Fluid Mechanics: viscous flow in pipes

[Feodalherren]

Homework Statement

Homework Equations

ρ=789, μ=.0012

The Attempt at a Solution

From the energy equation we get hf=0.9

We know that hf=f(L/D)(V^2)/(2g)
[sorry don't know how to use latex after they removed the bar on the right]

Now I can substitute V for Q, but I'm stuck with f and Q as unknowns and I only have one equation. The solutions manual did something weird :

128μLQ/πρgd^4

It looks like they substituted f for something, but I can't figure out what. I only know how to look up f in the Moody diagram or 64/Re for laminar flow. Did they iterate it somehow?

 

[Chestermiller]

I'm confused over what you did. I assume hf is the head at the bottom of the 2 mm tube minus the head at the top of the 2 mm tube, correct? Can you write an overall macroscopic force balance for the fluid in the 2 mm tube? Let's see.

Chet

 

[SteamKing]

Homework Equations

ρ=789, μ=.0012

Units, units, units! Always show the units!

The Attempt at a Solution

From the energy equation we get hf=0.9

We know that hf=f(L/D)(V^2)/(2g)
[sorry don't know how to use latex after they removed the bar on the right]

Now I can substitute V for Q, but I'm stuck with f and Q as unknowns and I only have one equation. The solutions manual did something weird :

128μLQ/πρgd^4

This expression is derived from Hagen-Poiseuille flow:

http://en.wikipedia.org/wiki/Hagen–Poiseuille_equation

It's this equation which covers laminar flow conditions.

It looks like they substituted f for something, but I can't figure out what. I only know how to look up f in the Moody diagram or 64/Re for laminar flow. Did they iterate it somehow?

If you calculate the Reynold's Number for the flow, Re < 2300 indicates laminar flow, Re > 4000 indicates turbulent flow, and in between there is a transition zone as indicated on the Moody Diagram.

For Laminar flow, there is a direct relationship between f and Reynolds Number.

 

[Chestermiller]

OK. I went through the derivation of the equations, and got the relationship you wrote down as well as the value of 0.9 m for hf. Now, for laminar flow, just substitute f = 64/Re into your equation, with Re = ρvD/μ, and then solve for v. This is basically what SteamKing is recommending. Then, as he says, once you know v, check the Reynolds number to see if the flow is laminar. If it is, then you're done. If not, then you are going to have to solve it by trial and error. To get you started, there are analytical approximations to f as a function of Re for the turbulent flow region that you can use to provide an initial guess.

Chet

 

[Feodalherren]

Thanks guys! I'm still doing something wrong though,
this is what I get:

.9 = (64)(.0012)V / (789)(.002)(2)(9.8)

V = 362

Gives a WAY too high Reynold's number. The book has it listed at 795.

 

[Chestermiller]

Thanks guys! I'm still doing something wrong though,
this is what I get:

.9 = (64)(.0012)V / (789)(.002)(2)(9.8)

V = 362

Gives a WAY too high Reynold's number. The book has it listed at 795.

You should have a D^2 in the denominator, not a D. You're off by a factor of 500. Of course, if you had carried the units along with your calculation, you would have seen that immediately.

Chet

 

[Feodalherren]

I usually carry units but it's just too much work when there are so many of them to take into account. Anyway, stupid mistake. Thank you again. I don't know what I'd do without this forum.

 

[Chestermiller]

I usually carry units but it's just too much work when there are so many of them to take into account. Anyway, stupid mistake. Thank you again. I don't know what I'd do without this forum.

I don't feel like it's too much work.

Chet

 

[SteamKing]

I usually carry units but it's just too much work when there are so many of them to take into account. Anyway, stupid mistake. Thank you again. I don't know what I'd do without this forum.

Of course, if this problem were on a test, and you missed getting the correct answer becuz units ...

 

[Feodalherren]

I always include units on tests ;). I was trying to blast through this homework as I have a dynamics quiz due. I'll re-do everything in the weekend.