Turbine Blade Cooling

Turbine Blade Cooling (propulsion/axial-compressor/turbine-blade-cooling)

Use when the task is the heat-transfer side of a turbine blade row design: how much cooling flow the hot section must bleed from the compressor to hold the blade metal below its allowable temperature. This leaf implements the standard cooling-effectiveness model in pure Python, stdlib only. The baseline model is internal convection; an optional film cooling term adds a fixed effectiveness gain at the leading edge. It pairs with propulsion/axial-compressor/turbine-stage for the aerodynamic design of the same blade row and with the gas-turbine-cycle leaves for the cycle cost of the bleed flow.

Domain quick reference

• Cooling effectiveness: phi = (T_gas - T_metal_allow) / (T_gas - T_coolant). The metal temperature is held a fraction phi of the way from the gas temperature down toward the coolant supply temperature.
• Coolant fraction from the energy balance: m_dot_c / m_dot_g = phi / (1 - phi) * CP_RATIO. A unit coolant-to-gas specific heat ratio (CP_RATIO = 1.0) is the documented simplification for conceptual design; the coolant heat capacity rate must offset the blade heat load implied by the effectiveness.
• Bleed limit: fractions up to BLEED_LIMIT = 0.20 are practical for a blade row; above that the cooling-air penalty on the cycle becomes prohibitive and the design must trade metal temperature, gas temperature or cooling scheme.
• Film cooling: film at the leading edge lifts the effectiveness by FILM_IMPROVEMENT = 0.15 over the internal-convection baseline, capped at PHI_CAP = 0.95. The achievable metal temperature is T_m = T_gas - phi_eff * (T_gas - T_coolant).
• Margin: margin_k = T_metal_allow - T_m. A positive margin means the cooling scheme holds the metal below the allowable temperature.
• Units are SI throughout: K, dimensionless ratios and fractions.

Workflow

1. Fix the operating point: hot gas total temperature t_gas_k, the allowable blade metal temperature t_metal_allow_k and the coolant supply temperature t_coolant_k.
2. Compute the required cooling effectiveness with effectiveness; the function rejects non-physical inputs with ValueError.
3. Convert the effectiveness into the required coolant-to-gas mass flow fraction with coolant_fraction.
4. Check the fraction against the practical bleed limit with bleed_verdict; a fraction above 0.20 flags a design that needs revisiting.
5. Add the film cooling option: metal_temp_with_film with film_cooling=True returns the achievable metal temperature with the leading-edge film effectiveness gain.
6. Pull the whole estimate together with analyze, which reports effectiveness, coolant_fraction, verdict, metal_temp_k and margin_k in one dict.
7. Confirm the deterministic checks with the contract test scripts/test_turbine_blade_cooling.py.

Worked example

Case 1: first blade row, t_gas = 1500 K, allowable metal 1200 K, coolant 800 K.

• Effectiveness: phi = (1500 - 1200) / (1500 - 800) = 0.4286.
• Coolant fraction: 0.4286 / 0.5714 = 0.75, far above the 0.20 bleed limit, verdict "exceeds bleed limit".
• With film: phi_eff = 0.4286 + 0.15 = 0.5786, so T_m = 1500 - 0.5786 * 700 = 1095.0 K, margin +105 K. Film cooling brings the row inside the allowable metal temperature.

Case 2: high pressure turbine blade with film, t_gas = 1600 K, allowable 1250 K, coolant 900 K.

• phi = 350 / 700 = 0.5, fraction 1.0, exceeds bleed limit.
• phi_eff = 0.65, T_m = 1600 - 0.65 * 700 = 1145 K, margin +105 K.

Case 3: t_gas = 1600 K, allowable 1350 K, coolant 900 K: phi = 250 / 700 = 0.3571, fraction 0.5556, still exceeds the bleed limit.

Sensitivity: at t_gas = 1500 K and coolant 800 K the required fraction falls below 0.20 only when the allowable metal temperature exceeds 1383.33 K, about 117 K below the gas temperature.

Verification

• Confirm effectiveness(1500, 1200, 800) returns 0.4286 and coolant_fraction of it returns 0.75.
• Confirm analyze(1500, 1200, 800, film_cooling=True) reports metal_temp_k 1095.0 K and margin_k 105.0 K.
• Confirm analyze(1600, 1250, 900, film_cooling=True) reports metal_temp_k 1145.0 K.
• Confirm rising allowable metal temperature lowers the required fraction monotonically and crosses below 0.20 at 1383.33 K.
• Confirm ValueError rejection: t_gas <= 0, t_gas <= t_coolant, t_metal_allow >= t_gas, t_metal_allow <= t_coolant, phi outside (0, 1), and negative coolant fractions.
• Run the contract test offline: python3 scripts/test_turbine_blade_cooling.py (35 tests, deterministic).

Related leaves

• propulsion/axial-compressor/turbine-stage: the velocity-triangle aerodynamic design of the same blade row, the sibling this leaf feeds with its allowable metal temperature context.
• propulsion/axial-compressor/multi-stage-compressor: the compression system that supplies the cooling bleed flow.
• propulsion/axial-compressor/compressor-map: bleed extraction points and their effect on the operating line.
• propulsion/gas-turbine-cycle leaves: the thermodynamic cycle cost of the cooling-air bleed.

Pitfalls

• Accepting a coolant fraction above the bleed limit as a closed design: in all three worked-example cases the required fraction (0.75, 1.0, 0.5556) far exceeds the 0.20 practical bleed limit, and only the film-cooling effectiveness gain brings the metal temperature inside - the bleed_verdict flags the trade, it does not validate it.
• Neglecting film cooling when the internal-convection fraction is prohibitive: adding FILM_IMPROVEMENT = 0.15 at the leading edge lifts the effectiveness (0.4286 to 0.5786 in case 1) and turns a 300 K over-temperature into a +105 K margin; reporting the baseline-only number understates what the row can achieve.
• Setting the allowable metal at or above the gas temperature: the effectiveness definition requires t_metal_allow < t_gas and above t_coolant; t_metal_allow >= t_gas or t_metal_allow <= t_coolant raises ValueError because the metal could not be held.
• Forgetting the film cap: film effectiveness is capped at PHI_CAP = 0.95, so stacking film on top of an already high effectiveness does not keep improving the metal temperature without bound.
• Confusing the margin direction: margin_k = t_metal_allow - T_m is positive when the scheme holds the metal below the allowable; a negative margin means the metal is too hot, not a safety surplus.
• Reading the coolant fraction without the cycle and model context: the 0.20 bleed limit exists because the cooling-air penalty becomes prohibitive above it (pair with the gas-turbine-cycle leaves), and CP_RATIO = 1.0 is a documented conceptual simplification - real cooling design needs 3D conjugate heat transfer analysis beyond this model.

Behavior contract (gate 3)

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

python3 scripts/test_turbine_blade_cooling.py

The test covers the three worked-example operating points (effectiveness within 1e-4, fractions within 1e-3, film metal temperatures within 1.0 K), the bleed-limit verdicts, the film effectiveness cap at 0.95, the analyze dict outputs and margins, the sensitivity boundary crossing at 1383.33 K with the monotonic trend, and ValueError rejection of every non-physical input.

Compliance

• Standards referenced, not reproduced: FAR-33 is US government work (public domain) and covers engine type certification; the hot-section cooling relations above are simplified conceptual-design correlations, summary-only per standards-map.yaml. Real cooling design needs 3D conjugate heat transfer analysis beyond this model.
• compliance: STANDARDS-REF, gated: false.