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Rockoons and Propellent Mass Fractions.

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klugesmith

Sat Feb 24 2018, 03:51AM

Registered Member #2099 Joined: Wed Apr 29 2009, 12:22AM
Location: Los Altos, California
Posts: 1714

Patrick, I wasn't sure whether to use this thread or your earlier one about mostly the same thing:
"Orbital mechanics and Gravity Well".

Near the end of that thread, some of us regulars agreed that numerical simulation was the best way to get a handle on things.

I just spent some play time, making a good start on a rocket flight simulation built from scratch.
It's in MS Excel with 16 columns of formulas, using numbers for a Falcon 9 v1.1 rocket with 1000 kg payload.
Mostly following in footsteps of the amateur presentation I cited in the earlier thread.
For more simplification, I'm firing the rocket straight upward, and stopping soon after first stage engine cutoff.
77 timesteps of between 1 s and 5 s, amounting to 190 seconds, closer together where interesting things happen.
Anyone want to review the model, play with variables, translate it to another language, or extend it to 2-dimensional flight?

Haven't yet compared any waypoints with those in documented launches, but the numbers seem to make sense.
Mass flow is constant 2128 kg/s for 180 s (no throttling), as total mass drops from 518 to 135 metric tons. That's 36 tons for the almost-dry first stage, and 99 tons for the rest of the rocket. Specific impulse ramps from 282 s (sea level) to 311 s (vacuum), as the atmospheric pressure falls off. (The 2nd stage engine has a bigger nozzle, optimized for vacuum, and gets 340 s.)

Atmosphere model is standard, with formulas that change at heights of 11 and 25 km:

Drag coefficient changes dramatically with Mach number, copied from a chart in that student presentation I cited. The resulting drag forces are important for airframe loading, but don't appear to take a gigantic bite out of the acceleration (gray curve in this semilogarithmic chart).

I'm seeing Mach 1 at 75 s, 8.7 km altitide. Max Q of 23.7 kPa at 81 s, 10.7 km, 353 m/s.
Peak acceleration 3.6 g, just before engine cutoff, at 115 km altitude and 2.1 km/s.

It would be trivial to turn off the atmospheric drag (just set the frontal area to zero) or to start at high altitude. Any requests? Any guesses?

Patrick

Sat Feb 24 2018, 06:20PM

Registered Member #2431 Joined: Tue Oct 13 2009, 09:47PM
Location: Chico, CA. USA
Posts: 5639

klugesmith wrote ...

Patrick, I wasn't sure whether to use this thread or your earlier one about mostly the same thing:

I didnt realize i had two till last week ! dam.

thanks for going to all that trouble ! im looking at the axis units, on the left and right of the first graph. im re-reading everything, very cool.

Over expansion from a nozzle seems like a gain for the rockoon, but properly stage chosen nozzles mitigate that advantage i guess.

Uspring

Sat Feb 24 2018, 06:32PM

Registered Member #3988 Joined: Thu Jul 07 2011, 03:25PM
Location:
Posts: 711

Nice work klugesmith

I tried an analytical approach for an into orbit launch with constant acceleration in order to get a feel for the advantage of an high altitude launch, but got stuck with a nasty integral.

I used a "flat earth" approximation, i.e. I split up the velocity into a vertical and horizontal part vv and vh and then reduced the gravitational acceleration by vhÂ²/R (R is the earth radius). The term is the centrifugal acceleration. I think the approximation is not too bad since the rocket does not travel very far compared to the earth radius during the launch. Maybe an idea to incorporate launches into orbit within your spreadsheet.

I don't think the gravitational angling in the link you quoted is a sensible concept. I don't understand why that should be optimal wrt fuel requirements.

Uspring

Tue Feb 27 2018, 12:23PM

Registered Member #3988 Joined: Thu Jul 07 2011, 03:25PM
Location:
Posts: 711

Here's a simple spreadsheet for launch calculations.
]rocket.zip[/file]
Column A is the acceleration (taken to be constant), B horizontal velocity component (m/s), C vertical velocity component, D velocity, E horizontal position (m), F altitude. The first line, except D1, are input values. Thrust is applied in the direction of the velocity. I've given some (low) initial velocity to the rocket in order aim the rocket into an orbit. The calculation uses the gravity assisted turn. See here

. I've used this way of specifying the trajectory, since it makes the trajectory direction dependent on only one parameter (initial horizontal velocity).

Shown is a launch from 30 km altitude. The rocket reaches a speed of 8 km/s horizontally (=orbital speed) at about 351 seconds. For a launch from ground, the initial horizontal speed has to be adjusted to 1.03 m/s to reach the same speed at a similar altitude. Burn length is then 354s. Not much of a difference.

But:
I have not included air drag. From klugesmiths diagram, it can be seen, that above supersonic speeds air drag takes a jump upward. In my calculation, this speed is reached already at 4 km altitude for a ground level launch, so it will likely be of great impact. For a 30 km altitude launch that will be much less.

Wrt to fuel consumption, the ideal situations would be

a) No atmosphere, unlimited acceleration:
High acceleration horizontally, basically a kick to about 8 km/s speed. Then wait until the rocket has reached max altitude, then another kick along its flight direction to make the orbit circular (Hohmann orbit). Used e.g. when getting from earth orbit to Mars orbit.

b) No atmosphere, limited acceleration:
Most thrust horizontal and a small part vertical to keep the rocket just skimming the surface. Used e.g. for the moon ferry to get back into moon orbit.

c) With atmosphere, limited acceleration:
Complicated, first get out of the atmosphere and then accelerate horizontal to acquire orbital speed. When to do that, depends on air drag.

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