Up, up, a little bit higher

In honour of today's shuttle launch, I thought I'd start off with a bang, or at least a roar of fire: rocket fuel!

The shuttle uses two different styles of rockets with two very different types of fuel: liquid and solid. The shuttle itself has three big rocket engines that run on liquid hydrogen and liquid oxygen, stored in the big orange external tank, while the rest of the thrust is provided by solid fuel in the solid rocket boosters, a mixture of ammonium perchlorate (NH₄ClO₄), aluminum, iron oxide, and some binders to hold it in its moulded shape.

The liquid fuel engines fire first; they are the controllable ones. If there's a problem, they can be shut down. Once the solid fuel starts burning, they only stop by running out of fuel - and the solid rockets provide the majority of the boost to get the shuttle off the ground.

And since this is a chemical engineer's blog, I'm going to do some calculations around the chemistry of those engines. Of course, this won't be nearly enough to design your own rocket engine from, but then I'm just doing this for fun, I'm not a rocket scientist.

We'll start with the relatively simple (chemically speaking) liquid fuel reaction:

\[\mathrm{H}_2 + {1\over 2}\mathrm{O}_2 \rightarrow \mathrm{H}_2\mathrm{O}\]

The heats of formation of H₂ and O₂ are both zero by definition: those are the elemental forms of those compounds. The heat of formation of H₂O is -241.826 kJ/mol. For every mole of water produced in this reaction (about 18 grams) 241 kJ of heat energy is released. In stark contrast, the heat capacity for water is about 33.5 J/mol.K. So every mole of hydrogen/oxygen rocket fuel that makes water heats its own reaction product up by:

\[e = C_pnT\] \[T = {{241,826 \mathrm{J/mol}} \over {(33.5 \mathrm{J/mol.K})(1 \mathrm{mol})}} = 7218 \mathrm{K}\]

From cryogenic gases to very hot, in the blink of an eye. (Note that \(C_p\) actually varies with temperature; I'm treating it as a constant for simplicity even though that means my temperature will be a little off, as \(C_p\) doubles before it reaches 7000 degrees. It's still up in the mid-thousands. I'll use 5000K for the final temperature.)

The actual rocket thrust is caused by all that hot water vapour trying to squeeze out the rocket nozzle at once. Not only is it heated up, which makes the volume increase, it's also gone from liquid to gas, which means even more volume increase. One mole of liquid hydrogen is 2g: at 71g/L and 20K (-253 C) this is about 28mL or roughly 2 tablespoons. Half a mole of liquid oxygen is 16g: at 1.14 g/mL and 50K (-223 C) this is about 14mL or just about 1 tablespoon. Combined, that's 43mL - or would be, if they didn't react. The volume of one mole of water vapour at several thousand degrees and atmospheric pressure (on the other side of the nozzle, where it's trying to escape to) is:

\[V = {{nRT}\over P} = {{(1\mathrm{mol}) (8.314{\mathrm{L.kPa}\over\mathrm{mol.K}}) (5000\mathrm{K})} \over {101.3\mathrm{kPa}}} = 410\mathrm{L}.\]

So from 43mL (0.043L) to 410L. The exhaust is 10,000 times bigger than the fuel. That's going to do some pretty serious pushing. How much exactly will depend on how fast the liquid fuel is fed to the engines and on the nozzle design, neither of which I'm going to go into because I only want to look at the chemistry side of it.

Next, the solid rocket boosters (SRBs). Ammonium perchlorate and aluminum react according to the reaction

\[10\mathrm{Al} + 6\mathrm{NH}_4\mathrm{ClO}_4 \rightarrow 4\mathrm{Al}_2\mathrm{O}_3 + 2\mathrm{AlCl}_3 + 12\mathrm{H}_2\mathrm{O} + 3\mathrm{N}_2\]

Same deal as above. The heats of formation of each chemical are as follows:
Al = 0
NH4ClO₄ = -295.77 kJ/mol
Al₂O₃ = -1620.57 kJ/mol
AlCl₃ = -584.59 kJ/mol
H₂O = -241.826 kJ/mol
N₂ = 0

The overall heat of reaction is:

\[ \Delta H_r = (4(-1620) + 2(-585) + 12(-241)) - 6(-295) = -8772\mathrm{kJ} \]

This is of course per 6 moles of ammonium perchlorate. That's a lot more heat released than the hydrogen fuel rocket, and again, the heat capacities of the products are much lower than the heat of reaction. Al₂O₃ is about 192 J/mol.K, AlCl₃ is about 82 J/mol.K, H₂O as above is 33.5 J/mol.K, and N₂ varies from 29 to 39 J/mol.K.

This time we have a copy of the heat capacity equation \(e = C_pnT\) for each compound, plus a sum of the four energies which equals the overall heat of reaction, to give us 5 equations and 5 unknowns. Rather than waste time with algebra, I'll put them in a spreadsheet that takes one temperature for all 4 heat capacity equations, solves each of them separately for energy, then sums the energies. Then, play with the temperature to see how much energy it takes to get to that temperature. Or, faster, use the "goal seek" feature to let it play with the temperature until the sum of energies is 8772000J. This gave me a bit over 6000K of temperature rise. We're starting at ambient temperature which in Cape Canaveral is anywhere between 270K and 310K, so it's about 6300K when it's done reacting.

Back to the ideal gas law to find the volume. Remember that we're not dealing with 1 mole of product anymore.

\[V = {{nRT}\over P} = {{(21\mathrm{mol}) (8.314{\mathrm{L.kPa}\over\mathrm{mol.K}}) (6300\mathrm{K})} \over {(101.3\mathrm{kPa})}} = 10860\mathrm{L}.\]

The initial fuel volume, for ammonium perchlorate and aluminum (discounting the trace amount of binding agent) is 460mL. (10 moles of aluminum is 270g, at 2.7g/mL that's 100mL. 6 moles of ammonium perchlorate is 705g, at 1.95g/mL that's 360mL.)

The exhaust is about 30,000 times bigger than the fuel, all squeezing through the nozzle at a ridiculous rate.

Endeavour / STS134. Image credit: NASA

With all that pushing it skyward: Endeavour, happy travels.