Water Cooled Gravity Tubes

 The standard idea presented on this blog includes some of these following points, relevant to the argument I want to make here:

• A very large outer pressure vessel holds in air, and the space inside is micro-gravity as the outer pressure vessel does not rotate
• Tubes inside rotate to provide gravity
• The tubes are surrounded by nested flow dividers to make fluid forces workable
• People, cargo, and many utilities go in and out via 2 openings at either side of the tubes

On that last point, the movement of air removes the heat produced in the tube and the excess CO2. I wrote about the thermal limit that this imposes here:

https://gravitationalballoon.blogspot.com/2014/12/natural-circulation-heat-removal-from.html

Here, I will revisit that, and add a new idea that breaks the upper limit. For the numbers, I'm using this script:

https://github.com/AlanCoding/gravitational-balloon-mathematics/blob/master/thermal/gravity_tube_scaling_sweep.py

(I am using AI for many of my scripts and data generation now, but not having it write for me)

In all cases, that script assumes a heat production of 20 W/m^2 on the inner surface of the rotating tube. This is much lower than the 400 W/m^2 number for Earth equator, but still probably a dramatic under-estimate because it is not thought that people will grow food inside of these tubes (low-g tubes specifically for farming are more likely). Given that, a rough reproduction of the prior result would be this:

Air velocity through the access opening, acting as inlet/outlet

A couple of numbers and assumptions here:

• L/R, length over radius for cylinder, is 2.0
• q'', mentioned above, 20 W/m^2
• Air Delta T is 5 degrees F for comfort, 2.77 Kelvin
• 6 m/s is allowable speed for the tip of the access tunnel and air flow in/out

We don't really need to discuss population density as a real metric. Depending on the density you're interested in, you update that 20 W/m^2 number. It would obviously increase if you have people packed in living on 10 levels. Lower population levels might not _necessarily_ decrease it by much if you still expect the inside to be brightly illuminated.

So for these numbers, natural circulation is effective until about 2 km. And above, I used a lazy free jet condition. Natural circulation might only get you to 1km before it starts to heat up. This is very large. Keep in mind you have to increase the number of dividers almost linearly. So if R=250m requires 16 dividers, and R=500m requires 32 dividers, then R=1km requires 64 dividers, and R=2km requires 124 dividers. Knowing that these will require some form of stabilization, that seems like a tall order. But we don't know for sure! It could be active control by valve modulation, which scales very well, or it could even be the negative pressure design with some form of fully passive stabilization! No one really knows.

I came here to answer the absurd question of "can we go bigger?"

Let me start by rejecting forced air flow. This is probably a bad idea. Even if it allowed you to make things a little bigger (disrupting traffic a lot) that solution won't work for very long before the velocities get too big.

To go bigger, let's review the parameters that are holding us back. The 5 F temperature change is very stifling. That requires large volumes, plus the very low density of air. What could do better in both respects? Just do water cooling. For this, I'll invoke a prior post about water supply:

https://gravitationalballoon.blogspot.com/2016/03/water-sewage-service-between-space.html

You could have continuous water supply. So scale that up to supply a LOT more water, and use that water for cooling. And because people aren't in direct contact with the water, you can accept a much larger temperature difference.

Water-specific assumptions:

• water velocity 2 m/s (probably conservative)
• water temperature drop 20 K = 36 F

Now I'm going to give a very boring graph to ballpark the pipe size of a water pipe for different gravity tube sizes.

It's fully linear. And qualitatively, we have very large water pipes in real cities on Earth. About 0.5 meters is normal for water supply. But stormwater and other things will go to extreme sizes, I would ballpark at 5 meters which is a more relevant number here.

But how would this affect the rotating transfer joint for water? Well, we might sacrifice some of the center space to make this more viable because for the quantities involved it might get splashy. This might be an unnecessary change, I'm not sure.

Now for the conclusions. If a 5 meter diameter pipe is reasonable, where does that put the maximum radius under this scheme? Around 4km, which is substantially larger. Almost unthinkable. You could make it even bigger, you would just need a larger pipe. I have basically rejected the natural circulation outcome because it would make the water flow too fast. This, itself, would become a problem.

But to sanity check some things about a R=4km tube:

• population ~60,000 people
• L=8.6 km
• CO2 rise under stated assumptions, 0.97 ppm if 1 acre per person, 2 ppm for more reasonable 500 m^2/person

Even all other things being workable, the simple transit in and out of this starts to look absurd and/or impractical. The entire point of the shared atmosphere from a superstructure is to make transit fast, but you would wind up spending a lot of time in the shuttle just in and around the tube itself. That said, these sizes might still be physically plausible. Water cooling might, in general, be a good idea for all sizes. And because it is so viable, I have to change my position somewhat from 800m being an upper limit for a tube, to, I don't know what the size limit would be.

First Glimmers of Experimental Confirmation

I want to be clear that I am very bad at building things, and am humbled by the process of setting up a real experiment. A whole lot goes wrong with doing the things in practice. But I am still happy enough with how things are going to share this attempted 3-divider setup:

You can see staggered heights and radii. In my ideal world, I would have just put these carefully-measured plastic crafts into a bucket, one inside the other, and see the result I'm looking for. It hasn't quite gone so well, but I still think I'm tracking towards the data I'm looking for in the bigger picture.

Phases of Experimentation

There's a lot to think about for putting together the water-medium bucket-scale simulation of flow dividers. I usually don't really think about those things until it goes wrong when I try and actually spin it.

For the first overall geometry, I tried to do a floater-sinker combination on an open cylinder. So nothing holding the cylinder in place, just some things attached that keeps it vertical. This always went wrong because it's fundamentally not stable - it was interesting to measure significant decrease in drum speed, reflecting increased drag from the wobbling state. But even before I got far into this observation, I observed that my first choice for floater material dissolved in water. That went so badly it was almost funny.

Moving on, clearly it was time to experiment with some "spacer" approaches. I tried to Jerry-rig this with a few household items, such as toothpicks. These items were not up to the task. The one exception of note was ping-pong balls. I could believe those would work to hold a divider in place and not add too much extra friction. However, those things are maybe 1/100th the density of water, and I had more practical problem that the divider didn't extend out of the water, vertically, far enough. So with enough speed these just flew out. But a taller divider might still work. Partially sinking the balls was a cute idea, but if it works, it requires less-sticky materials than what I have.

Still not realizing exactly the extent that stickiness (general friction coefficient) is playing, I tried some cardboard spacers. In retrospect, why did I bother? It gave me some good numbers on what happens when something substantially increases the friction the motor is seeing.

This all led to the current approach, which is to cut a circle for the top and bottom of the cylinder that has a hole in the center just large enough to let the metal shaft through. This seemed to address the wobbling issues I was seeing. But the first time, I incorrectly made the thing too short.

In the meantime, the drum itself had a tremendous amount of wobble because I was just bad at drilling the holes. After getting some help with re-doing that, the reference speeds showed dramatic improvements and the fluid movements were a lot more gentle.

Finally, to what was hopefully my last critical oversight, after building the 3 color drums, I included a spacer on the bottom for the drums to "sit" on. I did not realize that they didn't need to sit at all. After a certain speed, there we get a significant lifting force. If you look at what I just wrote on the negative pressure designs, this is already foretold. Fluid forces will push the ends in if the dividers are open to the fluid. Well the connection I'm using is definitely not water-tight so we can be pretty sure this is happening. Because only 1 of the 2 cylinder's end are under water, this... pushes it up. There was nothing to stop its movement up, so the "hat" I created for them would all start flying off and everything went off the rails.

Pretty Sure Working just a Little Bit Now

To set expectations, I have ran a few references with water and no water in the bucket. This gives a sense of the fluid-dynamic specific contribution to the torque. The speed moving the drum (due to the bearings and stuff) was quite significantly less than the motor free spinning the shaft.

For adding a single divider, I have a best-case scenario of adding about 4 RPM speed. This is if the divider itself doesn't add additional friction (which it will) or generally cause chaos as all the other experiments have.

Running all 3 dividers nested in each other (before the hats flew off), it was hard to tell what was going on but speed measurements seemed to be at around parity with the reference. This was encouraging. I did another go with a single divider, and again, before the lifting problem it seemed to be touching parity or maybe sometimes just a little better. It was unclear when the lifting problem was becoming terminal so it's hard to say what range the effect was expected to be measurable.

So onto this configuration, where I might have corrected enough things for it to not be terrible.

Just running with the middle divider present (blue). Down at the bottom, there is a spacer both under and above the bottom circle of the drum. I obtained some numbers, and they're not much:

V	Reference	With blue divider
6	135.8	135.4
7	159.4	160
8	183.5	184
9	207.1	207.6
10	232	233

But we are now hitting around 1 of the 4 RPM. I'm using an automotive tachometer, and because it's doing a timing measurement, I actually believe it is accurate to the 0.1 RPM. The larger problem is the physical speed stability, and it tends to get worse before the dividers break apart. Otherwise it is >1 RPM roughly, in good conditions. So this still might be better to call it a statistical suggestion of a speedup.

I need to go higher voltage & speed, but still have a problem with the hat flying off... right around the 10 V mark, which is also the speed the expected 4 RPM speedup would apply to.

This is still extremely underwhelming, but these experiments have seen a gradual improvement from the dividers making the speed drop, to reaching parity with the reference, to now, maybe just dipping into the improvement territory. Still lots of build quality issues to address. Once I get a much more satisfying speedup, and generally ability to spin the thing faster, that will open up a number of theoretical questions to testing.

Taking Seriously Negative Pressure Designs

I need to take a moment to register one of the biggest shifts in thinking I've had the the flow dividers up until this point. In almost every case, I have taken the same approach to the problem that I'll call the "pressure walk" problem.

This problem involves first setting a boundary condition for pressure, establishing that you know pressure at some given point.

• Points where we know the pressure is equal to (or roughly equal to) ambient
• All points fully outside of the tube
• The open ends of the tube
• Basically the entirely centerline inside of the tube (the axis of rotation)
• Points where the pressure is higher than ambient
• The habitat surface, as the air above it is centrifuged in the "down" direction
• All points between flow dividers, as these need a positive pressure

Of course, I'll make a diagram here.

It is important to conceptualize that the connection point, at the opening at the end, has no real pressure difference from ambient except for what's necessary for the air bleed system. Also, positive pressure is needed to maintain shape, so extra positive pressure may be needed for this.

What is Different About Negative Pressure

The one key change we will make for negative pressure designs is that the dividers will be wide open. I imagine a porosity of 10% or more. So they are flow dividers, in that they present large surfaces to disrupt large flow patterns, but they don't hold air in. Because of this, the pressure walk goes straight through the dividers. Start from ambient and go to lower radii, and pressure decreases (for the same reason it increases going the other direction.

The challenge that you've created for yourself is now that you have a highly negative pressure region, and you somehow have to seal that over a rotating seal. We have pressure over a rotating self in the prior design, but this is different. The other design set no minimum of the pressure difference.

Here, I tried to give some illustration of the ingress over the rotating seal, due to those pressure differences. How much is the pressure difference?

I made a table and got numbers, but they are basically the same as the pressure of the stages in the old design. This is driving by the rotation rate and radius. Again, the difference is that this pressure is exposed to a rotating seal. There is a minor difference, because of accounting for the walk through the stages before entering the rigid part, but it's mostly this effect.

Recapping the above points, but for negative pressure designs:

• Points where we know the pressure is equal to (or roughly equal to) ambient
• All points fully outside of the tube
• The open ends of the tube
• Basically the entirely centerline inside of the tube (the axis of rotation)
• Points where the pressure is higher than ambient
• The habitat surface, as the air above it is centrifuged in the "down" direction
• Points where the pressure is lower than ambient
• Everywhere between the flow dividers

This is an odd juxtaposition of positive and negative pressures. In fact, the inner-most stage which will drive the design winds up being negative pressure by about the same amount that the habitat is positive.

This is very seriously and legitimately weird. It's so new to me that I'm still actually not sure if there's a better way to design it so that the pressure across the rotating seal can be further reduced. Having connection points at larger radii would probably help, but the innermost stage is still the design driver.

Also, there are very interesting structural implications of this design. The flow dividers wouldn't just be blocking flow, but would also have to contain members that could push back against the air pressure on the rotating end plates. Because nothing else is moving at the required speed, so it has to be integrated into the stages.

I am extremely unsure which approach is better. The pressure over a rotating seal is an argument against this. But it's also a very attractive feature to have porous flow dividers. I learn towards the negative pressure design being worse, but it's also entirely plausible that bizarre flow mechanics lead to discovery of new effects where the porous design allows it to perform dramatically better. So for a good while into the future, I expect these to be 2 valid and competing design paths and one shouldn't be dismissed for the other.

Speaking more practically, I can see the negative pressure designs working better for small scale experiments.

Flow Divider Mechanical Spacers - 6 Wheel Design

Following up from speculation via numerical simulation, and in anticipation of physical experiments, I intend to lay out a specific form of mechanical support for the flow dividers. The goal here isn't to hold them in place against gravity (there is no gravity), or even really to push back against external forces. The goal is just to keep things in place against fluid forces.

The canonical description I've given of the flow dividers assumes that they are flexible sheets, but they do hold pressure. That is important here, in predicting what happens if you push against it. Each layer doesn't have rigidity directly, but physically wants to hold its shape the same way a balloon does.

Logical Path to Wheels

Start from the beginning, we will assume (still speculative, awaiting experiment) that the sheets experience wobble in the xy-plane (both directions other than in the axis of rotation). Further, we assume that this wobble gets worse with the number of sheets until it becomes destructive.

Now the key challenge here is that you can't just add a structure that everything attaches to, otherwise you defeat the entire goal. One structure would experience a large (unacceptable) relative velocity to most layers. So you want to add tracks? Like roller coasters? Sure. But as we have sheets N=1, 2, 3,... you must have those tracks attaching 1-to-2, which are moving relative to each other, and 2-to-3, and so on. Transmission of force has to go over many moving interfaces, which looks bad. Thankfully though, in the neutral state we know that there is no fluid force expected, it's just that movements from neutral position tend to worsen, not dampen.

Some of the most obvious intuitions are to:

• Put balls in-between the layers
• Put cylinders in-between the layers

There are other paths I want to go down with these approaches, but they both share the same general flaw that they are obviously very material inefficient and might be relatively bad for worsening friction. It's hard to imagine how you hold balls in place.

Cylinders might have better options to hold them in place. Imagine a circular strut that holes the "handles" of the rolling pins. But again, the rolling pins (cylinders) have to fill almost the full volume in-between flow dividers.

So we come to the conclusion that what we want is something like a limited number of balls, but what we can hold in place. That geometry gets a little weird. Make it a wheel, and now it starts to make sense. On top of that, you don't need to fill the entire space. Maybe just the top and the bottom of the straight region of the friction buffers. And given that top & bottom specification (2 locations vertically), what's the minimum number of locations we need to "pin down"? 3 within a given plane to stabilize a circle. So 3x2=6.

For the rest of this I want to describe this 6 wheel pattern. This is still just a rough draft, but presenting something with some logical clarity is nice to have around, have documented as a starting reference. It might not be needed, no one really knows. Certainly not me!

Wheel Position Around Tube

We need a picture for what I said about the "straight region of the friction buffers". For the center part of the rotation, we want even geometry, and then beyond that we have a region I call the "taper" which presents all kinds of unique problems. Wheels will be placed in that specific joint between the straight region and the taper region.

In the diagram I have here, I cut the top taper off. No reason, it's just a lot to draw... but this illustrates where the wheels would be. It's notable that I've taken a 7 (actually 8, close enough) sheet design, and the wheels divide it up into 3 parts. This will be important.

Wheel Position in Global Geometry of Gravity Tube

Now let's look from a cross-section view, this time of a more simple design of 5 layers - 2 layers of wheel separators and 3 layers of flow dividers.

This distinction between layer "types" is very important. The layers with wheels are inherently porous. That's how wheels work, there are moving parts. This doesn't rule out getting some flow-dividing benefit from those layers (and I think you would), but these can't be quite as effective as the other layers because they can't hold pressure. Because they can't hold pressure, the tapering solution is unclear. As you go towards the center, there can't be any outward pressure pushing the material in the taper out (because the wheel openings are letting that pressure out).

It would also be coherent to have wheel layers that are strictly for mechanical support.

From the cross-section view, let's take both types of layers and unfurl them. These are meant to be something like construction templates.

I've added a reinforced section to the flow divider layers. But it is very important that these don't need to cover the entirely length along the straightaway section. It only needs to have some rigidity around where the wheels might touch them. This is somewhat of an alternative to having the entire circle lined with wheels. We just look to have 1 rigid circle bumping against 3 wheels, which has stability.

Wheel Structure

But what would the actual wheels be? Remember that the flow dividers are moving relative to each other. So in the following diagram picture that. The rotation of the wheel will be going along with the movement of the sheets and also roughly with the flow (the exact flow shape is a weird rabbit hole in Google Scholar).

We have every reason to believe that the wheels will give relative little penalty to the friction/drag of the overall structure. Because there's no relative velocity between the wheel and the flow dividers.

I, importantly, envision a "sheath" like structure along the strut. This is to maintain the orientation of the wheel, so it doesn't flip over to the side.

The wheels are a thing that I am, now, working on some preliminary 3D printing ideas. For bucket-scale experiments. These wheels could be real and relevant for both the bucket experiments and full-scale space habitats. This concept of the wheel spacer layers is also somewhat coherent as garage-scale craft, just made of some metal bars.

Brief Preview of Future Topics and Current Focus

Previously, I did an agenda post. Many things I went on to complete, but others I abandoned because I took a different direction.

I'm going to do the same thing here. Why?

The pivot to cislunar space by NASA and SpaceX has happened. I believe they will accomplish their goal, and the revolution in space development will happen. Artificial gravity won't come into play anytime soon, but the SpaceX plan was a 1 million city on Mars. I would ask the question of what the equivalent accomplishment between Earth and the Moon would look like, and that involves artificial gravity. Because of that, I could see a scenario (even if not imminent) where these writings become relevant.

This scenario requires an approach where flow dividers in an integrated atmosphere shows an obvious advantage both at small scales and at large scales, in addition to demonstration of basic physical viability.

I started writing about the Bare Metal Sphere Habitat (BMSH) to talk about things that could be built in cislunar space. The overlap of thermal & shielding functions are really the basic things to work out there, and I'm not entirely finished either. So with this, I will break things into sections with brief abstracts of unwritten work. Finished posts will have pictures, but this will not.

Cislunar Habitat Design Reference

The BMSH is a thing to build out technical specifics of what it would look like to use the friction-buffer idea, but without the self-gravitating walls. This imagines rotating tubes inside of a larger pressurized vessel that is mostly stationary (probably sun-tracking), and built out of materials from Lunar sources. This might seem self-obvious, but thermal management is still a thing, as well as other considerations.

A Complete Description of Internal Shielding

The last post on this gave a throw-away geometry for the porous internal shielding. However, I want to get something that:

• a) Isn't just intended to be awful but might be a competitive design and
• b) Have specific numbers and materials for the shielding, pinning down the specific penalty for making it porous

For (b) I already have some numbers. It's 2.1x the mass usage. Realistically, I still believe it will come out to less when using full, proper, radiation shielding calculations. I did my calculations with really basic attenuation mechanics, which are still non-trivial.

With porous internal shielding, air can pass through it. I envision this working with the "apple core" flow pattern. A large flow goes through the center of the sphere, and turns around to return flow in the other direction along the edges (behind the shielding).

Umbrella Protection and Thermal Control of Space Habitats

A bare sphere still seems very unprotected. So I need to take time to write a post that covers due diligence for operating in cislunar space. This needs to answer the question of radiation embrittlement (of Aluminum probably) and impact risk. The embrittlement question is just a matter of running the numbers. But the impact risk issue gets weird in the way I like.

For such a large station, it would be hard to do maneuvering to avoid debris in space. Even if you did so, your threshold for moving the station would have to be awful high. So you would have to accept impact, and in our case that would hit the structure directly (same could be said for other designs). Because of that, I had a wacky image in my head of a catchers glove held by an arm that could move in front of debris. However, as I thought more about this wacky idea, it quickly becomes apparent that accurately predicting the impact time is much easier (basically already solved) than the impact location. So we could have a time-based control to go into temporary "turtle mode" where the station pulls its head into its shell. What would this look like?

If you look at what debris shielding looks like, its goal is spallation and tends to be multiple layers separated by distance. This works well with our approach, because we can use plenty of space for "turtle mode" because it's not frequently deployed. So this system would extend to potentially a large radius, multiple sheets, and physical separation between those sheets. It is also very interesting to note the similarity between missile shielding and thermal radiation blockage, which is layers of multiple sheets (in vacuum) with separation between them. We do have a need, already, for thermal control. So it makes sense to have a single structure that solves both of these problems. This would likely take the appearance of multiple umbrellas around the sphere, partially deployed for thermal control, and sometimes fully expanding for missile protection (turtle mode). We already have some spacecraft references that so this type of folding and unfolding, although not on the scale or speed involved. However, the idea is simple and effective, and mechanical & engineering needs modest.

To get full coverage, we might divide up the sphere into a platonic solid, where each face would correspond to an umbrella. The umbrellas might overlap with each other at slightly different radii, or their shapes might interlock fairly exactly at the same radius. Even when collapsed, we would have the poles of the folded umbrellas extending outward like a spiky space station.

This is a tremendously fun design, which I think would be generally useful. There is still a specific articulable motivation for the BMSH, since it will use mongrel Alumnium from the moon, and we are worried about tiny impacts tearing up its surface, causing large cracks, potentially compromising. The good thermal conductivity casts some doubt on how good spot-welding would be for fixing impacts. The umbrella won't fully stop impacts, but will dramatically decrease the depth disturbed which could easily get us into a safe-forever territory.

Bare Sphere Habitat with External Shielding

The umbrella seems to be a shoe-in for the reference design. However, the combined structural & heat rejection function are highly suspect. Using the wall as your radiator requires exposing the Aluminum directly to space, and this might be unwise. It also strains our ability to remove heat at large sizes at which point we'll need a new method anyway.

So I should describe the obvious solution of putting heat pipes into the wall. However, the specifics of what the heat pipe connects (thermally) is an important question. We want a leak to not be a big deal, but also want to use the heat pipe physics to spread out heat as much as possible on the radiator surface. Because of this, I envision the heat pipes expanding into an external radiator structure, but possibly not going into the habitat itself. If it just comes close enough to the internal wall of the pressure vessel, simple radiator fins can do the rest of the job, and no internal leak becomes plausible.

Philosophically, we need to salvage a principle from the overall BMSH approach, which is to reduce the distance the heat travels. I have been running some numbers, and long-distance heat transfer quickly gets insane.

Vacuum Long-Distance Heat Transfer

After thinking and writing about this topic for years, I've become convinced that long-distance heat transfer would have to be done in vacuum. That introduces all kinds of new problems for the heat transfer from vacuum to atmosphere, and becomes a bit of a nightmare with extremely non-obvious solutions. No matter though, for basically all sizes of BMSH we are interested in, heat pipes work well. That is, we just have to keep the pipes limited to traversing from one side of the wall to the other, and no more. This requires a very large number of heat pipes, and they would have to be mass-manufactured.

To give spoilers, I envision a space "train" with blocks that look like the monolith from 2001 Space Odyssey. As these go through the radiating region, they would be spread out. However, as they approach the habitat to dump their coldness, they would need to fold up multiples into a single unit and then physically touch the habitat to conduct heat. The need for folding is a consequence of scale. It also appears to be operationally awful and something to be avoided if at all possible. I don't think you can avoid it. At some scale, you would have to use this.

Reference Design

The discussion up to this point is about the design space, not just one specific design. Provided I can ever finish talking out the details of how such a habitat would be built in the abstract, we need to move onto pinning down certain scales and describing what operation would be like in practice. So, moving on from physics to more practical concerns... ultimately getting a named reference that people care about and will reference. One day.

Habitat Molting

I have some images sitting around I want to share of how I envision this approach to going from one size to a bigger size. In this conception, you start with a sphere, and build a bigger sphere around it. For multiple reasons, I think this is a bad idea, but I still need to put some pictures on my blog to have something to point to.

Joining of habitats

A better idea that habitat molting is to join spheres together. This means that there will be a circular opening on the surface of the sphere which is docked to another sphere. With 2 equal sizes, you have an internal volume in a barbell like space.

My original idea is that you would mass produce identical spheres with docking ports, eventually creating a potentially regular 3D pattern. However, this is god-awful for heat removal. Doing this would require transferring heat through vacuum. However, getting the heat pipeline to move heat out of the habitat in any sane way was worse than I thought. The 3D lattice of identical spheres is an idea I thought on for a long time, and eventually rejected.

So what's the alternative? Connecting spheres of different sizes. This means that volume additions are slightly less incremental, and more exponential. Every time you make a new sphere, maybe that new sphere has double the volume of the last sphere. Continue this until you block heat transfer from the prior spheres significantly... I'm thinking 4 spheres. So you have spheres at radii of 1km, 1.3km, 1.5km, and 2km. At this point, the 1km sphere would only be 7% of your total volume. So you would tear it down. Throw it away and recycle the materials of the smaller sphere. This would open up space to add a new sphere of 2.5km. And so on, this pattern can be continued forever to continue growing space. The rest of the details, like the heat transfer, the umbrellas, and everything will remain. There will be some de-construction happening as needed. However, new additions are always significantly larger than the parts being tore down, so this adapts well to exponential growth. It answers the big questions about the lifecycle of the habitat, while not pinning people down to an original decision about the number of inhabitants. This is fundamentally a very growth-oriented approach.

More broadly, as long as you can do permanent docking of spheres, the ability to do incremental growth is a massive argument in favor of the integrated rotation, enabled by flow dividers. This is even more-so a powerful argument in cislunar space.

Back to design of flow dividers

I'm happy to write all of that, and there's much more I want to share, but right now I need to keep my focus on the truly novel part of this blog which is the flow dividers. All above topics will be shelved for a while.

Consideration of Flow Divider Spacers

Referring back to the simulation results (and its associated theory) we don't have a good claim to think that the flow dividers will be naturally held in place, and we do have some reason to believe that fluid forces will want to either cause a wobble or break it apart. Even if we have an acceptable wobble (likely IMO) there can be resonance modes that break it apart.

Instead of accepting wobble or designing around resonance modes, we would be much more likely to add another physical structure to prevent them (very speculative all-around here). So I need to describe some options for that. This need to be a tour of just all possible geometries.

You could have simple scaffolding put between the sheets. If they touch the sheets, they bounce off (with friction, sure). The more contact, the more drag penalty you incur, so you minimize the area of contact. This might work well making limited parts of the flow divider sheets rigid. The minimum contact points would be simply 2. Maybe 1 but that seems dicey. 2 contact points would also help the axial thrust. You would place these at the end of the cylindrical region, right where the tapering starts. This might be all that you need - 2 hoops on each flow divider and scaffolding in the shape of hoops in between layers.... I will really need to draw this out sometime.

Next, we will get fancier. Instead of scaffolding that occasionally touches neighbor sheets at velocity (like 5-10 m/s) you could put the scaffolding in the middle of the channel (0 relative velocity to the fluid) and attach rotating structures to the scaffolding. This could be wheels which have an axis in the same direction as the whole tube, or they could be cylinders. At the extreme, you could have cylinders running the entire cylindrical geometry section. This would provide extremely good stability but it is unclear how much flow would be negatively impacted.

The next geometry would be to throw some spheres into it, and let them move around. The obvious geometry selection is that there sphere's diameter is the channel width (approximately), although this incurs some risk of bounce, and corresponding stress on the dividers. This seems wacky, but too early to rule things out.

Serious Discussion of Negative Pressure Designs

For probably the last year or two, I have been extremely bound to the idea that each flow divider much hold some pressure. Otherwise, you can't have a coherent boundary condition where positive pressure is maintained up until the connection point at the access tunnel.

However, at the point that we are adding spacer structures, this notion deserves to come under attack. If we have a network of cylinders or spheres floating around in-between the flow dividers, why not just get rid of the flow dividers? What are the consequences of that?

I can very easily tell you the elephant-in-the-room consequence. You can't taper in the same way. This dramatically changes the image of what we're building, but maybe not for the worse. You can have the poofy ballon type shape as it goes up the edges. There wouldn't be a nice 45 degree angle from the surface where people live up to the access point, but there might be a -45 degree angle in its place. Let me explain.

There will be a free-air condition from a microgravity point up to the outside of the tube's hull. Combined with the fact that the air is rotating, this means we will be at less pressure there than in the microgravity. This means that the "ends" of the flow divider region will be sucking. This can be accomplished by using rigid sheets for the ends, or by using a flexible sheet attached to a ring. This flexible sheet will bow inward.

Related, I've also accepted that what I call the "negative pressure" designs might actually be easier to test that the flow dividers themselves, and so they might be the first things to actually post results.

Experimental Designs

For real experiments, I don't have anything to show yet. I am gaining better clarity on the "bucket" scale of experiments, which I always envisioned as the widest variety of experiments. I have multilple sizes, and many flow divider configurations in mind. Towards the end of this, some end taper designs should be getting tested, which is far from where I am now.

As for the purpose of the experiment, I'm becoming increasingly focused on the near-term goal of showing some effect at all. That is, lower power input for the same speed, enabled by passive flow manipulation. Any result of this kind proves stability uncompromised by fluid forces exists. What it doesn't show is that the particular flow divider design will scale.

Like I was arguing in the negative pressure designs, the number of flow divider solutions might be more numerous than I had thought. I have been studying the Taylor-Couette flow charts more, and realize that the flow patterns are donut-like for the regime I care about but also extremely chaotic. I still believe that the viability of a divider structure is almost a direct function of Reynolds number. It's possible that one will work at the bucket scale but not at larger scale. Developing multiple viable solutions now will help speed up future work.

Thoughts on a Space Station Test

In my experimental design plan, I penciled in 2 versions of air experiments. I am leaning towards saying this is overkill. There is the possibility of a scenario where we go straight from some University-level work at the room scale (ceiling fan stuff) straight to experiments on a space station. I think this, in large part, because activity on space stations (and particularly commercial ones) will probably increase in frequency in the next few years.

This is extremely appealing because the materials you need to make it happen are dramatically less than what you need on Earth. To minimize work up there, you would have to combine the motor and speed/torque measurements, and the motor would be the heaviest part by far. Aside from the motor, everything else would be inflatable, taking up relatively little mass. Then the size you can deploy for the experiment would be a large fraction of the habitat module where they conduct the experiment. The more I think about it, this isn't actually that crazy.

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.