aerothermodynamic analysis of heat recuperation by … · the exhaust jet of a modern turbofan...
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Proceedings of GPPS Forum 18 Global Power and Propulsion Society
Montreal, 7th-9
th May 2018
www.gpps.global
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License CC-BY-NC 4.0
GPPS-2018-0143
AEROTHERMODYNAMIC ANALYSIS OF HEAT RECUPERATION BY A NOZZLE-INTEGRATED THERMOELECTRIC GENERATOR
IN A TURBOFAN
Dragan Kožulović Department Automotive and
Aerospace Hamburg University of
Applied Sciences dragan.kozulovic@haw-
hamburg.de Hamburg, Germany
Christoph Bode Institute of Jet Propulsion and
Turbomachinery Technische Universität
Braunschweig [email protected]
Braunschweig, Germany
Jens Friedrichs Institute of Jet Propulsion and
Turbomachinery Technische Universität
Braunschweig [email protected]
Braunschweig, Germany
ABSTRACT
To harvest the waste energy from the engine exhaust, a
thermoelectric generator is applied to the core nozzle of a
turbofan engine. In this way, a heat flux is generated from
the inner to the outer side of the core nozzle. To assess the
corresponding effect to the boundary layers at both sides and
to the engine performance, RANS-simulations with
prescribed heat flux at both core nozzle surfaces are
conducted. For the cruise and top of climb conditions, heat
flux of 30 kW and for the takeoff of 45 kW were applied.
Only 10% of the heat flux is converted into electrical power.
These simulations have been compared to simulations with
adiabatic walls. The velocity profiles of both boundary layers
(inner and outer side of the core nozzle) and in the shear
layer downstream the nozzle remain almost constant,
whereas density and temperature profiles change
considerably. The overall impact at thrust and propulsive
efficiency is negligible.
INTRODUCTION
The exhaust jet of a modern turbofan engine still has a
very large temperature (500 − 800 𝐾), providing an
opportunity for energy harvesting. In the present approach,
this is done by a TEG (thermoelectric generator), which is
placed in the core nozzle, cf. Figs. 1 and 2. A sketch of the
TEG and its installation in the nozzle is given in Fig. 3. The
TEG redistributes the heat from the core to the bypass flow
and directly converts a part of the corresponding heat flux
into electrical energy. Due to large velocities and large
temperature difference, remarkable amounts of heat flux can
be achieved, as demonstrated in (Bode et al. 2017a). To
assess the TEG performance in future engines, a successor of
current mid-size engines (as used for Airbus A320neo and
Boeing 737 MAX) with an entry into service in 2030 has
been selected as reference. Reasonable assumptions for the
corresponding technology improvements have been made for
both the engine and TEG parameters. The assessment of
TEG has been done on engine cycle and whole-aircraft level
in previous work, cf. (Bode et al. 2017a). Further, some
detailed study on TEG design has been conducted in
Ziolkowski et al. 2016) and (Bode et al. 2017b). Now, the
effect of heat transfer at the nozzle boundary layers and
engine performance is investigated in more detail.
Figure 1: Velocity contours in the meridional plane
As a key parameter of previous investigations, the power
density of the TEG system has to be larger than 149 𝑊/𝑘𝑔
in a favourable scenario and larger than 379 𝑊/𝑘𝑔 in an
unfavourable case, in order to provide fuel savings for a
typical flight mission. In the present work, maximum electric
power output is 4.5 𝑘𝑊 at take-off, which translates into
2
maximum TEG system weight of 30.2 𝑘𝑔 or 11.9 𝑘𝑔 per
engine, respectively. Commercially available TEGs have
power densities up to 500 𝑊/𝑘𝑔 at module level. This value
is reduced at system level, but it still fits well into the
aforementioned range, making the TEG a viable solution for
engine exergy harvesting.
Figure 2: Pressure contours and core nozzle
Figure 3: TEG sketch and installation in the nozzle
As an example for redistribution of heat from the core
exhaust to the bypass flow, the intercooled-recuperated
engine can be mentioned. The beneficial effect at the engine
efficiency is well known, however, large amounts of heat
flux (order of magnitude: 1 − 10 𝑀𝑊) are required, leading
to large and heavy heat exchangers. In contrast, the present
study is focused at much smaller amounts of heat flux
(< 50 𝑘𝑊) in combination with thermoelectric generation,
hence avoiding the excessive heat exchanger weight. On the
one hand, much smaller impact at the engine performance is
expected, due to the small amount of heat flux. On the other
hand, the heat flux is applied to the outer boundary layer of
the core nozzle, which is exposed to adverse pressure
gradients and large thickening. Heating this region, even with
small amounts of energy, may have significantly different
effects than an intercooler, which is heating the freestream
region of the bypass flow.
The research on boundary layer heating has attracted
much attention, especially for high speed flows, where
compressibility leads to considerable increase of
temperature. In general, heating leads to a reduction of skin
friction, which may be a rather small effect, but also a very
large one, depending on many parameters: Mach number,
Reynolds number, pressure gradient, turbulence intensity,
beginning of the thermal boundary layer (with respect to the
velocity boundary layer), etc. Following authors give a
comprehensive overview of corresponding methods, where
the last one by White seems to address the most elaborated
approach:
Shapiro (1954, pp. 1107-1117) only flat plates
Hoerner (1965, chapter 17, pp. 3-7)
Schlichting and Gersten (2006, pp. 616-623)
Kays and Crawford (1987, pp. 286, 305-310)
White (1999, pp. 539-562)
Boundary layer heating has been successfully applied to
different objects by (Kramer et al. 1999): a wind tunnel
model, a flying testbad and the fuselage of a business jet.
Although a considerable drag reduction was achieved, the
heating equipment was heavy and it required an extra source
of energy. This disadvantage is not present for the case of jet
engines, since the core flow already provides a large amount
of (wasted) heat.
METHODOLOGY
CFD solver
The parallel CFD-solver TRACE of DLR Cologne has
been applied, cf. Nürnberger (2004), Kügeler (2005) and
Becker et al. (2010). In this solver, the three-dimensional
Reynolds-averaged Navier-Stokes equations are numerically
solved on multi-block meshes by a finite volume technique.
Only structured meshes are used in this study. In addition to
the Navier-Stokes equations, the equation of state for ideal
gases and Fourier’s law for heat conduction are applied.
Here, air with constant heat capacity is used in the complete
computational domain.
The convective fluxes are discretized by the 2nd
order
TVD upwind scheme of Roe (1981) and the diffusive fluxes
by a central differencing scheme. The cross derivatives,
which appear when converting from curvilinear to Cartesian
frame of reference, are taken into account (full Navier-Stokes
approach). Only steady simulations have been performed,
using an implicit predictor-corrector time integration scheme.
The turbulence is modelled by the two-equation SST
model of Menter et al. (2003), where the excessive
turbulence production is prevented by limiting the production
term of the turbulent kinetic energy equation to be less than
10 times the corresponding destruction term. This model is
3
based on the Boussinesq approximation, hence the turbulent
shear stress and turbulent heat flux are calculated by eddy
viscosity and turbulent Prandtl number, the latter being
assumed constant. Due to the high Reynolds number and
high turbulence intensity in both internal engine flows, the
boundary layers have been assumed to be fully turbulent.
Hence, transition has not been modelled.
Non-reflecting boundary conditions by Saxer and Giles
(1993) have been applied to the inlet and outlet boundaries of
following types:
1) Farfield with constant stagnation pressure and
stagnation temperature as well as flight Mach number
according to the investigated operating points, together with
turbulence intensity of 0.1% and turbulent length scale of
0.1 𝑚. Inlet distortion is not present and the angle of attack
was set to zero. Besides the inlet plane above the nacelle (as
shown Fig. 1), the same farfield condition is also prescribed
to the lateral and outlet surfaces (not visible in the Figure).
2) Bypass and core inlet are prescribed as one-
dimensional radial distributions of stagnation pressure,
stagnation temperature, turbulent kinetic energy and
dissipation rate, taking into account both the hub and the
casing boundary layers. These distributions are taken from
previous RANS simulations, where the fan and the low-
pressure turbine have been investigated in more detail. The
integral values are adapted to the previous engine cycle
calculations at corresponding stations. Further, the radial and
circumferential angles are set to zero.
Boundary layers of all no-slip boundaries are resolved
with a dimensionless wall distance of the wall adjacent cells
of 𝑦+ ≈ 3, which is, for the aim of the current study, well
enough for the low-Reynolds formulation. Usually, adiabatic
condition is used for the Stokes-walls, except for the both
sides of the core nozzle (cf. Fig. 2), where constant area-
specific heat flux �̇�𝑎 is prescribed to mimic the heat
redistribution by the TEG. Due to the TEG efficiency of
10%, the �̇�𝑎 values are set to extract 100% of the required
heat flux �̇� from the inner side and to add only 90% of it to
the outer side:
�̇�𝑜 = −0.9 �̇�𝑖 (1)
with: �̇�𝑖 = �̇�𝑎,𝑖 𝑎𝑖 and: �̇�𝑜 = �̇�𝑎,𝑜 𝑎𝑜
The remaining 10% of the heat flux are converted into
electric power and are removed from the computational
domain. Following the standard convention, the extracted
heat flux is negative and the added heat flux is positive,
hence the minus sign in Eq. 1. Further, the slightly larger
outer nozzle area 𝑎𝑜 (as compared to the inner one 𝑎𝑖), is
also taken into account when setting the �̇�𝑎,𝑜 and �̇�𝑎,𝑖 values.
a) 3D view
b) Meridional plane of nozzle region
Figure 4: Mesh of the test case
Mesh
The 3D-mesh was generated by the commercial tool
ANSYS ICEM CFD 17.0, and it consists of about 0.9 million
nodes, cf. Fig. 4. The boundary layers are resolved by ca. 30
cells. Since the flow is axisymmetric, only a 45° segment
with 10 equidistant cells in circumferential direction was
meshed. The extension of the computational domain is 10
diameters in both directions (axial and radial). Previous
investigations of grid resolution show that further increase of
node number has only negligible effect at the results, in
particular at the thrust and propulsive efficiency, cf. Bode et
al. (2017b).
Simulation Quality
Depending on the test case, the convergence of steady
simulations was achieved after less than 10 000 iterations,
and it was characterized by a density residual drop of at least
three orders of magnitude (see Fig. 5) and a relative
difference of in- and outlet massflow ≤ 10−3. A typical
simulation is conducted in ca. 4 hours on a commodity
cluster with 8 CPUs (Intel 𝑖7 − 3770, each with 4 cores).
𝑈∞
nozzle region
see below
4
Figure 5: Density residual, cruise condition with heat transfer
Besides the aforementioned grid sensitivity study, two
additional measures have been done to increase the
simulation quality and credibility. First, the turbulence model
has been changed from SST to k− of Wilcox (2006),
together with the Kato-Launder (1993) fix for the stagnation
point anomaly. Second, instead of constant turbulent Prandtl
number, it has been correlated to the Peclet number,
according to Kays and Crawford (1987, p. 228). Results from
both studies show some significant deviations in boundary
layer prediction when compared to the SST model and a
constant Prandtl number. However, the difference between
adiabatic and heat transfer simulations, which is of major
interest in the present investigation, was very similar to
original solver settings.
Post-processing
As part of the post processing, the integral boundary
layer parameters are determined by integration of the flow
field perpendicular to the blade surface up to a point where
99% of the total pressure deficit is reached, cf. Kozulovic
(2007) for more details. Further, several planes have been
selected for the result analysis, cf. Figs. 1 and 2:
IS and IE: start and end of inner core nozzle side
OS and OE: start and end of outer core nozzle side
OPP: peak pressure at outer core nozzle side
W1, W2 and W3: wake planes downstream the core
nozzle
RESULTS AND DISCUSSION
Scope of Investigation
Three different operating points have been investigated:
cruise – CRZ, top of climb – TOC and take-off – TAKO.
Corresponding engine and flight conditions are known from
previous investigations. For each case, both adiabatic and
heat flux simulations have been conducted. The applied heat
fluxes are given in Tab. 1, together with the flow
temperatures in Tab. 2. As an additional study, the heat flux
has been varied for the cruise condition.
Heat flux �̇� in [kW] CRZ TOC TAKO
Inner side -30 -30 -45
Outer side 27 27 40.5
Difference
= electric power
3 3 4.5
Tab. 1: Heat flux of investigated configurations
Temperature 𝑇 [K] CRZ TOC TAKO
Core flow 605 623 693
Bypass flow 238 238 290
Tab. 2: Free-stream temperatures of investigated configurations
Due to the hollow structure of the clean nozzle (cf. Fig.
3), which consists of two thin plates with air in between, a
very small amount of heat flux will establish between the
core and bypass flow. Preliminary calculations provide a heat
flux which is less than 10% of the value with TEG’s installed
in the nozzle. For this reason, the heat flux is neglected for
the configuration without TEG, that means adiabatic walls
have been applied.
Most analysis work is done with the cruise condition:
boundary layer analysis, shear layer development and
integral values. The boundary and shear layer analysis has
also been done for the other operating points, but is not
presented here, due to very similar results as cruise
condition. For this reason, only integral values are provided
for TOC and TAKO.
Cruise – Outer Side of Core Nozzle
The development of different parameters along the outer
side is shown in Figs. 6 and 7. According to the pressure
coefficient 𝑐𝑝, this side is exposed to an adverse pressure
gradient between the TEG start position (OS) and peak
pressure position (OPP). Only a small region downstream the
OPP position shows flow acceleration. Both the adiabatic
and heat transfer simulation show almost identical pressure
distribution. In other words, the heat transfer effect is only
restricted to a thin viscous region near the wall, without
influencing the potential flow.
All other parameters show a significant influence of the
heat transfer. As expected, the skin friction coefficient 𝑐𝑓 and
momentum thickness 𝛿2 are reduced, which is in agreement
to the literature on heated walls. As a different behaviour
between 𝑐𝑓 and 𝛿2, the skin friction is reduced very quickly
from the beginning of the heated region, whereas the
momentum thickness is diverging only slowly from the
adiabatic solution. At the end of the nozzle, the momentum
thickness is reduced by 4%. In contrast, the displacement
thickness 𝛿1 is increased by 11%. The wall temperature 𝑇𝑊
and wall density 𝜌𝑊 are considerably changed by heating,
both in a very rapid way at the beginning and keeping the
difference further downstream.
0 2000 4000 6000 8000 1000010
-7
10-6
10-5
10-4
10-3
10-2
10-1
100
Residual L1
Residual max.
| |
5
Fig 6: Development of pressure, skin friction and temperature along the outer side of core nozzle
To study the effect of heating at turbulence, the
maximum turbulence intensity across the boundary layer
𝑇𝑢𝑚𝑎𝑥 is also displayed. Obviously, the turbulence is very
quickly reduced, but after reaching the minimum value of
12%, it remains almost constant and ends up with the same
value as adiabatic result. In other words, the very rapid
decrease, which is induced by heating, is suddenly stopped at
this value. In boundary layers, turbulence balance is
dominated by the source terms (production and destruction)
of the 𝑘- and -equation. These terms find a balance at the
above mentioned turbulence value, leading to the sudden
change of the trend. The reasons therefore are not understood
yet.
Fig 7: Development of different values along the outer side of core nozzle
-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6-1.10
-1.05
-1.00
-0.95
-0.90
-0.85
adiabatic
heat transfer
OS
x [m]
cp
[-]
OE
OPP
-0.1 0 0.1 0.2 0.3 0.4 0.5 0.60.000
0.002
0.004
0.006
0.008
0.010
adiabatic
heat transfer
cf[-]
x [m]
-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6200
250
300
350
400
450
adiabatic
heat transfer
TW
[K]
x [m]
-0.1 0 0.1 0.2 0.3 0.4 0.5 0.60.0
0.1
0.2
0.3
0.4
0.5
adiabatic
heat transfer
W
[kg/m3]
x [m]
-0.1 0 0.1 0.2 0.3 0.4 0.5 0.60.000
0.001
0.002
0.003
0.004
0.005
1
[m]
2[m]
x [m]
2
1
-0.1 0 0.1 0.2 0.3 0.4 0.5 0.60.0
0.1
0.2
0.3
0.4
0.5
adiabatic
heat transfer
Tumax
[-]
x [m]
6
Cruise – Inner Side of Core Nozzle
The opposite effects occur at the inner side, where the
heat is extracted from the flow, cf. Fig. 8. Only the pressure
distribution remains almost constant. Contrary to the outer
side, the inner boundary layer is accelerated, which results in
a much smaller momentum thickness. However, the increase
of momentum thickness due to heat transfer is roughly the
Fig 8: Development of different values along the inner side of core nozzle
same as the reduction at the outer side (+0.095 𝑚𝑚 vs.
−0.102 𝑚𝑚). This indicates that the beneficial and
detrimental effects at viscous losses are nearly compensated,
giving a very small overall effect at the engine performance.
Cruise – Boundary Layer Profiles
The profiles of last position OE are displayed in Fig. 9,
since the complete upstream influence is agglomerated there.
The effect of heat transfer at the velocity is almost negligible.
Hence, the change of the momentum thickness must be a
result of density change, solely. Indeed, both the density and
temperature profile are considerably altered by the heat
addition. Further, the turbulent kinetic energy 𝑡𝑘𝑒 is only
slightly reduced.
Fig 9: Boundary layer profiles, outer side
Looking at the boundary layer profiles of the inner
nozzle side (position IE), opposite observations are found, cf.
Fig. 10: density is increased, temperature is decreased and
turbulent kinetic energy is slightly increased. Only the
velocity profile is almost unchanged by the cooling, which is
consistent to the case of the heating of the outer side.
-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6-0.4
-0.2
0.0
0.2
0.4
0.6
adiabatic
heat transfer
cp
[-]
x [m]
IS IE
-0.1 0 0.1 0.2 0.3 0.4 0.5 0.60.000
0.002
0.004
0.006
0.008
0.010
cf[-]
x [m]
-0.1 0 0.1 0.2 0.3 0.4 0.5 0.60.000
0.001
0.002
0.003
0.004
0.005
1
[m]
2[m]
x [m]
2
1
0 100 200 300 400-0.01
0.00
0.01
0.02
0.03
0.04
0.05
adiabatic
heat transfer
d[m]
u [m/s]0.25 0.30 0.35 0.40 0.45
-0.01
0.00
0.01
0.02
0.03
0.04
0.05d
[m]
[kg/m3]
200 250 300 350 400-0.01
0.00
0.01
0.02
0.03
0.04
0.05d
[m]
T [K]0 100 200 300 400
-0.01
0.00
0.01
0.02
0.03
0.04
0.05d
[m]
tke [m2/s
2]
7
Fig 10: Boundary layer profiles, inner side
Cruise – Downstream development
Since the velocity profiles are not affected by the heat
transfer, the same can be observed in the downstream
development, planes W1, W2 and W3, cf. Fig. 11. Only
density and temperature show smaller gradients, according to
the corresponding boundary layer profiles. This is clearly
visible in W1, but, to a smaller degree, also in W2 and W3.
From this point of view, the heat redistribution starts the jet
mixing procedure (of some parameters) prior to the nozzle
exit.
Fig 11: Downstream development
0 100 200 300 400-0.01
0.00
0.01
0.02
0.03
0.04
0.05
adiabatic
heat transfer
d[m]
u [m/s]0.12 0.14 0.16 0.18 0.20
-0.01
0.00
0.01
0.02
0.03
0.04
0.05d
[m]
[kg/m3]
450 500 550 600 650-0.01
0.00
0.01
0.02
0.03
0.04
0.05d
[m]
T [K]0 200 400 600 800
-0.01
0.00
0.01
0.02
0.03
0.04
0.05d
[m]
tke [m2/s
2]
-100 0 100 200 300 4000.20
0.25
0.30
0.35
0.40
adiabatic
heat transfer
ux
[m/s]
r [m]
W1
W2
W3
0 0.1 0.2 0.3 0.4 0.50.20
0.25
0.30
0.35
0.40
[kg/m3]
r [m]
W1
W2
W3
200 300 400 500 600 7000.20
0.25
0.30
0.35
0.40
T [K]
r [m]
W1
W2
W3
8
Cruise – Variation of heat flux
In principal, the abovementioned phenomena are
attenuated or intensified if the heat flux is lowered or
increased, respectively. However, this mechanism is not
linear, as indicated by the outer momentum thickness in Fig.
12. The heat flux increase seems to have less impact than a
decrease of the same size. Probably a saturation takes place
at large heat transfer rates.
Fig 12: Impact of heat flux variation at momentum thickness
Integral Values As a final investigation, the engine thrust 𝐹 and the
propulsive efficiency 𝑝𝑟𝑜𝑝
are determined, cf. (Kožulović
2010). Here, the radial variation of different values is taken
into account, cf. Fig. 13. In this way, there is a contribution
of force 𝑑𝐹(𝑟) and of rate of change of kinetic energy 𝑑�̇�(𝑟)
at each radial position 𝑟. When integrated in radial direction,
they provide 𝐹 and 𝑝𝑟𝑜𝑝
:
𝐹 = ∫ 𝑑𝐹(𝑟) 𝑅
0 (2)
𝑑𝐹(𝑟) = 𝑑�̇�(𝑟)(𝑢𝑒(𝑟) − 𝑢0) + 𝑑𝐴 (𝑝𝑒(𝑟) − 𝑝0) (3)
𝑑�̇�(𝑟) = 𝜌𝑒(𝑟) 𝑢𝑒(𝑟) 𝑑𝐴 (4)
𝑑𝐴 = 2 𝜋 𝑑𝑟 (5)
𝑝𝑟𝑜𝑝
=𝑡ℎ𝑟𝑢𝑠𝑡 𝑝𝑜𝑤𝑒𝑟
𝑟𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒 𝑜𝑓 𝑘𝑖𝑛𝑒𝑡𝑖𝑐 𝑒𝑛𝑒𝑟𝑔𝑦=
𝐹 𝑢0
�̇� (6)
�̇� = ∫ 𝑑�̇�(𝑟) 𝑅
0 (7)
𝑑�̇�(𝑟) =1
2 𝑑�̇�(𝑟) [𝑢𝑒
2(𝑟) − 𝑢02] (8)
Fig 13: Engine control surface
In this derivation, the fuel mass flow �̇�𝑓 has been
neglected. Further, the integration includes only the engine
mass flow. As integration plane, a distance of roughly one
nacelle diameter downstream the engine has been selected,
since the lateral velocities were almost diminished at this
position. Further downstream positions yield very similar
results and almost the same trends.
For all investigated cases, the thrust is only slightly
increased and the propulsive efficiency is only slightly
decreased by heat transfer, cf. Tab. 3. Both effects are mainly
due to density change, which is also accounted for in the
integration. As already mentioned, the velocity distributions
are almost unchanged, hence their impact is negligible.
Operating point 𝐹 [𝑁] 𝑝𝑟𝑜𝑝
[−]
CRZ-adiab.
CRZ-heat
21910.4
21922.4
0.87207
0.87199
TOC-adiab.
TOC-heat
22257.5
22268.4
0.86422
0.86412
TAKO-adiab.
TAKO-heat
54585.8
54578.1
0.50328
0.50322
Tab. 3: Thrust and propulsive efficiency
CONCLUSIONS
By installation of TEG in a core nozzle of a turbofan, a
heat flux is generated from the core to the bypass flow. The
corresponding effects at the inner and outer boundary layers
of the core nozzle have been investigated by RANS
simulations. Following key conclusions can be drawn:
By heating the outer boundary layer, the momentum
thickness is reduced by 4%.
By extracting the heat from the inner boundary
layer, its momentum thickness is increased by
roughly the same amount as the reduction in the
outer boundary layer.
Velocity profiles remain almost constant, whereas
the density and temperature profiles are
considerably changed. This holds true for both
-0.1 0 0.1 0.2 0.3 0.4 0.5 0.60.0008
0.0012
0.0016
0.0020
0.0024
0.0028
Q'=2 kW
Q'=3 kW
Q'=4 kW
2
[m]
x [m]
𝑟
9
nozzle boundary layers as well as for the shear layer
downstream the nozzle.
Turbulent kinetic energy is slightly reduced by heat
addition and slightly increased by heat extraction.
Thrust is slightly increased and propulsive
efficiency is slightly decreased by the heat transfer.
Both parameters show a change of the 4th
significant
figure, indicating an almost negligible influence.
NOMENCLATURE
Latin Symbols
𝐴 area
𝑐𝑓 =𝜏𝑊𝜌
2𝑢0
2 skin friction coefficient
𝑐𝑝 =𝑝−𝑝0𝜌
2𝑢0
2 pressure coefficient
𝐹 thrust
𝑓 fuel
�̇� rate of change of kinetic energy
�̇� massflow
𝑝 pressure
�̇� heat flux
�̇� =�̇�
𝑎 area-specific heat flux
𝑅 radius
𝑟 radial coordinate
𝑇 temperature
𝑇𝑢 turbulence intensity
𝑡𝑘𝑒 turbulent kinetic energy
𝑡𝑠𝑓𝑐 thrust-specific fuel consumption
𝑢 velocity
𝑊 wall
𝑥, 𝑦, 𝑧 Cartesian coordinates
Greek Symbols
𝛿1 = ∫ (1 −𝜌(𝑦) 𝑢(𝑦)
𝜌0 𝑢0)
∞
0𝑑𝑦 displacement thickness
𝛿2 = ∫𝜌(𝑦) 𝑢(𝑦)
𝜌0 𝑢0(1 −
𝑢(𝑦)
𝑢0)
∞
0𝑑𝑦 momentum thickness
𝑝𝑟𝑜𝑝
propulsive efficiency
𝜌 density
Abbreviations
CRZ cruise
RANS Reynolds-averaged Navier-Stokes (equations)
TSFC thrust-specific fuel consumption
TAKO take-off
TEG thermoelectric generator
TERA Thermoelectric Energy Recuperation for Aviation
TOC top-of-climb
TRACE Turbomachinery Research Aerodynamic
Computational Environment
ACKNOWLEDGMENTS
Financial support from the German Federal Ministry of
Economic Affairs and Energy is gratefully acknowledged for
funding the TERA-project (grant: 20E1303). We thank the
colleagues from this project for their support and fruitful
discussions: Dr. J. Sieber from MTU Aero Engines, F.
Ahrendts from TU Braunschweig, Dr. P. Ziolkowski from
DLR Cologne and Dr. K.-D. Büchter from Bauhaus
Luftfahrt. Further, the TRACE software package was
provided by the DLR-Institute of Propulsion Technology,
Cologne. We highly appreciate the outstanding TRACE-
support by Dr. E. Kügeler and his colleagues.
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