frictional loss calculation of a fluid between a rotating disc and a wall

[Cheetos]

I'm looking into the frictional loss of fluid between a rotating disc and a wall. Many works have been done for rotating cylinder inside a stationary tube, so I was trying to use that to come up with an equation to estimate the frictional loss.

s - the uniform gap between the disc and the wall
r2, r1 - OD and ID of disc
w - constant disc rotation

Not sure if anyone has done this before, but would love to get feedbacks on my calculation. I'm not so sure about my double integral and also my last integral.

 

[btrueblood]

Your equations appear to assume laminar flow, which may or may not be correct.

There is a famous paper by Theodore Von Karman, written in 1921 (in Aachen, so it's in German), that covers the topic of drag on a rotating disc, both as a numerical solution (one of the few problems involving the full Navier Stokes equations that can be directly solved with classical methods), and with comparison to test data. An English translation of that paper should be available in NASA archives.

Or you can look at the wikipedia article here:

https://en.wikipedia.org/wiki/Von_K... flow is,for centrifugal fans or compressors.

 

[Cheetos]

Thanks @btrueblood for the direction. This was helpful. I was reading Saari's Thermal Analysis of High-speed Induction Machines. He referenced Daily and Nece 1960, which addressed laminar and turbulent friction coefficient. I was wondering if you have come across this before. Regime I and laminar equation given in von Karman are pretty similar. Regime III and turbulent equation and von Karman are a little different.

For Saari's reference, I'm a little confused with s/r1 term in the equations. In the figure description and y-axis of the graph, it uses s/r2 as spacing ratio, but the equations all use s/r1. I wonder if that's a typo.

 

[btrueblood]

I think I agree that it's a typo, and would trust the equations until proved otherwise.

The turbulent b.l. equations will vary depending on the turbulence model used (and there are a plethora of them). For all of the research over the years since Von Karman, there is a lot of conflicting information in the literature, and things get weirder as you discuss shrouding the disk, and heat transfer. Confusing, at the least.

There are also a lot of papers written by a prof. Owens, J., at Bath University in England regarding shrouded disks, dating from the 80's up to the early 2000's at least. He's likely retired by now, but may have a replacement continuing his work (he collaborates (-ed?) quite a bit with Rolls Royce). Was helpful when I contacted him back in the early 2000's, though we didn't have sufficient funds at the time to hire him as a consultant.