Exercises: Chemical Rocket Engines

Work these with a calculator and $g_0 = 9.81\ \text{m/s}^2$ (the code problems use $g_0 = 9.80665$ to match the book's astrotools convention — a difference below rounding). Difficulty: ⭐ foundational, ⭐⭐ intermediate, ⭐⭐⭐ challenging. Worked solutions to the daggered (†) and odd-numbered problems are in appendices/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. Every engine spec here is approximate and version-dependent (Tier 2).

Part A — Warm-ups: propellants, parts, and cycles (⭐)

17.1 † Which of the four workhorse propellant combinations has the highest specific impulse, and in one sentence, why?

17.2 A stage carries $60\ \text{t}$ of methalox with a bulk density of $830\ \text{kg/m}^3$. What propellant tank volume does it need?

17.3 † Name the engine cycle from the clue: "burns a few percent of the propellant in a small separate combustor to spin the turbine, then dumps that gas overboard."

17.4 An engine's fuel pump raises $50\ \text{kg/s}$ of RP-1 (density $810\ \text{kg/m}^3$) by a pressure of $300\ \text{bar}$. Estimate the hydraulic power the pump delivers, in megawatts.

17.5 † Match each engine to its cycle: Merlin, RS-25, RL10, Raptor — (a) expander, (b) gas-generator, (c) full-flow staged combustion, (d) fuel-rich staged combustion.

17.6 State the two operational things a solid rocket motor cannot do that a liquid engine can.

Part B — Propellants and the density trade (⭐⭐)

17.7 † A hydrolox stage and a kerolox stage each carry $200\ \text{t}$ of propellant. Using bulk densities of $360\ \text{kg/m}^3$ (hydrolox) and $1020\ \text{kg/m}^3$ (kerolox), compute each propellant tank volume and their ratio. In one sentence, why does this push designers to put hydrogen on the upper stage?

17.8 Explain, in terms of exhaust molecular weight, why LOX/LH2 gives a higher specific impulse than LOX/RP-1. (Conceptual — you will prove it in Chapter 19.)

17.9 † Methane can be synthesized on Mars from atmospheric CO₂ and water ice. Explain why this one fact favors methalox for a Mars-return vehicle, and give two further reasons methane is attractive compared with hydrogen or kerosene.

Part C — Engine cycles (⭐⭐)

17.10 † A gas-generator engine diverts $4\%$ of its total propellant flow to the gas generator, contributing negligible thrust. If the chamber specific impulse is $340\ \text{s}$, estimate the engine's effective $I_{sp}$ and the loss in seconds.

17.11 Explain why the expander cycle is practical only for low-thrust (upper-stage) engines. (Hint: compare how wall heat transfer and engine thrust each scale as the engine gets bigger.)

17.12 † Full-flow staged combustion uses two preburners rather than one. List three concrete advantages this buys over a single-preburner staged-combustion engine.

Part D — Solid motors and burn rate (⭐⭐)

17.13 † A solid propellant follows $r = a\,p_c^{\,n}$ with burn-rate exponent $n = 0.4$. If chamber pressure rises by $15\%$, by what percentage does the burn rate rise? Explain why $n < 1$ makes the motor stable rather than explosive.

17.14 A solid motor burns propellant of density $\rho_p = 1800\ \text{kg/m}^3$ at a burn rate $r = 0.9\ \text{cm/s}$ over a burning surface area of $A_b = 25\ \text{m}^2$. Compute the mass flow rate $\dot m = \rho_p A_b r$. If its sea-level $I_{sp} = 265\ \text{s}$, what thrust does it produce? (Check the units carefully — convert cm/s to m/s.)

Part C′ — Implement it in Python (⭐⭐)

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

17.15 † Write bulk_density(rho_ox, rho_fuel, of_ratio) returning the mixture's bulk density $\rho = 1/\big(\tfrac{f_{ox}}{\rho_{ox}} + \tfrac{f_{fuel}}{\rho_{fuel}}\big)$, where the oxidizer mass fraction is $f_{ox} = \text{OF}/(1+\text{OF})$ and $f_{fuel} = 1/(1+\text{OF})$. Trace it for hydrolox: bulk_density(1141, 71, 6.0).

17.16 Write gg_effective_isp(isp_chamber, gg_fraction) returning $(1 - f)\,I_{sp}$. Trace it for gg_effective_isp(330, 0.03).

17.17 † Write solid_thrust(rho_p, area, burn_rate, isp) that returns thrust in newtons via $\dot m = \rho_p\, A_b\, r$ and $F = \dot m\, I_{sp}\, g_0$. Trace it for solid_thrust(1800, 25, 0.009, 265).

Part E — Back of the envelope, find the error, why can't you just… (⭐⭐⭐)

17.18 † (Back of the envelope.) Super Heavy clusters 33 Raptor engines, each flowing on the order of $650\ \text{kg/s}$ of propellant, and burns for roughly $150\ \text{s}$. Estimate the total propellant mass to one significant figure, and compare it with the quoted ~3,400 t capacity. State your assumptions.

17.19 Why can't you just throttle a solid rocket motor down for a gentle landing, the way a Falcon 9 throttles its Merlins? Answer in physical terms — what would you have to control, and why can't you?

17.20 † (Find the error.) A student writes: "Liquid hydrogen has the highest specific impulse, so every stage of every rocket should use hydrolox for maximum performance." Identify the flaw, and name a real first stage that deliberately does not use hydrogen.

Part F — Design it and interleaved synthesis (⭐⭐ / ⭐⭐⭐)

17.21 † (Mission project.) For your chosen mission (Track A/B/C/D), write a short propulsion concept note for each propulsive stage of your spacecraft: the propellant class, the engine cycle (or "solid"/"pressure-fed"), and the one-line reason (driven by thrust, restart, throttle, and storability). Add it to your MDR.

17.22 (Design.) You must choose the engine for a reusable first stage that lands propulsively. Which propellant and which cycle would you choose, and name two engine features from §17.6 that make propulsive landing possible? Justify each choice.

17.23 † (Ch. 16 interleave.) An engine has mass flow $\dot m = 300\ \text{kg/s}$ and effective exhaust velocity $c = 3{,}300\ \text{m/s}$. Compute the momentum thrust. If the nozzle pressure term adds another $5\%$, what is the total thrust? (Use the thrust equation of Chapter 16.)

17.24 (Ch. 3 interleave.) A spacecraft's hypergolic apogee engine ($I_{sp} = 320\ \text{s}$) must supply $\Delta v = 1.8\ \text{km/s}$ to circularize at GEO. What mass ratio $m_0/m_f$ does the spacecraft need for that burn? (Rocket equation, Chapter 3.)

17.25 † (Synthesis.) The RS-25 has a substantially higher specific impulse than the Merlin, yet SpaceX built the "worse" engine and transformed the launch industry. Using this chapter's themes, explain how a lower-$I_{sp}$ engine can yield a better vehicle and a better program.

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: a correctly named propellant/cycle, an explicit reason tied to thrust/restart/storability, and units plus a sanity check on every computed number.