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Turbomachinery Aero-ThermodynamicsAero-Thermodynamics 0D-2D

Alexis. Giauque1

1Laboratoire de Mecanique des Fluides et AcoustiqueEcole 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 progressPropulsion – Contra-Rotative Open Rotors (CRORs)


http://www.aeronewstv.com/fr/industrie/motoristes-aeronautiques/1373-quand-snecma-developpe-le-moteur-du-futur.html

Table of Contents

1. Vues and Analysis surfacesMeridional viewCascade view

2. ThermodynamicsRelative total/stagnation variables

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


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


Meridional view

The definition of the meridional view is best understood by the nextpicture which represents an axial turbine stage


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


Meridional view in a centrifugal compressor

In the case of a centrifugal compressor, the meridional view is a bit morecomplex


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.


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


Cascade view in axial machines

The following picture presents the notation used in axial machines


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


Cascade view in a centrifugal compressor

In the case of a centrifugal compressor, the cascade view is a bit morecomplex


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.


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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



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


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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



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.



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


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.


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.


Transformation representation – h0-s diagram

The following picture shows iso-pressure curves in the h0-s diagram orentropy diagram.



Identifying the expression of the isobare curves.

T0|(p0=cst) = Kes0|(p0=cst)/Cp



Showing that isobaric curves are divergent.

Obtaining

[T02 − T01](p0=cst) = [T02 − T01](p0=cst),s0=s∗0e(s0−s∗0 )/Cp



The following picture shows iso-pressure curves in the h0-s diagram orentropy diagram.


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 ofthe total enthalpy is accompaniedby 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 enthalpyis constant, only the entropyincreases.


Transformation representation – analysis

Let’s come back to the compression in the rotor. Two cases here:

In this case the available energy islimited. The compression ratio in 2 islower than the isentropic one becauseof the presence of losses.

In this case the target compressionratio is reached. Yet to do so, morework is necessary because of the losses.


Transformation representation – analysis – Compressor

Summing up our findings, for a compressor


Transformation representation – analysis – Turbine

Summing up our findings, for a turbine


Evolution of the main variables duringcompression/expansion

The following table summarizes the evolution of the main variables duringadiabatic compression/expansion

Comp. Rotor Comp. Stator Turb. Rotor Turb. Stator

Total enth. h0 ↑ → ↓ →Relative total enth. h0r → x → x

Enth. h ↑ ↑ ↓ ↓Total Pres. P0 ↑ → or ↓ ↓ → or ↓

Relative total Pres.P0r → or ↓ x → or ↓ x

Static Pres.P ↑ ↑ ↓ ↓Kinetic ener. V 2/2 ↑ ↓ ↓ ↑

Total Temp.T0 ↑ → ↓ →Relative total Temp.T0r → x → x

Static Temp.T ↑ ↑ ↓ ↓Entropy s ↑ ↑ ↑ ↑


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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




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


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 ?


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


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)


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.


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


Polytropic efficiency for a compression

Let’s define the polytropic efficiency. Let’s discretize the transformationinto infinitesimal steps as below



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


The polytropic efficiency therefore compares the work used to change thetotal pressure (

∫ 21 dp0/ρ0) to the actual work used (

∫ 21 dh0)


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

)


Link between Polytropic efficiency and polytropic exponentfor a compression

Obtaining

γ − 1

ηpγ=

n − 1

n


Link between Polytropic and isentropic efficiency for acompression

Obtaining

ηc =Π

(γ−1γ

)− 1

Π

(γ−1ηpγ

)− 1


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.


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)


Polytropic efficiency and aerodynamics for a compression

Obtaining

ηp =dwu − dwd

dwu



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.


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