Polytropic compression is commonly assumed for dynamic (centrifugal and axial) compressors.
The previous discussion of the adiabatic process showed that its relationships need mathematical corrections to make credible predictions. The corrections are compromises between theory and actual gas deviations, and they do not always yield sufficiently accurate predictions for some types of applications. Unfortunately, even this process requires adjustments to account for the non-ideal behavior of many gases.
Polytropic Relationships
The polytropic compression process is described mathematically as follows.
where:
n = polytropic exponent
where:
np = polytropic efficiency
where:
Hpoly = polytropic head, ft.
In Equation 100-30, k is ordinarily taken at the average compression temperature by most compressor manufacturers. Therefore, when estimating overall flange-toflange performance, use k at average flange-to-flange temperature to yield results very close to those of stage-by-stage calculations. In the case of single-stage machines, the difference between k at inlet temperature and average temperature is generally very small. Accordingly, in this manual, k at average compression temperature will be used.
A thermodynamic diagram can be used for a polytropic calculation by first determining the adiabatic head Had using Equations 100-26 and 100-28. Polytropic head Hpoly can then be determined by:
The relationship between polytropic and adiabatic efficiencies is:
This relationship is graphically represented by Figure 100-8.
From the foregoing discussion, it should be obvious that k is not equal to h. In some of the early compressor publications, the k and h exponents were erroneously treated as the same value. This error may have been one of nomenclature. At any rate, it is important to recognize that k is associated with the adiabatic process, and h with the polytropic process.