Thursday, December 18, 2014

Simplified Fully Turbulent Argument for Flow Dividers

Recently thinking about the core technology for a gravity balloon habitat (the friction buffers in my prior terminology), I realized that I have yet to confirm that I have convinced anyone of its function. Further deconstructing the disconnect, I'm not sure how many people have been convinced of the problem (a few, at least), and only that subset of people are eligible to understand the proposed solution in the first place. My current guess is that time is just needed to process it. This concept took me quite some digesting at first.

The underlying equations can be as simple or as complicated as you like. A true freely rotating cylinder isn't nice to work with, and for the laminar solution it even has a paradox for it. Since my prior post which handled the issue was relatively general over all flow regimes, it was also less easy to understand. Here, I'll simplify things to show the scaling that will most likely be relevant.

Extremely Simplified Argument

If you'll indulge only the most basic dimensional analysis intrinsic to almost all turbulent drag calculations, the motivation and operation behind the friction buffers becomes clear very quickly. Imagine a baseball flying through the air. Most physicists wouldn't hesitate to quickly categorize the force as being proportional to v^2. If you want to think in terms of units of power, it's a very quick step to say that (power) = (velocity) x (force). That means that power has a form of v^3 in the same way that we said that force has a form of v^2. With this, we can get a dependence on the number of sheets.

Variables Needed:
  • N: number of sheets
  • tau: shear pressure on outer wall of habitat (Pascals)
  • f: Darcy friction factor
  • rho: density of air
  • v: velocity of air within a single channel
  • V: outer velocity of the habitat itself
  • P: power per unit area

The argument is accomplished here in its most simple form. Our equation for shear pressure (the retarding force per area on the cylinder) only involves the velocity out of all variables impacted by the number of sheets. Since the relevant channel velocity is divided by the number of stages, power ultimately goes with 1/N^2. Clearly, adding more layers of sheets will reduce friction.

Including Friction Factor

But why does this not involve the width of the channel? Because to a first approximation, the width of the channel doesn't matter, and that is hidden as a dependency of the friction factor "f". Of course this is a bad assumption. Intuitively the friction between two sheets moving parallel depends on the distance between them. The root of the prior formulation is that fluid velocity only increases logarithmically from the distance to the wall. Given high enough Reynold's number ln(x) is kind of a little bit flat-ish.

Clearly this is unsatisfactory, so to remedy this we merely rewrite things with a form for the friction factor - which is where this dependency lies. This is commonly illustrated in the Moody Chart. Here I have reproduced that chart from the Colebrook-White equation.


Many approximations exist for the friction factor, but we know almost exactly what regime we'll be in. I plotted those points for different numbers of sheets, along with an approximation for the reference tube design. We'll plug that approximation into the prior format.

As a minor detail, Moody's chart has the roughness in it as well. This will penalize adding more sheets technically, but it doesn't matter in the end. The reason is because decreasing the channel width increases the size of the roughness to channel size. However, even if we have 20 sheets, that leaves something like 5 meters. The sheet itself should only be a few milometers in my view, so clearly the roughness can't be more than this. Comparing to the values displayed here that's off the chart. Might as well consider it smooth.

New Variables:
  • d: half-width of a single channel
  • D: width of entire friction buffer region

With this specific approximation, we can plug back into the previous equation in order to have a very definite form for the shear stress. To go from this to power, multiply by the overall V. This involves a few steps and a little bit of math, but at the end we'll have a very useful form.


Let me stress that this is an accurate representation of our situation as long as we're in the turbulent regime. What is this constant on the top though? That's the power you would have in the case of N=1 for a specific reference case. That still doesn't represent a rotating cylinder in a free atmosphere, because it utilizes a standard channel width of about 100 meters.

I find it entirely arguable that you would have to install a flow arrester like this if you built any kind of in-atmosphere artificial gravity tube... because if you don't you'll have very high air currents to deal with. Using this flow arrester will increase drag somewhat from the scenario where you don't use it. Anyway, it forms a reference case that I need in order to make this math comprehensible.

So the basic proposition rehashed is:

The economic case seems pretty clear. Even in Manhattan people average something like 30 square meters of land per person. In the reference case, that would amount to 15 kW per person, which would likely dominate energy consumption. This doesn't mean it's impossible. You could do this, but if you did, thermal and energetic constraints would be setting your engineering limits at just about every turn. Clearly you wouldn't go this route if a better alternative existed - and it does. The formula is quite simply to scale the number of flow divider sheets to achieve the necessary power level.

Power Calc Stage-by-Stage

One might also be confused by the calculation of power. Indeed, for the fluid dynamics calculations we used "v", lower case v, which is the half-velocity within a channel. But to get total power consumption, we multiply shear stress by "V", the total velocity shift over the entire friction buffer region. This is justified because the torque to keep the tube rotating is applied relative to the stationary reference frame, but it is dragged on by the fluid forces present in only one layer.

But I don't expect everyone to believe that right away, so I'll show the other approach as well. Let the index _i mean that a quantity is for a single stage. To sum up power, we have to add a factor of 2 because these aerodynamic forces exist on both sides of the sheet.

Variables:
  • P_i : power for a single stage
  • P : total power summed for all stages
  • tau : shear stress, there is only one value of this, because force is transmitted over all layers

I don't think this is particularly profound, but perhaps it needed to be stated.

Appendix

The expression for "tau" uses 4 f in it. This is done in order to employ the Darcy friction factor. If you used the Fanning friction factor instead, you would just use "f".

Reynolds number contains "d" for the width of the channel. We know there are N channels, however, the equation for shear stress doesn't apply directly to Couette flow, but instead a channel with one free end. This is mathematically the same as half of one of our channels, so the operative d would be D/2. However, hydraulic diameter in Reynold's number must be found by using 4Ax/Dx, where Ax and Dx are the flow area and diameter. This comes out to 2 times the width of the channel in question. Combining these two factors, we see that the linear dimension to be used in Reynold's number is D/N.

Wednesday, December 17, 2014

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.

Tuesday, December 16, 2014

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.

Tuesday, December 9, 2014

Transit within Space Mega-Cities

This is intended to be the first post in a series on transport issues applicable to moving between one artificial gravity cylinder and another (not necessarily specific to a gravity balloon). The growing list of topics under this umbrella, new and old, have grown so much that a full series is needed. This post acts as a summary of those topics and establishes a central claim.

In particular, I want to talk about a billion person city. That is, a 3D city in space or inside an asteroid which has on the order of a billion people. For the purposes of this topic, I will use the term colony to denote a rotating artificial gravity cylinder in such a cluster, and the word city to refer to the entire cluster. In previous posts, I outlined how 1 billion people is just on the elbow of population constraints shifting from packing density to heat removal. Even that latter constraint isn't absolute, but it's not important because transportation times will put larger sizes in a different stratum anyway. The level of cultural connection is fundamentally different between occasional travel versus daily service.

Just that word "daily" introduces all kinds of qualifiers in the first place. Any concept of a day/night cycle would be artificial. My position is that this is only slightly different from modern cities on Earth since artificial light became rife, except that there's no natural suggestion from nature. The specifics of how a society manages the biological need for a day/night cycle is one thing I don't care to take a position on. The only kind of assumption I'm interested in here is how many trips per day someone takes between seperate colonies. Ideally this wouldn't include grocery trips or other relatively fungible activities. However, the entire draw of such a city is the connectedness, so I expect the relatively routine days of living in one colony and working in another would be balanced by extremely active collaborators. If you only go between your work and home colony that will obviously result in 2 trips. To balance things out, I'm saying 4 on average per person per day, for 4 billion A->B trips each day.

How you accomplish this volume, and the specifics of the physical manifestation is what I'm interested in here.

1. The Comparison

As an alternative hypothesis, this discussion will entertain the idea of lots of conventional artificial gravity habitats packed closely together. That conventional concept is an independently pressurized colony that rotates to create gravity. The problem I'm interested in here is inter-colony transport. Inter-city transport would be very different, the demand for it isn't well established, and there's not much I could say anyway. Comparing to the gravity balloon, the one obvious difference is that the space between the colonies is vacuum.

This still accomplishes the argument that a larger number of smaller colonies will have more surface area than a large single colony with the same volume. Transit is a lot easier with this system. Transportation within a colony is not much easier than transportation on Earth since it is a terrestrial-like environment. With no convenient fossil fuels you'll find public transportation more appealing too.

It should also be noted that these colonies would need to be tethered together in some way to avoid eventual collisions and resist destruction due to any perturbation. For instance, if a transport ship lost its pressure boundary, that air will then impart an impulse to the nearby colonies. It's obviously necessary to prevent those disturbances from causing millions of people's death by slow drift and collision of their colonies.

My reason for introducing this dichotomy is to identify very stark different between the gravity balloon and the conventional scenarios. Starting out, we don't want to prejudice ourselves to one or the other. Indeed, I think there are scenarios where the conventional approach might be better. It all comes down to what it is that you value. My predictable insinuation, however, is that if you desire a hyper-connected community of a billion people, the gravity balloon is almost certainly your best option... mostly because it doesn't need airlocks for intra-colony transportation.

But first, we need to formalize why airlocks matter.

2. Transport Network Topologies

Transportation can either be on-demand or mass transit. On-demand can entail either a personal or a shared vehicle. It also doesn't have to be the case that an on-demand vehicle is for a single person, but if it's not this will impose additional constraints which I don't want.

My perspective is that whether a trip is serviced by on-demand or mass transit depends on the volume of demand for the trip, measured in people per second. While the ideal volume for mass transit is disputable, I put it roughly in the range of 1.0 people per second. However, the minimum acceptable volume also depends on the number of transfers someone has to take during their trip. Doing 5 transfers at a cost of 1 minute each is taken to be equivalent to waiting 5 minutes for direct service.

Quantifying the difficulty of a transfer isn't precise but is still obvious. Going through an airlock is going to carry a higher time cost than stepping from one platform to another.

 2.1 Spoke-hub

On one extreme, the spoke-hub network topology is heavy on transfers. If all colonies only connected to one hub by their own dedicated route (or "spoke"), then even to get to your neighbor colony you'd have to travel to the very center of the city, transfer, and travel to the destination.

This complexity buys you a reduction in the total number of routes that have to be planned and maintained. Since the number of travelers is a constant, this increases the volume that each route sees. The 1 central hub is the extreme example, because there's only 1 route per colony.

Such extremism is not necessary. If you turn this into a hierarchy then the more central routes and hubs will have even higher volume than the local routes. As such, this method lends itself nicely to mass transit. The local routes will inevitably see usage rates 8 times the population of the colony per day. Since this will be in the 1,000s at least, any literal hub-and-spoke topology could be serviced entirely with mass transit.

Relaxing the extremism a little bit more, you would likely add cross-links either between colonies or regions. Since the volume is so high, this can be done while keeping mass transit viable, but still reducing the number of transfers.

Our tradeoff for extremely high volume is a larger number of transfers. I'll call the number of hierarchy levels I, and the number of regions/colonies serviced by a hub N. Then the total number of colonies is N^I. This established, the maximum number of transfers would be 2*I-1. If you used the smallest possible value of N, N=2, then you could be facing on the order of 30 transfers.

Such transfers would have to be virtually instant in order for the transportation system to be practical.

2.2 Point-to-Point

By putting people into colonies, we are dividing up our billion people into groups which are probably 10,000s to millions of people. Even the smallest realistic number of colonies is around 1,000. Even with this number of people, mass transit could only be minimal in a point-to-point system. This is because the number of routes in a full point-to-point system goes with the number of colonies squared. You'd be looking at 2-3 people for every route every minute.

This still forms a decent alternative hypothesis. By making the colonies large (although practically so), regular flights between colonies can be possible with the ridership being a handful of people.

2.3 Penalties for Distance and Transfers

A space city might find either one of these topologies favorable depending on the type of city and available transportation technologies.

Broadly, larger colonies are associated with conventional colonies because the task of reducing air drag in a gravity balloon becomes more onerous. Larger colonies are also associated with greater propensity to point-to-point transport topology.

More transfers are also disassociated with conventional colonies. The self-evident facts are that all colonies maintain their own atmosphere, and that colonies are stationary relative to each other. That means that you will use a minimum of 2 airlocks. If find it unlikely that any system designer would want to do any more than this. Any mid-range transfers would likely avoid merging the atmospheres of the vehicles if at all possible. Airlock cycling is slow and expensive.

3. Physical Mechanisms

Routes and transfers have been addressed in the abstract sense up until this point. That is, a route connects a hub or a colony to another hub or colony, while a hub is a point where one can exit one vehicle and enter another.

Even without any statements regarding the possible or likely network topology, a great deal can be inferred about the transportation system of a future space city from the underlying technologies. Popular illustrations of space colonies with artificial gravity sometimes add one or two details of ships entering and exiting... but if you start increasing the expected volume things start to get real weird real fast.

3.1 Means of Docking

By docking, I mean mean moving in and out of a colony or a transportation hub. Docking entails getting on a shuttle, getting off, and even the process of finding your next shuttle.

3.1.1 Conventional

Ports are most commonly depicted on the axis of a rotating space colony. For a cylinder, these are the two ends. For a sphere, these are the two poles on the axis. This rotates along with the space station, but the rotation rate is slow, and the accelerations mild.

But this is inherently one-lane. You could dock off-center but this will use propellant which I find to be incompatible with a large space city because the propellant demand for billions of trips would be huge. The only alternative would be to surround the city in a barrier and recover and recycle propellant. As such, I find the only valid solution to be sending larger ships through the airlock. This will likely be a ship that carries other ships in order to satisfy the point-to-point topology.

3.1.2 Gravity Balloon

A colony inside of a gravity balloon has open ends. This means that, hypothetically, someone could just walk off the edge. For our (more serious) purposes, we need some kind of architecture that gets them somewhere else after leaving. Since most of my reference designs have ends 30 m in diameter or more, this will entail collecting the people into a smaller space.

The solution I've been leaning toward is to have people's path through the colony mirror that which the air flow takes. Come in at the edge of the inlet end. Either take an escalator, shuttle, or a slide down to the terrestrial environment. Once they decide to leave, a building which spans the full diameter of the colony rises to the center-line. Here, they enter a tube where the low gravity and air flow move them out.

Individuals still need to make their own decision about what route to take, so you would essentially need a lot of handles and rails so they can move through the hub to their transport, which will be mass transport. Other than this, there's not much else to figure out. The atmosphere is continuous, so the only reason for doors in a transit shuttle is to keep people from falling out. A similar logic will apply for the transit hubs.

3.2 Means of Moving

The primary difference between the gravity balloon and conventional cities is the existence of an atmosphere in the former. This is a detriment to power consumption and speed limits in transit. However, it also can be a nice thing to have something to push against.

My position is that whether an atmosphere is a merit or a disadvantage is a function of the total size we're talking about. Since a gravity balloon can more easily host an extremely high density, these attributes somewhat make sense in conjunction. Alternatively, some methods might be available to get extremely high speeds in less dense clusters of conventional colonies.

3.2.1 Conventional

In the conventional space city concept, we are forced to move relatively small shuttles through the vacuum between colonies. I'll start out with the assumption that these shuttles are mostly free-flying and I'll classify the issues into 3 major parts, and only briefly touch on them here.

After departing the docking facilities of a colony, the shuttle obviously has to first accelerate. Since we're not very interested in multiple km/s speeds, this can almost certainly be accomplished mechanically. But we also must keep in mind Newton's law of motion. The transit volume almost certainly rules out reaction engines, so the shuttles are necessarily pushing against either the colony or some ancillary structure. I would favor the latter. A non-rotating envelope could surround the colony to provide easy access and also transfer momentum to and from the colony through axial bearings. So this non-rotating structure could launch shuttles through mechanical tracks or even mechanical arms. However, each shuttle is going in a different direction. The shuttle departure and arrival rate is also fairly high, so you would likely have many of these systems serving a single colony.

Once you get going, there will necessarily have to be some avoidance software, if we assume a mostly crude method of moving in the direction of the destination colony. If I assume a shuttle is about a square meter in cross-section, then random walks through my reference space city would result in roughly 1 collision every week. This is probably most surprising in illustrating the enormity of the 3D space. This is even after subtracting the volume of the colonies themselves. Avoidance might be one thing for which propulsive maneuvers makes sense because they are both infrequent and random.

Not only do you have to avoid other shuttles like yourself, but the lattice of colonies needs to be navigated around, but I classify this differently in the category of steering. Because you'll be going on movingly a preplanned route, it's thinkable that your redirects can be done by stationary infrastructure along your way. I would imagine that magnetic forces would be the most ideal for this because frequent catching and relaunching would make for a very uncomfortable ride.

Lacking a good vision of implementation of a system that handles these issues, I'm tempted to say that free flying shuttles won't exactly be the architecture that a conventional space city winds up with. I think something much more akin to roads are likely, but very different space roads. I would imagine these are steel guiding rails which the shuttle only makes contact with during acceleration, braking, and steering. This road network, thus, would still somewhat resemble a hierarchical network topology, although certainly not spoke-hub. I envision something much closer to an interstate system. For the large volume routes, the shuttles might even combine themselves in something resembling a mass transit system, but still while avoiding connecting via an airlock.

3.2.2 Gravity Balloon

Moving through a continuous microgravity atmosphere is the same as planes on Earth, except without the added complication of gravity. Thus, no novel technologies are needed and we can concern ourselves only with the economics of the system. If we use an extremely hierarchical transit system, the basic unit of transportation would likely be the individual and they would make decisions about where to go by grabbing onto guiding anchors as I argued before. Starting from that point, we only need to further identify the technology options for high-volume routes. These could use several mechanisms to minimize air drag, but the logistical implications of these approaches are also important.

One approach might be to just not worry about drag. You could even use a rope tow to transfer people relatively locally between colonies of the local hub. That is essentially a simple pulley in zero gravity. Many of these trips would be around 1 km in length, but they would need to be completed quite fast and very safely. The biggest problem with a person directly holding on to a fast rope tow would be the they can't breathe. If everyone wore aviator masks, the speed could potentially be rather high. That raises another complication - of how to grab on in the first place. This could likely be solved by a series of rope tows in steadily increasing speed, but all within arm's reach to the next stage. This also sounds somewhat dangerous, so a simple alternative might be a shuttle moving along a rope. It could accelerate with its own wheels, or it could just be on a pulley system. The only drawback is the relatively inflexibility since these approaches very strictly connect either two places or a loop.

The rope tow might be the most simple system I can think of. However, if we want to go in the extreme of maximizing logistical prowess while still not worrying about energetic efficiency, even a pneumatic tube starts to look like a valid option. If a traveler's destination was obvious (as in a true spoke-hub topology), the system planner can avoid a transit center where the person chooses where to go, and simply opt to suck the people up with a current of air. Yes, like the Jetsons. Actually, this could boost a much higher throughput with much greater simplicity than any other alternative. On the other hand, it can't be very fast (without killing people), and it would be highly inefficient.

Fixing the problems with the pneumatic tube by adding greater complexity would, in fact, be possible. You could actually invent a system that borrowed the physical principle from the friction buffers that I've discussed for the rotation of the colonies themselves. This concept was obvious to me a while ago, but I found it completely ridiculous. However, after studying the nature of hierarchical transit systems, it seems that the most central transportation routes can quite possibly get used by a large fraction of all the trips taken in a day. I mean, one route might service 2 billion trips per day. This is 1,000s of people per second. Even at 100 miles per hour, you could have 100s of people per meter. Any conventional system would involve massive docks or massive ships. The logistics of getting people in and out of a shuttle would be mind-boggling at these volumes. So a pneumatic tube isn't absolutely a terrible idea.

Considering these economic pressures, let me propose a ridiculous idea which would feasibly service the ridiculous populations of a space mega-city. Take a closed tube that goes in a large loop - a torus. Now surround this tube in another tube. Do this over many tubes. Have the inner tube rotate about its perpendicular axis at the desired speed. Each enveloping tube will move slightly slower. Let's say 5 mph over 20 tubes for a total of 100 mph travel speed. Then, of course, you would have doors in each level of tube so that someone could eventually make their way to the central tube. This could handle extremely large volumes, would be 100% continuous, and the power requirements wouldn't be all that high because the flow is controlled carefully. The main problem, I would imagine, would be training people to use it.

For people desiring methods which are more familiar, borrowing from planes, both jet and propeller craft are possible. You would only exchange the wing for another control surface or two. For the biggest legs of a mass transit system, air-breathing shuttles could reduce energy consumption and increase speed by increasing their scale. This is a coherent vision, but I doubt that the air-breathing engine would be the ideal choice for acceleration at the start of the trip since a simple mechanical boost at the start would be fairly trivial. Direction changes can be handled slowly over the course of the trip by control surfaces, so any launcher wouldn't need to aim either.

We don't necessarily have to consider on-demand transportation, but that is also relatively easy. It might also be a preference under certain circumstances, or just plain fun.

4. Gravity Balloon Connectivity Argument

Here's the point:
Why build a gravity balloon space city instead of the alternative?
I've been as general and generous as I can with the conventional type of space city. Indeed, I'm not trying to argue that any particular reference design is bad, just different. The attribute where a gravity balloon is appealing is transit connectivity. One real reason for pursuing a gravity balloon is desire for an interesting, vibrant, and interconnected city in space. I did not go into numbers here, although I hope to get to those in later posts, so you'll have to take my word on matters of degree. I wouldn't be saying any of this if I didn't have some empirical basis ready.

The desire to mingle among a billion person community daily might be somewhat difficult to reconcile with conventional space colonies. Even if one favored the extreme over-sized space habitat of a McKendree cylinder, it would take a heroic technology effort to make daily transit among the entire pseudo-terrestrial area viable.

As we try to get design convergence in the reference conventional I mention, several issues pop up. Smaller colonies result in greater transit network challenges and shorter average travel distances. However, larger colonies raise some serious safety concerns due to the speeds of the rotation. Even on the large side of ideal design range, I'm absolutely sure that the space between colonies would involve a large amount of clutter in order to make the formation of the movement of shuttles workable and comprehensible.

A gravity balloon, on the other hand, has really weird possibilities for the transportation routes, and considering the size that we're talking about these might even be relevant. Either way, having air to push against is a huge benefit. Also, no separate life support system is needed for every trip and people can live without constant airlock cycling. Those issues with conventional colonies are not deal-breakers, but the gravity balloon clearly has much greater desirability in this context.

Thursday, December 4, 2014

Rockfill Pressure Boundary in Asteroids

Context of Void Use

This post expands on concepts introduced in a previous post about living in the voids in asteroids, and it is the writeup of the fractal image I presented in the post right before this one. I've come to call this scenario "under-pressure" because it entails using a pressure below what's necessary for the rocks to fully float against the atmosphere pressure. In the present context, the gravitational balance is really the upper limit to the pressure and radial extent for using the voids in the way I'm talking about here.

In that previous point I laid out a very vague idea of how a collection of smaller rocks (within a lattice of larger rocks) might form the backstop for the airtight lining. I'll look into this with a little more detail in this post. Rubble piles are likely to be a collection of (former) asteroids which are resting in contact with each other, held by gravity.  We don't know much about these structures yet, but the general idea is something like this, from a 1999 Nature article, "Survival of the weakest"

 
Visual Conception of Rubble Pile Interiors

As the different scenarios in the above image suggest, this illustration is essentially a guess at the interior of asteroids.  The constituent parts could be small, large, or a combination of them all.  Furthermore, there are a large number of candidate asteroids.  There could be a great diversity in their different interiors.

In order to build a space habitat in a cavity within one of these rubble piles, I would imagine that the ideal interior structure would be composed of rocks on the 1 km scale.  This is approximately the minimum scale needed to produce artificial gravity at acceptably low rotation rates.  It is reasonable to believe that an ideal candidate asteroid exists out there somewhere, although we can't say which asteroid that is.  It's certain that many of them have unusable interiors for this type of habitat, so the problem is just narrowing down the sample space.

Another issue is how one might seal off the pressure boundary around the empty space to be transformed into a habitat.  If you plan to ship in material to cover a kilometer scale gap, then you've obviously defeated the impetus of the concept altogether. We are obligated to think of approaches more clever than that.

Lattice Structure of Unmovable Rocks

As a best guess, we might as well imagine the interior to be some lattice structure.  We might also imagine that the rubble is roughly spherical rock.  The 3D packing of spheres is a well-studied problem, particularly in material science.  There are several configurations that an infinite lattice of this sort might have, but the most efficient structures are only 2, which are "face centered cubic" and "hexagonal closed packed".  For the purposes of this study, there is no reason to distinguish between them.  They are different geometries, but the important numbers such as coordination number and packing density are all the same.  The FCC structure looks like:



packing density = 0.740

Compare this packing density to the measured asteroid macro-porosity values.  You can see that asteroid porosity is all over the place, but seems to be roughly constrained by the FCC density as a maximum.


This makes a lot of sense.  Since FCC is has the most efficient packing density, and we expect that not all asteroids have differing degrees of porosity.  It also makes it clear that if you want to find the kind of ideal body I'm referencing, you really will have to cherry pick.  Of course, the above data set isn't comprehensive either, so there's plenty of space to find a body which approximates an FCC structure of the desired size.

Movable Rock for Boundary Formation

So we have our perfect candidate, what now?  The habitat (or at least the early version of it) will go in the interstitial space.  That space is an odd one, because it doesn't start out enclosed.  That leaves the builders with the burden of enclosing it.  This is a self-obvious reality of the gravity balloon idea.  If no excavation is necessary to get to the center, they the center obviously shouldn't be expected to have a well-defined cavity.  From there, we have to make it into a well-defined empty space.  That means sealing off a boundary within this contiguous interstitial space.  That would be very difficult if you used manufactured materials.  In fact, it would defeat the entire point.

The obvious solution is to seal off the surrounding pathways between the rocks with other rocks.  Refer to the FCC diagram.  The interstitial space has 4 close-by rocks, for which the centers are arranged like a tetrahedron.  Looking toward one "exit" of this space, we see 3 rocks arranged in a triangle.  This strange triangle space is what needs to be blocked. It's not hard to set a minimum amount of rock needed to accomplish this, because we could just fill it in in 2D. I took a fractal approach:


Specifically, I used a program to place circles in the empty spaces between the previous level of circles. This has an obvious branching of 1->3. With each level you'll increase the number of circles you're adding by a factor of 3, but not all of these will be the same size circle. Because of this, I found it easier to add circles in order of their size. Then, using the handy svg markup, place them on a palate. The presentation of this graphic does falter a little bit, because the circles can only be specified by integer values for their location and radius. This causes some of the smaller ones to be displaced a bit from where they should be.

This is useful because it provides a long list (as long as you want it) of circles that allows us to map the connections between volume and area. Importantly, this theoretically lifts the requirement for the liner which hermetically seals the air inside the habitat to have any material strength as the number of rocks diverge to infinity. In practice, the areas very quickly fall off to imperceptibly small values, which is expected in these fractal scenarios.

But now how do we resolve the problem of carrying this out in the real world? In the previous illustration you would have to commit more resources to secure these rocks in place. In practice, you would prefer to use a rock which is larger than than the hole, and sufficiently large so that its compressive strength holds up against the atmospheric pressure. This is a function of the contact angle and more complicated structural engineering, which I will not get into. For a vague idea of what we would do, I just multiplied the size of all the circles by a constant factor, which would seem to be mathematically sound. So collapsed to a 2D view, it would be something like this:


Now returning  to the volume vs. area correlation, we can produce some graphs from the realistic over-sized rock scenario. It will then tell us essentially what volume of rock we need to move into place, based on the assumption that we can find any size of perfectly spherical rock just laying around inside the asteroid voids.

Obviously that last assumption was stretching, so let me talk about where I intended to go with this.

Blasting or Cutting or Shattering?

Originally, I had wanted to get a figure in terms of tons TNT that would be needed to blast rock in order to create the size distribution of rock that I worked with in the above figures and graphs. However, once it was finished, it became apparent that this was a fool's errand. Firstly, how the heck do we quantify a metric for mass of TNT per fracture area of rock? Actually I found some useful references for this (pdf link).

All kinds of variables are involved, but you can still get a metric for mass of TNT per square foot of the fracture you're creating. At first, the reference gives mass of TNT per foot of borehole, but this is a function of the spacing of the boreholes and the burden above it. Obviously, combining the linear mass density of TNT of the borehole with the distance between the boreholes gives a area mass density over the fracture area. But what exactly should those parameters be? The overburden is completely non-applicable in the space application we're looking at.

Nonetheless, I picked some parameters just to see. My estimate was about 12 kg TNT / m2 of fracture. However, if you used this to cut all of the needed rock, you will quickly find that you'll be sending more explosives than what you would otherwise be sending as a pressure vessel to cover a similar volume. This would make the proposal absurd and clearly uneconomic.

Are there other ways to fashion rocks in the way you want without explosives? I picture space probes with chainsaw-like things attached. This would have complications too, since most of their cutting would be done on the center piece, they would either need really long cutters or they would need to excavate a very wide fracture so they could fit into it. Alternatively, a more simple and primitive method may suffice. Why not just bang rocks together in order to break them up? Nothing about this proposal actually required a well-controlled shape. So sure, why not? Although this assumes movable large monolithic rocks available to begin with. Plus, how do you shatter the largest rock if it's the biggest one around?

Finding out how much area needed to be cut turned out to be disappointingly easy. This is because the 2D cross section is a straightforward multiple of the spheres that would occupy it. It's just 4 Pi r2 / 2 divided by Pi r2, you get 2. But now what is the total area that we're dealing with in the first place? This requires the advanced science of calculating the area of an equilateral triangle.


For reference, the ballpark figure I'm looking at is D=1 km. Maybe more. In order to get the area of the rocks with the slight over-sized method I mentioned previously, you'd multiply it by that over-sizing factor. Again, none of this was particularly difficult or insightful, and it's not clear what the application would be in the first place, since smashing rocks together doesn't have an easily quantifiable cost.

Inside-out Regolith, Carried by a Million Robots?

In spite of all this, we're still not lacking for an argument that voids can be sealed with a hierarchical distribution of rocks. Why? Because nature already did it. We are relatively confident that many asteroids have large voids inside, and we are also confident that they have something kind of like a contiguous (even if dusty) surface on the outside. At first these facts seem contradictory. But it makes sense when you consider that 1) first large rocks congeal into a pile and then 2) smaller rocks continue to fall onto this pile. While some small rocks will fall between the larger rocks, some will get stuck. As more and more get suck, the probability that the next rocks that fall onto it will get stuck increases, up until the point when the surface is effectively sealed. Not sealed against gases by any means, but against particles falling on the asteroid.

As a bit of a troll argument, you could simply rearrange the rocks from the outside regolith into the structure to seal a void which I have discussed here. That would require nothing more than pushing them around. But this would be a complicated low-gravity maneuver of moving many rocks around the surface and interior of an asteroid.

But we have another point to grapple with - loose material still exists within the interior of the asteroid. Hypothetically (and somewhat realistically) you could run an inventory of all the rocks around the asteroid, and chances are that you'll have a good shot at finding a sufficient size distribution to make your seal without any blasting. Some will only need to be moved a small distance, for instance, from one void into the next, but many others will need to be dragged through the byzantine path through the asteroid pores. The largest rocks will be the worst.

In fact, the very largest one, the center stone, is the problematic one here. By the very fact that it can't FIT out of the void's entrance-way, we would expect that it can't fit INSIDE in the first place if we're hauling it from somewhere else. This isn't strictly geometrically true, but it's close enough. This leads to a dilemma. Maybe you cut the center-stone, and then glue the pieces back together. Thankfully, no glue is strictly required. If the forces balance correctly, then it can just "sit" there, held by the outward force from the habitat's pressure. I'm not very concerned about that detail.

The only real problem is that you'll have to move a lot of rocks. Exactly how many depends on the convergence between the price of moving stuff and the strength of the pressure liner. I would assume that the robots would move non-propulsive in a method similar to lead climbing in the outdoor pastime of recreational climbing. By attaching anchors and ropes at various points, there should be enough degrees of freedom to get where you want to go.

A Big Dump Truck of Tiny Qualifiers

I don't think this discussion is pointless. I think there is a compelling argument, but there are far more unknowns than what there are certainties. Also, an analytical approach isn't nearly as helpful as what I had hoped.

But none of this suggests that we should toss the idea out. We don't know if this method will be viable, but our best understanding lightly suggests it will work. This is still saying nothing of the vastly complex structural engineering problem that comes along with potential re-seating of the structural rocks. Once again, this doesn't appear as a game changer to me, it just seems like a tremendously complicated analytical task.