# Dual Cycle > Use when you must compute the operating point of a reciprocating high-speed compression-ignition aircraft powerplant: the air-standard dual-cycle thermal efficiency from the compression ratio, the specific-heat ratio, the pressure ratio of the constant-volume heat-addition phase and the cutoff ratio of the constant-pressure phase, the isentropic compression temperature ratio and state temperatures, the four-stroke indicated and brake power from the indicated mean effective pressure, displacement, crankshaft speed and mechanical efficiency, and the brake specific fuel consumption on Jet-A. Produces the single-point summary dict with the dual-cycle efficiency, heat additions, mean effective pressure, BSFC in both units, thermal efficiencies, volumetric fuel flow and the reference-only compression-ignition band verdict, reported and never enforced. Trigger: dual-cycle, air-standard-dual-cycle, sabathe-cycle, limited-pressure-cycle, mixed-heat-addition, pressure-ratio-of-heat-addition. - Skill: `ashfordeou/dual-cycle` (Agent Skill, multi-file: 3 files) - Install (CLI): `npx skillmds@latest add ashfordeou/dual-cycle` - Raw SKILL.md: https://api.skillmd.com/api/skills/ashfordeou/dual-cycle/raw - Safety review: pending - Works with: Claude Code, Claude.ai, OpenAI Codex - Category: Coding & Dev Tools - License: Apache-2.0 - Author: ashfordeOU (https://skillmd.com/u/ashfordeou) - Updated: 2026-09-17 - Page: https://skillmd.com/skills/ashfordeou/dual-cycle --- # Dual Cycle (propulsion/reciprocating/dual-cycle) Use when the task is a reciprocating high-speed compression-ignition aircraft powerplant operating-point analysis, an aircraft diesel of the direct-injection class: the air-standard dual (Sabathe / limited-pressure) cycle thermal efficiency under the mixed constant-volume-then-constant- pressure heat-addition model, the five-state temperature bookkeeping and both heat additions, the ideal-cycle mean effective pressure, the four-stroke indicated and brake power from the mean effective pressure bookkeeping, and the brake specific fuel consumption and thermal efficiencies on Jet-A. This leaf implements the standard textbook air-standard dual cycle and PLAN indicated-power method in pure Python, stdlib only. It pairs with propulsion/reciprocating/piston-engine-cycle for the spark-ignition Otto cycle prime mover (the rho to 1 limit of this model), with propulsion/reciprocating/diesel-cycle for the pure constant-pressure Diesel cycle prime mover (the alpha to 1 limit of this model), with propulsion/turboprop/free-turbine and propulsion/turboprop/turboprop-cycle for the other aircraft prime-mover classes, and with propulsion/engine-airframe for installed-behavior bookkeeping once the powerplant side is fixed. ## Domain quick reference - Air-standard dual thermal efficiency (mixed heat addition): eta = 1 - (1/r^(gamma-1)) * ((alpha*rho^gamma - 1)/((alpha - 1) + gamma*alpha* (rho - 1))), with r the compression ratio (r > 1), gamma the specific-heat ratio (gamma > 1, air-standard default 1.4), alpha the pressure ratio of the constant-volume heat-addition phase (alpha >= 1, alpha = P3/P2) and rho the cutoff ratio of the constant-pressure phase (rho = V4/V3 = T4/T3, 1 < rho <= r; heat addition must end before bottom dead center). This is the ideal efficiency ceiling of the mixed heat-addition cycle, not a real-cycle prediction. - Limit identities: alpha to 1 recovers the air-standard Diesel closed form owned by propulsion/reciprocating/diesel-cycle exactly, and rho to 1 recovers the air-standard Otto closed form owned by propulsion/reciprocating/piston-engine-cycle exactly, so the dual ceiling interpolates between the two landed siblings' ceilings at the same compression ratio. - Isentropic compression temperature ratio: T2/T1 = r^(gamma-1); the state bookkeeping continues T3 = T2 * alpha (constant-volume heat addition 2 to 3), T4 = T3 * rho (constant-pressure heat addition 3 to 4) and T5 = T1 * alpha * rho^gamma (isentropic expansion 4 to 5 to V5 = V1, closed form). The temperature-form identity eta = 1 - q_out/q_in holds exactly. - Mixed heat addition per kg of air: q23 = (cp/gamma) * T2 * (alpha - 1) across the constant-volume phase, q34 = cp * T2 * alpha * (rho - 1) across the constant-pressure phase, q_in = q23 + q34; the alpha and rho round trips 1 + q23/(cv * T2) and 1 + q34/(cp * T3) are the exact inverses. - Constant-volume heat rejection 5 to 1: q_out = (cp/gamma) * (T5 - T1). - Ideal-cycle mean effective pressure: MEP = eta * q_in/(v1 - v2), with v1 = R * T1/p1, returned in Pa; MEP is exactly linear in the state-1 pressure p1. - Four-stroke indicated power (SI): P_i = IMEP * V_d * (rpm/60)/2, with IMEP in Pa and V_d in m3. One power stroke occurs per two crankshaft revolutions, so the revolution rate halves in the bookkeeping; the bar/L convention gives the identical value. - Brake power: P_b = P_i * eta_m, with eta_m the mechanical efficiency in (0, 1]; eta_m = 1.0 returns the indicated power exactly. - Brake specific fuel consumption (family shaft-power convention): BSFC = m_dot_fuel * 3600 * 1000 / P_b in kg/(kW h), also reported in lb/(hp h) through the exact unit bridge KG_PER_KWH_PER_LB_PER_HP_HR. - Thermal efficiencies on the Jet-A lower heating value: eta_i = P_i / (m_dot_fuel * LHV); eta_b = P_b / (m_dot_fuel * LHV) = eta_m * eta_i. The air-standard eta_dual is the ideal ceiling and sits above eta_b for any real point. - Volumetric fuel flow: V_dot = m_dot_fuel / rho_fuel, reported in m3/s, L/h and US gal/h. - Reference-only compression-ignition band verdict: brake thermal efficiency against 0.3262 to 0.3806 and BSFC against 0.35 to 0.42 lb/(hp h), the same published band class the diesel sibling's worked example reports for a compression-ignition aircraft engine of the Centurion/AE300 class. The verdict reports the point's position and never enforces it. - Units are SI throughout except the reporting conversions (hp, lb/(hp h), L/h, US gal/h) that the module carries as named outputs. ## Workflow 1. Fix the high-speed compression-ignition operating point: compression ratio, pressure ratio of the constant-volume phase, cutoff ratio of the constant-pressure phase and gamma, then compute the air-standard dual-cycle thermal efficiency with dual_efficiency and the isentropic compression temperature ratio with isentropic_temperature_ratio. 2. Compute the five-state temperature bookkeeping with cycle_state_temperatures (or the individual compression_temperature, cv_phase_temperature, cp_phase_temperature and expansion_temperature steps), the constant-volume heat addition with cv_heat_addition_j_per_kg, the constant-pressure heat addition with cp_heat_addition_j_per_kg, the total mixed heat addition with heat_addition_j_per_kg, and the constant-volume heat rejection with heat_rejection_j_per_kg. 3. Compute the ideal-cycle mean effective pressure from the state-1 pressure and temperature with mean_effective_pressure. 4. Compute the four-stroke indicated power from the indicated mean effective pressure, the displacement and the crankshaft speed with indicated_power. 5. Compute the brake power at the mechanical efficiency with brake_power. 6. Compute the brake specific fuel consumption with brake_specific_fuel_consumption, convert it to lb/(hp h) with bsfc_lb_per_hp_hr, and round-trip back to the fuel flow with fuel_flow_from_bsfc when checking a reported BSFC. 7. Compute the indicated and brake thermal efficiencies on the Jet-A lower heating value with indicated_thermal_efficiency and brake_thermal_efficiency. 8. Compute the volumetric fuel flow in m3/s, L/h and US gal/h with volumetric_fuel_flow. 9. Get the reference-only compression-ignition band verdict with ci_band_verdict, reporting the point's position without enforcing it, or call dual_cycle once for the full single-point summary dict that chains steps 1 through 9. 10. Confirm the deterministic checks with the contract test scripts/test_dual_cycle.py. ## Worked example High-speed direct-injection compression-ignition aircraft diesel at a cruise rating: compression ratio r = 16, pressure ratio of the constant-volume phase alpha = 1.35, cutoff ratio of the constant-pressure phase rho = 2.0, gamma = 1.4 (air standard), inlet temperature T1 = 288.15 K, state-1 pressure p1 = 1.5e5 Pa, IMEP = 1.8e6 Pa (18 bar), displacement V_d = 2.0e-3 m3 (2.0 L), crankshaft speed 2300 rpm, mechanical efficiency eta_m = 0.86, fuel flow m_dot = 3.9e-3 kg/s Jet-A at the reference density 800 kg/m3 and LHV 43.2 MJ/kg. - Ideal cycle: eta_dual = 1 - (1/16^0.4) * ((1.35 * 2.0^1.4 - 1)/((1.35 - 1) + 1.4 * 1.35 * (2.0 - 1))) = 0.622604338055, sitting between the same-r bounds 0.613804581712 (Diesel closed form at cutoff 2.0, the pure constant-pressure limit alpha = 1) and 0.670123022307 (the constant- volume ceiling 1 - 1/16^0.4), the interpolation identity; the isentropic compression temperature ratio is T2/T1 = 16^0.4 = 3.03143313302. - State bookkeeping from T1 = 288.15 K: T2 = 873.50745728 K, T3 = T2 * 1.35 = 1179.23506733 K (end of the constant-volume phase), T4 = T3 * 2.0 = 2358.47013466 K (end of the constant-pressure phase), T5 = T1 * 1.35 * 2.0^1.4 = 1026.58375212 K (end of the isentropic expansion back to V5 = V1), and the mixed heat addition q23 = 219468.748642 J/kg across the constant-volume phase plus q34 = 1185131.24266 J/kg across the constant-pressure phase gives q_in = 1404599.99131 J/kg (about 1.40 MJ per kg of air); the temperature-form identity 1 - q_out/q_in with q_out = 530089.943487 J/kg reproduces eta_dual to 1e-12 relative, and the alpha and rho round trips return 1.35 and 2.0 to 1e-15 relative. - Mean effective pressure: MEP = eta_dual * q_in/(v1 - v2) = 1691937.3034 Pa (about 16.9 bar) at the 1.5 bar state-1 pressure, the order of magnitude of the 18 bar IMEP documented input; doubling the state-1 pressure doubles the MEP to 3383874.60681 Pa, verified to 1e-12 relative (MEP is exactly linear in p1). - Pressure-ratio sensitivity at r = 17, rho = 2.2 (the receipt magnitudes): alpha 1.2 gives 0.619491257553 and alpha 2.0 gives 0.628441566104, so a longer constant-volume phase raises efficiency toward the constant- volume ceiling at fixed r and rho, the dual-cycle signature that separates this leaf from the pure constant-pressure sibling. - Cutoff-ratio sensitivity at the worked point: rho 1.8 gives 0.632550215211 and rho 2.2 gives 0.613014625635, so a longer constant-pressure phase costs efficiency at fixed r and alpha, the compression-ignition signature. - Indicated power: P_i = 1.8e6 * 2.0e-3 * (2300/60)/2 = 69000.0 W = 92.5305241821 hp. - Brake power: P_b = 69000.0 * 0.86 = 59340.0 W = 79.5762507966 hp. - BSFC: m_dot * 3600 * 1000 / P_b = 0.236602628918 kg/(kW h) = 0.388971600206 lb/(hp h), inside the published 0.35 to 0.42 lb/(hp h) compression-ignition band. - Thermal efficiencies on LHV: eta_i = 69000.0/(3.9e-3 * 43.2e6) = 0.409544159544; eta_b = 59340.0/(3.9e-3 * 43.2e6) = 0.352207977208 = eta_i * eta_m exactly, inside the published CI brake-thermal-efficiency window 0.3261878583333333 to 0.3805525013888889 and below the ideal eta_dual (0.35221 < 0.62260), as it must be. - Volumetric fuel flow: V_dot = 3.9e-3/800 = 4.875e-06 m3/s = 17.55 L/h = 4.63621951889 US gal/h. - Band verdict: eta_b_position "inside" and bsfc_position "inside" with enforced False. ## Verification - Confirm dual_efficiency(16.0, 1.35, 2.0, 1.4) returns 0.622604338055, and that efficiency rises as the pressure ratio grows at fixed r and rho (r = 17, rho = 2.2: 0.619491257553 at alpha 1.2, 0.628441566104 at alpha 2.0) while it falls as the cutoff ratio grows at fixed r and alpha (r = 16, alpha = 1.35: 0.632550215211 at rho 1.8, 0.613014625635 at rho 2.2). - Confirm the alpha to 1 limit reproduces the Diesel closed form 0.613684212124 at r = 17, rho = 2.2, and the rho to 1 limit reproduces the constant-volume Otto ceiling 0.678026275475 at the same r, so the dual ceiling sits strictly between the two landed sibling ceilings at every interior alpha and rho. - Confirm isentropic_temperature_ratio(16.0, 1.4) returns 3.03143313302 and compression_temperature(288.15, 16.0, 1.4) returns 873.50745728 K. - Confirm cv_heat_addition_j_per_kg and cp_heat_addition_j_per_kg round trip the pressure ratio and cutoff ratio to better than 1e-9 relative, and that the temperature-form identity 1 - q_out/q_in reproduces dual_efficiency to 1e-10 relative. - Confirm mean_effective_pressure(288.15, 1.5e5, 16.0, 1.35, 2.0) returns 1691937.3034 Pa and doubling the state-1 pressure doubles the MEP. - Confirm indicated_power(1.8e6, 2.0e-3, 2300.0) returns 69000.0 W and brake_power(69000.0, 0.86) returns 59340.0 W. - Confirm fuel_flow_from_bsfc round-trips the fuel flow used to build a BSFC value to better than 1e-15 kg/s. - Confirm every non-positive or non-finite compression ratio, pressure ratio, cutoff ratio, gamma, temperature, pressure, IMEP, displacement, rpm, power, fuel flow, density or LHV raises ValueError, that every mechanical efficiency at or below 0 or above 1 raises ValueError, and that a cutoff ratio above the compression ratio raises ValueError. - Confirm ci_band_verdict never raises for an out-of-band point; it only rejects non-finite or non-positive inputs. - Run the contract test offline: python3 scripts/test_dual_cycle.py (deterministic, no network). ## Related leaves - propulsion/reciprocating/piston-engine-cycle: the spark-ignition Otto cycle reciprocating prime mover, the pure constant-volume heat-addition sibling that this leaf's rho to 1 limit recovers exactly. - propulsion/reciprocating/diesel-cycle: the compression-ignition Diesel cycle reciprocating prime mover, the pure constant-pressure heat-addition sibling that this leaf's alpha to 1 limit recovers exactly. - propulsion/turboprop/free-turbine: the turboshaft power-turbine prime-mover slot, the shaft-power alternative to this reciprocating engine. - propulsion/turboprop/turboprop-cycle: propeller (Froude) efficiency and shaft-power-to-thrust bookkeeping once a prime mover, turbine or reciprocating, delivers shaft power. - propulsion/engine-airframe: installed thrust and drag bookkeeping for the airframe integration once the powerplant operating point is fixed. ## Pitfalls - Reading the air-standard dual efficiency as the achievable brake thermal efficiency: eta_dual (0.6226 in the worked example) is the ideal cycle ceiling; the real point's eta_b (0.3522) sits well below it because of heat transfer, combustion, pumping and mechanical losses the air-standard cycle does not model. - Confusing the pressure ratio alpha with the cutoff ratio rho: alpha = P3/P2 = T3/T2 describes the constant-volume heat-addition phase (alpha >= 1) and rho = V4/V3 = T4/T3 describes the constant-pressure heat-addition phase (1 < rho <= r); passing a cutoff ratio above the compression ratio raises ValueError because the heat addition cannot outlast the power stroke back to bottom dead center. - Treating the alpha = 1 or rho = 1 boundary as a claimed operating mode: those exact limits are test bounds that reproduce the Diesel and Otto sibling ceilings; this leaf's own claim is the interior mixed heat-addition case, alpha > 1 and rho > 1. - Forgetting the two-revolutions-per-power-stroke factor: the four-stroke indicated power formula divides by two after the rpm/60 conversion (one power stroke per two crankshaft turns); omitting it doubles the indicated power. - Treating the BSFC and brake-thermal-efficiency bands as a pass/fail gate: ci_band_verdict and dual_cycle carry enforced: False and never raise for an out-of-band point; the bands are a published reference range for compression-ignition aircraft engines, not a certification limit. - Mixing the BSFC reporting units: the module's native BSFC is kg/(kW h); the lb/(hp h) figure is a separate conversion through KG_PER_KWH_PER_LB_PER_HP_HR, not a unit-label swap on the same number. - Applying this leaf to spark-ignition, pure constant-pressure Diesel, gas-turbine, turboshaft or turboprop powerplants: the mixed heat-addition dual cycle, IMEP bookkeeping and CI bands here are specific to high-speed direct-injection compression-ignition reciprocating engines; the pure Otto cycle belongs to propulsion/reciprocating/piston-engine-cycle, the pure Diesel cycle belongs to propulsion/reciprocating/diesel-cycle, and shaft-power turbine cycles belong to propulsion/turboprop/free-turbine and propulsion/gas-turbine-cycle. ## Behavior contract (gate 3) Run the deterministic contract test (stdlib unittest, offline): python3 scripts/test_dual_cycle.py The test covers the air-standard dual-cycle efficiency at the worked-example and receipt compression, pressure and cutoff ratios, the pressure-ratio and cutoff-ratio sensitivity orderings, the alpha to 1 and rho to 1 limits that recover the Diesel and Otto sibling ceilings and the gamma to 1 limit that approaches zero, the five-state temperature bookkeeping and the temperature-form efficiency identity, the constant- volume and constant-pressure heat additions with their round trips and the constant-volume heat rejection, the ideal-cycle mean effective pressure and its linear scaling with the state-1 pressure, the four-stroke indicated power receipt target and its linear scaling with rpm and displacement, the brake power receipt and worked-example targets, the brake specific fuel consumption and its lb/(hp h) conversion with the round trip back to fuel flow, the indicated and brake thermal efficiencies and their ordering against the ideal dual efficiency, the volumetric fuel flow in three reporting units, the reference-only band verdict at an inside and an artificial out-of-band point, the full dual_cycle single-point summary dict and its determinism, and ValueError rejection of every non-physical input including a cutoff ratio above the compression ratio and a pressure ratio below 1. ## Compliance - Standards referenced, not reproduced: 14 CFR Part 33 (far-33, Airworthiness Standards for Aircraft Engines) historically covers reciprocating and turbine aircraft engines, including certified compression-ignition aircraft diesels of the high-speed direct-injection class; the relations above are standard air-standard-cycle and PLAN-bookkeeping engineering methodology, summary-only per standards-map.yaml. - compliance: STANDARDS-REF, gated: false.