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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.

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