If the Saturn V rocket along with its Apollo spacecraft was miniaturized, for example to 1/72 scale so it was five feet tall, could it still perform a moon landing like the Apollo missions and get back to Earth? The rocket equation only involves the percent mass of the rocket as propellant for a certain delta-v, so it seems that a miniature Saturn V would have the same total delta-v as the real thing.
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4$\begingroup$ This raises the question, "Can a miniature Moon pie get to Saturn and back?" $\endgroup$– B. Clay Shannon-B. Crow RavenJan 27, 2016 at 22:39
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11$\begingroup$ I don't see why you couldn't put a small marshmallow cookie sandwich on a Saturn flyby-and-return mission, though I'm skeptical of the scientific value thereof. $\endgroup$– Russell BorogoveJan 27, 2016 at 23:46
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3$\begingroup$ Keep in mind that the rocket equation is only valid if no other external forces (such as friction from air resistance) act on the rocket. It essentially assumes that the rocket is already in space. So yes to a theoretically perfect 1/72 scale model having the same performance characteristics in space. But no to it having the same capabilities if launched from Earth. If NASA could put anything on the moon using 1/373248th the fuel, they'd probably be doing that all day long. $\endgroup$– arothJan 31, 2016 at 1:32
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6 Answers
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If you could miniaturize each component uniformly, you're correct that the rocket equation terms would all balance out and the rocket would be capable of the same delta-v performance.
However, the impact of atmospheric drag would be much worse; drag force is proportional to cross sectional area, not to volume. A Saturn V loses less than 1% of its delta v potential to drag; a five foot version would probably lose so much to drag that it couldn't reach orbit.
I imagine there would be other square-versus-cube-scaling problems involved in ridiculously tiny fuel pumps, injectors, and combustion chambers as well, probably making the engines vastly less efficient or preventing them from working altogether, but since I don't know much about those issues we will assume some sort of elfin magic is taking care of that end of things.
answered Jan 27, 2016 at 7:26
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13$\begingroup$ I think the (pressurized) fuel tanks would suffer from the square vs. cube scaling problem too. With half the diameter you could make the walls half the thickness but you’d only have a quarter of the volume. $\endgroup$– MichaelJan 27, 2016 at 14:10
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5$\begingroup$ I am not a mechanical engineer, but this thread suggests that for pressurized tankage, mass runs proportional to volume: yarchive.net/space/launchers/fuel_tank_scaling_laws.html $\endgroup$ Jan 27, 2016 at 16:05
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10$\begingroup$ @RussellBorogove I am a chemical engineer, and mass of a tank does run proportional to volume and pressure - if you assume no corrosion allowance or surface coatings. So if you use uncoated stainless steel you are correct, otherwise not. Your link also mentions insulation, which obviously scales by surface area. $\endgroup$ Jan 27, 2016 at 20:45
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7$\begingroup$ Saturn V may have only suffered 1% atmospheric drag, but you can bet the drag the engine nozzles gave to the escaping exhaust was a lot more. Scale those engines down by a factor of 5 leaving everything else equal, and they're going to perform a lot worse. I wouldn't be surprised if it never got off the ground. $\endgroup$ Jan 27, 2016 at 20:49
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6$\begingroup$ The full scale Saturn V holds about 3.6 million liters of fuel and oxidizer. A 1/72 scale version would hold about 10 liters of same. If it were even remotely possible to get anything to the moon using only 10 liters of fuel+oxidizer, I think everybody would be doing it. $\endgroup$– arothJan 31, 2016 at 1:30
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You've got a few problems:
When a model is scaled to 1:2 it's size, its area drops by 1/4 but its volume by 1/8. This is known as the square-cube scaling problem. So you will have one quarter the drag but one eighth the impulse. Russell expands on this, so I'll address the other issues.
The Reynolds number of the air does not scale with the model. So you not only have the square-vs-cube scaling problem (which could be mitigated overcome by making the model a bit longer), you've got probably an order of magnitude more drag per unit of area. You can think of this as the air being thicker from the craft's point of view. At extreme (nano) scaling the air molecules can even act as discrete particles colliding with the model!
If acceleration is scaled then the model will spend much more time in the atmosphere. Each second not in orbit will use up to 9.81 m/s of delta-v on gravity drag, no matter how much your craft weighs or how much propellant you brought with you.
answered Jan 27, 2016 at 13:37
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$\begingroup$ Hmm, regarding the "1/8th problem": The main or almost-only thing the propellant in a rocket is lifting - is the propellant. So I believe that "just scales" as Russell says. (Of course 2 & 3 are correct.) $\endgroup$– FattieJan 27, 2016 at 18:43
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9$\begingroup$ Regarding 3, to a first approximation, acceleration wouldn't be scaled -- 1/8 the fuel flow rate produces 1/8 the thrust divided by 1/8 the mass. The cross-sectional drag and aero viscosity issues are the real show-stoppers. $\endgroup$ Jan 27, 2016 at 19:04
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In addition to the other answers, you'd also have to take temperature into the equation.
Your cryogenics tanks have much thinner thermal insulation, primarily LH2 (LOX was "insulated" by the tank hull + ice, yes, really) but the cryogenic is still at the same low temperature. Much more ice build-up on the outside in relation to the Saturn's total weight. Much more pressure build up in the tanks due to increased boiling. Instead of a smooth lift-off I'd expect some crackling, rupturing and hydrogen/oxygen rivers on the pad...
But you'd definitely need not worry about pogo or combustion instability in the F-1 engines. :-)
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Even setting aside the other problems, rocket engines themselves do not scale indefinitely.
You decrease the scale of a rocket motor, you decrease the thickness of the rocket chamber walls, which in turn decreases the bursting pressure of said rocket chamber.
If you scale the rocket down enough, this will become a limiting factor.
At that point you will have to drop the pressure of the rocket chamber to avoid bursting it, which, since the maximum specific impulse achievable by a chemical rocket is limited by (among other things) chamber pressure, means that there will come a point where your specific impulse has to drop as you decrease the scale.
And sooner or later (read: sooner), it will reach a point where the specific impulse is low enough that the effective delta-v is too low to reach orbit.
You have analogous problems with thrust (pressure limited, again), temperature inside the rocket chamber (surface area to volume ratio increases as the scale decreases, causing the percentage heat loss to rise), and tankage (pressure limited, yet again).
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Also don't forget that the tanks would be thinner in general, making it much more likely to break through the tank, and suffer problems. The tanks need to be able to withstand the pressure and the weight of the payload and rocket. Thin walls would probably still hold the weight, but they probably wouldn't hold the pressure.
Overall, it'd be nice if this would work, but there are a lot of problems in terms of miniaturization, and all orbital rockets are big for a reason.
answered Jan 27, 2016 at 15:30
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1$\begingroup$ They're big because payloads are big. $\endgroup$ Jan 27, 2016 at 17:01
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2$\begingroup$ @MichaelKjörling welcome to Kerbal Space Program. $\endgroup$– user5892Jan 28, 2016 at 20:12
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2$\begingroup$ Actually there are a few rockets whose payload fairing (can) have a larger diameter than the upper stage. $\endgroup$– MSaltersJan 29, 2016 at 10:32
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1$\begingroup$ @MichaelKjörling the Falcon Heavy also has a larger diameter for payload than the upper (or lower) stages. Or even the Atlas. $\endgroup$– user5892Jan 29, 2016 at 15:24
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While you can miniaturize most of the brute-force parts (insulation, no. Engine walls???) that's not all that's in the rocket. There's a bunch of electronics on board and almost all of it is as small as they could build already, no miniaturization possible.
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2$\begingroup$ Saturn V instrument unit: 2000 kg. Saturn Five Foot equivalent mass budget: 5 grams. I bet the core chips in your phone are probably less than 5 grams: a much more capable computer, with gyroscopes, accelerometers, GPS, etc. Now, the battery may be a problem... $\endgroup$ Jan 29, 2016 at 2:10
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$\begingroup$ @RussellBorogove The computing power I agree about--but not the wire and antennas. $\endgroup$ Jan 29, 2016 at 3:32
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$\begingroup$ @RussellBorogove Add to the wire & antennas the high power transmitter stages--and their power sources. $\endgroup$ Jan 29, 2016 at 3:52
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$\begingroup$ A joke pointed that if the Almighty decided to shrink the universe to one tenth of its original size, the sausage shops will immediately notice it, all sausages will drop down, as if it were hung on a cord, the weight a string can hold is proportional to an exponential of its size, reducing cord diameter in a half will reduce strength in much more than half. $\endgroup$– UrquiolaJun 13, 2017 at 18:23