It's still quite hard to do with one stage, the main problem being drag.
First of all, I'm going to ignore the problem of guidance. The burn will
probably be over before it gets to an altitude where fins are ineffective,
and if not perhaps the rocket can be spin stabilized in the launcher
without adding any mass to the rocket.
I'm going to assume the 0.90 propellant mass fraction is for the motor
only, and imagine a notional minimum diameter 98 mm rocket with 8 kg of
motor (7.2 kg of propellant) and 2 kg of other stuff (body, fins,
recovery). Furthermore I'm going to assume a 10 second burn time and
constant thrust. That seems to be around the upper end of COTS long burn
motors. If we know the quantity of propellant, Isp (200 s), and burn time,
then the average thrust is determined to be 1411 N. I'll also assume a Cd
of 0.75, kind of a moderate value for a well built rocket.
Air launched from 12.2 km, this rocket gets to 64 km.
Now let's do some sensitivity analysis. What if we could reduce the other
mass from 2 kg to 1? Then the rocket only gets to 61 km. This rocket is in
the regime where the drag is so high that the rocket is better off being
heavier.
If you can work really hard to reduce the Cd, you have to get it down to
0.51 for the rocket to get to 100 km.
If you leave the Cd at 0.75 and double the burn time to 20 seconds (and
halve the thrust) then the rocket goes to 107 km.
This would be much easier with two stages, not to increase the impulse or
reduce the dry mass, but just to put a delay between the burns to simulate
a longer burn time. You can see why Sugarshot to Space tried so hard to
make a single chamber, two burn motor.
Or, you can beat the problem into submission with the square/cube law. If
you go back to the original design (Cd 0.75, 10 second burn) and triple the
mass of everything and increase the diameter by the cube root of 3 (new
diameter ~150mm) then it gets to 99 km.
So that's my final answer to the original question. The smallest
conventional single burn HPR with the given propellant fraction and Isp
without any exotically long burn time that can get to 100 km from an air
launch at 40,000 ft is a 150 mm minimum diameter rocket with a total mass
of about 30 kg.
On Sat, Feb 27, 2021 at 1:58 PM Dr Edward Jones <RocketPioneer@xxxxxxxxx>
wrote:
That's to ask, how small might a solid propellant rocket of 0.90 mass
fraction and Isp of 200 be to reach 100-km from a balloon launch at 40k
feet?
Let's think an apogee of 100-km.
Thanks
Edward