Open-Channel Flow – Critical depth in a rectangular channel Water flows at Q = 10 m^3/s in a horizontal rectangular channel of width b = 3 m. Compute the critical depth of flow (in metres).

Introduction / Context:
Critical depth is the flow depth at which specific energy is minimum for a given discharge. In rectangular channels it is a key parameter for classifying flow (subcritical, critical, supercritical), locating hydraulic jumps, and designing control sections and flumes.

Given Data / Assumptions:

• Discharge Q = 10 m^3/s.
• Rectangular channel width b = 3 m.
• Steady, prismatic channel; gravity acceleration g = 9.81 m/s^2.
• Critical depth relation for rectangular channels applies.

Concept / Approach:

For a rectangular channel, let discharge per unit width be q = Q / b. The critical depth y_c satisfies q^2 = g * y_c^3. Therefore y_c = (q^2 / g)^(1/3). This comes from the condition Froude number = 1 and minimizing specific energy with respect to depth.

Step-by-Step Solution:

Verification / Alternative check:

Check sensitivity: if width were slightly larger (smaller q), y_c would drop. The calculated 1.04 m is consistent with a moderate unit discharge of about 3.33 m^3/s·m.

Why Other Options Are Wrong:

1.13 m and 1.45 m correspond to using g ≈ 10 or arithmetic errors; 2 m is far too large for the given q and would not satisfy q^2 = g y_c^3.

Common Pitfalls:

Forgetting to divide by width to get unit discharge, or using g = 9.81 incorrectly (e.g., mixing units). Some also mistakenly apply triangular or trapezoidal relations to rectangular sections.

Final Answer:

1.04 m