Exercises: Combustion and Propellants

Work these with a calculator, $g_0 = 9.81\ \text{m/s}^2$, $R_u = 8.314\ \text{J/(mol·K)}$, and the two-knob relation $v_e \propto \sqrt{T_c/\mathcal{M}}$. Difficulty: ⭐ foundational, ⭐⭐ intermediate, ⭐⭐⭐ challenging. Worked solutions to the daggered (†) and odd-numbered problems are in the appendix answers-to-selected.md — try each cold first. For the "implement it" problems, do not run the code: hand-trace it and write the result in an # Expected output: comment, exactly as the chapter does. Treat all flame temperatures, mixture ratios, and molecular weights as approximate (Tier 2/3); the reasoning is what is graded.

Part A — Stoichiometry and energy (⭐)

18.1 † Balance the combustion of methane, $\text{CH}_4 + \_\,\text{O}_2 \rightarrow \_\,\text{CO}_2 + \_\,\text{H}_2\text{O}$, and use molar masses ($\text{CH}_4 = 16$, $\text{O}_2 = 32\ \text{g/mol}$) to find the stoichiometric mixture ratio (O/F).

18.2 Do the same for hydrogen, $2\,\text{H}_2 + \text{O}_2 \rightarrow 2\,\text{H}_2\text{O}$ (with $\text{H}_2 = 2.016\ \text{g/mol}$). Confirm the stoichiometric O/F is about 8.

18.3 † Methane's heat of combustion is about $50\ \text{MJ}$ per kg of $\text{CH}_4$. Using the stoichiometric O/F from 18.1, find the energy released per kilogram of propellant mixture (fuel + oxidizer). Compare it to the $13.4\ \text{MJ/kg}$ of hydrogen–oxygen and comment.

18.4 For a hydrogen–oxygen engine running at mixture ratio $r = 5$, use $\mathcal{M} = 2(1+r)$ to find the mean exhaust molecular weight. Is that heavier or lighter than the pure-water value at stoichiometric?

18.5 † An engine has $\gamma = 1.20$, $T_c = 3{,}500\ \text{K}$, exhaust $\mathcal{M} = 0.020\ \text{kg/mol}$, and expands from $p_c = 100\ \text{bar}$ to $p_e = 0.1\ \text{bar}$. Use the boxed formula of §18.2 to estimate $v_e$ and the corresponding $I_{sp}$. Which propellant family is this?

18.6 Propellant A: $T_c = 3{,}600\ \text{K}$, $\mathcal{M} = 22$. Propellant B: $T_c = 3{,}300\ \text{K}$, $\mathcal{M} = 13$. Without a full calculation, which has the higher exhaust velocity? Justify with the two-knob relation.

Part B — The two knobs (⭐⭐)

18.7 † Using $v_e \propto \sqrt{T_c/\mathcal{M}}$, compute the ratio of exhaust velocities for LOX/LH$_2$ ($T_c = 3{,}300\ \text{K}$, $\mathcal{M} = 13$) and LOX/CH$_4$ ($T_c = 3{,}500\ \text{K}$, $\mathcal{M} = 20$). By roughly what percentage does hydrogen beat methane, and which knob is responsible?

18.8 For hydrogen–oxygen, tabulate the knob $\sqrt{T_c/\mathcal{M}}$ at $r = 8, 6, 4, 3$ using $\mathcal{M} = 2(1+r)$ and the approximate temperatures $T_c = 3{,}300, 3{,}300, 2{,}950, 2{,}450\ \text{K}$. At which mixture ratio is the knob largest, and why does it eventually turn over as $r$ falls further?

18.9 † Derive the relation $\mathcal{M}(r) = 2(1+r)$ for hydrogen–oxygen exhaust (complete combustion to water plus leftover hydrogen, $r \le 8$, frozen composition). (Hint: work per gram of H$_2$; count the moles of each species in the products and the total product mass.)

Part C — Implement it in Python (⭐⭐)

Write each function, then hand-trace it for the given inputs and record the result in an # Expected output: comment. Do not run it.

18.10 † Write exit_velocity(gamma, Tc, M, pe, pc) returning the boxed formula of §18.2 (with $R_u = 8.314$, $M$ in kg/mol). Trace it for exit_velocity(1.20, 3300, 0.013, 0.1, 200), and convert the result to $I_{sp}$ with $g_0 = 9.80665$.

18.11 Write mean_molar_mass(r) returning $2(1+r)$ and use it inside knob(Tc, r) returning $\sqrt{T_c / \text{mean\_molar\_mass}(r)}$. Trace knob(3300, 6).

18.12 † Write specific_energy(dHf_gas_kJ_per_mol, moles_water, mass_kg) that returns the energy released per kilogram (in MJ/kg) for a hydrogen–oxygen reaction forming moles_water moles of water. Trace it for $\Delta H_f = -241.8\ \text{kJ/mol}$, 2 moles of water, and a propellant mass of $0.03603\ \text{kg}$.

Part D — Find the error (⭐⭐)

18.13 † An engineer writes: "Kerosene–oxygen burns hotter than hydrogen–oxygen, and the exhaust velocity formula has $T_c$ in it, so kerosene must give a higher specific impulse." Identify the flaw and correct the conclusion.

18.14 A student claims the best mixture ratio is always the stoichiometric one, "because that extracts the maximum chemical energy and therefore the maximum temperature and thrust." Explain, using both knobs, why real engines deliberately run fuel-rich instead.

Part E — Design it (⭐⭐ / ⭐⭐⭐)

18.15 † You are choosing propellant for a communications satellite's onboard propulsion (Track A): it must survive years in orbit and reignite reliably for station-keeping, with modest total impulse. From the four families of §18.3, choose a combination and defend it in three sentences, naming the two properties that dominate the decision and the one you sacrifice.

18.16 (Mission project) Write the propellant-choice note for your mission (Track A/B/C/D) as described in the Mission Design Checkpoint: your chosen combination, the two or three properties that drove it, and what you gave up. Save it to your MDR.

18.17 † A hydrogen upper stage could run at its $\sqrt{T_c/\mathcal{M}}$ optimum near $r = 3.5$ for maximum $I_{sp}$, but is flown at $r \approx 6$ instead. Explain the vehicle-level trade that pushes the flown mixture ratio oxidizer-ward of the chamber optimum, and name the mass the richer setting would penalize.

Part F — Back of the envelope & "why can't you just…" (⭐⭐⭐)

18.18 † A hydrogen upper stage loses roughly $1\%$ of its propellant per hour to boiloff on the pad. Estimate how long it could sit fueled before losing a tenth of its hydrogen, and explain in one sentence why this drives cryogenic vehicles to a late-load countdown. (Order-of-magnitude reasoning is the point.)

18.19 Why can't you just run every engine at the stoichiometric mixture ratio to get the most energy out of the propellant? Frame your answer in terms of flame temperature, exhaust molecular weight, and hardware survival.

18.20 † Why can't you just pick the single highest-$I_{sp}$ propellant (LOX/LH$_2$) for every stage of every rocket? Give two distinct physical reasons a designer often chooses a lower-$I_{sp}$ propellant instead.

Part G — Interleaved & synthesis (Chapters 3, 16, 17) (⭐⭐ / ⭐⭐⭐)

18.21 † (Ch. 3) Your combustion audit gives a vacuum $I_{sp} = 452\ \text{s}$ for a hydrogen upper stage. Convert it to exhaust velocity, then use the rocket equation to find the mass ratio needed for a $\Delta v = 4.1\ \text{km/s}$ trans-lunar injection burn.

18.22 (Ch. 16) Chapter 16 defined effective exhaust velocity rigorously. In one or two sentences, explain how the combustion quantities of this chapter ($T_c$, $\mathcal{M}$) become the $v_e$ that Chapter 16's thrust and $I_{sp}$ definitions depend on.

18.23 † (Ch. 17) A staged-combustion engine's fuel-rich preburner produces turbine-drive gas at about $800\ \text{K}$, far below the ~$3{,}500\ \text{K}$ main-chamber temperature. Using this chapter's ideas, explain how running the preburner extremely fuel-rich achieves that low temperature, and why it is necessary.

18.24 (Synthesis) State the Rayleigh criterion for combustion instability in your own words, and explain why an injector baffle and a tuned acoustic cavity attack the problem in two different ways.

Solutions to † and odd-numbered exercises are in appendices/answers-to-selected.md. Reference code for the "implement it" problems is in code/exercise-solutions.py. For design problems, the rubric rewards: an argument in the chapter's own terms (the two knobs, mixture ratio, storability), explicit units, and a sanity check on every number.