Water Rocket Science Project by Tanguy: Reach 11,2k m/s!

[Tanguy]

Homework Statement

I have to do a presentation about water rocket and the theoretical pressure needed to send me (a mass of 60kg) up in space

Relevant Equations

Simplified Newton's seconds law : Sum(F) = m.a
Bernouilli's theorem : ½.v^2+g.z+(roh).P=constant
Kepler's Laws

Hi ! I'm a French student in Terminale (12th grade equivalent) and I have to do an oral presentation of 5min (Grand Oral) for my baccalauréat linked with my professional project. As I want to become an engineer in astrophysics, I want to talk about water rocket and the theoretical possibility to send a human being up in space.

With a bunch of approximations and some equations found on the Internet, I came up with an approximation of the acceleration of the rocket at liftoff : m.a = -m.g + (volumic_mass_of_water).(area_of_the_neck).[(water_ejection_speed)]^2,
with (water_ejection_speed)=sqrt[2.([inside_pressure] - [atmospheric_pressure])/(volumic_mass_of_water)]

Using 5cm as the diameter of the neck, 9.81 as and 60kg as m, then by integrating the expression of a, I come up with
velocity(t) = v(t) = 6,5*10^(-5)*P*(t)-9,81*(t), with P the searched initial air pressure in the rocket.

I know that a rocket needs a velocity of 11,2*10^(3)m/s in order to espace earth attraction.
But I have absolutely no idea about how to link the expression of velocity (with 2 variables) I found and this given velocity of reference.
Does someone have any idea about it ? Should I simplify the expression of velocity by choosing a value for (t) ? If so, which value should I take ?
Thanks your for your answers,
Have a good day
Tanguy

 

[erobz]

Homework Statement: I have to do a presentation about water rocket and the theoretical pressure needed to send me (a mass of 60kg) up in space
Relevant Equations: Simplified Newton's seconds law : Sum(F) = m.a
Bernouilli's theorem : ½.v^2+g.z+(roh).P=constant
Kepler's Laws

Hi ! I'm a French student in Terminale (12th grade equivalent) and I have to do an oral presentation of 5min (Grand Oral) for my baccalauréat linked with my professional project. As I want to become an engineer in astrophysics, I want to talk about water rocket and the theoretical possibility to send a human being up in space.

With a bunch of approximations and some equations found on the Internet, I came up with an approximation of the acceleration of the rocket at liftoff : m.a = -m.g + (volumic_mass_of_water).(area_of_the_neck).[(water_ejection_speed)]^2,
with (water_ejection_speed)=sqrt[2.([inside_pressure] - [atmospheric_pressure])/(volumic_mass_of_water)]

Using 5cm as the diameter of the neck, 9.81 as and 60kg as m, then by integrating the expression of a, I come up with
velocity(t) = v(t) = 6,5*10^(-5)*P*(t)-9,81*(t), with P the searched initial air pressure in the rocket.

I know that a rocket needs a velocity of 11,2*10^(3)m/s in order to espace earth attraction.
But I have absolutely no idea about how to link the expression of velocity (with 2 variables) I found and this given velocity of reference.
Does someone have any idea about it ? Should I simplify the expression of velocity by choosing a value for (t) ? If so, which value should I take ?
Thanks your for your answers,
Have a good day
Tanguy

Try to learn latex to present the mathematics that describe the physics. The link is in the bottom right left hand corner of the reply box.

 

[haruspex]

by integrating the expression of a

Please show your working for this. What function are you using for for the pressure? Are you assuming adiabatic expansion?
I assume "volumic mass of water" means its density.

a rocket needs a velocity of 11,2*10^(3)m/s in order to espace earth attraction.

You don't need "escape velocity". That is the velocity needed to go infinitely far away. You just need to get into orbit.

 

[jbriggs444]

But I have absolutely no idea about how to link the expression of velocity (with 2 variables) I found and this given velocity of reference.

So you have two variables: Pressure and time, right?

But you also have a problem. Finite fuel.
And you have another complication. A vehicle mass that is reduced over time.

There is a way forward. It involves shifting from thinking about pressure to thinking about exhaust velocity. And it involves thinking about mass ratios. How much of your craft is payload compared to how much is fuel?

How much experience do you have with differential equations? We can do this with a differential equation. Or we can do it with a simulation. Perhaps with Excel.

Note that compressed air is a horrible power source. An isothermal expansion can do OK but is unrealistic. An adiabatic expansion is what you will actually have. Much worse. So far, you are completely glossing over what happens to pressure as the air expands.

I have an expectation for the result of a correct calculation here. It will not be pretty.

 

[kuruman]

Another practical matter to consider is whether the water propellant will stay in the liquid phase as the rocket goes higher and higher.

 

[jbriggs444]

Another practical matter to consider is whether the water propellant will stay in the liquid phase as the rocket goes higher and higher.

If we keep it under the pressure from the compressed air above, it should not boil until after it is exhausted.

If it does boil before being exhausted, what you have would be a cold gas thruster operating with a propellant with a molecular weight of 18. That would give an $I_\text{sp}$ of something worse than gaseous Nitrogen ($I_\text{sp} = 80 \text{ sec}$). In other words, you'd do better by getting rid of the water and using compressed Nitrogen without additional reaction mass. But of course the tradeoff then is increased volume.

Ice build up is not completely impossible, but is unlikely to be a huge issue.

 

[hutchphd]

This is a good but quite difficult learning project. It will reward your (probably considerable) effort. You will not be nearly able to orbit with this technology from earth but that is not the point.
I like the idea trying to produce a simulation if you are at all computer savvy. My choice would be an excel spreadsheet that would conscutively calculate velocity, thrust, distance, remaining tank pressure, water, etc vs time for a given set of motor parameters and projectile dry weight. Each line will be one small increment of time
The most difficult thing to estimate may be the speed at which water will eject for a given nozzle aperture and tank pressure. Make a guess (as a changeable input) and make a model. You can modify it as necessary. Spo get started! You will get too much help here.

 

[erobz]

Note that compressed air is a horrible power source. An isothermal expansion can do OK but is unrealistic. An adiabatic expansion is what you will actually have. Much worse. So far, you are completely glossing over what happens to pressure as the air expands.

Given the mass of water the rocket would have to carry to get a person into orbit, it would probably be closer to an isothermal expansion. For a toy 2-liter rocket, the whole process is over in a moment and the accelerations are way beyond what a human could withstand. They toy model rocket is definitely adiabatic. The real deal would need to be much more controlled and far less rapid in its expansion IMO.

If a person were on the bottle rocket in the video (scaled up obviously) their intestines would probably be on the floor beneath them...

 

[haruspex]

it would probably be closer to an isothermal expansion.

Given the -gt term, time is of the essence. An ascent slow enough for isothermal is out of the question. Instead, a ridiculously high pressure would be needed.
There is also the problem of how even a slow ascent could be isothermal given the lapse rate.

 

[erobz]

Given the -gt term, time is of the essence. An ascent slow enough for isothermal is out of the question. Instead, a ridiculously high pressure would be needed.
There is also the problem of how even a slow ascent could be isothermal given the lapse rate.

A slow expansion has time to absorb heat, doesn't it? An expansion like in the model bottle rocket is also out of the question if the astronaut is to survive. My gut thinks maybe we could be in between a rock and a hard place, but these are all things the OP should explore.

 

[haruspex]

A slow expansion has time to absorb heat, doesn't it? An expansion like in the model bottle rocket is also out of the question if the astronaut is to survive. My gut thinks maybe we could be in between a rock and a hard place, but these are all things the OP should explore.

It has to absorb heat from the surroundings. As you ascend, those surroundings get colder.

 

[erobz]

It has to absorb heat from the surroundings. As you ascend, those surroundings get colder.

Yeah, I did think that would be an issue too in the long term. Like I said, rock and a hard place. I think the bigger issue though is the amount of mass of fuel you would have to carry to get into orbit in under 5 g's (average is about 3g for a real launch). That is "the rocket problem" at its core.

Just about every variable is in flux over these scales.

 

[haruspex]

Yeah, I did think that would be an issue too in the long term. Like I said, rock and a hard place. I think the bigger issue though is the amount of mass of fuel you would have to carry to get into orbit in under 5 g's (average is about 3g for a real launch). That is "the rocket problem" at its core.

Just about every variable is in flux over these scales.

The ideal would be to achieve the human factors limit, a 4g acceleration. Since the mass declines, the thrust would have to decrease similarly, so the standard constant thrust rocketry equation does not apply.
Of course, this is unlikely to match the rate at which the pressure drops, so we'd need the ability to adjust the nozzle aperture.

 

[erobz]

I think any serious attempt at modeling the thrust could be modified from what @haruspex and @Chestermiller came up with for unsteady flow in a draining tank:

velocity-of-efflux-out-of-a-water-tank

by accounting for the work done in the expansion and vicious effects in the nozzle. Obviously the complexity of a model would be further increased by the fact that "short of us holding it still", the frame is accelerating in the "actual launch".

That being said, that is (some presently unknown amount) beyond 12th grade, 5 minute presentation! ( As much as I'd like to, and probably will try to explore the prior scenario) The OP sticking to Bernoulli's and an isothermal expansion should be plenty challenging as it is.

 

[erobz]

The ideal would be to achieve the human factors limit, a 4g acceleration. Since the mass declines, the thrust would have to decrease similarly, so the standard constant thrust rocketry equation does not apply.
Of course, this is unlikely to match the rate at which the pressure drops, so we'd need the ability to adjust the nozzle aperture.

I agree, controlling the thrust as a function of mass (or time) to maintain constant acceleration is likely inevitable due to human physiology. Very rich problem indeed.