Tuesday, December 13, 2005

Basic Sizing of a Rocket

Now – how big does your rocket have to be to get there? The following are pertinent equations for sizing a rocket:

For a rocket stage with a certain payload mass, propellant mass fraction, delta v, and engine Isp, the mass of the fuel, inert mass, and everything else falls into place. Tsiolkovsky’s (spelling on slide – I are engineer) rocket equation relates the mass ratio of the rocket stage to the amount of dv needed.

Isps (specific impulses) are typically around 300-420 sec for liquid propellants, and around 300 or below for solid propellants. Hydrogen/Oxygen – the best chemical propellant usable (there are better ones involving fluorine – but they produce poisonous gas as an exhaust!) has around 420 sec. 440 sec in vacuum with a good engine. The specific impulse, as a measurement, is defined as the total impulse (integral of thrust with time – the total change in momentum of the spacecraft) divided by the total weight of the propellant on earth. Neglecting atmospheric effects, it ends up being equal to the exhaust velocity * earth gravity. The faster your exhaust velocity is, the less fuel mass you need to make a certain change in momentum. (Thrust = mass flow rate * exhaust velocity, Impulse = mass expended * exhaust velocity).

Propellant Mass fraction is the ratio of fuel mass to the mass of the full stage – the payload. Chemical rocket stages have propellant mass fractions ranging from 0.7 to 0.9. It is a good idea to conservatively estimate 0.8, even though some rockets may achieve better performance when all is said and done. If you don’t end up having enough mass budgeted for your engines, you’ll have to iterate your calculations again.

For multi-stage rockets, one of the things that must be decided is what fraction of the total delta v each stage will take.

MRstage1 * MRstage2 * MRstage3 = exp((dv1/Isp1 + dv2/Isp2 + dv3/Isp3)/g0).

For stages of equal capability (equal Isp and propellant mass fraction), the lightest rocket is the one that has equal mass fractions (and equal fractions of total delta-v) for each stage.

The payload to total mass ratio helps round out the equations to find the total mass. Note – if the mass ratio is too low (dv too high) and the pmf is too great, it means the rocket stage can’t carry enough fuel to loft both the payload and the structural mass to the delta v. The payload fraction goes to zero, and then negative. This means you’ll have to stage your rocket further, or slim down on structural mass.

Example – you want to loft a 10000 kg payload to geostationary orbit. You decide to use a two stage rocket to accomplish this – both stages use hydrogen and oxygen propellant and a decent engine (420 sec Isp). Because you want your rocket to have minimized mass requirements, you equalize the dv required between stages.

Total mission dv = 9000+3580 = 12580 m/sec.
Dv for stage 2 = 6290 m/sec
Dv for stage 1 = 6290 m/sec
Propellant mass fraction for each stage = 0.8
Isp for each stage is 420 sec

The massfraction for each stage is 0.2169.
The payload fraction for each stage is 0.02116. This means that payload is only 2.1% of the weight of each stage!

The payload mass for the second stage is 10000 kg. The total mass of the first stage is 472546 kg. The fuel mass is 462546 kg. The inert mass is 92509 kg.

The payload mass for the first stage is the fully loaded mass of the second stage. The total mass of the first stage (payload included) is 22332000 kg. The fuel mass is 1717496000 kg. The inert mass is 4373000 kg.

So your total rocket mass is 22,332 tons. Umm, how realistic is this exactly? If you needed it to accelerate at 1.1 g forces (for takeoff purposes) you would need 241 MN of thrust. That equates to 129 space shuttle main engines. This is one gigantic rocket! I don’t think we would want to build anything this big. If you were constrained to a propellant mass fraction of 0.8, you would have to use more stages.

Let’s start over with a 3 stage rocket:

Dv for each stage = 4193 m/sec
Propellant mass fraction = 0.8
Isp = 420
MR = 0.361
Payload fraction = 0.20125

Notice that now your payload is 20% of the mass of each stage, and 8% of the total rocket mass. This is far better than the previous case, where your payload fraction was only 0.04%. Your rocket is going to end up being much smaller.

Stage 3:
Total Mass: 49689 kg
Fuel: 31,751 kg
Inert Mass: 7937 kg

Stage 2:
Total Mass: 246901 kg
Fuel: 157700 kg
Inert Mass: 39442 kg.

Stage 1:
Total Mass: 1,226,837 kg
Fuel: 783,949 kg
Inert Mass: 195,987 kg

A 1226 ton rocket is 20 times smaller than our previous 22332 ton rocket. To take off at 1.1 g forces, this rocket will only have to generate 13 MN of thrust. It would only take the equivalent of 7 SSMEs (52360 kg of engine there, leaving 143 tons left over for 1st stage structure).

This is still a sizeable rocket, but that is because 10 tons to GEO is a large payload for that sort of orbit.


Anonymous cheap viagra said...

if the mass ratio is too low (dv too high) and the pmf is too great, it means the rocket stage can’t carry enough fuel to loft both the payload and the structural mass to the delta v. The payload fraction goes to zero, and then negative.

Monday, May 02, 2011 9:15:00 AM  

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