Turbomachinery Aero-Thermodynamics Aero-Thermodynamics 0D-2D Alexis. Giauque1 1 Laboratoire de Mécanique des Fluides et Acoustique Ecole Centrale de Lyon Ecole Centrale Paris, January-February 2015 Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 1 / 48 And now what are the stakes and technologies? Sustainable progress Propulsion – Contra-Rotative Open Rotors (CRORs) Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 2 / 48 Table of Contents 1. Vues and Analysis surfaces Meridional view Cascade view 2. Thermodynamics Relative total/stagnation variables 3. Transformations Transformation types Transformation representation Evolution of the main variables during compression/expansion 4. Efficiency Isentropic efficiency Polytropic exponent Polytropic efficiency Link between Polytropic and isentropic efficiency for a compression Polytropic efficiency and aerodynamics Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 3 / 48 Views and Analysis surfaces In order to understand and design compressors and turbines, it is necessary to simplify the flow representation without losing the main physical features. Views and Analysis surfaces Two types of views are most commonly used: the meridional view the blade-to-blade view Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 4 / 48 Meridional view The definition of the meridional view is best understood by the next picture which represents an axial turbine stage Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 5 / 48 Meridional view – How we use it Let’s consider an axial stage. rm is the mean radius and h is the blade height. Obtaining ṁ = 2πρVz hrm Assuming uniform ρ and Vz , the mass flow rate can therefore be obtained directly by using informations available from the meridional view in the case of an axial stage Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 6 / 48 Meridional view in a centrifugal compressor In the case of a centrifugal compressor, the meridional view is a bit more complex Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 7 / 48 Difference between meridional view and meridional surface It should not be confused with the meridional surface which is a plane projection of this surface. Note that in axial machines, both view and surface are identical. Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 8 / 48 Cascade view The cascade view or blade-to-blade view is very important because it comprises all the necessary informations related to the work exchange in the machine. The definition of the cascade view is best understood by the next picture which represents an axial turbine stage Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 9 / 48 Cascade view in axial machines The following picture presents the notation used in axial machines Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 10 / 48 Cascade view in axial machines The following table gives the name of the angles used in axial machines c g i δ? γ φ β β0 Alexis Giauque (LMFA/ECL) corde pas angle d’incidence angle de déviation angle de calage angle de cambrure angle flux angle métal Turbomachinery Aero-Thermodynamics II chord pitch incidence angle deviation angle stagger angle camber angle flux angle blade angle Ecole Centrale Paris 11 / 48 Cascade view in a centrifugal compressor In the case of a centrifugal compressor, the cascade view is a bit more complex Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 12 / 48 Difference between cascade view and cascade surface It should not be confused with the meridional surface which is a plane projection of this surface. Note that in axial machines, both view and surface are identical. Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 13 / 48 Table of Contents 1. Vues and Analysis surfaces Meridional view Cascade view 2. Thermodynamics Relative total/stagnation variables 3. Transformations Transformation types Transformation representation Evolution of the main variables during compression/expansion 4. Efficiency Isentropic efficiency Polytropic exponent Polytropic efficiency Link between Polytropic and isentropic efficiency for a compression Polytropic efficiency and aerodynamics Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 14 / 48 Relative total/stagnation variables Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 15 / 48 Relative total/stagnation variables The relative velocity is defined as the velocity relative to a moving framework. ~ and Let’s consider a fluid particule in a rotor that rotates at the velocity U ~. as an absolute velocity V ~ Relative velocity W ~ is defined as The relative velocity W ~ =V ~ −U ~ W Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 16 / 48 Relative total/stagnation variables The total temperature in the relative frame is the temperature measured by a sensor that rotates with the blades so that T0R = T + W2 2Cp The total pressure and the total density in the relative frame are related to the total temperature because they are obtained by decelerating the flow isentropically. They write as γ T0R γ−1 P0R = P T 1 T0R γ−1 ρ0R = ρ T Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 17 / 48 Table of Contents 1. Vues and Analysis surfaces Meridional view Cascade view 2. Thermodynamics Relative total/stagnation variables 3. Transformations Transformation types Transformation representation Evolution of the main variables during compression/expansion 4. Efficiency Isentropic efficiency Polytropic exponent Polytropic efficiency Link between Polytropic and isentropic efficiency for a compression Polytropic efficiency and aerodynamics Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 18 / 48 Transformation types Transformations in compressors and turbines are considered adiabatic. Thermal inertia characteristic time : τth = ρCp V /hS Convective characteristic time : τcv = L/U Obtaining τcv << τth Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 19 / 48 Transformation types Without heat exchange, the following relation holds Dh0 Dwu = Dt Dt The change in total enthalpy corresponds to the effective work exchanged between the fluid and the machine. Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 20 / 48 Transformation types By using the conservation equation of the total enthalpy we have 1 Dp0 Ds0 1 Dh0 = + T0 ( ) Dt ρ0 Dt Dt The effective work exchanged between the machine and the fluid will therefore serve two purposes change the total pressure, increase the entropy. Since we know that the entropy always increases in adiabatic transformations, for a given effective work, the change in total pressure will be decreased by the creation of entropy in the system. 1 Here we recall that the entropy of the stagnation state is the same as the one of the static state Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 21 / 48 Transformation types – Rotor In a rotor, the effective work is positive for a compressor and negative for a turbine. effect of effective work in rotor wheels In a compressor rotor wheel, the total pressure will increase. Usually it is related to an increase in static pressure and in kinetic energy. In a turbine rotor, the total pressure decreases. Usually both static pressure and kinetic energy decrease at the same time. Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 22 / 48 Transformation types – Stator In a stator the effective work is zero so that the total enthalpy is 0 conserved. Since Dh Dt = 0, the total temperature is conserved. effect of effective work in stator wheels Usually, the total pressure decreases due to the existence of losses. In a compressor stator, the static pressure usually increases and the kinetic energy decreases. In a turbine stator, the static pressure usually decreases and the velocity increases. Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 23 / 48 Transformation representation – h0 -s diagram The following picture shows iso-pressure curves in the h0 -s diagram or entropy diagram. Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 24 / 48 Transformation representation – h0 -s diagram Identifying the expression of the isobare curves. T0 |(p0 =cst) = Ke s0 |(p0 =cst) /Cp Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 25 / 48 Transformation representation – h0 -s diagram Showing that isobaric curves are divergent. Obtaining ∗ [T02 − T01 ](p0 =cst) = [T02 − T01 ](p0 =cst),s0 =s ∗ e (s0 −s0 )/Cp 0 Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 26 / 48 Transformation representation – h0 -s diagram The following picture shows iso-pressure curves in the h0 -s diagram or entropy diagram. Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 27 / 48 Transformation representation – analysis As said, the entropy diagram is useful to identify work and heat exchange. Let’s consider a compression stage. 1 represents the inlet of the rotor. Between 1 and 2, the increase of the total enthalpy is accompanied by an increase of entropy. 2 represents the outlet of the rotor. It is also the inlet of the stator. 3 represents the outlet of the stator. In the stator, the total enthalpy is constant, only the entropy increases. Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 28 / 48 Transformation representation – analysis Let’s come back to the compression in the rotor. Two cases here: In this case the available energy is limited. The compression ratio in 2 is lower than the isentropic one because of the presence of losses. Alexis Giauque (LMFA/ECL) In this case the target compression ratio is reached. Yet to do so, more work is necessary because of the losses. Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 29 / 48 Transformation representation – analysis – Compressor Summing up our findings, for a compressor Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 30 / 48 Transformation representation – analysis – Turbine Summing up our findings, for a turbine Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 31 / 48 Evolution of the main variables during compression/expansion The following table summarizes the evolution of the main variables during adiabatic compression/expansion Total enth. h0 Relative total enth. h0r Enth. h Total Pres. P0 Relative total Pres.P0r Static Pres.P Kinetic ener. V 2 /2 Total Temp.T0 Relative total Temp.T0r Static Temp.T Entropy s Alexis Giauque (LMFA/ECL) Comp. Rotor ↑ → ↑ ↑ → or ↓ ↑ ↑ ↑ → ↑ ↑ Comp. Stator → x ↑ → or ↓ x ↑ ↓ → x ↑ ↑ Turbomachinery Aero-Thermodynamics II Turb. Rotor ↓ → ↓ ↓ → or ↓ ↓ ↓ ↓ → ↓ ↑ Turb. Stator → x ↓ → or ↓ x ↓ ↑ → x ↓ ↑ Ecole Centrale Paris 32 / 48 Table of Contents 1. Vues and Analysis surfaces Meridional view Cascade view 2. Thermodynamics Relative total/stagnation variables 3. Transformations Transformation types Transformation representation Evolution of the main variables during compression/expansion 4. Efficiency Isentropic efficiency Polytropic exponent Polytropic efficiency Link between Polytropic and isentropic efficiency for a compression Polytropic efficiency and aerodynamics Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 33 / 48 Isentropic efficiency If we represent the compression and expansion on the entropy diagram, this gives us a graphical interpretation of the isentropic efficiency Compression Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Expansion Ecole Centrale Paris 34 / 48 Isentropic efficiency Isentropic efficiency The isentropic efficiency compares the actual transformation work with the work of an hypothetical isentropic transformation As we have seen the isentropic change in total enthalpy is lower than the actual one in a compression. The isentropic efficiency is therefore defined as ηc = h02is − h01 h02 − h01 higher than the actual one during an expansion. The isentropic efficiency is therefore defined as ηt = Alexis Giauque (LMFA/ECL) h02 − h01 h02is − h01 Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 35 / 48 Use isentropic efficiency Together with the pressure ratio, the isentropic efficiency is most often used to obtain the resulting stagnation temperature of a compression/expansion. Let’s imagine a compressor stage takes air in at T01 = 300K and P01 = 1bar . It’s compression ratio is Π = 1.5 and it’s isentropic efficiency is ηc = 0.8. γ = 1.4 What will be the resulting stagnation pressure and temperature ? Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 36 / 48 Use isentropic efficiency Together with the pressure ratio, the isentropic efficiency is most often used to obtain the resulting stagnation temperature of a compression/expansion. Let’s imagine a compressor stage takes air in at T01 = 300K and P01 = 1bar . It’s compression ratio is Π = 1.5 and it’s isentropic efficiency is ηc = 0.8. γ = 1.4 What will be the resulting stagnation pressure and temperature ? How to use isentropic efficiency? p02 = T02is T01 = ηc = T02 = Πp01 = 1.5bar γ−1 γ γ−1 P02 =Π γ P01 T02is − T01 T02 − T01 T02is − T01 T01 + ηc Alexis Giauque (LMFA/ECL) T02 = T02 = T02is /T01 − 1 T01 1 + ηc ! γ−1 Π γ −1 T01 1 + ηc T02 ≈ 346K T02is ≈ 337K Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 36 / 48 Why isentropic efficiency is not the ultimate tool? As we have seen from the previous slides, the isentropic efficiency is quite useful. But imagine that you have two compressor stages with different pressure ratio that you want to compare. Compressor stage 1 has a pressure ratio of Π1 and compressor stage 2 has a pressure ratio of Π2 > Π1 . Obtaining ∆s = Cp ln 1 + Alexis Giauque (LMFA/ECL) Π γ−1 γ ηc −1 ! − r ∗ ln(Π) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 37 / 48 Polytropic efficiency Because the entropy creation is a complex function of Π, the isentropic efficiency ηc will also depend on Π even if the relative mechanical dissipation (aerodynamic quality) is the same for the two stages. We therefore need a tool to compare the aerodynamic quality of a compression stage without the interference of the pressure ratio it provides. Such a tool is the polytropic efficiency ηp . Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 38 / 48 Polytropic exponent The polytropic exponent is defined through the following relation p0 = constant ρn0 n is the polytropic exponent and describes the type of transformation that the fluid undergoes. We recall that in case of an isentropic transformation, we have p0 = constant ργ0 Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 39 / 48 Polytropic efficiency for a compression Let’s define the polytropic efficiency. Let’s discretize the transformation into infinitesimal steps as below Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 40 / 48 Polytropic efficiency for a compression The polytropic efficiency of a transformation is defined as P limi−>∞ i ∆h0i is ηp = ∆h0 P From the previous sketch, one can see that ∆h0is ≤ i ∆h0i is ≤ ∆h0 so that ηp ≥ ηc . Using Gibbs equation, the previous expression for the polytropic efficiency can be rewritten as follows: R2 R2 dp0 /ρ0 1 dh0is ηp = R 2 ηp = 1R 2 1 dh0 1 dh0 Polytropic efficiency for a compression The polytropic Refficiency therefore compares the work R 2used to change the 2 total pressure ( 1 dp0 /ρ0 ) to the actual work used ( 1 dh0 ) Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 41 / 48 Polytropic exponent for a compression Using the definition of the polytropic efficiency for an infinitesimal T02 transformation to express the ratio T 01 Obtaining T02 T01 Alexis Giauque (LMFA/ECL) = Π γ−1 ηp γ Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 42 / 48 Link between Polytropic efficiency and polytropic exponent for a compression Obtaining n−1 γ−1 = ηp γ n Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 43 / 48 Link between Polytropic and isentropic efficiency for a compression Obtaining ηc = γ−1 γ γ−1 ηp γ Π Π Alexis Giauque (LMFA/ECL) −1 −1 Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 44 / 48 Link between Polytropic and isentropic efficiency for a compression The following picture presents in the case of a compressor the evolution of the isentropic efficiency with the pressure ratio for different values of the polytropic efficiency. As one can see the isentropic efficiency is always smaller than the polytropic one. The difference between the two efficiencies increases with the pressure ratio. Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 45 / 48 Polytropic efficiency and aerodynamics for a compression Let’s now come back to our initial problem of two compressor stages having different pressure ratios. How do we compare their aerodynamic quality? Obtaining ∆s = Alexis Giauque (LMFA/ECL) 1 − ηp r ∗ ln (Π) ηp Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 46 / 48 Polytropic efficiency and aerodynamics for a compression Obtaining ηp = Alexis Giauque (LMFA/ECL) dwu − dwd dwu Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 47 / 48 Polytropic efficiency for a compression polytropic efficiency The polytropic efficiency of a transformation can also be defined as ηp = dwu − dwd dwu where dwu is the infinitesimal effective work and dwd is the elementary mechanical dissipation. We see from this expression that it correctly represents the effect of aerodynamics losses without introducing any thermodynamic bias. Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 48 / 48
