Lacey’s silt factor from regime velocity and hydraulic radius: If V is the regime mean velocity (m/s) and R is the hydraulic mean depth (m), the Lacey silt factor f can be expressed as

Introduction / Context:
Lacey’s regime theory relates channel geometry, velocity, slope, and bed material through empirical equations. Besides the well-known relation f = 1.76 √m (m in mm), Lacey also linked hydraulic radius R and regime velocity V to the silt factor f. This question asks for that direct expression of f using V and R.

Given Data / Assumptions:

• Regime (quasi-stable) alluvial channel transporting its silt load.
• V in m/s, R in m.
• Lacey regime relationships apply.

Concept / Approach:

One of Lacey’s regime relations can be written as R = 5 V^2 / f. Rearranging gives f = 5 V^2 / R. Although f is also tied to bed material size, this form is useful when R and V are known from observations or preliminary design.

Step-by-Step Solution:

Verification / Alternative check:

Design examples in regime theory often back-calculate f from gauged R and V using the same rearranged formula. The value can be cross-checked against f = 1.76 √m to see if it aligns with expected bed material size.

Why Other Options Are Wrong:

(b), (c), and (d) are incorrect rearrangements lacking empirical support. (e) is a correct relation for f versus sediment size, not for f directly from V and R.

Common Pitfalls:

Mixing unit systems and constants; confusing the sediment-size definition of f with the hydraulic relation using R and V.

Final Answer:

f = 5 V^2 / R