Flow Rate for a Closed Pipe

Problem solution was updated to correctly equation for angle of flow level in partially filled pipe, theta.

The pipe pictured below has an inner radius r of 18 inches and a flow depth h of 9 inches. Given a slope S of 1% and Manning’s constant n of 0.025, determine the flow rate Q in cubic feet per second. 

A) 7.4 cfs

B) 15.0 cfs

C) 4.8 cfs

D) 23.2 cfs


Answer: C) 4.8 cfs

Flow Depth:
$$ h=9in $$

$$ r=18in $$

Calculate the angle theta for a partially filled pipe:
\(\sout{\theta=2{\ast\cos}^{-1}\left(\frac{r-h}{h}\right)=2.09\ radians}\) 

\(\theta=2{\ast\cos}^{-1}\left(\frac{r-h}{r}\right)=2.09\ radians\)

Calculate area of flow:
$$ A={\frac{r^2\left(\theta-\sin{\theta}\right)}{2}}=1.38{\rm ft}^2 $$

Calculate wetted perimeter: 
$$ P=r\theta=37.7in $$

Calculate hydraulic radius:
$$ R=\frac{A}{P}=0.440 $$

Calculate flow based on Manning’s equation:
$$ Q=\frac{1.49}{n}{AR}^{2/3}S^{1/2}=4.76\frac{{ft}^3}{s} $$