Rectangular Notch (Sharp-Crest) — Head–Discharge Relation In a standard sharp-crested rectangular notch, how does discharge Q vary with the head H measured above the crest, ignoring submergence and applying an appropriate coefficient?

Introduction:Weirs and notches are primary measuring devices for open-channel flow. For a sharp-crested rectangular notch, the discharge–head relation is a well-known power law that enables simple flow metering with head measurements.

Given Data / Assumptions:

• Free, fully aerated nappe conditions.
• Velocity of approach correction small or included in Cd.
• Rectangular, sharp crest; head H measured above crest.

Concept / Approach:The theoretical discharge through a strip of width b at depth y is dQ = b * sqrt(2gy) dy. Integrating from 0 to H, Q_theoretical = (2/3) * b * sqrt(2g) * H^(3/2). Real flow uses Q = Cd * (2/3) * b * sqrt(2g) * H^(3/2). Thus Q is proportional to H^(3/2).

Step-by-Step Solution:Set up elemental discharge using Torricelli speed sqrt(2gy).Integrate across depth 0 to H to obtain H^(3/2) dependence.Apply discharge coefficient Cd to account for contraction and viscosity.

Verification / Alternative check:Empirical calibrations confirm the 3/2 power for sharp-crested rectangular weirs under free overflow. Deviations arise mainly from approach velocity and submergence effects.

Why Other Options Are Wrong:Inverses with H are physically incorrect for free overflow; H^(5/2) appears with some combined relations but not the basic rectangular notch formula.

Common Pitfalls:Measuring head from the wrong datum; neglecting velocity of approach; using the formula under submergence without correction.

Final Answer:directly proportional to H^(3/2)