Friday, April 29, 2011

Shooting Gumballs

So, this is a recycle in a way. In the dim and distant past before I had a blog (read: last summer), a friend and I spent an interesting afternoon working out the answer to a scenario we came up with, and I thought it was fun, so I'm posting it here at Engineer in Progress. Let's say NASA and commercial comes through, the whole shebang. 4 commercial crew vehicles funded to completion by CCDEV (SpaceX Dragon, Boeing CST-100, SNC Dreamchaser, and Blue Origin's....thing). Between NASA and commercial, 4 man-rated US launchers (SLS, Atlas V, Falcon 9, Falcon Heavy). ISS is fully utilized, Bigelow gets things going on a commercial station, Astrobotic goes to the moon and wins the Google Lunar X-Prize (or some other comparable team does).

But you don't work for any of those companies. You don't build launchers, or capsules, or stations, or unmanned landers. You work for Space Supply Corp, Inc, Ltd and your division is bidding on a vending machine contract. Specifically, you're bidding to refill these critical pieces of equipment:
That is, gumball vending machines. Well, a version of them that doesn't use gravity feed. This isn't 2001, we don't have artificial gravity! This is near-term gumball-machine resupply contracting, after all. much does an astronaut need to feed in to get out his gumball? Glad you asked!

The first question is shipment method. The average 1" gumball consists of a spherical shell about 1/8" thick, surrounding a central cavity that is approximately atmospheric pressure. This central cavity poses a problem: on the ground, the gumball does not experience a significant net pressure across the shell, since the air inside and outside are pretty similar (off by maybe a few kPa depending on the weather and the altitude of the ambient conditions of manufacture and measurement). In orbit, though, the gumball center has a full atmosphere inside, and a vacuum outside. Thus, it's acting like a pressure vessel, with the pressure inside being like being 30 feet underwater. If the gumball shell is not capable of taking the stresses of this, then the shell will fail, and either simply rupture or pop like popcorn (hence the highly technical term "popcorning"). The Space Shuttle External Tank's anti-ice insulation (which is made up of similar closed cells of foam) would do the same after time in space, which is why no studies for a Skylab-style wet lab were ever really considered seriously--huge risk for orbital debris.

So, if the gumballs aren't strong enough, they pop. How do you know if they are? Engineering! The material stresses on a spherical thin-walled pressure vessel are as below.

Sigma (the o with the line on the left) is the stress on the material. If this exceeds the maximum material properties of un-chewed gumball (chewed gum is over course deformable, which would change things in ways I'm not entirely trained to handle at the moment, but thankfully Space Supply Corp. Inc. Ltd does not supply NASA with ABC gum), the shell will break. In the equations, p is the pressure, so here it's 101 kPA (atmospheric in kiloPascals, the metric unit for pressure). The t is thickness, and the radius is r. Keep these units consistent, and they'll cancel out leaving just a ratio. (For the record, the radius-to-thickness ratio for 1" gumballs is slightly too large for the thin-wall approximation according to rules of thumb. It's 4:1, when 10:1 is supposed to be the max. But you get free engineering secrets for gumball shipment, you take my baseless assumptions.).

Calculating, the material stress is 404 kPa. So...does the gumball pop? Unfortunately, I can't find any research on the material properties of unchewed gum (attention scientists!), but this is a required strength something like 1/2th the minimum strength of rubber, so I feel confident in saying it has a good chance of working. This is nice, it means you don't need to use precious volume inside a pressurized transport like Dragon, Cygnus, CST-100, Soyuz, Progress, whatever, you just ship up your gumballs in a bag in Dragon's trunk, or the HTV external pallet bay, or something like that. You'll need an astronaut to grab it with a robot arm or snag it on an EVA to bring it in, but hey, it works.

SpaceX Dragon Approaching ISS
Unpressurized cargo like gumballs can go inside the cylindrical trunk section.
So, now we have the shipping requirements (unpressurized transport, some minor details on receiving). What's the cost? The current Falcon 9 carries 10.5 tons to orbit, but the version coming with the Merlin 1D upgrade is supposed to increase that to about 16 tons. Both are to cost about $56 million. This is a price of either $5,333 per kg or $3500. Wikipedia says Russia's Proton costs about $4400/kg. I'm not running numbers for the Atlas V or Delta IV, I can tell they will be worse since they cost more than Falcon by a factor of 4 or more for similar base payloads, and the pricing doesn't get better for Heavy versions. Falcon Heavy, on that note, is supposed to get down to $2200/kg.

So, what does this all mean for our gumballs? A 16 lb bag of 1" gumballs on Amazon costs roughly $30 (though the actual specs at the bottom say 17 lbs...odd), and contains 850 balls. Thus, each masses about 8.75 grams. Since the bag comes included in this mass and gumballs are probably vacuum-rated, there's no further shipping mass. So, cost to fly is easy if you completely ignore any cost-sharing or free-rides with other customers: Multiply the per-kg launch cost by the mass in kg of one gumball, and find the cost to user on orbit. For Falcon 9 currently, this would be $46.66, or $30.63 on the Merlin 1D variant that's supposedly coming soon. The Russian Proton workhorse would cost you $37.6 dollars per gumball, while the Space Shuttle is in the range of $291.67 per ball. Falcon Heavy's goal price would see a gumball representing a cost of $19.25. However, while the astronaut is carefully feeding those 77 quarters into the little slot and cranking the handle, note the shipping markup: the cost to Space Supply Corp, Inc, Ltd is only $30 a bag for 850, or about 3.5 cents per gumball, or a shipping and handling markup of about 5400%. For the record, this also means that the 25 cent gumballs you see so often are selling for a retail markup of 200%.

However, the gumball isn't a great metric for spacecraft themselves, just an example of the costs of shipping even a relatively tiny but imaginable payload--this same economics applies to every T-shirt, can of tuna, and bag of wet-wipes sent to the ISS crew and any potential future space exploration. Cheap to supply, expensive to ship. Spacecraft are a bit different, since even a small spacecraft (a bit heavier than a backpack with a few engineering texts in it) might contain several million dollars of specialized components, plus the spreading out of the cost of integration, testing, design and development before selection of a final's easy for me to design something like this costing $10 million that costs no more than $50,000 to launch to orbit where spacecraft cost is only half a percent of total cost. However, any base needs some kind of supplies, and these will include a large portion of the cheap-to-buy, expensive-to-ship variety I mentioned. ISS uses a good amount of food and water, and when HTV-2 burned up on return from the ISS earlier this year, a large part of the trash it carried was foam that had ridden to orbit wrapped around science experiments.
Astrobotic Lander
110 kg of payload to surface available--for a price
As a final exercise for the reader, the Google Lunar X-Prize team Astrobotic has a publicly posted payload planning guide that says a base price of $1.8 million per kg. Find the cost of a gumball delivered to the lunar surface. Highlight for answer:  $15,700 or so. A lunar gumball costs about the same as four tons of supply-cost gumballs, 62.8 thousand Earth gumballs, or about 424 orbital gumballs.

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