Chezy’s Formula — Correct Expression for Discharge in Open Channels According to Chezy’s formula, which expression correctly gives the discharge Q through an open channel, where A = area of flow, C = Chezy’s constant, m = hydraulic mean depth (hydraulic radius), and i = bed slope?

Introduction:
Chezy’s relation is one of the earliest uniform-flow formulas. It connects the mean velocity in an open channel to the hydraulic radius and bed slope through an empirical constant C, enabling designers to estimate discharge for given geometry and slope.

Given Data / Assumptions:

• Uniform, steady open-channel flow.
• Prismatic channel with constant roughness and slope.
• Hydraulic mean depth m = A / P (also called hydraulic radius R).

Concept / Approach:
Chezy’s velocity formula is V = C * sqrt(m * i). Multiplying by area yields the discharge: Q = A * V = A * C * sqrt(m * i). This expression shows that Q scales with the square root of the product of hydraulic radius and slope, not linearly.

Step-by-Step Solution:
Start with V = C * sqrt(m * i).Compute discharge: Q = A * V.Therefore, Q = A * C * sqrt(m * i).

Verification / Alternative check:
Manning’s formula V = (1/n) * m^(2/3) * i^(1/2) also implies a square-root dependence on slope, consistent with Chezy’s structure.

Why Other Options Are Wrong:
Linear forms with m * i (options A, B, D) contradict the square-root relationship and have incorrect dimensions for velocity before multiplying by A.

Common Pitfalls:
Confusing hydraulic mean depth with flow depth; treating C as constant across roughness/scale without calibration.

Final Answer:
Q = A * C * sqrt(m * i)