Open-channel measurement with a venturi flume: If D is the upstream depth above the throat sill and B is the throat width, what is the theoretical maximum discharge Q for critical flow in the throat?
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AQ = 1.71 B D
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BQ = 1.71 B D^{1/2}
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CQ = 1.71 B D^{3/2}
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DQ = 1/71 B D^{2/3}
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EQ = 1.71 D^{3/2} / B
Answer
Correct Answer: Q = 1.71 B D^{3/2}
Explanation
Introduction / Context:Venturi flumes accelerate flow to near-critical or critical conditions at the throat, enabling discharge measurement from upstream depths. Under critical flow, discharge can be expressed as a function of throat width and upstream depth above the throat sill.
Given Data / Assumptions:
- Venturi flume with throat width B.
- Depth above sill at approach section is D (energy losses neglected for theory).
- Critical flow occurs at the throat.
Concept / Approach:
For critical flow, specific energy is minimized and the Froude number equals 1. The theoretical discharge per unit width for critical depth y_c is q = (g)^{1/2} * y_c^{3/2}. Using practical constants in metric units, this yields Q = C * B * D^{3/2}, with C ≈ 1.705–1.71 for the idealized case.
Step-by-Step Solution:
Start with q = (g)^{1/2} * y_c^{3/2} (per unit width).For a rectangular throat: Q = B * q = B * (g)^{1/2} * D^{3/2}.Using g ≈ 9.81 m/s^2 gives coefficient ≈ 1.705–1.71.Hence Q ≈ 1.71 * B * D^{3/2} (theoretical).Verification / Alternative check:
Standard flume handbooks list similar coefficients for the ideal case; field coefficients may differ due to contraction and energy losses.
Why Other Options Are Wrong:
Linear or square-root dependence on D does not reflect critical-flow scaling; division by B is incorrect for rectangular throats; 1/71 is a spurious factor.
Common Pitfalls:
Confusing upstream gauge depth with critical depth; ignoring calibration coefficients for real flumes.
Final Answer:
Q = 1.71 B D^{3/2}