# Multi Stage Compressor > Use when you must design or match a multi-stage axial compressor for a gas turbine: compute the overall pressure ratio as the product of the stage pressure ratios, size the stage count required to reach the target pressure ratio from a design stage pressure ratio, account for the reheat factor that inflates the actual work split, size the annulus area from the mass flow, the axial velocity, and the density, distribute the stage work with equal or rising schemes, and correct the rotor speed to the reference temperature for off-design matching. Produces the stage count, overall pressure ratio, reheat factor, annulus area, and work distribution in SI units that gate the FAR-33 compressor architecture review. Trigger: multi-stage compressor, stage matching, overall pressure ratio, stage count, reheat factor, annulus area, work distribution. - Skill: `ashfordeou/multi-stage-compressor` (Agent Skill, multi-file: 3 files) - Install (CLI): `npx skillmds@latest add ashfordeou/multi-stage-compressor` - Raw SKILL.md: https://api.skillmd.com/api/skills/ashfordeou/multi-stage-compressor/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/multi-stage-compressor --- # Multi-Stage Compressor (propulsion/axial-compressor/multi-stage-compressor) Use when the task is multi-stage axial compressor design and matching: overall pressure ratio from the stage pressure ratios, the stage count to meet a target ratio, the reheat factor, the annulus area layout, the stage work distribution, and the corrected speed for off-design matching. ## Domain quick reference - Overall pressure ratio: PR_overall = product of the stage pressure ratios, PR_overall = pi_1 * pi_2 * ... * pi_n, dimensionless. Each stage pressure ratio must exceed 1. Example: 6 stages at 1.45 give 1.45**6 = 9.29. - Stage count: n = ceil(ln(PR_overall) / ln(pi_stage)), with ln the natural logarithm. The result is the number of identical design stages and is always at least 1. Example: PR_overall 40 with 1.20 per stage gives ceil(20.23) = 21 stages. - Reheat factor: RF = W_actual / W_ideal_sum, dimensionless, always at or above 1. W_actual is the total work absorbed by all stages, W_ideal_sum the sum of the ideal (isentropic) stage works. Each stage re-compresses the reheat loss of the previous stage, so the actual total work exceeds the ideal sum; typical values run from 1.01 to 1.06 and grow with the stage count. - Stage work distribution: W_total = sum of the stage works w_k. Equal scheme: w_k = W_total / n for every stage. Rising scheme: w_k = W_total * 2*k / (n*(n+1)) for k = 1 to n, a linear ramp that puts about twice the first-stage work on the last stage, matching the rising back pressure along the flow path. - Annulus area: A = m_dot / (rho * V_ax) in m^2, with mass flow m_dot in kg/s, density rho in kg/m^3, axial velocity V_ax in m/s. Density rises through the compressor, so for a constant axial velocity the annulus area shrinks toward the rear stages. - Corrected speed: N_corr = N * sqrt(t_ref / t) in rpm, the rotor speed referred to the reference temperature for off-design matching. Standard day t_ref = 288.15 K; at t = t_ref the corrected speed equals the physical speed. - Units: pressures dimensionless, work in J/kg (or J), area in m^2, flows in kg/s, velocities in m/s, densities in kg/m^3, speeds in rpm, temperatures in K. - FAR-33 (14 CFR Part 33) sets the engine type certification context; it calls for demonstrating compression system operability but does not prescribe the matching relations above, which are common turbomachinery design practice. ## Workflow 1. Fix the design point: mass flow, overall pressure ratio target, inlet temperature, axial velocity, and density. 2. Choose the design stage pressure ratio (or a per-stage ratio list) and compute the stage count with stage_count(total_pressure_ratio, stage_pressure_ratio). 3. Compute the achieved overall pressure ratio from the stage ratio list with overall_pressure_ratio and confirm the product meets the target. 4. Distribute the total work with stage_work_distribution using the equal or rising scheme; prefer rising when the rear stages would otherwise be under-loaded at the rising back pressure. 5. Estimate the reheat factor with reheat_factor(actual_work, ideal_work_sum) and check it stays at or above 1. 6. Size the annulus at each station with annulus_area(mass_flow, axial_velocity, density), letting the local density grow along the flow path so the area shrinks. 7. For off-design matching, refer the rotor speed with corrected_speed(physical_speed, t_ref, t) before comparing with map data. ## Pitfalls - Adding stage pressure ratios instead of multiplying: the overall pressure ratio is the product, so 6 stages at 1.45 give 9.29, not 8.70. - Rounding the stage count down: the count is the ceiling of the log ratio; 6 stages at 1.45 reach only 9.29, so a target of 10 needs 7 stages at that loading. - Trusting a reheat factor below 1: the actual total work can never sit below the ideal stage work sum, so such a value is a data error; the function raises ValueError instead of returning it. - Mixing corrected and physical speed: corrected speed equals the physical speed only at the reference temperature; comparing raw speeds measured on a different day shifts the matching point. - Passing a stage or total pressure ratio at or below 1: no compression happens, and the functions raise ValueError rather than divide by zero or return a nonsense count. - Assuming an equal work split everywhere: rising back pressure unloads the rear stages of an equal-scheme machine; use the rising scheme or accept the matching penalty. - Mixing units: keep everything SI (kg/s, m/s, kg/m^3, K, Pa); a density in kg/L or a flow in lb/s breaks the annulus area. - Sizing the annulus with the inlet density at every station: the area must shrink as density grows, or the axial velocity falls and the rear stages stall. ## Behavior contract (gate 3) The design and matching relations are exercised by the gate 3 contract test: scripts/test_multi_stage_compressor.py against scripts/multi_stage_compressor.py (stdlib unittest, offline). Run: python3 scripts/test_multi_stage_compressor.py ## Compliance - Standards referenced, not reproduced: FAR-33 is US government work (public domain) and covers engine type certification, not stage matching methods; the pressure ratio, reheat factor, annulus, and work distribution relations are common turbomachinery methodology, summary-only per standards-map.yaml. - compliance: STANDARDS-REF, gated: false.