Axial Turbine Stage (propulsion/axial-compressor/turbine-stage)
Use when the task is single axial turbine stage velocity-triangle
analysis: specific work output, stage loading, flow coefficient,
degree of reaction, blade row losses, and total-to-total or
total-to-static efficiency from the blade speed, axial velocity, and
flow angles.
Domain quick reference
- Dixon convention: flow angles measured from the axial direction,
all angles in RADIANS. Absolute angle alpha, relative angle beta.
- tan(beta) = u/ca - tan(alpha) at the same axial station; beta1,
beta2 are the relative angles at rotor inlet and outlet.
- Specific work output w = uca(tan(alpha1) - tan(alpha2)) in J/kg.
Positive for a work-extracting stage (alpha1 > alpha2).
- Flow coefficient phi = ca/u, dimensionless.
- Stage loading psi = w/u**2, dimensionless; turbine stages run near
psi = 1 to 3, higher than compressor stages.
- Degree of reaction r = 1 - ca/(2u)(tan(alpha1) + tan(alpha2)),
dimensionless; r = 0 is an impulse stage, r = 0.5 is a symmetric
50% reaction stage with equal enthalpy drops across rotor and
stator rows.
- Row losses: nozzle loss = zeta_nc1**2/2 and rotor loss =
zeta_rw2**2/2 in J/kg, with c1 = ca/cos(alpha1) and
w2 = ca/cos(beta2); zeta_n and zeta_r are the row enthalpy loss
coefficients (air-standard defaults 0.05 each).
- Total-to-total efficiency eta_tt = w/(w + nozzle_loss + rotor_loss),
total-to-static efficiency eta_ts = w/(w + nozzle_loss +
rotor_loss + c32/2) with the exit absolute angle alpha3 (axial
exit alpha3 = 0 loses the full ca2/2).
- Units: u, ca in m/s; angles in rad; w and losses in J/kg; phi, psi,
r, eta dimensionless.
Workflow
- Fix the design point: blade speed u, axial velocity ca.
- Set the absolute flow angles alpha1, alpha2; derive beta1, beta2
from tan(beta) = u/ca - tan(alpha).
- Compute the work output with specific_work.
- Compute the dimensionless loading with flow_coefficient and
stage_loading.
- Compute the degree of reaction with degree_of_reaction; check the
impulse (r = 0) and 50% reaction (r = 0.5) design limits.
- Budget the row losses with blade_row_loss and compute the
efficiencies with total_to_total_efficiency and
total_to_static_efficiency.
- Assemble the full assessment with stage_properties and gate the
turbine stage design review on it.
Pitfalls
- Degrees instead of radians: entering 45 for alpha1 instead of
pi/4 changes the tangent and the work by a large factor; convert
first.
- Angles from the tangential direction: this convention flips the
tan terms and the sign of the work; keep angles from axial.
- Confusing alpha with beta: alpha is the absolute flow angle, beta
the relative one; passing beta into specific_work double-counts
the blade speed.
- Sign of work: alpha1 < alpha2 adds work to the flow (compressor
sense); a turbine stage must show alpha1 > alpha2 for output.
- Negative loss coefficients: zeta_n or zeta_r below zero raise
ValueError; do not catch and continue, the efficiency is
meaningless.
- Non-physical inputs: u <= 0 or ca <= 0 raise ValueError.
Behavior contract (gate 3)
The velocity-triangle logic is exercised by the gate 3 contract test:
scripts/test_turbine_stage.py against
scripts/turbine_stage_logic.py (stdlib unittest, offline). Run:
python3 scripts/test_turbine_stage.py
Compliance
- Standards referenced, not reproduced: FAR-33 is US government work
(public domain) and covers engine type certification, not stage
analysis methods; the velocity-triangle relations are common
turbomachinery methodology, summary-only per standards-map.yaml.
- compliance: STANDARDS-REF, gated: false.
1---2name: turbine-stage3description: Use when you must analyze a single axial turbine stage: compute the specific work output, the stage loading, the flow coefficient, the degree of reaction, the total-to-total efficiency, and the blade row losses from the blade speed, the axial velocity, the absolute flow angles, and the relative flow angles. Produces the velocity-triangle performance parameters in SI units that gate the turbine stage design review in the FAR-33 engine context. Trigger: axial turbine stage, stage loading, degree of reaction, flow coefficient, blade row loss, total-to-total efficiency, specific work output, turbine rotor, turbine stator.4license: Apache-2.05---67# Axial Turbine Stage (propulsion/axial-compressor/turbine-stage)89Use when the task is single axial turbine stage velocity-triangle10analysis: specific work output, stage loading, flow coefficient,11degree of reaction, blade row losses, and total-to-total or12total-to-static efficiency from the blade speed, axial velocity, and13flow angles.1415## Domain quick reference1617- Dixon convention: flow angles measured from the axial direction,18 all angles in RADIANS. Absolute angle alpha, relative angle beta.19- tan(beta) = u/ca - tan(alpha) at the same axial station; beta1,20 beta2 are the relative angles at rotor inlet and outlet.21- Specific work output w = u*ca*(tan(alpha1) - tan(alpha2)) in J/kg.22 Positive for a work-extracting stage (alpha1 > alpha2).23- Flow coefficient phi = ca/u, dimensionless.24- Stage loading psi = w/u**2, dimensionless; turbine stages run near25 psi = 1 to 3, higher than compressor stages.26- Degree of reaction r = 1 - ca/(2*u)*(tan(alpha1) + tan(alpha2)),27 dimensionless; r = 0 is an impulse stage, r = 0.5 is a symmetric28 50% reaction stage with equal enthalpy drops across rotor and29 stator rows.30- Row losses: nozzle loss = zeta_n*c1**2/2 and rotor loss =31 zeta_r*w2**2/2 in J/kg, with c1 = ca/cos(alpha1) and32 w2 = ca/cos(beta2); zeta_n and zeta_r are the row enthalpy loss33 coefficients (air-standard defaults 0.05 each).34- Total-to-total efficiency eta_tt = w/(w + nozzle_loss + rotor_loss),35 total-to-static efficiency eta_ts = w/(w + nozzle_loss +36 rotor_loss + c3**2/2) with the exit absolute angle alpha3 (axial37 exit alpha3 = 0 loses the full ca**2/2).38- Units: u, ca in m/s; angles in rad; w and losses in J/kg; phi, psi,39 r, eta dimensionless.4041## Workflow42431. Fix the design point: blade speed u, axial velocity ca.442. Set the absolute flow angles alpha1, alpha2; derive beta1, beta245 from tan(beta) = u/ca - tan(alpha).463. Compute the work output with specific_work.474. Compute the dimensionless loading with flow_coefficient and48 stage_loading.495. Compute the degree of reaction with degree_of_reaction; check the50 impulse (r = 0) and 50% reaction (r = 0.5) design limits.516. Budget the row losses with blade_row_loss and compute the52 efficiencies with total_to_total_efficiency and53 total_to_static_efficiency.547. Assemble the full assessment with stage_properties and gate the55 turbine stage design review on it.5657## Pitfalls5859- Degrees instead of radians: entering 45 for alpha1 instead of60 pi/4 changes the tangent and the work by a large factor; convert61 first.62- Angles from the tangential direction: this convention flips the63 tan terms and the sign of the work; keep angles from axial.64- Confusing alpha with beta: alpha is the absolute flow angle, beta65 the relative one; passing beta into specific_work double-counts66 the blade speed.67- Sign of work: alpha1 < alpha2 adds work to the flow (compressor68 sense); a turbine stage must show alpha1 > alpha2 for output.69- Negative loss coefficients: zeta_n or zeta_r below zero raise70 ValueError; do not catch and continue, the efficiency is71 meaningless.72- Non-physical inputs: u <= 0 or ca <= 0 raise ValueError.7374## Behavior contract (gate 3)7576The velocity-triangle logic is exercised by the gate 3 contract test:77scripts/test_turbine_stage.py against78scripts/turbine_stage_logic.py (stdlib unittest, offline). Run:79python3 scripts/test_turbine_stage.py8081## Compliance8283- Standards referenced, not reproduced: FAR-33 is US government work84 (public domain) and covers engine type certification, not stage85 analysis methods; the velocity-triangle relations are common86 turbomachinery methodology, summary-only per standards-map.yaml.87- compliance: STANDARDS-REF, gated: false.
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