nozzle-design

name	nozzle-design
description	Nozzle design — convergent-divergent (De Laval) theory, isentropic relations, area-Mach relation, design Mach number, shock in nozzle (normal shock), thrust coefficient (C_F), specific impulse (I_sp), nozzle efficiency (ηn), bell nozzle vs. conical, method of characteristics (MOC), industrial spray nozzles (ASME PTC 19.5), and rocket/gas turbine afterburner applications.
metadata	{"priority":7,"promptSignals":{"phrases":["nozzle design","convergent divergent nozzle","De Laval nozzle","nozzle Mach number","thrust nozzle","nozzle throat"],"minScore":3}}

Nozzle Design — Complete Skill

Compressible Flow Fundamentals

Isentropic Relations

Stagnation (total) conditions: T₀/T = 1 + (γ-1)/2 × M² [T₀ = total temperature; T = static; M = Mach number; γ = ratio of specific heats] p₀/p = (1 + (γ-1)/2 × M²)^(γ/(γ-1)) [total pressure ratio] ρ₀/ρ = (1 + (γ-1)/2 × M²)^(1/(γ-1)) [total density ratio]

For air (γ = 1.4): T₀/T = 1 + 0.2M² p₀/p = (1 + 0.2M²)^3.5 ρ₀/ρ = (1 + 0.2M²)^2.5

Speed of sound: a = √(γRT) [m/s; R = gas constant; T = static temperature [K]] At M = 1 (throat): a* = √(γRT*); T* = T₀ × 2/(γ+1) = T₀ × 0.833 (air)

Critical pressure ratio: p*/p₀ = (2/(γ+1))^(γ/(γ-1)) = 0.5283 (air) [minimum p₀/p_exit to choke flow]

Area-Mach Relation

Area ratio: A/A* = (1/M) × [(2/(γ+1)) × (1 + (γ-1)/2 × M²)]^((γ+1)/(2(γ-1)))

Simplified (air): A/A* = (1/M) × [1/6 × (1 + 0.2M²)]^3 [γ = 1.4]

Area-Mach table (subsonic branch: M < 1; supersonic branch: M > 1 for same A/A):*

M	A/A*	p/p₀	T/T₀
0.5	1.340	0.843	0.952
1.0	1.000	0.528	0.833
1.5	1.176	0.272	0.690
2.0	1.687	0.128	0.556
2.5	2.637	0.0585	0.444
3.0	4.235	0.0272	0.357
4.0	10.72	0.00659	0.238
5.0	25.00	0.00189	0.167

Newton-Raphson iteration for M from A/A:* f(M) = A/A* - g(M); f'(M) = -g'(M) M_{n+1} = M_n - f(M_n)/f'(M_n) [converges in 3–5 iterations]

Convergent-Divergent (De Laval) Nozzle

Operating Regimes

Regime 1 — No flow / subsonic throughout: p_exit/p₀ = 1 → no flow; p₀ = p_exit

Regime 2 — Subsonic, unchoked: p*/p₀ < p_exit/p₀ < 1 → subsonic throughout; M_throat < 1; not choked M_exit < M_throat < 1

Regime 3 — Choked (M_throat = 1; subsonic divergent section): p_exit/p₀ = p*(M_design_subsonic)/p₀ → isentropic; first choked solution (subsonic exit)

Regime 4 — Normal shock in divergent section: p_exit between regime 3 and design → normal shock in divergent section at some location x_shock Exit flow: subsonic after shock

Regime 5 — Normal shock at exit plane: p_exit slightly above design supersonic → shock at exit; sharp pressure rise

Regime 6 — Overexpanded (oblique shocks): p_exit < p_design_pressure; overexpanded → oblique shocks or Prandtl-Meyer expansion at exit

Regime 7 — Design condition (fully expanded): p_exit = p_e_design → isentropic; no shocks; maximum thrust; nozzle pressure ratio NPR = p₀/p_exit

Regime 8 — Underexpanded: p_exit > p_e_design → underexpanded → Prandtl-Meyer expansion fans at exit

Throat Design

Mass flow rate: ṁ = A* × p₀ × √(γ/(RT₀)) × (2/(γ+1))^((γ+1)/(2(γ-1))) For air: ṁ = 0.6847 × A* × p₀ / √T₀ [SI; ṁ [kg/s]; A* [m²]; p₀ [Pa]; T₀ [K]]

Throat radius R_t: R_t = 0.5 × R_chamber (wall curvature at throat); bell nozzle R_throat = (0.4–0.6) × R_t Converging angle: 20–45° (half angle from axis); too steep → separation Throat curvature (upstream): R_c = 1.0–1.5 × R_t (bell) or 0.5–1.5 × R_t

Normal Shock in Nozzle

Normal shock relations (upstream M₁; downstream M₂): M₂² = (M₁² + 2/(γ-1)) / (2γM₁²/(γ-1) - 1) [(γ-1)=0.4 for air] p₂/p₁ = (2γM₁² - (γ-1)) / (γ+1) T₂/T₁ = [2γM₁² - (γ-1)] × [2 + (γ-1)M₁²] / [(γ+1)M₁]² p₀₂/p₀₁ = [(2γM₁²/(γ-1)-1)^(-1/(γ-1)) × ((γ+1)M₁²/(2+(γ-1)M₁²))^(γ/(γ-1))] [total pressure loss]

Shock location for given back pressure: Use isentropic table to find M₁ at cross-section; compute p₀₂ after shock; find exit M from subsonic branch; match to back pressure

Rocket Nozzle Design

Thrust and Performance

Thrust equation: F = ṁ × V_e + (p_e - p_a) × A_e [V_e = exit velocity; p_e = exit pressure; p_a = ambient; A_e = exit area]

Exit velocity: V_e = √(2γ/(γ-1) × R × T₀ × [1 - (p_e/p₀)^((γ-1)/γ)]) [isentropic; R = specific gas constant]

Specific impulse: I_sp = F / (ṁ × g₀) [seconds; g₀ = 9.81 m/s²; higher I_sp → more efficient propellant use] Vacuum I_sp for ideal nozzle: I_sp_vac = V_e_opt / g₀ [at optimal expansion p_e = p_a = 0] Typical: LOX/LH₂ → I_sp ≈ 450 s; LOX/RP-1 → I_sp ≈ 350 s; solid motors → I_sp ≈ 260 s

Thrust coefficient: C_F = F / (p₀ × A*) [dimensionless; accounts for geometry and expansion ratio] C_F = √(2γ²/(γ-1) × (2/(γ+1))^((γ+1)/(γ-1)) × [1 - (p_e/p₀)^((γ-1)/γ)]) + (p_e - p_a) × A_e / (p₀ × A*) Ideal C_F range: 1.2–2.0 depending on γ and expansion ratio

Characteristic velocity: c* = p₀ × A* / ṁ [m/s; depends only on combustion chamber conditions, not nozzle geometry] c* = √(γRT₀/γ) × [(γ+1)/2]^((γ+1)/(2(γ-1))) / √γ [theoretical]

Expansion Ratio Selection

Optimal expansion: p_e = p_a (ambient pressure) → maximize thrust ε = A_e/A* from area-Mach relation at M_e; M_e from p_e = p₀ × (p_e/p₀) = p_a

Sea-level vs. vacuum trade-off: Sea level: p_a = 101.3 kPa → small ε (lower M_e) optimized; ε ≈ 4–8 Vacuum: p_a = 0 → maximum expansion; ε = 40–200 for upper-stage engines

Altitude compensation: Aerospike / plug nozzle: ambient pressure acts as outer boundary → self-compensating Linear aerospike: SSME alternate; RVX-1; efficient across altitude range

Bell Nozzle (Rao Optimum)

Bell vs. conical comparison:

Parameter	Conical (15°)	Bell (80% Rao)	Parabolic Bell
Length	L_ref	0.80 × L_ref	0.75 × L_ref
Divergence loss (%)	1.7	< 0.5	< 0.7
Uniformity at exit	Poor	Excellent	Good
Design complexity	Simple	Complex (MOC)	Moderate

Thrust correction factor (conical): λ = 0.5(1 + cos α) [α = half-angle of divergence; λ = 0.983 at 15°; applies to exit momentum term]

Bell nozzle contour (Rao parabola): Initial expansion angle θ_i from throat (25–55°); exit angle θ_e (5–12°) Parabola connecting inflection point to exit; parameterized by expansion ratio ε and Ln/L_15° ratio

Method of Characteristics (MOC)

For supersonic nozzle contour design: Characteristic lines: C+ and C- families along which Riemann invariants are constant R+ = ν(M) + θ = const along C+ (forward Mach line) R- = ν(M) - θ = const along C- (backward Mach line) ν(M) = Prandtl-Meyer function = √((γ+1)/(γ-1)) × arctan(√((γ-1)/(γ+1) × (M²-1))) - arctan(√(M²-1))

MOC procedure:

1. Divide symmetry plane of throat (uniform M = 1 line) into N points
2. For each point pair, apply compatibility relations to find interior field
3. Apply wall BC (θ_wall = 0 at centerline; θ_wall = nozzle contour angle elsewhere)
4. Trace wall contour that produces uniform exit flow at design M_e

MOC result: wave-cancellation nozzle; optimal Rao bell; uniform exit flow without reflections

Industrial Spray and Flow Nozzles

Flow Measurement Nozzles (ASME PTC 19.5)

ASME/ISO long-radius nozzle: Coefficient of discharge: C_d = 0.9900 (Re_D > 10⁶; well-characterized) ṁ = C_d × A_throat × √(2ρ_f × ΔP) [incompressible; ΔP = upstream-throat pressure drop] Expansibility factor Y: Y = 1 - (0.41 + 0.35β⁴) × ΔP/(γ × p_upstream) [compressible correction; β = d_throat/D_pipe]

Venturi nozzle: C_d = 0.985 (smooth inlet); lower pressure drop than orifice plate; preferred for clean fluids ASME MFC-3M standard

Spray Nozzles (Industrial)

Types and droplet sizing: Full-cone: θ = 25–120° spray; D_SMD = C × (σ/ρ_l)^0.6 × Q_l^0.5 / (ΔP^0.4 × D_orifice^0.2) [Sauter mean diameter] Flat fan: narrow angle; uniform coverage; cleaning, coating Hollow cone: thin-walled spray; combustion atomization; pressure-swirl type Air-blast atomizer: D_SMD ≈ 585 × √(σ/(ρ_l × V_r²)) × (μ_l/(σ × ρ_l)^0.5)^0.45 × (1 + 1/ALR)^0.5 ALR = air-to-liquid mass ratio; higher ALR → finer droplets

Pressure drop: Q = C_d × A × √(2ΔP/ρ) [for liquid; ρ = liquid density] C_d = 0.60–0.90 depending on geometry (sharp orifice: 0.61; well-rounded: 0.90)

Nozzle Thermal Analysis

Heat flux at throat (rocket): q_throat = 0.026 × (k_gas^0.6 × ρ_gas^0.8 × V_e^0.8) / (D_t^0.2 × μ^0.6) [Bartz correlation; W/m²] Maximum temperature on wall: T_w = T_aw - q × t_wall / k_wall [T_aw = adiabatic wall temperature] T_aw = T₀ × (1 + r × (γ-1)/2 × M²) / (1 + (γ-1)/2 × M²) [r = recovery factor ≈ 0.9 turbulent]

Regenerative cooling channel sizing: q = h_coolant × (T_w - T_coolant) [h from Dittus-Boelter: Nu = 0.023 Re^0.8 Pr^0.4] Coolant velocity ≥ 10 m/s for adequate cooling (LOX/LH₂ propellant systems)

Standards and References

Standard	Scope
ASME PTC 19.5	Flow measurement nozzles
ISO 5167-3	Nozzle flow measurement
AIAA-S-080A	Solid propulsion nozzle design
NASA SP-8120	Liquid rocket engine turbopump nozzle design
NASA TN-D-4689	Method of characteristics for nozzle contour
SAE AS4770	Aerospace fluid nozzle requirements

Output

Provide: application (rocket/gas turbine/industrial flow/spray), working fluid (γ, R [J/kg·K], T₀ [K], p₀ [Pa]), design Mach number M_e at exit, area ratio ε = A_e/A* from isentropic table, throat area A* [m²] from ṁ and choking relation, mass flow rate ṁ [kg/s], exit conditions (T_e [K], p_e [Pa], V_e [m/s]), thrust F [N] and I_sp [s] (if rocket), C_F (thrust coefficient), nozzle contour type (conical θ [°] / bell Rao parabola / MOC contour), bell nozzle key angles (θ_i [°] and θ_e [°]), operating regime (choked? shock-free at design back pressure?), NPR_design = p₀/p_e, overexpanded/underexpanded boundary pressures, thermal heat flux at throat q [W/m²] (if high-temperature), cooling requirement, and applicable standard (ASME PTC 19.5, NASA SP-8120, ISO 5167-3).