Matched at one speed and at no other

Work out of a change of swirl drew a box round one rotor and read its work off two velocity triangles, and what a turning frame keeps split the pressure rise inside that box into a part a boundary layer limits and a part the radius gives for nothing. The stress that picks the aerodynamics then capped the radius from outside the fluid altogether. All three were about one stage at one operating point.

A gas-turbine compressor is eight, ten or fifteen of those stages in a row on one shaft, and the method that made each of them tractable — a box with nothing assumed about the inside — does not survive being repeated. The reason is not that the boxes interact through their pressure fields, although they do. It is that they share one number, and the number means something different in each of them.

The machine here is eight identical stages at a mean blade speed of 300 metres a second, each drawn for a flow coefficient of 0.5 and a work coefficient of 0.35. At design speed it passes 19.97 kilograms of air a second at a pressure ratio of 6.94.

One mass flow, eight densities #

A stage’s flow coefficient is its axial velocity divided by its blade speed, $\varphi = C_x/U$, and it is the number the whole of a stage’s behaviour hangs on. It fixes the angle at which the air meets the blades, and so whether the blades turn it cleanly, stall because the angle is too steep, or pass so much flow that the stage stops compressing.

Every stage passes the same mass flow, and mass has nowhere to go:

$$\dot m = \rho_i\,A_i\,C_{x,i}\qquad\text{for every stage } i.$$

The designer chooses the annulus area $A_i$ behind each stage so that at the design point every stage sees its design flow coefficient. That means cutting each area for the density the air will have when it arrives there, and the air arrives denser at every stage than at the last, so the annulus shrinks along the machine — here from 0.120 square metres at the inlet to 0.035 at the eighth stage.

Once cut, the areas are fixed, and the blade speed is the same for every stage because they share a shaft. Divide stage $i$’s flow coefficient by the first stage’s at any operating condition, and the mass flow, the areas’ absolute sizes and the speed all cancel:

$$\frac{\varphi_i/\varphi_d}{\varphi_1/\varphi_d} \;=\; \frac{\rho_{i,d}/\rho_{1,d}}{\rho_i/\rho_1}.$$

A stage is mismatched against the first by exactly the ratio of the density rise the machine was drawn for to the density rise it is actually making. That is the whole of stage matching in one line, and it contains no blade, no characteristic and no efficiency.

The consequence of slowing the shaft follows directly. The work a stage does is its work coefficient times the blade speed squared, so at 70 per cent speed each stage does about half the work, the temperature and pressure climb much less along the machine, and the density at the rear falls far short of what the annulus was cut for. The rear stages must pass the same mass flow through areas cut for air 3.42 times as dense as the inlet’s, carrying air only 1.88 times as dense, so the axial velocity there rises against a blade speed that has fallen. The front stages, by the same identity, are pushed the other way.

And the identity has a converse that is the best evidence it is the right one. A machine whose density does not rise can never be mismatched: every stage’s flow coefficient moves together at every speed, because the ratio on the right is one. Setting the work coefficient to a hundred-thousandth of its value leaves the eight stages within a thousandth of each other at half, eight-tenths and 1.2 times design speed. That is why a multistage water pump has no matching problem at all. Water barely changes density across a pump, and incompressibility is a property of what a flow does to a parcel’s volume, which here is exactly the property that matters.

What each end of the machine’s blades see #

A flow coefficient is an angle in disguise, and the angle is what a blade actually responds to.

At design speed both triangles are identical: 150 metres a second of axial velocity against 300 of blade speed, so the air arrives at the rotor 63.4 degrees from the axis, which is the angle the blades were cut to meet. At 70 per cent speed on the operating line the first stage’s blade speed has fallen to 210 metres a second and its axial velocity to 67, and the air now arrives at 72.2 degrees. The blade meets it nearly nine degrees further round than it was drawn for, which on a compressor blade is well past the incidence at which the suction-side boundary layer separates.

The last stage has the same blade speed and an axial velocity of 123 metres a second, and the air arrives at 59.7 degrees — nearly four degrees less than design. A blade met at negative incidence does little turning, the flow through the row accelerates rather than diffuses, and the stage stops raising the pressure and starts passing flow as a restriction does.

So the two ends of one machine are failing in opposite directions at the same moment, and no single change to the inlet flow can help both: taking in more air relieves the front stage and drives the rear one further into choke, and taking in less does the reverse.

A window for each stage, and whether they overlap #

That last sentence can be made exact. Each stage alone tolerates a range of flow coefficient, stated here as 0.82 to 1.30 of its design value, and at a given shaft speed that range corresponds to a range of inlet mass flow.

The bars step to the left along the machine because each stage’s mismatch is compounded by everything ahead of it. At 90 per cent speed they still share a stretch of mass flow, from 0.772 to 0.867 of the design value, and a throttle set anywhere in that stretch keeps all eight stages working. It is a narrow stretch — a tenth of the design flow — and it no longer contains the flow the nozzle wants to pass.

At 75 per cent speed the stretch has gone. The first stage stops stalling only once the inlet flow is above 0.654 of the design value, and the last starts choking once it is above 0.607. There is no mass flow that satisfies both, and it is not a question of finding the right throttle setting: every setting that rescues the front of the machine drowns the back.

This is the same shape of argument that the group with no head in it made about shaft speed for a single pump: two constraints moving in opposite directions turn a free choice into a window, and a window can be empty. There the two constraints were a machine type and cavitation. Here they are the two ends of one machine.

The speed below which nothing works #

Following the overlap across the whole speed range turns the two windows into a boundary.

The overlap is widest a little below design speed, not at it. At design speed every stage sits at the centre of its own range, so every one of them limits the overlap equally; slightly below, the front stages have moved towards stall and the rear towards choke by amounts that partly open the stretch between them before the mismatch grows enough to close it.

Below that the stretch narrows steadily and vanishes at 80.06 per cent of design speed. Below that speed no mass flow keeps every stage of this machine working, and it is found by bisection on whether the overlap exists, checked against the intersection of the eight single-stage ranges computed separately.

The operating line is the flow the machine actually runs at, set by the nozzle downstream of it, and it leaves the overlap sooner, at 85.09 per cent speed, when the first stage on it reaches the stall limit. Between the two speeds the machine could still be kept working, but only by moving its flow off the line the nozzle sets — which is precisely what a variable-area nozzle or a bleed valve does, and is the first sign that the fixes are about changing the flow rather than the blades.

How much of this is the blade that was assumed #

The limits 0.82 and 1.30 are an assumption about the blades, and a number that decides where a machine stops working ought to be tested against its own assumptions.

The closing speed moves with the limits, and by a great deal: from 69.7 per cent of design speed for a blade that tolerates a flow coefficient down to 0.72 of design and up to 1.4, to 86.6 per cent for one that tolerates only 0.88 to 1.2. Where the overlap closes is a property of the blades. That it closes is not. Every combination drawn closes somewhere, and none can fail to, because the identity says the mismatch between the two ends grows without bound as the density rise falls towards nothing, while any real blade’s tolerance is finite.

That split between what the model decides and what it borrows is worth being explicit about. The identity is exact given the mean-line picture. The characteristic that turns a flow coefficient into work, the efficiency curve and the stall and choke limits are stated, not computed from a blade, and they are the same for all eight stages; a real design gives each stage its own. The shape of the result — front starving, rear choking, a closing speed — is the identity’s. Its position is the characteristic’s.

Throwing air away before the rear stages #

If the rear of the machine is carrying more volume than its annulus was cut for, the direct remedy is to give it less air, and the only way to do that without changing the front is to take air out partway along.

Bleeding a fifth of the flow before the fourth stage lowers the closing speed from 80 to 65 per cent of design. The earlier the air is taken the more it helps, because every stage behind the bleed point is relieved of flow its under-dense air could not pass, and more stages are behind an early bleed than a late one.

The curves also say where bleeding stops helping, and it is not where the bleed runs out. Bled before stage 3, the closing speed falls to 0.643 with a fifth of the flow taken and then barely moves — 0.634 at a quarter, 0.629 at 35 per cent — because the stage immediately behind the bleed is now being starved by the bleed itself: at the closing speed with 35 per cent taken, stage 3 sits exactly on its stall limit while stage 8 still sits exactly on its choke limit. Bled before stage 6 the curve flattens at 0.666 from a quarter of the flow onwards, and for the opposite reason: the five stages ahead of the bleed are now a machine of their own that cannot match, with stage 1 on its stall limit and stage 5 on its choke limit at the plateau, and no bleed behind them can reach either. Only a bleed in the middle of the machine keeps paying, down to 0.504 of design speed before stage 4 with 35 per cent taken. A bleed moves the mismatch; it does not remove it, and taken too early it simply relocates the starvation to the stage behind it.

It is not free, and the model shows the cost directly. The bleed air has been compressed and is then thrown overboard, so the work done on it is lost. And a bleed open at design speed wrecks the match it was not needed for: with a quarter of the flow bled before stage four, the overlap at design speed shrinks to a sliver between 1.064 and 1.088 of the design flow, and with 35 per cent there is none at all. A bleed valve is a part-speed device that must be shut at full speed, which is why engines schedule their bleeds against shaft speed and why a bleed stuck open is an engine that will not reach its rated thrust.

Asking the front stages for less #

The other remedy acts on the other end. The front stages are starving because the air meets their blades at too steep an angle, and the angle can be changed without changing the flow by turning the stator row ahead of each one.

Closing a row of variable stators gives the air swirl in the direction of blade motion before it reaches the rotor, which reduces the angle the rotor sees it arrive at for the same axial velocity. In effect it lowers the flow coefficient the stage wants. Turning the first three rows so that the first stage wants a fifth less lowers the closing speed to 64.8 per cent of design, about as much as bleeding a fifth of the flow before stage four; turning only the first row achieves 73.4 per cent.

The single row flattens almost at once. Turned further than f = 0.9, the first row alone buys nothing: the closing speed sits at 0.738, 0.734 and 0.729 as f falls to 0.65, because the untouched second stage has become the front of the machine’s starvation — at the closing speed it is the stage on its stall limit, with the first stage comfortably inside its range at 1.10 of its new design value. Turning three rows by graded amounts keeps working down to 0.492 of design speed at f = 0.65, which is why variable stators on real compressors come in sets of rows moved together by one linkage rather than as a single inlet row.

The two remedies act on opposite ends and neither substitutes for the other. Variable stators rescue the front stages from stall and do nothing for the rear stages’ choking; bleed rescues the rear and leaves the front as it was. A compressor that must run over a wide range of speed is given both — rows of variable stators at the front and bleed valves behind the middle — and the schedules that move them with shaft speed are among the most important numbers in an engine’s control system.

There is a third remedy that the model does not draw and that engine design adopted early: split the stack. Put the front stages on one shaft and the rear on another, let each run at its own speed, and the rear shaft slows less than the front one at part power, which reduces the mismatch at source. That is a two-spool engine, and its invention was a response to exactly this problem.

Why a box round each stage is the wrong box #

Every earlier turbomachine argument succeeded by drawing a box and refusing to look inside it. Here the box round each stage is perfectly valid — Euler’s equation holds for each of the eight — and the method fails anyway.

It fails because each box’s behaviour depends on a number, its flow coefficient, that is set by the boxes ahead of it through a quantity none of the boxes controls: the density the air has reached. A box round the whole machine fails for the opposite reason. It gives the overall work and pressure ratio correctly and has no way to say that the first stage inside it is stalled.

The failure is specific to compressibility. The identity says so directly — no density rise, no mismatch — and the same observation explains the direction of everything above. Slowing the shaft lowers the density rise, which starves the front and floods the rear. Speeding it up raises the density rise beyond the design value, which floods the front and starves the rear, and the first figure shows exactly that reversal at 110 per cent speed. A compressor’s stall line has a kink where the stage that stalls first changes from the front at low speed to the rear at high speed, and this is where it comes from.

It is a curious fact that the choking at the rear is the same limit a nozzle throat has: a passage that cannot pass more flow however the conditions downstream change. In a single duct that limit is set by the speed of sound. In a compressor it arrives long before sound speed, as the stage stops turning the flow, and it arrives because upstream stages have left the air less dense than the passage was built for.

What the mean line leaves out #

Every stage has the same characteristic. Real stages are designed individually, the front ones usually with more tolerance for low flow because they are known to be the first to stall at part speed. That moves the closing speed, as the margins figure shows, and does not remove it.

The stall limit is a line, and stall is an event. A stage that crosses its limit does not simply produce less pressure. Its flow breaks into cells rotating round the annulus at a fraction of shaft speed, or the whole machine reverses its flow in a surge cycle, and which of those happens depends on the volume of the ducts downstream as much as on the blades. Nothing here is dynamic.

The flow is a mean line. Radius does not appear. A real stage’s hub and tip sections stall at different flow coefficients, and the casing boundary layer thickens along the machine and eats into the effective area — which is itself a mismatch that grows towards the rear.

The first stage is transonic and has no shock in it. With 150 metres a second of axial velocity against 300 of blade speed and no inlet swirl, the air meets the first rotor at 1.005 times its own speed of sound. A real front stage running there carries a passage shock whose loss grows steeply with incidence, so its tolerance at part speed is worse than a single efficiency curve says, and the stall limit assumed for it is generous. That moves the closing speed up, not down.

No swirl enters the stages. The triangles were drawn with the air arriving axially, which makes the angle arithmetic simple and is not how a multistage machine with stators between rotors runs. The angles would change and the direction of every change would not.

The operating line is a choked nozzle’s. An engine’s compressor is followed by a combustor and a turbine, whose own flow capacity sets the line; a choked nozzle is the simplest downstream condition that produces a definite line and it is a fair model of a turbine’s first nozzle row running choked.

Who worked it out #

Mismatch at part speed was met the moment axial compressors reached pressure ratios where the density rise was large, in the jet engines of the 1940s, whose early designs were notorious for stalling at low speed. The stage-by-stage bookkeeping of the kind drawn here — stacking a stage characteristic stage after stage with the density carried forward — became the standard way to predict it in the 1950s, and it is still how a new compressor’s part-speed behaviour is first estimated.

The remedies arrived in the same decade and in the same order as the reasoning. Blow-off valves came first; the two-spool engine followed, splitting the stack so its halves could run at different speeds; and variable stator rows at the front of single-spool compressors followed that. Modern high-pressure compressors carry all three.

Still open: what happens when the line leaves the window #

Everything here decides whether a steady operating point exists with every stage inside its limits. It does not say what the machine does when there is none. A compressor pushed past its stall limit either develops rotating stall — a steady state with a cell of stalled flow travelling round the annulus — or goes into surge, a violent oscillation of the whole flow through the machine and its ducts, and which one it chooses is decided largely by a single dimensionless group comparing the compressibility of the downstream volume with the inertia of the air in the compressor. That is a question about a dynamic system rather than about a sequence of boxes, and its answer is a growth rate rather than a window.

Beside it is a second problem, already named: a blade passing a stator row is excited at the frequency the blade counts set, and the variable stators drawn here change the wakes the rotor passes through as they turn — so the part-speed fix and the part-speed resonance are adjusted with the same hardware.