $$

Therefore, when the crest length of the weir equals the width of the channel, the number of end contractions 'n' in the Francis formula is 0.