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?

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:

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}