Turbines: Stage Aerodynamics

Turbine velocity triangles, why a turbine stage does far more work than a compressor stage, Zweifel loading, choked vanes, HP and LP turbines

The Jumo 004, 1944

The Me 262's Junkers Jumo 004 had an eight-stage axial compressor and a one-stage turbine with hollow blades. The prototype engines had used scarce nickel, cobalt and molybdenum in quantities Germany could not afford in production. So the production 004B's turbine blades were folded and welded from Cromadur — 12 % chromium, 18 % manganese, 70 % iron, an alloy developed by Krupp — and cooled by air bled from the compressor. Given the lower-quality steels, the engines had a service life of only 10–25 hours, perhaps twice that in the hands of a careful pilot (the Jumo 004). One turbine stage drove eight compressor stages. This lesson explains how a turbine stage can do so much more work than a compressor stage — and why it pays for it in temperature.

A turbine stage runs the other way round from a compressor stage. Its fixed row comes first: the nozzle guide vanes accelerate the hot gas and throw it into the rotor with a large swirl. The rotor takes the swirl out again, and by Euler's equation the work it extracts is

w = U\,(C_{\theta 2} - C_{\theta 3}).

The difference from a compressor is the pressure gradient. Through a turbine the pressure falls and the relative flow accelerates; boundary layers in accelerating flow stay attached even when the blades turn the flow through 100° or more. A turbine stage can therefore take out a far larger swirl than a compressor stage can put in.

📐A high-pressure turbine stage

Blade speed 420 m/s, axial velocity 180 m/s, vane exit angle 72°, axial exit from the rotor (Lab 1):

Swirl into the rotor: $C_{\theta 2} = 180\tan 72° = 554$ m/s; out: zero.
Work $w = 420 \times 554 =$ 232.7 kJ/kg, a stagnation temperature drop of 201 K; loading $\psi = 1.32$ .
The rotor turns the relative flow from +36.7° to −66.8° — over 100° — while the relative velocity doubles ( $W_3/W_2 = 2.04$ ).

That one stage does the work of six compressor stages like the one in Lesson 6-1.

💡Impulse, reaction and Zweifel

An impulse stage ( $R = 0$ ) does all its expansion in the vanes; the rotor only turns the flow. A 50 % reaction stage shares the expansion. High-pressure turbines are often low-reaction (about 0.3–0.4) to do a lot of work per stage; LP turbines are closer to 0.5.
Zweifel's coefficient sets how far apart turbine blades can be: the ratio of the actual tangential force on a blade to an ideal one. Blades too far apart overload their suction surfaces; too close together add friction, weight and cooling air. A value around 0.8 is a common choice.
The first vane row usually runs choked (Lesson 2-1), so it fixes the core's corrected flow: $\dot m\sqrt{T_{t4}}/p_{t4}$ = constant.

Z_w = 2\,\frac{s}{b}\,\cos^2\beta_3\,\left(\tan\beta_2 + \tan\beta_3\right)

Zweifel's loading coefficient for a turbine blade row with pitch s, axial chord b, and relative inlet and exit angle magnitudes β2 and β3 measured from axial (inlet and exit on opposite sides of the axial direction).

Derivation

Zweifel's coefficient

Per unit span, the tangential force on one blade equals the rate of change of tangential momentum through one passage:

F_\theta = \rho C_x s\,(W_{\theta 2} + W_{\theta 3})

, with

W_{\theta 2} = C_x\tan\beta_2

and

W_{\theta 3} = C_x\tan\beta_3

for angle magnitudes on opposite sides of axial.

Zweifel compared this with an ideal blade loaded by the full stagnation-to-exit pressure difference over its whole axial chord

b

, with no suction-side diffusion:

F_{ideal} = (p_{t2} - p_3)\,b \approx \tfrac12\rho W_3^2\,b

for incompressible flow. With

W_3 = C_x/\cos\beta_3

Z_w = \frac{F_\theta}{F_{ideal}} = \frac{\rho C_x^2 s(\tan\beta_2 + \tan\beta_3)}{\tfrac12\rho C_x^2 b/\cos^2\beta_3} = 2\frac{s}{b}\cos^2\beta_3(\tan\beta_2 + \tan\beta_3).

Given the angles from the velocity triangles, choosing

Z_w

fixes the pitch-to-chord ratio — and hence the number of blades.

📐The turbine with real gas

The HP turbine of Lesson 4-2's turbofan must give its compressor 541.8 kJ per kilogram of gas. With real combustion products from Cantera (Lab 2):

Burning the same fuel flow gives a turbine entry of 1766 K, not the 1700 K the constant- $c_p$ burner balance assumed: the products' real $c_p$ is lower than 1156 over most of the burner's temperature rise.
Extracting the work drops the gas to 1343 K; a constant $c_p$ of 1156 would have predicted 1297 K — 45 K too cold.

Forty-five kelvin at a turbine's second stage is the difference between a blade that lasts and one that does not (Lesson 6-4).

📐LP turbines and geared fans

The LP turbine of the same engine must drive the fan: 436 kJ per kilogram of core gas (Lab 3). Its blade speed is set by the fan, whose tips must not go too far past the speed of sound. At a loading of 2.0:

Direct drive, the LP turbine turning at fan speed — mean blade speed about 220 m/s: 5 stages.
A geared fan, letting the LP shaft turn about three times faster — about 380 m/s: 2 stages.

The gearbox costs weight and needs cooling, but it shortens the turbine — one of the arguments behind geared turbofans (Lesson 10-2).

If I have seen further it is by standing on ye sholders of Giants.

– Isaac Newton, letter to Robert Hooke (1676), in his own spelling (Wikiquote)

💡HP and LP turbines

A high-pressure turbine has one or two stages, very high loading, small blades, the hottest gas and intensive cooling. A low-pressure turbine has three to seven larger, uncooled stages turning slowly with the fan (fewer behind a geared fan). The two are matched through the flow they pass: both are close to choking at their inlets, which links them to each other and to the nozzle (Lesson 8-1).

💡Where you will use this

Lesson 6-4 keeps these blades alive. Lesson 8-1 uses the choked turbine vanes to match the engine. Lesson 9-2 asks what holds a turbine disc together at 10,000 rpm.

📚Further reading

S. Larry Dixon and Cesare A. Hall, Fluid Mechanics and Thermodynamics of Turbomachinery (7th ed., Butterworth-Heinemann, 2014), on axial turbines.
H. I. H. Saravanamuttoo et al., Gas Turbine Theory (7th ed., Pearson, 2017), on axial-flow turbines.
The Jumo 004 on Wikipedia.

↓But there is a problem

The gas that reaches a modern HP turbine is hotter than the melting point of the metal its blades are made of. They survive anyway — for years. How?