Rectangular weir – Bazin’s formula constant m According to Bazin’s formula, the discharge over a rectangular sharp-crested weir can be written in the form Q = m * L * sqrt(2 * g) * H^(3/2). In this expression, what is m equal to (in terms of the discharge coefficient Cd)?

Introduction:
Empirical weir formulas express discharge as a constant times L * sqrt(2 * g) * H^(3/2), where L is the effective crest length and H is the head over the crest. Bazin’s form isolates a constant m that depends on hydraulic coefficients and edge geometry. This question asks for m in terms of the discharge coefficient Cd for a rectangular sharp-crested weir under free flow.

Given Data / Assumptions:

• Sharp-crested rectangular weir under free (not submerged) conditions.
• Velocity of approach neglected in the basic form.
• Standard definition H measured above crest.
• Cd encapsulates contraction and velocity effects for the crest.

Concept / Approach:

Starting from the elemental strip integration for weir flow and incorporating the discharge coefficient Cd yields Q = (8/15) * Cd * L * sqrt(2 * g) * H^(3/2). Comparing with Q = m * L * sqrt(2 * g) * H^(3/2) shows m = (8/15) * Cd. Many handbooks later package constants numerically (e.g., Francis’ 3.33 in US units), but in SI symbols Bazin’s decomposition cleanly separates the coefficient multiplier (8/15) and Cd.

Step-by-Step Solution:

Verification / Alternative check:

Dimensional factors L * sqrt(2 * g) * H^(3/2) carry the correct units of discharge; coefficients such as (8/15) arise from integrating velocity over depth assuming an ideal nappe shape and then adjusting by Cd.

Why Other Options Are Wrong:

(3/5) * Cd or 1.84 * Cd: Not the standard coefficient from the strip integration.(2/3) / Cd and Cd/(8/15): Invert Cd incorrectly and give wrong scaling.

Common Pitfalls:

Confusing the Bazin form with Francis’ empirical constants in specific unit systems; always compare symbolic forms to identify m.

Final Answer:

m = (8 / 15) * Cd