Chapter 19 — Key Takeaways (Nozzle Theory and Thermodynamics)

A one-page reference. Reread this before an exam, or before you size any nozzle or pick a propellant.

The core relations

Relation Form Use it to find
Area–velocity $\dfrac{dA}{A} = (M^2 - 1)\dfrac{dV}{V}$ why the nozzle converges then diverges
Isentropic temperature $\dfrac{T_c}{T} = 1 + \dfrac{\gamma-1}{2}M^2$ static $T$ at any Mach number
Isentropic pressure $\dfrac{p_c}{p} = \left(1 + \dfrac{\gamma-1}{2}M^2\right)^{\frac{\gamma}{\gamma-1}}$ exit pressure from exit Mach
Choked mass flow $\dot m = A_t p_c \sqrt{\dfrac{\gamma}{R T_c}}\left(\dfrac{2}{\gamma+1}\right)^{\frac{\gamma+1}{2(\gamma-1)}}$ throat size ↔ mass flow
Ideal exhaust velocity $v_e = \sqrt{\dfrac{2\gamma}{\gamma-1}\dfrac{R_u T_c}{\mathcal{M}}\Big[1 - (p_e/p_c)^{\frac{\gamma-1}{\gamma}}\Big]}$ jet speed from chamber state
Area (expansion) ratio $\dfrac{A}{A_t} = \dfrac{1}{M}\Big[\dfrac{2}{\gamma+1}\big(1+\dfrac{\gamma-1}{2}M^2\big)\Big]^{\frac{\gamma+1}{2(\gamma-1)}}$ $\epsilon$ from exit Mach (and vice-versa)

The one relationship to memorize

$$ v_e \propto \sqrt{\frac{T_c}{\mathcal{M}}} $$

Hotter chamber and (especially) lighter exhaust molecules mean a faster jet. Molar mass is the denominator — the biggest lever in chemical propulsion.

Symbols and units

Symbol Name Units Notes
$M$ Mach number — $M = V/a$; $a = \sqrt{\gamma R T}$; $M=1$ only at the throat
$\gamma$ ratio of specific heats — rocket exhaust $\approx 1.15$–$1.25$
$T_c, p_c$ chamber (stagnation) temp, pressure K, Pa the reservoir the nozzle draws on
$\mathcal{M}$ exhaust molar mass kg/mol LH2 $\approx 0.013$; RP-1 $\approx 0.022$
$R = R_u/\mathcal{M}$ specific gas constant J/(kg·K) $R_u = 8.314\ \text{J/(mol·K)}$
$A_t, A_e$ throat, exit area m² $\epsilon = A_e/A_t$
$v_e$ gas exit velocity m/s the jet speed
$c$ effective exhaust velocity m/s $= v_e + (p_e-p_a)A_e/\dot m$; $= v_e$ if perfectly expanded
$\epsilon$ expansion ratio — bigger → lower $p_e$ → higher $v_e$ (but heavier)

The de Laval nozzle in one picture

Section Flow Rule (from area–velocity)
Converging subsonic, $M<1$ area down → speed up (garden hose)
Throat sonic, $M=1$ $dA=0$, minimum area, choked
Diverging supersonic, $M>1$ area up → speed up (the bell)

A purely converging nozzle can never exceed $M=1$; the diverging section is what makes exhaust supersonic.

Choked flow — what it means

  • $M=1$ at the throat fixes $\dot m$ from $p_c$, $A_t$, $T_c$ alone — independent of back pressure.
  • Critical ratios ($\gamma=1.2$): $p^*/p_c = (2/(\gamma+1))^{\gamma/(\gamma-1)} \approx 0.56$; $T^*/T_c = 2/(\gamma+1) \approx 0.91$.
  • Throttle a rocket by changing $p_c$ (the throat area is usually fixed).

Over- / under-expansion (decision aid)

Condition Name Where it happens Effect
$p_e > p_a$ under-expanded vacuum bell in air / any bell in vacuum plume swells; leaves some thrust on the table
$p_e = p_a$ perfectly expanded one altitude only maximum thrust for the conditions
$p_e < p_a$ over-expanded big bell at sea level thrust penalty; risk of flow separation

Altitude compensation: $p_a$ falls from ~1 bar to 0 during ascent, so a fixed nozzle is optimal at only one altitude. Sea-level bells are small ($\epsilon \sim 15$–$25$); vacuum bells are large ($\epsilon \sim 50$–$200$).

Numbers worth memorizing

  • Best chemical $I_{sp}$: LH2 ≈ 450 s (vacuum); methane ≈ 370–380 s; kerosene ≈ 330–350 s.
  • Critical pressure ratio $p^*/p_c \approx 0.5$–0.6; sea-level $\epsilon \sim 16$, vacuum $\epsilon \sim 70$–165.
  • RS-25 (worked in Case Study 1): $p_c = 206$ bar, $\epsilon = 69$, vacuum $I_{sp} \approx 452$ s.
  • $R_u = 8.314\ \text{J/(mol·K)}$; $g_0 = 9.81\ \text{m/s}^2$.

Why LOX/LH2 beats LOX/RP-1 (the payoff of Ch. 18)

Even though kerosene burns hotter, hydrogen's exhaust is about half the molar mass (light $\text{H}_2$ + $\text{H}_2\text{O}$ vs heavy $\text{CO}_2$ + $\text{CO}$), and $v_e \propto 1/\sqrt{\mathcal{M}}$ more than makes up the difference: ~$443$ s vs ~$339$ s. The penalty is hydrogen's terrible density (~$70$ vs ~$810\ \text{kg/m}^3$) → huge tanks → why first stages often prefer dense RP-1 or methane.

Common pitfalls

Pitfall Reality
"A converging nozzle can go supersonic with enough pressure." No — supersonic flow needs a diverging section; converging caps at $M=1$.
"Hotter fire always means higher Isp." $v_e \propto \sqrt{T_c/\mathcal{M}}$; molar mass usually dominates (LH2 wins while cooler).
"Lower back pressure means more mass flow." Once choked, $\dot m$ is fixed by $p_c, A_t, T_c$; back pressure is irrelevant.
"Bigger bell is always better." Only up to flow separation and mass; a big bell at sea level separates.
Confusing $v_e$ (gas) with $c$ (effective). $c = v_e + (p_e-p_a)A_e/\dot m$; equal only when perfectly expanded.

Mission / astrotools additions this chapter

  • MDR: chose your in-space stage's propellant and computed its $v_e$/$I_{sp}$ → propellant mass.
  • propulsion.py: exit_velocity(gamma, Tc, molar_mass, pe, pc) and expansion_ratio(gamma, pe, pc) (extending thrust from Ch. 16).