Answer:
10.22 m^3/s
Explanation:
To estimate the pressure drop in a pipe we use the Darcy-Weisbach equation
[tex]\frac{(deltaP)}{L} =\alpha *(\frac{density}{2})*(\frac{v^{2}}{D})[/tex] (equation 1)
With:
[tex]\alpha[/tex] = Darcy-Weisbach friction coefficient
L = length of duct or pipe
v = velocity of fluid
D= hydraulic diameter
Also flow rate is:
[tex]Q=v*A[/tex]
Where v is:
[tex]v=\frac{Q}{A}[/tex]
Area as a function of the diameter is:
[tex]A=\pi *\frac{D^{2}}{4}[/tex]
So
[tex]v=\frac{4*Q}{\pi*D^{2}}[/tex] (equation 2)
For a laminar regime the the Darcy-Weisbach friction coefficient is function of the Reynolds number (Re) as:
[tex]\alpha=\frac{64}{Re}[/tex]
[tex]Re=\frac{density*v*D}{u}[/tex]
With: v =velocity, D= diameter if the pipe and u= viscosity.
With this information alpha would be:
[tex]\alpha=\frac{64*u}{density*v*D}[/tex] (equation 3)
Replacing equation 3 in equation 1 we have:
[tex]\frac{(deltaP)}{L} =\frac{32*u*v}{D^{2}}[/tex]
And finally replacing the value for v in this equation we have:
[tex]\frac{(deltaP)}{L} =\frac{128}{\pi}*\frac{u*Q}{D^{4}}[/tex]
Clearing for Q we get an expression to estimate the expected flow rate in the pipe.
[tex]Q=\frac{deltaP}{L}*\frac{\pi}{128}*\frac{D^{4}}{u}[/tex]
We know
Delta P = 1.84 psi or [tex]\frac{lb}{in^{2}}[/tex]
L= 75 ft or 900 in
D for a 2 nominal schedule 40 PVC is 2.047 in. In tables you find External diameter and internal diameter. For calculations you use internal diameter (ID)
U for water at 20°C is [tex]2.034*10^{-5}\frac{lb*s}{ft^{2}}[/tex] or [tex]1.4113*10^{-7}\frac{lb*s}{in^{2}}[/tex]
[tex]Q=\frac{1.84\frac{lb}{in^{2}}}{900in}*\frac{\pi}{128}*\frac{(2.047in)^{4}}{1.4113*10^{-7}\frac{lb*s}{in^{2}}} = 6.24*10^{3}\frac{in^{3}}{s}[/tex]
So the flow expected for this pipe is [tex]6.24*10^{3}\frac{in^{3}}{s}[/tex] or [tex]10.22 \frac{m^{3}}{s}[/tex]
