In Francis’s rectangular weir formula with end contractions, the effective length is reduced by how much per end contraction when the head over the sill is H?

Introduction / Context:
When measuring discharge over a rectangular sharp-crested weir with end contractions, the jet converges laterally, effectively reducing the length through which flow passes. Francis’s empirical formula corrects the crest length to account for this contraction.

Given Data / Assumptions:

• Rectangular sharp-crested weir with end contractions.
• Head over crest (sill) is H.
• Standard, fully aerated nappe; no submergence.

Concept / Approach:
Francis proposed using an effective length L_e = L − 0.1 n H, where L is the physical crest length and n is the number of end contractions (typically n = 2 for a weir contracting on both sides). Thus each end contraction reduces the length by 0.1 H.

Step-by-Step Solution:
Identify number of end contractions: n = 0, 1, or 2 depending on abutment conditions.Compute L_e = L − 0.1 n H.Therefore, the reduction per end is 0.1 H.

Verification / Alternative check:
Standard hydrometry references reproduce the same reduction term in Francis’s formula and compare favorably with laboratory data for common head ranges.

Why Other Options Are Wrong:
0.2 H, 0.3 H, 0.4 H, 0.5 H exaggerate the contraction effect and would underpredict discharge.

Common Pitfalls:

• Forgetting to set n properly (e.g., suppressed weir has n = 0).
• Using the reduction with a drowned nappe; different corrections apply.

Final Answer:
0.1 H