This article demonstrates various aspects of polytropic efficiency calculations method for performance simulation of gas turbines. A recursive algorithm for integrating the incremental classic definition can easily be derived in a manner that any equations of state v(p, T) can be used. The Constant Dissipation Rate Algorithm (CDRA) is recommended for all calculations of polytropic changes with elevated accuracy requirements but especially if high pressure or pronounced real gases are involved. All required formulas are indicated in the paper. There is also a simple test for the achievable convergence or accuracy included. The comparison of the CDRA with two conventional formulas indicates astonishing deviations even for dry air.

## Article

Polytropic efficiency calculations are the preferred method for (one dimensional) performance simulation of Gas Turbines. The basics of these methods have been defined by Mr. Dzung who was working for Brown Boveri & Cie in the forties of the last century. Since then formulas were used, which describe the (average) energy conversion in compressor and turbine bladings in gas turbines. Dzung defined the expression “polytropic efficiency” for an incremental ratio of output power divided with an input power, where a part of the input power is lost by friction and similar effects. Let us call this the “classic definition of the polytropic efficiency”. But this self-evident definition could not be integrated for real gases because at Dzung's time neither computers nor a large wealth of equations of state were available.

Instead Dzung assumed that the air and the combustion gases are ideal. Thus he could derive analytical formulas for the polytropic efficiency for compression or expansion. The first one is indicated here as an example:

Polytropic efficiency for compressing an ideal gas (formula “IGF”):

p_{1}, T_{1}, s_{1} = pressure, temperature and specific entropy in state 1 (inlet condition) p_{2}, T_{2}, s_{2} = pressure, temperature and specific entropy in state 2 (outlet condition)R: Gas constant η_{p}: Polytropic efficiency

This formula applies exactly only for an ideal gas with the equation of state pv=RT and fulfilling the caloric equation for a specific enthalpy increment dh=Cp(T)dT (with the specific heat Cp being independent of the pressure). It is still frequently used.

But in spite of the fact that the flue gases in gas turbines are considered being “near-ideal” they behave “real” such that the polytropic efficiencies calculated with the mentioned formulas are rather inaccurate regarding the strict classic definition. But in the practical application for gas turbine performance applications this inaccuracy is normally satisfactorily compensated in the two typically combined tasks as follows:

Calculating a reference polytropic efficiency for a well measured blading section from the known inlet and exit conditions (pressure and temperature)

Calculating a missing pressure or temperature (or a derived quantity like a specific power) for either inlet or exit of a blading section with a polytropic efficiency value from an assumed reference.

If these two tasks are done for nearly similar conditions and with the same formula a mistake in the polytropic efficiency does not matter, because this is just a reference number with no absolute significance. Around these two basic tasks there are many applications in the gas turbine industry. Such calculations are used for defining guarantee conditions, for interpreting acceptance tests, for corrections to standardized ambient conditions and last but not least in the control and protection systems for determining the firing temperature from measured turbine exhaust data.

But in many other cases such polytropic efficiencies are taken from publications and other sources as reference for an achievable blading performance. The mentioned compensation of mistakes will not be on hand if the conditions for task 2 are not nearly equal to the ones for task 1 or if the used formula for both tasks is not the same. In publications often polytropic efficiencies are indicated without any information on the determination method behind it although there are many different methods which could have been used.

In such cases and also if applied for higher pressure levels above 50 bar or for pronounced real gases like CO_{2}, the ideal gas related formulas do not deliver a satisfactory accuracy. And there we stumble additionally over another kind of trap: The definition of a “polytropic change of state” has altered over time. Nowadays most thermodynamic books define it with the formula pv^{n}=constant. (p= pressure v = specific volume and n= polytropic exponent). Let us call this the polytropic exponential definition. Many books do not even mention that this formula only applies for ideal gases and that it can lead to large mistakes with pronounced real gas behavior. Nevertheless with this definition another polytropic efficiency formula has been generated, which is sometimes recommended as a good approximation even with real gas behavior. Calculations with this formula are designated with “PEF” in Figure 1 on the following page.

The author was faced in recent years with several cycle analysis problems with pronounced real gas behavior while the result should comply with the requirements of the classic definition of the polytropic efficiency was required. This was motivation enough for an in-depth inquiry for appropriate methods. The result was an abundance of methods and all of them based on the (questionable) polytropic exponent definition with some additional corrections. No simple satisfying solution applicable to any (empirically defined) equation of state was found.

Looking into the oldest textbook references on definitions of polytropic changes of state from Zeuner (1905) and Stodola (1922) brought enlightenment. Both authors introduced the adiabatic and polytropic change of state by simply keeping the expression vdp/dh constant. This corresponds to the “classic definition” as indicated above. Of course this definition is incremental and both Zeuner and Stodola solved the related integration problem by assuming the ideal gas equation (of state) pv=RT and the caloric equation dh=cpdT and after some derivations they both ended with the above mentioned exponential definition.

But in our computer time a recursive algorithm for integrating the incremental classic definition can easily be derived in a manner, that any (and also empirically defined) equations of state v(p,T) can be used. This has been done in the paper IMECE2014-36202. Following the classic definition this method is named the Constant Dissipation Rate Algorithm or CDRA. There is also a simple test for the achievable convergence or accuracy included. Two sample formula comparisons shown in the paper are in Figure 1 and 2 below.

The comparison of the CDRA with two conventional formulas indicates astonishing deviations even for dry air. Figure 1 shows the deviations in terms of the calculated exit temperature according to task 2 as indicated above. Figure 2 shows the deviation in terms of the calculated polytropic efficiency according to task 1 as indicated above. More figures also for other gases are shown in the above mentioned paper.

As a conclusion the Constant Dissipation Rate Algorithm is recommended for all calculations of polytropic changes with elevated accuracy requirements but especially if high pressure or pronounced real gases are involved. All required formulas are indicated in the above mentioned paper. The author is happy to receive feedback (hans.e.wettsten@bluewin.ch) on experiences made with this method. It can also not be excluded that recursive methods like the CDRA above have already been used. But no corresponding publication has been found yet.