Internal Shielding for Space Habitats

 This is a direct followup to my last post, where I ask "what specifically do we get in the yellow region?" That is, what works for the "small" habitats?

Further fueling this, I got a good comment from the internet on the material-specific shielding needs.

"But metals are actually about the worst possible GCR shielding material, they create enough secondary spallation radiation that two tonnes per square meter of solid aluminum is ... actually worse than no shielding!"

and the given pdf link

To recap, I gave 3 regions of space habitats based on the physics.

1. yellow - so small the pressure vessel by itself is insufficient for shielding
2. green - pressure vessel is sufficient(?) for shielding and thermal conduction
3. blue - pressure vessel is so big that heat removal becomes the limiter

It is most likely to be made out of Aluminium just due to what we know is available on the moon, and in that (realistic) case, the yellow region intrudes into larger R values. What I thought was the "green" region actually has insufficient shielding. The solution in both cases is the same - to add more shielding!

The Problem with Adding More Shielding

My goal was to describe a habitat where the shell covers a number of functions, those being:

• (p) pressurization of the habitat
• (h) heat removal from the habitat (the shell _is_ the radiator)
• (r) radiation shielding from all kinds deep space radiation **needs solution**

The problem with adding additional (r) to the outside is that (h) is then blown! This is a philosophical engineering contradiction. It makes me think of the baseball "hand over hand" game. Shielding wants to be on the outside, problem! Now the radiator wants to be on the outside! This isn't just limited to my weird ideas, but a real general strangeness, I asked about it for _micrometeroid protection_ here, which again, is another thing that wants to be on the outside!

To re-state the obvious, if you surround the radiator with shielding, the radiator no longer works.

Alternative - use some kind of fluid (heat pipes) or active heat transfer to go from the surface of the habitat, past the shielding, to a separate radiator on the outside of the shielding. This just loops us right back around to (p), (r), (h) all being separate functions. Can we do any better?

Introduction of Internal Shielding

So let's keep the radiator on the outside, which is the same as the pressure vessel, which is (p) + (h). And we will separately solve the problem of radiation shielding (r), because this does have more flexibility. It physically matters where we put the radiator, but it doesn't exactly matter where we put the shielding. As long as radiation gets stopped sometime before it gets to the humans, we have done our job.

Put the shielding inside the pressure envelope. This was also the size range in which we said that we introduce a _new problem_ of _thermal_ shielding, so both of these structures will be inside, one enveloping the other. It's not obvious which would go in the most-inner location. However, I drew a nuclear power plant in various pictures, so I will put the shielding on the outside, so that the plant can get a lower temperature heat sink without a higher dose. In general, not exposing people working on the thermal barrier to radiation would be nice.

You could also partially coat the _habitat exterior_ in a thermal barrier, since the problem is basically that the radiator works _too well_ in this configuration. Actually, that might help to declutter some of this diagram, so I'll give that snapshot here. It's also useful, because, conceptually, if you get this, you get the overall thing going on with thermal.

That really does help to declutter things. The "thermal blocker" might just be white paint, which is sufficient to decrease the emissivity of the surface. Better, would be "blinds" that could increase or decrease the thermal emissivity on-demand. This would give a thermostat for the space station. The design for this (even though in vacuum) is dramatically more simple than thermal insulation in the atmosphere, so it's hard to see how anyone would choose the in-atmosphere idea. After all, people keep talking about how the vacuum is such a good insulator... this leverages physics more effectively.

This doesn't yet fill in the details of what we would have at the radiation barrier. Because if nothing else, we need heat removal to go through the radiation barrier - bringing us to the central problem here. Can you block radiation, but not block air and movement of people and other things? Yes.

Geometries for Optical Blocking

I know that radiation is more complicated than just this, but to a first approximation you can think of radiation as traveling in a straight line. So bear with me, and let's first focus on the purely "optical" problem. For heat removal, we need air to move freely through the shielding, so it needs to be porous. But no straight line should be able to go through this. This isn't actually fancy or difficult to a first approximation.

First draft internal shielding design - shielding materials are solid lines

This works. In the diagram, the radiation shielding is illustrated with straight lines. There are 2 layers here. One layer has gaps in it, with shielding going perpendicular to the wall, but not all the way to the next layer. The next layer just has gaps and shielding. As long as you can not draw a line that goes straight from one side to the other, this accomplishes the basic concept. This is a 2D sketch, and the 3D version would just require symmetry going out of the page, and that should work.

This comes with additional costs in terms of:

1. constricts flow area of air, which is also our coolant, meaning higher head or higher Delta T
2. needs more shielding material

Exactly _how much_ extra it costs in both cases is interesting, and I have ballparked some numbers. For (1) I got that we preserve 33.3% of the flow area (losing 2/3rd). Before I tried this, I tried it with holes, with a 3-layer design.

holes-and-patches concept for porous shielding

This holes-and-patches idea seemed to get 20% flow area, which is worse. But it is also interesting that the better you do for (1), the worse you might do for (2), and I think this is a legitimate tradeoff in these two geometric examples. That might not necessarily hold for all designs, and truly, I believe some designs may be better in both categories than the ideas I've given here.

I found some form of prior art to say, yes, you can do better. I got some weird places with the AI suggestions. The names get truly weird, like "Gyroid Triply Periodic Minimal Surface". And it was hard to tell if they were still obeying the "optical" blocking idea. But for this one I have relatively good confidence.

https://kenbrakke.com/evolver/examples/periodic/periodic.html

specifically

https://kenbrakke.com/evolver/examples/periodic/gyroid/gyroid.html

This is said to get >80% flow area. Intuitively, I strongly believe that the boxy diagram I drew above can be beaten.

Conclusions

But in any case, we are very good here to say positively that this is possible. Let's bring it back to answer some likely questions - are these segments I drew just floating in air? Mostly. This is microgravity. It just needs some minimal tethers to hold them in place. Next - could people (and possibly cargo trucks) just float through this? Yes, that's the idea. I'd assume there will be hand-holds, or guide tethers to help them.

Next big question - where does the optical assumption break down? Well your shielding would still be primarily basaltic oxides and glass-ceramic from the moon. But in addition to that you need an air gap, plus some low-Z material, and then a thin layer of special neutron absorbers. These are all still relatively abundant from lunar sources. The only real potential conflict with this design is the large size needed for the gap. However, you could just repeat the shielding pattern twice (different materials in each) to give yourself an abundantly large gap. So I would say these constraints could hurt our metrics (flow area, shielding size), but don't conflict fundamentally.

So, I like it. I want to keep it as a decent reference design.

What sizes? Referring to the prior diagrams, starting at sizes of R=0.1 km = 100 meters, it seems kind of plausible. You might not have full flow-dividers for gravity modules, but I expect you could have some kind of gravity module in some sense to stave off bone loss, and I could see this still fitting inside of internal shielding. We haven't really ruled windows, but getting light through the shielding will give extra challenges.

Then for max size, I think it's best to assume some Aluminum given lunar materials for the structure. Going by the argument that it's just not good for shielding at all, we might still need internal shielding, even at R=30km, and maybe even bigger.

Bare Metal Sphere Habitat

 This post will show "convergence of functions" for a space habitat, presenting a design that is surprisingly simpler than expected - the bare metal sphere habitat! The first thing I have to address here is the categorization of functions that a habitat must provide.

Space Habitat Requirements

Hard requirements are non-negotiable things with a physical basis. In this classification, there are 5 of these, aside from the noted location-specific caveats. For any particular design, we want labels for (p), (r), (e), (h), and (g). Otherwise we don't really know what we're looking at. The payoff is that some of these functions converge onto the same structure in some designs.

• (p) Pressurization to 20 to 100% of sea level so humans can breathe
• Much agriculture requires N2 air content, pointing to the upper end of that range (out of scope here)
• (r) Radiation shielding of 2 to 10 tons per square meter
• caveat: some LEO locations can meet requirements with much less
• (e) Provision of energy, could be from external solar, or internal like nuclear
• (h) Heat removal, all energy from prior point plus any radiative ingress from sunlight
• (g) Artificial gravity via rotation

The key distinction is when multiple requirements are served by the same structure, and where they are not. Then, for my own original ideas to be presented with that categorization applied. Why? Because this illustrates the whole point, as functional groupings are *different* in different designs, and I have a very different grouping (compared to prior art) that I want to present to the world.

Habitat Design Families and Lewis One

I'm a little obsessed with the Lewis One space station concept. If you compare to other ideas like Island Three, Lewis One separates functions that might otherwise have been integrated. Specifically, Lewis One separates the shielding into an outer envelope. If you look at its literature, you'll find computer graphics from 1991. It's a bit of a shame that I couldn't find any updated drawings. The internet has maybe three images, not enough to understand what's going on at a glance. So here is my humble redraw.

Later, many of the same people pitched Kalpana One, which goes back toward the classical “everything rotates” design, like 2001's Space Station V. We haven’t built rotating habitats at all, so there is no path dependence yet. Any commercial station currently taken seriously is selling microgravity, not the opposite. At some future point I am convinced we add back in artificial gravity. At this point, absolutely nobody knows how that will happen because it is not anywhere in major current space priorities, private or public. You might start with bearings between the rotating part and other station modules, like Nautilus-X, which was a serious idea. Even assuming that type of thing, it's not easy to say what the next step is. That's why it's a great time to talk about this now - before the industry is ready to talk about this in the first place. That begs the question of what we should be assuming. What is scarce, and what is valuable? By the time we are able to build these... maybe mass-optimization isn't as big of a deal. My philosophically more complex motivation is that, if ASI arrived tomorrow, do we have something worth asking them for? Maybe everything we have yet seriously imagined is too modest. Maybe.

Now, put Lewis One and a classical fully-rotating design into this requirements language. I’ll use “Kalpana One” as the name tag for the classical design family.

Comments directly on these diagrams:

• Whether or not the "external solar" is co-rotating or not depends on design. The Kalpana One writeup makes a surprising choice of remote power transmission.
• Also note that both designs require extra shielding. In Lewis One, this just happens to be separated & stationary.
• Exactly how heat removal and power transmission in Lewis One makes it into the rotating pressure envelope is not spelled out in the Lewis One writeup.
• Lewis One is providing an extra pressurized microgravity habitat inside the shielding. You could argue Kalpana One and others have a near-zero-gravity environment in the center. I don’t buy that as equivalent.
• This is all as-reduced-as-possible, only containing hard requirement features for the most part. Mentally picture these being shiny space habitats crawling with robots and spaceships.
• The "grav module" wording comes from Lewis One. I call the analog structure in this blog just "tubes". I switched my wording to that here. Just temporarily.

Bare Metal Sphere Functions

Now let's get to the "what if" of this all. What if we throw everything away and start all over from the start. We need to hold in pressure. Physics students will make a sphere, so do that. Easiest to assume steel, if not aluminum or something else you can send from a mass driver on the moon. Those have good heat conduction.

We have covered (p), and next up, we ask: do we even need (h), or (r)? Specifically, is there a parameter space where the metal sphere we already imagined (because we have atmosphere) can take care of these functions? This is not obvious, because they go in opposite directions - thicker walls, better shielding, but worse conductivity.

For a first-pass baseline to keep the math clean, assume:

• deep-space thermal conditions (think Pluto vicinity), i.e., minimal radiative ingress from sunlight
• an internal power source, so (e) is satisfied by a nuclear reactor

Now for the hard part, we have one item left out - (g), provision of gravity. Well, that is the subject of this blog. Read my introduction post (link on right) for a basic description of the mechanism, but the idea is that you can rotate a tube inside of the microgravity atmosphere, but you need to add multiple shrouds to have the flow be managed and well-behaved. There are open questions related to how you maintain placement of those shrouds as I described in some recent posts, but I am very serious about proving solutions with experiments. I have little doubt that it is possible, and that is what I am here to convince the world of. So, that's where (g) is satisfied, and the completed diagram is below.

This differs from the pie-in-the-sky idea I've presented before where pressurization (p) is satisfied by rock weight around the sphere.

So with this formalization, I will acknowledge the advantage the bare metal sphere can have over a gravity balloon. The wall thickness is ~10 meters for bare metal sphere, but ~10 KILOMETERS for the gravity balloon. You can get away with conduction in the first case (numbers given below) but not for the second case in a million years. This requires that gravity balloons have some form of active heat removal. See the Orion's Arm article on gravity balloons, which shows a radiator. The large walls are why this is the canonical (and fair) portrayal.

Numerical Analysis of Viability Range

None of what is written here is from AI, but I am now using AI significantly to more quickly arrive at the answers I'm looking for. So here is my folder with details for the analysis, made by Codex / ChatGPT and my prompting.

https://github.com/AlanCoding/gravitational-balloon-mathematics/tree/master/bare_metal_sphere

Referring to the above diagram, there are 2 questions we are asking.

• At what point will the wall be thick enough to cover radiation shielding all on its own?
• At what point will the inhabitants be generating so much heat (due to increasing volume-to-surface area ratio) that the heat cannot be rejected fast enough?

To accomplish this, we have to start putting in specific numbers. Some are simple hand-waves, like using 0.8 for emissivity. Possibly the most complex one to pin down is the heat produced per volume, which comes from assumptions about the society that lives there. I will not go too far into justification, but here is where my spitball number comes from.

$$ q''' = \frac{23{,}000\ \mathrm{W}}{\mathrm{capita}} \cdot \frac{1{,}000\ \mathrm{capita}}{\mathrm{km}^3} $$

$$ 1\ \mathrm{km}^3 = 10^9\ \mathrm{m}^3 \quad\Rightarrow\quad q''' = 0.023\ \mathrm{W/m^3} $$

This is to say: 23 kW per person and 1,000 people per cubic kilometer. Convert units and it becomes 0.023 W/m^3, which is the `q_expected` parameter in the scripts. Shielding is set at 2 tons / m^2.

Put these into the scripts (python -m hab_sphere.numeric_summary --epsilon 0.8 --q_expected 0.023 --mu_req 2000), and getting specific numbers:

• steel
• shielding min: 0.63 km
• thermal max: 26.6 km
• Al
• shielding min: 2.0 km
• thermal max: 36.7 km

This is our first good news! It was not obvious at all that the constraints would "agree" with each other at all. The first number didn't have to be smaller than the second, but it is. Al has better thermal conductivity which is mostly the reason for the difference according to materials.

This is the literal convergence: (p) and (r) are served by the same metal wall at around the kilometer scale.

To give better sufficiency for this analysis, here are the "good" and bad regions plotted:

Neither the yellow or blue regions are fully idea-killers. If the thing is too small, you just need to add extra shielding - and this is exactly what Lewis One is doing (with some other differences). If you are in the thermally-limited zone, then you either need to generate less heat, or make the sphere bigger. Making the sphere bigger in this case might be "wasteful" of materials, but this would be judging prematurely, not understanding the true constraints of our future (possibly post-abundance) society. The ultra-large scales start to describe something more like the world of Virga, where distances between tubes become vast by necessity of heat balance.

Variations on the Bare Metal Sphere

This still needs additional scrutiny. In the good region, we find that the radiator is actually too good. In this case, we would need to add an insulator so that the temperature of the air does not drop too low.

Thermal power plants, inside the atmosphere, would prefer to exchange heat with the walls for efficiency, no matter how uncomfortably cold that is.

What if you wanted to use solar instead? It would be fairly straightforward to add penetrations to the sphere to run wires. After all, this is a non-rotating structure, and you would probably align its orientation with the light source. However, this in any configuration other than perfectly shielded from the sun will decrease the heat rejection capability. If you do extend a radiator outward, it would then be in the penumbra direction. The graphs and numbers here are kind of best-case, if around Pluto or something.

Yes, the temperature of the inner surface of the sphere must be slightly lower than the air due to convective losses, and this should be accounted for in a more accurate script (it is not now). However, I have published many blog posts on how to "globally" circulate the air, so I believe this is only a local problem and solvable in-atmosphere, making it vastly easier and "ordinary" engineering which is what we want.

The next predictable concern is whether increasing to mega-scale sizes might actually decrease the total amount of heat you can produce in the interior due to increasing wall thickness at some point. This does not appear to be the case after running the numbers.

You can see you can make it bigger and bigger, and still put more people in the habitat. The trend appears to continue forever, and breach at least the PW level. Breaching the TW level happens at only (lol) ~200 km radius. Considering the number of people this could house (43 million), that might not be unreasonable.

As a technical note, the mass of both air and metal wall scales linearly with the volume. This is because structural support (assuming some strength value) scales with the (pressure)x(volume) product. That means that, given the material, the habitat requires a constant ratio of air to metal, which is on the order of ~3 for steel with engineering margin. Once you start to think about this, it reveals that if we have mass drivers sending mass from the moon, the air materials, specifically nitrogen, quickly becomes the limiting factor. This is an interesting detail, but not in scope here.

The next objection I'd predict is that my math is wrong. Well, let's look at the breakdown and ask if it looks intuitive.

What we're seeing here is that the low-R regions are under-powered compared to what the steel can handle as a radiator (surprised me). But once we reach the thermal-limited range, we have to allow more temperature drop across the wall due to its increasing thickness. This hits hard due to the T^4 term. At the extreme values of R > 200 km, it would become extremely profitable to add some other heat rejection methods. However, these do not necessarily have to be active. A passive means to increasing heat rejection would be to mix some thermally conductive materials with the structural materials at the cost of a bit more materials.

But overall, I rate the overall idea as almost trollishly effective. Like the gravity balloon itself, I'm sure the reason other people haven't seriously put it forward is because of the apparent uselessness of a large volume of air-filled microgravity. To this, I have a very simple answer, which is to use the flow-dividers to add whatever gravity tubes you want inside of it. This is flexible and evolvable. By using metal structural pressurization, we allow a bare metal sphere hab to be built in cislunar space from mostly lunar materials, which can open up days-scale travel time to a place that has an actual shot at offering an experience, in the long-term, better than suburban existence on Earth, to put it in summary.

We also shouldn't ignore the "vibes" factor of it all. A great big metal sphere with a question mark for what goes inside feels very messy in a good way, similar to a cell of biology. This allows for multiple layers of governance, which is something you want when multiple millions of people are involved. Gravity structures are possible in engineering & economics-wise via flow-dividers, which is important due to human biology. This is more exciting IMO than other designs which say "made your world, here you go!" Starting with an atmosphere and re-arranging the interior ad-hoc feels more like Alpha from Valerian than a sterile space stations. Even extending beyond the pressure envelope, the idea has decent resilience to revisions. Put in literal windows? Should be possible. Docking should be taken for granted, which is large penetration, and might need material reinforcement around it, but that's all.

Broad Strokes of a Physical Test Plan

Simulations I presented in last post are grasping at something that can not really, realistically, be achieved. Even if I were to put in a turbulent model, account for momentum correctly, it would be very difficult to allow deformations of the flow-dividers.

I believe the answer to this is physical experiments, which are already fairly common in the adjacent research space. As I've gotten further into the topic, I've realized that the Russian doll type velocity staging is weirder than I originally thought, and actually non-trivial in its implementation. The core arguments hold, but the lack of similar applications on Earth leave us with a surprisingly empty engineering space, in terms of background literature. So the next logical step is to start experiments.

Reynolds Number Ranges

Before we even add flow dividers, we are going to pretend that we are doing basic Taylor–Couette flow, which is just a rotating drum inside a larger cylinder. In all numbers I'm giving here, I will not do anything fancier than that.

For the table, I am going to select (describe) a particular physical thing, like a bucket. I know what size bucket I can buy, so I will start with an outer radius, $r_o$ from the available product, I can potentially get. Define the gap $g$ as the difference between the outer radius $r_o$ and the inner radius $r_i$.

$$ g = r_o - r_i $$

The gap-based Reynolds number $\mathrm{Re}$ uses the relative tangential speed $\Delta U$, gap $g$, and kinematic viscosity $\nu$.

$$ \mathrm{Re} = \frac{\Delta U \, g}{\nu} $$

Angular speed $\omega$ is the tangential speed $\Delta U$ divided by the inner radius $r_i$.

$$ \omega = \frac{\Delta U}{r_i} $$

Rotation rate in revolutions per minute (rpm) is angular speed $\omega$ converted from radians per second.

$$ \mathrm{rpm} = \frac{\omega}{2\pi}\,60 = \frac{\Delta U}{2\pi r_i}\,60 $$

Solving for $\Delta U$ gives tangential speed from rpm and $r_i$.

$$ \Delta U = \frac{2\pi r_i\,\mathrm{rpm}}{60} $$

A simple turbulent wall-drag scaling relates available shaft power $P$ to steady-state speed $\Delta U$ using density $\rho$, friction factor $C_f$, inner radius $r_i$, and active length $L$.

$$ P \approx \pi \,\rho\, C_f \, r_i \, L \, (\Delta U)^3 $$

A smooth-turbulent closure for the friction factor uses $C_f$ as a function of Reynolds number $\mathrm{Re}$.

$$ C_f \approx 0.079\,\mathrm{Re}^{-0.25} $$

The power model is coupled to the flow state through the same Reynolds definition $\mathrm{Re}=\Delta U g/\nu$.

$$ \mathrm{Re} = \frac{\Delta U \, g}{\nu} $$

This is all a little scatter-shot, but it gives enough background to fairly simply fill in the remaining columns after we have selected some bounding inputs from the hardware store. Those inputs are:

• Outer radius $r_o$ set by the given container we have available or the maximum extent we're willing to build at that moment
• Length, L, also constrained by container. In most cases, by the vertical dimension.
• Available power, P, this is set by the motor we expect to use.

These are the numerical inputs for rows in the literal table below. However, you might note that a motor doesn't just have a power. You also need to get it such that it provides the correct speed. The approach I'm taking (assuming will be taken) is that for a given experiment, from this data, we basically find out how fast the motor needs to go. Then that feeds into what kind of motor we get. This likely requires some gearing, and later experiments might swap out gearing as needed.

Experiment	Outer R (m)	Length (m)	Power	$\mathrm{<u>\$\backslash Delta\; U\$\; (m/s)</u>}$	Re	drum rpm
Bucket-water	0.140	0.30	100 W	7.40	4.15e5	842
Pool-water	1.524	1.10	1 kW	4.69	1.41e6	36.6
Backyard-air	1.524	1.10	1 kW	42.3	8.46e5	330
Lake-water	10.0	15.0	15 kW	2.91	5.83e6	3.48
Hangar-air	10.0	15.0	15 kW	26.3	3.50e6	31.4
Space hab 250m	—	—	—	49.5	6.60e6	1.89

I've put simple names on the experiment scales. The first row comes from what kind of 5 gallon bucket you can get from the hardware store. The second row comes from some basic searching on what kind of above-ground pool (low quality would be sufficient) I can buy.

Then the power numbers are partly speculation, and another part, what motor would have a cost commensurate to the cost of the other stuff in the experiment.

Air has a convenience factor for experiment scaling - it ups you to a Reynolds number that you wouldn't otherwise counter except at a much larger scale. Compare lake-water to the space habitat and you get the point. This lake-level experiment would provide an appropriate level of validation before you went and launched something into orbit for real... at least in some senses.

The biggest drawback of water is that it is incomprehensible, and air is the ultimate objective. So it makes sense to run the experiment with air as the medium.

This leaves the big question of "how" you would conduct such an experiment. And that's something I have a few ideas on.

Driving Shaft and Half Scale

Return to the basic thing that we need. I like to illustrate with simple conical pinched ends. And in case there was any doubt, flow-dividers go inside other flow dividers. Dotted lines are to mark what wouldn't be seen from the outside.

This isn't very practical. Once you finish constructing one of the layers, you will have de-construct it to ever take it apart again. So I fully anticipate a half-scale kind of experiment where you would lob off one of the two end tapers. A shaft in the middle would be applying torque in any case, which I'll illustrate here.

You also have to hold them in place, the axial stability problem isn't really particularly interesting academically, so it would be better to isolate that factor and just investigate the wedge-effect type stability. Here is where another property of water is helpful. You can use the half-scale setup to also helpfully hand-wave the axial stability. My proposal for this is to add floaties to all of the flow dividers. These floaties would be circular (very thing donuts), made with Great Stuff or something similar.

Lately, I have been racking my brain on whether or not this can be a valid setup. Like, if it fails, would it be failing due to a reason that is meaningful? I think so, but it seems important to articulate why. As I've done many times here, you have to start from the access opening, and work your way out for each stage. As a result of this "walking", each stage is expected to hold some amount of pressure. This should still work starting from the bottom opening.

My challenge is to consider whether this can be compatible with the idea of adding floaties for the half-scale experiment. After all, the air above the water has an effectively constant pressure, so this would seem to violate the pressure differential on each stage. But not necessarily so. As these are rotating, water behaves as you would expect, with the surface demonstrating a slope. The little bit of rise-up of water on the inside should maintain the pressure differential.

Oh, things can go wrong with this. The rise-up could knock over some of the divider or the floatie, and that would be a failure. Or it could spill over. In all of these cases, however, it should be a fairly obvious failure mechanism. With this mental picture, I feel relatively good about the theory for moving forward with this solution for water experiments.

Air experiments have a different challenge. Because we do not live in micro-gravity, we would need a new solution. I believe that would not be the half-scale experiment details here. Instead, you would likely add a circular Helium bladder to keep each stage up. Doing things in air should technically require keeping both end tapers in place. That sure sounds hard, but it's a problem for another day.

Objective

So, what would we expect to get from this? The theory, however imperfect, does give us some ideas. Firstly, we want to replicate the instability that we predict. If we can't... that would be very notable. Astonishingly, I still don't really have an answer here. So they'll collide or not and I don't know.

But beyond that, we should absolutely not quit with unstable behavior, but try some stabilizing approaches. One would be to get some neutral buoyancy balls that match the gap distance, and then just throw them in and see how it goes. They probably won't self-sort, but I would want to see this play out. Predicting the most obvious outcome - we would want to add some sort of brace that holes in a cylindrical shape spacers. Spherical balls won't work for this... maybe at that point we would need wheels. At maybe somewhere around there 3D printing parts will help.

So, starting with the bucket-water experiment, we want stability demonstrated, with or without aids. Probably, ideally, with more than one solution to maintain stability. Then with this, prove some confidence to continue scaling up to larger sizes, with the idea that we can still get stability. Then, ultimately, we can get Reynolds number parity with what we would launch into space, and some well-developed corrections for in-compressible cases.

First Simulation Stability Results

In broad terms, if the friction-buffer idea is not "complete", it might not be really worth it to get excited about. I could be "wrong" in a certain sense, but only in the deeper more detailed engineering. So I'm trying to open the book on that, and here I will talk about (working name) wobble stability. This is as opposed to axial stability, so let me first give a roadmap of the context to put us in the right place.

Roadmap of Concepts

On the basic claim that flow dividers will reduce drag by the amount already quantified - I don't really have any doubts. Lately I have been using AI to re-do the numbers, and it comes to the same conclusion.

Beyond "flow dividers reduce friction" observation, probably the next most important observation is that you obviously should use inflatable dividers. If not, you create the need for large struts in a space they won't fit. There are pitfalls here, and I have many times had wrong-headed ideas on this. No, they can not be made of saran wrap. Each stage needs to hold an incremental amount of pressure out of the centrifugal pressure (which is much less than 1 bar), which is significant for large sizes. Their material needs are still self-evidently much less _even in total_ than the inner hull where people stand on. Then, _that hull_ is still much lower-need that a space habitat rotating in vacuum. The second-order consequence of this is that you have to walk the pressure profile up to the taper point, which is the subject of quite a few posts here.

These considerations of pressure profile between the friction-buffers not only reinforce the need for stages to hold a pressure difference (mPa range), but also the need for bleed air. You need to continuously have air entering each stage so that the connection point sees a constant flow of air out. This maintains the shape of each stage, which is required to a fairly tight tolerance in terms of fraction of its overall dimensions.

Now since we've arrived at the subject of the connection point, it's clear that the connection point needs to serve some purpose to maintain axial stability. Thus, what I called buttressing, described in a prior post.

As an addendum to that concept, some form of _contact_ is needed to hold axial position (the axial stability problem) because fluid dynamics is not doing you any favors there. It will... dampen movements. But eventually the air will move out of the way, up to the point of allowing contact. Still, there is not any particular _driving_ force we expect from the fluid. Things need to be held in place in some sense, for external air currents things like that.

What happens if contact occurs? Good question! This goes into the unsolved problems list. I might call this the rebound problem. It could depend on surface roughness, it could depend on a lot. Clearly important for safety and needs some results. Any physical experiments will probably hit this relatively fast. You know, by accident.

Returning from that digression, we will label axis of x, y, and z. The axial stability problem is the problem of stability in the z-axis. Now we get to our destination here - what about stability in the (x, y) plane?

That's a big question - and probably the _biggest_ question for the overall viability. Almost everything in this blog hinges on a satisfying resolution to this problem, which I will call the wedge stability. I have code for numerical investigation here, written mostly by AI.

Displacement Formalization

To clarify what it is we even mean to ask, we start with a cross-sectional view. At the center you have the hull (floor where people stand). It is rotating, and it has stages surrounding it rotating at various intermediary speeds.

Now the perturbation should be pretty clear. I find it much easier conceptually to move an arbitrary friction-buffer stage, as opposed to the hull or the outer stage. We can just assume the outer stage is stationary for simplicity and keep the hull stationary too, which is relatively massive making this a fine approximation. So here is a displacement of one stage into the negative x-axis.

In simulations I always displace one stage in the positive x-axis to start. I don't know why I drew it like this. Anyway, this shows the (x,y) axis and a starting displacement, the displacement is just to get the simulation going. Eventually, ultimately, we want it to be stable, so we want the displacement to go away.

What are the concerns? As we get into the wedge effect fundamentals, we can make a prediction.

The Wedge Effect

The wedge effect is what allows rotors of journal-bearing systems to levitate on a thin film of oil for some designs. These days, something called "tilt-pad bearings" makes it not relevant for many large machines. Nonetheless, there is plenty of literature out there on the subject, and in the simulations I have, I'm just re-producing the same integral the literature has.

This key integral measures the "squeeze" as a rotor with an offset pushes fluid into the point of lowest clearance. Thinking of it, still, mostly in the terms the literature does, you have gravity pulling the rotor down, and a fluid force vector that balances gravity which it must do exactly.

The biggest conceptual difficulty I have with this is ignoring the torque imbalance. You can see that the center of pressure must exert a force that is directly "up". That creates a torque because it is not acting in the line of gravity. This... doesn't do anything though. Because the rotor is being spun by a motor. The imbalance might make a spin a little bit slower, but that's it. There's a different equation in the system to account for the changes in angular speed, and that should be covered there. In any case, it's true that the net fluid force does not act in the direction of the center of the circle. Two things are going on:

• pressure buildup causes a large force in radial direction
• depending on the channel width, shear force increases or decreases

These factors are accounted for in the equations used. The shear force comes fairly directly from the separation distance, the retarding torque comes from a fairly simple integral given the offset from center of the rotor. The pressure buildup requires integrating to get the pressures in the first place, and then summing those via another integral, so pretty much a double-integral. I think it's most helpful to look at the pressure over the angles (making a full circle) before the _2nd_ integral so we know what we're dealing with. You can get these from running the script in the git repository:

python sim_250.py

The pressure at any given point pushes the rotor in the direction of its center. But after integrating over all the points, that will push the rotor at some angle that does not align with the offset. At the extreme offset, all of the pressure buildup is right at the choke point, so it _almost_ aligns with the direction of offset, and pushes back to restore its original position. But at the other extreme - a tiny offset, the push is almost entirely to the _side_.

From here, you can think of either a ratio of Fr/Ft, or the angle at which the force acts. Since we are free from gravity, this is the only force acting. The friction-buffer in practice has very little mass (even compared to the air), so the force accelerates mass that I just get from the mass of air in the stage's annular space (I know, this has problems). The repository has some graphs of these, but I will not include them here because it is already over-crowded with images.

In any case, just from the basic wedge effect equations from literature we reveal the core concern - given a small displacement we get pure Ft. If you look at a graph where the stage is displaced in positive x-axis, the force acts in positive y-axis. This is _strange_. Right away, it is neither stabilizing or destabilizing. It will cause acceleration, but orthogonal to the displacement. But of course, as it accelerates, it moves, and the force moves with it. Still mostly in the Ft direction, we can logically predict the force to "chase" the rotor around the circle, and here is where the worry begins. It might likely speed up in this case. However, at some point, Fr will grow in magnitude and it is possible to imagine a stable promenade around the circle.

Results

As given, which the most-correct type of numbers, we have an unstable system. This gives the center-line of various stages, and they spiral out. Eventually they hit, and the simulation ends with a violation of geometry.

Were there ways to make it stable? Kind of, yeah. Going back to the wedge effect itself, journal-bearings levitate on a film of oil precisely _because_ of a displacement. So what if we add a steady-state displacement? This is enforced by boundary conditions - meaning the outer stage would be held in place against the motor spinning the null. This is somewhat of a novel concept, that the math leads us to.

Results from those:

This did lead to a stable simulation. I pre-selected x-axis values, but did not so for y-axis values. And if you look back at the wedge effect, again, it doesn't act directly along the line of displacement, so the y-axis values are actually finding their new stable positions. Cool!

Unfortunately, I had to use a fairly extreme displacement. It's not obvious, but this _could_ undermine the entire idea by increasing the drag in the choke point. But these are very imperfect simulations, let's move on.

One thing the AI told me repeatedly, is that there's no dampening added to this simulation in particular. So it could be a forgone conclusion that this is always unstable, with no connection to the physics. This is a hard problem that still needs noodling on. However, I did experiment by adding a "fudge factor" of the Fr and Ft ratios.

Seeing this is kind of comforting... but also disconcerting. I am fairly darned sure that there is some envelope where this can work with constant "wobbles" shown here, but where the stages do not rest at center-line. If reality falls in this regime (which I still don't know) that means that, simultaneously, you CAN use the scheme in the unsupported Russian doll configuration and it will work, but you very well might not WANT to, because it is going to cause persistent movement and you would rather it not.

Take-Aways So Far

In the repo, where most of the content is written by AI, there are some limitations of these simulations outlined. And they are a LOT. Firstly, like oil journal-bearings, we are only even attempting to do laminar simulations. Well, if flow was laminar, the friction-buffers would make no sense to begin with. We are extremely very obviously concerned with turbulent. And that is going to give wildly different results as the channel width expands. So this is very absurdly limited.

Other flaws in the methodology are almost too numerous to count. The simulation does an integral, but in that neglects all momentum of the fluid in the discretized volumes of integration. So we tried very hard to just track the variables for each stage, but tracking variables for each element of fluid in each stage would be more appropriate... just more variables. This would also allow somewhat faithful dampening which are very likely to change the result. Even with these changes, however, I'm not convinced that we can obtain full center-line stability.

But possibly the biggest single flaw, which is unique to this problem, I think is treating any friction divider as a rigid structure to begin with. Even with inflation making giving it shape, it will be fundamentally deform-able. It's just a sheet with a fluid on each side, with tension in the sheet. That is going to give wildly different behavior from the simulation and this is the effect that I am most interested in. As wedge effect literature focuses on pressure buildup as you approach the point of narrowest clearance, I'm not convinced that happens at all in our case, because the sheets can move. Locally.

Overall, I remain cautiously optimistic, but have also accepted that there is likely going to be some space for an engineered solution for stability - for instance, a ring of separators inside of each stage. That can have a negligible impact on drag reduction, and also keep the stages from center-line divergence. The persistent wedge is endlessly fascinating, and shows how more understanding will probably open up the space for more elegant engineered solutions, and I am realizing how deeply of an not-understood problem this is that I'm looking at.

Taper-nested With Slightly More Space

Starting out, let's walk though a graph of the style given in a prior post. The question we are asking is "where can we terminate sheet number N", because they don't come to center-line. Somewhat arbitrarily, a relative speed of 3 m/s is allowed, as a seal must be established over that relative speed. The two series of air-relative and tube-relative give the maximum possible radius where the sheet will be at 3 m/s relative to the tube or to air.

In this case I am using a 500 meter radius tube, because I'm also interested in exploring those larger sizes here.

An easy observation is that, on the sides, relatively few sheets are allowed to have a relatively large opening. Because these are relatively few in number, I've draw in a "design" line where I'll simply use less than the maximum for the edges. I believe this will help both sanity (for now) and possibly logistics in practice.

The meaning of taper-nested is that the sheets come in towards center-line as much as needed. This graph means to quantify. Using these specific numbers (with 32 sheets, chosen somewhat arbitrarily), I want to give a better scaled illustration of what a "larger" tube of 500 meter radius might look like in practice, as a cross-section. You should note that the access opening here is more like 20 meters in radius which is somewhat less than the opening for the sheets around the center. For scale:

Going from big to small, with this being an end-on view:

• Grey outer circle: 500 meter radius of actual tube itself
• Green circle: the center opening of the outer-most friction-buffer
• Yellow circle: the center opening of the middle friction-buffer, the smallest center opening of the center openings
• White center: the access opening for moving people, goods, and air, in and out of the tube

This helps to illustrate the R^2 factor, as an increase in linear dimension of 2 is a difference in area of 4. So it certainly seems worth it to use any extra buffer space possible, as opposed to having a long access tube where the access opening is.

Aside: there is a weirdness here, which I need to give in specific terms:

• Sheet 16
• moving +3 m/s relative to the tube
• moving -3 m/s relative to the outside (stationary) air
• The actual hull of the tube itself, buttressed against sheet 16
• moving 3+3 = 6 m/s relative to the stationary outside air

This is a oddball kind of situation. At first look, it seems to me the most reasonable thing would be add 2 additional sheets around the access opening that divide the relative speed between the tube and outside air. This would only cover the distance about 20 meters to 40 meters in radius. Dreamily, I wonder if these might be retractable for cases where traffic is higher or large items have to be moved in or out.

Wrapping up this thought, I want to give an accurate sketch of the 500 meter case, at last.

The somewhat incredible density is why I'm always talking about a 250 meter reference, instead of this, which is twice as large and has twice as many sheets. This is more material-intensive, and less space-efficient. Nonetheless, I must acknowledge value in human factors of slower rotation (thus Coriolis forces) and and more wide-open interior space.

I also needed this to articulate the side view of taper-nested in mostly the standard sense but just adding some very detailed-oriented tweaks.

The Bleed System

The diagram above makes it very hard to believe that the sheets could resist any external force. Again, these will be inflated like a balloon. Each sheet hits either a rotating structure or an externally-connected static structure at an angle near to 90 degrees, maybe 80 or 85 degrees. This means that it lacks any kind of backstop, or buttressing, for the flow dividers as I suggested in another post.

In other posts I have mentioned another design element that I assume is present in basically all my writings. In essentially any case you need a way to put additional air into the sheets to keep and outward pressure to help maintain the shape. The rotation itself does this to some degree, but probably not enough, and not for the entire shape. These prior pictures here mention a small flow of air. Let me redo that same thing, but with the 500 meter reference in mind, and more granularity, and introducing a new thing - soft segmenting of the shape.

Again, return to the fact that there is no surface for the friction-buffers to "push" against to maintain position, at least in the axial direction. What's a way around that? Well you can just puff air into one end of the friction-buffer. Because of the small channel size, this is likely enough on its own. This is a big construction, so we probably going to want at least a handful of these in every sheet, add some sensing, and control systems to respond if needed.

But there's one more way we can do better. I added dotted lines to suggest that we can introduce additional very suggestive flow dividers between two of the sheets. These would not be a tight seal (they have no need to be), but would just lightly reduce the amount of air flow your valving or pumping would need to move to affect the shape at a particular place.

Bleed System Design Space

Similar to the previous pictures I've drawn on this, the above illustration implies that the air bleed system is a network of valves. These are passive devices that let air flow from an inside sheet to the next out-most stage depending on how far they open. Going all the way to the tube itself, pressure is always higher in the more inward stages, because they are rotating faster.

Even if the valves are passive, they may or may not have any controls applied to them. Without computer control, they are simply holes in the sheet... probably holes of carefully selected areas. These would simply be holes that let some air through, taking advantage of the pumping that is already happening by the main rotor. This air can is then used to maintain the inflation of the sheets, and the air ultimately leaks out the sides. The reason this has started to interest me is that I've realized that this could be relevant for early experiments. As those experiments are mainly going to be concerned with wedge effect and wobble, shape-keeping is likely to play into that. Sacrificing a teensy tiny amount of efficiency could mitigate potentially larger problems of contact and failure.

For extremely large tubes, these are extremely large operations, and if we can do better we would. The problem with passive valving is that it puts the air in a free-jet condition where it loses a LOT of its energy, because it's more than what's needed. The alternative is that you could have pumps go from the outside in, as opposed to inside out. The pressure difference you need to deal with is relatively small, and these are technology-level of computer case fans. But again, this could feed into the control system to prevent sheet-to-sheet contact which could be a very big deal.

The (lack of) Prior Art on Annular (Cylindrical) Flow Dividers

Myself, using old Google Scholar, I previously struggled to find any literature that might hint at the fluid dynamics mechanism I've proposed on this blog, which I tend to call the "friction-buffers". Let me be super clear what this describes:

• Layman:
• A tube in air is rotating, and it is surrounded by cylinders of increasing size. The cylinders rotate at slower speeds, because they are dragged by air movements caused by the tube. The cylinders allow us to spin the tube more easily than if they were not there.
• Kind of fancy:
• Multi-Annulus Taylor–Couette system with freely-moving intermediate cylinders
• Another:
• Segmented annular flow with passive staggered rotation enabled by flow dividers that suppress Taylor Vortices

I want to be abundantly clear about this - I do not want this to be my original idea. That just makes it harder to defend. Over and over again I tell myself that somebody must have come up with this idea before. But if I search I come back empty handed.

Gemini Research Output

That hasn't changed. But what has changed is that we have AI now, so at least I can prove that the AI can't find comparable background literature either. In case anyone was going to ask, although I am a major AI power-user, absolutely none of what's written here is from AI. Here's output from the latest Gemini:

https://gemini.google.com/share/0c90b978750a

Let me highlight the 2 key conclusions

1. It agrees with me. It didn't even take convincing by follow-up prompts, which other AIs have.
2. It can't find background literature either

On the first point, here's an evidentiary quote

The investigation into using intermediate, free-floating concentric cylinders to reduce viscous drag and suppress turbulence in high-speed annular centrifuges confirms the viability of this hydrodynamic control technique as a compelling alternative to vacuum operation.

I want to call out the bold, and really emphasize that this goes way further with the claims than I ever would have. This is probably not useful for Uranium enrichment or any other lab centrifuge applications. If drag matters, you can probably solve it by pulling a vacuum, and if you can reasonably pull a vacuum, that is obviously superior to the method I've proposed. I wouldn't call that "compelling", but if I was in academic I would probably write that anyway... but that's just an academia thing.

In any case, if this could be useful and practical for centrifuges, it's almost guaranteed to work for the much larger and slower case of free-floating tubes in microgravity. This is a full-throated endorsement. All the AIs agree, for whatever that's worth, which isn't much.

And as for (2), I'll just have to show you what it did find.

Rotating Half-Discs Drag Reduction

The best reference by far is:

2021

Reduction of turbulent skin-friction drag by passively rotating discs

https://eprints.whiterose.ac.uk/id/eprint/209042/1/2106.12824v1.pdf

This is abundantly clear that it proposes to have discs on a surface with a fluid moving over it, where half of the disc is kept under a divider. This allows the disc surface to be closer to the velocity of the fluid as opposed to the surface. It is very intuitive how this could work with the discs being rotated passively by the fluid, just like it is intuitive that you could reduce drag by putting a treadmill on it. These are just slightly more practical variations of a passive treadmill (if I am to tell it).

You see half of the discs because the other half is covered

Ok but this is still a bit abstract without a use case. I like this sketch, because it looks like an airplane wing.

Airplane wing maybe, my own interpretation

Could this method improve the fuel economy of an airplane? Yeah, that's totally physically possible. This idea does share many properties of the friction-buffers proposed in this blog.

• Introduce a flow divider (in my case) or a skin friction attacher (their case), which is a "sheet" in all cases
• Allow that sheet to move passively, meaning, it is moved by the flow itself
• There is an expectation that the flow becomes less turbulent, and the energy lost due to viscous forces decreases

So almost-check, check, and check. The other difference we might point out is the entire geometry is different - a surface vs. annular flow.

It looks like this paper also covered the same thing

2013

Turbulent drag reduction through oscillating discs

https://www.researchgate.net/publication/258796176_Turbulent_drag_reduction_through_rotating_discs

However, that was much harder to follow because none of the pictures made it entirely clear what it was showing.

Other Close Misses

This paper describes a passive mechanism for drag reduction in Taylor-Couette flow. Seems promising!

2024

Research on the Sealing Performance of Segmented Annular Seals Based on Fluid–Solid–Thermal Coupling Model

https://discovery.researcher.life/article/research-on-the-sealing-performance-of-segmented-annular-seals-based-on-fluid-solid-thermal-coupling-model/aabc017bc11133f9a41c1f81a3ec4b33

However, that is very clear that it optimized using a groove design. And looking at the pictures further, it might not even be talking about the general topic at all.

Looking further into papers on Taylor-Couette flow is an exercise in madness. One takes two cylinders (tubes) and places them next to each other and has them spin.

I'm tempted to believe that somebody wouldn't write a paper on this friction-divider concept. Let me explain why in 2 scenarios:

• Do not include the effect of oscillations with movement of the cylinders (global stability problem), you've effectively made an undergrad-level problem, not worthy of CFD or of a paper. It's too easy and the effect is obvious
• Include the movement of cylinders, you've added mechanical boundary movement to your CFD at which point you've made a problem that's too hard and give up

So basically, to get progress, I, or you (the reader) need to do it ourselves.

CFD or Experiment

Experiment.

This brings me back to my prior dichotomy. It's either "CFD is useless" or "CFD is impossible". Also, looking towards the specific application, the predictable objection should be a tiny bit more nuanced than "you can't divide the flow in half". Problems you're most likely to hit probably won't occur until you're at a high number of sheets. Not just 1. I'm thinking 4+ sheets to really get some value out of it.

I have some material to write on global oscillations, and maybe it's correct in its approach. But there's no natural confidence in it. Actually doing the experiment, showing the friction-buffers stay in place, and demonstrating a performance gain, would get miles and miles further confirmation that eigenvalues from the coefficients that you came up with on your own. Garbage in, garbage out.