Monday, January 16, 2006

Slow Launch

Every time I see a rocket vault off the pad, I wonder why the engineers didn't make the propellant tanks bigger. Adding propellant to any stage always increases the delta-V delivered, so long as you ignore gravity losses.

It takes a lot of thrust and energy expended just to get the rocket to hover stationary above the pad. The delta-V lost is called the gravity loss. As the rocket gets closer to orbital velocity, gravity losses drop. At low velocities, gravity losses are just time * acceleration by gravity.

Gravity losses, as it turns out, are a big deal. As you add propellant, you logarithmically increase the delta-V, but you linearly increase the gravity losses. At some point, gravity losses overtake the increased delta-V.

The tradeoff between the two is different for different kinds of rockets. In a solid-fuelled rocket, the entire stage is a big combustion chamber, that must contain the gas pressure used to accelerate the vehicle upward. Large pressure vessels are heavy, and so the fuel container is a large fraction of a solid rocket's mass. Pressure-fed liquid-fuelled rockets don't actually have the combustion chamber in the tanks, but the tanks must hold higher pressure than the combustion chamber, so the mass penalty is similar.

Pump-fed liquid-fuelled rockets hold their propellants at a small fraction of the combustion chamber pressure, and so their tanks are a small fraction of the weight of a similarly-sized solid rocket.

Let's take a look at the two extremes. First, a kerosene/LOX liquid-fuelled rocket, like the Saturn V or Falcon 9. I've made up a table to show the decreasing performance return of steadily larger and larger tanks. Here I've presumed a base rocket with an Isp of 290, a first-stage thrust of 750,000 kg, a Gross Lift-Off Weight (GLOW) of 500,000 kg, and a first stage burnout weight (this includes the upper stages) of 150,000 kg. This rocket has an initial acceleration of 1.5 G (but remember you lose 1 G to earth). Incremental tankage weighs just 2.5% of the incremental propellant stored. I'm assuming that the engine thrust stays fixed. Delta-V numbers are in meters/sec. Tower clearance times are a little high, as they assume no acceleration beyond the initial acceleration.

Isp 290
Ve 2842
Tankage 0.025
Thrust 750000
Burn rate 2586
Tower 50
inc inc delivered tower
G Mf Me delta-V time G-loss delta-V clear
2 375000 146875 0 0 0 0 3.2
1.9 394737 147368 136 7 73 63 3.4
1.8 416667 147917 143 8 81 62 3.6
1.7 441176 148529 151 9 91 60 3.8
1.6 468750 149219 159 10 102 57 4.1
1.5 500000 150000 169 12 115 53 4.5
1.4 535714 150893 179 13 132 47 5.1
1.3 576923 151923 191 16 152 39 5.8
1.2 625000 153125 205 18 178 27 7.1
1.1 681818 154545 221 21 210 11 10.1
1 750000 156250 240 26 252 -12 N/A


You can see why the Saturn V initial acceleration was just 1.13 Gs. You can also see why launching the Saturn V in a strong wind could have been a problem: it takes a long time to get past the tower.

Now let's look at the other extreme, a solid rocket first stage, like the Titan 4 or the Stick proposal. (Because most of the Shuttle's liftoff thrust is from it's solids, it fits in this category too.) Here the Isp is a bit lower, but more importantly the tankage fraction is far higher: 12%.

Isp 242
Ve 2371.6
Tankage 0.12
Thrust 750000
Burn rate 3099
Tower 50
inc inc delivered tower
G Mf Me delta-V time G-loss delta-V clear
2.5 300000 126000 0 0 0 0 2.6
2.4 312500 127500 69 4 35 34 2.7
2.3 326087 129130 71 4 38 33 2.8
2.2 340909 130909 73 4 41 32 2.9
2.1 357143 132857 75 5 45 30 3.0
2 375000 135000 78 5 50 28 3.2
1.9 394737 137368 80 6 55 25 3.4
1.8 416667 140000 83 6 61 22 3.6
1.7 441176 142941 86 7 68 18 3.8
1.6 468750 146250 90 8 77 13 4.1
1.5 500000 150000 93 9 87 6 4.5
1.4 535714 154286 97 10 99 -3 5.1


You can see why a Titan 4 gets off the pad with nearly 1.5 Gs of initial acceleration.

Solid rockets are a good match for first stage engines. It takes relatively little engineering work to produce a large amount of thrust from a solid motor, which is the biggest cost driver for first stage engines. But because the casing weighs so much, and also because solid propellants have lower Isp, the delta-V of a solid first stage is never going to be as good as a liquid-fuelled analog. That leaves more work for the second stage, which means thinner engineering margins and more desire for high-Isp propellants, like liquid hydrogen. And so, cheap solid rocket first stages drive more cost into the upper stage.