Cipolletti (trapezoidal) weir — Francis discharge formula For a Cipolletti weir of crest length L and head H, which expression gives the discharge Q?

Introduction / Context:
The Cipolletti weir is a trapezoidal sharp-crested weir with side slopes chosen so that the discharge coefficient is nearly constant and the end-contraction effect is compensated. Francis provided a convenient empirical discharge relation widely used in practice.

Given Data / Assumptions:

• Free, fully aerated nappe; sufficient approach condition corrections applied.
• Head H measured above crest; crest length L between nappe ends.
• Standard Cipolletti slope (1 horizontal : 4 vertical).

Concept / Approach:
For the Cipolletti weir, Francis-type formula is Q = C * L * H^(3/2) with C ≈ 1.84 (SI units when Q in m^3/s, L and H in m). The exponent 3/2 arises from the velocity head integration across the depth over the crest.

Step-by-Step Solution:

Verification / Alternative check:
Limit case: If side slopes vanish (rectangular weir), the constant differs and end-contraction corrections are needed; Cipolletti geometry removes the correction, keeping 1.84 applicable.

Why Other Options Are Wrong:

• H^(1/2), H, H^(5/2): incorrect exponent.
• Divide by L: in a weir, discharge grows with crest length, not inversely.

Common Pitfalls:
Using the rectangular weir coefficient directly for Cipolletti; measuring H too close to the crest (should be upstream at a recommended distance).

Final Answer:
Q = 1.84 * L * H^(3/2)