Optimisation d'échangeurs de chaleur: Condenseur à calandre ...

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

u

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

CU CU

20 20

HI H2

78 103

30 50 3

CU CU

20 20

HI H2

78 103

30 50

Z

s £ D. O

ed

U

< a

<D

.c >-> C 3

«2 cfl

E o aj W)

PJ

E i

\—■1 Cu O

rn

3 s H

£ >> o ,—,

• r i t/3 u r3

y ;

^ S — u. u s

c o 'J

1 3 — ■J-. s: ed

c ■ a c^

O '3 E

C 3

-9 J=

•a°.S

X l

<l

PQ î£

g'Si H 8

sC m es d

sC

o d 2

5.2

0

19.2

3

— -SÎ -st TT - - — "sf

"o c C « 5 2 2

c o

U u

g a S S u c c u

Ja u u ja

«n _) J «N

c u u

c o

u u c c o q

X Î X)

en NO

~ -~ 1—1 •—' OO 00 00 oo

n i n t n »n 00 OC

ua OO OO

O C o c e n c<-

c c e»"

o o en

O i n m

o o o o

m cn

X3 X> X> XI X) _ç X I X S S. J J

< < < < I 1 1 1

< 1

< 1

< < 1 1

1 1 1 1 tn vi vi tsi

Q Q Q û o m m o oo ON as oo

1

Q o o o

1 s-,

Q c oc

1 1 (Z) IA

Q Q m o ON 0 0

O C c o d d d d o o o o

T 3 " d 7 3 " O

O O O O

o

O

c

-o C

o o

O O O C c o d d d d

o o o o " O "O " O " O

O O O O

o

O

o ■o O

o o

O O O C c o d d d d

CS ( N ( S es es ( S m «n es es

0.3

D

0.2

D

Q (S C

û d

Û m m d

Q d 0

.4D

0.5

5D

o c os o>

C ON

o ON ■st

o o ON ON

o o o o " O "O "X3 T3 ■«T m < o w-i

-o o T3 >n

Q Q

es «n

- ( N cn Tl - m VO r- oo

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

* * ■ '

C uo I

•A, r- i

• i »

/ i

* * ■ '

/ /

• l t

1 I I

I I 1 I l 1

, ' • " ' : y ; i

/ - ■ ^ /

• t . , ' i

, ' é * — f

' - - * *

' — " "" "

/ / /

: / 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

Z U X

c o

es * tfl

*—» ■s: OJ

< O 

Q, C NC m'

h-

>> ^^ ê-J

V ■S o Efl

r 3 r ï C rrl bi) D U

•—

c: JM

0 o r,

U y.

- C 3 3 Z 4—1 3

-r) Cfl

<•> _ 3 U JO U.

J J

-o°.S

PQ ^

 O O O ^ o o v O N O t ^ ~ N O N D T t e n e n

I

Q

< l I

's:

Q

< < d > d > c i d > d > d > d > d > < d > d >

^3 "Q " ^ *Q 'O "Q *Q ^3 *0 "O "O o o o o o o o o o o o d d d d d d d d d d d *Q *0 ""O *0 *0 *T3 "3 'O *0 ""O *0 o o o o o o o o o o o d d d d d d d d d d d i f i i n i n i n i Y i i n i i n m i n i r i i r ) e s e s e s e s e s e s e s e s e s e s e s

Q Q Û Q Q Q Q Û Q Q Q

- s t ^ m m m " ! m e s o o o o o o o o

o o o o o o o o o o o ON O N O N O N O N O N O N O N O N O N O N

o o o o o o o o o o o "O "O *0 "O "O ""O "O "O *0 "O "O u-,«/-,eS"stio>n>n>n>/-.>/-iU-,

• ^ e s c n T t i o s D r ^ o o o N O —i

oo m O C ON o d d

2 2 "o "o o o

X X

-1 J

oo >n

o o i/-) o ND m

X I X en tfj

O O T t ON

m T t

es en

oo -st r-u-i ~ oo O T t T t —i m m

—I T t T t

H ° s U . e J3

XI XI X) J _) _) NO NO NO

0 O O ^ >n m

< n es es es

Q Q Q </-, i n i n m m -st o d d

o o o ON O N ON

o o o "O "O T3 es in in

T t U", NO

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

Bibliographie

[I] Communiqué présenté par Didier Coulomb, Directeur de l'Institut International du Froid. Institut International du Froid. Conférences des Nations Unies sur les changements climatiques. Poznan, Pologne, 1-12 décembre 2008.

[2] Développement durable : progrès et défis du secteur du froid. Institut International du Froid. 2003.

[3] Hydro-Québec. Pourquoi investir dans un système de réfrigération efficace. http://hydroquebec.com/affaires/appui_pmi/mesures/pop_refrigeration.html. 20janvier2010.

[4] L'Énergie pour construire le Québec de demain : La stratégie énergétique du Québec. Ministère des Ressources Naturelles et de la Faune. 2008-2015.

[5] R. Smith. Chemical Process Design and Integration, Wiley. New York. 2005.

[6] A. Karno, S. Ajib. Effect of tube pitch on heat transfer in shell-and-tube heat exchangers - new simulation software. Heat and Mass Transfer 2006; 42: 263-270.

[7] U.C. Kapale, S. Chand. Modeling for shell-side pressure drop for liquid flow in shell-and-tube heat exchanger. Int. J. Heat Mass Transfer 2006; 49: 601-610.

[8] Z.-Z. Xie, J-.F. Zhang, XL. Luo, Y.K. Chen, D.W. Ji. Modelbase of tube-and-shell heat exchangers and its application to simulation of heat exchanger networks. Journal of Systems Simulation 2005; 17: 2882-2887.

[9] ZH. Ayub. A new chart method for evaluating single-phase shell side heat transfer coefficient in a single segmental shell and tube heat exchanger. Applied Thermal Engineering 2005; 25: 2412-2420.

[10] R. Yahyaabadi. Fluid flow: analysing F-shell heat exchangers. Petroleum Technology Quarterly 2005; 10: 99-108.

[II] M. Serna, A. Jimenez. A compact formulation of the Bell-Delaware method for heat exchanger design and optimization. Chemical Engineering Research and Design 2005; 83: 539-550.

[12] D. Gulley. More accurate exchanger shell-side pressure drop calculations. Hydrocarbon Processing 2004; 83: 71-76.

[13] SA. Mandavgane, MA. Siddiqui, A. Dubey, SL. Pandharipande. Modeling of heat exchangers : using artificial neural network. Chemical Engineering World 2004; 39: 75-77.

[14] RK. Shah, DP. Sekulic. Fundamentals of Heat Exchanger Design. Wiley. New Jersey. 2003.

[15] GF. Hewitt, GL. Shires, TR. Bott. Process Heat Transfer. CRC Press: Boca Raton. 1994.

http://hydroquebec.com/affaires/appui_pmi/mesures/pop_refrigeration.html

95

[16] P. Wildi-Tremblay, L. Gosselin. Minimizing shell-and-tube heat exchanger cost with genetic algorithms and considering maintenance. International Journal of Energy Research 2007; 31: 867-885, DOI: 10.1002/er.l272.

[17] CR. Houck, J A. Joines, MG. Kay. A genetic algorithm for function optimization: a Matlab implementation. NCSU-IC Technical Report 1995.

[18] D. Eryener. Thermoeconomic optimization of baffle spacing for shell and tube heat exchangers. Energy Conversion and Management 2006; 47: 1478-1489.

[19] BK. Soltan, M. Saffar-Avval, E. Damangir. Minimizing capital and operating costs of shall and tube condensers using optimum baffle spacing. Applied Thermal Engineering 2004; 24: 2801-2810.

[20] M. Serna M, A. Jimenez. An efficient method for the design of shell and tube heat exchangers. Heat Transfer Engineering 2004; 25: 5-16.

[21] YA. Kara, O. Gùraras. A computer program for designing of shell-and-tube heat exchangers. Applied Thermal Engineering 2004; 24: 1797-1805.

[22] RD. Moita, C. Fernandes, HA. Matos, CP. Nunes. A cost-based strategy to design multiple shell and tube heat exchangers. J. of Heat Transfer 2004; 126: 119-130.

[23] R. Selbas, O. Kizilkan, M. Reppich. A new design approach for shell-and-tube heat exchangers using genetic algorithms from economic point of view. Chemical Engineering and Processing 2006; 45: 268-275.

[24] TW. Botsch, K. Stephan. Modelling and simulation of the dynamic behavior of shell-and-tube condenser. Int. J. Heat Mass Transfer 1997; 40: 4137-4149.

[25] JL. Alcock, DR Webb. An experimental investigation of the dynamic behavior of a shell-and-tube condenser. Int. J. Heat Mass Transfer 1997; 40: 4129-4135.

[26] MW.Browne, PK. Bansal. An overview of condensation heat transfer on horizontal tube bundles. Applied Thermal Engineering 1999; 19: 565-594.

[27] RK. Sinnot. Coulson and Richardson's Chemical Engineering, Butterworth-Heinemann: Stoneham, 1996.

[28] FP. Incropera, DP. Dewitt, TL. Bergman, AS. Lavine. Fundamentals of Heat and Mass Transfer, Wiley: Hoboken, 2007.

[29] J.C. Chato. Laminar condensation inside horizontal and inclined tubes. ASHRAE J. 1962; 4: 52-60.

[30] BR. Munson. Fundamentals of Fluid Mechanics, Wiley: New York, 2005. [31] IDR. Grant, D. Chisholm. Two-phase flow on the shell side of a segmentally baffled

shell and tube heat exchanger. Int. J. Heat Transfer 1979; 101: 38-42.

[32] National Institute of Standards and Technology. Thermophysical properties of fluid systems, http://webbook.nist.gov/chemistry/fluid/, 2007.

[33] I. Dinçer. Refrigeration Systems and Applications, Wiley: New York, 2003.

http://webbook.nist.gov/chemistry/fluid/

96

[34] RK. Green, SA. Tassou. A mathematical model of the heat transfer process in a shell and tube condenser for use in refrigeration applications. Applied Mathematic Modeling 1981;5:29-33.

[35] M. Markowski. Reconstruction of a heat exchanger network under industrial constraints - the case of a crude distillation unit. Applied Thermal Engineering 20 (15-16) (2000) 1535-1544.

[36] V. Briones, A.C. Kokossis. Hypertargets: a Conceptual Programming approach for the optimisation of industrial heat exchanger networks - Part III. Industrial applications. Chemical Engineering Science 54 (5) (1999) 685-706.

[37] G. Hall Stephen, B. Linnhoff. Targeting for furnace systems using pinch analysis. Industrial Engineering Chemical Research 33 (12) (1994) 3187-3195.

[38] N. Vaklieva-Bancheva, B.B. Ivanov, N. Shah, C.C. Pantelides. Heat exchanger network design for multipurpose batch plants. Computers Chemical Engineering 20 (8) (1996) 989-1001.

[39] J.-K. Kim, R. Smith. Cooling water system design. Chemical Engineering Science 56 (12) (2001) 3641-3658.

[40] L.E. Savulescu, M. Sorin, R. Smith. Direct and indirect heat transfer in water network systems. Applied Thermal Enineering 22 (8) (2002) 981-988.

[41] M. Ebrahim, A. Kawari. Pinch technology: an efficient tool for chemical-plant energy and capital-cost saving. Aplied Energy 65 (1-4) (2000) 45-49.

[42] K.C. Furman, N.V. Sahinidis. A critical review and annotated bibliography for heat exchanger network synthesis in the 20th century. Industrial & Engineering Chemistry Research 41(10) (2002) 2335-2370.

[43] T.K. Pho, L. Lapidus. Topics in Computer-Aided Design: Part II. Synthesis of Optimal Heat Exchanger Networks by Tree Searching Algorithms. AlChE Journal 19 (6)(1973)1182-1189.

[44] S. Bittanti, L. Piroddi. Nonlinear Identification and Control of a Heat Exchanger: A Neural Network Approach. Journal of Franklin Institute 334B (1) (1997) 135-153.

[45] T.F. Yee, I.E. Grossmann, Z. Kravanja. Simultaneous optimization models for heat integration-I. Area and energy targeting and modeling of multi-stream exchangers. Computers Chemical Engineering 14 (10) (1990) 1151-1164.

[46] S. Frausto-Hernandez, V. Rico-Ramirez, A. Jimenez-Gutierrez, S. Hernàndez-Castro. MINLP synthesis of heat exchanger networks considering pressure drop effects. Computers & Chemical Engineering 27 (8-9) (2003) 1143-1152.

[47] G.T. Polley, M.H. Panjeh Shahi and F.O. Jegede. Pressure drop considerations in the retrofit of heat exchanger networks. Transactions of the IChemE 68 (3) (1990) 211-220.

[48] M.L. Silva and R.J. Zemp. Retrofit of pressure drop constrained heat exchanger networks. Applied Thermal Engineering 20 (15) (2000) 1469-1480.

97

[49] MASS. Ravagnani, A.P. da Silva, A.L. Andrade. Detailed equipment design in heat exchanger networks synthesis and optimization. Applied Thermal Engineering 23 (2)(2003) 141-151.

[50] G.T. Polley and M.H. Panjeh Shahi. Interfacing heat exchanger network synthesis and detailed heat exchanger design. Transactions of the IChemE 69A (1991) 445-457.

[51] M.C. Roque and L.M.F. Lona. Synthesis of heat exchanger networks considering stream splitting, loop breaking and the rigorous calculation of the heat transfer coefficient according to the Bell-Dellaware method. Computers and Chemical Engineering 24 (2) (2000) 1349-1354.

[52] MASS. Ravagnani, J.A. Caballero. Optimal heat exchanger network synthesis with the detailed heat transfer equipment design. Computers & Chemical Engineering 31 (11) (2007) 1432-1448.

[53] B. Allen, L. Gosselin. Optimal geometry and flow arrangement for minimizing the cost of shell-and-tube condensers. International Journal of Energy Research 32 (10) (2008) 958-969.

[54] MASS. Ravagnani, A.P. Silva, P.A. Arroyo, Constantino AA. Heat exchanger network synthesis and optimisation using genetic algorithm. Applied Thermal Engineering 25 (7) (2005) 1003-1017.

[55] F.C. McQuiston, J.D. Parker, J.D. Spitler. Heating, Ventilating and Air Conditioning. Wiley. New York. 2005.

[56] Y.-C. Chang. Optimal chiller loading by evolution strategy for saving energy. Energy & Buildings 39 (4) (2007) 437-444.

[57] J.M. Gordon, K.C. Ng. Predictive and diagnostic aspects of a universal thermodynamic model for chillers. International Journal of Heat and Mass Transfer 38(5)(1995)807-818.

[58] J.R. Khan, S.M. Zubair. Design and performance evaluation of reciprocating refrigeration systems. International Journal of Refrigeration 22 (3) (1999) 235-243.

[59] H.T. Chua, K.C. Ng, J.M. Gordon. Experimental study of the fundamental properties of reciprocating chillers and their relation to thermodynamic modeling and system design. International Journal of Heat and Mass Transfer 39 (11) (1996) 2195-2204.

[60] J.M. Gordon, K.C. Ng, H.T. Chua. Centrifugal chillers: thermodynamic modelling and a case study. International Journal of Refrigeration 18 (4) (1995) 253-257.

[61] J.M. Gordon, K.C. Ng. Thermodynamic modeling of reciprocating chillers. Journal of Applied Physics 75 (6) (1994) 2769-2774.

[62] M.W. Browne, P.K. Bansal. An elemental NTU-6" for vapor-compression liquid chillers. International Journal of Refrigeration 24 (7) (2001) 612-627.

98

[63] H. Bechtler, M.W. Browne, P.K. Bansal, V. Kecman. New approach to dynamic modeling of vapor-compression liquid chillers: artificial neural networks. Applied Thermal Engineering 21 (9) (2001) 941-953.

[64] D.J. Swider, M.W. Browne, P.K. Bansal, V. Kecman. Modelling of vapour-compression liquid chillers with neural networks. Applied Thermal Engineering 21 (3)(2001) 311-329.

[65] J.C. Chen. Correlation for boiling heat transfer to saturated fluids in convective flow. l&EC Process and Design Development 5 (3) (1996) 322-329.

[66] R.L. Webb, N.S. Gupte. A critical review of correlations for convective vaporization in tubes and tube banks. Heat Transfer Engineering 13 (3) (1992) 58-81.

[67] N.G. Ng, H.T. Chua, W. Ong, S.S. Lee, J.M. Gordon. Diagnostics and optimization of reciprocating chillers: Theory and experiment. Applied Thermal Engineering 17 (3) (1997) 263-276.

[68] R. Selbas, O. Kizilkan, A. Sencan. Thermoeconomic optimization of subcooled and superheated compression refrigeration cycle. Energy 31 (12) (2006) 2108-2128.

[69] J.M. Gordon, K.C. Ng, H.T. Chua. Optimizing chiller operation based on finite-time thermodynamics: universal modeling and experimental confirmation. International Journal of Refrigeration 20 (3) (1997) 191-200.

[70] F.W. Yu, K.T. Chan. Strategy for designing more energy efficient chiller plants serving air-conditioned buildings. Building and Environment 42 (10) (2007) 3737-3746.

[71] B. Allen, L. Gosselin. Optimal geometry and flow arrangement for minimizing the cost of shell-and-tube condensers. International Journal of Energy Research 32 (10) (2008) 958-969.

[72] S.G. Kandlikar. A general correlation for saturated two-phase flow boiling heat transfer inside horizontal and vertical tubes. Journal of Heat Transfer 112 (1) (1990) 219-228.

[73] P.B. Whalley, J.G. Collier, J.R. Thome. Convective Boiling and Condensation, third ed., Clarendon Press. Oxford. 1996.

[74] L. Gosselin, M. Tye-Gingras, F. Mathieu-Potvin. Review of genetic algorithms utilization in heat transfer problems, International. Journal of Heat and Mass Transfer 52 (9-10) (2009) 2169-2188.

[75] H. Grollius, C. Meyer, M. Rautenberg, M. Bailey McEwan. Computer modeling of the performance of centrifugal water chillers in mine refrigeration installations. International Journal of Refrigeration 10(1) (1997) 49-52.

[76] F.W. Yu, K.Y. Chan. Optimization of water-cooled chiller system with load-based speed control. Applied Energy 85 (10) (2008) 931-950.

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)


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


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


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;


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 ) ; ■ : ■


119

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;

http://Tsat_ref_.cn

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


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 ) ) ;

125

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

126

% 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

127

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

Optimisation d'échangeurs de chaleur: Condenseur à calandre ...

Documents

Transcript of Optimisation d'échangeurs de chaleur: Condenseur à calandre ...