Practice Question: Flow in a Partially Filled Drain Line

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Solution

Answer: A) 3.8 cfs

Manning’s Roughness Coefficient for a Cast Iron Pipe:
\[n=0.013\]
Depth of Flow:
\[y=8 \ in\]
Radius:
\[r=6 \ in\]
Unfilled Segment Height:
\[h=2*6 \ in-8 \ in=4 \ in\]
\[θ=2*cos^{-1}⁡(\frac{r-h}{r})\]\[θ=2*cos^{-1}⁡(\frac{6 \ in- 4 \ in}{6 \ in})\]\[θ=2.46 \ rad\]
Calculate Area of Flow:
\[A=πr^2- \frac{r^2 (θ-sin⁡ \ θ)}{2}\]\[A=π(6 \ in)^2- \frac{(6 \ in)^2 (2.46 -sin⁡ \ 2.46)}{2}\]
\[A=0.25 \ ft^2\]

Calculate Wetted Perimeter:
\[P=2πr-rθ\]
\[P=2*3.14*0.5ft-0.5 \ ft*2.46 \ rad=1.91 \ ft\]
Calculate Hydraulic Radius:
\[R=\frac AP\]
\[R= \frac{0.25 \ ft^2}{1.91 \ ft}=0.13 \ ft\]

Calculate Flow Based on Manning’s Equation:
\[Q=\frac{1.49}{n} AR^{2/3} S^{1/2}\]\[Q=\frac{1.49}{0.013} (0.25 \ ft^2 )(0.13 \ ft)^{2/3} (0.01)^{1/2}\]
\[Q=0.73 \ cfs\]

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