Section 4: Soil and Water Conservation Engineering
Fluid Mechanics
Subtopic: Flow in open channels
Question Statement:
Two rectangular channels, A and B, join to form a large rectangular channel, C. Each rectangular channel is lined with the same material and has the same bottom slope. The bottom widths of Channels A and B are 2 m and 1 m respectively. If the flow depth in each channel is 2 m, the bottom width of Channel C (in m) is ________. (Rounded off to two decimal places)
Official Answer Key:
2.42 to 2.50
Step-by-Step Solution & Explanation
Same lining ⇒ same Manning’s n; same bottom slope ⇒ same S. From Manning’s equation, Q ∝ A R^(2/3). Continuity requires Q_C = Q_A + Q_B, so:
**(A R^⅔)_C = (A R^⅔)_A + (A R^⅔)_B**
Channel A (b = 2 m, y = 2 m):
A = 2 × 2 = 4 m²; P = b + 2y = 2 + 4 = 6 m; R = 4/6 = 0.6667 m
A R^⅔ = 4 × 0.6667^⅔ = 4 × 0.7631 = 3.052
Channel B (b = 1 m, y = 2 m):
A = 1 × 2 = 2 m²; P = 1 + 4 = 5 m; R = 2/5 = 0.400 m
A R^⅔ = 2 × 0.400^⅔ = 2 × 0.5429 = 1.086
Required for C: 3.052 + 1.086 = 4.138
Channel C (b = ?, y = 2 m): A = 2b; P = b + 4; R = 2b/(b + 4)
2b · [2b/(b + 4)]^⅔ = 4.138
Trial and interpolation:
b = 2.45 → 4.080
b = 2.50 → 4.198
Interpolating for 4.138: b ≈ 2.47 m
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