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Scale hierarchy, exergy maximization and urban efficiency
Salat, S. & Bourdic, L.
Proceedings of the 2nd International Exergy, Life Cycle Assessment and Sustainability Workshop & Symposium (ELCAS-2) under the frame of the "European Cooperation in Science and Technology" and UNEP/SETAC Life Cycle Initiative
Cities are complex open systems; they are almost entirely fed by flows coming from the outside. As such, approaches based on the second law of thermodynamics and entropy maximisation that only apply to isolated physical systems fail to properly address urban complex issues. Cities are historically driven by external flows, which push them away from classical closed-systems thermodynamic equilibriums. According to the thermodynamics of dissipative systems, urban systems respond spontaneously with the emergence of structure and order and the ‘steady-states’ reached correspond to exergy maxima. Elaborating on the World Bank’s forms and flows approach, this paper investigates the relationships between exergy and urban structures, especially by considering the role of scale hierarchy within urban elements. It shows how scale hierarchic structures are the only relevant way to maximise urban exergy production, and thus cities’ ‘effectiveness’ in that they make more effective use of the resources allocated, such as energy or land. Authors show that structures with appropriate scale hierarchies are the key for efficient urban planning, be it for energy use or for economic value creation. Starting from Alexander’s famous thought on cities and trees, authors eventually scrutinize the concepts of efficiency and resilience within urban networks.
Urban world is experiencing a never before seen growth. When put into perspective with climate change mitigation, fossil energy scarcity and poverty issues, this growth highlights the crucial need for more urban efficiency, be it on the energy or socio-economic side. A World Bank initiative  insists on the role of synergies and interdependences within urban structures, of a coordination between urban forms and urban flows. Taking this coordination between forms and flows as a starting point, this paper aims to investigate the relationships between urban structures, energy use and economic value creation. At the crossing between thermodynamics, industrial ecology and urban morphology, this paper summarises the lessons that can be drawn from each field and applied to urban analysis. It aims though to a pragmatic objective, keeping in mind that urban development is in the end primarily decided by policy-makers and urban authorities. In the first part, authors present a brief overview of thermodynamics, stressing the difficulties in applying it to complex open systems like cities. Investigating these issues further, authors highlight the relationships between thermodynamic aspects and organisation within complex systems. Discussing the influence of order and complexity on urban efficiency, this paper eventually stresses the role of scale hierarchic structures within cities.
The hard-to-grasp concept of entropy for urban development
Numerous papers have been investigating cities through the prism of thermodynamics, through entropy or exergy analysis. Some have been basing urban population models on entropymaximising ideas  and put it into perspective with urban fractal structures [3,4]. Salingaros and West argue their size-distribution rule for 'living cities' based on entropy considerations as well . Others have been introducing entropy [6,7] or exergy  as indicators for the sustainability of urban areas. Thermodynamics certainly offers a sound framework to analyse cities and understand the relationships between urban structures and urban efficiency, be it about the energy, social or economic issues. Entropy though remains a hard-to-grasp concept and is full of slip-ups when applied to urban analysis. To make a long story short, the concept was first coined thanks to Rudolf Clausius in 1865 in the context of the second law of thermodynamics, stating that entropy can only increase in isolated systems for an arbitrary adiabatic process. Both Boltzmann and Gibbs gave further strength to the concept of entropy, the former with a statistical interpretation, the latter with the introduction of chemical potentials. These scientific breakthroughs were at the basis of the industrial revolution to come. Second law of thermodynamics became one of the universal laws, as attested by Sir Arthur Eddington's famous sentence in the 30s: 'if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation'.
But building on Onsager's work, Prigogine brought some novelty and threw some doubts on the universality of the second law. With his dissipative structure theory, he opened a new era for thermodynamics with non-equilibrium thermodynamics. This major breakthrough marked the start of both scientific and philosophical inquiries into self-organizing systems analysis and the emergence of order. The major principle arising from Prigogine's work stands a minimization of entropy production (MinEP)  to identify possible steady-states for far-from-equilibrium systems. The best example to grasp Prigogine's MinEP statement is the Bénard cells. We just remind here that the classic second law of thermodynamics for closed adiabatic systems predicts that the most probable thermodynamic state is the one with the higher entropy, in other words the greater disorder. In the case of open flow-driven systems, the most probable state though is not the one with the higher entropy. The Bénart cells are an example of an open system driven by thermal flows. Let us take a thin layer of fluid between two planes, the bottom one being heated. The second law of thermodynamics for a closed system would predict a rise in entropy, and thus a greater disorder. On the contrary, order emerge when the temperature goes beyond a critical threshold, and regular-pattern convective currents appear inside the fluid: disorder and entropy drop. This is the most common example of flow-driven organised steady-states, far from the thermodynamic equilibrium predicted by the second law. For flow-driven open systems, the steady-state (convective currents) can be far from thermodynamic equilibrium (greater disorder): it has to be a minimum in entropy production. In a nutshell, entropy and thermodynamics are no easy-to-handle concepts, given that it depends on a wide range of specific assumptions: linear versus non-linear, far from equilibrium versus close to equilibrium, closed versus flow-driven systems. That is why any attempt to apply thermodynamics to cities should be considered very carefully as long as cities are no basic, simple and decomposable thermodynamic systems. Cities cannot be restricted to a sum of simple elements, but have to be considered as extremely complex and organised systems. Following Kay's approach , cities could be described as self-organised hierarchical organism: they are open dissipative systems, far from equilibrium, non linear. As such, analyse cities with thermodynamics only is no easy task.
Furthermore, it should be kept in mind that cities efficiency eventually relies on policy-makers and local authorities decisions. Entropy is an abstract object, a creation of mind, but has no tangible and sensible value. It is moreover a pretty obscure concept for a non-scientific audience. Stakeholders may not be fully convinced and follow policy recommendations that are based on incomprehensible pundit thoughts. This paper aims to conciliate the hard-core thermodynamic science applied to cities with another much more meaningful and easy-to-handle concept: Orderand organisation within urban structures.
What is order, and how is a city organised?
Regarding thermodynamics and entropy considerations, urban form and order emergence can be considered just as the other side of the coin. Emergence of order and drop in entropy are two expressions of the exact same phenomenon, the best example being Bénard cells, often quoted by Prigogine himself: the steady-state can be described as a minEP, or with the convective patterns. When it comes to cities, order and urban structure replace the convective patterns, and are still the expression of the thermodynamic state of the complex flow-driven urban system. Special attention has thus to be paid to the relationships between urban morphology and thermodynamic consideration. But unlike thermodynamics, morphology has the main advantage to be much more meaningful to the man in the street, and pragmatic for policy recommendations. These two aspects of the same problem are especially relevant when considering urban efficiency. Which are the most efficient urban tissues regarding energy consumption, be it related to transportation, heating or cooling? Interestingly, historical cities provide an amazing scope to investigate these issues, which should not be neglected when searching new paths toward efficiency. Throughout history, following abstract religious and then social complex codes, cities have been organised and structured given the external conditions: climate, energy inputs, economic inputs, etc. Concomitantly, inefficient structures have been naturally wiped out by what could be compared to a survival of the fittest process. Historical cities are thus the superimposition of two slow but powerful phenomenon: a socio-cultural design process coupled with a constructive-destructive process, letting only the most adapted urban structure subsist given the external conditions.
This analysis is aso transferable in some aspects to economic value creation, which is intrinsically related to urban morphology. Mangin and Panerai's analysis of the relationships between Manhattan's urban grid economic aspects is outstanding . Manhattan's initial structure is pretty simple, and inspired by 19th century European cities. But this structure has allowed an extraordinary intensification and complexification of the urban structure over time, and made possible one of the most intense economic value creation of the two last centuries. The initial extremely well thought out plan has been able to support a wide range of variations even the most unexpected ones: cars and lifts (and thus skyscrapers) were barely conceivable in 1811 when Manhattan's plan was being drawn . The lack of initial order able to allow changes over time has lead to the inefficient urban structures in some emerging countries. Additionally, the steady growth rate of urban areas makes it impossible for the useful constructive-destructive process to take place. The dramatic rise in external energy inputs due to low energy prices has allowed the emergence of energy greedy structures. Crazy urban sprawl has been coupled with a drop in urban density and compacity: both transport and building need more energy. The historical urban structures that have survived until now are among the most efficient ones [13,14]. They are pedestrian friendly and require less energy for transport. They are compact and require less energy for heating and/or cooling. Furthermore, partly because of the reasons described by Mangin and Panerai for Manhattan, these urban structures often have the higher
economic land value. On the other hand, there is a growing recognition that a lot of modernist urban structures fail in optimising energy use for transport especially, but also for buildings. Some of them, for a wide range of reasons, have already been destroyed after no more than 40 years.
Let's hope what will replace them will be more efficient.
Interestingly, this modernist failure related to urban sprawling and large-scale inefficiencies can be put into perspective with science history. From Newtonian gravity to Einstein's relativity, there has been a natural tendency for science towards reductionism . Every single problem has to be reducible to its simple elements, and equals the sum of its elements. Science has been developing alongside the inherent human difficulty to grasp and apprehend complexity. As such, modernist urbanism is a major outcome of this encompassing thought: cities are built with elementary blocks added on the existing urban fabric. But both on economic and energy aspects, the addition of a new subsystem does not necessarily lead to an overall optimisation, even though each addition is locally optimal . The optimisation under a wide range of constraints (transport, buildings, socio-economic aspects) mathematically leads to a sub-optimisation in every sector. The former destructive-constructive process mitigated this phenomenon and drove urban tissues towards efficiency. To compensate this process which is not happening anymore in these times of steady growth, a crucial attention has to be paid to urban structures efficiency as a whole and not only to subsystems.
How can 'order' help to make cities more efficient
The question is then how to improve urban structures' efficiency as a whole, and not only for urban subsystems. Industrial ecology, which provides a framework to analyse complex organised systems, can provide some interesting insights. Industrial ecology is moreover closely related with the thermodynamic issues discussed earlier in the paper. Kay's contribution to industrial ecology is especially relevant when considering urban systems. Mature ecosystems are highly organised and complex steady states far from equilibrium. These steady states are exergy production maxima, which is in compliance with Prigogine's MinEP principle described here above: there is a negative sign before the entropic term in the exergy definition. Following Kay's statements , an 'open system with exergy pumped into it is moved away from equilibrium, but nature resists movement away from equilibrium[...]. When the input of exergy and material pushes the system beyond a critical distance from equilibrium, the open system responds with the spontaneous emergence of new, reconfigured organized behaviour that uses the exergy to build, organize and maintain its new structure.' Order and organisation appear as a way for open systems to counteract exergy inputs, by increasing their exergy dissipating capabilities.
High levels of organisation and maximal exergy dissipation are thus closely related. But how is it related to urban efficiency? To avoid a vocabulary inaccuracy, Kay introduces the term of 'effectiveness' , corresponding to the ability to dissipate exergy. In other words, it describes how able the system is to fully use the high quality energy (high exergy) and only reject a low quality energy. This exergy aspect is very fundamental in industrial ecology. Coming back to cities efficiency, with similar energy inputs, a highly organised system will make a much more effective energy use than a less organised one, by extracting much more useful exergy from the same initial amount of energy. One major question remains: what is the right organisation given the various inputs?
The tree or the leaf?
Based on thermodynamic arguments, Adrian Bejan proposed in 1996 a 'constructal law' which provides an interesting insight on the former question. It stands that 'for a finite-size flow system to persist in time (to live) it must evolve such that it provides greater and greater access to the currents that flow through it'. In other words, it is a powerful tool to predict the emergence of structures, given the initial conditions. Heitor Reis proves the equivalence between minimal entropy production and Bejan's 'constructal law' for two simple specific cases . For point-toarea and point-to-volume flows, both thermodynamics and the 'constructal law' predict that the most efficient structure is a tree-like structure. This has been verified for a wide range of applications, explaining the emergence of power-laws both in inanimate (river deltas) and animate (trees, blood vessels, pulmonary airways...) systems . Let us consider a tree. There is one trunk, some large branches, more medium-size branches, and a large number of small-size branches. The relationship between the size of the branch and the number of branches of this size follows a mathematical relationship, which is called power-law distribution. This point highlights the crucial role of scale hierarchy within urban systems: as in natural structures, urban structures (networks, buildings, entities) have to follow specific size-multiplicity distributions to be efficient.
However, according to Alexander's famous paper , a city is not a tree! In his paper, he opposes the concept of semi-lattice to the concept of tree to describe cities. In order not to get into too obscure details, let us just compare a semi-lattice to a leaf. Whereas in a tree, a small branch only belongs to one bigger branch, in a leaf, a thin vein is linked to several broader veins (e.g. fig 1 and 2). The leaf is a much more subtle and complex structure than the tree. Alexander stands that artificial cities are trees instead of semi-lattices because of the inherent human tendency to reorganise every complex organisation into a tree structure. He claims though that cities have to be semi-lattices (leaves) in order to be living cities.
Interestingly, three recent papers [21,22,23] have drawn analogies between leaves and urban networks, giving a facelift to Alexander's arguments. Corson notably shows that the redundancy within leaves' venation networks improves the tolerance to damage. When a major vein is cut, smaller parallel veins take over and irrigate the structure almost as usual. This resilient phenomenon cannot happen with a tree structure: if a branch is cut, all the small branches belonging to the larger one will die. Transposed to urban networks, semi-lattice structures could then be a way to improve resilience, and reduce the damages caused by local changes. Let us consider an underground network. The more redundancy there are in the network (leaf structure), the less influence an incident on a track has. Whereas in the extreme case of a tree-like network, an incident on a single track could paralyse a part of the city.
To sum up this point, a tree structure tends to maximise the efficiency (in thermodynamic arguments), whereas a leaf structure improves the resilience and the resistance to damage. There is certainly a trade-off to be done between the tree structure and the leaf structure, depending on which aspect is regarded as primary: energy and efficiency on the one hand, or risk and resilience on the other hand.
The city is the particular arena where both the climate and the development challenge will take place in the century to come. Most of the current urban development is not sustainable, mainly because most of the new urban structures are intrinsically energy greedy. Building low energy buildings and 'eco-districts' is a part of the solution. It is not sufficient though. Cities are large and complex systems. As such, the overall optimisation will not be achieved by adding up locally optimised subsystems. On the contrary, the overall optimisation is achieved for sub-optimised subsystems. Building an eco-district outside from a city is a non-sense: sure it can perform well in heating energy consumption, but because of its distance from the existing city, it will structurally need a lot of energy for people transportation.
As in every complex open system, forms and flows play a major role within cities. Thermodynamics of dissipative structures provides a relevant and robust framework in order to address cities' efficiency issues. However, theory is pretty hard to grasp. This could jeopardize the chances thermodynamics-based tools being used for urban policy making. Authors propose an innovative approach that is based on urban order and structures considerations. This paper has scrutinised the relationships between exergy and entropy within urban systems on the one hand, and urban complexity on the other hand. It has notably highlighted the crucial role of scale hierarchic structures to maximise cities' efficiency. The main advantage of this approach is to be easier to handle in an urban policy making process. The pictorial tree-leaf example does not aim to provide any policy recommendation about urban structure. At least it has the virtue to highlight the very different implications of each kind of structure. Cities cannot be limited to a point-to-volume flow optimisation problem. Cities are the superimposition of numerous flow structures (point-to-volume, area-to-volume, volume-tovolume...) connected one with the others. As we discussed here above, the optimisation of every layer though will not lead to the overall optimum. On the contrary, the overall optimum is reached for a sub-optimisation of every element. City efficiency has to be analysed as a complex problem and thus as a whole. Thermodynamics insights are crucial. But a morphology approach, based on scale hierarchy and form optimisation, may be a powerful steering tool for urban policy.
 World Bank, "Eco2 Cities: Ecological Cities as Economic Cities" (2010).
 R. Bussiere & F. Snickers, "Derivation of the negative exponential model by an entropy maximizing method," Environment and planning A, 2 (1970) 295–301.
 Y.C. Chen, "A wave-spectrum analysis of urban population density: entropy, fractal, and spatial localization," Discrete dynamics in nature and society, 22 (2008).
 M. Batty, "Cities as complex systems: scaling, interactions, networks, dynamics and urban morphologies", CASA Working Papers, 131 (2008) Centre for Advanced Spatial Analysis (UCL), London, UK.
 N.A. Salingaros & B.J. West, "A universal rule for the distribution of sizes", Environment and Planning B: Planning and Design, 26, (1999) 909-923.
 C. Balocco & G. Grazzini, "Thermodynamic parameters for energy sustainability of urban areas", Solar Energy, 69, issue 4 (2000) 351– 356.
 Y. Zhang, S.F. Yang & W. Li, "Analyses of urban ecosystem based on information entropy", Ecological Modelling, 8, issue 10 (2006) 1-12.
 C. Balocco, S. Papeschi, G. Grazzini & R. Basosi, "Using exergy to analyze the sustainability of an urban area", Ecological Economics, 448 (2004), 231-244.
 Sir A. Eddington, The nature of the physical world (MacMillian, New York, 1930).
 I. Prigogine, Introduction to non-equilibrium thermodynamics (Wiley-Interscience, New York, 1962).
 J.J. Kay, "On complexity theory, exergy and industrial ecology: some implications for construction ecology". in C. Kibert, J. Sendzimir & B. Guy, Construction Ecology: Nature as the basis for green buildings, (Spon Press, 2002) 72-107.
 D. Mangin & P. Panerai, Projet Urbain (Eupalinos, Parenthèses, 1999).
 S. Salat, Cities and Forms, Sustainable Urbanism (Techne Press, 2011).
 APUR, "Consommations d'énergie et émissions de gaz à effet de serre liées au chauffage des résidences principales parisiennes" (2007).
 E. Morin, Science avec conscience (Fayard, 1982).
 D.J. Fisk & J. Kerhervé, "Complexity as a cause of unsustainability", Ecological complexity, 3 (2006) 336-343.
 A. Bejan & S. Lorente, "The constructal law of design and evolution in nature", Philosophical transactions of the Royal Society B, 365 (2010) 1335-1347.
 A. Heitor Reis, "Constructal Theory: From Engineering to Physics, and How Flow Systems Develop Shape and Structure", Applied Mechanics Reviews, 59 (2006) 269-282.
 C. Alexander, "A city is not a tree," Design, 206 (1965) 46-55.
 P. Portoghesi, Natura e architettura. (Skira,Milano, 1999).
 P.D. Dodds, "Optimal form of branching supply and collection networks", Physical Review Letters, 104 (29 January 2010).
 F. Corson, "Fluctuations and redundancy in optimal transport networks", Physical Review Letters, 104 (29 January 2010).
 E. Katifori, G.J. Szöllösi & M.O. Magnasco, "Damage and fluctuations induce loops in optimal transport networks", Physical Review Letters, 104 (29 January 2010