Turbomachinery Aero-Thermodynamics Aero-Thermodynamics 0D-2D Alexis. Giauque 1 1 Laboratoire de M´ ecanique 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
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3. TransformationsTransformation typesTransformation representationEvolution of the main variables during compression/expansion
4. EfficiencyIsentropic efficiencyPolytropic exponentPolytropic efficiencyLink between Polytropic and isentropic efficiency for a compressionPolytropic 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 necessaryto simplify the flow representation without losing the main physicalfeatures.
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 nextpicture 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
m = 2πρVzhrm
Assuming uniform ρ and Vz , the mass flow rate can therefore be obtaineddirectly by using informations available from the meridional view in thecase 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 morecomplex
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Difference between meridional view and meridional surface
It should not be confused with the meridional surface which is a planeprojection of this surface. Note that in axial machines, both view andsurface 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 itcomprises all the necessary informations related to the work exchange inthe machine.The definition of the cascade view is best understood by the next picturewhich 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 corde chord
g pas pitch
i angle d’incidence incidence angle
δ? angle de deviation deviation angle
γ angle de calage stagger angle
φ angle de cambrure camber angle
β angle flux flux angle
β′ angle metal blade angle
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 11 / 48
Cascade view in a centrifugal compressor
In the case of a centrifugal compressor, the cascade view is a bit morecomplex
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 planeprojection of this surface. Note that in axial machines, both view andsurface are identical.
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 13 / 48
Table of Contents
1. Vues and Analysis surfacesMeridional viewCascade view
3. TransformationsTransformation typesTransformation representationEvolution of the main variables during compression/expansion
4. EfficiencyIsentropic efficiencyPolytropic exponentPolytropic efficiencyLink between Polytropic and isentropic efficiency for a compressionPolytropic 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 movingframework.Let’s consider a fluid particule in a rotor that rotates at the velocity ~U andas an absolute velocity ~V .
Relative velocity ~W
The relative velocity ~W is defined as
~W = ~V − ~U
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 measuredby a sensor that rotates with the blades so that
T0R = T +W 2
2Cp
The total pressure and the total density in the relative frame are related tothe total temperature because they are obtained by decelerating the flowisentropically. They write as
P0R = P
(T0R
T
) γγ−1
ρ0R = ρ
(T0R
T
) 1γ−1
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 17 / 48
Table of Contents
1. Vues and Analysis surfacesMeridional viewCascade view
3. TransformationsTransformation typesTransformation representationEvolution of the main variables during compression/expansion
4. EfficiencyIsentropic efficiencyPolytropic exponentPolytropic efficiencyLink between Polytropic and isentropic efficiency for a compressionPolytropic 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 = ρCpV /hSConvective 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
Dt=
Dwu
Dt
The change in total enthalpy corresponds to the effective work exchangedbetween 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
Dh0
Dt=
1
ρ0
Dp0
Dt+ T0
Ds0
Dt(1)
The effective work exchanged between the machine and the fluid willtherefore serve two purposes
change the total pressure,
increase the entropy.
Since we know that the entropy always increases in adiabatictransformations, for a given effective work, the change in total pressurewill be decreased by the creation of entropy in the system.
1Here we recall that the entropy of the stagnation state is the same as the one of thestatic 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 aturbine.
effect of effective work in rotor wheels
In a compressor rotor wheel, the total pressure will increase. Usually it isrelated to an increase in static pressure and in kinetic energy.
In a turbine rotor, the total pressure decreases. Usually both staticpressure 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 isconserved. Since Dh0
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 kineticenergy decreases.In a turbine stator, the static pressure usually decreases and the velocityincreases.
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 orentropy diagram.
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Transformation representation – h0-s diagram
Identifying the expression of the isobare curves.
T0|(p0=cst) = Kes0|(p0=cst)/Cp
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 25 / 48
3. TransformationsTransformation typesTransformation representationEvolution of the main variables during compression/expansion
4. EfficiencyIsentropic efficiencyPolytropic exponentPolytropic efficiencyLink between Polytropic and isentropic efficiency for a compressionPolytropic 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 Expansion
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Isentropic efficiency
Isentropic efficiency
The isentropic efficiency compares the actual transformation work with thework 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 efficiencyis therefore defined as
ηc =h02is − h01
h02 − h01
higher than the actual one during an expansion. The isentropicefficiency is therefore defined as
ηt =h02 − h01
h02is − h01
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 35 / 48
Use isentropic efficiency
Together with the pressure ratio, the isentropic efficiency is most oftenused to obtain the resulting stagnation temperature of acompression/expansion.
Let’s imagine a compressor stage takes air in at T01 = 300K and P01 = 1bar . It’scompression 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 oftenused to obtain the resulting stagnation temperature of acompression/expansion.
Let’s imagine a compressor stage takes air in at T01 = 300K and P01 = 1bar . It’scompression ratio is Π = 1.5 and it’s isentropic efficiency is ηc = 0.8. γ = 1.4What will be the resulting stagnation pressure and temperature ?
How to use isentropic efficiency?
p02 = Πp01 = 1.5bar
T02is
T01=
(P02
P01
) γ−1γ
= Πγ−1γ
ηc =T02is − T01
T02 − T01
T02 = T01 +T02is − T01
ηc
T02 = T01
(1 +
T02is/T01 − 1
ηc
)T02 = T01
(1 +
Πγ−1γ − 1
ηc
)T02 ≈ 346K
T02is ≈ 337K
Alexis Giauque (LMFA/ECL) 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 quiteuseful.But imagine that you have two compressor stages with different pressureratio that you want to compare. Compressor stage 1 has a pressure ratioof Π1 and compressor stage 2 has a pressure ratio of Π2 > Π1.
Obtaining
∆s = Cp ln
(1 +
Πγ−1γ − 1
ηc
)− r ∗ ln(Π)
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 37 / 48
Polytropic efficiency
Because the entropy creation is a complex function of Π, the isentropicefficiency ηc will also depend on Π even if the relative mechanicaldissipation (aerodynamic quality) is the same for the two stages.
We therefore need a tool to compare the aerodynamic quality of acompression 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
ρn0= constant
n is the polytropic exponent and describes the type of transformation thatthe fluid undergoes.
We recall that in case of an isentropic transformation, we have
p0
ργ0= constant
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 transformationinto 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
ηp =limi−>∞
∑i ∆h0i is
∆h0
From the previous sketch, one can see that ∆h0is ≤∑
i ∆h0i is ≤ ∆h0 sothat ηp ≥ ηc .Using Gibbs equation, the previous expression for the polytropic efficiencycan be rewritten as follows:
ηp =
∫ 21 dh0is∫ 21 dh0
ηp =
∫ 21 dp0/ρ0∫ 2
1 dh0
Polytropic efficiency for a compression
The polytropic efficiency therefore compares the work used to change thetotal pressure (
∫ 21 dp0/ρ0) to the actual work used (
∫ 21 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 infinitesimaltransformation to express the ratio T02
T01
Obtaining
T02
T01= Π
(γ−1ηpγ
)
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 42 / 48
Link between Polytropic efficiency and polytropic exponentfor a compression
Obtaining
γ − 1
ηpγ=
n − 1
n
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 43 / 48
Link between Polytropic and isentropic efficiency for acompression
Obtaining
ηc =Π
(γ−1γ
)− 1
Π
(γ−1ηpγ
)− 1
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 44 / 48
Link between Polytropic and isentropic efficiency for acompression
The following picture presents in the case of a compressor the evolution ofthe isentropic efficiency with the pressure ratio for different values of thepolytropic efficiency.
As one can see the isentropic efficiency is always smaller than thepolytropic one. The difference between the two efficiencies increases withthe 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 stageshaving different pressure ratios. How do we compare theiraerodynamic quality?
Obtaining
∆s =1− ηpηp
r ∗ ln (Π)
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 46 / 48
Polytropic efficiency and aerodynamics for a compression
Obtaining
ηp =dwu − dwd
dwu
Alexis Giauque (LMFA/ECL) 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 elementarymechanical dissipation.We see from this expression that it correctly represents the effect ofaerodynamics losses without introducing any thermodynamic bias.
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics II Ecole Centrale Paris 48 / 48