12: COMPRESSORS AND TURBINES

Objective: To increase the pressure of the air before heat addition. We have seen previously that the cycle efficiency is strongly dependent on the ratio of the highest to the lowest pressure in an engine. Unless the pressure rise due to deceleration of the fluid is extremely high (because the fluid velocity is in the high supersonic range), a mechanical compressor is required to achieve the pressure rise.

Compressors usually operate in two steps (though in some designs it may not be possible to distinguish between these steps):

1. Increase the momentum of the air by doign work on it (using rotor blades, for example)

2. Decelerate the air to increase static pressure (using stator stages).

Often, the compression takes place through several compressor stages, with each stage increasing the pressure by a factor of, say, 1.8 or 2. Modern jet engines may have as many as 15 compressor stages, rotating on as many as three independent concentric shafts.

Both "Centrifugal" and "Axial" compressors are used.

CENTRIFUGAL COMPRESSOR

The tuboprop engine shown above uses two stages of centrifugal compression. Here the air enters the compressor close to the hub, and is then impelled outwards by the blades. It is then passed through an expanding duct at the periphery before being led back towards the hub for the next compression stage. The blades that push the air towards the periphery and increase its momentum are the rotor blades, while the expanding duct is the stator or diffuser passage.

Features of centrifugal compressors

1. Large pressure ratio per stage: stage pressure ratios can be as high as 3 or 4.

2. Few moving parts.

Such compressor stages are used on the low-pressure shaft of helicopter engines and turboprop engines. One famous application is in the Space Shuttle Main Engine turbopumps.

AXIAL COMPRESSOR

The compressor of the turbofan engine shown above is axial: the air moves primarily in a direction parallel to the axis of the engine. Each stage of the axial compressor consists of a rotor and a stator. Both rotor and stator are made up of a large number of individual blades, which are twisted airfoils, usually with a high degree of camber. The rotor blades add work to the air, so that the stagnation enthalpy rises, along with the stagnation temperature and stagnation pressure. This is usually accompanied by an increase in the velocity. The stator blade passages straighten out the flow and act as diffusers, slowing down the flow and thus increasing the static pressure and static temperature.

The flow through an axial compressor can be considered to consist of three types of flows:

1. Axisymmetric flow with work addition through an annular duct:

2. Flow over blade rows (cascades)

3. Secondary flows (recirculating flows, tip vortices, hub vortices, rotor/stator interactions, etc.)

FLOW THROUGH A SINGLE STAGE

Objectives:

Obtain expressions for

1. Torque and work per stage

2. Stagnation temperature rise

3. Stagnation pressure rise

4. Stage efficiency

5. Limiting pressure ratio per stage,

and relate these to the stage velocity diagrams.

Reference Frames:

The velocity of air at any station, measured with respect to the engine walls, is called the "absolute velocity", denoted by .

The velocity of air measured relative to a rotor blade is .

Thus,

where the blade speed at that radius is

and

with N being the shaft speed in revolutions per minute (RPM).

Velocity Diagrams for a Single Stage

Conservation of Angular Momentum gives:

Torque per unit mass flow rate = rate of change of angular momentum

Assuming that radial movement of the air is negligible within each stage. Thus, air entering the stage at radius r will leave at the same radius r.

Work done on the fluid per unit time per unit mass flow rate by the rotor is:

Torque on the stator =

Work done by the stator is zero, because the stator blades do not move with respect to the engine walls. Thus, there is no rise in stagnation enthalpy in the stator.

Assuming

a) uniform stagnation enthalpy per unit mass along the radius of the blades, and

b) Adiabatic conditions (heat transfer effects are negligible),

from which we get:

For the stator, since no work is done,

Thus, the changes in air properties through a compressor stage are related to the velocity vectors through the rotor and stator.

Stage Efficiency

Stage efficiency is defined as the ratio of the ideal work to the actual work.

Thus, the pressure stage pressure ratio is related to the stage temperature rise using the isentropic relations as before:

Limits on Stage Pressure Ratio

1. Compressibility:

As the relative Mach number becomes supersonic, shock losses can become substantial. In earlier compressors, the blade tip Mach number was kept below 1.0 to avoid the transonic drag rise. This imposed a severe limit on compressor shaft rpm, blade radius, and stage pressure ratio. In modern compressors, the rotor operates in the transonic regime, with shocks present in the rotor. While this causes some loss in stagnation pressure, much more work can be done by each rotor stage, and the shock provides a convenient way of increasing static pressure. As a result, stage pressure ratio is higher, and the overall weight of the compressor is reduced.

2. Flow Separation:

Usually, this is what limits the stage pressure ratio. Note that the flow in the compressor stage is moving against an adverse pressure gradient. Boundary layers thicken, and may separate if the pressure gradient becomes too large. This can be expressed using the pressure coefficient

for the rotor, and

for the stator.

The limiting value of the pressure coefficient is usually around 0.6 to avoid flow separation.

This shows why the stage pressure ratio is usually limited to about 1.4 for subsonic rotors. Under extreme conditions, transonic stages can reach stage pressure ratios as high as 2.2.

The effect of limiting stage pressure ratio on the number of stages in a compressor can be seen from the following:

Given a compressor pressure ratio of , and a stage pressure ratio of , the number of stages is:

Efficiency of Multistage Compressors

Usually, axial compressors have several stages. The efficiency of a compressor depends not only on the design of each stage, but also on the overall pressure ratio. In other words, given the same level of technology, a compressor with a higher pressure ratio will have a lower efficiency. This can be seen by relating the stage efficiency to the overall compressor efficiency.

Polytropic efficiency:

This is a useful concept to define the level of technology of the compressor. It is defined as the ratio of the ideal work required for a given differential pressure change to the actual work required.

Using this definition, simple thermodynamics can be used to show that the compressor efficiency becomes

Similarly, the stage efficiency becomes

The stagnation temperature ratio across a compressor with 'n' stages can be calculated as

where 2 and 3 refer to stations upstream and downstream of the compressor.

Illustration

An axial compressor has 16 stages, with an overall pressure ratio of 25. The stage efficiency is 0.93, and the pressure ratio is the same across each of the stages. Calculate the compressor efficiency.

Using the above expressions, the stage pressure ratio is 1.22284. The polytropic efficiency is 0.932, slightly higher than the stage pressure ratio as expected, and the compressor efficiency is obtained as 0.8965.

Degree of Reaction

The Degree of Reaction of a turbomachine stage is defined as the ratio of the static pressure change in the rotor to the static pressure change through the whole stage. Thus, for example, a compressor stage with a degree of reaction of 0.5 would share the pressure rise about equally between the rotor ans stator. This is desirable in the case of a compressor, where the pressure gradient is the major concern. However, turbine stages can be designed with extreme degrees of reaction

Solidity

The solidity of a stage is defined as the ratio of the blade chord to the blade spacing. If the solidity is low, there is less friction in the flow, but the blades have to work harder, and thus the pressure gradient is worse. If the solidity is high, the frictional losses are greater, but the machine can operate over a wider range of inlet conditions. Generally, the solidity is around 1.

AXIAL TURBINES

Differences between the flow field characteristics of turbines and compressors

Compressor	Turbine
Adverse pressure gradient	Favorable pressure gradient
Low stage pressure ratio (1.2 to 2)	High stage pressure ratio (> 4)
Limited by stall	Limited by choking and blade stress
Moderate temperature	High temperature, requiring cooling

The engine shown above has 10 fan and compressor stages, but only four turbine stages (2 on each spool) to provide enough work to run them. A turbine stage consists of a "nozzle", which is static with respect to the engine walls, and a "rotor". The nozzle is in fact a series of passages between aerodynamically-shaped surfaces (essentially airfoils). The rotor blades may be highly cambered, since flow separation is not as big a concern as in compressors.

Using the same nomenclature as for the axial compressor stage velocity diagram,

Work Output per unit mass flow rate

Also,

thus,

Note that here, T01 = T02 (no work in the nozzle).

Again, the work done per unit mass flow rate is proportional to the blade speed achieved, and the turning of the flow.

Impulse Stage

In a impulse turbine stage, the entire pressure drop occurs in the nozzle. The velocity diagram is shown below:

Note that for a turbine stage to operate, the C vector must be considerably longer than the U vector. In the case of an impulse stage, the flow gets turned by the impulse rotor, so that the static termperatures and the relative flow angles are:

50% Reaction Stage

In a 50% Reaction Stage, the static pressure drop is split between the rotor and the nozzle. Here, the velocity diagram is as shown below:

The velocity triangles become symmetric.