Head–discharge proportionality for notches Choose the correct statement(s) relating discharge Q to head H over a weir/notch under free-flow conditions.
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AFor a triangular (V-notch), Q ∝ H^(5/2)
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BFor a rectangular sharp-crested weir/notch, Q ∝ H^(3/2)
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CBoth (a) and (b)
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DNeither (a) nor (b)
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EOnly (b) is correct; (a) is false
Answer
Correct Answer: Both (a) and (b)
Explanation
Introduction / Context:Discharge measurement by notches and weirs is fundamental in hydraulic laboratories and field channels. The head–discharge exponent depends on the crest shape and flow contraction behavior.
Given Data / Assumptions:
- Free, fully aerated overflow without submergence.
- Standard coefficients near calibrated values; end-contraction corrections omitted for proportionality.
- Sharp-crested geometry for the rectangular case; V-notch with fixed angle (e.g., 90°) for the triangular case.
Concept / Approach:
Classical results: rectangular sharp-crested weir Q ∝ H^(3/2); triangular (V-notch) Q ∝ H^(5/2). The larger exponent for the V-notch reflects the narrowing flow area with decreasing head, increasing sensitivity to H.
Step-by-Step Solution:
Rectangular: Q = C_d * b * √(2 g) * H^(3/2).Triangular: Q = (8/15) * C_d * √(2 g) * tan(θ/2) * H^(5/2).Hence proportionalities in (a) and (b) are both correct.Verification / Alternative check:
Calibration charts and standard texts confirm these exponents under non-drowned, sharp-crested conditions.
Why Other Options Are Wrong:
Options denying either exponent contradict well-established discharge formulas.
Common Pitfalls:
Using rectangular exponents for V-notches; ignoring submergence or approach velocity corrections when high accuracy is needed.
Final Answer:
Both (a) and (b)