Question

If \(d=\) depth of flow, \(b=\) bottom width, \(\theta=\) angle of repose, then the most economical channel cross section for rectangular and trapezoidal section can be respectively expressed as _____.

• A. \(b=d^2;\; d=2b\tan\theta\)
• B. \(d=b/2;\; d=b\tan\theta/2\)
• C. \(b=2d;\; b=2d\tan\theta\)
• D. \(b=2d;\; b=d\tan\theta/2\)

Hint:

Important hydraulic efficiency conditions: For rectangular channel: \[ \boxed{ b=2d } \] For trapezoidal channel: \[ \boxed{ b=2d\tan\theta } \] Memory Trick: \[ \boxed{ \text{“Economical rectangle: width twice depth”} } \]

Answer 1

The Correct Option is C

Concept: In open channel hydraulics, the most economical channel section is the section that:
• Carries maximum discharge
• Has minimum wetted perimeter
• Minimizes frictional losses
• Provides maximum hydraulic efficiency The condition of maximum hydraulic efficiency is: \[ \boxed{ R = \frac{A}{P} \text{ should be maximum} } \] where:
• \(R\) = hydraulic radius
• \(A\) = area of flow
• \(P\) = wetted perimeter Different geometrical relationships exist for:
• Rectangular sections
• Trapezoidal sections These relationships are derived from hydraulic efficiency conditions.

Step 1: Most economical rectangular section. For a rectangular channel: \[ \text{Area} = A = bd \] and wetted perimeter: \[ P = b+2d \] For the most economical rectangular section: \[ \boxed{ b = 2d } \] This means:
• Bottom width equals twice the depth
• Hydraulic radius becomes maximum Thus: \[ \boxed{ \text{Rectangular section condition: } b=2d } \]

Step 2: Most economical trapezoidal section. For trapezoidal channels, the most economical condition is obtained when:
• Half the top width equals one sloping side
• Hydraulic radius is optimized For side slope angle \(\theta\): \[ \boxed{ b = 2d\tan\theta } \] This is the standard relation for the hydraulically efficient trapezoidal section.

Step 3: Writing the combined result. Thus: \[ \boxed{ \text{Rectangular: } b=2d } \] and \[ \boxed{ \text{Trapezoidal: } b=2d\tan\theta } \]

Step 4: Comparing with the options. Option (A): \[ b=d^2;\quad d=2b\tan\theta \] Incorrect relation. Hence: \[ \boxed{\text{Option (A) is incorrect}} \] Option (B): \[ d=b/2;\quad d=b\tan\theta/2 \] Not the standard economical section expression. Hence: \[ \boxed{\text{Option (B) is incorrect}} \] Option (C): \[ b=2d;\quad b=2d\tan\theta \] This exactly matches the standard hydraulic efficiency conditions. Hence: \[ \boxed{\text{Option (C) is correct}} \] Option (D): \[ b=2d;\quad b=d\tan\theta/2 \] Incorrect trapezoidal relation. Hence: \[ \boxed{\text{Option (D) is incorrect}} \] Final Conclusion: For most economical channel sections: \[ \boxed{ \text{Rectangular: } b=2d } \] and \[ \boxed{ \text{Trapezoidal: } b=2d\tan\theta } \] Hence the correct answer is: \[ \boxed{ (C)\ b=2d;\; b=2d\tan\theta } \]