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Centrifugal compressor speed

Q: What is the effect on the pressure change when the centrifugal compressor speed is to be increased? Are there ways we can increase the efficiency of a multi-staged compressor other than using interstage cooler?

A: An increase in centrifugal compressor speed will result in an increase in compressor polytropic head by a value equal to the square of the speed. The polytropic head is a function of pressure ratio across the compressor and therefore will result in a pressure rise, assuming the inlet pressure is not changed.

The addition of internal cooling reduces the power consumption of a centrifugal compressor because the compression cycle is changing from a polytropic cycle to a more isothermal cycle, which in turn is more efficient Achieving this is not easily accomplished as the change in temperature to the internal stages would require that some of the impellers be changed even if the compressor casing had the required inlet and outlet nozzles been provided. Depending  on the type of impellers used in the original design a more efficient set of impellers could possibly be retrofit, improving the overall efficiency.