Specific energy in a rectangular open channel: the minimum specific energy occurs at the critical depth hc. Which expression is correct for this minimum value?

Introduction / Context:
Specific energy E in open-channel flow is a key concept for analyzing transitions, critical flow, and controls such as weirs and flumes. For a rectangular channel, the minimum specific energy occurs at the critical depth hc, establishing an important design and diagnostic criterion.

Given Data / Assumptions:

• Rectangular channel of width b carrying discharge Q.
• Specific energy: E = y + V^2/(2 g) with V = Q/(b y).
• Critical depth hc gives minimum E for fixed discharge.

Concept / Approach:
Write E(y) = y + (Q^2)/(2 g b^2 y^2). Differentiate E with respect to y and set dE/dy = 0 to find hc. The condition leads to V_c^2 = g hc and hence hc = (Q^2/(g b^2))^(1/3). Substituting back into E gives Emin = (3/2) hc.

Step-by-Step Solution:
E(y) = y + (Q^2)/(2 g b^2 y^2).dE/dy = 1 − (Q^2)/(g b^2 y^3) = 0 at y = hc.Thus Q^2/(g b^2) = hc^3; and V_c^2 = g hc.Emin = hc + (V_c^2)/(2 g) = hc + (g hc)/(2 g) = (3/2) hc.

Verification / Alternative check:
Plotting E versus y at constant Q shows a U-shaped curve with its minimum at y = hc and E_min/Evaluated ratio of 1.5, confirming the formula.

Why Other Options Are Wrong:
hc, 2 hc, (1/2) hc, hc/3 do not satisfy the extremum condition derived from E(y); they misrepresent the balance between depth and velocity head at critical flow.

Common Pitfalls:

• Confusing specific energy with specific force (momentum function) used in hydraulic jump analysis.
• Using total head instead of specific energy (which is measured relative to the channel bottom).

Final Answer:
Emin = (3/2) * hc