Cipolletti (trapezoidal) weir — Francis discharge formula For a Cipolletti weir of crest length L and head H, which expression gives the discharge Q?
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AQ = 1.84 * L * H^(1/2)
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BQ = 1.84 * L * H
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CQ = 1.84 * L * H^(3/2)
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DQ = 1.84 * L * H^(5/2)
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EQ = 1.84 * H^(3/2) / L
Answer
Correct Answer: Q = 1.84 * L * H^(3/2)
Explanation
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:
Recognize trapezoidal weir with compensating side slopes.Select Francis relation: Q = 1.84 L H^(3/2).Confirm unit consistency and applicability (free nappe, ventilated).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)