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1
56th 3AF International Conference FP13-AERO2022-ruscio
on Applied Aerodynamics
28 – 30 March 2022, Toulouse - France
EXERGY ANALYSIS OF UNSTEADY FLOW AROUND AN
ADIABATIC CYLINDER IN VORTEX SHEDDING CONDITION
J P Ruscio (1) S Duplaa (2) M Aguirre (3) N Binder (4)
(1) ISAE-SUPAERO, Université de Toulouse, Toulouse, France, Email: juan.ruscio@isae-supaero.fr
(2) ISAE-SUPAERO, Université de Toulouse, Toulouse, France, Email: sebastien.duplaa@isae-supaero.fr
(3) ISAE-SUPAERO, Université de Toulouse, Toulouse, France, Email: miguel-angel.aguirre@isae-supaero.fr
(4) ISAE-SUPAERO, Université de Toulouse, Toulouse, France, Email: nicolas.binder@isae-supaero.fr
ABSTRACT
This paper shows the application of the aerodynamics
exergy analysis preliminary formulation developed for
unsteady flows for adiabatic and fixed bodies under low
subsonic flow conditions (no shock-waves). The
numerical implementation for URANS CFD cases of
the presented formulation is explained. The test case is a
circular cylinder in vortex shedding regime at Reynolds
number 140, at low subsonic flow, with no movement
with respect to the reference frame. The validation of
the developed formulation is done by comparing the
drag coefficient computed by the presented unsteady
exergy formulation versus the near field drag coefficient
taken as reference.
1. INTRODUCTION
Many of the new aircraft concepts require new tools for
aerodynamics analysis due to their more integrated
propulsion configuration in which the distinction
between drag and thrust may be ill-defined. Moreover, it
is of interest to know which is the energy available
(exergy) being wasted behind these complex
configurations that could be recovered. In this context,
Arntz [1] developed a first steady exergy formulation
for aerodynamic analysis taking as basis Drela’s [2]
power balance method. Then, Aguirre [3] proposed
further decomposition into irreversible and reversible
origin of the exergy quantities based on Meheut’s work
[4]; he also adapted the formulation for wind tunnel
testing (WTT).
Unsteady effects may take place in these new
configurations and the potential for energy recovery
from an unsteady process may not be negligible. Up to
these days no exergy analysis formulation was
developed for unsteady aerodynamics, therefore the
objective of this paper is to present and test such a
formulation for a first simplified case (adiabatic and
fixed body, with no shockwaves generation, and
negligible fluid body forces).
2. UNSTEADY AERODYNAMICS EXERGY
FORMULATION DEVELOPMENT
2.1. Procedure
In order to develop the formulation, an analogy with the
process followed by Arntz [1] will be used i.e. the
variation of exergy in a control volume will be the
starting point, and then a manipulation of this
expression will be done until being able to identify the
power taken (or provided) by the body and the different
types of exergy/anergy.
2.2. Formulation development
The control volume proposed is depicted in Fig. 1,
where 𝑆 is the outer surface, 𝑆 the surface enclosing
the body. The environment parameters pressure 𝑝,
temperature 𝑇, are taken from the infinite upstream ∞.
The generic velocity vector is 𝐕= (𝑢+𝑉) 𝒙 + 𝑣 𝒚 +
𝑤 𝒛 where 𝑢, 𝑣, and 𝑤 are the velocity perturbations
with respect to the infinite velocity vector components
(𝐕 is considered aligned with the axial axis 𝒙 of the
domain). The surface normal unitary vector is 𝐧.
Figure 1. Domain definition adapted from Arntz [1].
2
The mass specific flow exergy 𝜀 is defined as:
𝜀
=
(
ℎ
−
ℎ
)
−
𝑇
(
𝑠
−
𝑠
)
=
𝛿
ℎ
−
𝑇
𝛿𝑠
(1)
Where ℎ is the specific total enthalpy, and 𝑠 is the
specific entropy. The rate of change of the exergy in the
control volume could be computed integrating its
divergence; furthermore, it is possible to express this
quantity by surface integrals by means of the divergence
theorem:
∇
∙
(
𝜌
𝜀
𝐕
)
𝑑𝑣
=
𝜌
𝜀
(
𝐕
∙
𝐧
)
𝑑𝑠
+
𝜌
𝜀
(
𝐕
∙
𝐧
)
𝑑𝑠
(2)
The objective is to insert the well-known Eqs. 3-15 into
Eq. 2 and operate mathematically in order to obtain an
equivalent expression in which it would be able to
identify physical phenomena involved and their
contribution to exergy and anergy. Here below, they are
listed the equations used in the development of the
presented formulation:
The integral unsteady energy equation (gravitational
potential neglected):
𝜕
𝜕𝑡
𝜌
𝛿𝑒
+
𝛿
𝑉
2
𝑑𝑣
+ ∇∙(𝜌 𝛿ℎ 𝐕) 𝑑𝑣
=
∇
∙
𝜏
̿
∙
𝐕
𝑑𝑣
−
∇
∙
𝐪
𝐞𝐟𝐟
𝑑𝑣
(3)
Where
( ) denotes temporal derivative, 𝑒 the internal
energy of the gas, 𝜌 the density, 𝜏̿ the effective shear
stress tensor, and 𝐪𝐞𝐟𝐟 the effective heat flux.
The material derivative of the entropy:
𝑑𝑠
𝑑𝑡
=
𝜕𝑠
𝜕𝑡
+
𝐕
∙
∇
𝑠
(4)
The Gibbs equation for a fluid element:
𝜌
𝑇
𝑑𝑠
𝑑𝑡
=
𝜌
𝑑𝑒
𝑑𝑡
+
𝑝
𝑑
𝑑𝑡
1
𝜌
(5)
Where 𝑇 is the temperature, and 𝑝 the pressure.
The energy equation for a fluid element:
𝜌
𝑑𝑒
𝑑𝑡
+
𝑝
∇
∙
𝐕
=
𝜏
̿
∙
∇
∙
𝐕
−
∇
∙
𝐪
𝐞𝐟𝐟
(6)
The mass conservation:
𝜕
𝜕𝑡
(
𝜌
)
𝑑𝑣
=
−
𝜌
(
𝐕
∙
𝐧
)
𝑑𝑠
(7)
The effective dissipation rate Φ:
Φ
=
𝜏
̿
∙
∇
∙
𝐕
(8)
Where the effective shear stress tensor 𝜏̿ is defined
by:
𝜏
̿
=
(
𝜇
+
𝜇
)
𝑆
̿
(9)
In which 𝜇 , 𝜇 , are the turbulent and laminar dynamic
viscosities respectively, and 𝑆̿ is the mean strain rate
tensor.
The effective heat flux by conduction 𝐪𝐞𝐟𝐟 is:
𝐪
𝐞𝐟𝐟
=
−
𝑘
∇
𝑇
(10)
Where the effective thermal conductivity 𝑘 is:
𝑘
=
𝑐
𝜇
𝑃
+
𝜇
𝑃
(11)
In which 𝑃 , 𝑃 , are the Prandtl and turbulent Prandtl
numbers respectively, and 𝑐 is the specific heat at
constant pressure.
The integral momentum conservation for a generic
surface:
𝜕
𝜕𝑡
(
𝜌
𝐕
)
𝑑𝑣
=
−
𝜌
𝐕
(
𝐕
∙
𝐧
)
𝑑𝑠
−
𝑝
𝐧
𝑑𝑠
+
𝜏
̿
∙
𝐧
𝑑𝑠
(12)
Also knowing that reference pressure integrated over a
closed surface is zero:
𝑝
𝐧
𝑑𝑠
=
0
(13)
The total enthalpy difference definition:
𝛿
ℎ
=
𝛿𝑒
+
𝑝
𝜌
−
𝑝
𝜌
+
𝛿
𝑉
2
(14)
3
The energy height rate 𝑊𝛤̇of the body according to [5]
[6] defined by:
𝑊
𝛤
̇
=
−
𝐅
𝐀
∙
𝐕
(15)
If 𝑊𝛤̇ is positive, the body will be able to either gain
height or speed, whereas when it is negative it will of
course be the opposite.
The factor 𝐅𝐀 present in Eq. 15 is the resulting force on
the body either thrust or drag. It is actually the nearfield
drag/thrust that should be equivalent but opposite to the
resulting force in the outer surface of the domain 𝐅𝐎
obtained by far-field analysis which could be obtained
following the logic from Toubin [7] development who
obtained a far-field drag 𝐅𝐎 for unsteady flows.
Finally, operating Eqs. 1-15, also, applying the physical
considerations of non-power (so non-porous 𝑆),
motionless body with respect to the frame of reference
(𝐕∙𝐧 along 𝑆 is zero), adiabatic surfaces (𝐪𝐞𝐟𝐟 trough
surfaces is zero), and isolating 𝑊𝛤̇ we can obtain an
exergy method based drag power for unsteady flows
which is function of exergy and anergy quantitates:
−
𝑊
𝛤
̇
=
𝐷
𝑉
=
𝑓
(
𝑒𝑥𝑒𝑟𝑔𝑦
,
𝑎𝑛𝑒𝑟𝑔𝑦
)
(16)
The explicit definition of all the terms involved in Eq.
16 is provided in section 3, its numerical
implementation in section 4, and finally the results in
section 6.
3. EXERGY / ANERGY IDENTIFICATION
The Eq. 16 involves exergy and anergy terms with their
respective unsteady parts. This section is dedicated to
describe the physical phenomena related to them.
It will be used the prefix STE for the “steady” terms,
UNS for the “unsteady” terms, and TOT to refer to
“total” i.e. the addition of both STE and UNS terms. It
is worth to mention that the prefix STE invokes
“steady” in order to identify the terms that are also
present in the steady exergy development from Arntz
[1] (this does not mean that these terms have a steady
behaviour), analogously, the prefix UNS is used to in
order to identify the terms that are attached with the
presented unsteady formulation.
3.1. Anergy quantities
Viscous dissipation anergy rate:
𝑆𝑇
𝐸
̇
=
𝑇
𝑇
Φ
𝑑𝑣
(17)
Thermal conduction anergy rate:
𝑆𝑇
𝐸
̇
=
𝑇
𝑇
𝑘
(
∇
𝑇
)
𝑑𝑣
(18)
Unsteady anergy rate:
𝑈𝑁
𝑆
̇
=
−
𝑇
𝜕
𝜕𝑡
(
𝜌
𝛿𝑠
)
𝑑𝑣
(19)
Total Anergy rate:
𝑇𝑂
𝑇
̇
=
𝑆𝑇
𝐸
̇
+
𝑆𝑇
𝐸
̇
+
𝑈𝑁
𝑆
̇
(20)
3.2. Exergy quantities
Streamwise kinetic exergy rate:
𝑆𝑇
𝐸
̇
=
1
2
𝜌
𝑢
(
𝐕
∙
𝐧
)
𝑑𝑠
(21)
Transverse kinetic exergy rate:
𝑆𝑇
𝐸
̇
=
1
2
𝜌
(
𝑣
+
𝑤
)
(
𝐕
∙
𝐧
)
𝑑𝑠
(22)
Boundary pressure work rate:
𝑆𝑇
𝐸
̇
=
(
𝑝
−
𝑝
)
[
(
𝐕
−
𝐕
)
∙
𝐧
]
𝑑𝑠
(23)
Steady mechanical exergy rate:
𝑆𝑇
𝐸
̇
=
𝑆𝑇
𝐸
̇
+
𝑆𝑇
𝐸
̇
+
𝑆𝑇
𝐸
̇
(24)
Thermal exergy rate:
𝑆𝑇
𝐸
̇
=
𝜌
𝛿𝑒
(
𝐕
∙
𝐧
)
𝑑𝑠
+
𝑝
(
𝐕
∙
𝐧
)
𝑑𝑠
−
𝑇
𝜌
𝛿𝑠
(
𝐕
∙
𝐧
)
𝑑𝑠
(25)
Unsteady energy rate:
𝑈𝑁
𝑆
̇
=
𝜕
𝜕𝑡
𝜌
𝛿𝑒
+
𝛿
𝑉
2
𝑑𝑣
(26)
Unsteady streamwise momentum rate:
𝑈𝑁
𝑆
̇
=
−
𝑉
𝜕
𝜕𝑡
(
𝜌
𝑢
)
𝑑𝑣
(27)
Unsteady thermo-compressible work rate:
𝑈𝑁
𝑆
̇
=
−
𝜌
𝑝
𝜌
(
𝐕
∙
𝐧
)
𝑑𝑠
(28)
4
Unsteady exergy rate:
𝑈𝑁
𝑆
̇
=
𝑈𝑁
𝑆
̇
+
𝑈𝑁
𝑆
̇
−
𝑈𝑁
𝑆
̇
(29)
Total exergy rate:
𝑇𝑂
𝑇
̇
=
𝑆𝑇
𝐸
̇
+
𝑆𝑇
𝐸
̇
+
𝑈𝑁
𝑆
̇
(30)
3.3. Work rate (drag power)
Finally, the drag power is obtained by summing up the
total exergy and anergy quantities:
−
𝑊
𝛤
̇
=
𝐷
𝑉
=
𝑇𝑂
𝑇
̇
+
𝑇𝑂
𝑇
̇
(31)
3.4. Exergy and anergy Coefficients
The quantities expressed on Eqs. 17-31 could be non-
dimensionalized in order to obtain their contribution to
the drag (or thrust) coefficient by the Eq. 32:
𝐶
𝑑
̇
=
𝑥
̇
1
2
𝜌
𝑉
𝐴
(32)
Where 𝑥̇ is any quantity of exergy / anergy and 𝐴 a
reference surface.
4. FORMULATION NUMERICAL
IMPLEMENTATION
The unsteady exergy formulation presented is coded in
the exergy/far-field aerodynamics analysis tool from
ISAE-SUPAERO so-called “Epsilon” [8] which works
embedded in Paraview visualization tool [9]. The time
derivatives are computed with a finite forward
differentiation scheme, therefore, 2 consecutive time
steps of the CFD solution are needed. The surface and
volume integrations are done with an Epsilon
complement tool so-called “Epsilon integration tool”
[8].
The code performs the analysis along the body’s wake
through an array of given positions of survey planes, for
a given time step.
Surface integrals are computed at the given position,
whereas the volume integration is done from the given
position towards the infinite upstream as depicted in
Fig. 2.
Figure 2. Volume integration example, stripped area
(left). Surface integration example, vertical line (right).
5. TEST CASE DESCRIPTION
The formulation presented on section 3 was tested on a
flow around a subsonic 2D circular cylinder under
vortex shedding condition at Reynolds number Re of
140. The solver is URANS, and the turbulence model
used is k-w SST. The cylinder diameter is d = 0.01 m
The mesh contains 8x10E5 elements and has a refining
close to the walls in order to obtain a y+ value of 1
which allowed using a low y+ wall treatment. Due to
the fact that the results are going to be analysed by far-
field methods, special care was taken in the wake
refinement in order to capture the vortex shedding. The
time step size is t=1.4E-4 sec which gives as result a
resolution of approximately 2000 time steps per vortex
shedding cycle. A convergence study was done in time
step size, the convergence in nearfield drag and lift was
founded with 60 time steps per cycle, but a 2000 time
steps per cycle resolution was chosen in order to obtain
a convergence in exergy terms that involves temporal
derivatives, this is in line with the resolution per cycle
used by Toubin [7] in an analogue flow case, but for a
NACA0012 profile at high angle of attack. The
simulation was let to converge until the residuals
remained constant at the end of each iteration, the drag
and lift curves entered a periodic behaviour, and the
wake information reached the farfield survey sections.
The validation of the test case was done through 2
different criteria, the match of the Strouhal number
according Roshko model [10], and the nearfield drag
average. Regarding the vortex shedding period given by
the Strouhal number the error is 1.02%. Regarding the
nearfield drag, the average drag coefficient obtained
from the simulation is 𝐶=1.313 . This value is over
imposed in a 𝐶 vs Re plot [11] which includes the
results of compilation of experimental tests [12] [13]
[14]. This could be seen in Figure 3, in which our result
is the black dot pointed by the green arrow.
Figure 3. Drag coefficient vs. Reynolds number for a
cylinder from various experimental data [12] [13] [14]
adapted from [11]. The test case simulation result is the
single black dot pointed by the arrow.
6. TEST CASE RESULTS
In this section the results from the presented unsteady
exergy formulation will be evaluated firstly
.
5
qualitatively, afterwards quantitatively on the test case
described in section 5.
6.1. Qualitative results
The Figure 4 shows the steady axial kinetic exergy
𝑆𝑇𝐸̇ (integrand of Eq. 21 in dimensionless version).
This figure is a good opportunity to show that the
exergy is a concept related to the perturbations with
respect to the dead state (far upstream state in this case)
regardless its sense. To figure that, we can focus on the
vortex pointed by the white arrow (in black marked by
Q-criterion), in which there is an exergy available (red
spots) on the top and bottom of it, this is because its
clockwise vorticity induces a speed that is added to the
freestream speed on the top, and subtracted in the
bottom, but in both cases (top and bottom) the exergy is
positive.
Figure 4. Steady axial exergy 𝑆𝑇𝐸̇ drag coefficient
field. Vortex structures marked in black by q-criterion.
It is worth to remember that the field 𝑆𝑇𝐸̇ shown in
Figure 4 is the integrand of a surface integral, so it is to
be integrated along a line perpendicular to the infinite
upstream flow at the desired position as the example on
Figure 2 (right). This is only one example of all the STE
terms.
We continue with the qualitative evaluation but this
time for the unsteady UNS terms. In this category three
terms are shown instead of one as done for the STE
terms due to the fact that they are more relevant to the
development of the presented unsteady formulation.
The Figures 5, 6, and 7 depict the unsteady anergy
𝑈𝑁𝑆̇ (Eq. 19), unsteady energy 𝑈𝑁𝑆̇ (Eq. 26), and
unsteady thermo-compressible work 𝑈𝑁𝑆̇ (Eq. 28)
respectively (the vortex structures are marked in black
by q-criterion in all of the figures). It could be
appreciated in all of the 3 unsteady fields that close to
the body their value is close to zero, this is explained by
the fact that there, the unsteadiness is much less (two
jets of flow from the top and bottom part of the
cylinder) compared to the vortex street developed a
couple of diameters downstream which is highly
unsteady. The position along the wake of the peak in
unsteadiness is coincident for the anergy and thermo-
compressible work terms (Figure 5 and Figure 7
respectively), whereas for the energy term (Figure 6) it
is positioned further upstream closer to the cylinder.
Finally, the asymptotic values (far away downstream
values), for all of these 3 fields become zero as expected
due to the dissipation of any unsteadiness.
Figure 5. Unsteady anergy component field of the drag
coefficient (𝑈𝑁𝑆̇).
Figure 6. Unsteady energy component field of the drag
coefficient (𝑈𝑁𝑆̇).
Figure 7. Unsteady thermo-compressible work
component field of the drag coefficient (𝑈𝑁𝑆̇).
6.2. Quantitative results
In this section it will be analysed the evolution of the
anergy and exergy quantities along the wake computed
by means of the presented unsteady formulation. The
quantities are already dimensionless in order to show
their contribution to the drag coefficient. The cylinder
size is depicted respecting the abscise scale by the black
dot at the origin. The nearfield drag taken as reference is
the instantaneous nearfield drag.
6
Starting with the anergy, the Figure 8 shows the
evolution of the viscous dissipation anergy rate
𝑆𝑇𝐸̇, thermal conduction anergy rate 𝑆𝑇𝐸̇,
unsteady anergy rate 𝑈𝑁𝑆̇, and total anergy rate 𝑇𝑂𝑇̇,
these quantities corresponds to Eqs. 17, 18, 19, 20
respectively. The total anergy is zero before the body
and it converges to the instant nearfield drag value in
the far downstream. All the oscillatory behaviour of the
total anergy comes from the unsteady anergy as it could
be seen in the plot. The oscillations imposed by the
unsteady anergy term are mounted on the viscous
dissipation term which is monotonously increasing to
conform finally the total anergy.
Figure 8. Evolution of the anergy and its components
for a single instant along the wake.
Regarding the exergy terms, firstly, the evolution of the
steady mechanical exergy rate 𝑆𝑇𝐸̇ (Eq. 24), and its
components i.e. the streamwise kinetic exergy rate
𝑆𝑇𝐸̇ (Eq. 21), transverse kinetic exergy rate 𝑆𝑇𝐸̇
(Eq. 22), and boundary pressure work rate 𝑆𝑇𝐸̇ (Eq.
23) could be seen in Figure 9. There is a peak of 𝑆𝑇𝐸̇
close to the cylinder which is driven mainly for the
potential effect induced by the presence of the cylinder.
It is interesting to note that the peak of 𝑆𝑇𝐸̇ is located
at roughly 5 diameters downstream the cylinder and not
just after the body as one could expect for a steady case,
it is the predominant apart from the potential effect
already described. As expected, 𝑆𝑇𝐸̇ tends to zero
while moving downstream as it is being converted into
anergy.
Figure 9. Evolution of the steady mechanical exergy and
its components along the wake for a single instant.
Continuing with exergy, now we focus on the unsteady
part. The variation along the wake of the unsteady
energy rate 𝑈𝑁𝑆̇ (Eq. 26), unsteady streamwise
momentum rate 𝑈𝑁𝑆̇ (Eq. 27), unsteady
surroundings work rate 𝑈𝑁𝑆̇ (Eq. 28), and the result
of the summation of all of them to conform the unsteady
exergy rate 𝑈𝑁𝑆̇ (Eq. 29) is depicted in Figure 10. The
𝑈𝑁𝑆̇ and 𝑈𝑁𝑆̇ are not plotted separately but in a
single curve which is the sum of them, this is done in
order to show how the summation of both of these
quantities compensates the mean value of the 𝑈𝑁𝑆̇ to
form an 𝑈𝑁𝑆̇ which oscillates around zero.
Figure 10. Evolution of the unsteady exergy along the
wake for a single instant.
To finalize the exergy quantities analysis, the Figure 11
depicts the total exergy rate 𝑇𝑂𝑇̇ (Eq. 30) which is
conformed by the 𝑆𝑇𝐸̇ (Eq. 24), 𝑆𝑇𝐸̇ (Eq. 25), and
𝑈𝑁𝑆̇ (Eq. 29). It could be appreciated how the new
unsteady terms for the exergy modify the formulation in
order to be able to detect the peak of total exergy around
7 diameters downstream the cylinder. Again, as
expected, 𝑇𝑂𝑇̇ tends to zero while moving to the
infinite downstream. There are some regions in which
the exergy becomes negative e.g. at 6 diameters
downstream, according to [15], the exergy may take
negative values when an open system is being studied.
Figure 11. Evolution of the total exergy and its
components along the wake for a single instant.
Finally, The evolution of the total quantities of exergy
and anergy along the wake for a single instant, and their
combination to conform the unsteady exergy based drag
coefficient (𝑇𝑂𝑇̇+ 𝑇𝑂𝑇̇) that is capable to match the
reference value (i.e. the instant nearfield drag
7
coefficient) is depicted in Figure 12. To summarize, this
unsteady exergy based drag coefficient is expressed on
Eq. 33. It is worth to mention that the definition
“Unsteady exergy based drag coefficient” invokes the
fact that it is computed from the unsteady exergy
analysis of the system but it is actually composed by
both, total exergy and total anergy as shown in Eq. 31
and Eq. 33.
−
𝑊
𝛤
̇
=
𝐷
𝑉
=
𝑇𝑂
𝑇
̇
+
𝑇𝑂
𝑇
̇
=
𝑆𝑇
𝐸
̇
+
𝑆𝑇
𝐸
̇
+
𝑆𝑇
𝐸
̇
+
𝑆𝑇
𝐸
̇
+
𝑈𝑁
𝑆
̇
+
𝑈𝑁
𝑆
̇
−
𝑈𝑁
𝑆
̇
+
𝑆𝑇
𝐸
̇
+
𝑆𝑇
𝐸
̇
+
𝑈𝑁
𝑆
̇
(33)
Figure 12. Evolution of the total exergy, total anergy
and the unsteady exergy based drag coefficient along
the wake for a single instant.
7. CONCLUSIONS
A preliminary unsteady aerodynamics exergy
formulation was presented, numerically implemented
and tested successfully by obtaining an instantaneous
drag value that matches the instantaneous nearfield drag
value with a maximum error of -0.76% for any position
along the wake analysis from the trailing edge of the
body up to 30 diameters downstream. The tested
formulation admits one more degree of complexity not
present in the current test case which is the
compressibility effect.
Furthermore, the unsteady exergy formulation presented
is able to detect the position and quantify the peak of
exergy into the wake which is not trivially located at the
trailing edge of the body (as for steady cases). This was
even experimentally shown in [16] by a parametric
study varying the position of an energy harvesting
dispositive in the wake of a cylinder which acts mostly
in the transverse exergy.
8. REFERENCES
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[2] DRELA, Mark. Power balance in aerodynamic
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[3] AGUIRRE, Miguel Angel. Exergy analysis of
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