Critical depth definition — specific energy minimum in open channel flow The depth of water at which the specific energy is minimum is called the critical depth. Do you agree?
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AAgree
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BDisagree
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CAgree only for rectangular channels
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DAgree only when Froude number is zero
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EDisagree for wide channels
Answer
Correct Answer: Agree
Explanation
Introduction / Context:Specific energy E for open channel flow (per unit weight) is defined as E = y + v^2/(2g), where y is flow depth and v is mean velocity. The depth that minimizes E for a given discharge is the critical depth, a cornerstone concept in gradually varied flow and control transitions.
Given Data / Assumptions:
- Steady, prismatic channel with discharge per unit width q (for simplicity).
- Negligible energy losses across the control section.
- Hydrostatic pressure distribution.
Concept / Approach:For given q, E(y) = y + q^2/(2 g y^2) (rectangular channel). The minimum of E occurs where dE/dy = 0, yielding the critical condition. The associated Froude number Fr = v/√(g y) equals 1 at critical depth. The definition (depth at minimum E) is general and underpins control sections like weirs and flumes.
Step-by-Step Solution:
Write E(y) and set derivative to zero to find y_c.For rectangular channels, y_c = (q^2/g)^(1/3), and Fr = 1 at y = y_c.Thus, the depth corresponding to minimum specific energy is, by definition, critical depth.Verification / Alternative check:Plotting E vs. y shows a distinct minimum at y_c. Alternately, using Fr = 1 criteria confirms the same critical condition.
Why Other Options Are Wrong:
- Limiting to rectangular or wide channels is unnecessary; the definition of critical depth via minimum specific energy extends with appropriate geometric relations.
- Fr = 0 never occurs for flowing water; critical depth corresponds to Fr = 1, not zero.
Common Pitfalls:Confusing normal depth (from uniform flow) with critical depth; they coincide only under special conditions.
Final Answer:Agree