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
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And now what are the stakes and technologies?
Sustainable progress
Propulsion – Contra-Rotative Open Rotors (CRORs)
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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
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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
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Meridional view
The definition of the meridional view is best understood by the next
picture which represents an axial turbine stage
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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
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Meridional view in a centrifugal compressor
In the case of a centrifugal compressor, the meridional view is a bit more
complex
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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.
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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
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Cascade view in axial machines
The following picture presents the notation used in axial machines
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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
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corde
pas
angle d’incidence
angle de déviation
angle de calage
angle de cambrure
angle flux
angle métal
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chord
pitch
incidence angle
deviation angle
stagger angle
camber angle
flux angle
blade angle
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Cascade view in a centrifugal compressor
In the case of a centrifugal compressor, the cascade view is a bit more
complex
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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.
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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
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Relative total/stagnation variables
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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
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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
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Turbomachinery Aero-Thermodynamics II
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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
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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
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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.
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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
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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.
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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.
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Transformation representation – h0 -s diagram
The following picture shows iso-pressure curves in the h0 -s diagram or
entropy diagram.
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Transformation representation – h0 -s diagram
Identifying the expression of the isobare curves.
T0 |(p0 =cst) = Ke s0 |(p0 =cst) /Cp
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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
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Transformation representation – h0 -s diagram
The following picture shows iso-pressure curves in the h0 -s diagram or
entropy diagram.
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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.
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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.
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Transformation representation – analysis – Compressor
Summing up our findings, for a compressor
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Transformation representation – analysis – Turbine
Summing up our findings, for a turbine
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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
↓
↑
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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
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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
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Expansion
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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
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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 ?
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Turbomachinery Aero-Thermodynamics II
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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
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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)
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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 .
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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
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Polytropic efficiency for a compression
Let’s define the polytropic efficiency. Let’s discretize the transformation
into infinitesimal steps as below
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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 )
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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
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Link between Polytropic efficiency and polytropic exponent
for a compression
Obtaining
n−1
γ−1
=
ηp γ
n
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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
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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.
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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
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Polytropic efficiency and aerodynamics for a compression
Obtaining
ηp =
Alexis Giauque (LMFA/ECL)
dwu − dwd
dwu
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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.
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