Brayton Optimum Pressure Ratio

Brayton Optimum Pressure Ratio (propulsion/gas-turbine-cycle/brayton-optimum-pressure-ratio)

Use when you must select the pressure ratio of a simple gas-turbine-cycle that maximizes the net specific work at the cycle temperature limits, rather than evaluate the cycle at a ratio you already fixed. This leaf implements the max-work pressure ratio selection in pure Python, stdlib only, closed form: for the ideal cycle x_opt = sqrt(tau) with tau = T3/T1 and x = r**((gamma-1)/gamma), and with the compressor and turbine component efficiencies x_opt = sqrt(taueta_ceta_t). It also returns the zero-work limiting ratio at which the net specific work returns to zero (exactly the max-work ratio squared, the algebraic cross-check of the analysis) and issues the design verdict that the max-efficiency pressure ratio diverges from the max-work ratio. It pairs with propulsion/gas-turbine-cycle/gas-turbine-cycle, which evaluates the cycle at a supplied pressure ratio, and with the other gas-turbine-cycle pack leaves for the regenerative, intercooled and real-cycle variants.

Domain quick reference

• Cycle frame: T1 is the compressor inlet temperature, T3 the turbine inlet temperature, r the pressure ratio and x = r**KAPPA with KAPPA = (gamma - 1)/gamma = 2/7 the pressure exponent; gamma = 1.4, air standard, cp = 1005.0 J/(kg K) only for quoting work in J/kg.
• Nondimensional net specific work at tau = T3/T1: ideal w/(cpT1) = tau(1 - 1/x) - (x - 1); real w/(cpT1) = taueta_t*(1 - 1/x) - (x - 1)/eta_c, from w = cp*(w_t - w_c) with w_t = cpeta_tT3*(1 - 1/x) and w_c = cpT1(x - 1)/eta_c.
• Max-work condition d(w)/dx = 0: ideal x_opt = sqrt(tau), so r_opt = tau**(gamma/(2*(gamma-1))); real x_opt = sqrt(taueta_ceta_t), so r_opt = x_opt**(1/KAPPA). Component losses pull x_opt below sqrt(tau).
• Factored zero-work form: w = cpT1(x - 1)(A - x)/(xeta_c) with A = taueta_ceta_t, whose roots x = 1 (no compression) and x = A bound the power-producing band (1, r_zero); hence x_zero = taueta_ceta_t = x_opt2 and, because r = x(1/KAPPA) is a power, r_zero = r_opt**2 exactly.
• Efficiency arm of the verdict: ideal eta = 1 - 1/x rises monotonically with r (no finite maximizer); real eta_th = w/q_in with q_in = cp*(T3 - T2), T2 = T1*(1 + (x - 1)/eta_c), peaks at the smaller root of the quadratic b*(b - c + a)x**2 - 2abx + ac = 0 (a = eta_tT3, b = T1/eta_c, c = T3 - T1 + b), a ratio strictly above r_opt, and returns to zero at r_zero.
• Units SI throughout: K, J/kg, dimensionless pressure ratios.
• FAR 33 frames the powerplant certification context; the relations above are standard engineering methodology, summary-only.

Workflow

1. Fix the cycle temperature limits T1, T3 and the component efficiencies eta_c, eta_t; every function rejects non-physical inputs with ValueError (T1 <= 0, T3 <= T1, eta outside (0, 1], ratio at or below 1).
2. Select the ideal max-work pressure ratio with ideal_optimum_pressure_ratio(t1, t3), the d(w)/dx = 0 closed form x_opt = sqrt(tau) at unit component efficiencies.
3. Select the lossy max-work pressure ratio with optimum_pressure_ratio(t1, t3, eta_c, eta_t), the sqrt(taueta_c eta_t) closed form that degenerates exactly to the ideal selection at eta_c = eta_t = 1.0.
4. Cross-check the zero-work limiting ratio with zero_work_pressure_ratio(t1, t3, eta_c, eta_t): the factored form makes r_zero = r_opt**2 exact, and the ratio band (1, r_zero) spans r_zero/r_opt = r_opt, one decade above the optimum.
5. Evaluate the net specific work at the optimum with net_specific_work(t1, t3, r_opt, eta_c, eta_t) and confirm the peak against the log-symmetric neighbours of the ratio (factor sqrt(2) for the lossy cycle, factor 4 for the ideal cycle).
6. Verify the maximum by the derivative sign change: the closed-form d(w)/dx = (cpT1/eta_c)(A/x**2 - 1) is positive just below x_opt, zero at x_opt and negative just above it.
7. Issue the efficiency-versus-work verdict with design_verdict(t1, t3, eta_c, eta_t): the ideal efficiency rises monotonically with the ratio while the lossy-cycle efficiency peaks at the quadratic root r_eta_max above r_opt and returns to zero at the zero-work ratio, so only the specific-work criterion selects r_opt.
8. Confirm with the deterministic contract test: python3 scripts/test_brayton_optimum_pressure_ratio.py.

Worked example

T1 = 288.15 K, T3 = 1500 K (tau = 5.205622), eta_c = eta_t = 0.87 for the lossy cycle; all values are the real outputs of the module.

• Ideal cycle (eta_c = eta_t = 1): r_opt_ideal = 17.940 with w = 475639.9 J/kg (475.6 kJ/kg), above the log-symmetric neighbours w(r_opt_ideal/4) = w(4*r_opt_ideal) = 370621.4 J/kg; the zero-work ratio r_zero_ideal = 321.851 equals r_opt_ideal**2 (relative difference 7.06e-16) with w vanishing at scaled residual below 1e-9.
• Ideal efficiency arm: 0.002839 at r = 1.01, 0.561708 at the ideal max-work ratio, 0.640453 at 2*r_opt_ideal and 0.980693 at r = 1e6, still rising: no finite efficiency maximizer in the lossless cycle.
• Lossy cycle: x_opt = 1.984977, r_opt = 11.019 (down from 17.940), with w(r_opt) = 322937.1 J/kg above both neighbours w(r_opt/sqrt(2)) = w(r_opt*sqrt(2)) = 316453.3 J/kg.
• Zero-work cross-check: x_zero = 3.940135 = x_opt2 exactly and r_zero = 121.420 = r_opt2 (relative difference 2.34e-16); w at r_zero is 1.143e-10 J/kg (scaled residual 3.945e-16) and the band (1, 121.420) spans r_zero/r_opt = 11.019059 = r_opt.
• Derivative check: d(w)/dx = +66.583 at x_opt*(1 - 1e-4), 0 at x_opt and -66.563 at x_opt*(1 + 1e-4): the sign change confirms the maximum.
• Verdict: the real thermal efficiency at the max-work ratio is 0.362832; the closed-form efficiency optimum sits at x_eta_max = 2.564089, r_eta_max = 26.994 with eta_th = 0.400699, strictly above r_opt = 11.019, and the efficiency returns to 0 at r_zero = 121.420 where the net work is zero. The ideal-cycle efficiency at the lossy optimum (r = 11.019) is 0.496216, below the 0.561708 at the ideal optimum.
• Read-off: the engine maximizes net specific work near PR 11; PR 27 maximizes thermal efficiency (about 3.8 points higher) but at lower work per unit airflow; past PR 121 the compressor work exceeds the turbine output.

Verification

• Confirm ideal_optimum_pressure_ratio(288.15, 1500.0) returns 17.940 and optimum_pressure_ratio(288.15, 1500.0, 1.0, 1.0) equals it exactly (both tau**(1/(2*KAPPA))).
• Confirm optimum_pressure_ratio(288.15, 1500.0, 0.87, 0.87) returns 11.019, zero_work_pressure_ratio returns 121.420, and r_zero equals r_opt**2 within 1e-9 relative.
• Confirm net_specific_work at the module optimum ratio returns 322937.1 J/kg (475639.9 J/kg ideal), the log-symmetric neighbours are equal and below the peak, and w vanishes at both roots of the factored form to within 1e-9 scaled residual.
• Confirm the derivative sign change of d(w)/dx across x_opt and the verdict numbers (0.362832 at r_opt, r_eta_max 26.994, 0.400699 at the efficiency optimum, 0 at r_zero).
• Confirm every non-physical input raises ValueError: t1 at 0 and -1, t3 <= t1, eta_c and eta_t at 0, 1.5 and 1.01, and ratios at 1.0 and 0.5 on the work and efficiency evaluators.
• Run the contract test offline: python3 scripts/test_brayton_optimum_pressure_ratio.py (35 tests, deterministic, all numeric asserts order-safe tolerances).

Related leaves

• propulsion/gas-turbine-cycle/gas-turbine-cycle: the given-ratio cycle evaluation this leaf complements (efficiency, exit temperatures and specific work at a supplied pressure ratio).
• propulsion/gas-turbine-cycle/regenerative-cycle: the exhaust heat recovery variant whose recovery payoff boundary is a different optimum; it owns the regenerative trigger wording.
• propulsion/gas-turbine-cycle/intercooled-cycle: the inter-stage split of a fixed total ratio into compressor stages.
• propulsion/gas-turbine-cycle/real-cycle-effects: loss accounting beyond the constant component efficiencies used here.
• propulsion/gas-turbine-cycle/turbojet-cycle: the thrust-oriented variant; thrust and nozzle quantities are out of scope here.

Pitfalls

• Confusing the two optima of this pack: the max-work selection here (x_opt = sqrt(taueta_ceta_t), r_opt finite) is not the regenerative-cycle recovery optimum, and the two closed forms coincide only in the lossless case; never reuse the regenerative trigger wording for the recovery payoff boundary.
• Using the ideal closed form for a lossy engine: with eta_c and eta_t below 1 the max-work ratio falls (17.940 to 11.019 at 0.87/0.87), so sizing on the ideal value overstates the ratio and the work.
• Taking the wrong quadratic root: the efficiency stationarity quadratic has a larger unphysical root beyond the q_in = 0 pole at x = c/b; always take the smaller root x_eta_max.
• Equating max efficiency with max work: the efficiency optimum r_eta_max (26.994) sits above the max-work ratio (11.019) and the efficiency there is about 3.8 points higher at lower work per unit airflow; only the specific-work criterion selects r_opt.
• Calling design_verdict at unit efficiencies: at eta_c = eta_t = 1.0 the zero-work ratio coincides with the q_in = 0 pole, so eta_real_at_r_zero is not defined there; the verdict is for the lossy cycle.
• Extending beyond the model: air-standard gamma = 1.4 only, constant specific heats, and component losses limited to the constant efficiencies; station temperatures with pressure losses, thrust and augmentation quantities belong to the sibling leaves.

Behavior contract (gate 3)

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

python3 scripts/test_brayton_optimum_pressure_ratio.py

The test covers the worked-example anchors (ideal and lossy max-work pressure ratios 17.940 and 11.019, zero-work limiting ratio 121.420, net specific work 475639.9 and 322937.1 J/kg), the degeneracy of the lossy selection to the ideal closed form at unit efficiencies, the r_zero = r_opt**2 identity of the factored form with the scaled zero-work residuals, the log-symmetric neighbour peak property, the derivative sign change of d(w)/dx across x_opt, the ideal-efficiency monotone arm, the lossy efficiency quadratic-root optimum with the efficiency-versus-work verdict numbers, ValueError rejection of every non-physical input, and the determinism and stdlib-only checks.

Compliance

• Standards referenced, not reproduced: FAR 33 (14 CFR Part 33) frames the aircraft powerplant certification context; the max-work pressure ratio relations above are standard gas-turbine-cycle engineering methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.