Case Study: Three Engines, Three Propellants — Auditing RS-25, Merlin, and Raptor

"You can learn more from one honest engine test stand than from a shelf of brochures."

Executive Summary

Three of the most important rocket engines of the modern era burn three different propellants: the Space Shuttle Main Engine (RS-25) burns hydrogen, SpaceX's Merlin burns kerosene, and SpaceX's Raptor burns methane. Their specific impulses line up in a strict order — hydrogen highest, methane in the middle, kerosene lowest — and in this case study we audit that order using nothing but this chapter's two knobs. We will predict each engine's specific impulse from its chamber temperature and exhaust molecular weight, compare our reconstruction to the published numbers, and then answer the deeper question the $I_{sp}$ table cannot: if hydrogen wins on efficiency, why did SpaceX build Merlin around kerosene and Raptor around methane? The answer is that specific impulse is only one column of the ledger.

Skills applied

• Predicting the $I_{sp}$ ordering of propellants from the knob $\sqrt{T_c/\mathcal{M}}$ (§18.2).
• Reconstructing absolute $I_{sp}$ from chamber conditions with the exhaust-velocity formula (§18.2).
• Reading a propellant choice as a system trade among $I_{sp}$, density, and reuse (§18.3–18.4).
• Sanity-checking a computed number against a published one, and accounting for the residual.

Background

All three engines pair their fuel with liquid oxygen and run fuel-rich. Their approximate operating conditions (Tier 2 — rounded, representative values, not manufacturer data) are:

Notice immediately that the hottest-burning engine (Merlin, kerosene) has the lowest specific impulse, and the coolest (RS-25, hydrogen) has the highest. If your instinct still says "hotter should be better," this case study is the cure.

Phase 1: Predict the order from one knob

Before touching the full formula, compute the knob $\sqrt{T_c/\mathcal{M}}$ for each — it captures everything the propellant contributes to exhaust velocity:

$$ \text{RS-25: } \sqrt{\tfrac{3300}{13}} = \sqrt{254} = 15.9, \quad \text{Raptor: } \sqrt{\tfrac{3500}{20}} = \sqrt{175} = 13.2, \quad \text{Merlin: } \sqrt{\tfrac{3600}{23}} = \sqrt{157} = 12.5. $$

The order falls out at once: hydrogen (15.9) beats methane (13.2) beats kerosene (12.5). And the ratios are predictive. Hydrogen over kerosene is $15.9/12.5 = 1.27$; the published $I_{sp}$ ratio is $452/348 = 1.30$. Methane over kerosene is $13.2/12.5 = 1.06$; published, $360/348 = 1.03$. The single number $\sqrt{T_c/\mathcal{M}}$ — chamber temperature and exhaust weight, nothing else — reproduces not just the order but nearly the magnitude of the differences between three real engines. That is a remarkable payoff for one proportionality.

Phase 2: Reconstruct the absolute numbers

Now feed each engine's full conditions into the chapter's exhaust-velocity formula (using $\gamma \approx 1.20$ for the hydrogen and methane exhausts, $1.22$ for kerosene, and a small exit pressure for the large vacuum nozzles):

$$ v_e = \sqrt{\frac{2\gamma}{\gamma-1}\frac{R_u T_c}{\mathcal{M}}\left[1 - \left(\frac{p_e}{p_c}\right)^{(\gamma-1)/\gamma}\right]}, \qquad I_{sp} = \frac{v_e}{g_0}. $$

Carrying out the arithmetic (the code/ folder does each by hand):

Our reconstructions land within a couple of percent of the published values — extraordinary agreement for a back-of-envelope model. The residual is exactly what the model leaves out: the specific nozzle expansion ratio (RS-25's enormous 69:1 bell versus Merlin's), real-gas and dissociation-recombination effects, and combustion efficiency below the ideal. Those are the province of Chapter 19; the point here is that the propellant chemistry alone — $T_c$ and $\mathcal{M}$ — already fixes performance to within a few percent.

🔧 Engineering Reality: The one-percent residuals are not noise; they are money. On an upper stage, a single second of $I_{sp}$ can be worth tens of kilograms of payload. That is why real engine development chases the last few percent with equilibrium-chemistry codes, expansion-ratio optimization, and injector tuning — refinements that live on top of the chemistry ceiling this chapter sets, never past it.

Phase 3: The audit the $I_{sp}$ column hides — density and reuse

If hydrogen wins on efficiency by 27%, why does neither SpaceX engine burn it? Because $I_{sp}$ is one column, and two others decide vehicles.

Density. Bulk densities at these mixture ratios (computed from component densities) are roughly $0.36\ \text{g/cm}^3$ for LOX/LH$_2$, $0.83$ for LOX/CH$_4$, and $1.02$ for LOX/RP-1. Kerosene packs nearly three times the propellant mass into a given tank as hydrogen. On a first stage, burned quickly and low in the atmosphere, that density buys small, light, structurally efficient tanks and high thrust per unit frontal area — which is exactly why Merlin (and, historically, the F-1) burns kerosene. Hydrogen's bulk is tolerable only where its efficiency is carried a long way: upper stages and the RS-25, which the Shuttle bolted to giant solid boosters precisely to make up for hydrogen's feeble thrust density at liftoff.

Reuse. Here methane's advantage appears. A kerosene molecule is a long carbon chain; burned fuel-rich, it deposits soot and hard coke on injector faces and chamber walls, which must be cleaned between flights — a real obstacle to rapid reuse. Methane is a single carbon atom with four hydrogens; it burns almost cleanly, leaving little residue. For an engine meant to fly again within hours, that chemical cleanliness is worth more than the ~90 seconds of $I_{sp}$ that hydrogen would offer at the cost of bulk, boiloff, and handling misery.

Phase 4: Reading the three choices

Put the audit together and each engine's propellant is revealed as the correct answer to a different question:

• RS-25 / hydrogen: "Maximize efficiency, cost and bulk be damned." A reusable but expensive, refurbishment- heavy engine for a vehicle that accepted solid boosters to cover hydrogen's low thrust.
• Merlin / kerosene: "Maximize density and simplicity for a cheap, dense first stage." Accept a lower $I_{sp}$ and soot in exchange for compact tanks and a proven, storable-on-the-pad fuel.
• Raptor / methane: "Optimize for rapid, full reuse and eventual Mars operations." Take methane's balanced $I_{sp}$, high density, clean burn, and — the decisive future card — its manufacturability on Mars.

🔗 Connection: This is the Starship anchor's propulsion chapter. Every property we audited here — methane's clean burn, its density, its near-LOX storage temperature — is a reason Raptor exists, and we assemble the full Starship case study in Chapter 38. The engine cycle that lets Raptor reach 300 bar (full-flow staged combustion) is Chapter 17's story; the chemistry that makes methane the right fuel to run through it is this chapter's.

Discussion Questions

1. Merlin burns $300\ \text{K}$ hotter than RS-25 yet delivers 100 fewer seconds of $I_{sp}$. Explain the apparent paradox in one sentence, using both knobs.
2. If a future engine achieved a hydrogen exhaust molecular weight of $11\ \text{g/mol}$ (running richer) at the same $3{,}300\ \text{K}$, estimate its knob and the fractional $I_{sp}$ gain over the $\mathcal{M} = 13$ case.
3. Why does the density argument favor kerosene on a first stage but not on an upper stage?
4. Our reconstructions differed from published values by 1–2%. Name two physical effects, both deferred to Chapter 19, that account for the residual.

Your Turn: Extensions

• Option A (analysis). Add a fourth row for a hypergolic engine (N$_2$O$_4$/UDMH, $T_c \approx 3{,}300\ \text{K}$, $\mathcal{M} \approx 24$, $p_c \approx 15\ \text{bar}$). Predict its knob and $I_{sp}$ ordering relative to the three above, and explain why spacecraft still use it despite the low number.
• Option B (computation). Turn Phase 2 into a function audit(engine) that takes $(\gamma, T_c, \mathcal{M}, p_e, p_c)$ and prints $v_e$ and $I_{sp}$; reproduce the table. Do not run it — hand-trace and add # Expected output:.
• Option C (design). For your mission's stage, pick which of these three propellants you would use and defend it in the audit's terms (efficiency, density, reuse), then record the choice and the $I_{sp}$ you will carry into your rocket-equation sizing.

Key Takeaways

1. The $I_{sp}$ order of propellants is set by $\sqrt{T_c/\mathcal{M}}$, and the knob predicts real engines' ratios to a few percent — hydrogen wins on low molecular weight, not high temperature.
2. Chemistry fixes performance to within ~2%; the remaining percent is nozzle expansion and real-gas effects (Chapter 19), refinements on top of the ceiling, never past it.
3. Specific impulse is one column of three. Density (first-stage compactness, thrust per volume) and reuse (clean burn) decide propellant choices that the $I_{sp}$ table alone would get wrong.
4. Three engines, three right answers. RS-25 optimizes efficiency, Merlin density, Raptor reuse — the same physics, three different mission questions.