Using a very simple drag model... I used the data from page 16 of
http://www.jmrconline.org/Drag_Coefficient_Prediction.pdf built a table of mach number and Cd and interpolated. (This is actual data from a 5" rocket, were building a 6" rocket so it seems reasonable)
One of the interesting results is this graph:
It shows peak altitude achieved (y axis in meters) piloted against the equivalent peak drag. The peak drag units here are equivalent velocity at sea level in m/sec. The peak is at 50500m and 303m/sec (165Kft and 677mph)
Another interesting result (from a slightly different run) I'm not using any real fancy integrator and the results vs time steps don't change much:
| Time Step | Alt |
|---|---|
| 1 | 47458 |
| .1 | 49532 |
| .01 | 49576 |
| .001 | 49593 |
So this means for very crude integration steps you get reasonable results, thus allowing one to use the model for optimization seeking. Right now there are a number of limitations, the model assumes that the ISP does not change as the motor is throttled for peak drag limiting etc... so as I add more detail to the model it will be interesting to see what happens.
10 comments:
The Cd you are using would seem to include the base drag component, if the data follows the writeup in that report. When your engines are firing you will be pressuring your base region and a portion of the base drag will go to zero. You might want to break out the base drag and allow it to go to zero when the engines are firing. The rule of thumb is that the ratio of nozzle area over base area is the amount of base drag that 'goes' away due to base pressurization.
If have the motors on for some appreciable amount of time, this will make a difference.
From the graph it looks like any peak drag between 250 and 325 gets you above 50km, within 1% of optimal. Maybe it's not worth doing much more calculation.
not worth further optimization for this parameter... I hope to enhance the quality of the simulation to be a tool for evaluating a lot of stuff in the trade space, IE
Tank pressure and Tank mass vs ISP.
Optimum expansion ratio etc...
this is a simple 1d simulation, in parallel I'm working on a 6DOF sim to work on guidance and do some hardware in the loop simulations.
Cool graph man.
I did a similar study a few years ago, and the real take-away is that the optimum vertical speed (i.e. most energy efficient) for a given altitude is equal to the terminal velocity at that altitude. So to get the most altitude, start off slow and speed up as you get higher in altitude. It's the old-skool gravity-drag vs. aero drag fight.
Drag is= 1G + A Cd roe V^2
Lost Delta V is Drag *t
Distance is V *t
we want to maximize
V*t/(1G +A* Cd *roe *v *v)*t
So we want the maximum of
V/(1G + A* Cd * roe *v*v)
let k=1G
and a=A*Cd*roe
Maximum of
V/(k+aV^2)
1/(k/v _aV)
or find the minimum of
k/V +a*V
Complicated by the fact that Cd is actually a function of V
Regarding base drag with motor on/off:
A lot of programs, like AeroDrag and RASAero will give you curves of Cd vs Mach number with the motor on & off.
Hey, I was just curious to know how much you spent on unreasonable rocket compared to some of the competition. Also, I was also wondering what it takes to get a rocket into low earth orbit as compared to escaping the earth's orbit and getting it to the moon. I'm a Ron Paul fan too. I'm so glad to see a Ron Paul guy working to get things into outer space.
Paul, I'm trying to get an estimate for the required delta-V to orbit under these conditions:
1.)use a dense propellant such as kerosene/LOX; dense propellants are known to reduce gravity losses.
2.)use a moderate to high liftoff thrust/weight ratio, say, 1.4 and above; high liftoff T/W also reduces gravity losses.
3.)launch near equator to get the ca. 460 m/s tangential boost.
4.)only get to 100 km, the altitude considered space, to just launch satellites or make orbital transfers, not for long term orbits.
Can your program do that?
Bob Clark
Quite worthwhile info, much thanks for the post.
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