| name | rocket-propulsion |
| description | Rocket propulsion — specific impulse, thrust equation, rocket nozzle design (isentropic), ideal rocket equation, propellant types, chamber conditions, combustion stability. |
| metadata | {"priority":7,"promptSignals":{"phrases":["rocket propulsion","specific impulse","rocket nozzle","rocket equation","Tsiolkovsky","rocket thrust","propellant"],"minScore":3}} |
Rocket Propulsion — Complete Skill
Thrust Equation
F = ṁ v_e + (P_e - P_a) A_e
ṁ = propellant mass flow rate [kg/s]
v_e = exit velocity [m/s]
P_e = nozzle exit pressure [Pa]
P_a = ambient pressure [Pa]
A_e = nozzle exit area [m²]
Effective exhaust velocity:
c = v_e + (P_e - P_a) A_e / ṁ (for optimum expansion P_e = P_a: c = v_e)
Specific Impulse
I_sp = F / (ṁ g_0) = c / g_0 [seconds]
g_0 = 9.81 m/s² (standard gravity)
I_sp characterizes propellant efficiency:
| Propellant | I_sp [s] (vac) | Notes |
|---|
| Solid (HTPB/AP) | 260–295 | HTPB+ammonium perchlorate; high density |
| LOX/LH₂ | 450–460 | Best I_sp; cryogenic; Rocketdyne SSME |
| LOX/RP-1 (kerosene) | 350–370 | Falcon 9, Saturn V; dense, easy to handle |
| LOX/CH₄ (methane) | 375–380 | Raptor; good I_sp + storability |
| N₂O₄/UDMH | 315–330 | Storable; hypergolic; used for satellites |
| N₂O₄/MMH | 310–320 | Storable; spacecraft; Shuttle OMS |
| Cold gas (N₂) | 65–75 | Attitude control only |
Ideal Rocket Nozzle (Isentropic Flow)
Chamber to throat (choked):
P₀/P* = ((γ+1)/2)^(γ/(γ-1))
T₀/T* = (γ+1)/2
ρ₀/ρ* = ((γ+1)/2)^(1/(γ-1))
Throat condition: M = 1; P* ≈ 0.528 P₀ (for γ=1.4)
Exit Mach number from area ratio (isentropic):
A_e/A_th = (1/M_e) × [(2/(γ+1)) × (1 + (γ-1)/2 × M_e²)]^((γ+1)/(2(γ-1)))
Exit velocity:
v_e = √(2γ/(γ-1) × R/M_w × T₀ × [1 - (P_e/P₀)^((γ-1)/γ)])
R = 8314 J/kmol·K; M_w = molecular weight of exhaust gas
Mass Flow Rate
ṁ = P₀ A_th √(γ/R T₀) × (2/(γ+1))^((γ+1)/(2(γ-1)))
ṁ = P₀ A_th √(γ M_w / (R T₀)) × C_d_nozzle × (2/(γ+1))^((γ+1)/(2(γ-1)))
Characteristic velocity:
c* = P₀ A_th / ṁ [m/s]
Measures combustion efficiency; c* ≈ 1500–2400 m/s for typical propellants
Efficiency: η_c* = c_measured / c_theoretical ≈ 0.95–0.99
Tsiolkovsky Rocket Equation
Δv = I_sp × g₀ × ln(m₀/m_f) = c × ln(m₀/m_f)
m₀ = initial (wet) mass; m_f = final (dry) mass
Mass ratio: R = m₀/m_f = e^(Δv/c)
To LEO (~9200 m/s Δv including gravity losses):
With c = 3600 m/s (LOX/RP-1): R = e^(9200/3600) = 12.6 (very high mass ratio → staging)
With c = 4500 m/s (LOX/LH₂): R = e^(9200/4500) = 7.8 (more manageable)
Nozzle Design (Method of Characteristics)
Rao contour: optimized for maximum thrust with given length (minimum length nozzle)
Conical nozzle: simpler; half-angle typically 15°; slight loss from momentum divergence
θ_cone correction: η_div = (1 + cosθ)/2 ≈ 0.983 for θ = 15°
Expansion ratio ε = A_e/A_th:
Sea level: ε = 8–25 (compromise for atmospheric operation)
Vacuum: ε = 40–200 (maximize I_sp in vacuum)
Over-expanded (P_e < P_a): flow separation → instability if ε too large at low altitude
Combustion Chamber
Chamber pressure P₀: 2–25 MPa (low cost to high performance)
Higher P₀ → higher c*, higher I_sp, smaller nozzle → heavier chamber and pumps
Combustion stability:
Acoustic modes: longitudinal, tangential, radial
Baffles, acoustic cavities, injector pattern → prevent high-freq instability (chugging, screaming)
Low-frequency (chugging): propellant supply coupling → increase injection ΔP
L:* characteristic length L* = V_c/A_th (residence time proxy)
L* = 0.5–5 m (solids higher; liquids lower)
Solid Rocket Motor
Grain geometry: finocyl, star, wagon wheel (for regressive/progressive/neutral burning)
Burn rate: r = a P₀^n (St. Robert's law)
Typical: a = 0.01–0.05, n = 0.3–0.5 (n > 1 → unstable)
Output
Provide: F [kN], I_sp [s], c* [m/s], ṁ [kg/s], ε (area ratio), v_e [m/s], T₀ [K], P₀ [MPa], A_th [cm²], A_e [cm²], Δv budget from Tsiolkovsky for given mass ratio.
