Rectangular Weir with End Contractions – Francis Discharge Formula For a sharp-crested rectangular weir having n end contractions, the practical Francis form in SI units is Q = 1.84 * (L − 0.1 * n * H) * H^(3/2), where L is crest length and H is head over crest.
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AQ = 1.84 (L − 0.1 n H) H^(3/2)
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BQ = Cd (L − n H) 2 g H
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CQ = 1.84 (L − n H) H^2
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DQ = Cd (L − 0.2 n H) 2 g H^(5/2)
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EQ = 1.84 L H^(5/2)
Answer
Correct Answer: Q = 1.84 (L − 0.1 n H) H^(3/2)
Explanation
Introduction:The Francis formula is a widely used empirical relation for discharge over sharp-crested rectangular weirs. It corrects the crest length for end contractions and consolidates constants for SI usage.
Given Data / Assumptions:
- Rectangular, sharp-crested weir with free nappe and adequate aeration.
- Head H measured sufficiently upstream to avoid velocity-of-approach error.
- n end contractions (typically 2 for a contracted weir).
Concept / Approach:
The theoretical discharge per unit width from energy considerations scales with H^(3/2). Accounting for contraction and empirical coefficient Cd leads to a compact constant 1.84 in SI so that Q = 1.84 (L − 0.1 n H) H^(3/2).
Step-by-Step Solution:
1) Effective crest length L_e = L − 0.1 n H.2) Use Q ∝ L_e * H^(3/2) from sharp-crested weir theory.3) Apply consolidated constant → Q = 1.84 L_e H^(3/2).4) Substitute L_e to get the stated Francis form.Verification / Alternative check:
Dimensional analysis confirms m^3/s when L and H are in meters. Field calibrations show good agreement within the method’s range.
Why Other Options Are Wrong:
Options with H^2 or H^(5/2) do not match the sharp-crested exponent. Using n H instead of 0.1 n H overcorrects end contractions. Expressions with Cd 2 g are incomplete without integration and constants consolidation.
Common Pitfalls:
Measuring H too close to the crest, inadequate aeration of the nappe, or miscounting end contractions.
Final Answer:
Q = 1.84 (L − 0.1 n H) H^(3/2)