Exercises: Nozzle Theory and Thermodynamics

Work these with a calculator, $g_0 = 9.81\ \text{m/s}^2$, and $R_u = 8.314\ \text{J/(mol·K)}$. Difficulty: ⭐ foundational, ⭐⭐ intermediate, ⭐⭐⭐ challenging. Worked solutions to the daggered (†) and odd-numbered problems are in the appendix answers-to-selected.md — try each one 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. Unless told otherwise, model the exhaust as a calorically perfect gas and assume isentropic flow.

Part A — Warm-ups: ratios and conversions (⭐)

19.1 † For a rocket exhaust with $\gamma = 1.20$, compute the critical (throat) pressure ratio $p^*/p_c = \left(\frac{2}{\gamma+1}\right)^{\gamma/(\gamma-1)}$. Is the gas above or below half the chamber pressure when it reaches the throat?

19.2 A nozzle has a throat area $A_t = 0.05\ \text{m}^2$ and an exit area $A_e = 3.5\ \text{m}^2$. What is its expansion ratio $\epsilon$?

19.3 † Find the local speed of sound in an exhaust with $\gamma = 1.20$, specific gas constant $R = 520\ \text{J/(kg·K)}$, and temperature $T = 2000\ \text{K}$, using $a = \sqrt{\gamma R T}$.

19.4 The exhaust leaves a nozzle at $V = 3{,}600\ \text{m/s}$ where the local speed of sound is $1{,}200\ \text{m/s}$. What is the exit Mach number?

19.5 † Convert an exhaust velocity of $v_e = 4{,}200\ \text{m/s}$ to a specific impulse in seconds. Which propellant family (from Chapter 3's table) is this closest to?

19.6 For $\gamma = 1.30$, compute the front factor $\dfrac{2\gamma}{\gamma - 1}$ that appears in the exhaust-velocity formula.

Part B — Isentropic and choked flow (⭐⭐)

19.7 † For $\gamma = 1.20$ and chamber temperature $T_c = 3{,}600\ \text{K}$, use the isentropic relation $\frac{T_c}{T} = 1 + \frac{\gamma-1}{2}M^2$ to find the static temperature where the flow reaches $M = 2$.

19.8 (Implement it) Write choked_mass_flow(At, pc, Tc, R, gamma) returning $\dot m = A_t p_c \sqrt{\gamma/(R T_c)}\left(\frac{2}{\gamma+1}\right)^{\frac{\gamma+1}{2(\gamma-1)}}$. Hand-trace it for choked_mass_flow(0.06, 1.5e7, 3300, 600, 1.20) and record the result.

19.9 † Explain, in physical terms (not just "the equation says so"), why lowering the pressure downstream of a choked throat cannot push more mass per second through it.

Part C — Exhaust velocity and the molar-mass effect (⭐⭐)

19.10 † Compute the ideal exhaust velocity for $\gamma = 1.20$, $T_c = 3{,}500\ \text{K}$, exhaust molar mass $\mathcal{M} = 0.018\ \text{kg/mol}$, and pressure ratio $p_e/p_c = 0.01$. Convert to $I_{sp}$.

19.11 Propellant A has $T_c = 3{,}600\ \text{K}$ and $\mathcal{M} = 18\ \text{g/mol}$; propellant B has $T_c = 3{,}200\ \text{K}$ and $\mathcal{M} = 10\ \text{g/mol}$. Using $v_e \propto \sqrt{T_c/\mathcal{M}}$, which has the higher exhaust velocity, and by what factor? What does this say about "hotter is always better"?

19.12 † (Implement it) Write exit_velocity(gamma, Tc, molar_mass, pe, pc) for the ideal exhaust-velocity formula. Hand-trace exit_velocity(1.20, 3600, 0.013, 1e4, 1e7).

Part D — Expansion ratio and altitude (⭐⭐)

19.13 † A nozzle is designed to exit at $p_e = 0.05\ \text{bar}$. At roughly what ambient pressure is it perfectly expanded? Is it over- or under-expanded when fired at sea level ($p_a = 1.0\ \text{bar}$)? Which stage (first or upper) would use it?

19.14 A sea-level engine ($\epsilon = 16$) and a vacuum engine ($\epsilon = 165$) share the same chamber. Which produces a higher exhaust velocity in vacuum, and why is the small-$\epsilon$ engine nonetheless the right choice for a first stage?

19.15 † (Implement it) Write expansion_ratio(gamma, pe, pc) for the choked-throat area-ratio relation, then hand-trace it for $\gamma = 1.20$, $p_c = 200\ \text{bar}$, $p_e = 0.5\ \text{bar}$.

Part E — Design it (⭐⭐ / ⭐⭐⭐)

19.16 † Design a vacuum nozzle for a chamber at $p_c = 90\ \text{bar}$, $\gamma = 1.20$, that expands to $p_e = 0.06\ \text{bar}$. Find the exit Mach number and the expansion ratio $\epsilon$. Comment on the bell size.

19.17 (Mission project) For your mission's in-space stage (Track A/B/C/D), pick a propellant family, estimate its $T_c$, $\mathcal{M}$, and $\gamma$ (use the chapter's values), choose a chamber pressure and an exit pressure appropriate to where the stage fires, and compute $v_e$ and $I_{sp}$. Record the number in your MDR — it will size the stage's propellant mass.

19.18 † Size the throat: a stage needs a choked mass flow of $\dot m = 50\ \text{kg/s}$ with $p_c = 80\ \text{bar}$, $T_c = 3{,}400\ \text{K}$, $R = 500\ \text{J/(kg·K)}$, $\gamma = 1.20$. Find the throat area $A_t$ and the throat diameter.

Part F — Find the error & back of the envelope (⭐⭐⭐)

19.19 † Find the error. A designer says: "A longer nozzle always captures more energy, so I'll make the bell as long as manufacturing allows, on every stage." Identify what goes wrong, and on which stage the mistake is worst.

19.20 Back of the envelope. Estimate the percentage gain in exhaust velocity (hence $I_{sp}$) from switching an upper stage from an exhaust of molar mass $22\ \text{g/mol}$ to one of $13\ \text{g/mol}$, holding chamber temperature roughly fixed. Why is this the single biggest lever in chemical propulsion?

19.21 † Why can't you just raise the chamber pressure high enough that a purely converging nozzle spits out supersonic exhaust? Answer using the area–velocity relation.

Part G — Interleaved & synthesis (Chapters 3, 16, 18) (⭐⭐ / ⭐⭐⭐)

19.22 † (Ch. 16) Using the thrust equation $F = \dot m\, v_e + (p_e - p_a)A_e$, find the thrust of an engine with $\dot m = 300\ \text{kg/s}$, $v_e = 3{,}200\ \text{m/s}$, $p_e = 0.5\ \text{bar}$, and $A_e = 1.2\ \text{m}^2$, both at sea level ($p_a = 1.013\ \text{bar}$) and in vacuum. By how much does thrust rise from sea level to vacuum, and which term is responsible?

19.23 (Ch. 18) Kerosene burns hotter than hydrogen yet delivers a lower specific impulse. Explain this apparent paradox using this chapter's exhaust-velocity scaling.

19.24 † (Ch. 3) An upper stage with LH2 ($I_{sp} = 450\ \text{s}$) and one with RP-1 ($I_{sp} = 340\ \text{s}$) both have a mass ratio of $4$. Compute each stage's delta-v and the difference. Tie the result to the tyranny of the rocket equation.

19.25 (Synthesis) Despite hydrogen's higher $I_{sp}$, many first stages use RP-1 or methane. Give two physics/engineering reasons rooted in this chapter and Chapter 3 (think density, tank mass, mass ratio).

19.26 † (Ch. 16, synthesis) For the engine in Exercise 19.22, compute the effective exhaust velocity $c = F/\dot m$ at sea level and in vacuum, and compare each to the gas exit velocity $v_e = 3{,}200\ \text{m/s}$. Explain the sign of the difference in each case.

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: correct set-up of the isentropic/choked relations, explicit units, and a sanity check on the final number against a real engine.