Acta Polytechnica CTU Proceedings doi:10.14311/APP.2018.20.0065 Acta Polytechnica CTU Proceedings 20:65–72, 2018 © Czech Technical University in Prague, 2018 available online at http://ojs.cvut.cz/ojs/index.php/app ISENTROPIC EFFICIENCY OF CENTRIFUGAL COMPRESSOR WORKING WITH REAL GAS Jiří Oldřich Howden ČKD Compressors s.r.o., Klečákova 347/5, 190 00, Praha 9, Czech Republic correspondence: jiri.oldrich@howden.com, oldrich.jiri@seznam.cz Abstract. The contribution deals with calculation of isentropic efficiency and also with calculation of isentropic process of real gas or gaseous mixtures. The method is based on numerical solution of basic definitional equation of isentropic process and equation of isentropic efficiency with direct implementation of real gas equation of state (EOS). Keywords: Efficiency, isentropic, real gas, compressor. 1. Introduction Isentropic efficiency is an important parameter widely used in thermodynamic design of centrifugal compres- sor. Reversible adiabatic process identical to isen- tropic process is used as reference process. The isen- tropic process is a process to which great attention is devoted in literature. In some articles certain inaccu- racies are used repeatedly. For example dependence of heat capacities on temperature is ignored. Very often Poisson equations are used for calculation of real gases. Isentropic exponents that are generally dependent on temperature and pressure are often substituted by mean values. Adiabatic and isentropic processes are considered synonymous. These inaccuracies can gen- erate noticeable errors in calculations. The aim of this contribution is to clarify the above mentioned inaccuracies and to suggest thermodynami- cally correct procedures that are appropriate for PC’s. In the first part of contribution equation for some simple special cases is described very shortly. The second more extensive part deals with techniques for various combinations of input and output variables. All these techniques are based on behavior of real gases and are independent on real gases EOS used. 2. Isentropic efficiency Efficiency of compressor that does not exchange heat with the environment (compressor without intercool- ing or uncooled compressor section) is calculated by comparing the work that was actually spent on com- pression and the appropriate reference process. Refer- ence process is usually, depending on the compression process in the compressor, an isentropic process [1– 4], a polytropic process [2, 3, 5, 6], or a reversible isothermal process [2, 3, 7]. We will only deal with a reversible isentropic process and isentropic compres- sion in the following text. In the actual process that occurs in the compressor, losses occur in the flowing gas and, as a result of it, some part of the kinetic energy is converted into heat. This process leads to an increase in temperature and Figure 1. Isentropic (area ACDEFA) and actual work (area GBCDEFG) in T-S diagram when compressed from pressure p1 to pressure p2. entropy of compressed gas. A small part of the gener- ated heat is discharged from the compressor casing to the surroundings. The process that is going on in the actual compressor is irreversible. The final tempera- ture of the gas after its compression is always higher than temperature of gas after isentropic compression. In the diagram H-S in FIGURE 1 the reversible isen- tropic compression work is represented by the area ACDEFA and the actual work spent on compression from pressure p1 to the pressure p2 = p2a = p2s is rep- resented by the area GBCDEFG. The work actually spent is irreversible and can be divided into isentropic work, compression losses (area ABGFA) and increase of compression work due to losses (area ABCA) ∆Ha = H2a −H1. (1) Isentropic efficiency is defined as a relation between reversible isentropic work and actual work ηs = ∆Hs ∆Ha = H2s −H1 H2a −H1 . (2) Short repetition of basic principles [2, 3, 8]. • Adiabatic (isentropic) efficiency does not depend only on amount of losses. It depends on amount 65 http://dx.doi.org/10.14311/APP.2018.20.0065 http://ojs.cvut.cz/ojs/index.php/app Jiří Oldřich Acta Polytechnica CTU Proceedings Figure 2. Isentropic and actual processes in H −S diagram. of additional increase in compression work due to losses expressed by the area ACBA, too. • There are losses in gas flow in the actual process that occurs in compressor, and as a result, a part of the kinetic energy is converted into thermal energy and hence an increase in temperature and entropy of compressed gas. • At the same quality of compression mean value of additional increasing in compression work increases with increasing of total pressure. • At the same quality of compression the adiabatic efficiency will be at higher pressure ratio lower than at lower pressure ratio. • Temperature of gas after actual compression is al- ways higher than temperature after isentropic com- pression (see FIGURE 2). 3. Equation of isentrope Adiabatic process is defined by the equation dq = 0, (3) where q is heat input into system dS = dq T = 0. (4) Reversible adiabat is identical to isentrope. In irre- versible process is dS > dq T = 0. (5) Adiabat and isentrope are not identical. We rewrite equation (4) into form dS = ( ∂S ∂T ) V dT + ( ∂S ∂V ) T dV = = CV T dT + ( ∂p ∂T ) V dV = 0, (6) where p is the pressure of system and CV is heat capacity, which is derivative of internal energy with respect to temperature at constant volume CV = ( ∂U ∂T ) V . (7) That generally depends on temperature and on volume. For ideal gas the heat capacity CV = C0V depends only on temperature. Equation of isentrope is obtained by integrating the relationship (6). Its specific form depends on the chosen equation of state from which we determine (∂p/∂T)V and the chosen equation for the tempera- ture dependence of C0V . 4. Reversible process, ideal gas, constant C0V (perfect gas) This simplest case leads to the known Poisson equa- tions. Equation (6) we rewrite into form C0V T dT = − R V dV, (8) where derivative (∂p/∂T )V is found from equation of state written for one mole pV = RT. (9) After integration of Eq. (8) and after modification we receive well known Poisson equation in variables T −V TV k−1 = const., (10) where k = (C0V + R)/C 0 V . Poisson equations in vari- ables T−p and p−V we obtain by inputting of volume V or temperature T from equation of state (9) Tp 1−k k = const., (11) pV k = const. (12) 5. Reversible process, general equation of isentrope From Eq. (4) we obtain general equation of isentrope S(T,V ) = const. (13) Then the problem is reduced to standard calculation of entropy for known (chosen) equation of state and equation of temperature dependence of C0V (Figure 3). In the following text we will use the molar density ρ = 1/V instead of the volume as an independent variable. We will write the equation of the isentrope in the form S(T2,ρ2) −S(T1,ρ1) = 0, (14) where index 1 denotes inlet state and index 2 denotes end state after isentropic compression (Figure 2). 66 vol. 20/2018 Isentropic efficiency of centrifugal compressor working with real gas Figure 3. Dependence of C0p on temperature for several common used gases [9]. 6. Methods of calculation of isentropic process and isentropic efficiency Before we come to the description of the solution of each type of calculation, it will be useful to mention several methods of solving the isentropic process. • In the simplest case, we can assume that we are compressing the ideal gas. The calculation will be very easy and very inaccurate. • Entropy was often read from diagrams or tables in the past. The accuracy was limited by the size of the diagram. However, this method is completely inappropriate for calculating gaseous mixtures and cannot be used on PC. • Another of the approximate methods of calculating the isentropic process is the method based on the relations valid for the ideal gas, i.e. on the Poisson equations (10), (11) and (12). Isentropic exponent is calculated using of real gas EOS and general relationships. kp,V = − V p Cp CV ( ∂p ∂V ) T , (15) kp,T = 1 1 − p Cp ( ∂V ∂T ) p , (16) kT,V = 1 + V CV ( ∂p ∂T ) V . (17) Usually we do not know all parameters of end point of isentropic change (Figure 2). Thus we have to use value of isentropic exponent calculated for mean pressure and temperature. We estimate parameters in the point of interest from the behavior of the ideal gas. This method is suitable for PC and is appropriate for calculation of gas mixtures. Even in this case, we do not have enough accurate results. As an illustration of the inappropriate use of the relationships derived for the ideal gas in combina- tion with the isentropic exponent calculated by means of the real gas equation of state from the relation- ship (15), the calculation of the isentropic change from the relation can serve calculation of isentropic enthalpy from equation ∆Hs = p1V1 kp,V kp,V − 1  (p2 p1 )kp,V −1 kp,V − 1   . (18) There are a number of other methods described in literature but further we focus only on methods which use the solution of a system of nonlinear equations, one of which is always a chosen real gas equation of state and one the requirement for zero change of entropy in transition from state 1 to 2 (Figure 2). The following paragraphs provide algorithms of so- lution for individual types of calculations. In their design, particular emphasis was placed on working reliably and quickly. For every combination of input parameters the individual algorithm for solution of sys- tem of nonlinear equations was designed. Important parts of every algorithm are relations for estimation of the initial approximations. In all cases, the calcu- lation can be divided into a part independent of the real gas EOS and the equation for the C0p = C0p(T ) of the ideal gas and a part dependent on these equations. The suitable method for solution of this system is Newton’s method. Functions Fi that are used in the next detailed description of algorithms are shown in Table 3. To construct the Jacobi matrix of functions Fi that we need for solution by the Newton method, we need to know the partial derivatives of Fi. These derivatives are shown in Table 4. The abovementioned derivatives can also be obtained by numerical derivation of func- tions Fi. In Table 5 there are relations for calculation of entropy and enthalpy of real gas and relations for calculation of dimensionless quantities Q [10]. Any real gas EOS can be chosen for the following calculations. Dimensionless quantities Q for chosen EOS can be found in literature or can be derived from relations described in Table 5. 7. Calculations based on the real gas EOS In practical calculations, we need to find unknown parameters when we know the initial state and two parameters of discharge state. The combination of input and output parameters is shown in Table 1. In case that isentropic efficiency has value one i.e. ηs = 1 it is true p2s = p2a, T2s = T2a and ∆Hs = ∆Ha, then items 1, 2 and 3 from Table 1 show combination of input and output parameters of isentropic change of thermodynamic state see Table 2. 67 Jiří Oldřich Acta Polytechnica CTU Proceedings Input Output 1 p1, T1, p2s T2s, ∆Hs 2 p1, T1, T2s p2s, ∆Hs 3 p1, T1, ∆Hs p2s, T2s Table 1. Combination of output and input parame- ters for isentropic process. Input Output 1 p1, T1, p2a, ηs T2a, ∆Ha 2 p1, T1, p2a, ∆Ha T2a, ηs 3 p1, T1, T2a, ηs p2a, ∆Ha 4 p1, T1, T2a, ∆Ha p2a, ηs 5 p1, T1, p2a, T2a ∆Ha, ηs Table 2. Combination of output and input parameters. 7.1. Example of the solution by Newton’s method All combinations of parameters are mentioned in [4]. For one of them we will show the solution. We will compute isentropic temperature T2 = T2s and isen- tropic change in enthalpy ∆H = ∆Hs from known parameters p1, T1 and p2 (Figure 2). It is variant 1 from Table 1. In this case we assume that the isen- tropic efficiency is equal to one ηs = 1. Since this is the isentropic process, the original system of four equa- tions is now simplified to only two equations F1 = 0, F2 = 0. When we use general equation of isentropic process S1 − S2 = 0 (Figure 2) and general expression of real gas EOS we obtain the system of two nonlinear equations for two variables T2 and ρ2 F1 = S1(T1,ρ1) −S2s(T2s,ρ2s) = 0, (19) F2 = p2s ρ2sRT2s −z2s(T2s,ρ2s) = 0. (20) This system of equations will be solved by Newton’s method. Because it is known and very often used method, we will show only an matrix notation of linear equations for increments in ∆T and ∆ρ[ ∂F1 ∂T2s ∂F1 ∂ρ2s ∂F2 ∂T2s ∂F2 ∂ρ2s ] · [ ∆T ∆ρ ] = [ −F1 −F2 ] (21) Desired parameters T2 and ρ2 can be obtained from relationships T (k+1) 2s = T (k) 2s + ∆T, (22) ρ (k+1) 2s = ρ (k) 2s + ∆ρ. (23) The calculation is completed by fulfilling of the requiremen max(|δT|; |δρ|) ≤ � where δT and δρ are relative deviations and � is chosen value. Detail description of solution for this example: For inlet condition p1, T1 we determine values S1 and H1 from real gas EOS next we calculate C0V C0p = C 0 p(T1, −→x ), C0V = C 0 p −R. (24) Next step is solution of system of equations F1 = 0, F2 = 0 by using of Newton’s method. Initial approxi- mations are relations valid for ideal gas T (0) 2s = T1 ( p2s p1 )C0p −C0V C0 P , (25) ρ (0) 2s = p2s RT (0) 2s . (26) As the results we obtain T2s and ρ2s. From gen- eral form of EOS we obtain compressibility factor in point 2s (Figure 2) z2s = p2s RT2sρ2s (27) and enthalpy H2s = H2s(T2s,ρ2s,−→x ). (28) Finally we calculate change in enthalpy ∆H = H2 −H1, (29) where H2 = H2s. 7.2. Calculation of T2s, ∆Ha from known parameters p1, T1, p2a and ηs (figure 2, table 2 - item 1) We solve the system of equations F1 = 0, F2 = 0, F4 = 0 and F5 = 0 for unknown parameters T2s, ρ2s, T2a, ρ2a. Initial approximations are T (0) 2s = T1 ( p2a p1 )C0p −C0V C0 P , (30) ρ (0) 2s = p2a RT (0) 2s , (31) T (0) 2a = T (0) 2s −T1 ηs + T1, (32) ρ (0) 2a = p2a RT (0) 2a , (33) where C0p is C0p = C 0 p(T1, −→x ), C0V = C 0 p −R. (34) Change in enthalpy ∆Ha we obtain from equation ∆Ha = H2a −H1, (35) where H2a = f(T2a,ρ2a,−→x ). (36) 68 vol. 20/2018 Isentropic efficiency of centrifugal compressor working with real gas 7.3. Calculation of T2a, ηs from known parameters p1, T1, p2a and ∆Ha (figure 2, table 2 - item 2) We solve the system of four equations F1 = 0, F2 = 0, F5 = 0, F6 = 0 for unknown parameters T2s, ρ2s, T2a, ρ2a. Initial approximation T (0)2s we obtain from Eq. (30), ρ (0) 2s from Eq. (31) T (0) 2s = T1 + ∆Ha C0p (37) and ρ(0)2a from Eq. (33). Isentropic efficiency ηs we calculate from Eq. (2), where enthalpy H1 we know from calculation in initial point 1 and H2a = f(T2a,ρ2a,−→x ). 7.4. Calculation of p2a, ∆Ha from known parameters p1, T1, T2a and ηs (figure 2, table 2 - item 3) Calculation is based on solving of three equations F1 = 0, F4 = 0 and F8 = 0 for unknown parameters T2s, ρ2s and ρ2a. The first approximations are T (0) 2s = ηs(T2a −T1) + T1, (38) ρ (0) 2s = ρ1 ( T2s T1 ) C0V C0p −C 0 V , (39) ρ (0) 2a = T (0) 2s ρ (0) 2s T2a , (40) where C0p see Eq.(34). After every step of the Newton’s method the k-th approximation of the compressibility factor z(k)2a is found from real gas equation of state and then the k-th approximation of the pressure p(k)2a is calculated from equation p (k) 2a = z (k) 2a RT2aρ (k) 2a . (41) From equality p2s = p2a (see figure 2), we can calculate z (k) 2s = z (k) 2a T2aρ (k) 2a T (k) 2s ρ (k) 2s . (42) Change in enthalpy ∆Ha is determined from Eq. (1), where H2a = f(T2a,ρ2a,−→x ). 7.5. Calculation of p2a, ηs from known parameters p1, T1, T2a and ∆Ha (figure 2, table 2 - item 4) To calculate unknown parameters T2s, ρ2s and ρ2a system of three equations F1 = 0, F6 = 0 and F8 = 0 is solved. For the first approximation of temperature we choose T (0)2s = T2a. Density ρ (0) 2s we obtain from equa- tion (40), ρ(0)2a = ρ (0) 2s and z (0) 2a = z (0) 2s = 1. After every Figure 4. Ideal gas temperature T2s calculated from Eq. (30) – blue curve and real gas temperature T2s – red curve. Inlet conditions propane p1 = 550000 Pa, T1 = 293.15 K. Figure 5. Ideal gas isentropic work ∆Hs calculated from Eq. (43) – blue curve and real gas isentropic work ∆Hs – red curve. Inlet conditions propane p1 = 550000 Pa, T1 = 293.15 K. step of Newton’s method compressibility factors z(k)2a and z(k)2s are calculated similar way like in previous case i.e. by solution of real gas EOS and by using of Equations (41) a (42). Finally isentropic efficiency ηs we obtain from Eq. (2), where H1 is known from calculation of thermodynamic properties in point 1 (figure 2) and H2s = f(T2s,ρ2s,−→x ). 7.6. Calculation of ∆Ha, ηs from known parameters p1, T1, p2a and T2a (figure 2, table 2 - item 5) To calculate unknown parameters T2s, ρ2s and ρ2a system of three equations F1 = 0, F2 = 0 and F5 = 0 is solved. First approximation of temperature T (0)2s we ob- tain from Eq. (30), ρ(0)2s from Eq (31) and ρ (0) 2a from Eq. (33). Change in enthalpy ∆Ha we calculate by using of Eq. (1), where H2a = f(T2a,ρ2a,−→x ). Isentropic efficiency ηs we finally obtain from Eq. (2), where H2s = f(T2s,ρ2s,−→x ). 69 Jiří Oldřich Acta Polytechnica CTU Proceedings Figure 6. Relative deviation of isentropic work cal- culated for real and ideal gas. 8. Example As an example the calculation of temperature and isentropic work after isentropic change was chosen. In this example compression of propane from state 1 to state 2s (see Figure 2) is calculate by two methods. In the first case above described method based on the real gas is used and in the second case method based on an ideal gas. Parameters in point 1 are pressure p1 = 550000 Pa, temperature T1 = 293.15 K. The third parameter is gas pressure p2s which varies in the interval 800000 to 2500000 Pa. For calculation of real gas isentropic change the method described in paragraph 7.1 was used. This method is equivalent to method of paragraph 7.2 when ηs = 1. For calculation of temperature T2s of ideal gas the Equation (30) was for p2a = p2s = p2. Isentropic change of enthalpy was calculated from Eq. (18) in form ∆Hs = z1RT1 k k − 1 [( p2 p1 )k−1 k − 1 ] , (43) where z1 and k were calculated from BWR EOS. Re- sults of all calculations are on Figures 4, 5 and 6. 9. Conclusion The methods described in this contribution exactly solved problem of calculation of the isentropic effi- ciency for various input parameters. In case that efficiency is equal zero above described procedures are suitable for calculation of isentropic process. The above described methods advantageously use the di- mensionless variables defined in Table 5 and real gas EOS. It is evident from the work that some approximate procedures can lead to inaccurate results. Methods described here together with method de- scribed in [6] and in [7] and suitable real gas EOS form the basic tools for thermodynamic calculations of compressors or others flow machines. List of symbols T thermodynamic temperature [K] p pressure [Pa] R molar gas constant [J/(mol K)] z compressibility factor [–] ρ molar density [mol/m3] V molar volume [m3/mol] k isentropic exponent [–] CV molar heat capacity at constant volume [J/(mol K)] Cp molar heat capacity at constant pressure [J/(mol K)] H molar enthalpy [J/(mol)] S molar entropy [J/(mol K)] −→x vector of molar fractions of gas mixture [–] η efficiency [1] T0 standard temperature [K] p0 standard pressure [Pa] EOS equation of state INDEXES: s isentropic a actual 1 inlet, input 2 discharge, output id ideal gas 0 thermodynamic quantities of ideal gas (upper index) References [1] D. A. Kouremenos, X. K. Kakatsios. The three isentropic exponents of dry steam. Forschung auf dem Gebiete des Ingenieurwesens 51:117–122, 1985. doi:10.1007/BF02558416. [2] D. Misárek. Turbokompresory (Turbocompressors). SNTL, 1963. In Czech. [3] K. H. Lüdtke. Process Centrifugal Compressors. Springer, Berlin, Heidelberg, 2004. doi:10.1007/978-3-662-09449-5_9. [4] J. Oldřich, J. Novák, A. Malijevský. Adiabatický děj v technické praxi, (Adiabatic pro-cess in technical practice). Strojírenství 40(2):69–76, 1990. In Czech. [5] J. M. Schultz. The polytropic analysis of centrifugal compressors. Journal of Engineering for Power 84:69, 1962. doi:10.1115/1.3673381. [6] J. Oldřich. Advanced polytropic calculation method of centrifugal compressor. 2010. doi:10.1115/IMECE2010-40931. [7] J. Oldřich. Calculation of the isothermal efficiency of a cen-trifugal compressor compressing the real gas. In Proceedings of conference “Turbostroje 2016”. Plzeň, Czech Republic, 2016. [8] B. Eckert. Axialkompressoren und Radialkompressoren. Springer-Verlag, Berlin, 1953. doi:10.1007/978-3-642-52720-3. [9] P. J. Linstrom, W. G. Mallard. NIST Chemistry WebBook, NIST Standard Refer-ence Database Number 69. doi:10.18434/T4D303. [10] J. P. Novák, A. Malijevský, J. Pick. Použití bezrozměrných veličin při výpočtu termo-dynamických funkcí, (Use of dimensionless quan-tities for calculating of thermodynamic functions from equations of state). Chemické listy 73:1178–82, 1979. In Czech. 70 http://dx.doi.org/10.1007/BF02558416 http://dx.doi.org/10.1007/978-3-662-09449-5_9 http://dx.doi.org/10.1115/1.3673381 http://dx.doi.org/10.1115/IMECE2010-40931 http://dx.doi.org/10.1007/978-3-642-52720-3 http://dx.doi.org/10.18434/T4D303 vol. 20/2018 Isentropic efficiency of centrifugal compressor working with real gas F1 = S1(T1,ρ1) −S2(T2s,ρ2s) F2 = p2s ρ2sRT2s −z2(T2s,ρ2s) F3 = ∆Hs − [H2(T2s,ρ2s) −H1(T1,ρ1)] F4 = ηs[H2(T2a,ρ2a) −H1(T1,ρ1)] − [H2(T2s,ρ2s)− −H1(T1,ρ1)] F5 = p2s ρ2aRT2a −z2(T2a,ρ2a) F6 = [H2(T2a,ρ2a) −H1(T1,ρ1)] − ∆Ha F7 = ηS∆Ha − [H2(T2s,ρ2s) −H1(T1,ρ1)] F8 = z2(T2a,ρ2a)T2aρ2a −z2(T2s,ρ2s)T2sρ2s Table 3. Functions Fi. ∂F1 ∂T2s = R T2s (1 + 2QU2s + QC2s) − ∑ xiC 0 pi(T2s) T2s ∂F1 ∂ρ2s = RQT2s ρ2s ∂F2 ∂T2s = − QT2s T2s ∂F2 ∂ρ2s = − Qρ2s ρ2s ∂F2 ∂T2a = ∂F2 ∂ρ2a = ∂F6 ∂T2s = ∂F6 ∂ρ2s = ∂F7 ∂ρ2a = 0 ∂F3 ∂T2s = ∂F4 ∂T2s = ∂F7 ∂T2s = −R(QT2s − 2QU2s −QC2s− −1) − ∑ xiC 0 pi(T2s) ∂F3 ∂ρ2s = ∂F4 ∂ρ2s = ∂F7 ∂ρ2s = RT2s(QT2s −Qρ2s) ρ2s ∂F4 ∂T2a = ηs[R(QT2a − 2QU2a −QC2a − 1)+ + ∑ xiC 0 pi(T2s)] ∂F4 ∂ρ2a = ηs RT2a(Qρ2a −QT2a) ρ2a ∂F1 ∂T2a = ∂F1 ∂ρ2a = ∂F5 ∂T2s = ∂F5 ∂ρ2s = 0 ∂F5 ∂T2a = − QT2a T2a ∂F5 ∂ρ2a = − Qρ2a ρ2a ∂F6 ∂T2a = R(QT2a − 2QU2a −QC2a − 1)+ + ∑ xiC 0 pi(T2s) ∂F6 ∂ρ2a = RT2a(Qρ2a −QT2a) ρ2a ∂F8 ∂T2s = −ρ2sQT2s ∂F8 ∂ρ2s = −T2sQρ2s ∂F8 ∂T2a = ρ2aQT2a ∂F8 ∂ρ2a = T2aQρ2a Table 4. Derivative of functions Fi. 71 Jiří Oldřich Acta Polytechnica CTU Proceedings z = z(T,ρ) = p RTρ S(T,ρ,−→x ) = ∑ x1 [ S0i (T0) + ∫ T T0 C0pi(T) T dT ] − −R ln zRTρ p0 −R ∑ xi ln xi + R(ln z −QF −QU ) H(T,ρ,−→x ) = ∑ x1 [ H0i (T0) + ∫ T T0 C0pi(T)dT ] + +RT(z − 1 −QU ) Qρ = z + ρ ( ∂z ∂ρ ) T QF = ∫ ρ 0 (z − 1)d(ln ρ) QT = z + T ( ∂z ∂T ) ρ QU = T ( ∂QF ∂T ) ρ QC = T 2 ( ∂2QF ∂T 2 ) ρ Table 5. Relations for calculation of entropy and enthalpy of real gas and relations for calculation of dimensionless quantities Q [10]. 72 Acta Polytechnica CTU Proceedings 20:65–72, 2018 1 Introduction 2 Isentropic efficiency 3 Equation of isentrope 4 Reversible process, ideal gas, constant CV0 (perfect gas) 5 Reversible process, general equation of isentrope 6 Methods of calculation of isentropic process and isentropic efficiency 7 Calculations based on the real gas EOS 7.1 Example of the solution by Newton's method 7.2 Calculation of T2s, Ha from known parameters p1, T1, p2a and s (figure 2, table 2 - item 1) 7.3 Calculation of T2a, s from known parameters p1, T1, p2a and Ha (figure 2, table 2 - item 2) 7.4 Calculation of p2a, Ha from known parameters p1, T1, T2a and s (figure 2, table 2 - item 3) 7.5 Calculation of p2a, s from known parameters p1, T1, T2a and Ha (figure 2, table 2 - item 4) 7.6 Calculation of Ha, s from known parameters p1, T1, p2a and T2a (figure 2, table 2 - item 5) 8 Example 9 Conclusion List of symbols References