Blade Radial-Growth Worked Example
Calculation of the centrifugal and thermal radial growth of a turbine blade, showing the combined growth and the effect on tip clearance.
Problem
A turbine blade has a mean radius of 0.30 m, length 0.08 m, and is made of a nickel alloy (E = 180 GPa at temperature, rho = 8200 kg/m^3, CTE = 14 x 10^-6 /deg C). The blade operates at 11,000 rev/min at a temperature 600 deg C above ambient. Estimate the blade radial growth and the tip clearance change.
Given
- Mean radius r = 0.30 m
- Blade length L = 0.08 m
- Modulus E = 180 x 10^9 Pa
- Density rho = 8200 kg/m^3
- CTE alpha = 14 x 10^-6 /deg C
- Speed N = 11,000 rev/min
- Temperature rise Delta_T = 600 deg C
Step 1 — Centrifugal growth
For a thin ring approximation at the mean radius: omega = 2 * pi * 11000 / 60 = 1151.9 rad/s delta_c = rho * omega^2 * r^3 / E delta_c = 8200 * (1151.9)^2 * (0.30)^3 / (180 x 10^9) delta_c = 8200 * 1326870 * 0.027 / (180 x 10^9) delta_c = 293.8 x 10^6 / (180 x 10^9) = 0.00163 m = 1.63 mm Note: This is an approximate ring estimate. A blade is not a ring — the actual growth depends on the blade geometry and stress distribution.
Step 2 — Thermal growth
delta_t = alpha * r * Delta_T = 14 x 10^-6 * 0.30 * 600 = 0.00252 m = 2.52 mm This is the thermal expansion at the mean radius.
Step 3 — Total growth
delta_total = delta_c + delta_t = 1.63 + 2.52 = 4.15 mm The blade tip grows radially by approximately 4.15 mm from the assembled condition to the operating condition.
Result
The blade radial growth is approximately 4.15 mm (1.63 mm centrifugal + 2.52 mm thermal). The assembled tip clearance must exceed 4.15 mm plus the casing thermal expansion, plus a margin for tolerance and transient overshoot. This example is illustrative — the actual growth should be computed from FEA.
Assumptions and limitations
- The centrifugal growth uses a thin ring approximation — actual blade growth depends on geometry
- The thermal growth assumes uniform temperature — actual growth depends on the temperature field
- The casing thermal expansion is not included — it must be added separately
- Manufacturing tolerances and transient overshoot are not included
- FEA is required for accurate growth prediction