Chapter 18 — Key Takeaways (Combustion and Propellants)
A one-page reference. Reread this before an exam, or before you choose a propellant for your mission.
The one idea that organizes everything
$$ v_e \;\propto\; \sqrt{\frac{T_c}{\mathcal{M}}} $$
Exhaust velocity — and thus specific impulse, and thus, through the rocket equation, the whole vehicle — rises with flame temperature $T_c$ and falls with exhaust molecular weight $\mathcal{M}$. Only two knobs. Hotter and lighter is faster. Hydrogen wins on the lighter knob, not the hotter one.
The key equations
| Equation | Meaning / use |
|---|---|
| $\Delta H_\text{rxn} = \sum \Delta H_f(\text{prod}) - \sum \Delta H_f(\text{react})$ | Heat released by a reaction, from enthalpies of formation |
| $T_c \approx T_0 + \dfrac{\Delta H_\text{rxn}}{n\,\bar c_p}$ | Naive adiabatic flame temperature (overpredicts — no dissociation) |
| $v_e = \sqrt{\dfrac{2\gamma}{\gamma-1}\dfrac{R_u T_c}{\mathcal{M}}\left[1 - (p_e/p_c)^{(\gamma-1)/\gamma}\right]}$ | Ideal exhaust velocity (derived in Ch. 19; here $\gamma$ = specific-heat ratio) |
| $\mathcal{M}(r) = 2(1+r)$ | Mean exhaust molar mass for H$_2$/O$_2$, $r \le 8$ (frozen, complete combustion) |
| $r = \dot m_\text{ox}/\dot m_\text{fuel}$ | Mixture ratio (O/F); compare to stoichiometric |
| $\oint p'\,\dot q'\,dt > 0$ | Rayleigh criterion — combustion instability grows |
Definitions (the terms this chapter owns)
| Term | One-line definition |
|---|---|
| Combustion | Rapid exothermic fuel–oxidizer reaction releasing bond energy as heat |
| Stoichiometry | Atom-conserving accounting; stoichiometric O/F consumes both fully |
| Adiabatic flame temperature | Max $T_c$ if all released energy heats the products, none lost |
| Mixture ratio (O/F) | Mass of oxidizer per mass of fuel burned |
| Hypergolic propellant | Fuel + oxidizer that ignite spontaneously on contact |
| Cryogenic propellant | Gas at room temperature; must be stored below ~120 K |
| Combustion instability | Self-amplifying pressure/heat-release oscillation coupled to chamber acoustics |
| Green propellant | Low-toxicity, low-handling-cost replacement for hydrazine |
Propellant families — decision aid
| Combination | Vac $I_{sp}$ (s) | Bulk density | Storability | Choose it when… |
|---|---|---|---|---|
| LOX / LH$_2$ | 440–465 | ~0.36 (bulky) | cryo, fast boiloff | efficiency carried far — upper stages, kick stages |
| LOX / CH$_4$ | 355–380 | ~0.83 | mild cryo (near LOX) | reuse, balance, Mars ISRU — one engine both stages |
| LOX / RP-1 | 300–340 | ~1.02 (dense) | fuel storable | dense, high-thrust first stages |
| N$_2$O$_4$ / hydrazines | 285–320 | ~1.16 | storable for years | long waits + reliable restart (RCS, deep space) |
| Green monoprop (ASCENT, LMP-103S) | ~230–255 | high | storable, low-tox | small in-space thrusters; cut handling cost |
Rule of thumb: density matters most on first stages (burned low and fast); $I_{sp}$ matters most on upper stages (carried nearly to orbit); storability and restart matter most in space.
Why engines run fuel-rich
Excess light fuel drops exhaust $\mathcal{M}$ faster than it drops $T_c$, so $\sqrt{T_c/\mathcal{M}}$ peaks fuel-rich of stoichiometric (H$_2$/O$_2$ optimum near O/F ≈ 3.5; flown near 6 to shrink the hydrogen tank).
Combustion instability at a glance
| Type | Frequency | Couples to | Danger |
|---|---|---|---|
| Chugging | ~10–200 Hz | feed system / pumps | low, but disruptive |
| Buzzing | ~200–1,000 Hz | injector / manifold | moderate |
| Screech / screaming | >1,000 Hz | chamber acoustic modes | destroys engine in ms |
Fixes: injector baffles (break transverse modes — remove driving); acoustic cavities (tuned resonators — add damping); injector design (keep heat release out of phase — Rayleigh). Verified by testing (F-1: bombs in a running chamber).
Common pitfalls
| Pitfall | Reality |
|---|---|
| "Hydrogen wins because it burns hottest." | It burns cooler than kerosene; it wins on low molecular weight. |
| "Stoichiometric is optimal." | Engines run fuel-rich; $\sqrt{T_c/\mathcal{M}}$ peaks rich, and stoichiometric is too hot for hardware. |
| "Highest-$I_{sp}$ propellant is always best." | Density, storability, restart, reuse often outweigh $I_{sp}$. |
| "Green propellant = higher performance." | It means low toxicity/handling cost; $I_{sp}$ ≈ hydrazine. |
| "A better fuel could give unlimited $I_{sp}$." | Chemistry caps energy at ~10–13 MJ/kg mixture and $T_c$ by dissociation. |
Numbers worth memorizing
- Best chemical vacuum $I_{sp} \approx$ 450–465 s (LOX/LH$_2$); chemical energy ~10–13 MJ/kg of mixture.
- Stoichiometric H$_2$/O$_2$: O/F = 8 (flown near 6). Flame temperatures capped near ~3,600 K by dissociation.
- Boiling points: LOX 90 K, LCH$_4$ 112 K, LH$_2$ 20 K (why methane is "space-storable," hydrogen is not).
- $g_0 = 9.81\ \text{m/s}^2$; $R_u = 8.314\ \text{J/(mol·K)}$.
Mission / astrotools additions this chapter
- MDR: a propellant-choice note — chosen combination, the 2–3 properties that drove it, what you gave up.
propulsion.py:exit_velocity(gamma, Tc, M, pe, pc)— the $\sqrt{T_c/\mathcal{M}}$ core; feed its $I_{sp}$ into the Chapter 3 rocket equation to size propellant. (Chapter 19 wraps it with expansion-ratio optimization.)