Post on 05-Jan-2017
BENOIT ALLEN
OPTIMISATION D'ECHANGEURS DE CHALEUR : CONDENSEUR À CALANDRE, RÉSEAU D'ECHANGEURS
DE CHALEUR ET PRODUCTION D'EAU FROIDE
Mémoire présenté à la Faculté des études supérieures de l'Université Laval
dans le cadre du programme de maîtrise en génie mécanique pour l'obtention du grade de maître es science (M.Se.)
DEPARTEMENT DE GENIE MECANIQUE FACULTÉ DES SCIENCES ET GÉNIE
UNIVERSITÉ LAVAL QUÉBEC
2010
©Benoit Allen, 2010
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Résumé
La présente étude porte sur l'optimisation de systèmes thermiques servant soit à récupérer
de la chaleur ou à produire de la chaleur ou du froid. Essentiellement, le travail portera sur
les condenseurs de type tubes et calandre, sur les réseaux d'échangeurs de chaleur ainsi que
sur les systèmes de production d'eau froide. L'objectif ultime est de développer une
méthode permettant de déterminer le design minimisant les coûts reliés à l'achat et à
l'opération de ces systèmes thermiques. Pour atteindre cet objectif, on doit d'abord créer un
modèle mathématique permettant de calculer les surfaces d'échanges requises et les
puissances de pompage requises pour faire fonctionner un échangeur de chaleur. Basé sur
des relations analytiques et empiriques, le modèle doit tenir compte des variables design
considérées dans le problème, soit une dizaine de paramètres géométriques et le régime
d'opération. Il s'agit d'identifier les valeurs à accorder à chacune de ces variables de design
afin de faire le meilleur compromis entre la minimisation des surfaces d'échange de chaleur
requises et la quantité d'énergie requise pour faire fonctionner les systèmes. Autrement dit,
on cherche à minimiser le coût total, constitué du coût d'achat du matériel et des coûts
d'opération. Une fois cette démarche réalisée pour le condenseur à tubes et calandre, on
applique une méthode similaire pour optimiser une série d'échangeurs de chaleur dans le
cas des réseaux d'échangeurs de chaleur et finalement pour un cycle de réfrigération
composé de deux échangeurs, un condenseur et un évaporateur, ainsi qu'un compresseur.
Étant donné le nombre important de variables de design considéré pour chacun de ces
problèmes, le nombre total de design possible est trop élevé pour calculer le coût de chacun
d'entre eux et choisir le meilleur. Cela serait trop coûteux en temps de calcul. C'est
pourquoi nous ferons appel à l'utilisation d'algorithmes génétiques. Ces derniers nous
permettront d'identifier avec une excellente probabilité le design optimal et ce, dans un laps
de temps acceptable en pratique. La méthode est finalement validée grâce à des exemples
d'application.
Ill
Abstract
In this study, we work on three types of thermal systems: shell-and-tube condensers, heat
exchanger networks and refrigeration systems. These systems all have the common
characteristics to imply shell-and-tube heat exchangers. Our goal is to develop a method to
optimize the design of these systems. The optimal configuration must satisfy a given task at
a minimum cost, including purchase costs and energy costs. The development of this work
is separated under three scientific papers. In order to reach our main objective, we first
create a mathematical model to compute required heat transfer surface areas and pumping
power required for a given process for a condenser. This model accounts for the geometry
characteristics of the exchanger and the steady-state operating conditions. The condenser
model will be used in the two other parts of the work: heat exchanger networks
optimization and refrigeration systems optimization. For this last part, an evaporator model
is also created to complete the vapour-compression cycle. In each case, the performance of
the economic optimization is made using a genetic algorithm. These algorithms will enable
the determination of the best heat exchanger geometries and the best operating conditions.
The procedure developed in this work is validated with some test cases.
IV
Avant-propos
Je tiens tout d'abord à remercier mon directeur de recherche Louis Gosselin. Louis m'a non
seulement aidé à réaliser ce mémoire, mais tous les conseils qu'il m'a donné m'ont permis
de développer des méthodes de travail et une rigueur qui me seront utiles tout au long de
ma vie et de ma carrière d'ingénieur. Il a fait preuve d'une patience remarquable et il a été
un excellent guide dans tous les travaux auxquels j 'ai participé avec lui. Je suis également
reconnaissant envers Myriam Savard-Goguen, dont les travaux ont permis la publication
d'un des articles utilisés dans ce mémoire. Sans la contribution de Myriam, la réalisation de
cette étude n'aurait pu se concrétiser.
Je remercie aussi tous les étudiants du Laboratoire de Transfert Thermique et
d'Énergétique que j 'ai eu la chance de côtoyer au cours de mes travaux. Ces personnes ont
rendu mon séjour au LTTE plus qu'agréable et je conserve pour eux, un sentiment de
grande amitié. Je garderai toujours un excellent souvenir de mes journées au LTTE grâce à
ces personnes: Yohann Chataigner, François Mathieu-Potvin, John Niederreiter, Marie-
Andrée Julien, Maxime Tye-Gingras, Jonathan Dallaire, Jean-Michel Leblanc, Mai Thi Do,
Simon Bélanger, Axel Arnaud et Cassandre Nowicki.
Je ne pourrais continuer sans remercier ma conjointe, Anne-Marie. Tout au long de
ma maîtrise, elle a été à mes côtés pour m'encourager et me supporter. Je suis également
très reconnaissant envers mes parents qui m'ont toujours supporté dans mes études et
particulièrement pour mes travaux de recherche à la maîtrise. Ils ont toujours été près de
moi et ils ont toujours su me supporter et m'accompagner dans mes études. Ces personnes
ont toutes joué un rôle important dans l'atteinte de mes objectifs.
Finalement, la réalisation de ce travail a été rendu possible grâce au support
financier des Fonds Québécois pour la Recherche en Nature et en Technologies (FQRNT).
Table des matières
Résumé ii Abstract iii Avant-propos iv Table des matières v Liste des figures vii Liste des tableaux viii Nomenclature 1 Chapitre 1 4 Introduction 4
Problématique 4 Objectifs 6 Méthodologie 7
Chapitre 2 10 Article #1 10 Abstract 11
2.1. Introduction 12 2.2. Objective function 12 2.3. Heat transfer calculations 13
2.3.1 Heat transfer coefficients with in-tube condensation 17 2.3.2 Heat transfer coefficients with shell-side condensation 20
2.4. Pressure drop calculations 20 2.4.1 Pressure drop with in-tube condensation 21 2.4.2 Pressure drop with shell-side condensation 23
2.5. Design variables and procedure for determining the cost of a design 24 2.6. Optimization procedure with genetic algorithms 27 2.7. Test cases 27 2.8. Conclusion 31
Chapitre 3 33 Article # 2 33 Abstract 34
3.1. Introduction 35 3.2. HEN Problem formulation and design procedure 36 3.3. Description of the HE design problem and genetic algorithms 40
3.3.1 Objective function 40 3.3.2 HEs design variables 41 3.3.3 Optimization of HEs using genetic algorithms 45
3.4. Extension of Ref. [53] to HEs with partial condensation 46 3.5. Test cases 49 3.6. Conclusions 60
Chapitre 4 62 Article # 3 62 Abstract 63
VI
4.1. Introduction 64 4.2. Objective function and design variables 65 4.3. Condenser model 74 4.4. Evaporator model 74
4.4.1 Heat transfer calculations 75 4.4.2 Pressure drop calculations 78
4.5. Compressor model 80 4.6. Optimization approach 81 4.7. Test cases and results 82 4.8. Conclusions 90
Chapitre 5 91 Discussion et conclusions 91 Bibliographie 94 Annexe A 99
Calcul du coût d'un condenseur 99 Annexe B 108
Calcul du coût d'un évaporateur 108 Annexe C 115
Calcul du coût d'un compresseur 115 Annexe D 117
Optimisation d'un système de réfrigération 117 D.l Script d'optimisation 117 D.2 Calcul des coûts d'opération et d'achat du système 119
Annexe E 123 Optimisation d'un réseau d'échangeurs de chaleur 123
Vil
Liste des figures
Figure 2.1 The schematic representation of a straight-tube heat exchanger with one pass on tube shell and condensation on tube side 15
Figure 2.2 Schematic representation of a vertical heat exchanger when condensation occurs on shell side 16
Figure 2.3 Thermal circuits to determine for Twj and Twj 19 Figure 2.4 Geometrical design variables 26 Figure 3.1 Pinch analysis for heat exchanger network optimization 38 Figure 3.2 Overall procedure for total cost estimation 39 Figure 3.3a Temperature of cold and hot fluids in shell-and-tube heat exchanger without
condensation 43 Figure 3.3b Temperature of cold and hot fluids in shell-and-tube heat exchanger with
condensation of the hot fluid 44 Figure 3.4 Minimum heat exchanger network total cost as a function of minimum
temperature difference for test case #1 51 Figure 3.5 Optimal heat exchanger network design for test case #1 52 Figure 3.6 Minimum heat exchanger network total cost as a function of minimum
temperature difference for test case #2 56 Figure 3.7 Optimal heat exchanger network design for test case #2 57 Figure 4.1 Schematic representation of the vapor compression refrigeration cycle 66 Figure 4.2 Temperature-entropy diagram of an ideal vapor compression cycle 67 Figure 4.3 Schematic representation of total cost calculation procedure 73 Figure 4.4 Nucleate and convective boiling in evaporating two-phase flow 76 Figure 4.5 Constant quality separation of the evaporating refrigerant flow 77 Figure 4.6 Optimal cost distribution for test case #1 85 Figure 4.7 Optimal cost distribution for test case #2 86
vin
Liste des tableaux
Table 2.1 Process requirements for case study #1 (// = 5000 h year1, e = 0.1 SkW^If', n = 20 years, / = 0.05, Tjpump = 0.85) 28
Table 2.2 Process requirements for case study #2 (H = 5000 h year-1, e = 0.1 $kW~'lf \ n = 20 years, / = 0.05, rjpump = 0.85) 29
Table 2.3 Minimal cost heat exchanger geometries as found by the GA 30 Table 3.1 Process requirements for test case #1 50 Table 3.2 Stream data for shell-and-tube heat exchangers of the optimal heat exchanger
network for test case #1 53 Table 3.3 Optimal HE geometries as found by the GA for test case #1 optimal HEN 54 Table 3.4 Process requirements for test case #2 55 Table 3.5 Stream data for shell-and-tube heat exchangers of the optimal heat exchanger
network for test case #1 58 Table 3.6 Optimal HE geometries as found by the GA for test case #2 optimal HEN 59 Table 4.1 Design variables and their limit values for the two heat exchangers of the
problem 69 Table 4.2 Main features of the GA considered in this paper 82 Table 4.3 System requirements details for test case #1 and test case #2 84 Table 4.4 Optimal refrigeration cycle for test case #1 and test case #2 87 Table 4.5 Optimal refrigeration system characteristics for test case #1 88 Table 4.6 Optimal refrigeration system characteristics for test case #2 89
Nomenclature
A surface, m2
B coupe des chicanes, % CCU coût annuel du fluide de refroidissement (cold utility), $/an CHU coût annuel du fluide de chauffage (hot utility), $/an Co nombre de convection COST coût énergétique des fluides de refroidissement/chauffage, $/(kW-h) CP débit capacitif, W/K cp capacité calorifique à pression constante, J/(kg-K) D diamètre de calandre, m d diamètre de tube, m E puissance de pompage, W e coût de l'électricité, $/(kW-h) F facteur de différence de température effective Fr nombre de Froude / coefficient de friction g accélération gravitationnelle, m/s H période d'opération annuelle, h h coefficient de convection, W/(m2-K) HEAT chaleur, W I taux d'intérêt annuel, % i enthalpie spécifique, J/kg k conductivité thermique, WV(m-K) L longueur, m m débit massique, kg/s N nombre de tubes n durée de vie, an np nombre de passes de tubes OC coût d'opération annuel, $/an P pression, Pa p distance entre les tubes PC coût d'achat annualisé, $ Q taux de transfert de chaleur, W q " flux de chaleur, W/m R résistance d'encrassement, (m -K)/W Re nombre de Reynolds .s entropie spécifique, J/(kg-K) T température, °C, K t temps, h TC coût total annualisé, $/an U coefficient de transfert de chaleur global, W/(m2-K) V vitesse, m/s
w puissance, W We nombre de Weber x qualité
Symboles grecs
S facteur de coût <P facteur d'annualisation, an"1
n efficacité M viscosité dynamique, Pas p densité, kg/m3
a tension de surface N/m W rapport de pertes de pression
Indices
b chicane (baffle) c,h froid, chaud C condensation center centre cf zone d'écoulement perpendiculaire CU liquide de refroidissement (cold utility) comp compresseur cn condenseur ev évaporateur fg latent GO phase gazeuse seule GP vapeur surchauffée H élevé HU liquide de chauffage (hot utility) i, o entrée, sortie; intérieur, extérieur hg liquide, gazeux L bas lin logarithmique moyen LP liquide sous-refroidi LO phase liquide seule M matériel max maximum min minimum otl faisceau de tubes P pressure pump pompage ref réfrigérant s, t calandre (shell), tubes S,T initial (supply), objectif (target)
sat saturation T temperature TP biphasique w mur wf zone d'écoulement parallèle (window flow zone)
Chapitre 1
Introduction
Problématique
De nos jours, la valeur de l'énergie est en constante augmentation. Les besoins en énergie se
multiplient, conséquence de l'industrialisation et de l'augmentation de la population
mondiale. Les ressources énergétiques, quant à elles, ne se multiplient pas à un rythme
aussi important. Plusieurs sources d'énergie ne sont pas renouvelables et leur exploitation
engendre souvent pollution et rejets de gaz à effet de serre dans l'atmosphère. Depuis
maintenant quelques années, la population humaine prend justement conscience que cette
pollution est responsable de graves conséquences à l'échelle planétaire et que nous finirons
par épuiser les réserves de certaines ressources énergétiques. Plusieurs études scientifiques
ont d'ailleurs démontré que si l'humain continue à polluer comme il le fait actuellement, les
conséquences seront catastrophiques et irréversibles pour l'écosystème terrestre. Bien sûr,
plusieurs sources d'énergies renouvelables sont disponibles: l'énergie solaire,
l'hydroélectricité, l'énergie éolienne et la géothermie en sont quelques exemples. Toutefois,
ces sources d'énergies ont un potentiel limité et leur exploitation nécessite l'établissement
d'infrastructures particulières. Actuellement, l'énergie obtenue à partir de ressources
renouvelables ne peut répondre à tous nos besoins énergétiques. Économiquement parlant,
l'augmentation de la demande en énergie engendre directement une plus grande rareté et
donc une augmentation de ses coûts. Dans un tel contexte, il s'avère impératif de trouver
des solutions réalistes et efficaces afin d'utiliser l'énergie de manière intelligente. Il en va de
l'avenir de notre planète et de la survie de l'espèce humaine.
Une grande portion de l'énergie est utilisée dans l'industrie pour la production de
chaleur et la production de froid. Selon l'Institut International du Froid, il y aurait dans le
monde plus de 300 millions de mètres cubes d'espace réfrigéré et la production de froid à
elle seule est responsable de plus de 15% de la consommation d'électricité dans le monde
[1,2]. D'autre part, selon les études d'Hydro-Québec, le coût d'électricité des systèmes
thermiques utilisés dans les secteurs de l'alimentation, du plastique, de l'imprimerie et de
l'entreposage frigorifique peut atteindre 60% de la facture énergétique totale de ces
entreprises [3]. De plus, une grande partie de l'énergie utilisée par ces systèmes thermiques
est rejetée sous forme de chaleur dans l'atmosphère. On peut mettre en place des solutions
pour récupérer une partie de la chaleur perdue et on peut concevoir les systèmes de manière
à ce qu'ils répondent à un besoin en utilisant un minimum d'énergie dans le but
d'économiser cette dernière. Valoriser des solutions permettant un meilleur usage de
l'énergie fait d'ailleurs partie des priorités que s'est donnée l'Agence de l'efficacité
énergétique du Québec [4].
Pour être réalistes, les solutions envisagées doivent toutefois être conçues pour
produire ou récupérer un maximum d'énergie à un coût minimal. Pour une entreprise,
récupérer un maximum de chaleur ou avoir des systèmes thermiques consommant un
minimum d'électricité sont avantageux au niveau des coûts d'exploitation mais on doit
également tenir compte que plus les systèmes sont énergétiquement efficaces, plus leur coût
d'achat tend à augmenter. C'est un aspect du problème considérable car les entreprises
choisissent la plupart du temps leurs systèmes en fonction du coût total et non en fonction
de la consommation énergétique seulement. La réalité économique fait donc en sorte que
dans le choix d'un système, il faudra faire un compromis entre l'économie d'énergie et le
coût d'achat des systèmes afin d'avoir un coût global minimal.
Dans ce mémoire, nous allons nous intéresser à des systèmes thermiques
consommant beaucoup d'énergie. L'étude, divisée en trois articles scientifiques, portera
dans un premier temps sur la modélisation de condenseurs à tubes et à calandre, (prenez
note que les chapitres subséquents seront donc présentés en anglais, soit la langue dans
laquelle les articles ont été publiés. De plus, quelques éléments ont été insérés dans les
articles afin de clarifier certains aspects de la démarche.) Comme on le verra plus loin, on
utilisera dans les deux autres articles ce modèle. Dans un deuxième temps, nous étudierons
le design des réseaux d'échangeurs de chaleur. Ces systèmes sont utilisés pour réchauffer
plusieurs écoulements de fluides froids à partir d'écoulement plus chauds dont on doit
évacuer la chaleur. Finalement, nous nous intéresserons aux cycles de réfrigération. Tous
les systèmes étudiés ont la caractéristique commune d'utiliser des échangeurs de type tubes
et calandre. Nous tenterons dans chaque cas de minimiser le coût total des systèmes
incluant les coûts d'achat et d'opération. Un grand nombre de variables géométriques
caractérisent ces échangeurs et pour une application donnée, il est important de choisir les
valeurs accordées à ces variables de manière à minimiser les puissances de pompage
requises et maximiser le transfert de chaleur entre les fluides.
Objectifs
L'objectif principal de ce mémoire consiste à développer une méthode pour maximiser les
performances et minimiser les coûts des réseaux d'échangeurs de chaleur et des cycles de
réfrigération en optimisant la géométrie des échangeurs de chaleur utilisés et le régime
d'opération (pressions, débits, températures, écarts de température minimum).
Pour atteindre l'objectif principal de ce mémoire, nous devrons d'abord, dans le
deuxième chapitre, développer un modèle permettant de calculer les coûts reliés à l'achat et
l'opération d'un échangeur de chaleur avec condensation d'un fluide. D faudra alors trouver
une façon de quantifier le transfert de chaleur dans un écoulement à changement de phase.
Ce modèle sera utilisé pour accomplir la tâche des chapitres trois et quatre.
Par la suite, nous devrons identifier les méthodes de design des réseaux d'échangeurs
de chaleur et implanter ces méthodes numériquement. L'objectif est d'obtenir un modèle
permettant l'optimisation des échangeurs et des condenseurs du réseau d'échangeurs. Nous
pourrons alors déterminer pour une application donnée, la différence de température au
point de pincement offrant une combinaison d'échangeurs présentant un coût total minimal.
Le point de pincement est le point où la différence de température est minimale entre deux
écoulements. Dans le cas des réseaux d'échangeurs de chaleur, le point de pincement se
situe là où la différence de température est minimale entre la courbe composite des
écoulements de fluides froids et la courbe composite des écoulements de fluides chauds.
Une courbe composite représente la somme de plusieurs écoulements individuels avec en
abscisses l'enthalpie et en ordonnées la température. Pour un intervalle de température
donné, la courbe composite a un débit capacitif égal à la somme des débits capacitifs des
écoulements individuels dans cet intervalle [5].
La dernière partie du travail consistera à modéliser un cycle de réfrigération de
compression de vapeur. Pour ce faire, nous devrons d'abord développer un modèle
d'évaporateur puis avec le modèle de condenseur obtenu précédemment, nous combinerons
les deux échangeurs à un modèle de compresseur pour modéliser un cycle complet.
Ultimement, le but sera d'utiliser ce modèle afin de déterminer le cycle optimal pour une
application donnée.
Méthodologie
Le principal outil de travail utilisé pour atteindre les objectifs de ce mémoire est Matlab.
Tous les modèles permettant de calculer les coûts d'achat et d'opération des systèmes seront
implantés dans ce logiciel.
Afin d'atteindre les objectifs mentionnés précédemment, nous devrons d'abord
déterminer toutes les variables géométriques en jeu et établir les valeurs limites que nous
accorderons à ces variables. Pour l'étude du condenseur, les seules informations de départ
dont nous disposons sont les conditions d'opération. On identifiera donc les relations
permettant d'exprimer les valeurs des coefficients de transfert de chaleur et les puissances
de pompage par unité de longueur dans les échangeurs en fonction des variables
géométriques et des conditions d'opération qui varient selon le cas étudié. Les relations
empiriques disponibles dans la littérature seront utilisées. Les valeurs des coefficients de
transfert de chaleur et des puissances de pompage sont requises pour dimensionner
l'échangeur de chaleur et déterminer les coûts d'achat du matériel. Nous serons alors en
mesure d'implanter un modèle dans Matlab. Ce modèle pourra nous donner les coûts du
système pour une géométrie donnée. Étant donné que notre but consiste à optimiser cette
géométrie, nous couplerons le modèle obtenu à un algorithme génétique. Ainsi, à partir du
régime d'opération (débits massiques des fluides, températures d'entrée et de sortie,
pressions d'opération) associé à un cas spécifique, nous serons en mesure d'obtenir la
géométrie de l'échangeur de chaleur offrant un coût total minimal dans cette situation.
L'algorithme génétique sera ici utilisé puisqu'il offre l'avantage d'identifier la solution
optimale en ne calculant qu'une petite fraction de tous les designs possibles.
Pour l'étude des réseaux d'échangeurs, il faudra d'abord déterminer de quelle façon
nous couplerons les fluides dans les échangeurs de chaleur. Pour y parvenir, nous
utiliserons une méthode basée sur l'analyse de pincement. Cette méthode permet de
récupérer un maximum de chaleur tout en respectant un écart de température minimal entre
les fluides chaud et froid dans les échangeurs de chaleur. Nous implémenterons cette
méthode dans Matlab de telle sorte qu'à partir des températures d'entrée, des températures
cibles et des débits des différents fluides impliqués dans le système, notre code sera en
mesure de déterminer le nombre d'échangeurs de chaleur à utiliser ainsi que les fluides à
utiliser dans chacun de ces échangeurs. On pourra alors déterminer le design optimal de
chaque échangeur et de chaque condenseur avec un algorithme génétique et cumuler les
coûts des échangeurs pour un coût global minimal. Encore une fois, l'algorithme génétique
sera utilisé pour l'optimisation. Pour un cas donné, on déterminera à partir de cet algorithme
la combinaison d'échangeurs optimale, c'est à dire la géométrie et la dimension optimale de
chaque échangeur et ce pour chaque différence de température minimale considérée. On
pourra utiliser les résultats obtenus pour dire avec quel écart de température on obtient un
coût minimal.
Le dernier objectif consiste à optimiser un cycle de compression de vapeur. Ce type
de cycle thermodynamique sert à répondre à des besoins en réfrigération et utilise un
condenseur. Les trois composantes principales de ce système sont les deux échangeurs de
chaleur, un condenseur et un évaporateur ainsi qu'un compresseur. Nous débuterons d'abord
par créer un modèle pour l'évaporateur. En utilisant, une démarche semblable à celle
utilisée dans le chapitre 2, consacré à la modélisation du condenseur, nous devrons utiliser
des relations mathématiques permettant de calculer les taux de transfert de chaleur et les
puissances de pompage requises. Ces relations devront être exprimées en fonction de la
géométrie de l'échangeur et de son régime d'opération. À partir de ce modèle, du modèle
obtenu au chapitre 2 et d'un modèle de compresseur simplifié, nous aurons tous les
éléments nécessaires pour modéliser le circuit thermique. El s'agira alors de coupler les trois
éléments de façon à respecter le régime du cycle. En combinant une fois de plus notre
modèle à un algorithme génétique, nous obtiendrons un outil d'optimisation permettant
d'optimiser la géométrie des deux échangeurs du cycle, les pressions d'opération et les
débits des fluides afin d'atteindre un coût total minimal pour une application particulière.
10
Chapitre 2
Article # 1
Titre:
Optimal geometry and flow arrangement for minimizing the cost of shell-and-
tube condensers
Co-auteurs:
Benoît Allen, Louis Gosselin
Journal:
International Journal of Energy Research, Volume 32, Pages 958 à 969
11
Abstract
This paper presents a model for estimating the total cost of shell-and-tube heat exchangers
with condensation in tubes or in the shell, as well as a designing strategy for minimizing
this cost. The optimization process is based on a genetic algorithm (GA). The global cost
includes the energy cost (i.e., pumping power) and the initial purchase cost of the
exchanger. The choice of the best exchanger is based on its annualized total cost. Eleven
design variables are optimized. Ten are associated with the heat exchanger geometry: tube
pitch, tube layout patterns, baffle spacing at the center, baffle spacing at the inlet and outlet,
baffle cut, tube-to-baffle diametrical clearance, shell-to-baffle diametrical clearance, tube
bundle outer diameter, shell diameter and tube outer diameter. The last design variable
indicates whether the condensing fluid should flow in the tubes or in the shell. Two case
studies are presented and the results obtained show that the procedure can rapidly identify
the best design for a given heat transfer process between two fluids, one of which is
condensing.
12
2.1. Introduction
Shell-and-tube heat exchangers are widely used in industry, seizing as much as 65% of the
market [5]. Therefore heat transfer and fluid flow within these heat exchangers (HEs) have
been studied extensively [6-13], and many empirical relations are available to estimate their
performance [5,14,15]. With these models, the geometry of shell-and-tube HEs have been
optimized, mainly for minimizing their cost for a given process [16], either by testing all
possibilities or with other procedures, including genetic algorithms [6,17-23].
Most of the times, the geometry optimization of shell-and-tube HEs is made for
single phase flows. Despite their importance in several applications (e.g., vapor heating
systems, refrigeration, heat pumps, and power cycles), the modeling, design, and
optimization of shell-and-tube HEs in the presence of phase change (i.e., ebullition or
condensation) has received far less attention. Botsch and Stephan developed a model to
predict pressure drop and vapor temperatures in a shell-and-tube condenser [24]. This
model was developed from the experimental studies of Alcock and Webb [25]. Browne and
Bansal showed the influence of tube surface geometry and coolant velocity on the overall
heat transfer coefficient [26]. Nevertheless, an integrated modeling of the effects of detailed
geometrical features on heat transfer and fluid flow in condensers, and the cost
minimization of condensers by optimizing their geometry is yet to be addressed. In this
paper, we develop a model for estimating the cost of shell-and-tube condensers with one
tube pass, based on empirical correlations. We proposed an optimization procedure that
determines whether condensation should occur in the tubes or in the shell for minimal cost.
The procedure is adapted from a genetic algorithm which was initially developed for
optimizing single phase HEs [16].
2.2. Objective function
The purchase cost of a HE is mostly governed by its total heat transfer surface area A.
Different empirical relations are available to associate a cost with a given surface area. In
this paper, we used the following relation to evaluate the purchase cost PC [5]:
PC = 3.28xl04| I 80
A \ »
13
S M S P Ô T (2.1)
where PC is expressed in $, and the heat transfer area A in m . The dimensionless
correction factors Sp, ST and JM account respectively for the pressures and temperatures
of operation, and the materials considered. Their values could be found in Ref. [5].
In addition to the initial cost, the operating cost of the HE should be considered in a
life-cycle assessment of the device. The main contribution to the operation cost OC comes
from the pumping power required to drive the fluids [27]:
o c = (E, + E , ) x H x e 1000
where Es and E, are the pumping powers for the shell and tube sides respectively, H, the
annual operating period and e, the electricity cost. Finally, combining (2.1) and (2.2), the
total cost of the HE is expressed in terms of annuities:
/ ( l + / ) " TC = PC—± — + OC (2.3)
(l + / ) " - l
Our objective is to minimize the total cost, TC, by varying the condenser geometry. The
problem is similar to that reported in [16] for single phase HEs, but the very fact that one of
the fluids condense in the HE requires a new model for estimating the overall heat transfer
coefficients and pressure drops, as well as an adapted set of design variables.
2.3. Heat transfer calculations
In a design procedure, the heat transfer area A is unknown a priori, but it must satisfy the
process requirements in terms of heat transfer rate [14]:
A = — - — (2.4) U*TlmF
14
where Q is the specified heat transfer rate to exchange between the shell and tube fluids.
The correction factor F is used to account for the reduction of the effective temperature
difference for heat exchange when the number of tube pass is larger than 1. In this paper,
we consider HE with only one tube pass and one shell pass and consequently, F = \ .
In the present work, we are interested in two configurations of shell-and-tube HEs:
i) with horizontal in-tube condensation, as shown in Fig. 2.1 and ii) with condensation in a
vertical shell as shown in Fig. 2.2. Such HEs are commonly found in different installations
such as power plants [15].
15
Tc,i ♦ Th.o
h LGP -»+•-
Lc "LP
Tube length
Figure 2.1 The schematic representation of a straighttube heat exchanger with one pass
on tube shell and condensation on tube side.
16
T 1 c,o
Tu >1
v 7A,O
Figure 2.2 Schematic representation of a vertical heat exchanger when condensation
occurs on shell side.
As we consider straight-tube HEs with one pass on tube shell, the HE can be
separated into three sub-sections (see Fig. 2.1) according to the hot fluid phase: i) segment
with vapor phase (GP); ii) segment with condensation (C); and iii) segment with liquid
phase (LP). The area of each zone is given by
AJP ~ a CP
U G P ' * * Im.GP
A , o — ■ Q,
*i.p ^ L P A T , m . L P
(2.5)
and the total area A required in Eq. (2.1) for estimating the cost is simply the summation of
the surface area of each zone, hence A = AGP + Ac + Aw-
In reality, heat loss will occur between condenser and its environment. Since
information about environment is specific for a given case, here we assume no heat loss to
the environment. Moreover, it is a common assumption in literature to neglect heat loss to
17
the environment [14]. The heat transfer rate in each sub-section is thus easily computable,
either from the condensing (hot) fluid point-of-view
QCP = m ^ p h C P (Th i - Th5al ) Q c = mhifg Qu, = m h c p h L P (Thsa t - T h o ) (2.6)
or the cold fluid standpoint
QGP = K c p x (Tc,0 - Tca ) Q c = m c c p c (Tc<2 - TcA ) QLP = m c c p c [TcX - 7 \ ) (2.7)
The overall heat transfer coefficients (UGP, UC, and Uw) in Eq. (2.5) depend on the HE
geometry and on the fluids phase (i.e., liquid, vapor, mixture). The general expression for
the overall heat transfer coefficient based on tube outer diameter is given by [14]:
U = 1 _ d0 H d j d r ) n d „ 1 d„
h. ' 2k. d h. d. (2.8)
where hs and h, are respectively the heat transfer coefficients on the shell side and tube side,
Rs and R, are fouling resistances for both sides. The calculation of Eq. (2.8) requires the
knowledge of both the shell-side and tube-side heat transfer coefficients. These depend on
whether the condensing fluid flows in tubes or in the shell. Therefore, the determination of
hs and h, in each case is presented in the following two sub-sections.
2.3.1 Heat transfer coefficients with in-tube condensation
Consider that the cold fluid flows in the shell. We assume that the shell-side heat transfer
coefficient (hs) is fairly uniform throughout the HE, i.e. that hs is the same in each sub
sections of the exchanger. In other words, in Eq. (2.8) only the tube internal heat transfer
coefficient h, varies depending on the phase (superheated vapor, condensing fluid, sub-
cooled liquid) of the in-tube fluid.
The calculation of hs is based on the Bell-Delaware method [14]. For the sake of
concision, we do not repeat here the entire procedure for calculating the heat transfer
coefficient in the shell with this well documented method. Details can be found elsewhere
(e.g., Refs. [16] and [14]). The procedure relies on the calculation of an ideal heat transfer
18
coefficient for perfect cross-flow on tube bank, corrected for taking into account the various
bypasses and inherent imperfections.
We considered that the in-tube flow was turbulent (Re values are typically well
above the critical value for laminar-turbulent transition). The calculation of the tube-side
heat transfer coefficients for single phase flow (i.e., hucp, h,w) is straightforward as several
correlations for turbulent pipe flow are available [14,16]. We used that recommended by
Sieder and Tate [28]. For calculating the heat transfer coefficient in the condensation zone
of the tube (huc), we considered the correlation developed by Chato [29]:
l .C 0.555 SP h J (p h J -p K g )k l . ih h MhJ{Th,sal-Tw)di
1/4
(2.9)
with
' /« lfg + g Cp.h.l Vh.sal T„ J (2.10)
The calculation of the condensation internal heat transfer coefficient ht,c requires the
knowledge of the internal wall temperature Tw, which is unknown a priori. Furthermore, the
tube internal wall temperature Tw is a function of the position in the HE. Therefore, we
replaced Th,sat- Tw in Eqs. (2.9) and (2.10) by the logarithmic mean temperature difference:
AT \ * h.sat * w t \ ) \ * h,sat * w , 2 )
h,sat-w
In 1 h.sat 1 w , \
T - T V h.sat * w,2 )
(2.11)
where Twj and Twj are evaluated at the inlet and outlet of the condensation sub-section of
the HE (see Fig. 2.1). Eq. (2.11) provides an estimate of the wall-to-fluid temperature
difference in the condensation sub-section [28]. An iterative procedure allows to overcome
the difficulty introduced by the fact that wall temperatures are required to estimate ht,c, and
vice versa. Because we do not know initially the wall temperatures Twj and Twj, they are
first guessed. These guesses are used to make a first estimate of àThsal_w, Eq. (2.11). The
19
obtained value is inserted in Eqs. (2.9) and (2.10) to calculate huC- The overall heat transfer
coefficient Uc is then estimated, Eq. (2.8). Next, the wall temperatures are calculated from
the equivalent thermal circuits shown in Fig. 2.3.
T„,
IT**-
R ' + K
Figure 2.3 Thermal circuits to determine for Twj and Tw,2-
A simple thermal circuit analysis reveals that
\ * h.sat * c j j d-. 1
T =T -1 w . \ * h.sat
+R.+ d M d p / d , ) , d0
(
2k 1 \ (2.12)
with Tcj given by the right-hand side of Eq. (2.7). TWi2 is obtained similarly. The updated
Twj and Twj are used as new guesses in Eqs. (2.9) and (2.11). The procedure is repeated
until convergence, yielding an approximate value for Uc-
20
2.3.2 Heat transfer coefficients with shell-side condensation
Now, consider the case where the cold fluid flows in the tubes. We will assume that the
tube-side heat transfer coefficient (h,) is fairly uniform throughout the HE, i.e. that h, is the
same in each sub-sections of the HE. It is determined as in Section 2.3.1 for single phase
flow, i.e. with the correlation of Sieder and Tate [28]. It is now the shell-side heat transfer
coefficient hs that varies depending on the phase of the hot fluid.
The calculation of the shell side heat transfer coefficients for single phase flow (i.e.
hs,w, hSiGp) is determined using once again the Bell-Delaware method. For calculating the
heat transfer coefficient in the condensation zone of the shell, hs,c, we consider film-wise
condensation as recommended by Ref. [5]. The condensing vapor wets the surface of the
tubes forming a continuous film. Therefore, the heat transfer coefficient on shell side hSiC
corresponds to that of a condensing film. Following the procedure recommended by [5], we
combined Nusselt correlation for condensation
^ = 0 . 9 4 3 r k i P h . i \ 8 ^
kMh,AT (2.13)
(where AT is the temperature difference across the condensate film on the surface) to the
energy balance mhif =hsCxd0LcNAT, where N is the number of tubes in the bundle
yielding the following expression for the heat transfer coefficient:
hs,c=l.35k h.l
( P h l d o g N ^ (2.14)
2.4. Pressure drop calculations
The pressure drops on tube and shell sides are required to evaluate the pumping power
requirement. This information is needed to calculate the operating cost of HEs, Eq. (2.2).
Again, we perform the analysis separately for the case with in-tube condensation and the
case with condensation in the shell.
21
2.4.1 Pressure drop with in-tube condensation
For the case where the hot fluid flows in the tubes, shell side pressure drop (APS) is
computed using the Bell-Delaware method [14,16]. The total in-tube pressure drop is the
summation of three contributions (i.e., power dissipated by the vapor, liquid and mixture
phases), AP, = APt GP + APt c + APt LP . In-tube pressure drops APt Gp and APt LP are due to
single phase flows, and thus are straightforward to evaluate [5]:
^Pt .GP = s
( A C J \ p y 2 ( A S r \ r J h , G P y t .GP ^ J G P ^ i GP'-GP + 0.5 ^ t . L P = S
4fvk L P ' - ' L P . 1
di
2 Ph-LpV,LP (2.15)
where/G/> and/z./> are the friction factors for the single phase sub-sections (GP, LP) and are
calculated for turbulent flows b y / = (0.7901n(Re)-1.64)2 [28]. The factor 0.5 in the
expression of APt c p stands for the sudden contraction of the fluid at the tubes inlet and the
factor 1 in the expression of APt LP stands for the sudden expansion of the fluid at the tubes
outlet [30]. The pressure drop during condensation will be evaluated assuming an
equivalent homogeneous flow [15]. The two-phase density and two-phase viscosity are
defined respectively as [15].
— S M E M J_ = _^L + !z± (2.16) 0-*)/V f +*/>*.! Mh.C Ph.g Ph.,
where x is the quality (i.e., local fraction of the flow that is in the vapor phase). Introducing
the two-phase in-tube Reynolds number, Rec = Amhl\JCdlflhjC), it is possible to estimate
the local pressure gradient with [15].
dP 32 frhh2
= —h—- (2.17) d z X2Ph.c<li
where the friction factor is approximated by / = 0.046(Rec )"°2 for fully developed
turbulent flow inside smooth tubes. Eq. (2.17) can be integrated from the inlet of the
22
condensation zone to its outlet [15]. For simplicity, we assume that x varies linearly in tube
direction (x = z/Lc). Combining Eqs. (2.16) and (2.17) and changing the integration variable
dz for dx, the intube pressure drop in the condensation zone is approximated by:
32m„9/5 (p h M h , ) " 5 Lc \ ( \ x ) p . + x p h . ^ = °°46 / W , ft / V* (218)
4 x d, p K % P h , l [ ( l x ) M h , g + x f i h J ] v >
I The result of the integral / is
>"*,, 2Ph.gMh,,+Phj (2.19)
+W/ S
A,//M - W X . -4//w9/vM}
The pumping power for the shell side is:
j ^ P / n £5 = ^ ^ (2.20)
r e I pump
The total power dissipated on the tube side has three contributions (one contribution for
each segment of the flow), Et = EtGP + E lC + E lLP. E,,GP and EuLp being attributed to single
phase flow are calculated similarly to Eq. (2.20). Euc is due to condensing flow. As shown
in Eq. (2.16), the density / \ c is a function of the quality x. Integrating between the entrance
and the exit of the condensing zone assuming that the quality varies linearly with position,
an average density is achieved, p h C = 1ph gp h J \Ph g + p h l ) ■ Then, the power required for
driving the two phase flow is
_ ^ . .C^Ph^+Ph . , ) ' •
c =~ ~ y n rV~n
( } L J lpumpPh,gPh,l
23
2.4.2 Pressure drop with shell-side condensation
When condensation occurs in the shell, the total in-tube pressure drop is easily determined
with the following formula [5]:
AP t=np '±^+l.5l 4
(2.22)
where the factor 1.5 stands for the fluid contraction and expansion at the inlet and outlet of
the tube bundle.
The total shell side pressure drop is the summation of three contributions (i.e.,
power dissipated by the vapor, liquid and mixture phases):
APs=APsGP + APsC+APsLP (2.23)
Single phase sub-section pressure drops (APS,GP and APs,w) are calculated using
Bell-Delaware method [14]. Details can be found elsewhere [16]. This method cannot be
directly applied for the pressure drop in the condensing flow sub-section (APs,c)- Therefore,
we used the separated-flow model proposed by G.F. Hewitt et al. [15]. The condensation
sub-section pressure drop (APSyC) has two contributions: the cross-flow zone pressure drop
(APSyc,Cf) and the window-flow zone pressure drop (APs,c,wf) [5]:
* P , , c = à P , X M + à P , ^ (2.24)
For the cross flow sub-section, the pressure drop is obtained using the correlation
developed by Chisholm for turbulent flow in shell-and-tube heat exchanger [15,31]:
y r j =l + (Y2-l)(x-x2)0 M S+xU 3 1 (2.25)
where yrLo 2 represents the two-phase multiplier and Y 2 is the Chisholm parameter. They
are defined by the following expressions:
24
(%) r, f%)t (%L " (%), ^ 2 = A V T ^ = - /„ ,x (2-26)
The subscripts L0 and GO refer, respectively, to the total flow having liquid phase
properties and the total flow having the gas phase properties. (d/>/3z)to and {dP/dz)G0
can be determined by using directly the Bell-Delaware method because single phase flows
are considered.
We performed an integral on JC between 0 and 1 and combining Eqs. (2.25) and
(2.26), we finally obtained the following expression for the cross-flow zone pressure drop:
^ W = ( T ] L c \ [ l + (Y 2 - l ) (x-x 2 ) °™ + x l *}lx (2.27) V " z )LO 0
For the window-flow zone pressure drop, we used Grant correlation for turbulent flow in
shell-and-tube heat exchanger [15,31]:
"to = l + (K 2- l )x (2.28)
We applied the same procedure than for the cross-flow zone pressure drop and we obtained
the following expression:
^ P , . c ^ = [ ¥ ] L c \ [ l + (Y2-\)x}tx (2.29) V oZ ) L O o
Considering all these pressure drops contributions, the shell and the tube side pumping
powers were calculated in a way akin to that described in Section 2.4.1.
2.5. Design variables and procedure for determining the cost of a
design
The geometry of the HE has a strong effect on the overall heat transfer coefficient and
pressure drops, and thus on its total cost (purchase and operation). For a given process, the
geometry leading to the lowest cost is difficult to determine. Complex optimal tradeoffs
25
have to be found. Moreover, the side where the condensing fluid flows (tube or shell) also
influences the result. Here, we considered 11 design variables to optimize the geometry of
the shell-and-tube HE: 1) the tube pitch (p) can take four values: l.2d0, \.3d0, \Ad0 or
\.5d0; 2) the tube layout patterns can take three values: triangular (30°), rotated square
(45°) or square (90°); 3) the baffle spacing at the center (Lh,Cemer) can take eight values
ranking from 0.2D to 0.55D; 4) the baffle spacing at the inlet and outlet (Lt,,0 = Lbj) can also
take eight values ranging from lLh,Cenur to \.6Lb,cemer', 5) the baffle cut (B) can take eight
values: 25, 30, 40 or 45 %; 6) the tube-to-baffle diametrical clearance (A-b) can take four
values: Q.0\do, 0.04do, 0.07do or Q.\0do; 7) the shell-to-baffle diametrical clearance (As.b)
can take four values: 0.01D, 0.04D, 0.07D or 0.10D; 8) the tube bundle outer diameter
(D0,i) can take four values: 0.8, 0.85, 0.9 or 0.95 times the baffle diameter (D - As.b); 9) the
shell diameter (D) can take sixteen values ranging from 300 to 1050 mm, 10) the tube outer
diameter (d0) can take eight values ranging from 15.87 mm (5/8 po) to 63.5 mm (2.5 po). A
tube thickness is associated with each diameter value in accordance with the standards of
the Tubular Exchanger Manufacturer Associations (TEMA, 1988). Here, we considered
tubes with thickest walls, and 11) the HE side where condensation takes place: tubes or
shell. The first 10 design variables are as in Ref. [16], but it is important to remember that
the HE model (i.e., how the performance is affected by the geometry) used here is different,
as described in the previous sections. Furthermore, the last design variable (i.e., side of
condensation) is specific to this problem. Figure below is taken from a previous paper [16]
and shows geometrical design variables considered in this work.
26
K V L
S r M M M 1
b,i ^b.center ^ b . o V
30e
, -4 7d» \@©é ► ►
\ @ @ ^ 7 f ►
►
90e
G-O P
60e
OO o %e
Figure 2.4 Geometrical design variables.
An iterative procedure is required to determine the tubes length L and the heat
exchange surface A. For this problem, the tubes length is divided in three sections (two
single phase flow sections and one condensing flow section). These three lengths are
determined through the following expressions [14]:
l-T:v ~ J \ I C
7tdN
_ \ n d N
I -_ALP_ "LP
n d N (2.30)
AGP, AC and Aw are unavailable initially. Consequently, tubes length values are determined
through an iterative procedure. The values of AGP, AC and Aw are first guessed,
corresponding tube length of each zone (LGp, Lc and Lw) is calculated. With these values,
we can calculate the overall heat transfer coefficient of each sub-section. Values of AGP, AC
and Aw are finally updated with Eq. (2.5). These surface area values are used as new
27
guesses and the operation is repeated. After each iteration, the updated surface areas of each
sub-section are compared with those computed at the previous iteration. The procedure is
stopped when the relative difference between two successive iterations is less than 1% for
each section. We finally obtain values for AGp, Ac, Aw, LGP, LC and Lw- The total heat
transfer area A can now be calculated along with the purchase cost PC, Eq. (2.1).
2.6. Optimization procedure with genetic algorithms
With the eleven design variables considered, more than 134 million different HEs designs
can be considered. The time required to evaluate every possible combination is quite large
(see Section 2.7). Here, we used genetic algorithms (GA) to optimize the HEs for a given
heat transfer process. Genetic algorithms tend to converge to a global minimum solution by
evaluating only a small fraction of the design space so that the time of calculation required
to find the best design is decreased. The detailed GA optimization procedure is available
elsewhere [16-17].
2.7. Test cases
In this section, we consider two application cases. The specifications of the first test case
are presented in Table 2.1. A mass flow rate of 3 kg/s of hot water vapor at a pressure of
9.5 bars has to be cooled down from 182°C to 157°C. The fluid is under liquid state at the
outlet. The saturation temperature of water at this pressure is 177.66°C [32] so there is
condensation. We suppose that the condensing side pressure drop for this process is small
enough compared to the average pressure to neglect the effect of the pressure variations on
the condensing temperature. The validity of this approximation is verified later in this
section. This process is achieved with 16 kg/s of cold water entering the HE at a
temperature of 4°C to cool down the hot water. The outlet temperature of the cooling fluid
is not known a priori. There is no phase change on the cold fluid side. Stainless steel is used
as material of construction. For the considered materials, operating pressure and
temperature of operation, the capital cost correction factors in Eq. (2.1) take respective
28
values of ÔM = 2.9, ôp = 1.9, b r - 1.6 [5]. Processes similar to that of case study #1 are used
in chemical engineering and in HVAC (i.e., vapor heating systems).
Table 2.1 Process requirements for case study #1 (H = 5000 h year-1, e = 0.1
$kW~'lf', n = 20 years, / = 0.05, npump = 0.85).
Condensing fluid Cold fluid
Fluid - 2 , Pressure (N m~ )
Flow (kg s_1) Fouling resistance (m-2 K W-1)
Inlet Temperature (°C) Phase Density (kg m~3) Heat capacity (J kg-1 K"1) Dynamic viscosity (N s nf ) Thermal conductivity (W m-1 K"1)
Outlet Temperature (°C) Phase Density (kg m ) Heat capacity (J kg"1 K"1) Dynamic viscosity (N s m-2) Thermal conductivity (W nf ' K_1)
Condensation zone Saturation temperature (°C) Latent heat (J kg"1)
Water Water 9.5 x 105 1.013xl05
3 16 0.000275 0.000275
182 4 Vapor Liquid 4.8353 1000 2592.9 4207.5
1.51 x 10"5 0.0015672 0.036182 0.56867
157 Liquid
157 Liquid Liquid 910.58 1000 4325.1 4207.5
0.00017383 0.0015672 0.68093 0.56867
177.66 2 022 360
177.66 2 022 360
Table 2.2 contains the specifications for the second example. 6 kg/s of water enters
the HE at 4°C to cool down 1.5 kg/s of refrigerant R-134a entering at a pressure of 10.164
bars. The refrigerant enters the HE under vapor phase at 50°C and exits under liquid phase
at a temperature of 30°C. There is thus condensation of the refrigerant in the HE as the
phase change temperature at that pressure is 40°C [32]. Here again we consider constant
condensing temperature for the same reason as in case study #1 and the validity of this
29
approximation will be verified later. Stainless steel is once again used as material of
construction. The capital cost factors used in Eq. (2.1) for this example are ô\t = 2.9, 6p =
1.9, or = 1.6 [5]. Such process is typical of those encountered in refrigeration cycles
[33,34].
Table 2.2 Process requirements for case study #2 (H = 5000 h year"1, e = 0.1 SkW'h -1 ,
n = 20 years, / = 0.05, npump = 0.85).
Condensing fluid Cold fluid
Fluid -2-, Pressure (N m~ )
Flow (kg s_1) Fouling resistance (m~2 K W_1)
Inlet Temperature (°C) Phase Density (kg m-3) Heat capacity (J kg-1 K_1) Dynamic viscosity (N s m-2) Thermal conductivity (W nf ' K_1)
Outlet Temperature (°C) Phase Density (kg nf3) Heat capacity (J kg-1 K"1) Dynamic viscosity (N s ra" ) Thermal conductivity (W m_l K_1)
Condensation zone Saturation temperature (°C) Latent heat (J kg"1)
R-134a Water 1.064 xlO6 1.013 xlO5
1.5 6 0.000175 0.000275
50 4 Vapor Liquid 46.825 1000 1084.9 4207.5
1.28 xlO'5 0.0015672 0.016075 0.56867
30 Liquid
30 Liquid Liquid 1155.6 1000 1486 4207.5
0.00016581 0.0015672 0.075623 0.56867
40 163 030
40 163 030
The minimal cost designs found by the GA for the two case studies considered are
presented in Table 2.3. This table presents the optimal design parameters of the HEs and
the side where the condensing fluid must flow to obtain this optimal design. Five runs of
30
the program have been performed for each case study, and the algorithm found the designs
shown in Table 2.3 every time.
Table 2.3 Minimal cost heat exchanger geometries as found by the GA.
Case study #1 Case study #2
1. Tube pitch, p (mm) 2. Tube layout pattern (deg.) 3. Baffle spacing at centre, Lb_cenler (mm) 4. Baffle spacing at inlet/outlet, Lb,, = Lb,0 (mm) 5. Baffle cut, B (%) 6. Tube-to-baffle diametrical clearance, A,.b (mm) 7. Shell-to-baffle diametrical clearance, As.b (mm) 8. Tube bundle outer diameter, Doti (mm) 9. Shell diameter, D (mm) 10. Tube outer diameter, d0 (mm) 11. Number of tubes, N 12. HE side where condensation occurs
Tube length, L (m) Total surface area, A (m2) Pressure drop on shell side, APS (Pa) Pressure drop on tube side, AP, (Pa) Operating cost, OC ($ year-1) Initial cost including interest, IC ($ year-1)
Total cost, TC ($ year-1)
Number of evaluations Calculation time (s)
\-5d0 1.54, 90 90
0.55D 0.3D 0.55D 0.33D
28 25 0.01rfo 0.01^ 0.1D 0.01D
0.80(D-A,.fr) 0.80(D - As.b) 450 300 15.9 50.8 261 12 shell tubes
6.96 14.56 90.55 26.32
9.56 xlO3 2.89 x 104
2.84 x 104 1.11 xlO4
2371.10 213.59 25 241.98 6 810.30
27 613.08 7 023 .89
3120 2100 16 15
There is an important difference between the two test case solutions. The total cost
of the first design (27 613 $) is much higher than that of test case #2 (6 968 $). This is due
to the mass flow rates considered that are more important for the first case. Furthermore,
condensation occurs in the shell for test case #1 and in the tubes in case #2, which
demonstrates the optimization opportunity related to the flow arrangement.
31
Table 2.3 also contains the pressure drops of the optimal design for each case study
(AP,). For case #1, we have condensation in the shell. We obtain a value of 9.56 x 103 Pa,
which represents 1 % of the shell operating pressure. For case #2, the condensation occurs
in tubes and the pressure drop value is 1.11 x 104 Pa and it represents 11% of the operating
pressure. In each case, the pressure drop is considered small enough to approximate a
constant condensing temperature, validating the approximation described above.
As we said in Section 6, more than 134 million different HEs are possible with the
different values that can take the eleven design variables. In order to show the advantage of
the GA in this application, all the possible designs have been tested in order to find the best
one for a given process. The calculation time and design evaluations for test case #1 and
test case #2 are respectively 27 and 31 hours. These global tests led exactly to the optimal
designs found by the G A for cases #1 and #2 reported in Table III. Such conclusions were
also achieved in [16] for single phase HE. However, here the global testing of all possible
designs was much longer because the heat transfer and fluid flow calculations (Sections 2.3
and 2.4) are more complex, and require more iterative processes (e.g., for evaluating the
tube wall temperature or the length occupied by segments G,C,L, etc.). These results allow
us to conclude that the geometry found by the GA is the global minimum. The main
difference between the two optimization approaches is the calculation time needed to find
the best HE. It took 16 seconds to the GA versus about thirty hours for the global test. The
GA only had to test 3120 models to find the optimal design among the 134 217 728
possible. This represents only 0.0023% of all possible designs, which demonstrate the
usefulness of the GA for the resolution of condenser design problem.
2.8. Conclusion
In this paper, we presented an integrated model to evaluate the pressure drops and the heat
transfer surface area required for a shell-and-tube HE with condensation, either in the tube
or in the shell. These quantities allow us to calculate the initial cost and the operating cost
of the condenser. We considered eleven design variables regarding the geometry of the HE
and the side where condensation occurs (i.e., shell or tube). We studied two test cases
32
where we had to identify the optimal condenser architecture for a given process with
condensation. Our purpose was to find the HE with the lowest total cost in terms of
annuities. Optimization was performed using a genetic algorithm (GA). We compared the
solution found by the GA with the one found with a global test of every possible HE. The
comparison proved that the GA identified the global minimum in each of the cases studied,
and determine on what side the condensation should take place. Here we only considered
condensation of a fluid on one side of a HE with one tube pass. Further research could
focus on HE with ebullition or on more refined modeling of the condensation. The study of
multiple tube passes shell-and-tube HE with condensation could also be of interest.
33
Chapitre 3
Article # 2
Titre:
Optimizing heat exchanger networks with genetic algorithms for designing
each heat exchanger including condensers
Co-auteurs:
Benoît Allen, Myriam Savard Goguen, Louis Gosselin
Journal:
Applied Thermal Engineering, Volume 29, Pages 3437 à 3444
34
Abstract
The paper communication presents a procedure for the optimization of heat exchanger
network. The procedure first uses pinch analysis to maximize the heat recovery for a
given minimum temperature difference. Using a genetic algorithm (GA), each exchanger
of the network is designed in order to minimize its annual cost. Eleven design variables
related to the exchanger geometry are considered. For exchanger involving hot or cold
utilities, mass flow rate of the utility fluid is also considered as a design variable because
there is no restriction on utility outlet temperature. Partial or complete condensation of
hot utility fluid (i.e, water and vapor) is allowed. Purchase cost and operational cost are
considered in the optimization of each exchanger. Combining every exchanger
minimized cost with the cost of hot utility and cold utility gives the total cost of the HEN
for a particular ATmin. The minimum temperature difference giving the more economical
heat exchanger network is chosen as the optimal solution. Two test cases are studied, for
which we show the minimized total cost as a function of the minimum temperature
difference. A comparison is also made between the optimal solution with the cost of
utilities and without it.
Myriam Savard-Goguen contributed significantly to the realization of this paper.
She made a first version of the heat exchanger network design model in Matlab. This
model established a strong base to the realization of the final model. Redaction of the
paper and creation of the final model has been made by Benoit Allen and Louis
Gosselin.
35
3.1. Introduction
Heat exchanger networks (HEN) are required in applications that involve heat exchange
between two or more fluids [5]. They are found in many industries such as crude oil
distillation [35,36], furnace systems [37], multipurpose batch plants [38], cooling water
systems [39,40] and chemical plants [41]. These industries generally consume a large
amount of energy. In some batch plants, energy consumption can reach 10% of total
expenses of a company [38]. Well-designed HENs can significantly contribute to
decrease energy consumption. When designing a HEN, fluid match possibilities and
design options for each exchanger of the network are tremendously numerous.
Therefore, an efficient method must be used to design the best network in regards to the
purchase and operating costs as well as to the heat recovery, the primary purpose of a
HEN.
Many optimization techniques have been developed in the past for the heat
exchanger network problem. A review on the topic is available [42]. Pinch analysis is
one of the most prominent approaches to maximize heat recovery, even though other
methods exist (e.g., tree searching algorithm method [43], neural networks [44], mixed
integer nonlinear programming that allows any fluid match [45], etc.).
Once the HEN is designed, for example with the pinch analysis, its cost is often
calculated based on the required surface area for each heat exchanger with assumed heat
transfer coefficients. This approach has several limitations. For example, it does not
include the pumping power cost and provide no information relative to the design of the
heat exchangers (HEs) themselves.
Nevertheless, some authors have improved the approach. For example, Frausto-
Hernandez et al. [46], Polley et al. [47], Silva and Zemp [48] included a pressure drop
analysis to assess the pumping power cost. Optimization methods involving the design
of the heat exchangers of the HEN have been studied by Ravagnani et al. [49], Polley
and Panjeh Shahi [50], Markowski [35], Roque and Lona [51], and Ravagnani and
Caballero [52]. However, the number of design variables considered for these
36
exchangers are often fairly limited. Furthermore, boiling and condensation are not
considered.
In the present paper, we use pinch analysis with splitting to optimize HENs. As
the hot utility was assumed to produce water vapor, the HENs generated in this paper
include condensers. Then, a genetic algorithm (GA) designs in details each heat
exchanger for minimizing its cost (purchase and operation costs). The mass flow rates of
the utility fluids are also optimized. In the end, the optimal minimal temperature
difference, HEN and HEs are determined. Among the innovative aspects of this work
are the use of GAs, the level of details for HE optimization, and the consideration of
condensers in the HEN, and the optimization of utility fluid mass flow rates.
3.2. HEN Problem formulation and design procedure
The heat exchanger network (HEN) optimization problem is well documented in the
literature [5,35-52]. Therefore, we do not repeat here all the details related to this
problem. This paper relies on pinch analysis to determine the best fluid matches.
Each fluid involved must reach a target temperature (7V,/,, 7V,C) and is provided at
a supply temperature (TSih, TStC). The mass flow rates are also assumed to be known. The
network design is based on a minimum temperature difference (ATmin) such that the
temperature differences between hot and cold fluids in any HE of the network is equal or
greater than this value. Given a value of ATmin and the properties of the fluids, we can
match hot and cold fluids in order to maximize heat recovery. Matches are allowed only
between hot and cold fluids. The streams considered are divided into temperature
intervals constructed from supply and target temperatures of every fluid [5]. Hot and
cold pinch temperatures are then calculated using the problem table algorithm.
Essentially, this algorithm consists in achieving an energy balance considering all
streams present in each temperature interval. From the temperature difference across
each interval (AT), the heat balance (AHEAT,) is computed
A//£A7;=[ICPC-ICPW]A7; (3.1)
37
A negative AHEAT indicates a net surplus of heat and a positive AHEAT indicates a
deficit of heat. The excess heat is transferred from interval to interval, down the
temperature scale. Since a negative heat flow is infeasible, the minimum heat added to
ensure that heat flows are all positives is provided by the hot utility (HU), while the
remaining heat in the last interval is taken by the cold utility (CU) [5]. Starting from the
pinch, which is the most constrained point of the HEN, the appropriate matches are made
between the cold and hot fluids. Each match corresponds to a heat exchanger which will
have to be designed. Here, we considered shell-and-tube HEs. For the fluids that could
not reach their target temperature only by heat recovery, the cold and hot utilities are
used. Cold utility must not be used above the pinch nor the hot utility below the pinch.
This means that hot and cold streams must be cooled and heated to pinch temperature
only by heat recovery. Moreover, no heat exchange is allowed between a fluid below the
pinch and a fluid above the pinch. Stream splitting is allowed in order to increase match
possibilities. For example, a cold fluid with a relatively high heat capacity rate can be
split to be heat up by two hot streams with low heat capacity rates. This increases match
possibilities and consequently heat recovery. The whole design process is illustrated in
Fig. 3.1.
38
ABOVE PINCH
NO
NO
Place matches
Recombine splited fluids
Calculate
YES Split HOT stream
YES Split COLD stream
1 c,o I c,target ^ ^ NO . Use
1 c,o I c,target ^ ^ HOT utility
\ YES
cxrr»
BELOW PINCH
Split COLD stream
YES
Split HOT stream
YES
NO
NO
Place matches
Recombine splited fluids
Use , NO ^ ^ l h , o ^ 'h,target COLD utility ^ ^ l h , o ^ 'h,target
YES
t
CXTTV
Figure 3.1 Pinch analysis for heat exchanger network optimization.
A list of required heat exchangers with their corresponding duty (i.e., heat
transfer rate, fluids and mass flow rates involved) is established from that procedure.
Each of these HEs has to be designed so as to minimize the total global cost of the
network. The total cost of a HE includes its purchase cost and its operation cost
(pumping power). The total cost is expressed in this paper as an annualized cost. The
39
total cost minimization and the design of HEs is performed with a genetic algorithm as
described in the next section. The cost estimation procedure is summarized in Fig. 3.2.
Choose AT„i„
i f
Pinch analysis
i r
Design each HE with GAs
Cold-hot HEs
Hot-CU HEs
Cold-HU HEs
(condensation)
i '
Calculate total cost
Figure 3.2 Overall procedure for total cost estimation.
Furthermore, since ATmin of the HEN is usually not prescribed, we can vary its
value in order to minimize the global cost of the network. The optimal value of ATmj„
was found by designing networks with their HEs for several values of ATmin. We are thus
able to compute an annualized cost (i.e., cost of the HEs and cost of the utilities) for each
network and the more economical ATmin is identified by comparing each network total
annualized cost.
40
3.3. Description of the HE design problem and genetic algorithms
3.3.1 Objective function
Our objective is to minimize the heat exchanger network total cost. For a network with n
exchangers, the total cost is defined by
m
TC = Y\_PCj + 0 C j ] + C H U + C C U <3-2)
where PCj and OCj stand respectively for the annualized purchase and operational costs
of the HE of the HEN. PC is related to the required surface area of the HE which in turn
depends on the HE geometry. Details relative to its calculation are given in [14,16,53].
OC accounts for the shell side and the tube side pumping powers (pressure drops) and its
calculation can also be found elsewhere [14]. CHU corresponds to the total annual cost
of the hot utility used in the process (water vapor):
CHU=txCOSTH Ux1£ mHU,j{Cp,HU,Gp(*HU,i *HU,sat)~*~
l f g , H l j ( * ~ X o , j ' + Cp,HU,Lp(* HU.sat ~ * H U , o , j ' \ (3.3)
where a stands for the number of exchangers involving hot utility and t is the annual
operating period. We assumed that vapor was used as HU, and therefore, Eq. (3.3)
accounts for the possible condensation (partial or total). The three terms in the
summation in Eq. (3.3) represents the power given by the vapor to the HEs involving
HU, and the power given by the condensing mixture and the sub-cooled liquid if
applicable. Similarly, CCU stands for the total annual cost of the cold utility:
CCU=txCOSTcuxYJ y=i
mcU,j(Cp,CuVcU,i *CU,o,j))
1000 (3.4)
41
where b stands for the number of exchangers involving cold utility. Utility costs (i.e,
COSTcu and COSTHU) are expressed in $/kW-h. We used a cost of 0.015 $/kW-h for hot
utility and 0.010 $/kW-h for cold utility [5].
3.3.2 HEs design variables
Heat exchangers that make part of the HEN are separated in three categories: 1) cold
fluid to hot fluid heat exchangers, 2) heat exchangers with cold utility and 3) heat
exchangers with hot utility. Eleven design variables are common to every exchanger.
They are related to the shell-and-tube heat exchanger geometry [14]. For the HEN
problem considered, they can take the following values:
1) Tube pitch 0 ) : L2J0, \3d0, \Ad0 or \5d0
2) Tube layout pattern : triangle (30°), rotated square (45°) or square (90°)
3) Baffle spacing at center (LbiCenter) : eight values from 0.207) to 0.557?
4) Inlet / Outlet baffle spacing (Lb/Lb,0) : eight values from LbjCenter to 1.6LbiCenler
5) Baffle cut (B) : eight values from 25% to 45%
6) Tube-to-baffle diametrical clearance (A,.b) : 0.0ID, 0.047), 0.07D or 0.10D
7) Shell-to-baffle diametrical clearance (As.b) : 0.0ID, 0.04D, 0.07D or 0.10D
8) Tube bundle outer diameter (D„,/) : four values from 0.8(D - As.b) to 0.95(D - As.b)
9) Shell diameter (D) : 32 values from 0.300 m to 1.850 m
10) Tube outer diameter (d0) : eight values from 5/g in to 2.5 in
11) Number of tube passes : 1, 2 or 4 (for the heat exchanger with the HU only 1 pass
is considered) [53].
A twelfth design variable is added for the side (shell or tubes) where each fluid flows.
Since the outlet temperature of the cold utility stream in HEs of the second
category is not predetermined, its mass flow rate in each HE with CU can vary in order
to obtain a minimum annualized cost. Therefore, an additional design variable is added
to heat exchangers with CU: the mass flow rate of the cold utility fluid. The flow rate
must respect a minimum value ( mcu aiB ) in order to respect a minimum temperature
42
difference (ATmin) between the inlet temperature of the hot fluid (Thj) and the outlet
temperature of the cold utility ( Tcu o ). The maximum value of TCu.<? can be expressed by
T c u , o . ™ = T h . l - A Tm n (3-5)
With no heat loss to the environment, the heat transfer rate between the hot fluid and the
cold utility is determined by
Q = mhcP,h(Th.i -Th ,o) = ncijCp.cu(Tcu.o -Tcu, i ) (3-6)
The minimum flow rate of cold utility is calculated by combining Eqs. (3.5) and (3.6)
cP.hmh(.Th,-ThJ „ , .
m r „ m -= - (3.7) Cp,CU " h , . ~ ^ m i n ~ * CU j >
There is no physical restriction on the maximum CU mass flow rate ( mc l /ma ). However,
a mass flow rate interval had to be specified, so a maximum available value was chosen
and 128 possible values between (rhcu min) and (mCUnwi) were considered. We verified
that optimal mass flow rate lied within the specified interval.
For cold fluids that did not reach their target temperatures, a hot stream of vapor
is used as hot utility. It enters under overheated vapor and condensates inside the HE.
Since the modeling of heat exchanger with condensation has been developed in a
previous article entirely devoted to the subject [53], procedure to determine heat transfer
coefficients and pressure drop calculations for shell-and-tube condensers is not repeated.
However, we present later in section 4, an extension of [53] for the case of partial
condensation which was not considered in [53]. The advantage of using vapor as HU is
the high heat transfer coefficients that characterize a process involving phase change
[28]. As for the cold utility, the supply temperature (THu,s) of the hot utility is known but
there is no restriction on the vapor outlet temperature (THU.o)- Consequently, hot utility
mass flow rate (mHU) is also considered as a design variable. Limit values of this
parameter are established in order to ensure that condensation takes place. However, hot
utility fluid is not required to completely condensate. Fig. 3.3a and 3.3b illustrate
43
extreme situations from which mass flow rate limits are established. It must be greater
than a minimum value obtained when there is a difference ATmi„ between the outlet
temperature of the hot utility and the inlet temperature of the cold fluid as illustrated in
Fig. 3.3b.
Temperature
' HU.s
■ HU.sat
T ■ 1 C , l
Vapor phase
Heat exchanger length
Figure 3.3a Temperature of cold and hot fluids in shellandtube heat exchanger
without condensation.
44
Temperature
' HU.s
'HU.sat
THU.O
c,o
Vapor phase
Condensation
Liquid phase
Heat exchanger length
Figure 3.3b Temperature of cold and hot fluids in shell-and-tube heat exchanger with
condensation of the hot fluid.
T H U ^ = T c J + A T n m , (3.8)
From the energy balance between hot utility and cold fluid, we have:
m c C p ,c (*c ,o * e j ) m HU,min( C p.HU,Gp(*HU,i * HU ,sat >' "*"' fg ,HU "*Cp,HUU>(*HU .sat *HU\o'>
(3.9)
Combining Eqs. (3.8) and (3.9), the minimum mass flow rate can then be expressed as a
function of known parameters:
m m c C p . A T c . o - T c j )
HU. min Cp.HU,Gp(* HU.i *HU,sat' + lfg,HU + C p .HU.Lp(* HU.sat (*c j """ * " n i i i ) )
(3.10)
45
Maximum value occurs when the hot utility reaches its saturation temperature and just
starts to condensate. This is represented in Fig. 3.3a. An energy balance leads to:
mccp,ATc,o ~Tc,) = mHu.^THu.i T H U M ) (3.11)
. _ ^ p A T c , „ T c i ) mHU.max ~ ~ ~ (•*•l 2 >
*HU,i *HU,sal
3.3.3 Optimization of HEs using genetic algorithms
Genetic algorithms (GAs) are an optimization tool inspired by the Darwinian natural
selection. The procedure used in this paper has been well explained in previous articles
[16,53]. It has been proved that using genetic algorithms is a quick way to find the best
HE design among a large number of possibilities [16,53] and can also be used when
designing HENs [54]. Only a fraction of all possible designs needs to be calculated. This
results in an important economy of computational time.
The GA is used for the purpose of designing lowcost heat exchangers that
respect the heat duties imposed from the pinch analysis (section 3.2) minimizing the
global cost of the HEN. A priori the geometry leading to the lowest cost is not easy to
determine. An increase of the heat transfer area leads to a lower operating cost. On the
other hand, it leads to an increase of the purchase cost. Moreover, millions of possible
designs are feasible (see paragraph below). Hence, the GA significantly helps to identify
the optimal solution quickly. Considering the design variables listed above (Section 3.2),
the number of possible heat exchanger designs are:
1) Hot to cold fluid heat exchanger : 301 989 888
2) CU heat exchanger : 38 654 705 660
3) HU heat exchanger : 25 769 803 780
The main parameters of the binary GA used were as follows:
■ Number of individuals in the population = 30
■ Number of elites that propagate to the next generation = 5
46
■ Mutation rate = 4%
■ Number of crossover points = 3
■ Convergence criterion = 300 generations without improvement of the objective
function
GAs are probabilistic, and therefore 2 runs of the GA with the exact same setting could
lead to two different results. Therefore, for each HE the GA optimization was performed
7 times. Then the best result is taken as the best solution for this specific HE.
3.4. Extension of Ref. [53] to HEs with partial condensation
Calculation of the cost of a shell-and-tube condenser as a function of the design variables
listed in section 3.3.2 for a given heat duty was described in Ref. [53]. However,
complete condensation was assumed. Therefore, we extend in this section the procedure
to HEs with partial condensation.
Hot utility mass flow rate can take 128 possible values equally distributed
between mHU nùn and m H U max. We determine if condensation of the hot utility occurs
completely or partially by comparing the total heat transfer rate that is expressed by
Q = rhccp,(Tco-Tci) (3.13)
Two cases are possible. If
mHUCp.HU.Gp(THUJ - T H U . s a , ) + mHU ifg.HU ^ Q ( 3 1 4 )
Then condensation is complete. Otherwise, when
m HU C p .HU.GP^HU. i~^HU.sa t ) + mHU lfg .HU > & (3.15)
The condensation is partial.
For the first case, hot utility vapor will come out of the exchanger as sub-cooled
fluid. Then
47
rr rw, *£ mHUCp,HU.Gp(*HU.i *HU,sat) , _ . , . 7Wf/.0 = IHU.sa, 1 (3 .16 )
Method developed for optimization of shell-and-tube condensers [16] is directly applied
to design the corresponding heat exchanger.
For case #2, hot utility at the outlet of the exchanger will be a mix of gas and
liquid at saturation temperature (THu.o = Tmj.sat)- Heat transfer coefficient will be
calculated using correlation developed by Chato [29] if condensation occurs in tubes and
Nusselt correlation if condensation occurs on the shell side.
In order to be able to calculate pressure drop on the side where condensation
occurs, mix quality (x0) at the heat exchanger outlet has to be determined. Isolating the
quality from the energy balance on the HU side, we obtain:
y~~mHUCp,HU,GP\*HU,i~*HU,o> , ~ , ~ X o = — . (3-17)
lfg,HU
If condensation occurs in the tubes, total pressure drop can be separated in two terms
AP,=APlGP+APlC (3.18)
where AP,GP and AP,c stand respectively for the pressure drop in the vapor section and
for the pressure drop in the condensation section. We used the expression previously
developed [16] for the first term. For the condensation zone, it has been shown [53] that
pressure drop can be expressed by the following formula:
AP,C M 6 3 2 f ^ Z f m - ' r L c ) ( l ~ X ) P H U S + X P H U J (3-19)
/
The analytic resolution of the integral 1 yields to
4S
/ = 5(PnU.g XoPHU.g+XoPHU.l) [ ( 4 9 ) 0 „ + ( - 4 ^ + 4 ) ^ u
- \ f , l_ l , i. \2 L^ ° y ) P n U g ' U H U . l ^ y ^ A - o ^ ^ i P H U . g P H U . g JOV H-HU.g + P H U . I >
+(4xo +5)pHUJHm. t -^X„PHU.IPHUJ]
(3.20)
Pumping power is then calculated using following equation for tubes
p ^t.GP^HU , ^ . C % (XoPHU.g + fl ~ * g ) / W / ) „ , . . £., — 1 ( A ^ U
VpumpPnU.GP 2VpumpXo U — X0 )PHU,gPHU.l
The shell side pumping power is calculated as in [53].
If partial condensation occurs on the shell-side, total pressure drop is once again
separated in two parts
APs=APsjGP+AP1<c (3.22)
The pressure drop for superheated gas (APSIGP) is calculated using Bell-Delaware method
[11]. The entire procedure is explained elsewhere [14]. It has been shown in [15] that
condensation sub-section pressure drop has two contributions
APs,c=APJ,c,c/+AP^M/ (3.23)
where subscripts cf and wf respectively stand for the cross flow zone and the window-
flow zone pressure drop. From the Chisholm correlation [50], we obtained the following
expression to calculate the cross flow zone pressure drop
= [ ¥ ] ^ J [ l + (y2-1X*-*2)a8,5 + *U 7>* (3-24) V dz )LO 0
AP H -s.ccf
where Y2 is the Chisholm parameter and LO refers to the total flow having the liquid
properties. No analytical solution is found for the integral in Eq. (3.23). Consequently, a
numerical integral is performed to solve the problem.
Window-flow pressure drop is calculated with the following expression:
49
AP s.C.vtf ap dz
Lc\[\ + (Y2-\)x]lx LO 0
(3.25)
and the integration gives
, . C , I 3 z I C '1.0
Y 2 - l 2 (3.26)
Total pumping power for shell-side and tube-side is calculated as follows
AP sCPmHU AP s C m H U (x 0 p H U G P + ( l - x 0 ) p H U L P ) b , = h-
^IpumpPHU.GP 2 7 l p u m p X o ^ X O > P H U , G P P H U . L P
(3.27)
ATX l pump r e
(3.28)
3.5. Test cases
We considered two different test cases to show the ability and the versatility of the
proposed optimization procedure. Every heat exchanger designed for the two test cases
are assumed to operate 5000 hours per year. Electricity cost is 0.10 $/kWh and pump
efficiency is 85%. Moreover, each HE has a lifetime of 20 years and the annual interest
rate is 5%. Thermophysical properties (e.g., density, heat capacity, thermal conductivity
and viscosity) of the fluids used in the following examples are considered constant,
except water for which properties function of the average temperature of the fluid in the
exchanger.
We first considered a simple example that involves two hot streams and two cold
streams. Process requirements for test case #1 are shown in Table 3.1. A hot stream of
water (3 kg/s) and a hot stream of crude oil (7.2 kg/s) are available to heat up streams of
kerosene (3.6 kg/s) and of water (10 kg/s). Water vapor at 200°C is used as hot utility
and cold water at 20°C is used as cold utility. Similar case can be found in refinery to
preheat petroleum products. Optimization of the heat exchanger network has been
50
performed for 20 different values of the minimum temperature difference (ATmin) from
l°Cto20°C.
Table 3.1 Process requirements for test case #1.
Stream Stream fluid Supply temp. (°C) Target temp. (°C) Flow (kg/s)
30 7.2
50 3
140 10
140 3.6
HU Steam 200
CU Water 20
HI Crude oil 150 H2 Water 130
CI Water 100 C2 Kerosene 50
For each minimum temperature difference considered, a HEN has been designed
to recover as much heat as possible as explained in Section 3.2. Then, using the GA, we
found the optimal design for each exchanger of the network, and computed afterwards
the total annualized cost for the complete HEN. Costs of hot utility and cold utility are
also considered. Fig. 3.4 shows the annualized total cost, utility cost and HEs cost as a
function of ATmin.
51
220 000
200 000 -
180 000-
160 000
140 000
?,120 000
t5 100 000 o o
80 000
60 000
40 000
20 000
0.
T F
Utility cost
Cost of HEs
_l L 2 4 8 10 12
AT .(°C) 14 16 18 20
Figure 3.4 Minimum heat exchanger network total cost as a function of minimum
temperature difference for test case #1.
Utility costs are a way to gage heat recovery. The more hot and cold utilities are
solicited, the less heat is recovered. As mentioned above, 7 runs of the GA were
performed for each ATmin to identify the absolute minimal total cost. Considering the cost
of utilities, our results show that the optimal ATmin is 3°C for this test case. Not
considering utilities cost, the optimal solution is found at ATmin = 20°C. It is clear thus
that the cost of the utilities has an influence on the optimal solution.
Figure 3.5 shows a schematic representation of the optimal heat exchanger
network with the matches between cold and hot fluids as well as the points from which
utilities are used for each stream.
52
H2-
E | H I 55
C2
C1
' — Cold / Hot Sream Heat échanger Pinch
CU/HU Exchanger with cold / hot utility
cu.
cu.
/ / * I
/ / * /
- * — * ,VHU y / >
/ *
HU
20 40 60 80 100 Temperature (gC)
120 140 160
Figure 3.5 Optimal heat exchanger network design for test case #1.
Details about each heat exchanger inlet and outlet temperatures are given in
Table 3.2. Optimal design geometries of the eight HEs are listed in Table 3.3
Table 3.2 Stream data for shell-and-tube heat exchangers of the optimal heat
exchanger network for test case #1.
53
HE #
COLD stream
Tc, (°C)
' c.o
(°C) mc
(kg/s) HOT
stream Th.i Th,0 rhh
(°C) (°C) (kg/s)
1 CI 100 120.8 5.6 HI 150 103 7.2 2 CI 100 106 4.4 H2 130 103 2.2
3 C2 100 109.7 3.6 H2 130 103 0.8 4 C2 50 100 3.6 HI 102 78 7.2
5 CI C2
114.3 109.7
140
140 10 3.6
HU
HU
200
200 6 CI C2
114.3 109.7
140
140 10 3.6
HU
HU
200
200 CI C2
114.3 109.7
140
140 10 3.6
HU
HU
200
200
7 CU CU
20 20
HI H2
78 103
30 50
7.2 8
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55
A second test case involving more streams is studied. This example also involves
largest flow rates. Data is presented in Table 3.4. We need to heat up 4 streams: crude oil
(81 kg/s), water (35 kg/s), BPA (41 kg/s), LGO (26 kg/s). Three hot fluid need to be
cooled: kerosene (77 kg/s), water (47 kg/s), HGO (53 kg/s). Such processes are typically
found in petroleum industries.
Table 3.4 Process requirements for test case #2.
Stream Stream fluid Supply temp. (°C) Target temp. (°C) Flow (kg/s)
60 77 40 47
HI Kerosene 393 H2 Water 160
H3 HGO 354
CI Crude oil 72
C2 Water 62
C3 BPA 120 C4 LGO 147
60 53 356 81 210 35 370 41 284 26
HU Steam 372 CU Water 10
The minimal cost as a function of ATmjn is shown in Fig. 3.6. Once again, results
show a difference between the solution with and without the cost of utilities. The global
optimal solution is when ATmin = 4°C. Fig. 3.7 shows a representation of the optimal
network.
56
2 500 000
0 2 4 6 8 10 12 14 16 18 20 AT . (°C)
Figure 3.6 Minimum heat exchanger network total cost as a function of minimum
temperature difference for test case #2.
57
H3
H2
IHI 0)
co C4
C3
C2
C1
-
C Uo
i ■ 1
-
C Uo
i
— / — Cold / Hot stream - Heat exchanger Pinch Exchanger with cold / hot utility -
C Uo
i
CU/HU
Cold / Hot stream - Heat exchanger Pinch Exchanger with cold / hot utility -
C Uo
i
C Uo
n n
r- i
• i »
/ i
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C uo I
•A, r- i
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' — " "" "
/ / /
: / 1
°HU
1 1
«*'
°HU
1 1 1 i 1 1 1
50 100 150 200 250 Temperature (5
C) 300 350 400
Figure 3.7 Optimal heat exchanger network design for test case #2.
Heat exchangers temperature and optimal design geometries are respectively listed
in Tables 3.5 and 3.6.
58
Table 3.5 Stream data for shell-and-tube heat exchangers of the optimal heat
exchanger network for test case #1.
HE COLD Tc,i Tc,0 mc HOT Thji Th,0 mh # stream (°C) (°C) (kg/s) stream (°C) (°C) (kg/s)
1 C3 156 201.4 41 HI 393 369.3 77
2 C3 201.4 300.5 41 H3 354 277.4 53
3 C4 156 274.4 26 H3 277.4 219.8 53
4 C2 156 210 35 H3 219.8 160 53
5 CI 156 356 81 HI 369.3 160 77
6 C4 147 157 19.6 H2 160 150.1 12
7 C4 147 156 6.4 H3 160 150 6.4
8 CI 72 156 4 H3 160 99.2 5.6
9 C3 120 156 41 H3 160 123 41
10 C2 62 156 35 H2 160 65.2 35
11 CI 72 156 77 HI 160 75 77
14 CU 10
15 CU 10
16 cu 10
12 C3
C4
300.5
274.4
370
284
41
26
HU
HU
X l l
13
C3
C4
300.5
274.4
370
284
41
26
HU
HU XII
C3
C4
300.5
274.4
370
284
41
26
HU
HU
HI 75 60 77
H2 86.8 40 47
H3 123.8 60 53
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60
Figures 3.4 and 3.6 present the curves obtained from our simulations. When ATmin
increases, less heat is recovered and therefore, HU and CU are more solicited. This explains
why the utility costs increase with ATmin. We clearly see that utilities cost increase linearly
with ATmi„. On the other hand, the cost of the HEs themselves decreases when ATmin
increases. For the two cases considered, this decrease is greater for low values of ATmi„. As
a result, combination of HEs cost and utility costs presents an optimum.
For test case #1, a total of eight heat exchangers were optimized for each value of
ATmin. Hence, it is no surprise that the curves in Fig. 3.4 are smooth. On the other hand, for
the second example, the total number of exchangers in the network varies between 13 and
16 depending on ATmin. However, we did not notice any considerable step on the curves in
Figures 3.4 and 3.6. Nevertheless, curves in Fig. 3.6 are also smooth.
It is worth to recall that the optimization of each HE was performed 7 times (see
Section 3.3.3). The maximal variation of the HE cost between two runs of the GA was 5%
for the first test case and 1% for the second test case. Even though these variations were
relatively small, they were sufficient to disrupt the curves of Figs. 3.4 and 3.6 and create
"artificial" local minima, when only one run of the GA was performed for each HE. The
procedure proposed here with 7 runs of the GA per HE was found to be robust and
reproducible.
An interesting observation is the small number of HEs that were calculated to find
the best solution. For the combined optimization of case #1, 0.0000198% of every possible
design has been calculated to converge and this proportion is 0.0000193% for case #2. This
proves that using GA for the problem studied in this paper results in an important saving of
computational time.
3.6. Conclusions
A procedure is proposed for designing in details a HEN. For a given ATmin, an optimal HEN
was determined based on pinch analysis. Then, each HE (including condensers) of the
network was optimized with a GA. The optimal flow rates of the HU and CU fluids were
61
also optimized. The minimized total cost of the HEN was calculated. The procedure was
repeated for different ATmj„ in order to find the optimal value of ATmin. The procedure was
validated with 2 test cases. We found that the GA can rapidly identify the best design for
each HE, including for the condensers of the network. This yields a better estimate of the
total HEN cost, by including the pumping power in the total cost, and by providing a
detailed design for each HE.
Further research could include other types of HEs, such as plate heat exchangers,
and let the GA decide for each HE of which type it should be. The determination of the
optimal ATmi„ could also be performed by a GA or another optimization approach to speed
up convergence.
62
Chapitre 4
Article # 3
Titre:
Thermoeconomic optimization of components and operation of vapour-
compression refrigeration cycle with genetic algorithms
63
Abstract
This paper proposes a model for calculating the total cost of a refrigeration cycle including
a compressor and two heat exchangers. An optimization procedure based on a genetic
algorithm is used to minimize the annualized total cost of the system. The global cost
includes the energy cost (pumping and compression) as well as the initial cost of the
compressor, the evaporator and the condenser. A total of 24 design variables are considered
for this problem. Ten are related to the geometry of each heat exchanger. Two additional
design variables characterize the condenser (i.e., side (shell or tubes) of the refrigerant flow
and mass flow rate of water in which heat is rejected in the condenser). Finally, the
compressor inlet and outlet pressures represent two more design variables. Two case
studies are presented to show the potential of the approach to find the best solution for
different situations and the ability of the genetic algorithm to identify the best design for
this specific problem.
64
4.1. Introduction
Cold water is widely used to provide air-conditioning in large buildings [55] and many
industrial processes requiring refrigeration. Air conditioning is responsible for about 30%
of energy consumption in commercial buildings and this proportion reaches 50% in warm
climate regions [56]. With growing costs of energy and needs for more efficient systems,
the optimization of refrigeration systems represents potential savings in terms of money
and a potential for reducing energy consumption and green-house gas (GHG) emissions.
However, a lot of parameters must be considered for designing refrigeration systems.
Numerous designs are possible which makes the identification of the best system (i.e.,
optimal) a difficult task.
Modeling of the different parts of a refrigeration system has been extensively
studied and numerous thermodynamic modes have been developed. Gordon et al.
developed a relation between the coefficient of performance and the cooling rate of a chiller
[57]. Khan and Zubair developed a method to quantify irreversibilities in a vapor-
compression chiller [58]. Chua et al. led experimental study in order to show the impact of
different parameters on the COP of chillers [59] and Gordon et al. propose a diagnostic
model to predict chiller performance from few measurements [60] as well as a
thermodynamic model with adjustable parameter for a particular chiller [61]. Browne and
Bansal proposed a NTU based model [62]. In the last years, numeric tools allowed the
elaboration of vapor-compression chillers models with neural networks [63,64]. Methods to
calculate different parameters (heat transfer coefficients and pressure drops) required to
predict the performance of a chiller have been the subject of many works in particular for
the prediction of two-phase flow heat transfer. Chen developed a correlation from 600 data
points for in-tube convective boiling [65]. Webb and Gupte reviewed different correlations
to predict convective heat transfer in tubes and in tubes banks [66]. All these developments
led to the development of methods to determine performance and costs of refrigeration
systems in order to optimize their design. Ng et al. developed a diagnostic method to
65
establish optimal operating conditions for reciprocating chillers [67]. Selbas et al. worked
on an exergy-based thermodynamic optimization procedure [68]. Gordon et al. proposed an
optimization approach based on finite time thermodynamics model [69]. Finally, Yu and
Chan optimized the number and size of chillers to satisfy a refrigeration demand at a
minimized cost [70]. These studies show the optimization opportunities of refrigeration
systems, but did not optimize the complete geometry of the components of the systems.
In this paper, we develop a thermoeconomic model for estimating the total cost of a
complete chiller including a shell-and-tube condenser, a shell-and-tube evaporator as well
as a reciprocating compressor. The model accounts for the geometry of the two heat
exchangers. In order to identify the system with the minimal cost, we propose an
optimization procedure that determines the best geometry for each heat exchanger and the
best operating conditions of the vapor compression refrigeration cycle.
4.2. Objective function and design variables
We consider a classical vapor compression refrigeration cycle, see Fig. 4.1. The refrigerant
pressure and temperature are increased by a compressor (point 1 to 2). The vapor
refrigerant is cooled down in a condenser (point 2 to 3) before going through a valve in
which its pressure and temperature are decreased (point 3 to 4). Then, it absorbs heat from a
heat source (point 4 to 1) in an evaporator, and so on. The combination of these
components (i.e., compressor, evaporator, and condenser) forms a "chiller". The
corresponding temperature-entropy diagram of the cycle is given in Fig. 4.2.
66
comp
Figure 4.1 Schematic representation of the vapor compression refrigeration cycle.
67
Figure 4.2 Temperature-entropy diagram of an ideal vapor compression cycle.
For a given design heat load ( QL ), we want to design the best components of the
refrigeration cycle. The objective function to minimize is the overall cost of the project.
This cost is dominated by the purchase costs of the condenser, of the evaporator and of the
compressor, the cost for pumping the fluid through the heat exchangers, and the cost related
to the compression of the refrigerant. The purchase costs are initial costs while pumping
and compressing costs are recurrent costs. Therefore, we annualize the purchase costs by
considering an interest rate I and a number of years n for the project, in such a way that the
total cost (TC) can be written as
TC = (PC +PC +PC )d>, +(OC +OC ) \ cn ev comp ) T I,n \ pump comp)
(4.1)
with:
tl,n = (l + 7 ) n - l (4.2)
68
PC values represent purchase costs (i.e., initial costs) and OC, annual recurrent costs.
Details about their calculations are given in the following sections.
Equation (4.1) will be minimized by varying a certain number of design variables
that characterize each component of the chiller system. The list of design variables and
their possible values for the geometry of the two shell-and-tube heat exchangers (condenser
and evaporator) are available in Table 4.1. Ten design variables characterize the condenser
geometry, plus one more design variable for the side where the condensing refrigerant fluid
flows. Ten design variables characterize the evaporator geometry.
69
Table 4.1 Design variables and their limit values for the two heat exchangers of the
problem.
Possibilities Condenser Evaporator
Shell diameter (D) 16 300 to 1050 mm 300 to 1050 mm
Tube diameter (d0) 8 15.87 to 63.5 mm 15.87 to 63.5 mm
Tube bundle
configuration 3 90°, 30°, 45° 90°, 30°, 45°
Tube pitch (p) 4 1.24, to L54 1.24, to 1.54
Shell-to-baffle
spacing (As.b) 4 0.014, to 0.14, 0.014, to 0.14,
Tube-to-baffle
spacing (A,.b) 4 0.014, to 0.14 0.014 to 0.14
Baffle cut (B) 25 25 to 45 % 25 to 45 %
Center baffle spacing
yj-ib.center) 8 0.2D to 0.55D 0.27) to 0.55D
Inlet/outlet baffle
spacing (LbJLb,0) 8 l-ib.center tO Y.OL,bcenter '-'b.cemer tO 1 .OL, b c e n t e r
Tube bundle diameter
(Doll) 4
0.8(7)-ZU) to 0.95(D
-A-b)
0.8(D - A-b) to 0.95(D
- A-b)
Refrigerant flowing
side 2 Tubes, shell
Refrigerant flowing
side 2 Tubes, shell
Refrigerant operating
pressure 256
Depends on heat sink
temperatures
Depends on chilled
water temperatures
Condenser water
mass flow rate 256
Minimum to maximum
available
Condenser water
mass flow rate 256
Minimum to maximum
available
70
Furthermore, we consider 3 operating parameters: the mass flow rate of the heat
sink fluid in the condenser ( mc ), and the refrigerant operating pressures in the condenser
(Pad and in the evaporator (Pev). The possible values of these parameters are chosen in
order to avoid temperature crossing in the heat exchangers. In a counter-flow condenser,
temperature crossing occurs when we calculate a cold fluid temperature that would be
higher than hot fluid temperature at any point in the heat exchanger. Such condenser is
physically impossible to realize.
The maximum and minimum possible values of the operating pressures as well as
the water flow rate (heat sink) are established from the system requirements. The minimum
refrigerant operating pressure in the evaporator is chosen in order to avoid the freezing of
water inside the exchanger. For this reason, we choose the minimum operating pressure for
which the corresponding saturation temperature is 0°C (P@T =«,(.)• With the same
reasoning, the maximum value is the pressure for which the saturation temperature of the
refrigerant corresponds to the supply temperature of the chilled water (P@T =r ). For the
condenser, the minimal operating pressure is the pressure for which the saturation
temperature corresponds to the water outlet temperature when the maximum water mass
flow rate (rhcma) is used:
m. ( h - h ) T = ref v z—^- + T (4 3) 1 ref.sat.min . ^ 1 c , i V*—V f.max p.c
The maximum operating pressure is the maximum pressure at which the compressor and
the heat exchanger can operate. Finally, the minimum water mass flow rate circulating in
the condenser will depend of the refrigerant operating pressure in the condenser. It is
calculated independently for each design considered. It is chosen in order to have a water
outlet temperature equal to the saturation temperature of the refrigerant:
m ref{h 'h) . . . . m r min = r - —-r (4.4) c ,min
p,c \ ref,sat c,i )
71
The maximum value is simply the maximum mass flow rate available.
The operating pressures will influence the compressor size and therefore, its
purchase and operating costs that are taken into account in our optimization. Note that the
refrigerant mass flow rate ( mref ) follows from the knowledge of QL, P n , Pcn and that of
the cycle (See Fig. 4.2), and therefore it is not considered as a design variable here, but
rather as an optimization result . Hence, a total of 24 design variables are taken into
account. All of these variables will have an effect on the overall heat transfer coefficients,
the pressure drops and on the power input to the system, and consequently on its total
annualized cost.
Figure 4.3 shows the methodology that we use to determine the total cost for a
specific set of design variables. The grey upper boxes represent the chilled water mass flow
rate ( mh ) and its inlet and outlet temperatures (7/,,,, Tn,o) as well as the values given to the
24 design variables for the specific design considered (geometry, condenser mass flow rate
and operating pressures). The first step consists in determining the refrigerant
thermodynamic properties at different point of the cycle. These values depend on pressure
and temperature. A Matlab function has been created in order to interpolate thermodynamic
properties from a database containing values of required properties for many temperatures
and pressures. Thermodynamic properties of refrigerant used in this paper are taken from
the National Institute of Standards and Technology (NIST) [32]. Since operating pressures
are taken as design variables, refrigerant thermodynamic specific properties at the four
points in the cycle are computed using our interpolation functions. Hence, for each specific
design, the following properties are interpolated: specific entropy (s), specific enthalpy (/),
specific heat capacity (cp), density (p), thermal conductivity (k), dynamic viscosity (//).
This procedure is required since heat transfer coefficients, pressure drops, compression
work and heat transfer rates are calculated from these specific properties. Next, the
refrigerant mass flow rate (mref ) is computed. Then the required power input (Wcomp), the
heat load (QL) and the rate of heat rejected (QH) are calculated. These first steps are
72
represented inside the bold square in Fig. 4.3. From there, the problem is separated in three
parts (condenser, evaporator, and compressor). The next steps consist in calculating heat
transfer coefficients and pressure drops in order to determine the length of the two heat
exchangers as well as the compressor size. The left part of the diagram in Fig. 4.3 accounts
for condenser calculations while the right part accounts for the evaporator calculations.
Compressor costs can be computed directly from refrigerant mass flow rate and refrigerant
specific enthalpies at point 1 and 2 (Fig. 4.2). The following sections provide more details
about the models used to calculate the required parameters, i.e. how to relate Eq. (4.1) to
the design variables via an appropriate modeling of the system and of its components.
Given the 24 design variables, the refrigerant mass flow rate and its specific
thermodynamic properties, the three main components of the system (i.e. compressor,
condenser and evaporator) are considered as three independent design problems.
73
Condenser Geometry
Condenser water mass flow rate
Operating Pressures |
Cond. / Evap. Sat. Temperatures
Chilled fluid properties n»ti, T*,, Tu» c^,
Condenser water temperatures
Condenser sub
sections heat transfer area first guesses
R152a Saturation Specific properties
s2 = s, t4 = ts
1. ï_
AT»T)
* Qi =<,(', 0
Q H = ' " r . f ( >2 - ' , ) *
ATLJ A7„x
Wcaup " » V ( ' J ' I )
Condenser sub
sections: desuperheating (d)
condensing (c)
&»../= » V ('2 ~'s)
Q t r ,= '" ,A ' ' h)
Tube/shell side pressure drop calculation
Pumping power
Evaporator Geometry
Evap. Separation
X, = T .
X = 1
Ar„
C>„,,=à/iooo
Evap. heat transfer area first guess Ao
1000
Shellside heat transfer coefficient
Cfor i =1.1000;
I 1 ■„.,=&., IA
Intube heat transfer coefficient
T A=Q^^T^,/u, ;
1000
Tube/shell side pressure drop calculation
Pumping power
Figure 4.3 Schematic representation of total cost calculation procedure.
74
4.3. Condenser model
In a precedent paper [71], we developed a model of condenser relating its
geometrical features to its purchase cost and its pumping power cost. Therefore, we will
only summarize this approach here as details are available elsewhere. First, the shell-and
tube heat exchanger is separated in two sections: (1) the desuperheating region where the
refrigerant passes from superheated vapor to saturated vapor and (2) The condensing region
where condensation brings refrigerant to saturated liquid state. The desuperheating section
corresponds to the line between points 2 and 5 in Fig. 4.2 while the condensing section is
represented by the line between points 5 and 3 in the same figure. Heat transfer coefficients
and pressure drops are calculated separately for each sub-section. On the shell-side, we rely
on Bell-Delaware method to calculate those parameters [14]. Heat transfer coefficient of
the two-phase condensing flow is given by an empirical correlation proposed by Chato [29]
for the case where the refrigerant flows in tubes. It is given by the Nusselt correlation [5] if
the refrigerant flows in the shell side. Pressure drops are calculated using a homogeneous
two-phase flow model proposed in [14]. Heat transfer coefficients and pressure drops
calculation requires the value of the heat exchanger surface area. Since heat transfer
coefficients are needed to determine the surface area, an iterative procedure is needed.
Hence, we first suppose a heat transfer surface area value for each sub-section, a global
heat transfer coefficient is then computed and from the heat transfer rate of each sub section
as well as the logarithmic mean temperature difference established from known heat sink
properties, new heat transfer surface areas are computed. This procedure is repeated until
convergence. The procedure is once again schematically represented on the left side of Fig.
4.3.
4.4. Evaporator model
The approach that we developed for modeling the evaporator is similar to that used for the
condenser. This sub-section presents calculation of the heat transfer coefficients in the
75
evaporator as well as the pressure drop required to maintain the mass flow rate of
refrigerant in the heat exchanger.
4.4.1 Heat transfer calculations
In our analysis, the water (heat source) is forced to flow on the shell-side and the
evaporating refrigerant flows on the tube side. A previous article [16] demonstrates the
approach used to calculate the heat transfer coefficient on the shell side for a single phase
flow. Here, we only explain the method that we built to predict heat transfer coefficient in
the evaporating refrigerant that flows in tubes.
Since the model must be able to predict the heat exchange surface area of numerous
different designs, a general correlation is required to predict the heat transfer coefficient in
different conditions. Correlations used for the single phase flow cannot be used here
because the refrigerant is evaporating and the flow structure is characterized by complex
physic phenomenon [28].
In this paper, we use an empirical correlation proposed by Kandlikar [72]:
fc =C , l - x \ 0 * f n \ 0.5
X J \ r i J
f 4 0 0 < / > +c
n d 2 ^
V mref lfg ) f l (4.5)
Coefficients Ci to C5 depend on the dimensionless convection number defined as:
Co = \ \ 0 * ' „ \0-5
l-*ï PA ' i J
(4.6)
Heat transfer coefficient can be calculated for any given conditions by changing the
constants values [72]. Evaporation can be convective, i.e. for Co < 0.65, or nucleate, i.e. for
Co > 0.65. Nucleate boiling occurs in first stages of the evaporation process, for small
values of quality (x). For higher quality values, nucléation disappears, which is why Eq.
(4.5) considers nucleate boiling and convective boiling zones separately as illustrated in
Fig. 4.4. Ffl is a correction factor that depends on the refrigerant fluid used in the cycle. Fjj
76
is defined for nine different refrigerants in [72]. Many of these refrigerants have been
phased out due to their high ozone depletion factor. R-152a being the only refrigerant still
accepted from those proposed by [72], we only consider the utilization of this refrigerant in
our analysis. Further work could focus on other refrigerants provided that a correlation such
as Eq. (4.5) is available for them.
! I QL
" V J NUCLEATE
.r = 0
Figure 4.4 Nucleate and convective boiling in evaporating two-phase flow.
It can be seen from Eq. (4.5) that the value of the heat transfer coefficient hev
depends on x, the quality of the two-phase flow. Here we cannot consider the quality as
constant. In fact, refrigerant comes out of the evaporator with a 100% quality ( x - \ )
whereas it enters the heat exchanger with a low quality. To solve this problem, tubes are
virtually separated in n small sections and inside each of these sections, quality is
considered as constant. Hence, the correlation of Eq. (4.5) can be applied to each section
separately. This idea is shown in Fig. 4.5 in which one tube of the evaporator tube bundle is
represented. Taking a higher number of sub-sections will bring more accuracy but longer
calculations.
77
m ref
V A
QL
i 1 A A
■S- L.
\
r\\
m, ci
-v
X , - . Y 4 3 ( Y , - . Y 4 ) 5( .Y , - .Y 4 )
m in 2« (2/7l)(.r1.x4)
2/7
Figure 4.5 Constant quality separation of the evaporating refrigerant flow.
Equation (4.5) shows that the heat transfer coefficient also depends on the heat flux
(q n„) . Since q"ev depends on the heat transfer coefficient, here again, an iterative approach
is used to determine the heat transfer coefficient. The tubes subsections in Fig. 4.5 are such
that the heat transfer rate is equal in every subsection (Fig. 4.5). Therefore, the heat
transfer surface area of each subsection is different. Considering m sections, calculation of
the heat transfer rate is given by:
Qev = ™m l X :
m v /g«p„ (4.7)
For a given section of constant quality, we suppose a heat transfer surface area Ao,
that is our first guess. The heat flux based on our first guess is determined using the
following relation:
" ev A,
(4.8)
Quality and heat flux being now fixed in a given subsection, the heat transfer
coefficient in the evaporating zone (hev) is calculated using Eq. (4.5). Knowing the heat
78
transfer coefficient on shell-side of the evaporator (hs) and the tubes thermal conductivity
(kw), a global heat transfer coefficient is calculated:
U = (4 9) " 1 d, / 0 l n K / 4 ) , 1 ( ' }
hs d0 2kw h„
and an updated heat transfer surface area (Aj) is obtained from the following relation:
A = ^ " (4.10) ev Im
The new surface value is taken as the new guess and back to Eq. (4.7), the
procedure is repeated until convergence of the heat transfer coefficient. Calculation of the
log mean temperature difference is required for each section:
I T _ T \ - ( T - T ) \ 5.1 ref .sat ) \ s.o ref .sal ) ( 4 1 1 1
l m ~ ^ [ ( T s . , - T r e f , s a , ) l ( T s , o - T r e f , s a t ) ]
TSfi and TSi0 account for the shell temperature associated with the section considered. These
temperatures can easily be computed from the value of the heat transfer rate Qev.
The procedure is summarized in the right hand side of Fig. 4.3 and is repeated for
each section of constant quality. The required evaporator heat transfer surface area is
obtained by summing the area calculated for each of the m sub-section.
4.4.2 Pressure drop calculations
It has been mentioned that the refrigerant is forced to flow inside the tubes. Pressure drop in
the two-phase flow has to be calculated in order to determine the pumping power input to
the system. For a given mass flow rate and a given tube diameter, Friedel correlation is
used [14]:
79
VLO2 = R + 3.24ZJ 0.045ii/ 0.035 Frum>We
(4.12)
Friedel correlation is considered as an accurate correlation when (//, / / O < 1000 [73]. y/i LO
is the ratio between the pressure drop of the two-phase flow and the pressure drop of the
same mass flow rate of the same refrigerant at a saturated liquid state:
V, LO jdp/dz)
{dpldz)L0 (4.13)
where
R = ( \ -x) 2 +x 7
Z = x078(l-;c)
( ~ t \ Pifg
0.24
( V " ( , , \019 ( J =
\pg J
M,
U i J
1 - ^ V Pi J
.0.7
(4.14)
The two dimensionless numbers Fr et We are the Froude number and the Weber number of
the flow:
Fr = '— gdiPTP
rh r}d W e ^ ^ y ^
PTP a
(4.15)
The two-phase density of the flow is defined by:
PTP
( . V x l - x — + IP» J i
(4.16)
The first step consists in calculating the pressure drop of the refrigerant mass flow rate as a
saturated liquid:
dp^ dz JLO d.
80
(4.17)
We make the assumption that the flow is turbulent since we wish to have a turbulent flow
in order to increase heat transfer coefficients. Hence, the friction coefficient can be
calculated by the following correlation for turbulent fully developed flow inside smooth
tubes:
-0.2 / „ = 0 . 0 4 6 ( 1 ^ )
Then combining Eqs. (4.12) and (4.13), the two-phase flow pressure drop is found:
(4.18)
dP fdP dz dz
R + 3.24ZJ 0.045ti/ 0.035
LO Fr^Wé (4.19)
And from the heat exchanger length (Lev) obtained from heat transfer surface calculation,
we obtain the pressure inside the tubes:
AP = s dz ) w
t „ 3.24Z/ R + ^ 0 . 0 4 5 ^ 0 . 0 3 5 (4.20)
Once again, pressure drop depends on the quality of the liquid-vapour refrigerant mixture.
For this reason, tubes are separated in sections of constant quality. Hence, pressure drop is
calculated for each of these sections and the total pressure drop is obtained by summing
pressure drop of every section.
4.5. Compressor model
The most expensive component of a refrigeration (purchase and operation) system is the
compressor. The compressor capacity required is determined by the pressure ratio of the
refrigerant between the condenser and the evaporator as well as the mass flow rate of
refrigerant circulating through the cycle. The operation cost is thus
81
mr^ (i,—i) H e OCcomp=
refK2 x ) " * (4.21) comp i comp
where /, and i2 are the refrigerant specific enthalpies at the compressor inlet and outlet
respectively. 77 is the annual operating time of the system, e is the electricity cost and ncomp
is the compressor efficiency.
Determination of compressor purchase cost is made using the following relation
proposed by Smith [5]:
PCcomp =98400 / ■ s 0.46
comp
v250000, (4.22)
where W is the required power of the compressor:
W c o m p = r e f K l l } (4.23) " c , f comp
4.6. Optimization approach
Again, our objective is to identify the set of design variables (see Table 5.1) that will lead
to the cheapest system and yet respect the problem specifications. In order to accomplish
this task, we use genetic algorithms (GAs). GAs are a probabilistic method that relies on
the principles of natural selection to improve a population of designs over generations. We
will not repeat here all the details on this procedure that is becoming more widely used and
that is well documented in literature. A recent review on the use of GAs in heat transfer
problems is available [74].
In this paper, the GA that we used is binary and elitist. Each population is made of
150 individuals. Each individual is represented by a 79 bits vector. Mutation probability is
4 %. Four crossover points are considered and crossover occurs at a 90% probability for
82
each of these points. Table 4.2 summarizes the main features of the GA. Considering all the
design variables, this gives 6.045xlO23 possible designs. Computing each of these designs
would take too much time to be efficient.
Table 4.2 Main features of the GA considered in this paper.
Number of individuals 150/generation
Vector length ( 1 design) 79 bits
Mutation probability 4 %
Crossover points 4
Crossover probability 90%
Convergence 300 generations without evolution
For every generation in the process, the cost of each design of the population is
calculated following the procedure illustrated in Fig. 4.3 and explained in the previous
sections. The optimal solution (lowest total cost design) is considered to be identified when
the best solution remains the same for 300 consecutive generations of the GA.
GAs are probabilistic processes and the optimal solution found by the algorithm can
vary from one run to another. That is why for each test case considered, 5 runs of the GA
are performed with the same settings and we considered the optimal solution to be the best
design among the 5 runs. The following section shows the results obtained.
4.7. Test cases and results
We consider in this section two test cases to show the ability of the methodology developed
in this paper to find the optimal solution for different systems. For the first case, we
consider a large capacity water chiller. The system must be designed to cool down 26.5
kg/s of water from 13°C to 4°C in steady state. This represents a cooling load of 1 MW.
83
The complete data of the problem as well as the economical parameters are given in Table
4.3. The example is taken from the refrigeration requirements of a gold mine [75].
For the second case, we consider a typical water chiller designed to supply cold
water for air-conditioning systems [76]. The mass flow rate of water to cool down is 72
kg/s. Its discharge temperature is 12.5°C and its supply temperature is 7°C. This represents
a cooling load of 1.66 MW. Once again, the problem is summarized in Table 4.3. Here
again we choose the operation pressure for which the corresponding saturation temperature
is 0°C as the minimum accepted pressure in the evaporator to avoid icing of the chilled
water.
84
Table 4.3 System requirements details for test case #1 and test case #2.
Test case#l Test case #2
Condenser
Available water mass flow rate 56.4 kg/s 87 kg/s
Inlet water temperature 24°C 33°C
Fouling Resistance 0.000275 (K-m2)/W 0.000275 (K-m2)/W
Evaporator
Chilled water mass flow rate 26.5 kg/s 72 kg/s
Chilled water supply temperature 13°C 12.5°C
Chilled water return temperature 4°C 7°C
Fouling resistance 0.000275 (K-m2)/W 0.000275 (K-m2)/W
Economic considerations
Lifetime 20 years 20 years
Operation period 5000 h/year 5000 h/year
Electricity cost 0.10$/(kW-h) 0.10$/(kW-h)
Interest rate 5 % 5 %
Compressor efficiency 85% 85%
Refrigerant R152a R152a
Refrigeration load 1MW 1.66 MW
85
For the first test case, the annualized minimal cost obtained by the genetic algorithm
is 152 526 $. Purchase accounts for 43.8 % of the cost while energy consumption accounts
for 56.2 %. More details about cost distribution are available in Fig. 4.6. This distribution is
comparable to a typical industrial refrigeration system cost distribution [3]. The
characteristics of the optimal vapor-compression cycle and the geometry of the two heat
exchangers are presented in Tables 4.4 and 4.5. It is interesting to see that only 2 of the 5
GA runs gave the exact same solution. However, variations between the solutions achieved
by the five runs are very small. In fact, the relative difference between the highest cost and
the lowest cost obtained is only 0.04 %. The entire procedure took more than 61 hours in
calculation time and an average number of 822 generations was required for each run of the
GA on a Intel Pentium 4 3.2 Ghz. A total of 616 200 systems have been modeled. This
represents only 1.02xl0~16 % of every possible system design.
IPC 1 * ' " comp
n o c c o m p 61%
Figure 4.6 Optimal cost distribution for test case # 1.
86
The minimal annualized cost obtained for test case #2 is 277 627 $. 35.6 % accounts
for the purchase of the system while 64.4 % accounts for the energy consumption. Here
again, details about cost distribution is available in Fig. 4.7. Again, this distribution is
comparable to a typical industrial refrigeration system cost distribution [3]. The
characteristics of the optimal cycle and the geometry of the two heat exchangers are
presented in Tables 4.4 and 4.6. Unlike test case #1, the genetic algorithm was able to
identify the same optimal solution (i.e. geometry, operating pressures, and mass flow rates)
for the 5 runs of the GA. An average of 1 159 generations, representing 173 850 system
designs, was required by the GA to identify the best system. 70 hours of calculation time
was needed to proceed this example.
IPC comp
4%
uoc comp
53%
uoc pump
4%
Figure 4.7 Optimal cost distribution for test case #2.
Table 4.4 Optimal refrigeration cycle for test case #1 and test case #2.
87
Test case #1 Test case #2
Operating pressures
Evaporator 264 kPa 282 kPa
Condenser 773 kPa 975 kPa
Refrigerant mass flow rate 4.05 kg/s 7.16 kg/s
Condenser water mass flow rate 56.4 kg/s 87 kg/s
System capacity 998.4 kW 1657.7 kW
Work input 158.6 kW 335 kW
COP 6.3 4.9
Costs
Purchase 98 964$ 67 444$
Energy 178 663 $ 85 083 $
Total 277 627 $ 152 527 $
Table 4.5 Optimal refrigeration system characteristics for test case #1.
88
Condenser Evaporator
Length 6.20 m 5.66 m
B 25% 25%
D 550 mm 850 mm
do 15.9 mm 22.2 mm
di 10.3 mm 15.4 mm
P 19 mm 27 mm
Tube configuration 90° 90°
t-ib.center 303 mm 468 mm
Lb,ilLb_0 303 mm 608 mm
A-b 0.2 mm 0.2 mm
A-b 5.5 mm 8.5 mm
D o t l 517 mm 799 mm
Refrigerant side tubes tubes
Tube side Reynolds 6 238 2 352
Shell side Reynolds 28 220 4 780
89
Table 4.6 Optimal refrigeration system characteristics for test case #2.
Condenser Evaporator
Length 6.88 m 5.03 m
B 25% 25%
D 1050 mm 850 mm
do 22.2 mm 15.9 mm
di 15.4 mm 10.3 mm
P 27 mm 21 mm
Tube configuration 90° 90°
L-ibxenler 578 mm 468 mm
L b , j / L b o 751 mm 514 mm
A-b 0.2 mm 0.2 mm
A-b 10.5 mm 8.5 mm
Do l , 988 mm 799 mm
Refrigerant side tubes tubes
Tube side Reynolds 4 427 3 783
Shell side Reynolds 19 032 7 733
For test cases #1 and #2, Reynolds numbers show that flow is turbulent in each heat
exchanger. This confirms the turbulence assumption we made earlier. Coefficients of
performance of the optimal solutions for test cases #1 and #2 are respectively 6.3 and 4.9.
These values are large even if the optimization has not been performed on the criteria of the
best COP. However, these COP values are comparable to those presented in papers from
which test cases data has been taken. COP is 7% higher than COP obtained in [75] for test
case #1 and 14% lower than value obtained in [76] for test case #2.
The results obtained show that the utilization of GA represents an effective solution
to identify the best refrigeration system. The strong advantage of this method is the rapidity
90
and robustness with which the optimal solution is found. In fact, 60 and 70 hours represent
acceptable amount of time for an optimization problem. Considering every possible
solution would be an impracticable approach. It has been shown that the GA did not always
converge on the same solution. However, differences between solutions are relatively
small, hence we can conclude that GA is able to identify nearly optimal solutions but not
always the absolute minimal cost system.
4.8. Conclusions
In this paper, we presented a complete model to evaluate the performance and the cost of a
refrigeration system. 24 design variables were considered regarding geometry of the heat
exchangers, compressor size (operating pressures) and fluid mass flow rates. A model from
a previous article was used for the condenser. A model to evaluate heat transfer coefficients
and pressure drop in the evaporator was built and a complete method to evaluate
performance of the complete refrigeration cycle including the compressor was developed.
Genetic algorithm was used to perform optimization of two different test cases. The results
proved that the GA was able to identify an optimal solution with satisfying repeatability.
Here, optimization of an ideal vapor compression cycle and a constant refrigeration load
were considered. Further research could allow the addition of an HE for regeneration or
consider the utilization of phase change materials to store energy when refrigeration load is
not constant in time. Optimization was performed on the criteria of the minimum total
annualized cost in this paper. The optimization in regard of the COP and a comparison with
the results obtained here would be of interest for future work.
91
Chapitre 5
Discussion et conclusions
Le cœur de ce travail a porté sur l'optimisation de trois types de systèmes thermiques :
condenseur, réseau d'échangeurs de chaleur et systèmes de réfrigération. Dans un premier
temps, il a fallu créer un modèle mathématique permettant de calculer la surface d'échange
et les puissances de pompages requises pour faire fonctionner un condenseur de type tubes
et calandre pour satisfaire un échange thermique entre deux fluides dont un se condensant.
Le modèle a montré sa capacité à déterminer les valeurs recherchées à partir d'une
géométrie d'échangeurs donnée et des conditions d'opération données.
Ce modèle a ensuite été combiné à un algorithme génétique. Le résultat de cette
combinaison nous offre une méthode capable d'identifier avec une excellente probabilité et
dans un délai de temps raisonnable la géométrie minimisant le coût total de l'échangeur.
Effectivement, l'exécution répétée du code a mené à chaque fois au même design optimal.
De plus, les résultats obtenus avec l'AG ont été comparés avec les résultats lorsque tous les
designs ont été calculés. Dans les deux cas et pour deux situations différentes, les résultats
étaient concordants.
Les résultats obtenus ont pu être appliqués dans un deuxième temps pour
l'optimisation de réseaux d'échangeurs de chaleur incluant des condenseurs. Une procédure
basée sur l'analyse de pincement a d'abord été implantée afin d'automatiser la distribution
des fluides dans les échangeurs de manière a respecter un écart minimum de température.
Ensuite, un algorithme génétique permet une fois encore d'optimiser la géométrie de
chaque échangeur. La procédure a été répétée pour différents écarts de température. On est
donc en mesure de déterminer pour quel écart le système obtenu est le moins coûteux. Pour
les deux cas considérés dans cette étude, on obtient un système dont le coût est minimal
pour un écart de température minimal de 2 degrés et 4 degrés Celsius. Il serait intéressant
92
d'explorer la possibilité d'utiliser d'autres types d'échangeur. Ainsi, nous pourrions élargir
le domaine d'application de notre modèle mathématique et de faire un choix d'échangeur
judicieux.
Dans le quatrième chapitre, nous nous sommes concentrés sur les systèmes de
réfrigération. Nous nous sommes basés sur un cycle de compression de vapeur idéalisé
pour réaliser notre modèle. Une fois de plus, nous avons utilisé les algorithmes génétiques
pour optimiser la géométrie des deux échangeurs de chaleur dans le système. Il a fallu
implanter une méthode afin de modéliser le transfert de chaleur pour un réfrigérant
s'évaporant dans les tubes. En plus de la géométrie, le régime d'opération a été optimisé de
manière à obtenir un système offrant un coût minimal. Cette méthode nous a permis de
quantifier la répartition des coûts pour un système optimal. La méthode développée prend
en compte la variation des propriétés en fonction de la pression et de la température des
fluides. Afin de considérer cet aspect, la création de fonctions d'interpolation a été
nécessaire. Nous disposons donc, à la suite de ces travaux, d'un outil d'optimisation
efficace permettant d'optimiser dans un délai de temps réaliste, un système de réfrigération
idéalisé.
Les valeurs de coefficient de performance obtenues à partir de notre modèle sont
relativement élevées. Il est à noter que nous avons considéré un cycle thermodynamique
idéal et que plusieurs irréversibilités ont été négligées. Un système réel aura donc un
coefficient de performance plus faible que celui prédit par notre modèle. De plus, nous
avons négligé toutes pertes thermiques vers l'environnement dans les échangeurs de
chaleurs. Quoiqu'il s'agisse d'une approximation généralement acceptée, il existe des
corrélations pour quantifier ces pertes. Nos travaux futurs nous permettront d'intégrer plus
de phénomènes à notre modèle afin d'obtenir des valeurs de COP s'approchant plus des
valeurs obtenues pour des systèmes réels.
Les résultats obtenus dans ce mémoire sont très concluants et offrent une bonne base
et plusieurs ouvertures pour de futurs travaux. E sera éventuellement intéressant de pousser
nos travaux, particulièrement au niveau du système de réfrigération. Jusqu'à maintenant,
93
nous avons développé un modèle et nous avons performé une optimisation pour un régime
permanent et une demande de refroidissement constante. Les systèmes utilisés dans la
pratique doivent fournir une demande en refroidissement variable et les solutions pratiques
doivent contenir un système de contrôle adéquat pour prévoir ces variations. Les variations
de la demande peuvent également avoir une influence sur la géométrie optimale. D sera
donc intéressant d'intégrer la modélisation d'un système de contrôle au système afin de
performer éventuellement une nouvelle optimisation. Le stockage de chaleur dans des
matériaux à changement de phase serait également un aspect à considérer pour niveler la
consommation énergétique des systèmes, ce qui peut engendrer d'importantes économies.
De plus, certaines des corrélations utilisées nous permettent d'obtenir un modèle valide
seulement pour un seul type de réfrigérant. Il serait intéressant d'élaborer un montage
expérimental afin d'acquérir les mesures requises pour établir une corrélation pour d'autres
types de réfrigérants. Finalement, il sera aussi intéressant d'explorer l'utilisation des
réseaux de neurones. Cette option pourrait nous permettre d'accélérer la démarche et de
réduire les temps de calcul. En somme, toutes ces améliorations permettront de développer
une méthode nous permettant d'obtenir des solutions qui se rapprochent plus d'une solution
pratique optimale et ce dans un délai de temps rentable. Les travaux réalisés dans le cadre
de cette étude offre une excellente base pour l'atteinte de cet objectif.
94
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99
Annexe A
Calcul du coût d'un condenseur
condenseur.m Ce programme permet de calculer les coûts d'achat du matériel et les coûts d'opération pour l'utilisation à des conditions données. La fonction prend en entrée les paramètres suivants : cp_2 cp_3 cp_5 enthalpie_3 enthalpie_5 individu k_3 k_5 k_w kc m_ref mc_max mu_3 mu_5 mu_c mucw muw_3 muw_5 Pr_3 Pr_5 Pre Rc R_ref rho_3 rho_5 rhoc T_2 T_5 Tci Tsat ref cn
Heat capacity of refrigerant at point 2 (J/kgK) Heat capacity of refrigerant at point 3 (J/kgK) Heat capacity of refrigerant at point 5 (J/kgK) Specific enthalpy of refrigerant at point 3 (J/kg) Specific enthalpy of refrigerant at point 3 (J/kg) Vector containing design variables Thermal conductivity of refrigerant at point 3 (W/mK) Thermal conductivity of refrigerant at point 5 (W/mK) Thermal conductivity of tubes material (W/mK) Thermal conductivity of cold fluid (W/mK) Mass flow rate of refrigerant (kg/s) Maximum mass flow rate of cold fluid (kg/s) Dynamic viscosity of refrigerant at point 3 (Pa*s) Dynamic viscosity of refrigerant at point 5 (Pa*s) Dynamic viscosity of cold fluid (Pa*s) Wall dynamic viscosity of cold fluid (Pa*s) Wall dynamic viscosity of refrigerant at point 3 (Pa*s) Wall dynamic viscosity of refrigerant at point 5 (Pa*s) Prandtl number of réfrigérant au point 3 Prandtl number of réfrigérant au point 5 Prandtl number of cold fluid Cold fluid fouling resistance (m/v2*K/W) Refrigerant fouling resistance (mA2*K/W) Density of refrigerant at point 3 (J/kgK) Density of refrigerant at point 5 (J/kgK) Cold fluid density (J/kgK) Temperature of refrigerant at point 2 (K) Temperature of refrigerant at point 2 (K) Temperature of cold fluid at exchanger inlet (K) Saturation temperature of refrigerant (K)
Les arguments de sorties sont les suivants : ATC_cn L Be Ds do di
Total condenser annualized cost ($/an) Tubes length (m) Baffle cut (%) Shell diameter (m) Outiside tubes diameter (m) Inside tubes diameter (m)
100 pt bundleangle Lbc deltatb deltasb Dbaffle Dotl Lbi Lbo Cond_side mc Tco APE_cn OC_cn
Par: Benoît Allen
Tube pitch (m) Bundle angle (degrés) Baffle spacing at center (m) Tube to baffle spacing (m) Shell to baffle spacing (m) Baffle diameter (m) Tube bundle diameter (m) Inlet baffle spacing (m) Outlet baffle spacing (m) Condensing fluide flowing side (tubes or shell) Cold fluid mass flow rate (kg/s) Cold fluid outlet temperature (K) Annualized purchase cost ($/an) Annualized operating cost ($/an)
Date: 10 décembre 2009
Université Laval, Québec, Canada function [ATC_cn,L,Be,Ds,do,di,pt,bundleangle,Lbc,deltatb, deltasb, ...
Dbaffle,Dotl,Lbi,Lbo,Cond_side, mc, Tco,APE_cn,OC_cn]... = condenseur(cp_2,cp_3,cp_5,epe,enthalpie_3,enthalpie_5, individu, k_3,k_5,k_w,kc,m_ref,mc_max,mu_3,mu_5,muc,mucw,muw_3,muw_5, Pr_3, . Pr_5,Pre, Re, R_ref,rho_3,rho_5,rhoc,T_2,T_5,Tci,Tsat_ref_cn)
Economic data n = 20; H = 5000; fe = 0.10; intérêt = 0.05; eta = 0.85; factorm = 2.9; factorp - 1.9;
% Lifetime (year) % Annual operating period (hour) % Energy cost ($/Kwh) % Annual interst rate (%) % Pump efficiency % Material capital cost factor % Pressure capital cost factor
% Temperature capital cost factor if T_2 < 373
factort = 1; elseif T_2 > 373 && T_2 < 773
factort = 1.6; elseif T_2 > 773
factort = 2.1; end Heat transfer rates in heat exchanger sub-sections Qg = m_ref*((cp_5+cp_2)/2)*(T_2-Tsat_ref_cn);% Vapor sub-sec t ion (W) Qc = m_re f* (en tha lp ie_5-en tha lp ie_3) ; Q_cn = Qg+Qc;
Minimum cold fluid mass flow rate mc_min_l = Q_cn/((T_2-1)-Tci)/epe; mc_min_2 = Qc/cpc/((T_5-l)-Tci); mc_min = max(mc_min_l,mc_min_2);
% Condensing sub-section(W) % Total (W)
101
Decoding design variables [Be, D s , d o , d i , p t , bundleangle ,Xt , X I , C L , L b c , d e l t a t b , d e l t a s b , D b a f f l e , . . .
Dotl , Lbi, Lbo,Cond_side,mc] = design_condenseur(individu,mc_max,mc_min);
Cold fluid temperatures Tco ■ Tci+Q_cn/mc/cpc; % Outlet (K) Tc2 = Tci+Qc/(mc*cpc); % Between vapor and condensing sub-sections (K) Logarithmic mean temperature differences % Vapor sub-secion (K) dtlmg = ((T_2-Tco)-(Tsat_ref_cn-Tc2))/log((T_2-Tco)/(Tsat_ref_cn-Tc2));
% Condensing sub-section (K) dtlmc = ((Tsat_ref_cn-Tci)-(Tsat_ref_cn-Tc2))/log((Tsat_ref_cn-Tci)/...
(Tsat_ref_cn-Tc2)); Geometric parameters
Given parameters wp = 0.05*Ds; % Width of the bypass lane (m) Nss = 2 ; % Number of sealing strip pairs CTP =0.93; % Tube layout factor Np = 0 ; % Number of pass divider lanes parallel to the crossflow s = 1; % Number of tube pass Calculated geometric parameters [Aocr,Aot,Aow,F,Fc,Nrcc,Nrcw,Nssplus,Nt,rb,rim,rs] = géométrie(Be,...
bundleangle,CL,CTP,deltasb,deltatb,di,do,Dotl,Ds,Lbc,Np,Nss,pt,s,Xl,... Xt,wp);
Required heat transfer surface areas for SHELL-side condensation i f Cond_side == 1
In tube heat transfer coefficient % Flow velocity (m/s) Vt = mc/(Aot*rhoc);
% Reynolds number Ret = mc*di/(Aot*muc);
% Heat transfer coefficient (W/mA2K)
ht = 0.024/di*kc*PrcA0.3*Ret~0.8*(muc/mucw);
% Friction factor cfrict = (0.790*log(Ret)-1.64)A-2;
Shell side ideal heat transfer coefficient (vapor sub-section)
102 % Fluid mass velocity based on the minimum free area (kg/mA2s) Gs = m_ref/Aocr;
% Reynolds number Resg = m_ref*do/(mu_5*Aocr);
% Colburn factor coefficients [big,b2g,-,~,bg,alg,a2g, ~,~, ag] = coeffab(bundleangle,Resg);
% Colburn factor jg = alg*(1.33/(pt/do) )Aag*(Resg) Aa2g;
% Ideal heat transfer coefficient (W/mA2K)
hidg = (jg*m_ref*cp_5*Pr_5A-(2/3) )/Aocr;
% Ideal friction factor fidg = blg*(1.33/(pt/do))Abg*(Resg)Ab2g;
% Ideal pressure drop in crossflow section between two baffles (Pa) dpwidg = (2+0.6*Nrcw)*m_refA2/(2*rho_5*Aocr*Aow);
% Ideal pressure drop associated with ideal one-window section (Pa) dpbidg = 4*fidg*Gs
A2*Nrcc/(2*rho_5)*(muw_5/mu_5)
A0.25;
Shell side heat transfer coefficient (condensing sub-section) % Liquid Reynolds number Resl = m_ref*do/(mu_3*Aocr);
% Liquid Colburn factor coefficients [bll,b21,-,~,bl, -, -, -, -, ~] = coeffab(bundleangle,Resl);
% Liquid ideal friction factor fidl = bll* (1.33/(pt/do))Abl*(Resl)Ab21;
% Pressure drop in crossflow section between two baffles (Pa) dpwidl = (2+0.6*Nrcw)*m_refA2/(2*rho_3*Aocr*Aow);
% Pressure drop associated with an ideal one-window section (Pa) dpbidl = 4*fidl*Gs
A2*Nrcc/(2*rho_3)*(muw_3/mu_3)
A0.25;
% Condensing heat transfer coefficient (W/mA2K)
hsc = 1.35*k_3*(rho_3A2*do*9.807*Nt/mu_3/m_ref)A(1/3);
% Correction factors for heat transfer coefficient [Jcg, -, Jig, -, Jbg, ~,Jrg,~,ksibg,~,ksilg,~,ksisg,~,ksibcg,ksilcg,...
ksiscg,ksibcl,ksilcl,ksiscl] ■ corrections_condenseur(Fc,Lbc,.. Lbi,Lbo,Nssplus,Resg,Resl,rb,rim,rs);
Iterative loop on heat transfer surface areas flag = 1 Ags = 20 Acs = 20
% While loop interruptor % vapor sub-section transfer area first guess (mA2) % condensing sub-section transfer area first guess (mA2)
while flag > 0
103 Lg = Ags/(pi*do*Nt); % Vapor sub-section length (m) Lc = Acs/(pi*do*Nt); % Condensing sub-section length (m) L = Lg+Lc; % Total tubes length (m)
Nb = (L-Lbi-Lbo)/Lbc+1; % Total number of baffles Nbg = (Lg-Lbi)/Lbc; % Baffles in vapor sub-section Nbc = Nb-Nbg; % Baffles in condensing sub-section
% Correction factors on heat transfer coeff (number of baffles) if Resg > 100
Jsg = (Nbg-l+(Lbi/Lbc)A(1-0.6)+(Lbo/Lbc)A(l-0.6))/... (Nbg-l+(Lbi/Lbc)+(Lbo/Lbc));
else Jsg m (Nbg-l+(Lbi/Lbc)A(1-0.333)+(Lbo/Lbc)A(l-0.333))/...
(Nbg-1+(Lbi/Lbc)+(Lbo/Lbc)); end
% Vapor sub-section effective heat transfer coefficient (W/mA2K)
if Nbg >= 1 hsg = hidg*Jcg*Jlg*Jrg*Jbg*Jsg;
else hsg = hidg*Jcg*Jlg*Jrg*Jbg;
end
% Tube-side pressure drop (Pa) dpt = s*(4*cfrict*L/di+1.5)*rhoc*VtA2/2;
% Shell-side pressure drop for vapor sub-section (Pa) dpsg = ((Nbg-1)*dpbidg*ksibg+Nbg*dpwidg)*ksilg+2*dpbidg*...
(1+Nrcw/Nrcc)*ksibg*ksisg;
% Shell-side condensing sub-section gas phase pressure drop (Pa) dpscg = ((Nbc-1)*dpbidg*ksibcg+Nbc*dpwidg)*ksilcg+2*dpbidg*...
(1+Nrcw/Nrcc)*ksibcg*ksiscg;
% By unit tube length (Pa/m) dpfdzGO ■ dpscg/Lc;
% Shell-side condensing sub-section liq. phase pressure drop (Pa) dpscl = ((Nbc-1)*dpbidl*ksibcl+Nbc*dpwidl)*ksilcl+2*dpbidl*...
(1+Nrcw/Nrcc)*ksibcl*ksiscl;
% By unit tube length (Pa/m) dpfdzLO = dpscl/Lc;
% Chisholm parameter Y2 = dpfdzGO/dpfdzLO;
% Shell-side condensing sub-section cross-flow pressure drop (Pa) dpbc = dpfdzLO*(l.1527246+0.2275*Y2)*Lc;
% Shell-side condensing sub-section window pressure drop (Pa) dpwc = dpfdzLO*(0.625+0.375*Y2)*Lc;
104 % Shell-side condensing sub-section total pressure drop (Pa) dpsc = dpbc+dpwc;
Global heat transfer coefficients % Vapor sub-cesction (W/mA2K) Ufg = l/(l/hsg+R_ref+(do*log(do/di))/2/k_w+Rc*do/di+do/ht/di);
% Condensing sub-cesction (W/mA2K) Ufc = l/(l/hsc+R_ref+(do*log(do/di))/2/k_w+Rc*do/di+do/ht/di);
Required heat transfer areas Afg = Qg/(Ufg*dtlmg*F); % Vapor sub-cesction (mA2) Afc = Qc/(Ufc*dtlmc*F); % Condensing sub-cesction (mA2) Aft = Afg+Afc; % Total (mA2)
Pumping power equirements % Tube side (W) Et = dpt*mc/rhoc/eta;
% Shell side (W) Es = dpsg*m_ref/rho_5/eta+dpsc*m_ref*(rho_5+rho_3)/...
(2*eta*rho_5*rho_3); Convergence
if max(abs(Ags-Afg),abs(Acs-Afc)) < 0.01 flag - 0;
else flag = 1; Ags = Afg; % Heat transfer area new guess (mA2) Acs = Afc; % Heat transfer area new guess (mA2)
end end
end Required heat transfer surface areas for TUBE-side condensation i f Cond_side == 2
Shell-side heat transfer coefficient % Fluid mass velocity based on the minimum free area (kg/mA2s) Gs = mc/Aocr;
% Reynolds number Res = mc*do/(muc*Aocr);
% Colburn factor coefficients [bl,b2,-,~,b,al,a2,-,-, a] = coeffab(bundleangle,Res);
% Colburn factor j = al*(1.33/(pt/do))Aa*(Res)Aa2;
% Ideal heat transfer coefficicient (W/mA2K) hid = (j*mc*cpc*PrcA-(2/3))/Aocr;
105 % Ideal fricition factor fid = bl*(1-33/(pt/do))Ab*(Res)Ab2;
% Pressure drop in crossflow section between two baffles (Pa) dpwid = (2+0.6*Nrcw)*mcA2/(2*rhoc*Aocr*Aow);
% Pressure drop associated with an ideal one-window section (Pa) dpbid = 4*fid*GsA2*Nrcc/(2*rhoc)*(mucw/muc)A0.25;
% Correction factors for heat transfer coefficient [Jc,Jl,Jr,Jb,ksib,ksil] = corrections(Fc,Nssplus,Res,rb,rim,rs);
Tube-side vapor sub-section heat transfer coefficient % Flow velocity (m/s) Vtg = m_ref/(Aot*rho_5);
% Reynolds number Retg = m_ref*di/(Aot*mu_5);
% Heat transfer coefficient (W/mA2K) htg = 0.024/di*k_5*Pr_5A0.3*RetgA0.8*(mu_5/muw_5);
% Friction factor cfrictg = (0.790*log(Retg)-1.64)A-2;
% Condensing sub-section average massic volume (mA3/kg) amv = l/rho_5/rho_3*(-rho_5+rho_3)/2+l/rho_3;
Iterative loop on heat transfer surface areas Ags = 20 Acs = 20 flag = 1
% vapor sub-section transfer area first guess (mA2) % condensing sub-section transfer area first guess (mA2) % While loop interruptor
while flag > 0 Lg = Ags/(pi*do*Nt); % Vapor sub-section length (m) Lc = Acs/(pi*do*Nt); % Condensing sub-section length (m) L = Lg+Lc; % Total sub-section length
Nb = (L-Lbi-Lbo)/Lbc+l; % Total number of baffles
% Correction factors on heat transfer coeff (number of baffles) if Res > 100
Js = (Nb-l+(Lbi/Lbc)A(l-0.6)+(Lbo/Lbc)A(l-0.6))/... (Nb-1+(Lbi/Lbc)+(Lbo/Lbc));
else Js = (Nb-1+(Lbi/Lbc)A(1-0.333) + (Lbo/Lbc)A(l-0.333) ) / . . .
(Nb-1+(Lbi/Lbc)+(Lbo/Lbc)); end
% Shell side effective heat transfer coefficient (W/mA2K) if Nb >= 1
hs = hid*Jc*Jl*Jb*Js*Jr; else
hs = hid*Jc*Jl*Jb*Jr;
106
end
Iterative loop on condensing sub-section heat transfer coeff Twls = (Tci+T_5)/2; % Wall temperature first guess Tw2s = (Tc2+T_5)/2; % Wall temperature first guess flagl = 1; % While loop interruptor
while flagl == 1 % Log mean temperature difference (K) DeltaTsatw = ((T_5-Twls)-(T_5-Tw2s))/log((T_5-Twls)...
/(T_5Tw2s)) ;
% Moadified latent heat (kJ/kg) iifg_ref_cn = (enthalpie_5-enthalpie_3)+...
(3/8)*cp_3*DeltaTsatw;
% Condensing sub-section heat transfer coefficient (W/mA2K) htc = 0.555*((9.8065*rho_3*(rho_3-rho_5)*(k_3A3)*...
iifg_ref_cn)/(mu_3*DeltaTsatw*di))A0.25;
% Total thermal resistance (mA2K/W) Rtot = 1/hs+Rc+do*(log(do/di))/(2*k_w)+R_ref*do/di+...
(l/htc)/(do/di);
% Calculated wall temperatures (K) Tw2 = -(1/htc/Rtot)*(T_5-Tc2)+T_5; Twl = -(1/htc/Rtot)*(T_5-Tci)+T_5;
% Converge (While loop stop verification) if abs((Twls-Twl)) < 0.01
flagl = 0; end
Twls = Twl; % Wall temperature new guess (K) Tw2s = Tw2; % Wall temperature new guess (K)
end
% Correction factor on cross flow pressure drop (baffle spacing) if Res <= 100
ksis = (Lbc/Lbo)A(2-1)+(Lbc/Lbi)A(2-l); else
ksis = (Lbc/Lbo)A(2-0.2)+(Lbc/Lbi)A(2-0.2); end
% Shell side pressure drop (Pa) dps = ((Nb-l)*dpbid*ksib+Nb*dpwid)*ksil+2*dpbid*...
(1+Nrcw/Nrcc)*ksib*ksis;
% Tube side vapor sub-section pressure drop (Pa) dptg = s*(4*cfrictg*Lg/di+0.5)*rho_5*(VtgA2)/2;
% Tube side condensing sub-section pressure drop (Pa) dptc = ((0.046*(32*m_refA(9/5)*(mu_5*mu_3)A(1/5)*Lc)/(4A0.2*...
piA(9/5)*diA(24/5)*rho_5*rho_3)))*(-0.1388888889*(-4* . ..
107
m u _ 5 A ( 9 / 5 ) * r h o _ 5 - 5 * m u _ 5 A ( 9 / 5 ) * r h o _ 3 + 9 * m u _ 5 A ( 4 / 5 ) * . . . r ho_5*mu_3+9*mu_3 A (4 /5 )* rho_3*mu_5-5*mu_3 A (9 /5 )* rho_5-4* . . . mu_3 A (9 /5)*rho_3) / (mu_5 A 2-2*mu_5*mu_3+mu_3 A 2)) ;
Global heat transfer coefficients % Vapor sub-section (W/mA2K) Ufg = l/(l/hs+Rc+(do*log(do/di))/2/k_w+R_ref*do/di+do/htg/di) ;
% Condensing sub-section (W/mA2K) Ufc = l/(l/hs+Rc+(do*log(do/di))/2/k_w+R_ref*do/di+do/htc/di) ;
Required heat transfer areas Afg - Qg/(Ufg*dtlmg*F); % Vapor sub-section (mA2) Afc = Qc/(Ufc*dtlmc*F); % Condensing sub-section (mA2) Aft = Afg+Afc; % Total (mA2)
Pumping power equirements (W) Et = ( d p t g * m _ r e f / r h o _ 5 / e t a ) + ( d p t c * m _ r e f * a m v / e t a ) ; % Tube s i d e
(W) Es = d p s * m c / r h o c / e t a ; % S h e l l s i d e (W)
Convergence if max(abs(Ags-Afg),abs(Acs-Afc)) < 0.01
flag = 0; else
flag = 1; Ags = Afg; % Heat transfer area new guess (mA2) Acs = Afc; % Heat transfer area new guess (mA2)
end end
end Costs computation
Purchase cost ($) PE_foul_cn m 3 . 2 8 * 1 0 A 4 * ( A f t / 8 0 ) A 0 . 6 8 * f a c t o r m * f a c t o r p * f a c t o r t ;
Annualized purchase cost ($/year) APE_cn = (PE_foul_cn)*interet*(1+interet)An/((1+interet)An-1); Annual operating costs ($/year) OC_cn = (E t+Es )*H*fe /1000 ;
Total annual cost ($/year) ATC_cn = APE_cn+OC_cn ;
Published with MATLAB®
108
Annexe B
Calcul du coût d'un évaporateur
Evaporateur.m
Ce programme permet de calculer les coûts d'achat du matériel et les coûts d'opération pour l'utilisation à des conditions données. La fonction prend en entrée les paramètres suivants : cph enthalpie_l enthalpie_4 enthalpie_6 individu k_l k_6 k_w m_ref mh mu_l mu_6 mu h Pr_l Pr_6 Prh R_ref Rh rho_l rho_6 rhoh T_4 tension_6 Thi Tho
Heat capacity of refrigerated (hot) fluid (J/kgK) Specific enthalpy of refrigerant at point 1 (J/kg) Specific enthalpy of refrigerant at point 4 (J/kg) Specific enthalpy of refrigerant at point 6 (J/kg) Vector containing design variables Thermal conductivity of refrigerant at point 1 (W/mK) Thermal conductivity of refrigerant at point 6 (W/mK) Thermal conductivity of tubes material (W/mK) Mass flow rate of refrigerant (kg/s) Mass flow rate of refrigerated (hot) fluid (kg/s) Dynamic viscosity of refrigerant at point 1 (Pa*s) Dynamic viscosity of refrigerant at point 6 (Pa*s) Dynamic viscosity of refrigerated (hot) fluid (Pa*s) Prandtl number of réfrigérant au point 1 Prandtl number of réfrigérant au point 6 Prandtl number of refrigerated (hot) fluid Refrigerant fouling resistance (mA2*K/W) Refrigerated (hot) fluid fouling resistance (mA2*K/W) Density of refrigerant at point 1 (J/kgK) Density of refrigerant at point 6 (J/kgK) Refrigerated (hot) fluid density (J/kgK) Temperature of refrigerant at point 4 (K) Refrigerant surface tension at point 6 (N/m) Refrigerated (hot) fluid inlet temperature (K) Refrigerated (hot) fluid outlet temperature (K)
Les arguments de sorties sont les suivants ATC_ev L Be Ds do di Pt bundleangle Lbc deltatb deltasb Dbaffle Dotl
Total evaporator annualized cost ($/an) Tubes length (m) Baffle cut (%) Shell diameter (m) Outiside tubes diameter (m) Inside tubes diameter (m) Tube pitch (m) Bundle angle (degrés) Baffle spacing at center (m) Tube to baffle spacing (m) Shell to baffle spacing (m) Baffle diameter (m) Tube bundle diameter (m)
109
Lbi Lbo APE_ev OC_ev
Par: Benoît Allen
Inlet baffle spacing (m) Outlet baffle spacing (m) Annualized purchase cost ($/an) Annualized operating cost ($/an)
Date: 10 décembre 2009
Université Laval, Québec, Canada function [ATC_ev,L,Bc,Ds,do,di,pt,bundleangle, Lbc, deltatb, deltasb, . . .
Dbaffle, Dotl,Lbi,Lbo,APE_ev,OC_ev] =... évaporateur(cph,enthalpie_l,enthalpie_4,enthalpie_6,individu,k_l, . k_6,k_w,m_ref,mh,mu_l,mu_6,muh,Pr_l,Pr_6,Prh,R_ref,Rh,rho_l,rho_6, rhoh,T_4,tension_6,Thi,Tho)
Economic data n = 20; H = 5000; fe = 0.10; intérêt = 0.05; eta = 0.85; factorm = 2.9; factorp = 1.9;
% Lifetime (year) % Annual operating period (hour) % Energy cost ($/Kwh) % Annual interst rate (%) % Pump efficiency % Material capital cost factor % Pressure capital cost factor
% Temperature capital cost factor if Thi < 373
factort=l; elseif Thi > 373 && Thi < 773
factort=l.6; elseif Thi > 773
factort=2.1; end Refrigerant quality x_inlet =(enthalpie_4-enthalpie_6)/(enthalpie_l-enthalpie_6); x = linspace(x_inlet,0.95,100); % Sub-section Pre-allocating vectors (to speed up calculations) Thx = NaN((length(x)-l),1); dtlm = NaN(length(x),1); Q_ev = NaN(length(x),1); h_TP = NaN(length(x),1); Co = NaN(length(x),1); Uf = NaN(length(x),1); Ac = NaN(length(x),1); rho_TP = NaN(length(x),l); EE = NaN(length(x),1); FF = NaN(length(x),1) ; HH = NaN(length(x),1); Fr_TP = NaN(length(x),1) We_TP = NaN(length(x),l) dp_TP = NaN(length(x),1)
Inlet
110
E_TP = N a N ( l e n g t h ( x ) , 1 ) ;
Total heat transfer rate (W) Q = m_re f* ( en tha lp i e_ l - en tha lp i e_4 ) ;
Refrigerated fluid temperatures at sub-sections boundaries Thx(l) = Tho + Q/(mh*cph)/length(x); for section = 2 : (length(x)-1)
Thx(section) = Thx(section-1) + Q/(mh*cph)/length(x); end Logarithmic mean temperature differences for each sub-section dtlm(l) = ((Thx(l) - T_4) - (Thi - T_4))/log((Thx(1) - T_4)/(Thi - T_4)); for section = 2 :(length(x)-1)
dtlm(section) = ((Thx(section) - T_4) - (Thx(section-1) -... T_4))/log((Thx(section) - T_4)/(Thx(section-1) - T_4));
end dtlmdength(x) ) = ((Tho - T_4) - (Thx (length (x)-1 ) - T_4) ) /log ( (Tho -T_4)/(Thx(length(x)-l) - T_4)); Decoding design variables [Be,Ds,do,di,pt,bundleangle,Xt,XI,CL,Lbc,deltatb,deltasb,Dbaffle,Dotl,...
Lbi,Lbo] = design_evaporateur(individu); Geometric and physic parameters
Given parameters wp = 0.05*Ds; % Width of the bypass lane (m) Nss = 2 ; % Number of sealing strip pairs CTP =0.93; % Tube layout factor Np = 0 ; % Number of pass divider lanes parallel to the crossflow s = 1 ; % Number of tube pass f = 1.10; % Kandlikar parameter depending on refrigerant g = 9.81; % Gravitationnal acceleration (m/sA2) Calculated geometric parameters [Aocr,Aot,Aow,-,Fc,Nrcc,Nrcw,Nssplus,Nt,rb,rlm,rs] = géométrie(Be,...
bundleangle,CL,CTP,deltasb,deltatb,di,do,Dotl,Ds,Lbc,Np,Nss,pt,s,Xl,... Xt,wp);
Required heat transfer area calculation G_ref = m_ref/Aot; % Refrigerant mass velocity (kg/mA2s) G_h = mh/Aocr; % Hot fluid mass velocity (kg/mA2s) V_ref_L = m_ref/(Aot*rho_6); % In-tube velocity (m/s) Re_ref_L = (m_ref*di)/(Aot*mu_6); % Refrigerant liquid Reynolds number Re_ref_V = (m_ref*di)/(Aot*mu_l); % Refrigerant vapor Reynolds number Re_h = (mh*do)/(muh*Aocr); % Refrigerated fluid Reynolds number Fr_ref_L = (G_refA2)/(rho_6*g*di); % Refrigerant Froude number
% Colburn factor coefficients [bl,b2,-,-,b,al,a2,-,-, a] = coeffab(bundleangle,Re_h);
I l l
% Colburn factor j = al*(1.33/(pt/do))Aa*(Re_h)Aa2;
%facteurs de correction J [Jc,Jl,Jr,Jb,ksib,ksil] = corrections(Fc,Nssplus,Re_h,rb,rim,rs); Iterative loop on heat transfer surface areas flag = 1; % While loop interruptor As = 0.05*ones(length(x),1); % Heat transfer areas first guess (mA2)
while flag == 1 L = sum(As)/(pi*do*Nt); % Tubes length (m) Nb = (L-Lbi-Lbo)/Lbc+1; % Number of baffles
Shell side heat transfer coefficient % Ideal heat transfer coefficient (W/mA2K) hid = (j*mh*cph*PrhA-(2/3))/Aocr;
% Correction factors (for number of baffles) if Re_h > 100
Js = (Nb-1+(Lbi/Lbc)A(l-0.6)+(Lbo/Lbc)A(l-0.6))/(Nb-1+(Lbi/Lbc)+...
(Lbo/Lbc)); else
Js = (Nb-1+(Lbi/Lbc)A(1-0.333)+(Lbo/Lbc)A(l-0.333))/(Nb-l+... (Lbi/Lbc)+(Lbo/Lbc));
end
% Effective heat transfer coefficient (W/mA2K) if Nb >= 1
h_s = hid*Jc*Jl*Jb*Js*Jr; else
h_s = hid*Jc*Jl*Jb*Jr; end
Loop for tube side sub-sections heat transfer coefficient for i=l: length(x)
% Heat flux on each tube sub-section (W/mA2) Q_ev(i) = (Q)/(length(x)*As(i));
% Convection number on each tube sub-section Co(i) - ((l-x(i))/x(i))A0.8*(rho_l/rho_6)A0.5;
if Fr_ref_L <=0.0 4 if Co(i) > 0.65
% Nucleate boiling (N) heat transfer coeff (W/mA2K) h_TP(i) = 0.023*(Re_ref_LA0.8)*(Pr_6A0.4)*(k_6/di)*...
(0.6683*(((l-x(i))/x(i))A0.8*(rho_l/rho_6)A0.5)A... (-0.2)*(25*Fr_ref_L)A0.3+1058*(Q_ev(i)*pi*diA2/4/... m_ref/(enthalpie_l-enthalpie_6))A0.7*f);
else % Convective boiling (C) heat transfer coeff (W/mA2K)
112 h_TP(i) = 0.023*(Re_ref_LA0.8)*(Pr_6A0.4)*(k_6/di)*...
(1.136M((l-x(i))/x(i))A0.8*(rho_l/rho_6)A0.5)A... (-0.9)*(25*Fr_ref_L)A0.3+667.2*(Q_ev(i)*pi*diA2/4/... m_ref/(enthalpie_l-enthalpie_6))A0.7*f);
end else
if Co(i) > 0.65 % Nucleate boiling (N) heat transfer coefficient (W/mA2K) h_TP(i) = 0.023*(Re_ref_LA0.8)*(Pr_6A0.4)*(k_6/di)*...
(0.6683*(((l-x(i))/x(i))A0.8*(rho_l/rho_6)A0.5)A... (-0.2)+1058*(Q_ev(i)*pi*diA2/4/m_ref/(enthalpie_l-... enthalpie_6))A0.7*f);
else % Convective boiling (C) heat transfer coeff (W/mA2K) h_TP(i) = 0.023*(Re_ref_LA0.8)*(Pr_6A0.4)*(k_6/di)*...
(1.136*(((l-x(i))/x(i))A0.8*(rho_l/rho_6)A0.5)A... (-0.9)+667.2*(Q_ev(i)*pi*diA2/4/m_ref/(enthalpie_l...
-enthalpie_6))A0.7*f); end
end end
Global heat transfer coefficient for each sub-section (W/mA2K) Uf = l . / ( l /h_s+Rh+do*log(do/d i ) /2 /k_w+R_ref*do/d i+do. /d i . /h_TP) ;
Calculated heat transfer area for each sub section (mA2) Ac = (Q_ev.*As)./dtlm./Uf;
Convergence i f (max(abs(Ac-As)./As)) > 0.01
f l ag = 1; As = As+0.1*(Ac-As); % Heat transfer area new guess (mA2)
else flag = 0;
end end Required final heat transfer area (mA2) Atot = sum(Ac);
Shell side pressure drop % Idea l f r i c t i o n c o e f f i c i e n t f id = b l* (1 .33 / (p t /do) ) A b*(Re_h) A b2 ;
% Ideal pressure drop in crossflow section between two baffles (Pa) dpwid = (2+0.6*Nrcw)*mhA2/(2*rhoh*Aocr*Aow);
% Ideal pressure drop associated with an ideal one-window section (Pa) dpbid = 4*fid*G_hA2*Nrcc/(2*rhoh);
% Correction factor on cross flow pressure drop for baffle spacing if Re_h > 100
ksis = (Lbc/Lbo)A(2-1)+(Lbc/Lbi)A(2-1);
113 else
ksis = (Lbc/Lbo)A(2-0.2)+(Lbc/Lbi)A(2-0.2); end
% Shell side pressure drop (Pa) dps = ((Nb-1)*dpbid*ksib+Nb*dpwid)*ksil+2*dpbid*(1+Nrcw/Nrcc)*ksib*ksis; Tube side pressure drop % Vapor friction coefficients cf_ref_V = (0.790*log(Re_ref_V)-1.64)A(-2);
% Liquid friction coefficients cf_ref_L = (0.790*log(Re_ref_L)-1.64)A(-2);
% Tube side pressure drop with liquid phase properties (Pa/m) deltap_L0 = s*(4*cf_ref_L/di)*rho_6*(V_ref_LA2)/2;
for k = 1:length(x) % Sub section two-phase density (kg/mA3) rho_TP(k) = (x(k)/rho_l+(l-x(k))/rho_6)A-l;
% Friedel correleation parameters EE(k) = (l-x(k))A2+x(k)A2*(rho_6*cf_ref_V)/(rho_l*cf_ref_L); FF(k) = x(k)A0.78*(l-x(k))A0.24; HH(k) = (rho_6/rho_l)A0.91*(mu_l/mu_6)A0.19*(l-mu_l/mu_6)A0.7;
% Sub section two-phase Froude number Fr_TP(k) = G_refA2/(g*di*rho_TP(k)A2);
% Sub section two-phase Weber number We_TP(k) = G_refA2*di/(rho_TP(k)*tension_6);
% Sub-section two-phase pressure drop (Pa) dp_TP(k) =
(EE(k)+3.24*FF(k)*HH(k)/(Fr_TP(k)A0.045*We_TP(k)A0.035))*deltap_LO*Ac(k)/ (pi*do*Nt);
% Sub-section pumping power requirements (W) E_TP(k) = dp_TP(k)*m_ref/rho_TP(k)/eta;
end Total pumping power equirements E_t = sum(E_TP); % Tube side (W) E_s - dps*mh/rhoh/eta; % Shell side (W) Costs computation
Purchase cost ($) PE_foul_ev = 3.28*10A4*(Atot/80)A0.68*factorm*factorp*factort; Annualized purchase cost ($/year) APE_ev = (PE_foul_ev)*interet*(1+interet)An/((1+interet)An-1);
Annual operating costs ($/year) OC_ev « (E_ t+E_s )*H*fe /1000 ;
Total annual cost ($/year) ATC_ev = APE_ev+OC_ev;
114
Published with MATLAB®
115
Annexe C
Calcul du coût d'un compresseur
Comp_iso_s.m
Ce programme permet de calculer les coûts d'achat et les coûts d'opération pour l'utilisation d'un compresseur à des conditions données. La fonction prend en entrée les paramètres suivants : m_ref : Mass flow rate of refrigerant (kg/s) enthalpie_l : Specific enthalpy of refrigerant at point 1 (J/kg) enthalpie_2 : Specific enthalpy of refrigerant at point 2 (J/kg) eff_comp : Efficacité du compresseur
Les arguments de sorties sont les suivants : TC_comp_A : Total annual cost ($/an) CE_comp_A : Annual purchase cost ($/an) OC_comp_A : Annual operating costs ($/an)
Par: Benoît Allen
Date: 11 décembre 2009
Université Laval, Québec, Canada function [TC_comp_A,CE_comp_A, OC_comp_A] = comp_iso_s(m_ref,enthalpie_l, . . .
enthalpie_2,eff_comp) Economic data n = 20; % Lifetime (year) H = 5000; % Annual operating period (hour) fe = 0.10; % Energy cost ($/Kwh) intérêt = 0.05; % Annual interst rate (%) Required work input (W) W_comp = m_ref*(enthalpie_2 - enthalpie_l)/eff_comp; Costs computation
Purchase cost ($) CE_comp = 98400*(W_comp/250000)A0.46; Annualized purchase cost ($/year) CE_comp_A = (CE_comp)*interet*(1+interet)An/((1+interet)An-1); Annual operating costs ($/year)
116
OC_comp_A = ( W _ c o m p ) * H * f e / 1 0 0 0 ;
Total annual cost ($/year) TC_comp_A = CE_comp_A +OC_comp_A;
Published with MATLAB®
117
Annexe D
Optimisation d'un système de réfrigération
D.l Script d'optimisation
REF.m Ce script permet d'optimser un système de réfrigération pour une demande en réfrigération donnée.
Par: Benoît Allen
Date: 30 septembre 2009
Université Laval, Québec, Canada
Initialisation c l e a r c l c t i c ;
Data for interpolation of specific properties of refrigerant at point 2 load data; Pressions = data(:,l); entropies = data(:,2); Temperatures = data(:,3); Capacités = data(:,4); Enthal = data(:,5); PPPP = linspace(min(Pressions),max(Pressions),100); ssss = linspace(min(entropies),max(entropies), 100); [PPPP,ssss]=meshgrid(PPPP,ssss) ; TTTT = griddata(Pressions,entropies,Temperatures, PPPP, ssss); CCCC = griddata(Pressions,entropies,Capacités, PPPP, ssss) ; HHHH = griddata(Pressions,entropies,Enthal,PPPP,ssss); Fluid properties
Cold fluid in the condenser (HEAT SINK) mc_max = 25; % Maximum mass flow rate (kg/s) Re = 0.000275; % Fouling resistance (mA2K/W) Tci = 297; % Inlet temperature (K) rhoc = 982.3; % Density (kg/mA3) epe = 4186; % Heat capacity (J/kgK)
118 mue = 0.000453; mucw = 0.000453; kc = 0.656; Pre = cpc*muc/kc;
% Dynamic viscosity (Pa*s) % Wall dynamic viscosity (Pa*s) % Thermal conductivity (W/mK) % Prandtl number
Refrigerated fluid (HEAT SOURCE) mh = 10; Rh = 0.000275; Thi = 295; Tho = 275; rhoh = 982.3; cph = 4186; muh = 0.000453; muhw = 0.000453; kh = 0.656; Prh = cph*muh/kh; Refrigerant (R152a) R_ref = 0 . 0 0 0 2 7 5 ; T_re f_ l im = 2 7 3 . 1 5 ;
Material properties k_w = 6 0 . 5 ;
Optimization ag_systerne;
% Mass flow rate (kg/s) % Fouling resistance (m
A2K/W)
% Inlet temperature (K) % Outlet temperature (K) % Density (kg/m
A3)
% Heat capacity (J/kgK) % Dynamic viscosity (Pa*s) % Wall dynamic viscosity (Pa*s) % Thermal conductivity (W/mK) % Prandtl number
% Fouling resistance (mA2K/W)
% Minimum temperature (K)
% Thermal conductivity (W/mK)
% Genetic algorithm execution
% Optimal system design values (stocked in gagnant.txt) gagnant(phen_final,CCCC,epe, cph, HHHH, k_w,kc,mc_max,mh,muc,mucw,muh,PPPP, .
Prc,Prh,R_ref,Re, Rh, rhoc, rhoh, ssss,T_ref_lim,Tci,Thi,Tho,TTTT); Calculation time t = t o c ; d i s p l a y ( t ) ; ■ : ■
Published with MATLAB®
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D.2 Calcul des coûts d'opération et d'achat du système
System.m Ce programme est la fonction objectif de l'alorithme génétique. Elle permet de calculer le coût total annualisé de chaque système de réfrigération d'une population fourni par l'algorithme génétique. La fonction prend entrée les paramètres suivantes : phen CCCC cpc cph HHHH k_w kc mc_max mh mue mucw muh PPPP Pre Prh R_ref Rc Rh rhoc rhoh ssss T_ref Tci Thi Tho TTTT
lim
Population Refrigerant heat capacity data vector Condenser cold fluid heat capacity (J/kgK) Heat capacity of refrigerated (hot) fluid (J/kgK) Refrigerant specific enthalpy data vector Thermal conductivity of tubes material (W/mK) Thermal conductivity of condenser cold fluid (W/mK) Maximum mass flow rate of condenser cold fluid (kg/s) Mass flow rate of refrigerated (hot) fluid (kg/s) Dynamic viscosity of condenser cold fluid (Pa*s) Wall dynamic viscosity of condenser cold fluid (Pa*s) Dynamic viscosity of refrigerated (hot) fluid (Pa*s) Refrigerant pressure data vector Prandtl number of cold fluid Prandtl number of refrigerated (hot) fluid Refrigerant fouling resistance (mA2*K/W) Cold fluid fouling resistance (mA2*K/W) Refrigerated (hot) fluid fouling resistance (mA2*K/W) Cold fluid density (J/kgK) Refrigerated (hot) fluid density (J/kgK) Refrigerant specific entropy data vector Minimum refrigerant temperature (K) Temperature of cold fluid at exchanger inlet (K) Refrigerated (hot) fluid inlet temperature (K) Refrigerated (hot) fluid outlet temperature (K) Refrigerant temperature data vector
L'argument de sortie est le suivant : OBJECTIVE : Vector containing annualized cost of each design
Les points 1,2,3,4,5 et 6 correspondent aux états suivants du réfrigérant dans le cycle 2
/ /
CN
EV
point 1 : sortie de 1'évaporateur point 2 : entrée du condenseur
120 point 3 : sortie du condenseur point 4 : entrée de 1'évaporateur point 5 : état de vapeur saturée à la pression du condenseur point 6 : état de liquide saturée à la pression de l'évaporateur
Par: Benoît Allen
Date: 30 septembre 2009
Université Laval, Québec, Canada function [OBJECTIVE] =systeme(phen,CCCC,cpc,cph,HHHH,k_w,kc,mc_max,mh...
,muc,mucw,muh,PPPP,Prc,Prh,R_ref,Rc,Rh,rhoc, rhoh, ssss. . . ,T_ref_lim,Tci,Thi, Tho, TTTT)
%Efficacité du compresseur eff_comp = 0.85;
% décodaqe des variables de design [nind,~] = size(phen);
% vecteur contenant la valeur objectif (coût total) pour chaque individu OBJECTIVE = NaN(nind,1); Calculation loop for the population for iter = l:nind Design vector
individu = phen(iter,:); Refrigerant evaporator properties
% Saturation temperature (K) Tsat_ref_ev = T_ref_lim+((individu(24)-l)/256)*(Tho-T_ref_lim);
% Saturation pressure (Pa) P_ev = P_vs_Tsat(Tsat_ref_ev);
% Thermodynamic properties at points 1 and 6 [rho_l,rho_6,enthalpie_l,enthalpie_6,cp_l,cp_6,mu_l,mu_6,k_l,...
k_6,s_l,s_6,tension_6] = proprietes_sat(P_ev); Pr_l = cp_l*mu_l/k_l; Pr_6 = cp_6*mu_6/k_6;
Refrigerant condenser properties % Saturation temperature (K) Tsat_ref_.cn = Tci+( (individu(23)-1 )/256) * (380-Tci) ;
T_5 = Tsat_ref_cn;
% Saturation pressure (Pa) P_cn = P_vs_Tsat(Tsat_ref_cn);
% Thermodynamics properties at point 2 [T_2,cp_2,enthalpie_2] = point2(CCCC,HHHH,P_cn,PPPP,s_l,ssss, TTTT); cp_2 = cp_2*1000;
121 enthalpie_2 = enthalpie_2*1000;
% Thermodynamic properties at points 3 and 5 [rho_5,rho_3,enthalpie_5,enthalpie_3,cp_5,cp_3,mu_5,mu_3,k_5,k_3,...
s_5,s_3, tension_3] = proprietes_sat(P_cn); muw_3 = mu_3; Pr_3 = cp_3*mu_3/k_3; muw_5 = mu_5; Pr_5 = cp_5*mu_5/k_5;
% Thermodynamic properties at point 4 enthalpie_4 = enthalpie_3; T_4 = Tsat_ref_ev;
Required refrigerant mass flow rate (kg/s) m_ref = mh*cph*(Thi-Tho)/(enthalpie_l-enthalpie_4);
Condenser heat transfer rates % Condensation (W) Q_53 - m_ref*(enthalpie_5-enthalpie_3);
% Total (W) Q_23 = m_ref*((cp_2+cp_5)/2)*(T_2-T_5)+Q_53;
Condenser cold fluid minimum temperatures Tc5_min = Q_53/cpc/mc_max+Tci; Tco_min = Q_23/cpc/mc_max+Tci;
Annualized cost calculations % Temperature limits if T_5 > Tc5_min
if T_2 > Tco_min [ATC_ev,L_ev] = évaporateur(cph,enthalpie_l,enthalpie_4,...
enthalpie_6,individu, k_l,k_6,k_w,m__ref,mh,mu_l,mu_6,muh... ,Pr_l,Pr_6,Prh,R_ref,Rh,rho_l,rho_6, rhoh,T_4... ,tension_6,Thi,Tho);
[ATC_cn,L_cn] = condenseur(cp_2,cp_3,cp_5,cpc,enthalpie_3, ... enthalpie_5,individu,k_3,k_5,k_w,kc,m_ref,mc_max,mu_3, ... mu_5,muc,mucw,muw_3,muw_5,Pr_3,Pr_5,Prc,Rc,R_ref, ... rho_3,rho_5,rhoc,T_2,T_5,Tci,Tsat_ref_cn);
[ATC_comp] = comp_iso_s(m_ref,enthalpie_l,enthalpie_2, ... eff_comp);
% Exchanger length limits if max(L_ev,L_cn) > 7
ATC_ev = Inf; ATC_cn = Inf;
end else
ATC_ev = Inf; ATC_cn = Inf; ATC_comp = Inf;
end
122
else ATC_ev = Inf; ATC_cn = Inf; ATC_comp = Inf;
end Refrigeration system design total cost ($/year)
OBJECTIVE(iter) = ATC_ev+ATC_cn+ATC_comp; end
Published with MATLAB®
123
Annexe E
Optimisation d'un réseau d'échangeurs de chaleur
Optimisation_reseau.m Ce programme permet l'optimisation d'un réseau d'échangeur de chaleurs. Le coût total de chaque réseau est calculé pour chaque incrément d'écart de température minimal. La fonction prend en entrée la variable "table", un tableau contenant les données de départ du problème. Chaque rangée représente un fluide.
Première colonne : Température initiale du fluide (degrés C) Deuxième colonne : Température objectif du fluide (degrés C) Troisième colonne : Débit massique total du fluide (kg/s) Quatrième colonne : Identification du fluide
1 : air 2: vapeur d'eau 3: huile 4: rl34a 5: eau 6: kérosène 7: LGO 8: HGO 9: Naphta 10 : BPA 11 : Crude oil
Les deux dernières rangées de la table doivent correspondre à la vapeur d'eau et à l'eau froide. Leurs valeurs de débit massique et leurs températures finales sont à priori inconnues. On inscrit donc 0 aux colonnes 3 et 4. Le nombre de rangées de la matrice correspond au nombre de fluides dans le réseau. L'argument de sortie est "tcost", le coût total du réseau d'échangeur. Cela inclue les coûts d'achat et d'opération de TOUS les échangeurs de chaleur.
Afin de vérifier la validité de la méthode, ce script permet d'ajuster le nombre de simulations consécutives effectuées et le nombre de fois que TAG est exécuté pour l'optimisation de chaque échangeur.
Par: Benoît Allen
Date: 28 février 2010
Université Laval, Québec, Canada
124 function [tcost] = optimisation_reseau(table) Initialisation de l'affichage c l c ;
t i c -
Informations économiques H = 5000; % Période annuelle d'opération tarif_HU = 0.015; % Coût du hot utility HU ($/kWh) tarif_CU = 0.005; % Coût du cold utility CU ($/kWh) DT = 20; % Limite supérieure du deltaTmin (degrés C) Nombre de simulations simulations = 1; Nombre d'exécution de l'AG pour chaque échangeur iterations = 7; Optimisation for simul = 1 : s imula t ions
Pré-allocation des vecteurs tmin_vector = NaN(DT,l); n_HE = NaN(DT,1) , n_HU = NaN(DT,l) n_CU = NaN(DT,1) n_TOT = NaN(DT,l); tcost_vector = NaN(DT,l); Ucost_vector = NaN(DT,l); tcost_vector_sansU = NaN(DT,l);
% Vecteur des deltaTmin % Nombre d'échangeurs fluide-fluide % Nombre d'échangeurs fluide-HU % Nombre d'échangeurs fluide-CU % Nombre total d'échangeurs % Coûts totaux % Coût de CU et HU % Coûts totaux sans HU et CU
Optimisation du réseau pour chaque DT minimum for 1 = 1:DT
tmin_vector(1,1) = 1*1;
% Différence de température minimale Tmin = tmin_vector(1);
Distribution des fluides % La fonction design3 permet de distribuer les fluides dans les % échangeurs. À partir de la matrice "table" contenant % l'information relative aux fluides et Tmin, la différence de % température minimale à respecter, on obtient l'information % relative à tous les échangeurs requis dans TF, THU et TCU de
même % que les températures au pincement (THmin et TCmin) et les % quantités d'énergie requises pour réchauffer/refroidir les % fluides [TF,THU,TCU, N,QHmin,QCmin, THmin, TCmin] = design3(table, Tmin);
Nombre d'échangeur fluide chaud-fluide-froid n_HE(l , l ) = l e n g t h ( T F ( : , 1 ) ) ;
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Nombre d'échangeurs avec "Hot utility" n _ H U ( l , l ) = l e n g t h ( T H U ( : , 1 ) ) ;
Nombre d'échangeurs avec "Cold utility" n _ C U ( l , l ) = l e n g t h ( T C U ( : , 1 ) ) ;
Nombre total d'échangeurs n _ T O T ( l , l ) = n _ H E ( l , l ) + n _ C U ( l , l ) + n _ H U ( l , 1 ) ;
Initialisation des vecteurs Valeur objectif ObjVSelEX = NaN(n_HE(l,1), 1) ObjVSelHU = NaN(n_HU(l, 1),1) ObjVSelCU = NaN(n_CU(l, 1), 1)
Initialisation des matrices valeus objectif
% HE fluide froid - fluide chaud % HE avec "Hot utility" % HE avec "Cold utility"
% Chaque colonne correspond à une itération ITER_EX = NaN(n_HE(l,l) , 7) ITER_HU = NaN(n_HU(l,l),7) ITER_CU = NaN(n_CU(l,l),7)
% HE fluide froid - fluide chaud % HE avec "Hot utility" % HE avec "Cold utility"
Calcul des coûts
Aucun échangeur : impossible à résoudre, on impose un coût infini i f ( i sempty(TF) == 1 && isempty(TCU) == 1 && isempty(THU) == 1)
t c o s t _ v e c t o r ( 1 , 1 ) = I n f ; U c o s t _ v e c t o r ( 1 , 1 ) = I n f ; t c o s t _ v e c t o r _ s a n s U ( l , 1) = In f ;
e l s e
HE fluide froid - fluide chaud if isempty(TF) == 0
for k = l:n_HE(l,l) for iter = 1:iterations
aghex; % exécution de l'AG ITER_EX(k,iter) = ObjVSelEX(k, 1);
end % On pend la valeur minimum des itérations ObjVSelEX(k,1) - min(ITER_EX(k,:));
end sumEX = sum(ObjVSelEX);
else sumEX = 0;
end HE avec "Cold utility"
if îsempty(TCU) — 0 generationCU = NaN(n_CU(1,1),1); for k - l:n_CU(l,l)
for iter = 1:iterations aghex_CU2; % exécution de l'AG ITER_CU(k,iter) = ObjVSelCU(k,1);
end
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% On pend la valeur minimum des itérations ObjVSelCU(k,l) = min(ITER_CU(k, :)) ; % Cold utility en W (m*cp*deltaT) generationCU(k,l) = TCU(k,5)*calculcpCU(TCU(k,:))*...
(TCU(k,l)-TCU(k,2)); end sumCU = sum(ObjVSelCU)+sum(generationCU)*H/1000*tarif_CU; sumCU_sansU = sum(ObjVSelCU);
else sumCU = 0;
end HE avec "Hot utility"
if isempty(THU) == 0 generationHU = NaN(n_HU(l,1),1); for k = l:n_HU(l,l)
for iter = 1:iterations aghex_HU2; % exécution de l'AG ITER_HU(k,iter) = ObjVSelHU(k, 1);
end % On pend la valeur minimum des itérations ObjVSelHU(k,l) = min(ITER_HU(k,:)); % Hot utility en W (m*cp*deltaT) generationHU(k) = THU(k,6)*calculcpHU(THU(k, :))*.. .
(THU(k,4)-THU(k,3)); end sumHU = sum(0bjVSelHU)+sum(generationHU)*H/1000*tarif_HU; sumHU_sansU = sum(ObjVSelHU);
else sumHU = 0;
end Coûts totaux des échangeurs avec "utility"
t c o s t _ v e c t o r ( 1 , 1 ) = sumEX+sumCU+sumHU;
Coûts totaux des échangeurs sans "utility" tcost_vector_sansU(l,1) = sumEX+sumCU_sansU+sumHU_sansU;
Coût total du "utility" Ucost_vector(1,1) = t c o s t _ v e c t o r ( 1 , 1 ) -
t cos t_vec to r_sansU(1 ,1 ) ;
Graphique des coûts en fonction du deltaTmin figure((2+simul)) figure((2+simul)) plot(tmin_vector, tcost_vector,'MarkerFaceColor', [0 0
,'Marker','square','Color',[0 0 0] ) ; xlabel({'\DeltaT_m_i_n (°C)'},'FontSize'...
,12,'FontName','Arial'); ylabel(['',sprintf('\n'),'TC ($)'],'FontSize'...
,11,'FontName','Arial'); title('Total HEN cost (TC) vs. deltaTmin (\DeltaT_m_i.
hold on
,11, 'FontName', 'Anal' ) ; e('Total HEN cost (TC) vs. deltaTmin (\DeltaT_m_i_n)', 'FontWeight','bold','FontSize',12,'FontName','Arial'); on
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p l o t ( t m i n _ v e c t o r , t c o s t _ v e c t o r _ s a n s U , ' C o l o r ' , ' b l u e ' ) ; end
end
Coût minimum obtenu t c o s t ■ m i n ( t c o s t _ v e c t o r ) ;
Affichage display(n_T0T); display(tcost_vector); display(tcost_vector_sansU);
% Nombre total d'échangeurs % Coûts minimaux % Coûts minimaux sans "utility"
if tcost == Inf dispC ' ) ; dispC ' ) ; disp('RÉSOLUTION IMPOSSIBLE'); dispC ' ) ;
end end
toc; Published with MATLAB® 7.9