What Is the Minimum Diameter of a Duct to Limit Head Loss to 20 Meters?

[proponents]

Homework Statement

Heated air at 1 atmosphere and 35 deg. Celsius is to be transported in a 150 meter long circular plastic duct (smooth) at a rate of 0.35 cubic meters/sec. If the head loss in the pipe is not to exceed 20 meters, determine the minimum diameter of the duct.

Homework Equations

hlt (total head loss) = hl + hlm (major and minor head loss)

there is no minor head loss

hl=f*(L/d)*[(v^2)/2] - (friction factor times length over diameter times velocity squared over 2)

V=Q/A

Re= ro*v*d / meu

The Attempt at a Solution

Okay so first I tried to calculate diameter using a combination of the major head loss equation and the Q=VA eq. I got a huge mess involving large calculations to the 1/5th power. I don't that that was the right approach.

I also got the density and dynamic viscosity of air at 35 deg-

ro(air@35deg.C)=1.15kg/m^3
meu(air@35deg.C)=1.88E-5

Okay so then I thought I could get the Reynolds # but I am missing velocity and diameter.

NEXT- I figured- there is no head loss because (being that the pipe is smooth) there are no losses to friction and there are no components (valves, elbows etc) so there are no major or minor head losses. Where would they come from if not from friction? So now I am a little lost with this one. Am I applying the wrong formula? Not having the velocity OR diameter is really messing me up. Any help is appreciated. Thanks

 

[proponents]

I have also tried the Swamee-Jain formula... but that does not apply to smooth pipes. I can't think of any other way to get diameter when you only have Q, hl, L, and the properties of the fluid...

 

[MechanicalMan]

Do what you were doing in the first attempt, with V = Q/A. Keep in mind that A = (pi/4)*D^2

 

[proponents]

Well like I said earlier, I ended up using the Swamee-Jain equation form 3, and calculated the diameter. But, since the Swamee Jain eq is only accurate for pipes with a relative roughness greater than smooth, I needed to check this. I set up a spreadsheet with hundreds of diameters. Then calculated the Reynolds number and friction factor for each (friction factor using the Head Loss eq.). I then plotted these on the Moody Chart. Where this curve intersected with the curve for smooth pipes, I had my diameter. It matched the diameter I calculated courtesy of Swamee and Jain.

 

[MechanicalMan]

Seeing how you don't have velocity or friction factor leads me to believe that this might be an iterative solution.

Assume a value of velocity, use that to find your Reynolds, find your friction using the Colebrook equation. Find the diameter. Rinse and repeat until you get some sort of convergence.

 

[MechanicalMan]

My piping systems design textbook says that for a smooth pipe, you can take your roughness e = 0.0015mm.

Try that and see what kind of answer you come up with.