How much delta-v does a 2 liter bottle of soda have?

[CaelumEtAstra]

I was watching this video...

...and it got me thinking...how much delta-v (Isp & thrust as well) does a bottle of soda have? How much if we, say, added Mentos?

[ChrisSpace]

We'd need to know exactly how fast the stuff comes out the bottom.

[CaelumEtAstra]

We'd need to know exactly how fast the stuff comes out the bottom.

Good point, didn't think of that...I'm pretty sure there's some mythbusters videos out there with the stream flowing out next to a ruler.

[Gaarst]

If you have the soda effective exhaust velocity, getting the dV of a soda bottle is pretty straighforward.

- - - Updated - - -

So, using Mentos added into a coke bottle:

We can get the exhaust velocity of the coke using the height of the coke "geyser":

v_e = sqrt(2gh)

Assume h = 10m (pretty high for a Coke-Mentos reaction but still realistic), we know g = 9.81 m/s², then:

v_e = sqrt(2*9.81*10) = 14.0 m/s

For I_sp:

I_sp = v_e / g = 1.42 s

For dV:

dV = v_e*ln(m₀/m)

Consider a 2l bottle, then m₀ = 2055 g and m = 55 g;

dV = 14*ln(2055/55) = 50 m/s if all the coke is ejected at the same speed.

The thrust is given by:

F = dm * v_e

dm is the Coke flow rate in kg/s. Assuming the reaction lasts 5 secs, them dm = 0.4 kg/s

F = 0.4 * 14 = 5.6 N

Edited November 1, 2015 by Gaarst

[Scotius]

But...why would you waste perfectly good fizzy drink? You can't build useable rocket with it anyway - so better drink it

[RainDreamer]

I asked this a while ago on another thread, K^2 was nice enough to provide an answer:

What is the ISP of a vigorously shaken 1.5l bottle of carbonated drink? How much delta V does it have when strapped on to an average astronaut?

It's hard to give a solid estimate, because there are a lot of factors. But it's easy to put an upper limit on it. Soda bottle is pressurized to about 2 bar, or 200kPa. If we assume that CO2 is released at sufficient rate to maintain that pressure, the total energy liberated in 1.5L would be about 300J. For 1.5kg of propellant, we get 10m/s exhaust velocity. That's I_SP of 1.02s. Realistically, you'll probably get close to that early on, but it will quickly drop to a much lower value. So average I_SP is likely to be lower.

It might also be more useful to consider total impulse in this case. Again, considering an upper limit, the total impulse works out to 10m/s * 1.5kg = 15Ns. If we consider a typical astronaut in a space suit, which is about 130kg, we get delta-V 15Ns / 130kg = 0.12m/s. Which really isn't much to begin with, and will likely be even less in practice.

You could use rocket formula here, but since initial and final masses are almost identical, you should get the same result.

[CaelumEtAstra]

Very interesting, indeed. Soda's delta-v is even worse than a water rocket! And that's using compressed air and water!

[Starwaster]

So how many would we need to put one pound satellite into LEO?

[Scotius]

It would probably collapse into singularity before taking flight

[Gaarst]

So how many would we need to put one pound satellite into LEO?

1 pound = 450g

Dry mass is then 500g, wet mass is 2500g.

Taking K^2's ejection speed: v = 10 m/s

Rearranging the rocket equation for n bottles gives us:

M₀ / M₁ = e^{dV / v}

M₀ is the wet mass of n bottles plus one satellite, so: M₀ = n * m₀ + 0.450

M₁ is the dry mass of n bottles plus one satellite, so: M₁ = n * m₁ + 0.450

But, as it turns out, the limit of M₀ / M₁ as n goes to infinity is equal to m₀ / m₁ = 2.055 / 0.055 = 37.4

In other words, even with no payload and infinitely many bottles, you couldn't get more than 36.2 m/s of dV with a single stage rocket.

Now, consider many stages:

The delta v of the k-th stage is given by:

dV = 10 * ln(M_k,0 / M_k,1)

M_k,0 is the wet mass of n bottles plus k - 1 stages above, so: M_k,0 = n * m₀ + S_k-1

M_k,1 is the dry mass of n bottles plus k - 1 stages above, so: M_k,1 = n * m₁ + S_k-1

S_k-1 is the mass of the k - 1 stages above the k-th stage.

Then again, we can see that M_k,0 / M_k,1 tends to 37.4 as n goes to infinite.

So, each stage of the rocket can only give 36.2 m/s of dV.

If you're a good pilot, then you can get to LEO for 9000 m/s of dV. You'll then need 249 ideal stages to get to LEO. With each stage having infinitely many bottles, but more infinitely many than the previous one (this is actually possible, see transfinite numbers).

Having infinitely many bottles on each stage is not very practical, so we won't use ideal stages anymore, but finite stages, which are much more convenient (you'll see that "convenient" is relative here) to work with. We decide we only need 34 m/s of dV per stage, we'll then need 265 stages to get to orbit.

We already saw that the delta v of the k-th stage is :

dV = 10 * ln(M_k,0 / M_k,1)

We can consider S_k as a series (recall S_k is the total mass of the rocket at the k-th stage), then:

S_k = ∑ (n_k * m₀) = m₀ * ∑ n_k

S₀ = 0.45 kg

n_k is the number of bottles needed in the k-th stage to achieve 34 m/s of dV.

n₁ = 0.45 * C = 32.02, C = (e^3.4 - 1) / (m₀ - m₁ * e^3.4) = 71.169

n_k = C * S_k-1

Calculating the first few terms of n_k and S_k shows us the problem:

S₃ = 1 427 066 kg

In other words, only the first 3 stages of our coke rocket would weigh half a Saturn V.

Considering the first few terms of S_k allows us to make the following approximation:

S_k = S₀ * e^4.992k

While my calculator doesn't want to give me a result for our 265 stages, it tells me that the 14 first stages would weigh half the mass of our Sun.

Also, the first 20 stages would weigh more than our galaxy, the Milky Way; and the observable Universe is most likely somewhere around 25 stages.

TL;DR: we would need a lot of bottles.

Edited November 2, 2015 by Gaarst

[Frybert]

Most likely would impart a higher DV to someone who drinks it

[Scotius]

So. Like i wrote - instant black hole Better just have a drink. And on a related note - i LOVE this community. Where else someone would be willing to post a wall of math calculating dV of infinite amount of coke bottles?

[Gaarst]

Actually, using a few tricks to make my calculator cooperate, I could get the final numbers.

So, to put a 1 pound satellite in LEO with a coke rocket you'll need a rocket weighing around 10⁵⁷⁴ kg, made of approximately the same number of 2l coke bottles.

And that is assuming all the structural parts of the rocket and the mentos used are massless.

Edited November 1, 2015 by Gaarst

[ChrisSpace]

So, to put a 1 pound satellite in LEO with a coke rocket you'll need a rocket weighing around 10⁵⁷⁴ kg

How many observable universes would that weigh?

[fredinno]

Actually, using a few tricks to make my calculator cooperate, I could get the final numbers.

So, to put a 1 pound satellite in LEO with a coke rocket you'll need a rocket weighing around 10⁵⁷⁴ kg, made of approximately the same number of 2l coke bottles.

And that is assuming all the structural parts of the rocket and the mentos used are massless.

What about with a 5 stage rocket?

[WestAir]

How much mass would a bottle rocket need to escape a bottle rocket that has enough Dv to orbit the Earth?

Edited November 3, 2015 by WestAir

[Gaarst]

What about with a 5 stage rocket?

Approximately 31 million tons (or 10 000 Saturn V), but you won't get anywhere with 170 m/s dV

How much mass would a bottle rocket need to escape a bottle rocket that has enough Dv to orbit the Earth?

It depends on the distance from the second rocket to the centre of mass of the first one. And ultimately the latter's density.

[fredinno]

Approximately 31 million tons (or 10 000 Saturn V), but you won't get anywhere with 170 m/s dV

Holy .....

[CaelumEtAstra]

Well...now it seems silly why I even asked in the first place. 31 million tons for a 436 gram payload? And even then only 176 m/s? Curse you, rocket equation and your hopeless tyranny!

(Jeb, pack it up. Soda is not a viable means of spaceflight, and it's not worth getting the launchpad sticky.)

Edited November 4, 2015 by Mrsupersonic8

[RainDreamer]

Well now, I wonder if we can somehow liberate the energy content from that sugar water to propell itself instead of just using pressure...like maybe set fire to the thing by some method...how much more dV would we get?

[Gaarst]

Well now, I wonder if we can somehow liberate the energy content from that sugar water to propell itself instead of just using pressure...like maybe set fire to the thing by some method...how much more dV would we get?

You could calculate with the energy contained in the coke (kcal nutritional value) which, AFAIK, is basically how much energy you will get from drinking and digesting it. From then you could get dV, and all other stuff to make a rocket.

I may do the maths tomorrow, if anyone else doesn't do it in the meantime

Edited November 4, 2015 by Gaarst
Using actual words is better