Axial Compressors: Stage Design, Maps and Surge

The de Haller ratio and diffusion factor, stage loading and stage count, compressor maps, surge margin, stall and surge, variable stators

The Tomcat's engines

The F-14A Tomcat's Pratt & Whitney TF30 engines were found to be ill-adapted to the demands of air combat. They were prone to compressor stalls at high angle of attack if the pilot moved the throttles aggressively — and because the Tomcat's engines sat far apart, a stall on one side produced asymmetric thrust that could throw the aircraft into an upright or inverted spin, from which recovery was very difficult (the TF30). Nothing was wrong with the compressor on the test stand. In the aircraft, at high angle of attack, its inlet delivered distorted air, and a rapid throttle movement asked the compressor for more pressure ratio than its blades could hold. The flow broke down. This lesson is about how hard a compressor's blades can work, and about the margin that keeps them from giving up.

A compressor blade row works against a rising pressure. In the rotor's frame it slows the relative flow down — diffuses it — and the boundary layers on its blades, pushed back by the adverse pressure gradient, are always close to separating. The more each stage is asked to do, the more it must slow the flow, and the nearer it comes to that edge. Two simple measures tell a designer how near.

\frac{W_2}{W_1} \ge 0.72 \ \ \text{(de Haller)}, \qquad DF = 1 - \frac{W_2}{W_1} + \frac{|\Delta W_\theta|}{2\sigma W_1} \lesssim 0.45 \ \ \text{(rotor tip)}

The de Haller limit on the relative velocity ratio across a rotor, and Lieblein's diffusion factor with solidity σ = blade chord / blade spacing.

Derivation

Where the diffusion factor comes from

The suction surface of a blade accelerates the flow to a peak velocity and then decelerates it to the exit velocity

W_2

; separation depends on that deceleration. Lieblein estimated the peak as the inlet velocity plus a term from the blade's circulation: the circulation per blade is

s\,\Delta W_\theta

(blade spacing times the change in swirl), shared over a chord

c

, which raises the suction-side velocity by about

s\Delta W_\theta/(2c) = \Delta W_\theta/(2\sigma)

W_{max} \approx W_1 + \frac{\Delta W_\theta}{2\sigma}.

The diffusion from the peak to the exit, normalised by

W_1

, is then

\frac{W_{max} - W_2}{W_1} = 1 - \frac{W_2}{W_1} + \frac{\Delta W_\theta}{2\sigma W_1} = DF.

Cascade tests showed losses rising sharply above about 0.6, and at rotor tips above about 0.45. More blades (higher solidity) lower it — at the price of more friction and weight.

📐How far can the rotor turn the flow?

For the stage of Lesson 6-1, the relative flow arrives at

\beta_1 = 58.6°

(Lab 1):

De Haller allows turning it down to $\beta_2 = 43.6°$ at most — our 45° is just inside.
The diffusion factor is 0.428 with a solidity of 1.0, 0.395 at 1.25 and 0.373 at 1.5.

A compressor stage can turn the relative flow by only about 15° before it runs out of diffusion — the reason its work is limited to a few tens of kJ/kg.

📐How many stages?

A 27:1 high-pressure compressor, the GE9X's, from 400 K with a polytropic efficiency of 0.90 needs 742 kJ/kg and delivers air at 1139 K (Lab 2). The stage count depends on blade speed and loading:

$U = 350$ m/s, $\psi = 0.25$ : 25 stages.
$U = 400$ m/s, $\psi = 0.45$ : 11 stages — 11 stages needs $\psi = 0.422$ at 400 m/s.

GE raised the core's pressure ratio from 19:1 to 27:1 for the GE9X by using 11 stages instead of 9 or 10 (the GE9X). Each stage of a modern compressor is loaded far harder than in the 1950s, thanks to three-dimensional blade design.

💡The compressor map

A compressor's performance at every condition fits on one chart: pressure ratio against corrected mass flow

\dot m\sqrt{\theta}/\delta

, with lines of constant corrected speed

N/\sqrt{\theta}

(where

\theta = T_t/T_{ref}

and

\delta = p_t/p_{ref}

Each speed line rises to the left until the surge line, where the flow breaks down.
To the right, flow is limited by choking in the blade passages.
The engine's other components fix an operating line across the map (Lesson 8-1).
The surge margin is how far the operating point sits below surge — typically 15–25 % at design.

Derivation

Why corrected flow and speed

Dimensional analysis: for a compressor of fixed geometry and gas, the pressure ratio and efficiency can depend only on dimensionless groups of the flow, the speed and the inlet state. The natural ones are a flow Mach number and a blade Mach number:

\frac{\dot m\sqrt{R T_t}}{p_t D^2}, \qquad \frac{N D}{\sqrt{\gamma R T_t}} .

For one machine

D

R

and

\gamma

are fixed, so it is enough to track

\dot m\sqrt{T_t}/p_t

and

N/\sqrt{T_t}

, usually referred to sea-level standard conditions. Plotted this way, one map describes the compressor at any altitude and temperature.

📐Surge margin across the map

On an illustrative map with an operating line set by a choked turbine downstream (Lab 3, surge margin measured along each speed line):

At design speed: 18.7 %. At 80 % speed: only 5.0 % — part speed is where compressors are closest to surge.
A raised operating line, as when a nozzle closes: 11.3 % at design speed.
Inlet distortion that takes 10 % off the surge pressure ratio: 6.8 % — the TF30's problem.

💡Stall, surge and the cures

Rotating stall: one or more cells of stalled flow travel around the annulus; the compressor keeps running, roughly, but loses pressure ratio and its blades vibrate. Surge: the whole compressor loses its grip on the pressure behind it; the flow collapses and may reverse in seconds-long cycles, often with flames from the inlet and a loud bang. The cures widen the margin where it is thin:

Variable stators: in 1951 a GE team led by Gerhard Neumann built a test compressor with variable stators (the J79); the J79 that grew out of it put them into production.
Bleed valves that dump air from the middle of the compressor at low speed.
Multiple spools, so the front and rear stages can turn at different speeds.

The first principle is that you must not fool yourself — and you are the easiest person to fool.

– Richard Feynman, "Cargo Cult Science", Caltech commencement address (1974) (Wikiquote)

⚠Spin risk

On a twin-engined aircraft with widely spaced engines, a compressor stall on one side is not just a loss of thrust: the asymmetric thrust yaws the aircraft, and at high angle of attack that can start a spin. The F-14 was later re-engined with the General Electric F110.

💡Where you will use this

Lesson 8-1 puts the compressor map together with the turbine and nozzle to find the operating line; 8-3 follows the operating point through accelerations, where surge margin is spent fastest. Lesson 5-1 explained where the distortion comes from.

📚Further reading

Nicholas A. Cumpsty, Compressor Aerodynamics (Longman, 1989; Krieger, 2004), on loading limits, maps, stall and surge.
S. Larry Dixon and Cesare A. Hall, Fluid Mechanics and Thermodynamics of Turbomachinery (7th ed., Butterworth-Heinemann, 2014).
The TF30 on Wikipedia.

↓But there is a problem

The compressor takes about 740 kJ from every kilogram of core air. Something has to give it back — the turbine, working in the hottest gas in the engine. In 1944 the Me 262's Jumo 004 had one turbine stage, with hollow blades folded from sheet steel and cooled by air from the compressor. They lasted 10 to 25 hours.