Where is Manning’s formula applicable? Identify the correct application area for Manning’s equation relating velocity/discharge to roughness, hydraulic radius, and slope.

Introduction / Context:
Manning’s equation is a cornerstone of open-channel hydraulics, used to compute normal depth, discharge, and backwater profiles in rivers, canals, storm sewers running partially full, and lined channels.

Given Data / Assumptions:

• Steady, uniform (or gradually varied) flow in a prismatic open channel.
• Empirical roughness coefficient n represents boundary resistance.
• Hydraulic radius R = A/P; bed slope approximates energy slope for uniform flow.

Concept / Approach:

Manning’s formula: V = (1/n) * R^(2/3) * S^(1/2), Q = A * V. It is intended for free-surface flows where the hydraulic radius and slope characterize resistance. While sometimes adapted to pipes running partially full (treated as open channels), it is not the standard for full pressurized pipe friction (Darcy–Weisbach is preferred).

Step-by-Step Solution:

Verification / Alternative check:

Compare results with Chezy or Darcy–Weisbach adapted for open channels; consistent results validate the use of Manning in its empirical range.

Why Other Options Are Wrong:

(b) Manning gives velocity/discharge, not directly “head loss only.” (c) Pressurized full-pipe friction should use Darcy–Weisbach/Hazen–Williams, not Manning. (d) Compressible gas flow requires different models. (e) Unsteady surge needs waterhammer equations, not Manning alone.

Common Pitfalls:

Using an inappropriate n; assuming S equals bed slope in rapidly varied flow; applying Manning outside its empirical calibration range.

Final Answer:

flow in open channels