59967 - Survey of Calculation Methods for Polytropic Efficiencies 

Calculating polytropic efficiencies is a basic task used for quantifying performance of power cycles involving compression and/or expansion. Its root is probably the incremental definition of a “polytropic curve” of gases by Gustav Zeuner in 1905. This may be the oldest mention of the word “polytropic” in a thermodynamic context (the word is used in a biological context too, but then mostly spelled “polytrophic”).

In Turbomachinery blading, the typical changes of state are nearly adiabatic and polytropic. L. S. Dzung first defined an incremental polytropic efficiency in 1944. This has become the best thermodynamic quality measure of a blading.

Already Zeuner started his consideration with an incremental definition. However, he immediately integrated analytically assuming ideal gas data. This resulted in the well-known formula p*vⁿ=constant. His thoughts probably had roots in his earlier publications on thermodynamics. Obviously, he and the later thermodynamicists loved the analytical opportunities of this formula in spite of the fact that it hides the physical connection to notations like friction or dissipation. Thus, the majority of the thermodynamic community still considers this formula as “the definition of a polytropic change”.

However, the initial incremental definition survived. Stodola, Dzung and later scientists established it as another definition of a polytropic change of state. Thus, we face now two different definitions of a polytropic change, which are theoretically identical for ideal gases but different for real gases and vapors if used for a larger pressure difference. From both definitions, we can derive a definition of a polytropic efficiency, which I suggest to call:

·       Incremental Constant Dissipation Rate Definition (ICDRD)

·       Exponential Definition (ED)

These two definitions are identical if applied for an incrementally small change of state.

The ICDRD keeps the dissipation rate along any change path constant. This means that the polytropic efficiency for a finite change interval is the same for any part interval. This feature is frequently communicated in education for both definitions. However it applies stringently only for ideal gases in case of the Exponential Definition ED. In order to claim this feature for the ED applied to a finite interval and to a real gas, approximation methods are needed.

Thus, the thermodynamic community has developed libraries with such approximation methods in order to apply the ED for real gases instead of using a numeric integration of the ICDRD. In case of steam expansion, the isotropic efficiency definition has been preferred in spite of its additional dependency on the pressure ratio. A nearby reason for these historic decisions was the fact that most of these developments happened before the computer age, which nowadays facilitates iterative numeric integration.

This paper summarizes first the tasks supported by the calculation of polytropic efficiencies. It refers to the first pioneers Zeuner, Stodola and Dzung. We omit reviewing the (many) approximations used for the ED. Instead, we refer to recent review publications. However, the direct integration of the ICDRD has become attractive. It is theoretically sound and easy to understand. Only this definition allows a direct identification of the dissipation work. Its calculation core and some applications for steam expansion are shown below.

The current situation with the two different primary definitions creates confusion in the education step from fundamental to advanced thermodynamics. Thus, we suggest declaring the ICDRD to become the only definition of a polytropic efficiency and to use the ED just as example in a special case.