Propelling Nozzle

Propelling Nozzle (propulsion/gas-turbine-cycle/propelling-nozzle)

Use when you must size the convergent propelling nozzle of an air-breathing gas turbine at the conceptual level: deciding the choked or unchoked regime from the nozzle pressure ratio against the critical ratio, sizing the throat area from the design mass flow and total conditions under the choked relation, and returning either the choked exit state, exit velocity and gross thrust with the pressure term, or for an unchoked off-design point the exit Mach number and the actual mass flow the same throat passes. This leaf implements the standard convergent nozzle model in pure Python, stdlib only. It pairs with propulsion/gas-turbine-cycle/gas-turbine-cycle for the cycle context, propulsion/turbofan/turbofan-cycle which consumes the jet velocity as an input, and propulsion/rocket/nozzle-design which handles the chamber-anchored choked rocket nozzle. Propulsion afterburner and regenerative cycles set the nozzle entry conditions this leaf turns into thrust.

Domain quick reference

• Regime: the nozzle pressure ratio NPR = P0/Pa compares the total pressure upstream of the nozzle with ambient. The critical ratio is ((gamma+1)/2)^(gamma/(gamma-1)) = 1.851 at gamma = 1.33; a convergent nozzle chokes (sonic throat, Me = 1) when NPR >= critical, and runs subsonic unchoked below it.
• Choked throat sizing: m_dot = P0At/sqrt(T0) * sqrt(gamma/R) * (2/(gamma+1))^((gamma+1)/(2(gamma-1))). Rearranged it gives the throat area that passes a design mass flow at total conditions (P0, T0).
• Choked exit state at the throat: Te = T02/(gamma+1), Pe = P0(2/(gamma+1))^(gamma/(gamma-1)), Ve = sqrt(gammaRTe), with mach = 1.0.
• Gross thrust: Fg = m_dot*Ve + (Pe - Pa)*At. The second term is the nozzle pressure thrust; it is active whenever the exit static pressure Pe exceeds ambient Pa (an imperfectly expanded nozzle) and vanishes when the nozzle is fully expanded (Pe = Pa).
• Unchoked exit: NPR = (1 + (gamma-1)/2Me^2)^(gamma/(gamma-1)) solved for Me, then Te = T0/(1 + (gamma-1)/2Me^2) and Ve = Mesqrt(gammaR*Te).
• Unchoked mass flow through a fixed throat: m_dot = P0At/sqrt(T0) * sqrt(gamma/R) * Me * (1 + (gamma-1)/2Me^2)^(-(gamma+1)/(2*(gamma-1))).
• Constants: GAMMA = 1.33 (air-breathing nozzle products convention, matching the afterburner-cycle anchor), R_GAS = 287.0 J/(kg K), P_AMB_DEFAULT = 101325.0 Pa. Units are SI throughout: Pa, K, kg/s, m2, m/s, N.
• FAR-33 frames the airworthiness context; the relations above are standard engineering methodology, summary-only per standards-map.yaml.

Workflow

1. Fix the operating point: total pressure p0_pa, total temperature t0_k, ambient pressure p_amb_pa and the mass flow mdot_kg_s (nozzle_regime decides the regime from the nozzle pressure ratio against the critical ratio).
2. Confirm the design point is choked, then size the throat with throat_area from the design mass flow, P0 and T0; the choked relation is the sizing law for a convergent nozzle.
3. Get the choked exit state with choked_exit_state: Te, Ve, Pe, mach = 1.0 at the throat.
4. Form the gross thrust with gross_thrust from mdot, Ve, Pe, Pa and the throat area; the pressure term (Pe - Pa)At rides on top of the momentum thrust mdotVe.
5. Run the full design pass with nozzle_sizing, which returns the regime, throat area, choked exit state, gross thrust and the expansion verdict (FULLY_EXPANDED or PRESSURE_TERM_ACTIVE).
6. For an off-design point on the fixed throat, run off_design_nozzle: when the lower P0 keeps NPR below the critical ratio it returns the subsonic exit Mach number, exit velocity and the actual mass flow the throat passes; the unchoked_exit_state and unchoked_mass_flow helpers expose the same quantities standalone.
7. Check non-physical inputs: every function raises ValueError for non-positive pressures, temperatures, areas or mass flows, for a nozzle pressure ratio at or below one, and when the unchoked relations are called at a choked regime.
8. Confirm the deterministic checks with the contract test scripts/test_propelling_nozzle.py.

Worked example

Reference design point: P0 = 300 kPa, T0 = 900 K, mdot = 70 kg/s, Pa = 101.325 kPa; off-design at P0 = 140 kPa on the same throat.

• Regime: NPR = 2.961 >= critical 1.851, so the design point is CHOKED (nozzle_regime).
• Throat area: At = 70sqrt(900)/(300000sqrt(1.33/287)*0.5833) = 0.176305 m2 (throat_area).
• Choked exit: Te = 9002/2.33 = 772.5 K, Ve = 543.0 m/s, Pe = 300000(2/2.33)^(1.33/0.33) = 162109 Pa (162.1 kPa), mach 1.0 (choked_exit_state).
• Gross thrust: Fg = 70*543.03 + (162109 - 101325)*0.176305 = 48728.7 N, about 48.7 kN; verdict PRESSURE_TERM_ACTIVE because Pe > Pa (nozzle_sizing).
• Off-design unchoked: NPR = 1.382 < 1.851; Me = 0.7115, Ve = 400.6 m/s, and the same 0.176305 m2 throat passes 30.02 kg/s (off_design_nozzle).
• Continuity check: the unchoked flow evaluated just below the critical ratio equals the choked flow at the same total pressure within 1e-3 relative, so the two relations join smoothly at the choke boundary.

Verification

• Confirm nozzle_sizing(70, 300000, 900, 101325) returns regime choked, throat area 0.176305 m2, Te 772.5 K, Ve 543.0 m/s, Pe 162109 Pa, gross thrust 48728.7 N and verdict PRESSURE_TERM_ACTIVE.
• Confirm off_design_nozzle(0.176305, 140000, 900, 101325) returns Me 0.7115, Ve 400.6 m/s and an actual mass flow of 30.02 kg/s.
• Confirm the choked exit identities: Te = T02/(gamma+1) and Ve = sqrt(gammaRTe) hold exactly; with Pe = Pa the gross thrust reduces to mdotVe.
• Confirm the unchoked flow at the critical ratio matches the choked flow within 1e-3 relative (continuity at the choke boundary).
• Confirm every non-positive pressure, temperature, area or mass flow, a nozzle pressure ratio at or below one, and every call of the unchoked relations at a choked regime raises ValueError.
• Run the contract test offline: python3 scripts/test_propelling_nozzle.py (34 tests, deterministic).

Related leaves

• propulsion/gas-turbine-cycle/gas-turbine-cycle: the cycle analysis that sets nozzle entry total conditions.
• propulsion/gas-turbine-cycle/afterburner-cycle: heat addition upstream of the nozzle and the fully expanded ideal jet velocity.
• propulsion/gas-turbine-cycle/regenerative-cycle and real-cycle-effects: alternative cycle layouts and loss effects that change the nozzle entry state.
• propulsion/gas-turbine-cycle/combustor-design: the combustion chamber that delivers the entry gas.
• propulsion/turbofan/turbofan-cycle: consumes the jet velocity produced here as an input to the net thrust loop.
• propulsion/engine-airframe/engine-airframe-integration: installation drag and aircraft-level nozzle trade context.
• propulsion/rocket/nozzle-design: the rocket nozzle counterpart, chamber-anchored, always choked, with its own gas property model.

Pitfalls

• Sizing the throat with the unchoked relation: the throat area law above is the choked sizing relation and only applies when the design point is choked; feeding an unchoked design point to nozzle_sizing raises ValueError rather than returning a misleading area.
• Dropping the pressure term: at the worked example the momentum term is 70*543.03 = 38.0 kN while the pressure term (Pe - Pa)At adds about 10.7 kN, so gross thrust is 48.7 kN; quoting mdotVe alone understates the nozzle by more than 20 percent.
• Applying choked exit relations below the critical ratio: an unchoked nozzle has a subsonic exit with Pe = Pa and its flow follows the isentropic Mach relation; forcing Me = 1 inflates both velocity and thrust.
• Borrowing rocket nozzle logic: rocket nozzle-design is anchored to chamber conditions with combustion gas properties and always chokes; the air-breathing propelling nozzle sees total conditions from the engine cycle, can run unchoked over much of the flight envelope, and needs the pressure term whenever it is not fully expanded.
• Confusing total and static state: the nozzle works on total P0, T0 delivered by the cycle, but the exit static pressure Pe (not P0) is what appears in the pressure thrust term against ambient.
• Mismatching gamma: this leaf uses the 1.33 nozzle products convention that matches the afterburner-cycle anchor; using 1.4 cold air values shifts the critical ratio to about 1.893 and changes both the choke decision and the sized area.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_propelling_nozzle.py

The test covers the choked design point sizing (throat area 0.176305 m2, Te 772.5 K, Ve 543.0 m/s, Pe 162109 Pa, gross thrust 48728.7 N, verdict PRESSURE_TERM_ACTIVE), the unchoked off-design point (Me 0.7115, Ve 400.6 m/s, actual flow 30.02 kg/s), regime decisions at NPR 2.961 and 1.382 against the critical ratio 1.851, the choked exit identities Te = T02/(gamma+1) and Ve = sqrt(gammaRTe), the fully-expanded identity Fg = mdotVe at Pe = Pa, continuity of the unchoked and choked flow relations at the critical ratio, dict key sets, determinism, and ValueError rejection of non-positive pressures, temperatures, areas and mass flows, nozzle pressure ratios at or below one, and unchoked relations called at a choked regime.

Compliance

• Standards referenced, not reproduced: FAR-33 is named as the airworthiness frame for engine installation; the convergent nozzle relations above are standard engineering methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.