Optimization of drinking water distribution networks - Semantic Scholar

Computers, Environment and Urban Systems 36 (2012) 434–444

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Optimization of drinking water distribution networks: Computer-based methods and constructal design P. Bieupoude a,b,⇑, Y. Azoumah a, P. Neveu b a b

LESEE-2iE, Laboratoire Energie Solaire et Economie d’Energie, Institut International d’Ingénierie de l’Eau et de l’Environnement, 01 BP 594 Ouagadougou 01, Burkina Faso PROMES-CNRS UPR 8521, Laboratoire Procédés Matériaux et Energie Solaire, Université de Perpignan, Rambla de la thermodynamique, Tecnosud, 66100 Perpignan cedex, France

a r t i c l e

i n f o

Article history: Received 25 February 2011 Received in revised form 21 February 2012 Accepted 21 March 2012 Available online 4 May 2012 Keywords: Drinking water Pipe networks design Optimization Computer-based methods Urban systems Constructal theory

a b s t r a c t A well-known application of water engineering is drinking water distribution through pipe networks in urban and rural areas. The present work addresses this issue with a specific focus on the network design. First, the paper presents a brief review of computer-based design methods and shows that a significant number of efforts have been pursued. Secondly, it proposes the approach of geometric analysis of the distribution networks as complementary points of the former optimization methods. Finally, an original illustrative application is proposed. The geometric and multi-scale optimization known as the constructal design is used to analytically optimize T-shaped network architectures subject to an operational water quality constraint. This illustrative application leads to the determination of an optimal geometry of the network that minimizes head losses (factor of pumping energy). Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction The optimal design and management of urban networks is an interdisciplinary challenge touching environment, water, electricity and urban planning as noticed in many works (Christodoulou, Deligianni, Aslani, & Agathokleous, 2009; Ducrot, Le Page, Bommel, & Kuper, 2004; Kizito, Mutikanga, Ngirane-Katashaya, & Thunvik, 2009) and requires maximal computing skills (Akiba, 1982; Evatt, 1984; Ignizio, 1980; Keirstead & Shah, 2011; Miller, Hunt, Abraham, & Salvini, 2004; Moore & Kim, 1995). During last decades, numbers of researchers have put their interest on them in various aspects. This interest is understandable in many senses. First, these systems are huge economic infrastructures and their optimization is strongly needed in developing countries and even in western countries. Secondly, because of their very high importance, they require reliable design techniques for authorities to be assisted in investment decision making. According to literature, many progresses have been made in the study of water distribution systems (WDSs) and today, they are capable of serving rural and urban communities reliably, efficiently, and safely, both now and in the future (Chase, Savic, & Walski, 2001). Though the complexity and the size of WDS vary ⇑ Corresponding author at: LESEE-2iE, Laboratoire Energie Solaire et Economie d’Energie, Institut International d’Ingénierie de l’Eau et de l’Environnement, 01 BP 594 Ouagadougou 01, Burkina Faso. Tel.: +226 50 49 28 64; fax: +226 50 49 28 01. E-mail address: [email protected] (P. Bieupoude). 0198-9715/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compenvurbsys.2012.03.007

dramatically (from African rural areas to overpopulated cities in western countries), they have the same basic function of delivering water from sources or treatment facilities to customers (Chase et al., 2001). Technologies and researches on these systems have considerably evolved over time and through civilizations (Babbitt & Doland, 1931; Haestad methods, 1999). Today water distribution networks (WDNs) are the most known, the most well-tried and the most used systems worldwide (Chase et al., 2001) in providing water to populations. Being used for hot or cold water distribution, either for drinking water or agricultural irrigation, WDN as flow systems, are characterized by mechanical losses (head losses) that are factor pumping energy (Bejan & Lorente, 2007; Izquierdo, Montalvo, Pérez, & Herrera, 2008; Tondeur & Luo, 2004), chemical and biochemical reactions that refer to water quality management questions (Kerneïs, Nakache, Deguin, & Feinberg, 1995). To have a thorough understanding of these phenomena, in order to optimize and well manage WDN technically, economically and socially, an important amount of researches has been pursued during the last decades on the design (optimization and modeling). Various design methods have been developed, focusing on minimum cost objective (Alperovits & Shamir, 1977; Simpson, Dandy, & Murphy, 1994), on reliability aspects (Bai, Pei-jun Yang, & Song, 2007; Chiplunkar, Mehndiratta, & Khanna, 1990; Fujiwara & Khang, 1990; Todini, 2000; Wechsatol, Lorente, & Bejan, 2004), and on water quality (Bieupoude, 2011; Boulos, Rossman, & Vasconcelos, 1994) which is a critical environmental question

P. Bieupoude et al. / Computers, Environment and Urban Systems 36 (2012) 434–444

435

Nomenclature a Ci D Dk f h L Lg Lk Q Qk Sn ST t

friction losses parameter depending on the pipe material, Eqs. (9), (36) constraint i diameter (m) diameter of pipe k (m) geometric shape factor of the network head or head losses, Eq. (4) (m) pipe length (m) lagrangian function length of pipe k (m) volumetric flow rate (m3 s1) flow in pipe k (m3 s1) demand at node n (m3 s1) total surface of the area to be supplied (m2) water residence time taken as water quality constraint (s)

Greek symbols DH head losses (m) c optimization constant defined by c = (n + 1)/(m + 2) depending on flow regimes ki lagrangian multipliers

(World Health Organization, 1996). Because of the complexity of the problem (Savic, Walter, Randall-Smith, & Atkinson, 2000) most of these methods are computer-based (Alperovits & Shamir, 1977; Bhave, 1988; Gessler, 1985; Pierro, Khu, Savic, & Berardi, 2009; Simpson et al., 1994) and based on complex iterative calculations. The aim of the design is to find trade-off between objectives and design constraints and to predict the future working conditions of the system. The introduction of the latest technology which is modeling in 1980s was salutary in this field (Chase et al., 2001). Despite the last evolutions in modeling (due to the increase in computation skills), it remains a critical part of the design of water distribution systems. From gathering data, conception of the model itself (understanding, structuring and calibrating) to the implementation of the model, water researchers have provided sophisticated tools to reach the goals of rendering water systems reliable, efficient and safe (Augugliaro, Dusonchet, & Riva-Sanseverino, 1998; Baños, Bans, Gil, Reca, & Montoya, 2010; Bolognesi, Bragalli, Marchi, & Artina, 2010; Chu, Lin, Liu, & Sung, 2008; Eiben, Raué, & Ruttkay, 1994; Gupta, Bassin, Gupta, & Khanna, 1993; Gupta, Gupta, & Khanna, 1999; Keedwell & Khu, 2005; Klempous, Kotowski, Nikodem, & Ulasiewicz, 1997; Mustonen et al., 2008; Savic & Walters, 1997). Though WDN are becoming well-known systems (Chase et al., 2001) in both modeling and optimization studies, there is still much to know about the precision of models and some design or optimization techniques to increase the performances of these systems. Important reviews have been offered in literature. However these reviews included very little information on design methods based on geometric analysis of the networks architectures. This paper offers a brief review on WDN design and optimization methods and highlights the need of geometric optimization approaches for urban systems. Then it introduces the geometric and multi-scale optimization known as the constructal design (Bejan & Lorente, 2008) through an illustrative application. In this application, T-shaped network architectures for drinking water distribution are analytically optimized subject to an operational water quality constraint.

Subscripts 1, 2 scale index h, H direction index i element i, index of the ith element k index of the kth element Obj objective Opt optimized quantity Upscripts n, m parameters of friction losses equation Superscripts GA genetic algorithm HA heuristic algorithm MA memetic algorithm SA Simulated Annealing SS scatter search SSSA scatter search simulated annealing Syst system WDN water distribution network WDS water distribution system

2. WDN design problem and computer-based methods 2.1. Problem formulation The optimization studies of urban systems are multi objective and complex (Duh & Brown, 2007; Neema & Ohgai, 2010; Poelmans & Rompaey, 2010) because they are space and time-dependent. In the particular case of drinking water distribution networks, except the time and space-dependence and water demand questions, the fundamental technical challenge relies on the management of head losses due to friction and local losses in the network (Carlier, 1972) and the degradation of water quality through the distribution network (Rossman, Boulos, & Altman, 1993, Chase et al., 2001; Rossman & Boulos, 1996; Rossman, Clark, & Grayman, 1994). These points are related to the hydraulic performances of the networks. In addition, WDN are economic infrastructures requiring strong financial efforts in investment and operating costs (Keedwell & Khu, 2005; Savic & Walters, 1997). Therefore, optimization studies touching these aspects are very complex and require more and more computation efforts through robust algorithms (Bieupoude, 2011). Most of the time, the design problem formulates as follows: a certain population of density rp grouped in households, and distributed over a given area, needs to be supplied in drinking water through a distribution network (Bieupoude, 2011; Chase et al., 2001). A lot of technically acceptable solutions can be found to this problem. In Fig. 1, two possibilities to supply four users (or groups of users) are shown. For these two configurations of Fig. 1, the investment cost and pumping requirement may be different. Technical solutions to this problem may also differ in terms of total head losses, overall residence time, or total investment. The challenge relies on how to optimally design the network so that the pumping energy is minimal (Carlier, 1972) and other technical requirement on pressure and water residence time, are satisfied. In literature, this problem is solved in diverse directions: (a) cost optimization (Chu et al., 2008; Savic & Walters, 1997), (b) optimal water quality management criteria (Kerneïs et al., 1995; Kirmeyer, Friedman, Martel, Noran, & Smith, 2001), and (c) reliability aspects (Bai et al., 2007; Lansey & Mays, 1989; Todini, 2000).

436


(a)

User 4

User 1

ers a variety of WDN design options. A GA uses a population of individual solution (Keedwell & Khu, 2005) that iterates from one generation to the next as the research progresses. When they are applied to WDS design problems subject to cost and hydraulic constraints, they are capable of finding near optimal cost solutions (Keedwell & Khu, 2005). Literature explains that coupling GA with new evolutionary methods such as the ones (memetic GA) where local search is applied during the evolutionary cycle (Augugliaro et al., 1998; França, Mendes, & Moscato, 1999; Ozcan & Onbasioglu, 2006) can produce cost optimal WDN. Despite their success, genetic algorithms are generally criticized in two main areas of their operations: they are population based and therefore require a large number of objective function evaluations to solve a problem (Keedwell & Khu, 2005) and may find different solutions to the problem depending on their starting position in the search space.

Source

User 3 User 2

(b)

Source

User 2

User 4

User 3

User 1

Fig. 1. Two different networks for the same water distribution service.

In (a), the problem is formulated through the definition of a cost objective function (Fcost defined in Eq. (1)) that is related to the variables of the system: Dk diameters, Lk pipes length, N number of paths. N X F cos t ¼ b C k Lk Dak

ð1Þ

k¼1

Similar functions have been offered in literature (Baños et al., 2010; Bieupoude, 2011; Cunha & Ribeiro, 2004; Kirkpatrick, Gelatt, & Vecchi, 1983). The optimization consists in minimizing the cost function and checking the hydraulic conditions (required pressures, flow rates or flow velocities) of the systems. In (b), additional equations are considered in order to take into account water quality evolution through the network. A very well-known equation of them is the evolution of chlorine residuals in pipes presented in literature (Haestad methods, 2004):

@C i ¼ @t

Qi @C i þ hðC i Þ; Ai @x

i ¼ 1...P

ð2Þ

Ci is the concentration of chlorine residuals (or other water treatment agents), in the pipe i, at distance x from the origin. In (c), instead of supplying users through tree-shaped networks, loops are included in the network, making then it more reliable to overcome local damages such as pipe breaks or maintenance works. The inclusion of loops comes with additional hydraulic equations as shown by former researchers (Fujiwara & Khang, 1990; Fujiwara & Khang, 1991; Todini, 2000). Different computer-based optimization techniques are used in literature to solve the problem, and we propose to review some of them. 2.2. Computer-based methods 2.2.1. Design methods based on genetic algorithms A genetic algorithm (GA) is a computing search technique to find or to approximate solutions of optimization problems (Buckles & Frederick, 1992; Eiben et al., 1994; Goldberg, 1989; Li et al., 2004; Syswerda, 1989). It is a particular class of evolutionary algorithms in which evolutionary biology such as inheritance, selection, mutation and crossover techniques are used. Numerous advances have been registered in these algorithms which have benefited to the field of optimization, progressing from the early single objective algorithm to multi-objective algorithms (Fonseca & Fleming, 1995; Keedwell & Khu, 2005) that offer network design-

2.2.2. Design methods based on memetic algorithms Baños and his coworkers (Baños et al., 2010) offered a recent work on memetic algorithm (MA) applied to WDN showing that it can perform optimization of very complex systems such as looped WDS which are non-linear, constrained, non-smooth, non-convex, and hence multi-modal problems (Gupta et al., 1993). Like genetic algorithms, MA are methods inspired by models of natural systems that combine the evolutionary adaptation of a population with individual learning within the lifetimes of its members, called agents (Baños et al., 2010; Goldberg, 1989). Additionally, MA are inspired by the concept of meme, which represents a unit of cultural evolution that can exhibit local refinement (Dawkins, 1976). Its application to the least-cost optimization problem of looped WDN requires the mathematical formulation as shown in previous works (Baños et al., 2010; Kirkpatrick et al., 1983). The objective is to minimize the investment cost of the network subject to mass and energy conservation constraints and other operational conditions (minimum pressure requirements for users, minimum and maximum flow velocities, and pipe size restrictions). Tested in comparison with five other methods – of which four are heuristic-based (Simulated Annealing (SA) (Kirkpatrick et al., 1983), Mixed Simulated Annealing and Tabu Search (Gil, Ortega, Montoya, & Baños, 2002), Scatter Search (SS) (Marti, Laguna, & Glover, 2006) using SA as local searcher (SSSA), and Genetic algorithms) and the fifth one is Binary Linear Programming Method -, MA method obtained the best results while SSSA obtained good results (Baños et al., 2010); the other methods were slightly worse but vary less than 2.6% so remain acceptable. MA seems to be powerful and significantly better than the others, mentioned in this section when they are applied to large-scale networks but for medium-scale networks, meta-heuristic methods perform as well as MA (Baños et al., 2010). 2.2.3. Design methods based on heuristic approaches Design a network with an increased resilience (capability of overcoming stress or failure conditions (Todini, 2000)) calls robust computational efforts because dealing with it is not a sole question of a cost objective function minimization since a cost function does not incorporate adequately the concept of reliability. This kind of problem of overcoming resilience and minimum-cost optimization can be analyzed by heuristic method to identify reasonable solutions with limited computational requirements (Todini, 2000). Heuristics are typically used when there is no known method to find an optimal solution (Judea, 1984). The principle is based on successive iterations that depend upon the step before. It disregards or considers avenues by measuring how close the current iteration is to the solution. Therefore, some possibilities can never be generated as they are measured to be less likely to complete the solution. It is selective at each decision point; picking branches


that are more likely to produce solutions (Newell & Simon, 1976). This makes the heuristic approach suitable for WDN analysis even if some weaknesses remain: its less accurate results when compared with MA for example as explained in literature (Baños et al., 2010), or its computational time which may be very long for higher-scale networks. An important improvement point of HA is how from the current iteration one can move to the next iteration and the way potential solutions are managed. Tabu Search method (Cunha & Ribeiro, 2004) is one famous of heuristic approaches that performs very well by using memory structures: say another way, in Tabu search method, once a potential solution has been determined; it is marked as ‘‘tabu’’ so that the algorithm does not visit that possibility repeatedly. A corresponding model in WDN (Cunha & Sousa, 1999) where Tabu Search can be used is the following (Cunha & Ribeiro, 2004): Minimize F ¼

X

ck ðDk ÞLk Constraints ðFlow rate; Diameter sizes; HeadÞ

k2NP

ð3Þ In Eq. (3), F is the cost objective function, ck is a cost coefficient, Dk is the diameter of pipe k, and Lk is the length of pipe k. To well understand the method, analyze the neighborhood structures as applied by Cunha and Ribeiro (2004). The moves they considered first consist in reducing the diameter of one pipe at a time so that the neighbor solutions are characterized by configurations in which all pipes but one have the same diameter as in the current configuration. Then evaluating the cost of each configuration and solving the hydraulic equilibrium equations lead to the selection of the new current solution. In comparison with some other methods such as Split-pipe method (Lansey & Mays, 1989; Tospornsampan, Kita, Ishii, & Kitamura, 2007; Walski, 1985) or continuous diameters design (Fujiwara & Khang, 1990; Fujiwara & Khang, 1991; Varma, Narasimhan, & Bhallamudi, 1997), Tabu Search was found efficient (Cunha & Ribeiro, 2004) but among heuristic approaches it will be inappropriate to conclude that Tabu Search is the best or not. 2.2.4. Design methods based on perturbation method Methods that are presented above are iterative and then require a good initial estimate to reach solutions quickly without convergence problems. Perturbation methods permit to obtain a series of linear equations that can be solved easily using matrix methods when applied to a set of non-linear equations (Basha & Kassab, 1996). The advantage of this method is that the solution is obtained directly without iterations, initial estimates, and issues of convergence unlike the previous ones. The basis of the method is the following. The classical energy equation which states that the sum of head (H) losses and gains along a flow path must be equal to the difference of the end nodal heads DH. In particular, the sum of head losses and gains around a closed loop must be equal to zero (DH = 0).

X h ¼ DH

k ¼ 1; 2; . . . ; l þ nf 1

ð4Þ

k

The volumetric flow rate Q is related to the head losses h through

Q ¼ ah

x

ð5Þ

where a varies between 0.5 and 0.54 (Basha & Kassab, 1996) depending on the head losses equation, and a is function of the length L, Eq. (6); D in Eq. (6) is the diameter, and x is the pipe roughness. For example, for the Hazen-Williams friction losses equation (Chase et al., 2001), the parameter a of Eq. (5) writes:

a¼

0:849p C HW D2:63 41:63

L0:54

;

x¼

1 1:85

437

ð6Þ

Expressing Eq. (5) for every node results in a set of non-linear equations coupled with linear equations derived from Eq. (4) for every loop. It uses a perturbation technique called the delta expansion (Bender, Milton, Pinsky, & Simmons, 1989). This technique requires the replacement of the exponent x by d + 1 where d is the perturbation parameter (Basha & Kassab, 1996). The perturbation approach then consists in expanding the head losses in powers of d and determining analytically the terms of series, forming then equations system that is later solved recursively whereby at each step, the right-hand side is known from previous steps; the first-order solution is a function of the zeroth-order one, and the second-order solution depends on both the first-order and second-order elements. For more complex WDN including regulating valves, the approach follows the one earlier presented in literature (Jeppson & Davis, 1976). To have a practical application, readers are referred to examples offered in (Basha & Kassab, 1996). Results of this perturbation method are sufficiently accurate and it is shown that the smaller perturbation parameter causes series to reach an accurate solution with fewer terms (Basha & Kassab, 1996). Note finally that the perturbation solution is useful by itself or as a very good initial estimate for iterative methods if an exact solution is desired (Basha & Kassab, 1996). 2.3. Remarks This review is not exhaustive show that strong efforts that have been earlier pursued in developing methods/models for optimal design of WDN. Some models cover water quality aspects (Boulos et al., 1994; Dzialowski et al., 2009; Park & Kuo, 1996), leaks detection (Colombo, Lee, & Karney, 2009), water demand analysis (Dziegielewski & Baumann, 2011), and various other aspects (Christodoulou et al., 2009; Liao & Tim, 1994). This big amount of works explains the diversity of computer programs that are available on the topic of which a pioneering is EPAnet developed by the US environmental protection agency (Rossman, 2000). Their common points are the hydraulic calculations or minimization/maximization of an objective function (cost, pumping power, etc.) subject to operational constraints (pressure, velocities, water quality, reliability, etc.) in respect with given network architectures. The computer-aided design results in an optimal design of the given architectures with optimal diameters/ lengths of pipes. In some cases, when pipe lengths are regarded, the optimal structures can have different architectures from the initial (before optimization), but in a non-predictable way. According to authors, a complementary point to these design methods should be a full attention on the scales variability of the networks and the variation of the geometry (geometric analysis). In constructed areas, the number of degrees of the freedom for the variation of the geometry of the networks is limited. But when city planners are making new housing estates in new urban areas or city extension plans there is freedom to choose any configuration of urban networks (for drinking water or electric power distribution). Then the full geometric optimization of these structures should be more than required instead of empirical design approaches. The need of geometric optimization methods in engineering explains well the success of the constructal design that derives from constructal theory of Bejan (2000), Bejan and Lorente (2005) and Bejan and Lorente (2008). This approach does not aim to simulate ideally (perfectly) a real system but helps to predict the tendency of geometry variation, by solving a simplified problem (Tescari, Mazet, & Neveu, 2010). The results of this method cannot be used crudely to optimize the system but can be used as a system

438


3. Constructal theory, constructal law and constructal design

2011). The main point that retained our attention in these former studies is the scale by scale optimization (from lower order-constructs of the system to higher-order ones), or from maximal simplifications to increasing complexity of the designed system. In the following section an illustrative application of the multi-scale optimization is proposed.

3.1. Basis of the constructal approach

3.2. Constructal design of T-shaped water distribution network

The idea of optimal design of flow structures has been put forward in engineering many years ago and the geometric scales have been evocated in a pioneering work of Hess (1914), later developed by Murray (1926) who explained the optimal diameters ratio of blood vessels. The constructal theory was developed by Bejan (2000) and Bejan and Lorente (2008). ‘‘First developed in the late 1990, constructal theory holds that flow architecture arises from the natural evolutionary tendency to generate greater flow access in time and in flow configuration that are free to morph. It unites flow systems with solid mechanical structures, which are viewed as systems for flow of stresses. Constructal theory unites nature with engineering and helps us generate novel designs across the board, from high-density packages to vascular materials with new functionalities from tree-shaped exchangers to svelte fluid-flow and solid structures’’ (Bejan & Lorente, 2008). The constructal design that derives from constructal theory was developed based on a theoretical foundation (Bejan, Rocha, & Lorente, 2000; Lorente, Wechsatol, & Bejan, 2002). Today, design with constructal theory is a growing activity in thermal sciences (Azoumah, Bieupoude, & Neveu, 2012; Azoumah, Mazet, & Neveu, 2004; Eslami & Jafarpur, 2012; Lorenzini, Corrêa, dos Santos, & Rocha, 2011; Lorenzini, Garcia, dos Santos, Biserni, & Rocha, 2012; Tescari et al., 2010; Xiao, Chen, & Sun, 2011), chemical engineering (Lorente, Bejan, Al-Hinai, Sahin, & Yilbas, 2012; Mehrgoo & Amidpour, 2011; Tescari et al., 2010; Tondeur, Fan, & Luo, 2009) and fluids engineering (Bejan & Lorente, 2007; Bieupoude, Azoumah, & Neveu, 2011; Hart & Da Silva, 2011; Miguel, 2010). The statement of the theory which is called constructal law is explained by Bejan and Marden (2009) as follows: ‘‘the configuration and function of flow systems change over time in a predictable way that improves function, distributes imperfection, and creates geometries that best arrange high and low resistance areas or volumes’’. Through literature of the past decade on constructal theory, one can understand that the constructal law covers ‘‘natural design’’ phenomena across the board, from biology and geophysics to social dynamics and technology evolution; for example: tree-shaped architectures, river-basin and animal scaling laws, animal locomotion, the distribution of city sizes, dendrite crystals, vegetation, turbulent structures, the evolution of power and refrigeration plants, machine flights, etc. (Bejan & Lorente, 2011; Lage, 2008). In hands of an engineer, the constructal design is a multi-scale and geometric optimization method based on thermodynamical, technological, economical (. . .) criteria (Bejan & Lorente, 2008) proposing a regular repartition of imperfections in flow systems so that resistances met by the currents that are flowing through the system are minimized. Authors recognize that the present description of constructal theory is reductive, as noticed by reviewers. Indeed, instead of detailing the philosophy of constructal theory, of which a large description was previously offered in literature (Bejan, 2000; Lorente and Bejan, 2005; Bejan & Lorente, 2008; Bejan & Lorente, 2011), we presented it from an engineer’s point of view. This is in fact what we did in former studies that resorted to the constructal approach (Azoumah et al., 2004; Bieupoude et al., 2011, 2005, Azoumah et al., 2012; Tescari et al., 2010; Tescari, Mazet, & Neveu,

There are some attempts to the design of urban systems based on the philosophy of constructal theory (Bejan & Lorente, 2008). Our former works on urban systems that used the constructal approach dealt with the optimal design of WDN subject to an environmental constraint (Azoumah et al., 2012; Bieupoude et al., 2011) allowing to find the best way of doing agricultural irrigation (water quality independent) and tree-shaped drinking water networks (water quality dependent). Besides, some published attempts (Wechsatol, Lorente, & Bejan, 2001; Wechsatol, Lorente, & Bejan, 2002) on hot water distribution, urban constructions (Miguel, 2008), and electric power distribution (Arion, Cojocari, & Bejan, 2003a; Arion, Cojocari, & Bejan, 2003b) have shown that there is an unquestionable interest to resort to the constructal approach for the optimization of urban systems separately or coupling them. In this section we illustrate the constructal approach through an application to show its geometric and multi-scale abilities. Consider an area of surface ST (m2), to be supplied in water through a T-shaped network. The area ST may correspond to different geometries. On Fig. 2, three rectangular flow architectures are considered. These three networks influence the same surface ST. This means that if the population density is fixed, then these three networks will supply the same number of users. Though the total water demand will remain the same for the three structures (since the number of users is fixed), it is evident that other hydraulic parameters will not. An operational question that these three geometries inspire is how to design the network so that it minimizes the total head losses subject to both spatial constraint (due to the fixed surface ST to be occupied by users), and the overall residence time (t) constraint which is a water quality parameter according to former studies (Bieupoude, 2011; Bieupoude et al., 2011). In other terms, how to optimally allocate space to a Tshaped network, so that the network that fits this optimal space allocation is better in terms of head losses minimization subject to the constraint t.

pre-design (Bieupoude, 2011; Tescari et al. 2010). In the sections below, an illustrative application of the constructal design of Tshaped water distribution networks is proposed.

H

H h

(b) H h

(a)

h

(c)

Fig. 2. Three rectangular flow architectures of influence surface ST: (a) the area ST is alimented from the shortest side of the network (H > h), (b) h = H, (c) the area ST is alimented from the longest side of the network (H < h).

439


Different geometric scales of construction can be analyzed as shown in following sequence known as the constructal sequence (Bieupoude, 2011) with the objective of determining the better global geometry of the networks and optimal diameters distribution for the minimization of the total head losses. Remark that this design suits situations where users are not yet positioned, meaning that the surface ST is empty, allowing the designers to vary the configuration as in situations (a), (b) and (c) of Fig. 2. This situation is met when new housing estates are being established. So, to our understanding and in next sections, designers have freedom to change the configuration of their systems until they reach the solution that meets the optimization criteria.

This equation derives from the head losses created in a single pipe, evaluated as aQnL/Dm (Carlier, 1972; Bieupoude, 2011) where a is constant, Q (m3/s) is the volumetric flow rate, L(m) is the pipe length and D(m) is its diameter. The minimization of DH subject to the two constraints of Eqs. (7) and (8) can be performed by using symbolic mathematical computer systems such as MAPLE or MATHEMATICA (Zotos, 2007). We do not proceed this way. Rather, we use an analytical method by using the lagrangian function which is a linear combination of the objective function and the constraints. This function formulates as follows (Tondeur & Luo, 2004; Luo & Tondeur, 2005; Bieupoude, 2011)

3.2.1. Constructal optimization of the first construction The problem can be analyzed at starting point by considering two points in which the distribution service can be done (Fig. 3). This is the maximal simplification of the problem with the idea of geometry variation as known to the constructal approach (Bieupoude, 2011; Tescari et al., 2010). The constraint on the global geometry is expressed as follows

In Eq. (10), Lg is the lagrangian function; ki are the lagrangian multipliers, Ci is the ith constraint and Expr(Ci) is the literal expression of the ith constraint as function of the optimization variables. From Eq. (10) the lagrangian function of our problem writes as follows

Constraint 1 :

ST ¼ hH

ð7Þ

Lg ¼ F obj þ

Q nH H Q nh h pD2H1 H pD2h1 h Lg ¼ a m1 1 þ a m1 2 þ k1 þ t 4Q H1 21 4Q h1 22 DH 1 2 Dh1 2

t¼

pD2H1 H 4Q H1 2

1

þ

pD2h1 h 4Q h1 2

ð8Þ

2

DH1 is the diameter of the pipe of length LH1 shown in Fig. 3; and Dh1 is the diameter of the pipe of length Lh1 also shown in Fig. 3. As we notice in Fig. 3, h and H indicate the two dimensions of the rectangular surface that represents the influence zone of the network (H and H are also considered as directions that can be used to index lengths, flow rates and diameters). For example DH1 designates the first diameter in the direction of H. In the same manner, Q h1 designates the volumetric flow in the pipe of diameter Dh1 . Note that the relation between residence time and drinking water quality has been deeply evoked in literature (Boulos et al., 1994; Chase et al., 2001; Rossman et al., 1993; Bieupoude et al., 2011). The total head losses in the whole structure write from summing the head losses in each contributing pipe as follows (Carlier, 1972; Bieupoude, 2011)

DH ¼

Q nH a m1 DH 1

H 2

1

þ

Q nh a m1 Dh1

ð10Þ

!

þ k2 ðhH ST Þ

The second constraint upon the residence time is

Constraint 2 :

X ðki ðExprðC i Þ C i ÞÞ

h

ð9Þ

2

2

ð11Þ

By differentiating Lg with respect to its variables and cancelling all the derivatives, we obtain

Q nH1 H @Lg p DH H ¼ 0 ) a m mþ1 þ 2 k1 1 1 ¼ 0 1 @DH1 4 Q H1 2 DH 1 2

ð12Þ

Q nh1 h @Lg p Dh h ¼ 0 ) a m mþ1 þ 2 k1 1 2 ¼ 0 @Dh1 4 Q h1 2 Dh1 22

ð13Þ

Q nH 1 pD2H1 1 @Lg þ k2 h ¼ 0 ¼ 0 ) a m1 1 þ k1 @H 4Q H1 21 DH 1 2

ð14Þ

Q nh 1 pD2h1 1 @Lg þ k2 H ¼ 0 ¼ 0 ) a m1 2 þ k1 @h 4Q h1 22 D h1 2

ð15Þ

D2H H D2h h @Lg 4t ¼0) 1 1þ 1 2¼ @k1 Q H1 2 Q h1 2 p

ð16Þ

@Lg ¼ 0 ) hH ST ¼ 0 @k2

ð17Þ

From Eqs. (12) and (13), we can write Q nH

1 a m Dmþ1

H L

am

H1

H1 Q nh 1 Dmþ1 h1

H 21

DH

¼

h 22

2 p4 k1 Q H1

1

Dh

2 p4 k1 Q

1

h1

H 21 h 22

ð18Þ

This yields in

Dmþ2 h1 Dmþ2 H1 L h1 h L h1

¼

Q nþ1 h1

ð19Þ

Q nþ1 H1

And, noting that 2Q h1 ¼ Q H1 (because of the symmetry in the structure) we have

DH1 ¼ 2c Dh1 opt

with c ¼ ðn þ 1Þ=ðm þ 2Þ

ð20Þ

From Eq. (12) and (13), k1 can be expressed as follows

Fig. 3. Constructal network (first construction): users are alimented through two service points.

k1 ¼

nþ1 2a m Q H1

p Dmþ2 H1

¼

nþ1 2a m Q h1

p Dmþ2 h1

ð21Þ

440


By introducing k1 in Eq. (14) and in Eq. (15) we obtain nþ1 Q nH 1 2a m Q H1 a m1 1 þ p Dmþ2 DH1 2 H1

!

4Q H1 21

! nþ1

a

pD2H1 1

Q nh1 1 2a m Q h1 þ 2 p Dmþ2 Dm h1 2 h1

pD2h1 1 4Q h1 22

¼ k2 h

H

LH1

ð22Þ

LH2 ¼ k2 H

LH2

ð23Þ Lh2

This yields in

Lh2

n a m Q H1 1 aþ ¼ k2 h m 2 DH1 21

L h1

ð24Þ

h Lh2

n a m Q h1 1 ¼ k2 H aþ 2 2 Dm h1 2

ð25Þ

L h1

Lh2

By dividing member by member Eq. (24) by Eq. (25), we obtain

Q nH Dm h h ¼ 2 n 1 m1 H Q h1 DH1

ð26Þ

Considering Eq. (20) and the fact that 2Q h1 ¼ Q H1 , Eq. (26) gives

h ¼ 2nþ1mc ¼ fopt1 H

ð27Þ

Considering Eqs. (27) and (17), we have

(

hH ST ¼ 0 h ¼ 2nþ1mc ¼ fopt1 H

ð28Þ

ST fopt1

the former structure has been more ramified to reach these new service points. For this new system, the earlier variables change as follows:

Lh1 ¼ h=22 ;

ð29Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H ¼ ST ðfopt1 Þ1

ð30Þ

And then, with Eq. (27), we can determine h as follows

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h ¼ fopt1 ST ðfopt1 Þ1

ð31Þ

By combining Eq. (16) with Eqs. (20) and (28), we obtain Dh1 and DH1 1=2

W

ðDH1 Þopt ¼ 2 ð4tQ h1 =pÞ

1=2

ðDh1 Þopt ¼ ð4tQ h1 =pÞ c

1=2 1 1=2 1

W

ð32Þ ð33Þ

where

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi W1 ¼ ST ðfopt1 Þ1 ð22c2 þ 22 fopt1 Þ

ð34Þ

This optimization yields in the optimal shape factor fopt1 (defined in Eq. (27)) and optimal diameters distribution, Eqs. (32) and (33). 3.2.2. Constructal optimization of the second construction The water distribution service through the network of Fig. 3 is not practical since users need to be supplied only from two service points. In practice this means that the accessibility to pipes for an eventual connection of a new user is low and therefore, has to be improved. Then, from this structure of Fig. 3 we move towards a new one in which more pipes are added and the construction method is based on pairing the bifurcation nodes in T. Then a new architecture is obtained from the former (of Fig. 3) by assembling it four times, and the new structure is called the construct 2 (Fig. 4). Now, users are served through eight service points since

LH1 ¼ H=21 ;

LH2 ¼ H=22

ð35Þ

! ! Q nH1 1 Q nH2 1 Q nh1 1 Q nh2 1 a m 1þa m 2 Hþ a m 2þa m 3 h DH1 2 DH 2 2 Dh1 2 D h2 2

ð36Þ

And the former constraints t and ST become

t¼

or

Lh2 ¼ h=23 ;

The total head losses in the structure write

DH ¼

This allows writing, by eliminating h

H2 ¼

Fig. 4. Constructal network (second construction): adding more ramifications in the network.

pD2H1 1 4Q H1 21

þ

pD2H2 1 4Q H2 22

! Hþ

pD2h1 1 4Q h1 22

ST ¼ hH

þ

pD2h2 1 4Q h2 23

! h

ð37Þ ð38Þ

The mathematical resolution is quite similar to the previous and is detailed in Appendix. It yielded in the optimal geometric shape factor fopt2:

h 23n13mc þ 2nmc2 ¼ ¼ fopt2 H opt 23 þ 22n22mc

ð39Þ

By replacing c by its expression c = (n + 1)/(m + 2), we verify that

fopt1 ¼ fopt2 ¼ fopt ¼ 2nþ1mc

ð40Þ

The distribution of optimal diameters is shown in Appendix. From construct 2, one can move to construct 3, 4, . . . , N with corresponding optimal solution fopt3 ; fopt4 ; . . . ; foptN . This will be addressed in future works. 3.2.3. Results and significance From head losses equations found in drinking water literature (Bedjaoui, Achour, & Bouziane, 2005; Chase et al., 2001), fopt can be calculated as presented in Table 1. During implementations in practical operations, one can accept the approximation fopt 2. The significance of this result is that the way to supply optimally users spread over an area is to place them in the center of small rectangles (Fig. 5) in such a way that the total surface influenced by network obeys to the fopt relation. In other words, a T-shaped network with a unique entry flow is better (in terms of head losses minimization) when it fits a rectangle of surface ST = hH = 2H2 (where h and H are the geometric dimension of the surface influenced by the network) and when the diameters distribution in the network obeys to the relations offered in Eq. A.12. Note that

441

P. Bieupoude et al. / Computers, Environment and Urban Systems 36 (2012) 434–444 Table 1 Head losses equation parameters and optimal geometric shape factor obtained through the constructal optimization. Head losses formula

Poiseuille

Darcy-Weisbach

Hazen William

Manning–strickler

m n

4 1 0,33 1,59

5 2 0,43 1,81

4,87 1,85 0,41 1,78

5,33 2 0,41 1,76

c fopt

(a)

h

(Bejan & Lorente, 2008). A wide range of fields, from engineering to biology (Bejan, 2000), are in the broad coverage of constructal theory in terms of design (Bejan & Lorente, 2008). What is to be understood from this illustrative application is that the big unknown is the configuration to which constructal law draws attention (Bejan & Lorente, 2008).

H

H

(b)

h H

H

Fig. 5. Optimal architectures of T-shaped networks ST = hH = 2H2: (a) first element, (b) second element.

this result of fopt = (h/H) 2, applies only for turbulent flow regimes that are generally met in drinking water distribution (Bieupoude, 2011). 4. Conclusion

4.2. Coupling the design of two urban networks: electric power and drinking water In urban areas, these two resources (electric power and drinking water) have similar distribution techniques either from high voltage to lower or from high water head to lower at users scales. Though the control equipments and the transportation materials are completely different, the distribution network architectures involve same questions: flow sections, currents or flow rate, geometry of the network, total length of the network, and reliability of the network. In this view, the optimization problem that we evoked in this paper can be extended to a multi-disciplinary and multi-objective problem covering water and electric power distribution. These kinds of optimization have some first separate attempts for water networks in (Azoumah et al., 2012; Bieupoude, 2011; Wechsatol, Lorente, & Bejan, 2006) and for electric power distribution in (Arion et al., 2003a; Arion et al., 2003b). The novel suggestions proposed here rely on the coupling of both the two networks. Such study does not aim to perform a complete optimal design, but could highlight the trends of the optimal architectures of these urban systems that planners can refer to for their city planning decisions. In the same way, city street plans can be coupled (with water and electric power networks) in a geometric and multi-scale computer-based model for an integrated design as decision support tool for sustainable urban systems design.

4.1. Remarks on the illustrative application

4.3. Closing notes

In Section 3.2, DH representing head losses due to mechanical irreversibilities of the system, are minimized subject to a water quality constraint by keeping in mind that minimizing flow resistances under an operational constraint could result in better geometries, as stated by the constructal approach (Bieupoude, 2011). As optimization result, an optimal geometry of T-shaped networks is found for two levels of construction (construct 1 and construct 2). A geometric characteristic and diameter distribution laws of this optimal structure are found by using the Lagrangian multipliers method. Though this mathematical resolution can be used without evoking the constructal design, the idea to vary the geometry, from construct 1 to construct 2, or from fopt1 to fopt2 derives from the multi-scale design, promoted by the constructal method. This illustrative application permitted to introduce the geometric and multi-scale optimization method known as the constructal design. However authors do not envisage limiting the philosophy of constructal theory to this. It is in fact out of doubt that flow configurations as evoked here, (design, geometry), are handled in the direction of flow resistance minimization subject to given constraints. It is the view of constructal theory, that better flow configurations can be reach by the principle of maximizing ‘’global flow access in systems that are free to morph under global constraints’’

The operational experience of water distribution networks for more than fifty years and many recent studies have shown that analytical approaches coupled with computer-based methods have proven to be key routes for WDN optimization subject to mechanical, environmental and economic constraints. Extensive efforts have been pursued in the field of design and modeling to reach accurate optimal solutions through robust algorithms and to predict unknown working states of the WDN for chemical, biological and mechanical aspects. Maybe this big amount of researches on the domain has convinced engineers that water distribution networks are now very-well optimized systems in models and computer applications so that the limits of the optimization could not move ahead. Nonetheless, the need of good initial inputs for the iterative algorithms imposes to apply prior optimization methods capable of finding these inputs. The geometric, multi-scale and multi-criteria optimization method called the constructal design can help in finding pertinent inputs for these iterative computerbased methods. Therefore, next generations of water distribution network design models should consider the following points: (a) stressing on the geometry of the networks as proposed by the constructal approach; (b) coupling constructal method and other optimizing algorithms such as memetic algorithm, heuristic ones

442


or fuzzy optimization methods which seems to be non-negligible improving and promising routes in the design and optimization of WDN. Authors do not overlook certain reservations: commonly use of continuous variables in the objective function (Azoumah, Neveu, & Mazet, 2006; Azoumah, Neveu, & Mazet, 2007; Azoumah et al., 2012; Bejan & Lorente, 2008; Ghodoossi, 2004; Rocha, Lorente, & Bejan, 2009; Wechsatol et al., 2002; Wechsatol et al., 2006), freedom to morph, the non-guarantee for constructal design to improve the flow performance if the internal complexity of the flow area is increased (Ghodoossi, 2004) or its trend to not necessarily improve the flow performance if the internal branching of the flow field is increased (Kuddusi & Eg˘rican, 2008). But the strength of the constructal design has been experienced and found too much high (Bejan, 2000; Bejan & Lorente, 2008; Bieupoude, 2011) in engineering. Coupling it with computer based methods seems to be a promising future route for the optimization of urban systems such as drinking water or electric power distribution networks. Acknowledgements The International Institute for Water and Environmental Engineering 2iE, 01 BP 594 Ouagadougou 01, Burkina Faso (www.2ieedu.org), and its financial partners are gratefully acknowledged for their supports that permitted to successfully achieve this work.

DH1 Dh1 DH2 ¼ ¼ ¼ 2c Dh1 DH2 Dh2

By introducing these expressions of k1 (defined in Eq. (A.3)) in Eq. (A.2a) and in Eq. (A.2b), we obtain

8 Qn > > > a þ m2a Dmh1 < h1 n > Q H > > : a þ m2a Dm1 H1

h ¼ H

h1

ðA:1Þ

> > > Q nh pDh > 1 1 > ¼ 2k1 4Q 1 212 m a Dmþ1 > > 22 h1 > h > 1 > > > n > Q p D > h2 h 1 > ¼ 2k1 4Q 2 213 > m a Dmþ1 > 23 h2 > h2 > > > 2 2 > > Dh DH D2H > > 1 1 2 1 1 > H þ þ > Q H 21 Q H 22 Qh > 1 2 1 > > > : hH ¼ ST

D2h

þQ

2

h2

1 23

4t

h¼ p

nþ1 2m a Q H1

p Dmþ2 H1

¼

nþ1 2m a Q H2

p Dmþ2 H2

nþ1 2m a Q h1

p Dmþ2 h1

¼

p Dmþ2 h2

1

1 23

2

þ 212

D m Q n h H2 1 þ DH2 Qh 22 m 2Q n 2 1 21

Dh2 Dh 1

ðA:6Þ

h1

Qh

2

ðA:7Þ

fopt1 ¼ fopt2 ¼ fopt ¼ 2nþ1mc

0

ðA:7 Þ

By considering Eq. (A.2h), and Eq. (A.7), the expression of H can be obtained

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ST ðfopt Þ1

ðA:8Þ

And then

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ST ðfopt Þ1

ðA:9Þ

! ffi D2H1 Q h2 1 D2H2 Q h2 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ST ðfopt Þ1 þ 2 2 Q 1 2 Dh2 H1 2 D h2 Q H 2 2 ! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 D h1 Q h2 1 4tQ h2 1 þ 3 fopt ST ðfopt Þ1 ¼ þ 2 Q 2 Dh pD2h 2 h1 2 2

ðA:10Þ

2

From this, we can write

4tQ h2

ðA:11Þ

pD2h2

1=2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=2 1 ST ðfopt Þ ðð223c4 þ 22c3 Þ þ fopt ð222c4 þ 23 ÞÞ

ðA:12Þ

References

ðgÞ

nþ1 2m a Q h2

H1

Qh

We finally obtain Dh2, DH1 and DH2 from Eq. (A.4) and from Eq. (A.12).

ðhÞ

¼

h2

DH

ðA:2Þ

From Eq. (A.2) one can express k1 as follows

k1 ¼

D m Q n

Dh2 ¼ ð4tQ h2 =pÞ

ðf Þ 1 22

ðA:5Þ ¼ k2 h

And then

ðcÞ

ðeÞ

¼ k2 H

By replacing by its expression c = (n + 1)/(m + 2).we verify that

¼

ðbÞ

ðdÞ

1 22

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð223c4 þ 22c3 Þ ST ðfopt2 Þ1 þ ð222c4 þ 23 Þfopt ST ðfopt Þ1

ðaÞ

2

þ

By multiplying Eq. (A.2g) by the quantity Q h2 =D2h2 we obtain

By differentiating Lg with respect to DH1, DH2, Dh1, Dh2, h, H, k1 and k2, and cancelling the derivatives, we obtain

H2

1 21

Q nH 2 Dm H2

h2

1 23

h 23n13mc þ 2nmc2 ¼ ¼ fopt2 H opt 23 þ 22n22mc

h ¼ fopt

h2

2 8 n pDH 1 pD2H 1 QH Q nH > 1 1 2 1 1 2 > þ k ¼ k2 h a þ a þ m m > 1 1 2 1 2 4Q 4Q DH 2 DH 2 > H1 2 H2 2 > 1 2 > > > > > pD2h pD2h Q nh Q nh > 1 1 2 1 1 1 2 1 > > > a Dmh 22 þ a Dmh 23 þ k1 4Q h1 22 þ 4Q h2 23 ¼ k2 H > 1 2 > > > > > Q nH pDH > 1 1 > m a Dmþ1 ¼ 2k1 4Q H1 211 > > 21 > 1 H1 > > > > > Q nH p D > H 1 2 < m a mþ1 ¼ 2k1 4Q H2 212 D 22

þ Dm2

By considering Eq. (A.4), we finally obtain

H¼

! ! Q nH 1 Q nH 1 Q nh 1 Q nh 1 Lg ¼ a m1 1 þ a m2 2 H þ a m1 2 þ a m2 3 h DH 1 2 DH2 2 D h1 2 Dh2 2 1 0 2 2 pDH 1 pDH 1 1 2 B 4Q H1 21 þ 4Q H2 22 Hþ C C B þ k1 B 2 C þ k2 ðhH ST Þ A @ pDh 1 pD2h 1 1 2 h t þ 4Q 4Q 22 23

Q nh

1 22

From Eq. (A.5), when we eliminate k2, we can obtain by multiplying the numerator and denominator by the quantity (Dh2)m /(Qh2)n

Appendix A In the same manner as in Eqs. (10) and (11), the lagrangian function writes as follows

ðA:4Þ

ðA:3Þ

By combining terms of Eq. (A.3), one can write (given that QH1/Qh1 = 2, QH2/Qh2 = 2 et Qh1/QH2 = 2 because of the symmetry)

Akiba, H. (1982). Research in development of urban information systems. Computers, Environment and Urban Systems, 7, 41–51. Alperovits, E., & Shamir, U. (1977). Design of optimal water distribution systems. Water Resources Research, 13(6), 885–900. Arion, V., Cojocari, A., & Bejan, A. (2003a). Constructal tree shaped networks for the distribution of electrical power. Energy Conversion and Management, 44, 867–891. Arion, V., Cojocari, A., & Bejan, A. (2003b). Integral measures of electric power distribution networks: load-length curves and line-network multipliers. Energy Conversion and Management, 44, 1039–1051. Augugliaro, A., Dusonchet, L., & Riva-Sanseverino, E. (1998). Service restoration in compensated distribution networks using a hybrid genetic algorithm. Electric Power Systems Research, 46(1), 59–66.

P. Bieupoude et al. / Computers, Environment and Urban Systems 36 (2012) 434–444 Azoumah, Y., Bieupoude, P., & Neveu, P. (2012). Optimal design of tree-shaped water distribution network using constructal approach: T-shaped and Y-shaped architectures optimization and comparison. International Communications in Heat and Mass Transfer, 39, 182–189. Azoumah, Y., Mazet, N., & Neveu, P. (2004). Constructal network for heat and mass transfer in a solid-gas ractive porous medium. International Journal of Heat and mass transfer, 47, 2961–2970. Azoumah, Y., Neveu, P., & Mazet, N. (2006). Constructal design combined with entropy generation minimization for solid-gas reactors. International Journal of Thermal Sciences, 45(7), 716–728. Azoumah, Y., Neveu, P., & Mazet, N. (2007). Optimal design of thermochemical reactors basing on constructal approach. AIChE Journal, 53(5), 1257–1266. Babbitt, H. E., & Doland, J. J. (1931). Water supply engineering. New York, NY: McGraw-Hill. Bai, D., Pei-jun Yang, P., & Song, L. (2007). Optimal design method of looped water distribution network. Systems Engineering – Theory and Practice, 27, 137–143. Baños, R., Bans, R., Gil, C., Reca, J., & Montoya, F. (2010). A memetic algorithm applied to the design of water distribution networks. Applied Soft Computing, 10, 261–266. Basha, H. A., & Kassab, B. G. (1996). Analysis of water distribution systems using a perturbation method. Journal of Applied Mathematical Modeling, 20, 290–297. Bedjaoui, A., Achour, P., & Bouziane, M. (2005). New approach for the calculation of the economical diameter in the discharge pipe. Courrier du Savoir, 6, 141–145. Bejan, A. (2000). Shape and structure, from engineering to nature. Cambridge: Cambridge University Press. Bejan, A., & Lorente, S. (2005). La Loi Constructale. Paris: L’Harmattan. Bejan, A., & Lorente, S. (2007). Constructal tree-shaped flow structures. Applied Thermal Engineering, 27, 755–761. Bejan, A., & Lorente, S. (2008). Design with constructal theory. Hoboken: Wiley. Bejan, A., & Lorente, S. (2011). The constructal law and the evolution of design in nature. Physics of Life Reviews, 8, 209–240. Bejan, A., & Marden, H. (2009). The constructal unification of biological and geophysical design. Physics of Life Reviews, 6(2), 85–102. Bejan, A., Rocha, L. A. O., & Lorente, S. (2000). Thermodynamic optimization of geometry: T- and Y-shaped constructs of fluid streams. International Journal of Thermal Science, 39, 949–960. Bender, C., Milton, K., Pinsky, S., & Simmons, J. (1989). New perturbative approach to nonlinear problems. Journal of Mathematical Physics, 30, 1447–1455. Bhave, P. R. (1988). Calibrating Water Distribution Network Models. Journal of Environmental Engineering ASCE, 114(1), 120–136. Bieupoude, P. (2011). Approche constructal pour l’optimisation de réseaux hydrauliques. Thèse de doctorat, Université de Perpignan et Institut International d’Ingénierie de l’Eau et de l’Environnement : Soutenue le 16/12/ 2011 à Ouagadougou, Burkina Faso. Bieupoude, P., Azoumah, Y., & Neveu, P. (2011). Environmental optimization of treeshaped water distribution networks, Water Resources Management VI. In: Proc. of the 6th int. conf. on sustainable water resources management, Riverside, California 2011, WIT Press. Bolognesi, A., Bragalli, C., Marchi, A., & Artina, S. (2010). Genetic heritage evolution by stochastic transmission in the optimal design of water distribution network. Journal of Advances in Engineering Software, 41, 792–801. Boulos, P. F., Rossman, L., & Vasconcelos, J. (1994). A comparison of methods for modeling water quality in distribution systems. AWWA, Annual Conference and Exposition, New York, NY. Buckles, P., & Frederick, E. (1992). Genetic algorithms. Los Alamitos, CA: The IEEE Computer Society Press. Carlier, M. (1972). Hydraulique Générale et appliquée. Paris: Eyrolles. Chase, D., Savic, D., & Walski, T. (2001). Haestad methods. Water distribution modeling, Haested method, Heasted press, USA. Chiplunkar, A. V., Mehndiratta, S. L., & Khanna, P. (1990). Analysis of looped water distribution networks. Environmental Software, 5, 202–206. Christodoulou, S., Deligianni, A., Aslani, P., & Agathokleous, A. (2009). Risk-based asset management of water piping networks using neurofuzzy systems. Computers, Environment and Urban Systems, 33, 138–149. Chu, C., Lin, M., Liu, G., & Sung, Y. (2008). Application of immune algorithms on solving minimum-cost problem of water distribution network. Journal of Mathematical and Computer Modeling, 48, 1888–1900. Colombo, A. F., Lee, P., & Karney, B. W. (2009). A selective literature review of transient-based leak detection methods. Journal of Hydro-Environment Research, 2, 212–227. Cunha, M., & Ribeiro, L. (2004). Tabu search algorithms for water network optimization. European Journal of Operational Research, 157, 746–758. Cunha, M., & Sousa, J. (1999). Water distribution network design optimization: Simulated annealing approach. Journal of Water Resources Planning and Management ASCE, 125, 215–221. Dawkins, R. (1976). The selfish gene. New York: Oxford University Press. Ducrot, R., Le Page, C., Bommel, P., & Kuper, M. (2004). Articulating land and water dynamics with urbanization: an attempt to model natural resources management at the urban edge. Computers, Environment and Urban Systems, 28, 85–106. Duh, J., & Brown, D. G. (2007). Knowledge-informed Pareto simulated annealing for multi-objective spatial allocation. Computers, Environment and Urban Systems, 31, 253–281. Dzialowski, A. R., Smith, V. H., Huggins, D. G., deNoyelles, F., Lim, N., Baker, D. S., et al. (2009). Development of predictive models for geosmin-related taste and odor in Kansas, USA, drinking water reservoirs. Water Research, 43, 2829–2840.

443

Dziegielewski, B., & Baumann, D. D. (2011). Predicting future demands for water. Treatise on Water Science, chap. 1.10, 163–188. Eiben, A. E., Raué, P. E., & Ruttkay, Zs. (1994). Genetic algorithms with multi-parent recombination. In: Proceedings of the 3rd conference on parallel problem solving from nature, number 866 in LNCS, 78–87, Spring-Verlag. Eslami, M., & Jafarpur, K. (2012). Thermal resistance in conductive constructal designs of arbitrary configuration: A new general approach. Energy Conversion and Management, 57, 117–124. Evatt, B. S. Jr., (1984). New computer graphic tools for transportation planners. Computers, Environment and Urban Systems, 9, 21–32. Fonseca, C., & Fleming, P. (1995). An overview of evolutionary algorithms in multiobjective optimisation. Evolutionary Computation, 3(1), 1–16. França, P., Mendes, A., & Moscato, P. (1999). Memetic algorithms to minimize tardiness on a single machine with sequence-dependent setup times. In: Proceedings of the 5th international conference of the decision sciences institute. Athens, Greece. 1708–1710. Fujiwara, O., & Khang, D. (1990). A two-phase decomposition method for optimal design of looped water distribution networks. Water Resources Research, 26(4), 539–549. Fujiwara, O., & Khang, D. (1991). Correction to a two-phase decomposition method for optimal design of looped water distribution networks. Water Resources Research, 27(5), 985–986. Gessler, J. (1985). Pipe network optimization by enumeration. In: H. Torno, (Ed.), Computer applications in water resources. ASCE Water Resources planning and management division conference, Buffalo, NY. Ghodoossi, L. (2004). Conceptual study on constructal theory. Energy Conversion and Management, 45(9–10), 1379–1395. Gil, C., Ortega, J., Montoya, M. G., & Baños, R. (2002). A mixed heuristic for circuit partitioning. Computational Optimization and Applications, 23(3), 321–340. Goldberg, D. E. (1989). Genetic Algorithms in Search, Optimization and Machine Learning. Boston, MA, USA: Addison-Wesley Longman Publishing Co., Inc.. Gupta, I., Bassin, J. K., Gupta, A., & Khanna, P. (1993). Optimization of water distribution systems. Environmental Software, 8, 101–113. Gupta, I., Gupta, A., & Khanna, P. (1999). Genetic algorithm for optimization of water distribution systems. Environnemental Modelling and Software, 14, 437–446. Haestad methods (1999). Essential hydraulic and hydrology. Waterbury: Heastad Press. Haestad Methods (2004). Computer applications in hydraulic engineering (6th ed.). Waterbury, CT: Haestad Press. Hart, A., & da Silva, K. (2011). Experimental thermal-hydraulic evaluation of constructal microfluidic structures under fully constrained conditions. International Journal of Heat and Mass Transfer, 54, 3661–3671. Hess, W. (1914). Das Princip des kleinsten Kraftverbrauches im Dienste hämodynamischer Forschung. Archiv für Anatomie und Physiologie, 1–62. Ignizio, J. P. (1980). An introduction to goal programming with applications in urban systems. Computers, Environment and Urban Systems, 5, 15–33. Izquierdo, J., Montalvo, T., Pérez, R., & Herrera, M. (2008). Sensitivity analysis to assess the relative importance of pipes in water distribution networks. Mathematical and Computer Modelling, 48, 268–278. Jeppson, R. W., & Davis, A. L. (1976). Pressure reducing valves in pipe network analyses. Journal of Hydraulic division, 102(7), 987–1001. Judea, P. (1984). Heuristics: Intelligent search strategies for computer problem solving. Addison-Wesley. Keedwell, E., & Khu, S. (2005). A hybrid algorithm for the design of water distribution networks. Engineering Applications of Artificial Intelligence, 18(4), 461–472. Keirstead, J., & Shah, N. (2011). Calculating minimum energy urban layouts with mathematical programming and Monte Carlo analysis techniques. Computers, Environment and Urban Systems, 35, 368–377. Kerneïs, A., Nakache, F., Deguin, A., & Feinberg, M. (1995). The effects of water residence time on the biological quality in a distribution network. Water Research, 29, 1719–1727. Kirkpatrick, S., Gelatt, C. D., & Vecchi, M. P. (1983). Optimization by simulated annealing. Science, 220, 671–680. Kirmeyer, G. J., Friedman, M., Martel, K. D., Noran, P. F., & Smith, D. (2001). Maintaining distribution system water quality. Journal of American Water Works and Association, 93(7), 62–73. Kizito, F., Mutikanga, H., Ngirane-Katashaya, G., & Thunvik, R. (2009). Development of decision support tools for decentralised urban water supply management in Uganda: An action research approach. Computers, Environment and Urban Systems, 33, 122–137. Klempous, R., Kotowski, J., Nikodem, J., & Ulasiewicz, J. (1997). Optimization algorithms of operative control in water distribution systems. Journal of Computaional and Applied Mathematics, 84, 91–99. Kuddusi, L., & Eg˘rican, N. (2008). A critical review of constructal theory. Energy Conversion and Management, 49(5), 1283–1294. Lage et al. (2008). Professor Adrian Bejan on his 60th birthday. International Journal of Heat and Mass Transfer, 51, 5759–5761. Lansey, K., & Mays, L. (1989). Optimization model for design of water distribution systems. In L. R. Mays (Ed.), Reliability analysis of water distribution systems. New York, NY: ASCE. Li, Y., Ang, K. H., Chong, G. C., Feng, W., Tan, K. C., & Kashiwagi, H. (2004). CAutoCSD – Evolutionary search and optimization enabled computer automated control system design. International Journal of Automation and Computing, 1(1), 76–88. Liao, H., & Tim, U. S. (1994). Interactive water quality modeling within a GIS environment. Computers, Environment and Urban Systems, 18, 343–363.

444


Lorente, S., & Bejan, A. (2005). Svelteness, freedom to morph, and constructal multi-scale flow structures. International Journal of Thermal Sciences, 44, 1123–1130. Lorente, S., Bejan, A., Al-Hinai, K., Sahin, A., & Yilbas, B. (2012). Constructal design of distributed energy systems: Solar power and water desalination. International Journal of Heat and Mass Transfer, 55, 2213–2218. Lorente, S., Wechsatol, W., & Bejan, A. (2002). Fundamentals of tree-shaped networks of insulated pipes for hot water and exergy. Exergy, An International Journal, 2(4), 227–236. Lorenzini, G., Corrêa, R., dos Santos, E., & Rocha, L. (2011). Constructal design of complex assembly of fins. Journal of Heat Transfer. http://dx.doi.org/10.1115/ 1.4003710. Lorenzini, G., Garcia, F. L., dos Santos, E. D., Biserni, C., & Rocha, L. A. O. (2012). Constructal design applied to the optimization of complex geometries: T-Yshaped cavities with two additional lateral intrusions cooled by convection. International Journal of Heat and Mass Transfer, 55, 1505–1512. Luo, L., & Tondeur, D. (2005). Optimal distribution of viscous dissipation in a multiscale branched fluid distributor. International journal of thermal sciences, 44, 1131–1141. Marti, R., Laguna, M., & Glover, F. (2006). Principles of scatter search. European Journal of Operational Research, 169(2), 359–372. Mehrgoo, M., & Amidpour, M. (2011). Constructal design of humidificationdehumidification desalination unit architecture. Desalinaion, 271, 62–71. Miguel, A. F. (2008). Constructal design of solar energy-based systems for buildings. Energy and Buildings, 40, 1020–1030. Miguel, A. F. (2010). Dendridic structures for fluid flow: Laminar, Turbulent and constructal design. Journal of Fluids and Structures, 26, 330–335. Miller, E. J., Hunt, J. D., Abraham, J. E., & Salvini, P. A. (2004). Microsimulating urban systems. Computers, Environment and Urban Systems, 28, 9–44. Moore, J. E., II, & Kim, T. J. (1995). Mills’ urban system models: Perspective and template for LUTE (Land Use/Transport/Environment) applications. Computers, Environment and Urban Systems, 19, 207–225. Murray, C. (1926). The physiological principle of minimum work: I. The vascular system and the cost of blood volume. Proceedings of the National Academy of Sciences, 12, 207–214. Mustonen, S. M., Tissari, S., Huikko, L., Kolehmainen, M., Lehtola, M. J., & Hirvonen, A. (2008). Evaluating online data of water quality changes in a pilot drinking water distribution system with multivariate data exploration methods. Water Research, 42, 2421–2430. Neema, M. N., & Ohgai, A. (2010). Multi-objective location modeling of urban parks and open spaces: Continuous optimization. Computers, Environment and Urban Systems, 34, 359–376. Newell, A., & Simon, H. (1976). Computer science as empirical inquiry: Symbols and search. Communication of the Association for Computing Machinery, 19, 113–126. Ozcan, E., & Onbasioglu, E. (2006). Memetic algorithms for parallel code optimization. International Journal of Parallel Programming, 35(1), 33–61. Park, K., & Kuo, A. Y. (1996). A multi-step computation scheme: Decoupling kinetic processes from physical transport in water quality models. Water Research, 30, 2255–2264. Pierro, F., Khu, S., Savic, D., & Berardi, L. (2009). Efficient multi-objective optimal design of water distribution networks on budget of simulations using hybrid algorithms. Environmental Modeling and Software, 24, 202–213. Poelmans, L., & Rompaey, A. V. (2010). Complexity and performance of urban expansion models. Computers, Environment and Urban Systems, 34, 17–27. Rocha, L., Lorente, S., & Bejan, A. (2009). Tree-shaped vascular wall designs for localized intense cooling. International Journal of Heat and Mass Transfer, 52, 4535–4544. Rossman, L. (2000). EPANET 2 user’s manual. Water supply and water resources division. National Risk Management Research Laboratory. US Environmental Protection Agency, Cincinnati, OH 45268.

Rossman, L., & Boulos, P. (1996). Numerical methods for modeling water quality in distribution systems: A comparison. Journal of Water Resources planning and management, 122(2), 137–146. Rossman, L., Boulos, P., & Altman, T. (1993). Discrete volume element method for network water quality models. Journal of water resources planning and Management ASCE, 119, 505–517. Rossman, L., Clark, R. M., & Grayman, W. M. (1994). Modeling chlorine residual in drinking-water distribution system. Journal of Environmental Engineering, 120(4), 803–820. Savic, D., Walter, G., Randall-Smith, M., & Atkinson, R. (2000). Large water distribution systems design through genetic algorithm optimization. In: Proceeding of the ASCE joint conference on water resources engineering and water resources planning and management, Minncapolis, USA. Savic, D., & Walters, G. (1997). Genetic algorithms for least-cost design of water distribution networks. Journal of Water Resources Planning and Management ASCE, 123(2), 67–77. Simpson, A. R., Dandy, G., & Murphy, L. (1994). Genetic algorithm compared to other techniques for pipe optimization. Journal of water resources planning and management ASCE, 120(4), 423–443. Syswerda, G. (1989). Uniform crossover in genetic algorithms. In J. D. Schaffer. Proceedings of the third international conference on genetic algorithms. Morgan Kaufmann. Tescari, S., Mazet, N., & Neveu, P. (2010). Constructal method to optimize solar thermochemical reactor design. Solar Energy, 84, 1555–1566. Tescari, S., Mazet, N., & Neveu, P. (2011). Constructal theory through thermodynamics of irreversible processes framework. Energy Conversion and Management, 52, 3176–3188. Todini, E. (2000). Looped water distribution networks design using a resilience index based heuristic approach. Urban Water, 2, 115–122. Tondeur, D., Fan, Y., & Luo, L. (2009). Constructal optimization of arborescent structures with flow singularities. Chemical Engineering Sciences, 64, 3968–3982. Tondeur, D., & Luo, L. (2004). Design and scaling laws of ramified fluid distributors by the constructal approach. Chemical Engineering Science, 59, 1799–1813. Tospornsampan, J., Kita, I., Ishii, M., & Kitamura, Y. (2007). Split-pipe design of water distribution network using simulated annealing. International Journal of Computer Systems Science and Engineering, 1(3), 153–163. Varma, K., Narasimhan, S., & Bhallamudi, S. (1997). Optimal design of water distribution systems using an NLP method. Journal of Environmental Engineering ASCE, 123(4), 381–388. Walski, T. (1985). State-of-the-art: Pipe network optimization. Computer applications in water resources. New York: Buffalo. Wechsatol, W., Lorente, S., & Bejan, A. (2001). Tree-shaped insulated designs for the uniform distribution of hot water over an area. International journal of Heat and Mass Transfer, 44, 3111–3123. Wechsatol, W., Lorente, S., & Bejan, A. (2002). Development of tree-shaped flows by adding new users to existing networks of hot water pipes. International Journal of Heat and Mass Transfer, 45, 723–733. Wechsatol, W., Lorente, S., & Bejan, A. (2004). Tree-shaped network with loops. International Journal of Heat and Mass Transfer, 48, 573–583. Wechsatol, W., Lorente, S., & Bejan, A. (2006). Tree-shaped flow structures with local junction losses. International Journal of Heat and Mass Transfer, 49, 2957–2964. World Health Organization (1996). Guidelines for drinking-water quality, vol. 2, second ed., Health criteria and other supporting information. WHO, Mastercom/ Wiener Verlag. Xiao, Q., Chen, L., & Sun, F. (2011). Constructal design for a steam generator based on entransy dissipation extremum principle. Science china Technological Sciences, 54, 1462–1468. Zotos, K. (2007). Performance comparison of Maple and Mathematica. Applied Mathematics and Computation, 188, 1426–1429.