How weak can a propellant be and still get a rocket to space, theoretically?

Question

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?

Answer 1

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.)