Regime channel parameters: If m is the average silt particle size (in millimetres), Lacey’s silt factor f varies proportionally to which power of m?

Introduction / Context:
Lacey’s regime theory relates stable alluvial channel dimensions to discharge and sediment properties. The silt factor f captures the effect of sediment size on regime depth, velocity, and hydraulic radius. Correctly identifying how f scales with particle size is essential for preliminary canal sizing.

Given Data / Assumptions:

• Average sediment size m in millimetres (representative of bed load/suspended size influencing regime).
• Application of Lacey’s empirical relations.

Concept / Approach:
Lacey proposed f = 1.76 * sqrt(m) when m is in millimetres. Hence, f is proportional to m^0.5. This scaling reflects that coarser sediments require higher velocities and greater hydraulic radii for mobility without deposition or excessive erosion.

Step-by-Step Solution:
1) Recall Lacey’s expression: f = 1.76 * m^0.5.2) Identify proportionality: f ∝ m^0.5.3) Compare options and select m^1/2.

Verification / Alternative check:
Dimensionless trends in field canals support higher velocities (thus higher f) for larger sediment sizes. The square-root dependence is consistent with Lacey’s empirical datasets.

Why Other Options Are Wrong:

• m, m^3, m^2/3, m^1/3: do not match the empirically established square-root dependence.

Common Pitfalls:

• Confusing Lacey’s f with Shields parameter or fall velocity exponents from other theories.

Final Answer:
m^1/2.