Regime channel design (Lacey) — mean velocity expression A stable alluvial channel is to be designed to carry discharge Q (m^3/s) with silt factor f according to Lacey’s method. What is the expression for the mean regime velocity V (m/s)?

Introduction / Context:
Lacey’s regime theory provides empirical relationships for stable alluvial channels carrying uniform silt. Among these, the mean velocity is linked to discharge and silt factor, enabling preliminary sizing before detailed hydraulic checks. Knowing the correct form helps avoid misapplication of the regime equations.

Given Data / Assumptions:

• Regime (quasi-equilibrium) alluvial channel conveying silt uniformly.
• Silt factor f represents bed material size influence.
• Empirical Lacey relations apply (Indian practice).

Concept / Approach:
One of Lacey’s velocity relations is V = (Q f^2 / 140)^(1/6), connecting volumetric flow, silt factor, and mean regime velocity. Other Lacey equations relate wetted perimeter, hydraulic radius, and slope to Q and f, but the velocity expression above is the standard form used for preliminary design.

Step-by-Step Solution:
Identify the correct empirical relation among the options.Select V = (Q f^2 / 140)^(1/6) as per Lacey’s set.Reject dimensional or exponent mistakes in distractors.

Verification / Alternative check:
Cross-reference with other Lacey relations (e.g., P = 4.75 √Q, R = 0.47 (Q/f^2)^(1/3), S ∝ f^(5/3) Q^(−1/6)) shows internal consistency.

Why Other Options Are Wrong:
(Q f/140)^(1/3) and 0.48 (Q/f)^(1/3) are forms from other regime relations (e.g., hydraulic radius), not velocity.
(Q^2 f^2 / 140)^(1/6) and the reciprocal form distort exponents/dimensions.

Common Pitfalls:

• Mixing velocity and hydraulic radius equations due to similar-looking exponents.
• Using non-SI constants without unit consistency.

Final Answer:
(Q f^2 / 140)^(1/6)