Pipe-flow formulae (SI units): Which expression correctly represents the Hazen–Williams velocity relation for full pipes?

Introduction / Context:
The Hazen–Williams equation is an empirical relation widely used for turbulent water flow in full pipes. In SI form, it provides average velocity or discharge in terms of hydraulic radius and slope, using a material coefficient C.

Given Data / Assumptions:

• Water at ordinary temperatures; full circular pipe flow.
• Hydraulic radius R and energy-grade-line slope S are known.
• Coefficient C reflects internal roughness (dimensionless in SI usage).

Concept / Approach:
In SI units, a common form is V = 0.85 * C * R^0.63 * S^0.54, with V in m/s, R in m, and S dimensionless. The constant 0.85 reconciles unit systems from the original U.S. customary form. The equation is empirical and valid for water (not other fluids) in typical municipal ranges.

Step-by-Step Solution:

Verification / Alternative check:
Alternative discharge form: Q = 0.278 * C * A * R^0.63 * S^0.54, where A is area in m^2 and Q in m^3/s; both are consistent if V = Q/A.

Why Other Options Are Wrong:
Option b introduces an undefined “H”; option c changes the exponents; option e misplaces the constant as 1.85; “None” is invalid because a correct expression exists.

Common Pitfalls:
Mixing SI and US constants; applying Hazen–Williams to fluids other than water or outside recommended Reynolds number ranges.

Final Answer:
V = 0.85 C R^0.63 S^0.54