Representative velocity to use in hydraulic engineering calculations When solving flow problems in pipes and open channels, which velocity is typically used for continuity, energy, and momentum equations?
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Avelocity at the pipe centerline only
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Baverage velocity of flow over the cross-section
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Cmean of the centerline and wall velocities
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Dnone of these
Answer
Correct Answer: average velocity of flow over the cross-section
Explanation
Introduction / Context:Hydraulic equations (continuity A V, energy head V^2/2g, momentum ρ Q V) rely on a representative velocity. Because real velocity profiles vary across a section, a defined average is used to ensure mass and momentum balances are satisfied.
Given Data / Assumptions:
- Fully developed or developing internal/open-channel flows.
- Use of control-volume formulations in design practice.
Concept / Approach:The correct representative velocity is the cross-sectional average, defined as V_avg = Q / A = (1/A) ∫ V_local dA. This ensures that continuity, energy, and momentum equations remain consistent. Centerline or wall velocities do not conserve mass or momentum when substituted directly without profile factors.
Step-by-Step Solution:
Identify governing balances where V appears (A V, V^2/2g, ρ Q V).Define V_avg from Q = ∫ V dA = A V_avg.Use V_avg in hydraulic formulae; apply correction coefficients if high accuracy is required for energy or momentum (α, β).Verification / Alternative check:For laminar pipe flow, centerline velocity = 2 V_avg; using centerline value would double the implied discharge, violating continuity.
Why Other Options Are Wrong:Centerline/wall values are point velocities; their mean is arbitrary and not physically founded for balances.
Common Pitfalls:Ignoring kinetic energy/momentum correction factors in strongly non-uniform profiles.
Final Answer:average velocity of flow over the cross-section