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There was, briefly, a discussion recently of the use of Mentos and Coke to get to space. It wasn't a serious question but pointed to a way of understanding the core of how rockets work.

Last year I asked a variant of this and had enough trouble understanding the answers that I decided to leave it until I had seriously brushed up my math and physics. There was a link in one of the answers to this

With staging, the delta-v of each stage can be calculated via the rocket equation and summed:

$$\Delta v = \sum_{i=0}^{n-1} V_{\mathrm{e},i} \cdot \ln \left( \frac {M_{\mathrm{initial},i}} {M_{\mathrm{final},i}}\right)$$

Where $V_\mathrm{e}$ is the effective exhaust velocity, $M_{\mathrm{initial}}$ the initial mass, and $M_{\mathrm{final}}$ the mass of the rocket at the point of burnout of each stage.

When the $V_\mathrm{e}$ and mass ratios are the same for all the stages, this simplifies to:

$$\Delta v = n V_\mathrm{e} \cdot \ln (M_{\mathrm{ratio}})$$

and it can be seen that the delta-v is limited only by the n, the number of stages.

That bends my mind. It seems to say if you have essentially infinite fuel and stages, you can get to space under any conditions - really high gravity, really poor fuel, really bad engines, gigantic payload, whatever. But that can't be right. There must be some definable limit to all of the different elements.

So here on Earth, taking the propellant part, what is the very minimum expansion rate - if that is a decent plain English equivalent to specific impulse - needed to get a rocket to orbit?

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  • $\begingroup$ Let me know if there is a better way to frame the question, define it to be more answerable. Though it is a little late now and i won't get to it until the morning. $\endgroup$ – kim holder Apr 11 '15 at 3:22
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    $\begingroup$ Beyond just delta-V, if you're launching from a massive body like Earth, you also need thrust-to-weight ratio, because both gravity and atmospheric drag are "eating" your delta-V gains. I was a little annoyed that the Mentos rocket question got closed with one "not even theoretically possible" response that wasn't supported with a theoretical explanation. $\endgroup$ – Russell Borogove Apr 11 '15 at 3:48
  • $\begingroup$ @RussellBorogove yeah, that occurred to me vaguely, but i don't understand how the whole thing interrelates. I'd like to make sure this question is clearly answerable. I can understand that too many variables make it tough. From KSP i get that T/W needs to be over 1, but even with mentos and coke, maybe if your payload is small enough that is possible?? $\endgroup$ – kim holder Apr 11 '15 at 3:56
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    $\begingroup$ 1+ TWR is absolutely possible, as evidenced by videos of Mentos/coke rockets taking off. $\endgroup$ – Russell Borogove Apr 11 '15 at 4:04
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    $\begingroup$ Randall Munroe explored this in what-if.xkcd.com. $\endgroup$ – Hobbes Apr 11 '15 at 17:08
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A vertical Mentos-coke geyser through a not particularly optimal nozzle reaches an altitude of roughly 6 meters (by my eyeballing), implying exit velocity of a little over 5 m/s, equivalent to an Isp on the order of ~0.5s. (I also see references to 40-foot/12-meter Mentos geysers, implying that figures closer to 8m/s or 0.8s ISP are possible).

The existence of Mentos-coke rocket videos indicates that such a rocket can briefly attain greater than 1:1 thrust-to-weight ratios.

Let's assume for a moment that we can create Mentos rockets of any scale with the same structure-to-propellant mass ratios as a two liter bottle.

2L plastic soda bottles mass about 50 grams empty, and hold about 2kg of fuel+payload. Say the payload is 20% (i.e. stage-to-stage mass ratio of 5:1) -- 400g payload and 1600g of propellant.

So you're looking at... 5 ln(2050/450) = a whopping 7.5m/s of delta V per stage.

So you need fourteen hundred stages of Mentos rocket to get to orbit. And each stage is 5 times bigger than the previous stage. The ratio of the first stage mass to the last stage mass is thus 51400, a number with 979 digits in it. Sooooo... yeah, not even theoretically possible.

(With a stage mass ratio of 20:1 you could get it down to more like 1000 stages. A stage mass ratio of 1000:1 gives you a whopping 35m/s delta V per stage, bringing you down to about 300 stages. Optimize the nozzle, improve the Mentos injector, maybe you can get the exhaust velocity up to 50m/s, now you're down to 30 stages -- but the final stage is still 100030 times bigger, a 91 digit number.)

This is a rather dramatic way to illustrate that linear increases in delta-V require exponential increases in rocket size.

If you start with "what's the biggest rocket that could plausibly be built" you can work this logic backwards to find out what exhaust velocity you'd need to get a given payload to orbit.

Let's say we're building a 1 million ton rocket -- 300 times the mass of a Saturn V. To my mind, a rocket this size is neither "obviously possible to build" nor "obviously impossible to build". Again let's go with a 5:1 stage mass ratio, and a 2.5 ton payload - a manned capsule a bit smaller than a Gemini. That's an 8-stage launcher (2.5t x 58 = ~1x106t). Split the 10km/s orbital delta-V requirement evenly across the stages and you get 1.25km/s each. So:

$$1250 = V_\mathrm{e} \cdot \ln 5$$

Which works out to ~777 m/s exhaust velocity, or ISP of 79s. Interestingly, that's right in line with black powder model rocket engines! (I'm assuming that the XKCD What-If that Hobbes mentioned assumed ganging individual model rocket engines, rather than building a custom optimized rocket using comparable fuel, hence the different conclusion.)

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  • $\begingroup$ Great work @Russell $\endgroup$ – andy256 Apr 11 '15 at 6:22
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    $\begingroup$ The XKCD didn't do too good a job of handling the engines--he's assuming a lot more complexity than is needed. You don't use a E9-4, you use an E9-0 and simply tape it to the engine above. As the engine is spent it burns through the top of the fuel and momentarily thrusts backwards--right down the throat of the engine it's taped to. When the next stage lights the pressure is high enough to snap the tape and blow off the spent engine. $\endgroup$ – Loren Pechtel Apr 11 '15 at 19:37
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    $\begingroup$ Regardless, I'm pretty sure that doesn't scale to million-ton designs. :) $\endgroup$ – Russell Borogove Apr 11 '15 at 19:57

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