Critical depth and specific energy in open-channel flow At the critical depth of flow in a prismatic open channel, the specific energy (per unit weight) is at which extremum value?

Introduction / Context:Specific energy diagrams are central to open-channel hydraulics for analyzing transitions, controls, and critical flow conditions. Recognizing the extremum at critical depth enables design of flumes and weirs for flow measurement and control.

Given Data / Assumptions:

• Prismatic channel, steady flow, negligible energy losses over the control section.
• Specific energy E = y + V^2/(2g) for unit discharge q.
• Critical depth yc occurs when the Froude number Fr = 1.

Concept / Approach:

For a given discharge, the specific energy has a minimum at the critical depth. Mathematically, dE/dy = 0 yields the critical condition q^2 = g * A^3 / T for general sections, and for a rectangular section yc = (q^2/g)^(1/3). At this point, E is minimized, not maximized.

Step-by-Step Solution:

Verification / Alternative check:

On the specific-energy diagram, subcritical and supercritical branches meet at the lowest point where Fr = 1; any deviation in y increases E for the same discharge.

Why Other Options Are Wrong:

(b) Maximum occurs as y → ∞ for fixed q (E grows with y). (c) No averaging principle applies. (d) A definite extremum exists. (e) Bed slope does not set the extremum of E at a control section for a given q.

Common Pitfalls:

Confusing energy grade line (head losses) with specific energy; mixing critical depth with normal depth (which depends on slope and roughness).

Final Answer:

minimum