Standard rectangular weir discharge relation The empirical formula Q = m · L · √(2 g) · H^(3/2) for the discharge over a sharp-crested rectangular weir was suggested by which researcher?

Introduction / Context:
Sharp-crested weirs are among the most common flow-measurement devices in open channels. The classical discharge relationship for rectangular weirs is strongly associated with Francis, whose experiments established the widely used head–discharge law.

Given Data / Assumptions:

• Sharp-crested rectangular weir with crest fully aerated.
• Head H measured upstream sufficiently far from the crest.
• Coefficient m accounts for approach velocity and other corrections.

Concept / Approach:

The generalized form Q = m · L · √(2 g) · H^(3/2) is known as the Francis formula. It captures the 3/2 power dependence of discharge on head for a rectangular weir, with L as effective crest length and m as an empirical coefficient (close to unity with proper conditions).

Step-by-Step Solution:

Verification / Alternative check:

Cipolletti is linked to trapezoidal weirs with 1 horizontal to 4 vertical side slopes; Bazin and Rehbock provided other corrections, but the core 3/2 law for rectangular weirs is attributed to Francis.

Why Other Options Are Wrong:

(a), (c), and (e) are historically associated with different devices or correction schemes. (d) is incorrect since the attribution is well established.

Common Pitfalls:

Confusing Cipolletti’s trapezoidal weir with the rectangular Francis weir; neglecting aeration requirements near the nappe.

Final Answer:

Francis