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When I fly my water rockets as a hobby I use this formula to calculate the height of the flight so I get an idea of how high my water rocket will fly. enter image description here

My idea is to make a theoretical water rocket out of graphene (https://www.nanowerk.com/what_is_graphene.php). It would have around 50 - 100 layers of graphene and the air pressure would be brought up to 6000 bars. Then I would add some shampoo in the water to change the density (Best case scenario the density would be dropped by half but that is not fully possible). The rocket length would be around 4m and 16 cm in diameter. It would be filled up to 30% with water.

In this case then:

Mi = 400 cm / 3 * 8² * π = 26 808,257... cm³ = 26,808257 kg water

Mr = 0,16 * π * 4 * 50 * 0,00077 + 26,808257 = 26,885665 kg in total

Here 0,16 * π * 4 is the area made out of graphene. 50 is the numbers of layers of graphene. 0,00077 is the weight of a square meter. 26,808257 is the weight of the water.

Pi = 6000 bars = 600 000 000 pascal

p = 600 (kg/m³)

g = 9,81 m/s²

Then adding it to the formula: (26,808257 / 26,885665)² * (600 000 000 / 600 * 9,81) = 101 350,6604788 m =~ 101,35 km

This formula is leaving out the air resistance so the height it would fly is definitely lower, but if you increased the pressure by another 1000 bars or so then it would fly high enough to pass the Karman line.

Have I made any mistakes in my calculations or the method I used to calculate the height of the flight?

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  • $\begingroup$ Again, you can't use that formula to simulate your water rocket design, your chosen parameters are far outside the assumptions the formula makes. $\endgroup$
    – notovny
    Jul 30, 2022 at 10:51
  • $\begingroup$ @notovny then which formula can I use? $\endgroup$ Jul 30, 2022 at 11:00
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    $\begingroup$ Remember the maximum energy you can get from this system is its internal energy, i.e. the pressurized gas cools as it discharges and soon becomes liquid and loses all its pressure. BTW at 600MPa air is already supercritical at room temperature so this nitrogen-oxygen-water mixture doesn't behave the way you have imagined. $\endgroup$ Jul 30, 2022 at 11:28
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    $\begingroup$ I love the combination of "I'm going to make a 600MPa pressure chamber out of layered graphene" and "then I would add some shampoo to the water to change its density". Truly schizo tech. $\endgroup$ Jul 31, 2022 at 8:50

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The core of this question is the performance limits of a classic water rocket. For an incompressible fluid under pressure there reaches a point where the flow becomes 'choked', this is the point at which regardless of the upstream pressure the flow through the hole is constrained by the speed of sound in the material (this speed of sound limits applies to solids as well, preventing 'beat the speed of light with a rope' tricks).

This would make the peak exhaust velocity around 1500 meters per second, where chemical rockets are twice to three times that. Throwing those numbers into an online calculator gets a delta V of 8.7 kms, enough to theoretically make orbit. This is because we have a truly amazing mass fraction of 0.27% and holding the pressure constant. Saner values including fins and a nozzle still get a couple of kilometers a second, enough to reach the karmen line.

Where this gets far more complicated, and why mathing this more is moot is because water is not in fact incompressible at 6000bar, which gets an increase in the speed of sound up around 3000ms, which is good. Complicating things however is what else is happening inside the tank. The water may in fact 'freeze' into ice VI, with unclear results, further both Oxygen and Nitrogen will not be classical gases, making the contents of the tank a very complex supercritical fluid. This is has several results (and also means the mass of the gasses in the tank start to matter).

The liquid->gas phase change crossing the nozzle will give a performance boost, since X liters of fluid entering the nozzle will turn into Y cubic meters of gas increasing exhaust velocity. However this will also have a cooling effect, certainly turning the water to ice, and possibly the Oxygen as well for a performance loss and probably impacting smooth flow. Finally the actual contents of this tank is some ungodly highly reactive supercritical mix, which will be reactive with the carbon tank and metal fixtures.

All of this of course presupposes you can actually build a magic 50 graphine layer layup tank that has the properties claimed, suspect actual result would be better than carbon fiber (around 1km peak altitude) but not so much so as to get 100km.

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If you can build a ~6000 bar bottle you can afford something better in it- this is a purely academic exercise. That said, “something better” would be a non-polar, lightweight work fluid. Water’s high heat of vaporization is not your friend. And the numerous molecular modes will dissipate lots of the energy you put in. Shampoo makes it even worse, while being heavy.

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