Calculating polytropic efficiencies is a basic task used for quantifying performance of power cycles involving compression and/or expansion. The incremental definition of a “polytropic curve” of gases by Gustav Zeuner in 1905 may be the oldest mention of the word “polytropic” in a thermodynamic context [1].

In Turbomachinery blading, the typical changes of state are nearly adiabatic and polytropic. L. S. Dzung was probably the first defining an incremental polytropic efficiency in 1944 [3]. Recursive integration of this has become the best thermodynamic quality measure of a blading.

Both Zeuner and Dzung started their consideration with an incremental definition. However, they integrated analytically assuming ideal gas data. This resulted in the well-known formula
Most thermodynamic textbooks declare this the definition of a polytropic change of state. However, the incremental definition survived too. Stodola [2], Dzung and later scientists established it as another definition of a polytropic change of state.

Thus, we face now two definitions of a polytropic change of state, which are theoretically identical for ideal gases but different for real gases and vapors. In educational context, this is disturbing and forcing to a logical detour. We trace the historic roots and show that the initial incremental definition is the physically healthier one. Recursive integration allows direct application to turbomachinery with any finite pressure ratio and to any real fluid.

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