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Fractal Representation of Exergy

  • Laboratoire Énergétique Mécanique Électromagnétisme (LEME), Ville d'Avray, France
Fractal Representation of Exergy
Yvain Canivet 1,2,*, Diogo Queiros-Condé 1,* and Lavinia Grosu 1
1Laboratoire Energétique Mécanique Electromagnétisme, Institut Universitaire Technologique de
Ville d’Avray, Université Paris-Ouest Nanterre La Défense, 50 rue de Sèvres, Ville d’Avray 92190 , France;
2Chaire Economie du Climat, Université Paris Dauphine, Palais Brongniart, 28 Place de la Bourse, Paris
75002, France
*Correspondence: (Y.C.); (D.Q.-C.);
Tel.: +33-1-40-97-48-68 (Y.C.); +33-1-40-97-58-04 (D.Q.-C.)
This paper is an extended version of our paper published in the 7th International Exergy, Energy and
Environment Symposium (2015).
Academic Editors: Morin Celine, Bernard Desmet and Fethi Aloui
Received: 2 November 2015; Accepted: 29 January 2016; Published: 6 February 2016
Abstract: We developed a geometrical model to represent the thermodynamic concepts of exergy
and anergy. The model leads to multi-scale energy lines (correlons) that we characterised by fractal
dimension and entropy analyses. A specific attention will be paid to overlapping points, rising
interesting remarks about trans-scale dynamics of heat flows.
Keywords: exergy; entropy; scale-entropy; entropic-skins
1. Introduction
During the last decades, the formalism of thermodynamics has been applied to chaotic
phenomena, fractal, multifractal and many other complex systems. A thermodynamics of chaotic
systems, fractal signals and fractal networks has emerged; its history is detailed in the book of Beck
and Schlögl [1]. These intensive works on the concept of scale associated to a miniaturization of
components in electronics, have led to a new way of thinking. The general hypothesis of continuity is
invalid and heat transfer at micro- and nano-scales exhibits fractal features (e.g., [2,3]). These studies
have readdressed the difficulty of interpreting the notion of temperature, which sometimes needs a
new definition depending on the physical system under consideration. More generally, a geometry
of thermodynamics is needed. This is the goal of recent theories such as the constructal theory [46]
which first formalised the question of scale optimisation. In this framework, an elementary optimized
configuration is reproduced at every scale, from small to large, to maximize the overall efficiency.
Another approach to this question is made by the entropic-skins theory [79], which provides a
new geometrical framework for the study of scale-dependent fractal systems in terms of diffusion
of scale-entropy in scale-space.
Complex systems, and especially heat, are fundamentally fractal in their scale structure and we
believe that new tools and models must emerge to support the reflection and look further into this
“space into space”. This paper aims to provide a tool to investigate the fractal nature of heat. To do
so, we propose to “draw” a heat flow as a multi-scale line reflecting its internal scale complexity, or
in other words, its intrinsic quality. This concept of quality is now commonly related to exergy, also
called available energy in the early literature. Indeed, it is defined as the maximum extractable work
from a given system when bringing it to equilibrium with its surrounding environment by means of
ideal processes (of heat, work and eventually mass exchanges). In exergetics, considering a certain
Entropy 2016,18, 56; doi:10.3390/e18020056
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amount of fluid receiving some heat δQfrom its environment and providing some work δWto it, the
change in its internal exergy dξis given by Gouy-Stodola equation [6]:
TδQ+δWT0Π, (1)
where Tand T0are the temperatures (expressed in Kelvin) of the fluid and its environment
respectively, and Πis the total entropy generation. The first term on the right-hand side of
Equation (1) is called heat exergy dX =θδQ, where we have introduced the Carnot factor:
T. (2)
In this paper, we restrict our attention to heat exergy but we believe that the toy model presented in
the followings may be adapted to other forms of exergy. The supplement of exergy is called anergy,
it is the part of heat that cannot be extracted and is expressed as follows:
dA =(1θ)δQ=T0
TδQ=T0dS, (3)
where we used Clausius definition of entropy in the last equality. These two antagonist components
of energy are, however, related. To picture it, let us consider a turbine activated by a heated gas.
The exergetic part of the fluid may be seen as the one directly hurting the rotor blades, while the
anergetic one helps the former to reach the blades through interactions but ultimately misses them
(i.e., the velocity vector of the corresponding particles is not properly oriented). From this point of
view, anergy may be seen as a conductor, or more precisely, a support for exergy.
The Carnot factor (Equation (2)) holds in the case of heat transferred on macro-continua (e.g.,
solids, fluids) according to Fourier’s law. In the case of thermal radiations, in the cosmic background
medium for instance, the energy flux density Iscales as T4according to Stefan–Boltzmann law.
The expression for the ratio θ=J/I, where Jis the corresponding exergy flux density, may then
depend on the emissivity of the emitting body and the wave frequency of the beam, see [10,11] for
detailed analyses on the subject. For simplicity, in the followings, we restrict our discussion to the
case of heat transferred on macro-continua, but the results and interpretations will hold as long as
θ(or equivalently θ) will be concerned. The conclusions in terms of temperature will need more
careful attention.
Despite clarifying discussions (e.g., [12,13]), entropy, and moreover the distinction between
exergy and anergy, may be puzzling. We believe that the geometrical representation proposed in
this paper will provide insights to the question of the nature of heat as a trans-scale phenomenon.
2. Description of the Toy Model
This toy model has been motivated by the idea that a pure exergetic flow (of work or heat at
T) may be pictured as a straight line (as for laminar flows or directed bursts) while a degraded
heat flow (finite temperature) would exhibit more structural features (as for turbulent flows). In
order to draw a conceptual link from microscopic behaviour to large scale observables, we build the
trajectory of a heat quantum (in the sense of a small quantity of heat) placed on a plane lattice and
acting on a hypothetical piston by means of probabilistic evolution rules. The resulting large scale
path is called a correlon in the following.
To build up the correlon corresponding to a given heat load, we assimilate the energy content
Qto the total movement capacity of the heat quantum. We make this analogy in defining NQQ,
the total number of elementary steps at disposal to push the hypothetical piston. The quantum is
originally placed on the left side of a plane, at coordinate (i=0, j=0), right behind the hypothetical
piston. Step by step, the quantum may push the piston by one increment to the right (i,j)(i+1, j),
or diffuse on the side (i,j)(i,j±1), see Figure 1. In doing so, the quantum gives out the quantity
Entropy 2016,18, 56 3 of 9
q=Q/NQto every crossed lattice point to heat it up and will stop after the allowed NQsteps. The
distance between two lattice points lchave been arbitrarily set to 1. From a different point of view,
one can consider a heat source providing quanta of value q=Q/NQone after another, to heat up the
plane. In this picture, each heat quantum creates a displacement equal to lc, the minimal distance (i.e.,
between two lattice points). When the quantum is pure exergetic, the displacement is longitudinal
(i.e., going to the right if the heat source is on the left side on the plane). Conversely, if the quantum
is pure anergetic, the displacement is lateral.
p=θ+3(i+1, j)
(a) (b)
Figure 1. Algorithm for the building of a correlon (a) and diagram of a typical trajectory (b).
(a) Probabilities of diffusion of the heat quantum for j>0 (pout and pback are inverted otherwise);
(b) Diagram of a correlon obtain for θ=0.15 and NQ=60.
Since exergy is the available part of energy, the Carnot factor (Equation (2)) is taken to define the
probability pto push the piston further to the right:
p=θ. (4)
While the forward movement is driven by the exergy fraction of heat, the side diffusion is assumed
constrained by its anergy part. This comes down to consider anergy as a “structural strength” that
pulls back the diffusing heat quantum to the central line (j=0). This “exergy canal” may thus be
seen as plunged in an anergetic duct of size LA= (1θ)NQlc, which restrains the side diffusion. The
induced restoring force is considered spring like: further is the quantum from the central line, stronger
is the restoring force. Hence, the pull-back probability is defined as follows:
pback =1θ
LA/2 , (5)
where lj=|j|lcis the distance of the quantum from the central line. These probabilities are
summarized in Figure 1a and a schematic example of a resulting correlon is given in Figure 1b for
θ=0.15 and NQ=60. The red line traces the path followed by the quantum and the black squares
indicate the lattice regions successively covered.
3. Correlon Characteristics
The probabilistic algorithm of Figure 1a has been implemented in Matlab and four examples are
shown in Figure 2for a given heat load at various temperatures. First of all, concerning the global
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aspect, as expected (due to probability definitions) we see that the resulting length of the correlons
(i.e., the distance by which the piston has been pushed) is given by LX=NXlc'θNQlc, the analogue
for exergy in the model, where NXis the number of step the quantum has made to the right. Also,
the higher the temperature, the lower the side diffusion, i.e., the straighter the line. Table 1gathers
the characteristic quantities of the correlons shown on Figure 2and their corresponding analogous
energetic values. The exergetic (horizontal) lengths are not exactly equal to the corresponding θNQlc
because of the probabilistic behaviour of the heat quantum. However, when averaging over several
correlons we find the average hLXi=θNQlc.
(a) (b)
(c) (d)
Figure 2. Correlons obtained for a given heat load NQat four different temperatures. The central line
(j=0) and the hypothetical piston are represented by a black dashed line and a red one respectively.
(a) Correlon for θ=0.2 and NQ=1000, NX=189, LA/lc=800; (b) Correlon for θ=0.4 and
NQ=1000, NX=377, LA/lc=600; (c) Correlon for θ=0.6 and NQ=1000, NX=583, LA/lc=400;
(d) Correlon for θ=0.8 and NQ=1000, NX=813, LA/lc=200.
Table 1. Characteristic quantities of the correlons shown on Figure 2and their corresponding
analogous energetic values.
Figure θNum. of Quantum
Movement (NQ)
Length (LX)
Duct Length
2a 0.2 1000 1000 q189 lc200 q800 lc800 q
2b 0.4 1000 1000 q377 lc400 q600 lc600 q
2c 0.6 1000 1000 q583 lc600 q400 lc400 q
2d 0.8 1000 1000 q813 lc800 q200 lc200 q
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Secondly, the trajectories clearly exhibit a fractal behaviour since they present irregularities at
every scale; from the integral length l0=LXdown to the cut-off length lc. Therefore, we measured its
fractal properties by using two methods: the box-counting method and the Richardson’s method
of the ruler. Our fractal algorithms have been tested successfully on deterministic fractals. We
determined both these fractal dimensions in the scale range [lc,l0]. The box-counting method is fairly
straightforward to apply on a plane lattice and the fractal dimension sh is computed according to
the relation: N(l)lsh , where Nis the number of box of size lneeded to cover the correlon. The
Richardson method of the ruler consists in measuring the extended length (i.e., with all its foldings)
of the correlon, the fractal dimension int is then given by the relation: L(l)l1int . The distinction
between sh and int will be discussed in the next section. In a second time, in Section 3.2, we
investigate the question of entropy for the correlons. In Statistical Physics, the entropy of complex
systems is given by Gibbs–Shannon definition:
pjln pj(6)
where kBis the Boltzmann constant and pjis the probability of existence of the microstate j. Based on
Equation (6), we introduce a definition of entropy for the correlons where pjis the probability to find
the quantum at a distance jfrom the central line. Treating the overlapping distinctly or not will rise
interesting remarks consistent with the discussion about fractal dimension.
3.1. Fractal Behaviour
We first use the box-counting method to determine the fractal dimension of the space covered by
a given correlon. As mentioned previously, this technique is not sensible to the quantum overlapping
movements. Thus, it provides a dimension for the fractal external shape of the correlon, ignoring
its internal structure. Therefore, the dimension obtained in this way will be called the correlon shape
dimension sh, in blue as a function of θon Figure 3a. Second, to grasp the internal fractal features
(in red on Figure 1b) of the correlon, we use Richardson’s method of the ruler. The dimension hence
obtained will be called the correlon internal dimension int, in red as a function of θon Figure 3a.
(a) (b)
Figure 3. Fractal dimensions as functions of θ. The curves are obtained by averaging over
50 correlons for each temperature. (a) Internal and shape fractal dimensions of correlons as functions
of θ. (b) Internal fractal dimension int as a function of θfor various heat loads NQin the scale
range [lc,l0]750.
First, the correlon internal dimension obtained with Richardson’s method. It tends to the
embedding space dimension d=2 as θ0 and to the topological one d=1 when θ1. At
large temperature, this is not surprising since we built our model on the idea that a high exergetic
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flow of heat is a directed flow (as for laminar fluid flows). At low temperature the expectations where
not so obvious, nevertheless, it appears quite natural. When θ0, p0 so the quantum is highly
constraint in the range [0, NX], with NXNQ. It will diffuse many times (of order NQ/NX1) at
the same coordinate i, similarly to a random walk on a line, before jumping to the next column i+1,
hence “filling the space” on column after another.
Regarding the shape dimension, we naturally find the same limit at high temperature, but its
value is significantly lower than the one of the internal dimension as θdecreases. Its limit when
θ0 deserves to be emphasised. In fact, as θdecreases but still higher than about 0.05, it first tends
to the limit
θ0+sh =4
3, (7)
which is often found in the study of fractal interfaces (e.g., turbulent flames [9]) as well as in many
fields of research in physics, biology, etc. See for instance the book of Nottale et al. [14]. A worth noting
example is the external frontier of a Brownian motion on a plan [15,16]. See also the §15.4 in [17] for
a discussion on this multidisciplinary fractal dimension. However, when θ0 (for θ.0.05), the
probability palso tends to zero so that the quantum is strongly constraint on the very first few lines
and the fractal dimension abruptly drops down and tends to 1.
In a given scale range, the fractal dimensions introduced do not depend on the number of
quantum movement NQ. To see this, on Figure 3b, we have plotted the internal fractal dimension
int as a function of θfor various heat loads in the scale range [lc,l0]750. A similar graph would be
found for the shape fractal dimension sh. The range of study have been taken according to the
smallest correlon at NQ=750 so that every correlon is at least equal to its integral length and the
different realisations may be compared among them.
3.2. Entropies
In order to characterise further the global shape and internal structure of the correlons, we define
their entropy using a dimensionless expression of Equation (6):
Sint =
jln pint
j, (8)
where jmin and jmax are respectively the lowest and highest distances from the central line reached by
the quantum, and pint
jis the probability to find the quantum at a distance jfrom the central line:
NQ, (9)
where ni,jis the number of points of the correlon at coordinate (i,j).Sint will be called the correlon
internal entropy. It is plotted in red as a function of θon Figure 4a. Before discussing it further, let us
introduces what will be called the correlon shape entropy (plotted in blue as a function of θon Figure 4a):
Ssh =
jln psh
jpover ln pover, (10)
NQ, (11)
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with δi,j=1 if the quantum has passed at least once at the position (i,j)and 0 otherwise; and
pover =Nover
is the probability for the quantum to pass, at least a second time, through a lattice point already
covered. Nover is the total number of overlapping movement. The definition of Ssh clearly
distinguishes the first passage of the quantum at a given position from the others, which are
accounted all together in pover . Therefore, the information embodied in Ssh is limited to the fractal
shape induced by the quantum on the background medium, hence its name.
(a) (b)
Figure 4. Two Gibbs-Shannon based entropies as functions of θ. The curves are obtained by averaging
over 50 correlons for each temperature. (a) Internal and shape entropies as functions of θand their
difference. (b) Shape entropy Ssh as a function of θfor various heat loads NQ.
Let us now discuss the properties of the curves obtained for these entropies (Figure 4a), in the
context of our analogy between energy (exergy and anergy) and heat quantum movement. First,
the correlon internal entropy tends to a maximum when θ0. This seems fairly intuitive since
θ0TT0, which means that the heat quantity represented by the correlon tends to be in
thermal equilibrium with its surroundings. The quantum movements are very erratic and exhibit
multi-scale details, with increasing complexity as θdecreases.
At high temperature, the correlon shape and internal entropies display the same behaviour and
the difference of the two curves tends to 0 as θ1. As a matter of fact, in this limit, the probability p
tends also to unity, therefore pover tends to zero so that Equations (8) and (10) tend to the same value.
However, as the temperature decreases, there are more and more overlapping movements, and thus
less “efficiency” in the quantum movements. In other words, there are more “wasted” displacements,
i.e., that have not been exploited to heat a new lattice point. The green curve of Figure 4a shows
the difference
δS=Sint Ssh. (13)
The quantity δSmay be interpreted as a loss of information from the correlon to the background
medium. The shape entropy curve presents a maximum for θ=0.5. It separates the temperature
domain in two regions composed of:
1. a strongly oriented energy type, efficient to produce work,
2. a diffusive form of energy that tends to fill the space.
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The value θ=0.5 points out the existence of a particular temperature which appears as an
optimal one
Topt =2T0, (14)
which conciliates the two antagonist behaviours previously described and minimizes the information
loss from the correlon to the background medium. This optimum is often enhanced in the literature
on thermal engine optimisation in the context of finite time thermodynamics. It can be shown that the
maximal exergetic power is obtained for an exergetic efficiency of one half, whatever the operating
engine (e.g., [18,19]).
Figure 4b displays the shape entropy as a function of θfor different heat quantities. We have not
displayed the internal entropy not to overload the graph but for every NQ, it behaves as in Figure 4a,
with an increasing maximum when θ0 as NQaugments. We see that the higher NQQthe
higher the entropy, which is consistent with Clausius definition of entropy: dS =δQ/T. All exhibit a
wide plateau roughly centred around θ=0.5.
4. Conclusions
We developed a probabilistic toy model based on the main assumption that exergy is the
“directed” part of energy, and its complementary (anergy) is the support for exergy. The heat
correlons obtained display multi-scale structure with a cut-off length equal to the lattice spacing and
an integral length related to the exergy content of the represented heat quantity. We computed their
fractal dimensions as functions of θusing two methods:
1. the box-counting method gives a dimension related to the overall fractal shape of the correlons,
regardless of their internal details;
2. the Richardson’s method, sensible to the internal diffusion, allows a characterisation of
the overlapping.
The shape fractal dimension may be regarded as a sort of interface fractal dimension. The “loss”
of dimensionality with respect to the internal fractal dimension is actually a loss of details. From
this interpretation, we adapted the Gibbs–Shannon entropy and proposed two definitions in order
to investigate further the impact of overlapping. In considering these movements all together in a
single probability pover , the correlon shape entropy as a function of θ, naturally designates an optimal
θopt =0.5 which conciliate between directed exergy flow and supporting anergy content.
Acknowledgments: We wish to thank Michel Feidt for his encouragements and enlightening discussions. We
also thank Christian de Pertuis, Pierre-André Jouvet, Jean-René Brunetière and the team of the Chaire Economie
du Climat for their support in this project.
Author Contributions: Lavinia Grosu brought her expertise on thermal engine as a support for the reflection on
the nature of exergy. Diogo Queiros-Condé and Yvain Canivet developed the theoretical model. Yvain Canivet
programmed the model and deepened the fractal and entropy analyses. All authors have read and approved the
final manuscript.
Conflicts of Interest: The authors declare no conflict of interest.
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Cet ouvrage présente et analyse la signification de la structuration fractale de la Nature et des lois qui lui sont associées, notamment les lois log-périodiques, les structurations et les croissances fractales, les architectures du hasard de type arbres d’agrégation limitée par la diffusion où l’on rencontre le nombre d’or. Il propose une approche de ces systèmes par la théorie des peaux entropiques de Diogo Queiros-Condé qui y est intégrée avec celle la relativité d’échelle de Laurent Nottale. La théorie des peaux entropiques mêlant thermodynamique et fractales permet de reconsidérer la fractalité sous de nouveaux aspects comme, la fractalité dépendante d’échelle, l’entropie spatiale d’échelle, la diffusivité d’échelle dans le temps et les dynamiques trans-échelles, les fractales paraboliques , les multifractales et les constructales introduits par Adrian Bejan, ces réseaux multi-échelles optimisés par minimisation de la production d’entropie dont les applications industrielles sont considérables. Cette approche renouvelée débouche sur de nouveaux exemples de fractals, en cosmologie, dans les flammes turbulentes, dans les interactions fluide-structure via les problèmes de vibrations. Dans ce livre pluridisciplinaire sont également décrites des fractales inattendues dans les Sciences Humaines et Sociales avec leurs applications en anthropologie, en économie, en histoire, en démographie et dynamiques urbaines, en philosophie, en géographie sans oublier le domaine des arts. Des mouvements intellectuels majeurs comme celui des Subaltern Studies (Etudes subalternes) et de l’anthropocène sont ici analysés à l’aide de cet outil trans-échelles. Les évolutions et ruptures civilisationnelles sont présentées avec ce point de vue et des visions saisissantes du passé et de l’avenir de notre monde qui en sortent. Avec les travaux de Ivan Brissaud, notre ouvrage aborde de manière originale la dynamique temporelle des technologies à travers les âges, l’histoire des sciences, l’histoire des mouvements artistiques, l’évolution de l’empire romain, de la Russie et de la Chine et, plus fascinant encore, une histoire log-périodique des ordres monastiques. C’est à ce nouvel univers des fractales, des log-périodicités et des dynamiques trans-échelles, illustré par 215 figures, que les auteurs vous convient, en montrant comment la Nature s’est fait géomètre.
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A new geometrical framework describes the phenomenon of scale- and time-dependent dimensions observed in a great variety of multiscale systems and particularly in the field of turbulence. Based on the notions of scale entropy and scale diffusivity, it leads to a diffusion equation quantifying scale entropy and thus fractal dimension in scale space and in time. For a stationary case and a uniform sink of scale entropy flux, the fractal dimension depends linearly on the scale logarithm. Here, this is experimentally verified in the case of turbulent-flames geometry. Consequences for temporal evolution of scalar passive-turbulent interfaces are investigated and compared with experimental data. Finally, some aspects of dynamics exchange between spatial scales in a turbulent jet are also studied.
A geometry called entropic skins has been recently proposed to describe and interpret the phenomenon of intermittency in fully developed turbulence. It is shown that the entropic skins model represents the geometrical counterpart of the She–Lévêque's model but with a different interpretation for the main parameters. The comparison with the multifractal approach shows that this is a new geometrical framework of intermittency.
This book deals with the various thermodynamic concepts used for the analysis of nonlinear dynamical systems. The most important invariants used to characterize chaotic systems are introduced in a way that stresses the interconnections with thermodynamics and statistical mechanics. Among the subjects treated are probabilistic aspects of chaotic dynamics, the symbolic dynamics technique, information measures, the maximum entropy principle, general thermodynamic relations, spin systems, fractals and multifractals, expansion rate and information loss, the topological pressure, transfer operator methods, repellers and escape. The more advanced chapters deal with the thermodynamic formalism for expanding maps, thermodynamic analysis of chaotic systems with several intensive parameters, and phase transitions in nonlinear dynamics.
Let B be a two dimensional Brownian motion and let the frontier of B[0; 1] be defined as the set of all points in B[0; 1] that are in the closure of the unbounded connected component of its complement. We prove that the Hausdorff dimension of the frontier equals 2(1 \Gamma ff) where ff is an exponent for Brownian motion called the two-sided disconnection exponent. In particular, using an estimate on ff due to Werner, the Hausdorff dimension is greater than 1:015. 1 Introduction Let B(t) be a Brownian motion taking values in IR 2 which we also consider as I C. Let B[0; 1] be the image of [0; 1]. For any compact A ae I C we define the frontier of A, fr(A) to be the set of points in A connected to infinity. More precisely, fr(A) is the set of x 2 A such that x is in the closure of the unbounded connected component of I C n A. Take a typical point x 2 fr(B[0; 1]). Then locally at x the Brownian motion looks like two independent Brownian motions starting at x with the condition that x is...