8. a 6 - inch diameter pipeline has water flowing at a velocity of 2.6 fps. what is the gpm flow rate…

Explanation:

Step1: Convert diameter to radius in feet

The diameter $d = 6$ inches. Since 1 foot = 12 inches, $d=6/12 = 0.5$ feet, and the radius $r=\frac{d}{2}=\frac{0.5}{2}=0.25$ feet.

Step2: Calculate cross - sectional area of pipe

The cross - sectional area of a circle is $A=\pi r^{2}$. Substituting $r = 0.25$ feet, we get $A=\pi\times(0.25)^{2}=\frac{\pi}{16}\text{ ft}^2$.

Step3: Calculate volume flow rate in cubic feet per minute

The velocity $v = 2.6$ feet per second. Since there are 60 seconds in a minute, the velocity in feet per minute is $v=2.6\times60 = 156$ feet per minute. The volume flow rate $Q$ (in cubic feet per minute) is given by $Q = A\times v$. Substituting $A=\frac{\pi}{16}\text{ ft}^2$ and $v = 156$ ft/min, we have $Q=\frac{\pi}{16}\times156=\frac{39\pi}{4}\text{ ft}^3/\text{min}$.

Step4: Convert cubic feet per minute to gallons per minute

We know that 1 cubic foot is approximately 7.48 gallons. So the flow rate in gallons per minute (gpm) is $Q_{gpm}=Q\times7.48=\frac{39\pi}{4}\times7.48$. [ \begin{align*} Q_{gpm}&=\frac{39\times3.14\times7.48}{4}\ &=\frac{39\times23.4872}{4}\ &=\frac{916.0008}{4}\ & = 229.0002\approx229.0 \end{align*} ]

Answer:

$229.0$ gpm