Unit 3: Axial Flow Compressors - Subjective Questions

Working principle:

An axial-flow compressor raises the pressure of air while it flows approximately parallel to the compressor axis. A stage consists of one rotor blade row followed by one stator blade row.

• Rotor:
  • Rotates with angular velocity and transfers shaft work to the air.
  • Increases the absolute stagnation temperature and stagnation pressure.
  • Usually increases the tangential or whirl component of absolute velocity.
• Stator:
  • Does no shaft work because it is stationary.
  • Diffuses the high-velocity air, converting kinetic energy into static pressure.
  • Removes or modifies swirl and directs the flow to the next rotor at the required angle.

For an adiabatic stage, the stagnation enthalpy rise is

and the Euler compressor equation gives

where is blade speed and and are the whirl components at rotor inlet and outlet. Repetition of several stages produces the required overall pressure ratio.

The velocity triangle relates the following velocities:

• : absolute velocity of air, measured in the stationary frame.
• : relative velocity, measured by an observer moving with the rotor blade.
• : blade peripheral velocity, given by .

They are related by

At rotor inlet and outlet, the absolute velocity is resolved into:

where is the axial component and is the whirl component. If absolute angles are measured from the axial direction,

For relative angles measured from the axial direction,

Thus,

The inlet triangle determines the rotor incidence, while the outlet triangle determines flow deviation, work input, reaction, and the stator inlet condition.

Applying the angular-momentum equation to a rotor operating at radius , the torque exerted on the air is

The shaft power transferred to the air is

Since , the specific work input becomes

For an adiabatic rotor, this equals the stagnation-enthalpy rise:

Using for a perfect gas,

Therefore,

If axial velocity is constant and angles are measured from the axial direction,

so that

The result shows that compressor work is governed by blade speed and the change in absolute whirl velocity.

The degree of reaction, , is the ratio of the static enthalpy rise in the rotor to the static enthalpy rise in the complete stage:

For a repeating stage with negligible changes in axial velocity, the stage static enthalpy rise equals the stage stagnation-enthalpy rise:

In the rotor, rothalpy is constant for adiabatic flow:

At constant radius, is unchanged, giving

With constant ,

Substitution and simplification produce

Hence,

Using relative flow angles,

A larger means that a greater fraction of stage diffusion and static-pressure rise occurs in the rotor.

For a 50% reaction stage,

Therefore, ideally, one-half of the stage static-enthalpy rise occurs in the rotor and the remaining half occurs in the stator.

For constant axial velocity, the reaction relation is

Setting gives

The velocity triangles of a repeating 50% reaction stage are approximately symmetrical. Under the usual angle convention, this symmetry may be written as

in terms of angle magnitudes.

Advantages:

• Diffusion is shared nearly equally between rotor and stator.
• Rotor and stator blade shapes can be similar in a repeating stage.
• Excessive diffusion in either blade row is avoided.
• Good efficiency and practical stage loading can be obtained.

The symmetry is exact only under ideal assumptions such as constant axial velocity, repeating-stage conditions, and consistent angle definitions.

Incidence is the difference between the inlet relative flow angle and the blade metal angle at the leading edge:

For a fixed speed and inlet whirl,

Therefore, a reduction in mass flow generally reduces and increases the relative inlet angle . This produces positive incidence and can cause leading-edge separation and rotating stall. An increase in mass flow raises , decreases , and can produce negative incidence or eventually choking.

Consequences of operating away from design incidence:

• Increased profile and separation losses.
• Reduced pressure rise and efficiency.
• Altered outlet deviation and work input.
• Risk of rotating stall at low flow.
• Risk of choking at high flow.

Variable inlet guide vanes and variable stators are used to adjust inlet swirl and maintain acceptable incidence over a wider operating range.

A two-dimensional analysis at the mean radius is useful, but real compressor flow is three-dimensional because:

• Blade speed varies with radius as .
• Hub and casing boundary layers produce spanwise nonuniformity.
• Whirl velocity, work input, reaction, and pressure vary from hub to tip.
• Tip-clearance leakage, secondary flow, and radial migration occur.
• Annulus-area changes modify axial velocity.

For steady, axisymmetric flow with small radial velocity, radial force equilibrium gives

The outward centrifugal force associated with swirl is balanced by an outward increase in static pressure. In stagnation-enthalpy form, radial equilibrium can be represented as

for an axisymmetric flow with negligible radial velocity. This equation is used to determine spanwise distributions of axial velocity, whirl, pressure, and blade angles.

Important three-dimensional loss mechanisms include:

• Annulus boundary layers: Low-momentum layers grow along the hub and casing and distort the inlet profile.
• Passage secondary flow: Pressure differences between the pressure and suction surfaces drive cross-passage flow within the end-wall boundary layers.
• Tip-clearance leakage: Air leaks from the pressure surface to the suction surface through the rotor tip gap, forming a leakage vortex.
• Corner separation: Interaction of blade and end-wall boundary layers can separate near the suction-surface/end-wall corner.
• Radial migration: Centrifugal and pressure-gradient effects redistribute low-energy fluid along the span.
• Wake mixing: Blade wakes mix with the main stream downstream, causing stagnation-pressure loss.

These effects reduce pressure ratio and efficiency, alter deviation, and reduce stall margin. They are controlled through suitable blade twist, end-wall contouring, small tip clearances, controlled-diffusion profiles, boundary-layer management, and appropriate stage loading.

In a free-vortex flow, the circulation is constant along the span:

Therefore,

For a compressor rotor, the inlet and outlet distributions may be written as

The Euler work at radius is

Since ,

Thus, the stagnation-enthalpy rise is constant with radius. For approximately constant axial velocity, the absolute air angles are

and the relative angles are

Consequently, the blade angles must vary from hub to tip, requiring twisted blades. A major benefit is nearly uniform work input over the blade span.

For constant axial velocity, the degree of reaction is

Since , imposing a constant value of requires

A common constant-reaction design combines this condition with a prescribed constant stage work . From Euler's equation,

so

Solving the sum and difference equations gives

For constant , the absolute angles follow from

and the relative angles from

These relations determine the required blade twist. Unlike a simple free-vortex law, the whirl distribution contains terms proportional to both and .

Free-vortex design:

• Satisfies for each relevant station.
• With , it can provide nearly constant Euler work over the span.
• The degree of reaction generally varies from hub to tip.
• It naturally satisfies a simple radial-equilibrium condition when axial velocity is uniform.

Constant-reaction design:

• Maintains the selected value of along the span.
• At constant axial velocity, it requires
  
• If constant work is also imposed, the whirl distribution contains both and terms.
• It distributes rotor and stator diffusion more uniformly in a reaction sense.

Comparison:

Both designs require spanwise changes in air angle and hence twisted blades. Free-vortex design emphasizes uniform work and simple radial equilibrium, whereas constant-reaction design emphasizes a controlled division of pressure rise between rotor and stator. Practical compressors often use modified vortex laws to balance loading, Mach number, reaction, loss, and mechanical constraints.

A typical blade-row design procedure includes:

1. Specify requirements: Select mass flow, pressure ratio, rotational speed, efficiency, annulus dimensions, and operating range.
2. Choose stage loading: Select the work coefficient
   
   and flow coefficient
   
3. Construct velocity triangles: Determine , , , and the inlet and outlet flow angles.
4. Select reaction: Divide diffusion and static-pressure rise suitably between rotor and stator.
5. Apply radial equilibrium: Obtain spanwise velocity and angle distributions.
6. Select cascade geometry: Choose aerofoil profile, solidity, chord, pitch, camber, thickness, stagger, and aspect ratio.
7. Allow for incidence and deviation: Blade metal angles differ from the ideal air angles because of viscous effects.
8. Check aerodynamic limits: Evaluate diffusion, Mach number, losses, choking, stall margin, and tip-clearance effects.
9. Check structural constraints: Verify centrifugal stress, vibration, flutter, and manufacturing limits.
10. Refine and validate: Use cascade correlations, computational fluid dynamics, and experimental rig testing.

• Solidity: Ratio of blade chord to pitch :
  
  Higher solidity generally permits greater flow turning but increases wetted area and friction loss.
• Stagger angle: Angle between the blade chord line and a selected axial or tangential reference direction. It sets the blade-row orientation.
• Camber: Curvature of the blade mean line. Greater camber generally produces greater turning and aerodynamic loading.
• Incidence: Difference between inlet flow angle and inlet blade metal angle:
  
  Excessive incidence causes leading-edge separation.
• Deviation: Difference between the actual outlet flow angle and the outlet blade metal angle:
  
  It arises because the flow does not exactly follow the blade at the trailing edge.
• Turning: Change in flow direction through the cascade, commonly represented in magnitude by
  

These parameters determine blade loading, diffusion, pressure rise, loss, and stall margin. Their permissible values are commonly selected using experimental cascade correlations.

Pressure rise in a compressor blade row requires deceleration or diffusion of the flow. Excessive diffusion thickens the boundary layer and may cause separation.

The de Haller number for a rotor is commonly defined as

A low value indicates severe relative-flow deceleration. A traditional preliminary guideline is

although the acceptable value depends on blade profile and operating conditions.

A commonly used rotor diffusion factor is

where is solidity. The first term involving velocity ratio represents diffusion, while the whirl-change term represents blade loading.

Design significance:

• Large indicates high loading and increased separation risk.
• Values near or above roughly are generally avoided in preliminary subsonic design.
• Increasing solidity can reduce loading per blade, but excessive solidity increases friction and weight.

These criteria are empirical guidelines and must be supplemented by loss correlations, Mach-number checks, and three-dimensional analysis.

A centrifugal-compressor map normally plots pressure ratio or specific head against corrected mass flow for several corrected-speed lines.

Main features:

• At a fixed speed, pressure ratio usually rises as flow is reduced until the stable-flow limit is reached.
• The low-flow boundary is the surge line; operation to its left is unstable.
• At high flow, the impeller eye, diffuser, or another passage reaches choke, limiting mass flow.
• Efficiency contours form islands, with maximum efficiency near the design point.
• Increasing corrected speed generally increases attainable pressure ratio and flow capacity.

Corrected quantities account approximately for inlet-condition changes:

A compressor operating line is superimposed on the map. Safe design requires adequate separation between this line and the surge line while avoiding choke and excessive speed or discharge temperature.

An axial-compressor map plots total pressure ratio against corrected mass flow, with families of constant corrected-speed lines and efficiency contours.

• Speed lines: At each corrected speed, pressure ratio varies with corrected flow. Higher speeds generally produce greater pressure ratio.
• Stall or surge boundary: The low-flow ends of the speed lines form the surge line. Before complete surge, one or more blade rows may experience rotating stall.
• Choke region: At high corrected flow, a local blade passage reaches sonic velocity, and further reduction in downstream pressure produces little increase in mass flow.
• Efficiency islands: Peak isentropic efficiency occurs near the design flow and speed; efficiency falls because of incidence, separation, shock, leakage, and mixing losses away from design.
• Operating line: Engine and compressor matching establishes the normal path through the map.

A common total-to-total compressor efficiency is

For a perfect gas, the isentropic temperature ratio satisfies

The map is used to assess operating range, efficiency, stall margin, and component matching.

Axial compressors:

• Handle very large mass flow for a given frontal area.
• Produce a relatively small pressure ratio per stage, so many stages may be required.
• Can achieve high peak efficiency.
• Have a comparatively narrow stable operating range and are sensitive to incidence and stage matching.
• Are well suited to large gas turbines and high-flow propulsion systems.

Centrifugal compressors:

• Produce a larger pressure ratio per stage because of impeller work and radial diffusion.
• Usually have a wider stable operating range and greater robustness.
• Handle less mass flow for a given frontal area and have a larger diameter.
• Are commonly used in small gas turbines, turbochargers, and compact power units.

Both maps contain speed lines, efficiency contours, a low-flow surge boundary, and a high-flow choke boundary. The axial compressor is preferred when high mass flow and small frontal area are dominant, while the centrifugal compressor is attractive when compact stage count, ruggedness, and wide operating range are important.

Rotating stall:

• A local aerodynamic instability in which one or more stalled-flow cells travel circumferentially around an annulus.
• Usually caused by excessive positive incidence and blade-row separation at low mass flow.
• Produces pressure fluctuations, loss of pressure rise, vibration, and increased blade stress.

Surge:

• A system-level instability involving large oscillations of mass flow and pressure through the entire compression system.
• May include temporary flow reversal.
• Occurs when the compressor cannot sustain the pressure required by the downstream system.
• Can cause violent vibration, overheating, and mechanical damage.

Choking:

• Occurs when the Mach number reaches unity at a minimum effective flow area.
• Limits the corrected mass flow even if downstream pressure is reduced further.
• Commonly occurs at high flow in blade throats, the impeller eye, or diffuser passages.

Rotating stall can exist without full surge, but it may trigger surge. Stall and surge define the low-flow limit, whereas choking defines the high-flow limit.

In a multistage compressor, all stages pass essentially the same mass flow, but density rises progressively through the machine. Annulus area is therefore reduced, or velocity distributions are adjusted, to maintain suitable axial velocity.

At off-design speed, the stages do not retain their design flow coefficients simultaneously:

The front and rear stages may then experience different incidence and loading. At low speed, rear-stage capacity can restrict the front stages, causing excessive front-stage incidence and stall. Poor matching reduces pressure ratio, efficiency, and surge margin.

Variable geometry improves matching through:

• Variable inlet guide vanes that control rotor-inlet swirl.
• Variable stator vanes in the front stages that regulate downstream incidence and flow capacity.
• Compressor bleed valves that remove air during starting or low-speed operation.
• Multi-spool arrangements that allow compressor groups to rotate at different speeds.

These methods move the operating condition away from stall, reduce starting torque, and maintain acceptable incidence over a wider range.

1. Specific work input

From Euler's equation,

Therefore,

2. Stagnation-temperature rise

Thus,

3. Degree of reaction

For constant axial velocity,

Hence,

Thus, approximately of the ideal stage static-enthalpy rise occurs in the rotor.

4. Absolute flow angles

Therefore,