Mixed Flow Exhaust

Mixed-Flow Exhaust (propulsion/turbofan/mixed-flow-exhaust)

Use when the task is the two-stream turbofan design point with a MIXING (common) exhaust: the fan and core streams are traversed to the mixer entry plane, the constant-area mixer is closed by the energy and momentum balances over the two streams' total states, and the mixed stream expands through ONE common convergent nozzle. This leaf produces the mixed total temperature and pressure with the mixing loss ratio, the common-nozzle exit velocity and pressure term, the net thrust and TSFC of the mixed-flow configuration, and the same-traverse separate-exhaust baseline whose F and TSFC comparison gates the mixed-vs-separate exhaust decision at a low-BPR operating point. It is the exhaust-configuration partner of the separate-exhaust design point leaf in this pack: the sibling turbofan-design-point computes the same station chain but explicitly disclaims mixed-stream configurations, and this leaf returns the favor by using the separate-exhaust layout only as the comparison baseline, not as its output. Pure Python, stdlib only (math), constant-gamma isentropic component forms, single-layer ISA troposphere, deterministic.

Domain quick reference

• Ram traverse: v0 = machsqrt(GAMMA_CR_Ct0); tt0 = t0(1 + 0.5* (GAMMA_C-1)mach^2); pt0 = p0(tt0/t0)^(1/KAPPA_C).
• Diffuser: tt2 = tt0; pt2 = p0*(1 + eta_d*(tt0/t0 - 1))^(1/KAPPA_C).
• Cold compressor (fan, booster, HPC) at isentropic efficiency eta: tt_s = tt_inpr^KAPPA_C; tt_out = tt_in + (tt_s - tt_in)/eta; pt_out = pt_inpr.
• Burner: f = CP_C*(tt4 - tt3)/(eta_b*LHV); pt4 = pt3.
• HP spool per unit core air: CP_G*(tt4 - tt45) = CP_C*(tt3 - tt25).
• LP spool per unit core air: CP_G*(tt45 - tt5) = CP_C*((1 + bpr)* (tt13 - tt2) + (tt25 - tt13)).
• Turbine expansion at eta: tt_s = tt_in - (tt_in - tt_out)/eta; pt_out = pt_in*(tt_s/tt_in)^(1/KAPPA_G).
• Mass flows per unit core air: m_fan = bpr, m_core = 1 + f, m_total = bpr + 1 + f; the fuel mass rides the core through the mixer and common nozzle.
• Fan duct: tt16 = tt13; pt16 = pi_duct*pt13.
• Mixer entry at the equal static PRESSURE plane: p_s = pt16/(1 + 0.5* (GAMMA_C-1)M_fan_entry^2)^(GAMMA_C/(GAMMA_C-1)); the core entry Mach is the subsonic isentropic root at p_s; entry areas follow from continuity A = mRT_s/(p_sv).
• Mixed gas: cp_mix, r_mix mass-weighted; gamma_mix = cp_mix/(cp_mix - r_mix); energy gives tt_mix; the constant-area momentum balance m_totalv_exit + p_exitA = m_fv_f + m_gv_g + p_s*A with p_exit from continuity closes on the FIRST (subsonic) root by bisection.
• Mixing loss ratio: pt_mix/pt_tw below 1 when the entry velocities differ, with pt_tw the mass-weighted entry total pressure.
• Common convergent nozzle on the mixed stream: choked when npr >= ((gamma+1)/2)^(gamma/(gamma-1)); v_exit = cvsqrt(2cp*(tt_mix - Te)); a_by_mdot = rTe/(pev_exit).
• Net thrust: F = m_total*(v_exit_nozz - v0) + (pe - p0)A9; TSFC = f/F per unit core air. kg/(N s) converts to lbm/(lbf hr) via LBF_PER_HR = 3600G0 = 35303.94.
• Gas constants: GAMMA_C 1.4, CP_C 1005, R_C 287.142857; GAMMA_G 4/3, CP_G 1150, R_G 287.5. 14 CFR Part 33 (far-33) frames the engine certification context (reference-only); the relations are standard cycle-analysis methodology, summary-only.

Workflow

1. Fix the operating point: mach, altitude (ISA troposphere, 0-11000 m), OPR with opr = fprlpc_prhpc_pr, BPR, Tt4, the component efficiencies eta_d/eta_fan/eta_bst/eta_hpc/eta_b/eta_hpt/eta_lpt, the nozzle velocity coefficient cv, the duct pressure ratio pi_duct and the fan-stream mixer entry Mach M_fan_entry in (0, 1).
2. Get the atmosphere and freestream: isa_atmosphere then freestream_state for t0, p0, v0, tt0, pt0.
3. Traverse the station chain 0-2-13-2.5-3-4-4.5-5 with design_point_traverse (fan on both streams, booster and HPC on the core, burner, HPT, LPT) and confirm the closure identities opr == fprlpc_prhpc_pr, m_core == 1 + f and m_total == bpr + m_core.
4. Carry the bypass stream through the fan duct to the mixer entry: station 16 with tt16 = tt13 and pt16 = pi_duct*pt13.
5. Close the constant-area mixer with mixing_state on the two entry total states (fan stream cold, core stream hot, mass flows per unit core air) at M_fan_entry: read p_s, the core entry Mach, the entry velocities and areas, the mixed total temperature tt_mix, the mixed exit state on the first subsonic momentum-balance root, and the mixing loss ratio pt_mix/pt_tw.
6. Expand the mixed stream through the common convergent nozzle with common_nozzle using the mixed gamma/cp/r and cv: choked check, exit velocity and the exit-plane area per unit mass flow.
7. Compute the net thrust with net_thrust and the TSFC = f/F per unit core air for the mixed-flow configuration.
8. Build the separate-exhaust baseline from the same traverse: two per-stream convergent nozzles (fan nozzle on tt13/pt13 cold, core nozzle on tt5/pt5 hot), F_sep = F_fan + F_core, and report the F ratio and TSFC ratio that gate the mixed-vs-separate exhaust decision.
9. Confirm the deterministic checks with the contract test scripts/test_mixed_flow_exhaust.py (offline, stdlib unittest).

Worked example

Low-BPR military-style mixed-flow turbofan at Mach 0.85 and 10668 m: OPR 24 (fan 4.0, booster 1.6, HPC 3.75), BPR 0.6, Tt4 = 1750 K; eta_d 0.97, eta_fan 0.90, eta_bst 0.89, eta_hpc 0.88, eta_b 0.995, eta_hpt 0.90, eta_lpt 0.91, cv 0.985, pi_duct 0.98, M_fan_entry 0.3. All values are the real module outputs.

• Atmosphere and freestream: t0 = 218.808 K, p0 = 23835.9 Pa, v0 = 252.0946 m/s.
• Traverse per unit core air: tt2 250.4258 K, pt2 37724.0983 Pa; tt13 385.6541 K, pt13 150896.3931 Pa; tt25 447.9310 K, pt25 241434.2290 Pa; tt3 681.4885 K, pt3 905378.3586 Pa; tt45 1545.8911 K, pt45 519660.4508 Pa; tt5 1302.3820 K, pt5 242959.2421 Pa; f 0.025099; m_fan 0.600, m_core 1.025099, m_total 1.625099; hpc_pr 3.75.
• Fan duct: tt16 385.6541 K, pt16 147878.4652 Pa.
• Mixer: p_s 138927.3372 Pa at fan entry Mach 0.3000, core entry Mach 0.9486; fan stream T_s 378.8350 K, v 117.0737 m/s, A 0.004013 m2; core stream T_s 1132.5358 K, v 625.0169 m/s, A 0.003844 m2; duct area 0.007857 m2. Mixed state tt_mix 992.1523 K, cp_mix 1096.4648, r_mix 287.3681, gamma_mix 1.35517; exit v 356.7873 m/s, p_exit 155617.86 Pa, M_exit 0.591555 subsonic on the first branch, T_s_exit 934.1034 K; pt_mix 195867.88 Pa against pt_tw 207854.63 Pa, mixing_loss_ratio 0.942331 (below 1 because the entry velocities differ, v_f 117 vs v_g 625 m/s).
• Common nozzle: choked, Me 1.0, Te 842.5308 K, pe 104975.57 Pa, v_ideal 572.8084 m/s, v_exit 564.2162 m/s = cv*v_ideal, A/mdot 0.004088.
• Mixed-flow net thrust 1046.247 N per unit core air (106.7 lbf per kg/s core) and TSFC 2.40e-5 kg/(N s) = 0.847 lbm/(lbf hr).
• Separate-exhaust baseline from the same traverse: fan nozzle v19 354.044 m/s (choked), core nozzle v9 644.348 m/s (choked), F_sep 990.686 N, TSFC_sep 2.53e-5 kg/(N s) = 0.894 lbm/(lbf hr).
• Comparison: F ratio 1.056083 and TSFC ratio 0.946895: the mixed configuration gains about 5.6 percent thrust and 5.3 percent SFC at this low-BPR point, because mixing equalizes the very different jet velocities (354 vs 644 m/s).

Verification

• Confirm isa_atmosphere(0) returns (288.15, 101325.0) exactly and the freestream at sea-level static (mach 0) gives v0 = 0, tt0 = t0.
• Confirm the traverse closures: opr = fprlpc_prhpc_pr = 24, f from the burner energy balance closes tt4 = 1750 K, m_core = 1 + f and m_total = bpr + m_core.
• Confirm the mixer energy result tt_mix = 992.1523 K and the mass-weighted cp_mix/r_mix identities hold exactly; the momentum residual m_totalv + pA = m_fv_f + m_gv_g + p_s*A closes at M_exit = 0.591555 on the first subsonic branch.
• Confirm the mixing loss ratio 0.942331 stays below 1 (momentum mixing loss when the entry velocities differ).
• Confirm the common nozzle is choked at the worked point with Me = 1 and v_exit = cv*v_ideal (ratio 0.985 exactly).
• Confirm F and TSFC (1046.247 N, 2.40e-5 kg/(N s)) and the comparison ratios F 1.056083 and TSFC 0.946895 against the separate-exhaust baseline.
• Confirm every non-physical input raises ValueError: altitude outside [0, 11000] m, negative mach, efficiencies and cv outside (0, 1], pressure ratios below 1, tt4 <= tt3, opr < fpr, hpc_pr < 1, negative mass flows, m_fan_entry outside (0, 1), a core entry Mach reaching 1 (reduce m_fan_entry), a momentum balance with no subsonic root, and non-positive nozzle states or mass flow.
• Run the contract test offline under both interpreters: python3 scripts/test_mixed_flow_exhaust.py and ~/.pyenv/versions/3.13.12/bin/python3 scripts/test_mixed_flow_exhaust.py (56 tests, deterministic).

Related leaves

• propulsion/turbofan/turbofan-design-point: the separate-exhaust design point on the same station chain; the per-stream nozzle bookkeeping this leaf reuses only as the comparison baseline.
• propulsion/turbofan/turbofan-cycle: the one-stream cycle context and mass-flow/jet-velocity inputs.
• propulsion/turbofan/bypass-ratio-trade: the bypass-ratio thrust split at the velocity level for the trade sweep.
• propulsion/turbofan/turbofan-off-design: off-design matching and throttle behavior, explicitly outside this leaf's design-point scope.
• propulsion/gas-turbine-cycle/afterburner-cycle: core-only reheat, no fan stream or mixer.
• propulsion/gas-turbine-cycle/propelling-nozzle: single-stream choked convergent nozzle sizing; the common-nozzle step here uses the same choked/unchoked relations on the mixed stream.

Pitfalls

• Reporting the mixed total state without the momentum balance: the energy balance alone gives tt_mix; pt_mix must come from the constant-area momentum balance on the first subsonic root, and the mixing loss ratio pt_mix/pt_tw falls below 1 only when the entry velocities differ (0.9423 at the worked point).
• Feeding the entry total pressures into the loss bookkeeping as the mixed total pressure: pt_tw is the mass-weighted entry total pressure (207854.6 Pa at the worked point), and pt_mix (195867.9 Pa) sits below it by the momentum mixing loss, not above any single stream value.
• Assuming the mixer entry Mach on the core stream: the design input is the fan-stream entry Mach M_fan_entry; the core entry Mach is DERIVED from the equal static pressure plane (0.9486 at the worked point) and reaches 1 when the fan entry Mach is pushed too high, which raises ValueError (reduce M_fan_entry).
• Reading the separate-exhaust sibling's output as this leaf's model: turbofan-design-point owns the separate-exhaust per-stream nozzle output; here the per-stream nozzles appear only as the comparison baseline that produces the F and TSFC ratios.
• Treating the mixed nozzle as a second mixer stream: after the mixer there is ONE common convergent nozzle on the mixed stream with the mixed gamma/cp/r, sized by the same choked/unchoked relations as propelling-nozzle but never on two streams at once.
• Using variable specific heats or turbine cooling: all component forms are constant-gamma isentropic on the module gas constants only; real cycle effects and the nozzle throat AREA sizing for arbitrary entry states belong to other leaves.

Behavior contract (gate 3)

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

python3 scripts/test_mixed_flow_exhaust.py

The test covers the module constants, the atmosphere and freestream step at sea level and at the worked altitude, the diffuser and cold compression steps, the burner fuel-to-air ratio and its energy closure, the HP and LP spool work balances, the turbine expansion pressures, the full station traverse with the opr-product and mass-flow closure identities, the fan duct carry, the mixer entry plane with the derived core entry Mach and the continuity areas, the mixed gas properties, the energy balance, the momentum-balance subsonic root, the total pressure loss ratio, the common nozzle choked and unchoked regimes with the velocity coefficient round trip, the net thrust and TSFC at the worked point, the separate-exhaust baseline, the mixed-vs-separate F and TSFC ratio comparison, the full ValueError rejection list, and the determinism checks (no RNG, repeat runs identical).

Compliance

• Standards referenced, not reproduced: 14 CFR Part 33 (far-33) is named as the engine certification frame; the cycle relations above are standard engineering methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.