Just derived the volume flowrate through an elliptical tube with the Navier-Stokes equation. Fun.

[Unsickle]

Q=(-pi/4*mu)[(ab)^3/(a^2+b^2)]*(dP/dz)

Don't you just LOVE fluid dynamics class!!!

 

[Soybomb]

Thanks, now i'm sitting here twitching

 

[Adul]

Bzzzzwaap

the sounds of my brain loosing it.

 

[Capn]

I never really got that excited in my fluids class. But if it floats your boat dude.

 

[8008S]

At least define the variables for us!

 

[lowtech1]

Haven't had any use for fluid dynamics since I took it over 10 years ago.

But, hey! If it make your happy then congrats.

 

[Capn]

dp/dz is the pressure gradient, mu is the viscosity I believe. I'd figure that a and b are the semi-minor and semi-major axis of the ellipse?

 

[Unsickle]

pi == pi
mu == fluid kinematic viscosity
a == major axis of ellipse
b == minor axis of ellipse
dP/dz == difference in pressure between ends of the pipe, divided by the length of the pipe (pressure gradient along pipe)

Q == volume flowing past a cross section per unit time

if you let a=b=R then you get the equation describing flow through a circular pipe.

 

[Capn]

now just think what use this will ever have for you? Ah that's right none. If your boss wants you to calculate the flow rate through a elliptical tube then a) he assumes you're not too bright or b) it's not really an elliptical tube and he's telling you the customer needs a straw.

Take your pick!

 

[Unsickle]

The asymptotics are a little weird though...

if we let b=L and a=h, and allow L->infinity, then we should get something resembling flow between plates separated by the distance h. If you do the limits with l'opital it turns out that the flow rate predicted by the elliptical equation is (pi/4)[stuff] as compared to (2/3)[stuff]...

pi/4=.7854
2/3=.6667

That makes me think that the equation is wrong, since how could the flow through a superlong ellipse be more than the flow between two plates separated by distance h (which could essentially bound the ellipse.

The asymptotics for a circle are correct, however.

Anyways, the problem was a "what if" problem so it's not necessarily THe answer

 

[Capn]

Date of my thermal/fluids 2 exam ~ may 5th
Date I decided I didn't care about thermal/fluids ~may 6th

 

[Unsickle]

Ahh, I got it. Forgot to divide by 2 when I defined b=h, since b is only HALF of the effective height! That means that:

flow through infinitely elongated ellipse = .3927*[stuff]
flow through parallel infinite plates = 0.6666*[stuff]

which seems more realistic. What's the area ratio between an ellipse and a rectangle bounding the ellipse?

pi*ab/(2a*2b)=pi/4

what's the volume flowrate ratio between the elongated ellipse and the inifnite plates?

(pi/8)/(2/3)=3pi/16

Which is just enough to convince me that it's a good answer... the boundry effects of the ellipse as it closes in, even at infinite distance? shrug.

 

[Pretender]

And when will you ever need to calculate the volume flowrate? And why didn't you just ask some famous pipe-makers how their measure their flowrates.

 

[Capn]

Dude stop it already, you are going insane. Put down the text book and turn off your calculator. Call up your nearest wacked-out-engineer hotline.