Global Heat Transfer Without Alternating Layers

Here, I hope to offer an alternative to the previous method I discussed for exchanging hot air from the rotating tubes to the edges of the gravity balloon (from where they would need to go through another heat transport mechanism into the space radiator on the surface).

Because global heat transport is a relatively loosely constrained problem for gravity balloons of most conceivable sizes and densities, we are able to spend some of that margin in order to achieve an interior that may be more desirable to the inhabitants. The previous method is described here:

http://gravitationalballoon.blogspot.com/2014/12/global-air-heat-transport-in-gravity.html

The basic ideas of the first version are:

• The tubes are arranged in a regular lattice where each tube has a neighbor counter-rotating tube
• They are further arranged into cross-sectional layers where all tubes rotate in the same direction
• Sheets are placed spanning the space between similar-rotating tubes to block airflow going from layer to layer
• The flow makes a U-turn once it gets to the heat exchanger on the inner surface of the global gravity balloon wall and enters the neighboring sheet to continue a criss-crossing pattern

The potentially undesirable aspect of this is that the inner space has a great deal of clutter in the form of flow-dividing sheets. You could still have some holes in the sheets for travel of people and goods to go through, but the sheets would still be a nuisance. It would also be mostly mutually exclusive with large open spaces.

The alternative proposal is only marginally modified from that idea. My overall sketch is this:

Here, I am illustrating a pair of counter-rotating tubes. In the previous version of this idea, the air in the above diagram would be moving downward (by the directions in the sketch) within the space between the two tubes. The modification here is that we add a scaffolding around the surface of the tubes in this space. This achieves the goal of bulk flow in one single direction over a large volume within the gravity balloon.

QUALIFIER: Someone unfamiliar with the broader gravity balloon concept in this blog might find it easy to mistake the motion depicted here as directly corresponding to the outer surface of the artificial gravity tube itself. Instead, this is only the outermost friction buffer layer, traveling at a few m/s, instead of on the order of 100 m/s, which would be the outer hull of the artificial gravity tube itself.

What about the global circulation patterns? The air can't only move in one direction, it has to have a net loop in some sense. The best solution to couple with this design is to make the flow regions as large as possible, resulting in a "apple core" sort of circulation pattern. In this sense, there is one major river of flow going straight through the center, which fans out and goes along the outer regions to return to the other side and back again.

The main challenges, as I see them, are the management of the integrity of the shape of the outer layer friction buffer. It is true that I expect some natural wedge effect to help maintain stability, but there will be more deformation caused by the significant asymmetrical drag on the inner segment. Intentionally putting asymmetrical forces on the outer layer will also push them closer to the limit of the stability criteria (whatever that specific criteria may be) and may also increase the drag forces.

I started thinking about this topic more after seeing some artwork in the Accelerando blog. In future posts I do hope to provide more specific commentary and back-links those works. My intent here was just to put this concept out there.

Global Air Heat Transport in a Gravity Balloon

The complete scheme for heat transport within a gravity balloon habitat with artificial gravity cylinders can be segmented by following general outline. This tells the story of the path of heat as it's generated where people live through the point when it eventually gets emitted out into space.

1. Heat generation occurs inside a artificial gravity tube
2. Hot air flows out of the outlet window of the tube
3. Hot air flows to the walls of the gravity balloon
4. Heat energy passed through a heat exchanger to some non-air medium
5. Non-air medium passes through a space radiator as the ultimate heat sink

This post will focus on how the air currents transfer the heat from the outlet of a tube to the wall of the gravity balloon where the heat exchanger then picks up, #3 in the above list. The previous post outlined one valid scheme for #2. In the past I've entertained several bad ideas for a scheme that accomplishes #3. Notably, since the currents need to occur over such a large area and only need to move so slowly, I imagined an impossibly large fan many kilometers across. If possible, apparently absurd or extraordinary schemes should be rejected (unless the problem it solves is, itself, extraordinary). In this case, just like in the last part, a much more elegant solution presents itself.

Outer Sheet as the Air Movement Mechanism

Just like heat transport out of the artificial gravity tube, it would be preferable to reduce the number of parts, so if there's a device already called for by the design we would prefer to configure things so that one device solves multiple problems at the same time. Thankfully, that exact thing is possible. We will pump the air ultimately by using the driving force applied to keep the artificial gravity tubes spinning.

But first, I must specify that I imagine a stationary lattice which is connected to the asteroid rock and can accept forces. This is absolutely necessary for the motors to push against which keep the cylinders spinning, but they shouldn't be particularly difficult to build. A key distinction now comes in how we configure the outermost layer of the friction reducing flow dividers (friction buffers). That outermost layer could conceivably be connected to the stationary lattice structure. Up until this point my math has assumed this is the case, but that was only done for mathematical simplicity.

In fact, it would be best to allow the outermost layer to freely rotate. For some numbers, let's say that the speed of the habitat on the inner surface of the tubes is 100 miles per hour and there are 20 flow dividers. Each stage then sees roughly 5 mph relative speed to the next stage. What will be the velocity of the outermost layer? Answer: considerably more than 5 mph.

To understand this, I will offer a concept of "resistance to movement" between each layer of flow dividers. Given that the stages have constant spacing between them, this is roughly the same for them all (with some difference due to varying radii). However, the distance between the outermost layer and the bulk atmosphere isn't something which can be clearly defined. If we imagine the point of "r=infinity" to be another flow divider, the spacing between that and the outermost flow divider is clearly more than the spacing between the other dividers.

Given that the outermost sheet sees less resistance to its motion, its natural preference will be to couple more strongly to the speed of the habitat (100 mph) than to the bulk atmosphere (0 mph) than would be predicted by it's share of the speed divided evenly among the flow dividers (5 mph). But we have yet another pesky effect that we have to deal with. If multiple tubes are in the same general vicinity and their air currents compliment each other, this could increase the ultimate speed that the outermost layer equilibrates to. Those are potentially two factors which push it above the 5 mph prediction for this case.

For all of these reasons, I believe that the outermost layer will have some kind of "brakes" on it which prevents it from speeding up too much. Even better- if the motor is attached to the outermost layer this will improve efficiency somewhat, although it would require an additional motor to keep that layer rotating slightly relative to the stationary lattice. Even if you simply threw away the extra energy, the scheme will see a very small efficiency reduction and will work just fine. Accept a little extra energy consumption or a little extra complexity - the choice is yours.

All of this is only to say that the velocity of the outer layer can be selected to some degree.

Direction of Thermal Gradient

The ability to move flow in the local vicinity between tube would be pointless anyway if there wasn't some coherent path that takes the tube's hot air exahust to the wall's heat exchanger in order to ultimately dissipate the heat. Because of that, I'm making the obvious claim that every tube has its flow connected to two different channels. It obtains its intake air from one channel and exhausts hotter air into the other channel.

Picturing this takes a little bit of creativity, and my illustration skills might be lacking. The divider between the global channels (different from the friction buffers themselves) is a 2D sheet that cuts accross multiple artificial gravity tubes. For the most part, this plane cuts through the tube's axis of rotation. We just apply a slight skew or deviation in order to allow the exhaust and intake ends to connect to their respective channels. Here is my illustration of the situation:

Some attributes are exaggerated in this figure. The exhaust and intake windows are only 10% of the habitat's radius in most reference designs I've used. Thus, this slant we're working with might only be slight. Alternatively, the sheet could be completely parallel to the axis of rotation, and the windows will connect into small local depressions. Well "small" in this case would still be around 25 meters, but that's small compared to other stuff involved.

Hot Channel Calculations

Moving on, we need to figure out what limitations this scheme places on the overal gravity balloon size and/or population. I will return to my reference design in order to illustrate a pattern of air flow. Flow dividers must be added between a series of artifical gravity tubes, and then air flows in different directions on both sides. This produces an interlocking pattern of air currents.

Putting that in perspective of the entire balloon, I have the following image in mind. Here, I have included the presumed heat exchangers on the wall. I hope that makes it clear what kind of back-and-fourth pattern the flow is traveling in, and how it gets hottest right before it gets to the heat exchanger.

We can set the temperatures to whatever is desired at the start and endpoint for these flows. However, since the flow travels in a straight line, it seems fairly clear that the highest temperature change will be experienced by the line of colonies that goes straight through the center of the sphere. It is this row that sets our limit.

I would imagine this limit will be around 10 degrees C or Kelvin. Perhaps 20 degrees. If you refer to the heat transport within the tubes, that is certain to be on the order of 5 degrees, and some colony will experience the extremes of these temperatures. As such, it's probably best to keep it to 10.

As long as we're accepting my reference design, that has 22,000 people, and I'll stick to the claim that they're using 2 W each, for a total of 44 MW. On average, there is one colony per 1 km^3 lattice. The colony takes up a small fraction of the total volume (about 27%). Thus, the effective flow area is about 0.86 km^2. With all these ingredients, we can formulate the heat balance relationship for the balloon-level flows.

Heat Balance for Global Heat Transport Channels

Selecting the Delta_T value is related to biological and comfort limits. Heat transport within the tubes themselves is already known to require about 5 Kelvin of temperature change in order to employ natural circulation at the desired population density levels. If we add much more variation, then we could have undesirable large temperature swings. I would imagine that a number under 5 Kelvin would be acceptable.

As I previously argued, v is a tunable variable up to a certain limit, and that limit relates to the degree of friction reduction in maintaining spin. Presumably, this would be under about 3 m/s, but it could be a good deal more. With this piece of info, we have a fairly strong argument for what the bounds on these variables are.

For a given Qdot, we can find the number of colonies which can be served. Referring to the previous illustrations of the scheme, there is on average one colony per linear kilometer of flow chanel. That means that Qdot/(44 MW) will yield the maximum number of colonies it can serve. Since the hot channel's length is equal to the diameter of the gravity balloon, dividing by 2 can give us the radius of the maximum size (in km of radius) that the scheme can thermally support. I present those cases in this table:

Thermal Limitation of Gravity Balloon

Based on Global Heat Removal Channels

These are large sizes, and the assumptions about Delta_T and v are quite conservative.

Implications

This problem, in particular, would seem to be extremely easy to solve and pose little constraints on engineering of other related systems. There are multiple parameters that you could scale up in order to efficiently globally circulate air in a gravity balloon for just about any practical scale.

A large area for the flow channel was key in making this so easily solvable. Even if the generous parameters for maximum balloon size were not sufficient for someone's desires, there are multiple ways of pushing the envelope further. For instance, increasing the spacing between artificial gravity tubes or reducing the number of friction buffers. Even these methods would only need to be applied in a select regions which are subject to the extremes of the hot channel temperatures.

Another major benefit of this system is that the driving force is applied constantly. It is somewhat concerning that the flow path isn't completely straight, but I doubt that any really good solutions to this problem exist. You could conceive of extreme solutions, like flow divider sheets that are partially friction buffers for the tubes and partially friction buffers for the global flow channels, but this is certainly not necessary. The most important benefit of the continuous driving force as well as frictional losses is that no major pressure differential exists within the whole of the gravity balloon. This means that no hardened airtight doors will be necessary for people and goods passing between the different flow channels, which is sure to happen often.

Natural Circulation Heat Removal from Artificial Gravity Tubes

In a previous post I concluded that natural circulation of air was an attractive method of removing heat produced by a population living within an artifical gravity tube within a gravity balloon. This notion was still very vague, so I want to place some numbers on that, and also potentially define the scale at which it would be economical.
To summarize the idea, the airflow goes in at one end and out the other with no pumping. This is made possible by the fact that heat is produced by the inhabitants in their everyday lives, and also by the designed geometry of the cylinder which lets the hot air rise up to the outlet while preventing cold air coming from the inlet from reaching the center.

Forms loss (I'll sometimes call k-loss) is a means of grouping together various resistances to flow along a flow path. This is always referenced to a specific cross-section on the path, and I will reference it to the pinched open end. For a free jet condition, it is somewhere around 1.0 generally. In our case, we have a large stagnant atmosphere outside of the tube as well as a mostly stagnant atmosphere inside the tube. That causes both the inlet and outlet to be something close to free jet conditions... with a lot of qualifiers. Since the flow is expanding radially, it must also exchange a great deal of angular momentum with structures attached to the rotating tube, similar to the case of a centrifugal pump. This should substantially affect the k-loss value, but probably not by more than, say, a factor of 2. Given that we have 2 free jet conditions, I would most likely expect k to fall somewhere in the neighborhood of 2 to 4, but this is a highly imprecise science at this point. Thankfully, as long as it's somewhere close to that range it shouldn't critically wound our overall conclusions.

Tube radius, heat production, temperature range, end opening size, and air flow velocity are all important things which have very practical relevance to the design of an artificial gravity tube. Armed with some educated guesses for the k-loss factor, we can set constraints on these parameters. Firstly, I'll divide up these values which are absolute, fungible, and independent variables.

Parameters for air:

• density              rho0 ~ 1.3 kg/m3
• heat capacity    Cp ~ 1,005 J/(kg-K)

Relatively fixed variables

• temperature of the environment                    T0 ~ 293 K
• gravity in the living areas of the environment g = 9.8 m/s^2
• Power consumption per inhabitant    gamma ~ 2 kW
• Window edge radius relative to habitat radius  Rw/R ~ 0.1

Independent design variables

• Radius of the tube
• Change in temperature across the tubes
• Velocity of the air at the end seals
• Population of the society

Equations to Relate Variables

By definition, the k-loss equation is the following. This quantity represents the frictional pressure head fighting against the direction of flow.

Pressure Drop due to Friction

The essential idea of natural circulation is that heavy cold air flows down from the inlet to the surface habitat, and then less-dense warmed air rises from the habitat toward the center-line point. To find the change in density we must return to basic PV=nRT gas law concepts. Compared to the magnitude of the temperature change, the pressure changes very little relative to its environment value. Thus, to deal with the density change we can just imagine that it changes linearly with the temperature change.

Density Change Given Temperature Change

It is this density change which gives rise to the natural circulation driving force. This works by the analog of (Delta_P=rho g h) in constant Earth gravity. But gravity varies with radius in the case of artificial gravity. Since the driving force is the difference in hydrostatic pressure change with altitude, it goes with the change in density as opposed to absolute density.

Natural Circulation Driving Pressure

Driving force then exactly matches the frictional losses experienced over the flow path. Thus, we can set the two expressions to be equal. This constitutes the momentum balance for the natural circulation heat removal system.

Momentum Balance Final Form

Variables involved in design:

• Delta_T
• R
• v

With this relationship nailed down, we can consider the limitation on heat production. Along with this we have a litany other other supplementary relationships introduced. The mass flow rate through the tube is connected to the end window size. The area of the end window is related to the window's aspect ratio as well as the overall radius. Total heat production goes with total population as well as the per-capita energy intensity of the society (I call gamma).

Writing these all out and then combining them:

Population / Heat Relationship

Additional variables involved in design:

• P

So while we added another equation, we also added another free variable. In other words, this doesn't add any dimensionality to the problem, it's just an auxiliary equation that I'll use to calculate a population limitation given the other parameters.

Numerical Values

With more-or-less 3 variables and 1 equation, we have two degrees of freedom. The relationship is pretty straightforward but it's not very helpful in that form without comparing it to some reference designs or tangible speeds and sizes.

As a simple applcation of the equations, here are some values for 3 cases of radius, 3 cases of temperature, and 2 scenarios for the k-loss value. That is 3x3x2=18 total numbers. In each of those cases, we have dependent variables of "v" (the velocity at the end windows) and "P", the population.

For some further notes, I've included the velocity of the edge of the window for all the cases for different radii. These are assuming that the windows are 10% of the radius of the habitat surface. I've distinguished between that as "V edge" and the flow relevant to heat removal as "V flow". As you can see, the window edge velocities tend to be even higher than the outward and inward flow for the other parameters I've selected.  That, itself, might be a problem but it's a geometric consequence of the window size. The window could be made smaller while accepting some other sacrifices.

The population limit reported here is then divided by the livable area within a habitat. It is assumed (as in the reference design) that the length of the cylinder is equal to its diameter and no credit is taken for the are on the pinched ends.

I put NYC and Manhattan on this as well for a reference. Note that in my reference design for the artificial gravity tube, density is still incredibly high - about that of Manhattan. This applies for the scenario of Delta_T=5k, R=250m, and k=4, where the population constraint comes out to be about 20,000 people.

To me, this still seems to be about the most reasonable reference design. I will elaborate on that a bit more in the conclusion.

Carbon Dioxide Removal and Other Undesirables

The design principle of the gravity balloon is more-or-less to locate industrial facilities that don't need gravity (or strong gravity) within the open air microgravity environment between colonies. It is crucial that we can show that critical services (like heat removal) can be viably provided outside the gravity tubes. For heat removal, not only can this be done, but it can be done at incredibly low cost using natural circulation. But that's not all we have to worry about.

Possibly the most vital metric to control within a space habitat is carbon dioxide levels since this will cause negative health effects before lack of oxygen, however the limitation relative to the habitat's heat removal is less clear. Let's just look at the comparitative limits between these two. Consideration of the specifics of an artificial gravity tube isn't necessary. I'll just consider what temperature rise would also correspond to a dangerous rise in CO2 levels.

Certain specifiers are needed, but I'll consider the most active society possible in order to be conservative. A human doing normal work will emit 0.08 to 0.13 m3/h of CO2. Using this information as well as the scenarios I've defined, we can find the increase in CO2 parts per million (ppm) as the air flows from the inlet window to the outlet window. Here are my estimations:

Increase in CO2 Concentration

for a Given Rise in Temperature

• 2 K : 31 ppm
• 5 K : 78 ppm
• 10 K : 155 ppm

None of these are particularly deadly. Humans can easily tolerate increases this much or greater. However, these were only formulated based on the assumpting that people were consuming 2 kW on average. That was supposed to be a conservative assumption, but in this case lower values might put us in a bit of a bind. If that was reduced to a value closer to the biological limit of around 200 W instead, then the above temperature changes would correspond to a dramatically higher CO2 concentration rise. As such, it's plausable to create scenarios where CO2 removal would be the overriding constraint on the allowable population of the tube... but this probably wouldn't be likely under normal conditions.

Big Picture Conclusion

My pessimism in the last post on this subject is lessened substantially. We can state a number of relatively attractive combinations of parameters which would be economically desirable and physical plausible. However, there is still a bit of a tight design envelope to fit.

The heat production limit would likely constitute the gravity balloon's version of a "fire code". You could certainly pack more people into the tube, but the temperature would rise slowly. Except for some possibly extreme circumstances, it seems unlikely that CO2 removal would become more restrictive than heat removal.

I find it hard to argue against natural circulation as a means of cooling the tubes themselves. The benefits compared to the alternatives seem immense. The air flow rates are unlikely to surpass the speeds which will be encountered near the windows anyway, and being a fairly localized thing, I don't expect the end seals to have a dramatic impact on the overall drag anyway.

In fact, in some cases the air flow would be so low that in the center you couldn't rely on it to move out of the tube (starting at centerline) in a timely manner. For these cases, you would need a conventional transport system or elevator-like system. Since the heat production rate will vary throughout the day, this seems inevitable anyway.

Thermal Engineering of Free-Floating Artificial Gravity Tubes

This blog describes a concept of rotating tubes for artificial gravity, which are free-floating in a large zero-gravity atmosphere.  Specifically, my reference tube design fits completely inside of a 1 cubic-kilometer cube and could house 22,000 people in a familiar environment.  Obviously, the zero-gravity air environment would need to be much larger than this.

With this population, an obvious question follows: where does the heat go?  My intention is that these tubes will have "windows" on both ends which are open to the environment.  At this scale, no amount of conduction can be relied on, and air flow within zero gravity is notoriously stagnant.  The tube itself isn't zero gravity, so there will naturally be air currents as the inhabitants use energy locally, that heats the air, and that drives currents.  What kind of air flow rate (through the windows) would this drive?  Obviously that depends on the window size.

I previously imagined the window being about 20 meters in diameter, which limits its relative speed to a slow walking pace.  For a test case, consider "lateral flow", where air comes in one window and moves out the other.  Assume 2 kW of power consumption per inhabitant, and allow a 5 degrees Celsius temperature change as the air travels through.  The air speed at the edge of a window will then be 21 miles per hour.  That's a little bit too much.  Because of that, I'm altering the design.  The window will need to be about 40 or 50 meters in diameter.  I use 40 if it's double-flow (air comes in both windows) and 50 if one window is inlet and the other is outlet.  I choose these figures because it minimizes the air speed.  As you increase the diameter of the window, the speed of its edge relative to the zero-gravity atmosphere increases.  However, as you reduce it, the air flowing into the habitat has to be faster.  By making these two components roughly equal, you obtain the universal minimum.

Illustration of Wind Flow into an
Artificial Gravity Tube

With the parameters I've described here, the total wind speed comes out to about 13 miles per hour.  This isn't quite what I was hoping for, but it's still fairly permissive of easy travel into and out of the tubes.  To get more practical, however, it's obvious that what I've described as "lateral flow" won't exactly work.  To get this type of flow, you'll have to power it, but how?  People will be moving in and out constantly.  If you put a large fan there, you'll have to have an air-tight door for them to travel through, all the while avoiding being sucked up by the fan.  This might not sound challenging, but this is almost entirely a zero-gravity atmosphere (the window has about 1/10th Earth gravity).  A person's motion will be dominated almost entirely by the air currents, and not by their footing.  Because of this, it's worth while to compare the general desirability of flow schemes on both a technical and qualitative basis.

Advantages and Disadvantages of different Flow Schemes

In addition to "lateral flow", I'm also considering "centrifugal flow", where valves on the floor of the habitat allow some air flow through, into the friction buffers area.  It also follows that the friction buffers would have to allow some amount of air flow.  This could have an unrelated benefit of helping to hold the friction buffers into place.  The flow paths might be easy to engineer as well, since they could be macroscopic windows, which could even allow maintenance people to go into them.  This scheme is using the rotating tube as a centrifugal pump essentially, as I've described in prior posts.

Thinking more about the flaw of the first two, I came up with a natural circulation approach that is now my favorite.  Doing the calculations, you will find that the driving force from the centrifugal flow is too much by orders of magnitude.  This can be fixed by piping the flow back to the window of the tube.  With this setup, the flow is powered by the thermal difference of 5 degrees.  Cold air flows down and hot air flows up.  The only challenge is to them corral the hot air to the exit, and make sure the cold air makes it to the surface before going anywhere else.  This would muck up the aesthetics a little bit, to be sure.  The area of flow dividers would be massive.  Nonetheless, the problems of the other schemes seem to outweigh this.

Possibly the largest issue, and the one that concerns me the most, is vortex formation as the air travels from the windows to the surface and back again.  If these are not reduced or eliminated by radial flow dividers, then it could become a fairly bumpy ride.  This is important for intake as well as outtake - since the torque from both must balance each other.  To any degree that it doesn't, that could create big problems.

Numerically, there are several flows of energy at work.  Their relative scales are illustrated below.

• Population's Consumption - multiply the number of people by the power each uses, for microwaves and computers or whatever they use power for.
• Rotational Drag - this comes from the posts on friction buffers, it is the power necessary to power the motors that keep that station spinning against the drag of the atmosphere.
• Driving Force to the Baffles - power that would be released if the necessary air flow to cool the city were directed through the floor, into the friction buffers.  If it flows through valves, then it will increase the temperature of that air.  However, since this is a lower power level than the population's consumption, it will only be about 1 degree C, compared to the 5 degrees it has already heated by.
• Natural Circulation Driving Force - this is the power you would have to drive the flow in the natural circulation scheme against the all sources of friction
• Coolant Kinetic Energy at Window - using the lateral flow scheme (also the same for natural circulation), this is the power needed to accelerate the air flow at the windows to the free jet condition.  In other words, this is the power consumption to drive the flow if the hydrodynamic forms loss coefficient is equal to 1.0.

Returning to the last two quantities, it is obvious that natural circulation flow would NOT be sufficient to drive flow at the stated temperature difference.  This is because the effective forms loss coefficient for the entire process would certainly be greater than 1.0.  That means that, in practice, the window would need to be larger than 50 meters across, people would have to consume less power, or a higher change in temperature would have to be accepted.  I have not gone into detail for these calculations, because the above number is still a good ballpark.  It tells me the design is within engineering possibility.  Interestingly, another approach to improve natural circulation would be to make the diameter of the tube larger.  Even if this increased the population by an R^2 factor, it would increase the driving force and the area at the same time, so the total cooling capacity would follow with R^3.

Through this examination, I've established the following claims.

•  Heat removal limitations are not trivial.  Increasing population density by much could push thermal limitations.  Plus, very intentional designs to exchange heat with the surrounding air are required for my habitability constraints.
• Tossing air out into the friction buffer space would hypothetically work but is economically unrealistic for a 500 meter diameter tube, and gets worse as that gets larger.
• Flow dividers inside of the habitat are necessary to transport the air from the windows to the edges and back.
• Natural circulation is viable for the reference design, but only barely.
• Air movement will have an impact on accessibility in general, and has to be engineered together.

This is an important component of the overall proposal of gravity balloons, because the artificial gravity habitats need to be both economical and desirable.  They certainly seem to be.  Nonetheless, they will be complicated.

The subject of thermal management of these tubes also deserves mention of the analog on Earth: the Stack Effect.  A rotating artificial gravity of a given radius is extremely similar to an Earth building of that height.  Buildings on Earth also have to have HVAC systems, and this resembles one of those very strongly.  The stack effect for buildings on Earth essentially does the same thing that my natural circulation flow scheme does.  But on the other hand, those buildings don't have as many people, and also have more area (of windows and doors and such) exposed to the atmosphere.  So these two cases are similar, but one can't be called easier than the other.

Population Limits of Large Space Habitats

On the International Space Station (ISS), people are packed in like sardines.  For manned stations such as this, the constraint is generally taken to be that some bare minimum amount of volume per astronaut is need.  I'm not exactly sure what biological justification they use for this, but I'm pretty sure that not going insane is a big part of it.  My calculations put that volume in the ballpark of 100 cubic meters per astronaut.  That figure is complicated by the fact that it is not currently occupied by the intended number of astronauts.  There's also a lot of equipment.

For a distribution of artificial gravity rotating tubes in a zero-gravity atmosphere, I've made a reference design which also has a particular packing factor of people per unit volume.

In both of these cases, we're assuming some density constraint given to us from some other type of engineering (aside from the station design itself).  In the case of the ISS, human-factors engineering dictates a volumetric population density, and in the case of my reference tube design, various energetic constraints prevent sizes much larger (which I hope to write more about later).  Neither of these are very absolute, and compromises by the inhabitants would allow you to use a higher density.

At a certain size, however, there is a thermal limit based soley on how fast the station can radiate away its heat.  Assume a spherical habitat (I understand this is a bit of a physics meme, but in this blog it's literally the case).  With that, we very easily establish the maximum population that can be housed with a given sphere radius and a given radiator temperature.  This assumes the radiator covers the entire spherical surface and there is perfect heat transfer from the habitat's core to that surface.

Maximum Population Constraint Given

- Density Limitation and then

- Heat Transfer Limitation

It's obvious from the form of the equations that this concern is relevant for large sizes, but not for small ones.  The ISS still has a thermal management system, but it only needs a small radiator compared to the size of the entire station.  The density-based maximum population grows with R³ (radius cubed) and the heat radiation constraint grows with R².  That means there will be a cross-over point.

For the given parameters, these population constraints are plotted below.  I entertained two cases, one where people use 2 kW on average (which is consistent with the modern developed world electricity use), and one where they use 10 kW.  The electricity use alone doesn't account for all energy use, and then there is the issue that farming is very energy-intensive and may need to be conducted within that habitat.

Log-Log Graph of Habitat Diameter versus

Maximum Population that can be Sustained

The cross-over point is fairly obvious here.  To give real numbers, here is a table.

Radius of Crossover from

Density-Limited to Energy-Limited

and Maximum Population at that Radius

Assuming Maximum Packing Density of 22,000 people/km³

Per-capita energy use	Radius	Population
10 kW	5.7 km	17 million
2 kW	28.5 km	2.1 billion

Small asteroids-turned-habitat would not be limited by the heat removal, since they couldn't get much larger than a 10 km inner diameter.  But then again, a very large habitat size is quite desirable.

For someone designing a habitat of a size much greater than these crossover points, they'll have some decisions to make.  Imagine they're given population as a prior constraint.  They might simply choose to have a lower population density in order to avoid having to deal with the heat rejection issue.  On the other hand, to make use of air and asteroids as efficiently as possible, perhaps they'll implement some more exotic system.

The constraint that the radiator lies on the surface can be broken, and it wouldn't be particularly difficult.  A neat approach would be to use geosynchronous orbit to increase the effective radiating area.  This is logical because most asteroids rotate with relatively fast day-lengths, meaning that it wouldn't be difficult to build the needed "space elevator" from the surface to GEO, and also run large pipes over that distance.  Nonetheless, the material constraints would be much more significant in this case.  If you're only working between the inner edge of the habitat and the outer surface, there are lots of ways you can use the asteroid rock to help make building the coolant channels easier.  If you're "over-sizing" the entire thing with the ring radiator concept, then you're going to be tacking on a lot more material constraints.

So I sketched a brief image of what I'm talking about.

A Ring Style Radiator for Increased Population
in Large Gravity Balloons

Natural circulation would be desirable for this configuration, but it would not work beyond GEO.  Alternatively, your pumping force could be from natural circulation from GEO to the center, but it has to overcome the density difference pumping in the other direction from GEO to the outer edge of the radius.

Yet another problem is whether the ring approach would buy you any increased area in the first place.  For the concept to make sense, GEO would have to be much further than the radius of the gravity balloon.  With short rotation periods, this isn't quite the case.  Small asteroids rotate at a variety of rates, with some of them spinning so fast that GEO is below the surface.  Those are too small for the purposes of a gravity balloon in the first place.  In fact, most objects would have a much greater area covered in the GEO circle than by the outer surface area.  Additionally, this ratio will grow when the gravity balloon is inflated.  That process increases the moment of inertia, so the rotation rate slows and the GEO point moves outward faster than the surface itself.  If it is too far out, however, material limits will become a problem, as well as debris and other things.  Generally, the ring radiator would be a much more challenging concept to build.  But that makes sense, considering that it would never be built unless the size of the colony was gigantic to begin with.

I like to think of Virga as a world that is limited by heat removal.  The author even makes some references to ice formation on the inner side of its shell.  Heat removal limitations are actually a convenient explanation for the density of people in the sphere.  In the story, rotating cities are separated by great distances.  We can say this is because the atmosphere would get too hot if the density was much higher.  It's a neat idea.  It paints a picture of relatively empty and expansive gravity balloons when their diameter exceeds 10 km.  However, make no mistake that this is a constraint, not exactly a good detail.

Clover Orbits and Heat Removal

Without engineered solutions, any gravity balloon is thermally isolated.  The 10 km of rock would prevent any significant heat loss by conduction, and there is no other mechanism to get the heat out.  An engineered solution would have to entail some kind of matter exchange.  For small conceptions of the idea, this could probably be done in batch processes, which would also make more sense in the context of access tunnels and the airlock structure.  Once it gets much larger, however, there's a bit of a different challenge.  As the size grows, the habitable volume grows with the cube of the linear dimension.  In light of that, I want to talk about solutions for cooling that scale very well with large gravity balloons.

Pipelines in Zero Gravity

The idea of machines (and industrial society in general) spanning large swaths of volume in zero gravity is an exotic idea.  You can reliably have a process that involves huge exchanges of mass and even huge velocities while at the same time putting very little energy into it, and actually having very little rigidity to it.  The engineered systems in this environment could be large, complex, but fragile.  Imagine a normal train and the sheer impossibility of stopping such a thing, and multiply that concept by a thousand.  People won't even be worried about such huge huge things moving about because it would be so easy to move around it.  This is a true embodiment of the relative-ness of motion.  For a simple example, image a large pipeline of things moving in a giant circle.  This pipeline could be a simple ring, but with a thickness much smaller than its radius.

Moving from the inner surface to the outside surface requires doing work because there is a gravitational field.  If you're moving individual objects it also requires the energy to move in and out of the airlock, but we can imagine a smooth pipe that doesn't have this burden.  You don't necessarily have to put in the work against the field every time.  If the stuff you're moving is already moving sufficiently fast then it can make it out without much problem.  In a simplistic sense, we can imagine a set of tunnels in the rock, some going out and some going in.  After the cooling mass goes out a hole, then it makes its way to the "in" hole moving along the simple geodesic.  In this case, by geodesic, I mean a path traveled in free-fall.

Pragmatically, making a large pipeline of material going through an access tunnel would be quite an engineering feet, but mostly due to the challenge of keeping the air in.  In fact, the challenging of maintaining a pressure boundary over a moving surface is quite common in engineering and there is certainly no simple solution.  However, in the case of the shell of a gravity balloon, there would be abundant space over which to implement this solution.  Ultimately, this could provide an easier way for even people to get from the inside to the outside than by constantly messing with batch-process airlocks.  Another complication is that the shape would have to deform as the speed changes due to gravity.

Orbits Moving In and Out

Interested in how you would actually have to configure these holes, I asked about the problem in a limit case on Physics Stack Exchange.  To make the answer simple enough, we imagine that the rock is thin relative to the overall dimensions of the thing.  That now fits the same problem of a charged particle orbiting in and out of a spherical wire mesh at some voltage.  There was a very intriguing answer to that question.  Basically, there are huge number of ways to configure these odd orbits.  Not only can you have any given number of "dips" outside the shell, but for a given periodic number, you can have any degree of roundness you like.

The most practical case for industry in a gravitational balloon is obvious to me as the 2-exit orbit.  I've dubbed this a "2-gon" in the question on Stack Exchange because it exists within a family of chopped polygons.  This would be an ideal orbit for a pipeline into and out of a gravity balloon because it hits the walls at a high angle, limiting the distance of tunnels needed, and it can have a long path length within the center.  You can see in the question that for a larger number of dips it skirts the edge more, as opposed to making deep secant lines.

The 2-gon orbit image by Physics SE user JJ Fleck  

For gravity balloons of more moderate sizes, a large fraction of the orbit would be spent within the rock itself.  I tried to give some thought to the orbital shape within that rock.  I asked a question about a similar case of something orbiting within a field that was proportional in strength to the radius.  However, I realized that this doesn't fit the shell volume that I'm speaking of here.  The field is linear with radius but it is not directly proportional to radius.  That makes the solution for the orbit rather complicated.  Ultimately, it's sure to be a relatively small revision on what kind of shapes you can expect, but it needed to be mentioned.

The question remains of how practical this would be.  Particularly, how fast would such a cooling pipeline move?  We could use orbital dynamics, but there's another way to cheat and get an answer simply.  In the large case we know exactly what the gravitational field on the surface will be, provided we have the required internal pressure.  In this case the rock density doesn't even matter due to interesting mathematical conclusions.  The field is just:

$$g = 2 \sqrt{ \left( 2 \pi G P \right) } = 0.013 \frac{m}{s^2}$$

With this, I can generally answer the question of the orbit time for a coolant loop.  Again, for the large limit, there is a helpful simplification.  Just imagine the field on the surface to be constant.  Then we can get a time for the turn-around as well as a velocity.  The velocity should probably be kept well below the sonic velocity of the atmosphere, and ideally we would like to see round trip time less than a day or so.  To make this workable I assumed a generic velocity of 50 miles per hour, or 22.3 meters per second.  With that, the height a pipeline will protruce above the surface before naturally falling back down is calculated.

$$1/2 a t^2 = h \\ v = a t = sqrt( 2 h a ) $$

Numerically, plugging in the 50 mph figure:

$$ h = 1/2 v^2 / a \approx 20 km \\ h \approx \frac{1}{2} \frac{\left( 22.3 \frac{m}{s} \right)^2 }{ 0.013 \frac{m}{s^2} } \approx  20 km $$

This is a nice answer because it is on the order of the shell thickness in the first place.  That means that even if the case we're looking at isn't all that big, if the center bubble is just a few times greater than the shell thickness, the idea will be perfectly workable.  Now, with this speed we can imagine the travel times for certain things.  Even for a giant diameter of 250 km, it would still only take 3.1 hours to traverse the diameter at the speed.  You can then imagine multiplying this by 2 for the reverse trip.  Then, for the times spent outside the shell we'd be looking at about half the average velocity.  So credit about 10 km shell thickness, 20 km turn around distance, then a full orbit takes about 30 km of that distance.  This adds about .3 hours.  So for a full pipeline orbit through a 250 km gravity balloon, we'd be looking at a time on the order of about 7 hours.

There are several variables I haven't delved into, like the overall cross sectional size of such a space pipeline, but even with a normal "train" size, such a thing could be used to transport massive amounts of heat.  There are a lot of remaining complications, but it would scale very well.