# How do you calculate the air friction on a large magnetically levitated ring travelling within a toroidally shaped vacuum enclosure?

For this question, assume...

• That the inner wall of the enclosure and the surface of the ring are smooth.

• A high vacuum level, but not a perfect vacuum. Rarified gas density is $$\rho$$.

• That the ring has a circular cross section.

• That the ring is large, so that its inner radius $$\approx$$ its outer radius. Therefore, it is reasonable to estimate the ring's surface area with $$(2 \pi r) (\pi w)$$, where: $$r$$ is the average of the inner and outer radii, and $$w$$ is the difference between the inner and outer radii.

• That the gap, $$d$$, between the inner wall of the enclosure and the surface of the ring is small compared to $$w$$.

• That the ring is travelling at a very high speed, $$v$$, such as 20,000 m/s.

I'm assuming here that it is possible to estimate the air friction from first principals, for example, by mathematically describing the physics of how the remaining gas molecules in the vacuum will transfer momentum as they bounce back and forth between the ring and the inner wall of the enclosure. But I'm open to any and all approaches.

Will the result be independent of the shape of the ring's cross-section?

Will the result be independent of the volume of the space between the ring and the enclosure?

If you know of a publication that provides an experimentally verified solution to this problem, please do share that!

Additional info requested in comments...

Air friction is one of the energy loss mechanisms that needs to be worked out to determine the energy consumption and operating costs of an inertially supported megastructure concept, such as an orbital ring, Lofstrom loop, or tethered ring. The sketch below was copied from an early paper by Keith Lofstrom on Lofstrom loops. In these kinds of proposed structures, the velocity of the magnetically confined mass stream is significantly higher orbital speed.

• It would be nice to add some sketch to explain your problem. Can you explain where the information about the speed of 20,000 m per second came from? Commented Dec 25, 2023 at 7:18
• I have to award the bounty by 3pm PST on the 27th, so there's still time to submit a better answer. See link to my paper in the comment below for hints. Commented Dec 27, 2023 at 19:36
• Thank you for the link to your paper (+1). Such flows are usually described using the Knudsen number. We must first estimate this number in order to determine the rarefaction of a flow. Commented Dec 28, 2023 at 7:58

## 1 Answer

Sketch of an answer:

Unwrap the whole thing: ignore the fact it’s a ring, approximate it with a straight duct. Also ignore the cross section shape, approximate it with a rectangle with the same area and same narrow side.

Ignore the part where the walls are moving relative to each other, approximate it with stationary walls and a moving gas (at half the relative speed of the walls)

What is the Reynolds number? What is the flow regime? There’s probably some pretty good approximate formulas for drag of flow in a rectangular duct (or: in a thin layer between walls), but they would only be valid for specific ranges of Reynolds numbers

Lastly, what is the mean free path of gas molecules relative to the gap? If it is larger than the gap, there are probably really accurate formulas for drag from first principles.

• I believe the free path of the gas molecules is long relative to the gap. If there are accurate formulas for drag from first principles, finding them and including a reference to them would help to make the answer more useful. Commented Dec 26, 2023 at 7:49
• @phil1008 I'm sorry, I can't tell you what the formulas are, just that from first principles they should exist and be accurate in a way that normal drag formulas aren't (esp: no need for coefficient of drag). The basic idea is that any time a molecule hits a wall it flies of in a random direction with a relative velocity given by the thermal (Boltzmann) distribution, and doesn't stop until it hits the other wall, at which point it does that again; you can work out drag from how much momentum is transferred between the walls. Commented Dec 27, 2023 at 8:25
• @phil1008 It may be easiest to simulate this using a monte-carlo type piece of code than to find the analytical formulas; you just need to simulate a few million collisions to work out the momentum transfer pretty accurately Commented Dec 27, 2023 at 8:29
• I did attempt to work this out and explained the math in this paper: project-atlantis.com/wp-content/uploads/2023/07/…. See "Energy Losses Due to Moving Ring’s Air Friction". However, I was hoping that someone could independently derive these formulas so that we could compare notes. Commented Dec 27, 2023 at 19:33
• Yes, the rate and nature of spallation in the scenario where molecules glance off a smooth fast-moving surface is another topic of interest. For this question, I just want to get an estimate of the energy loss rate. That said, the topics may be interrelated. Commented Dec 27, 2023 at 22:44