modélisation mathématique du pressage à chaud des panneaux mdf

ZANIN KAVAZOVIĆ
MODÉLISATION MATHÉMATIQUE DU PRESSAGE
À CHAUD DES PANNEAUX MDF
Couplage du modèle mécanique avec le modèle couplé de
transfert de chaleur et de masse
Thèse présentée
à la Faculté des études supérieures de l’Université Laval
dans le cadre du programme de doctorat en mathématiques
pour l’obtention du grade de Philosophiae Doctor (Ph.D.)
DÉPARTEMENT DE MATHÉMATIQUES ET DE STATISTIQUE
FACULTÉ DES SCIENCES ET DE GÉNIE
UNIVERSITÉ LAVAL
QUÉBEC
2011
© Zanin Kavazović, 2011
ii
Résumé
Dans la présente thèse, nous nous intéressons aux phénomènes physiques se déroulant
durant le processus de pressage à chaud des panneaux de fibres de bois (MDF). La nonlinéarité et la forte interdépendance des phénomènes instationnaires de transfert de chaleur
et de masse et du pressage mécanique de l’ébauche de fibres rendent leur modélisation et
analyse non triviales. Dans un premier temps, nous avons effectué une étude de sensibilité
portant sur le modèle de transfert de chaleur et de masse proposé par Thömen et Humphrey
en 2006. Dans cette étude de sensibilité, nous avons déterminé l’impact de la variabilité des
propriétés matérielles, des modèles de sorption, des conditions aux limites et de la teneur en
humidité initiale sur les variables d’état et les résultats numériques du modèle
mathématique. Afin de mieux tenir compte des interactions complexes entre les différents
processus physiques, nous avons ensuite proposé un modèle mathématique global
tridimensionnel couplé modélisant le processus de pressage à chaud en lot (batch pressing).
Le modèle global est constitué de deux entités distinctes, soient le modèle mécanique et le
modèle couplé de transfert de chaleur et de masse. Dans cette première phase de
développement, la compression de l’ébauche est représentée par un modèle élastique
vieillissant que nous avons exprimé en formulation quasi-statique incrémentale. Les
variables d’état pour ce modèle sont l’incrément de déplacement et l’incrément de
contrainte. Tous les calculs se font sur une géométrie mobile dont la déformation
(compression) est une conséquence de la fermeture de la presse. Le développement du
profil de densité est ainsi calculé dynamiquement à chaque pas de temps. Quant aux
phénomènes de transfert de chaleur et de masse, ils sont modélisés par un système couplé
constitué de trois équations de conservation, notamment la conservation de la masse de l’air
et de la vapeur ainsi que la conservation de l’énergie. Les équations sont exprimées en
fonction de trois variables d’état, soient la température et les pressions partielles de l’air et
de la vapeur. Le modèle global est discrétisé par la méthode des éléments finis et les
systèmes résultant ont été résolus grâce au logiciel MEF++ développé au GIREF (Groupe
interdisciplinaire de recherche en éléments finis, Université Laval). Les simulations
numériques ont été menées aussi bien en deux qu’en trois dimensions. Les résultats
numériques de température et de pression gazeuse ont été comparés aux mesures prises au
laboratoire du CRB (Centre de recherche sur le bois, Université Laval) lors d’un procédé de
pressage en lot. Une bonne concordance entre les résultats numériques et expérimentaux a
été constatée. Afin d’enrichir le modèle proposé, les futurs développements devraient traiter
de la nature viscoélastique et plastique de l’ébauche soumise au pressage à chaud. Il
demeure néanmoins clair que la qualité des prédictions produites par des modèles
numériques dépendra toujours en grande partie de la disponibilité et de la qualité des
valeurs caractérisant les propriétés physiques et matérielles du produit à l’étude. Afin de
combler de nombreuses lacunes à ce chapitre, nous ne pouvons qu’encourager les
recherches menant à une meilleure connaissance de ces propriétés.
iii
Abstract
In this thesis, we study the physical phenomena involved during hot pressing process of
medium density fiberboard mats (MDF). Non linear nature and strong interdependency of
unsteady phenomena of heat and mass transfer and mechanical pressing of fiber mats make
the modeling and analysis of these processes difficult. Firstly, we have conducted a
sensitivity study on the heat and mass transfer model proposed by Thömen and Humphrey
in 2006. That sensitivity study helped to determine the impact of variable mat material
properties, sorption models, boundary conditions, and initial moisture content on the state
variables and on the numerical results of the mathematical model. To better take into
account complex interactions between different physical processes, we have further
developed a global coupled three-dimensional mathematical model aimed to simulate the
hot pressing process in a batch press. Our global model is made of two distinct entities: a
mechanical model and a coupled heat and mass transfer model. This thesis represents the
first phase of development where the compression of the mat is described by an aging
elastic model which was expressed in a quasi-static incremental formulation. The state
variables for this model are displacement and stress increments. All calculations were
performed on a dynamically changing geometry and deformation (compression) of the
geometry is a consequence of press closing dynamics. Hence, development of the density
profile is calculated at each time step. The heat and mass transfer phenomena are modeled
by a coupled system of three conservation equations: conservation of mass of the air and
the water vapor, and conservation of energy. The system is expressed as a function of three
state variables: temperature, partial air pressure and partial vapor pressure. The global
model is discretized in space by means of the finite element method and solved by the
MEF++ software developed by GIREF (Groupe interdisciplinaire de recherche en éléments
finis, Laval University). Numerical simulations were conducted in 2D and 3D. Numerically
predicted temperature and gas pressure were compared to experimental data obtained
during batch pressing experiments in the CRB (Centre de recherche sur le bois) laboratory.
A good overall agreement was found between numerical and experimental results. To
improve the proposed model, future developments should include viscoelastic and plastic
nature of the mat undergoing hot pressing process. It is however clear that the quality of
predictions delivered by numerical models will largely be dependent upon availability and
precision of parameters characterizing physical and material properties of the mat under hot
pressing conditions. Further research in that area is strongly recommended.
Avant-Propos
J’exprime ma gratitude aux membres du jury, professeurs Patrick Perré, Robert Guénette,
José Urquiza, Alain Cloutier et André Fortin, qui ont accepté d’examiner cette thèse malgré
un emploi de temps très chargé. Messieurs, vos commentaires constructifs et suggestions
ont grandement contribué à la qualité du document final. Je vous remercie pour vos efforts,
votre professionnalisme et le temps que vous avez consacré à la correction de la thèse.
Cette thèse de doctorat fait partie d’un projet stratégique du CRSNG portant sur la
modélisation numérique en sciences du bois. Le présent travail est le fruit de la
collaboration scientifique entre deux centres de recherche de l’Université Laval, soient le
Groupe Interdisciplinaire de Recherche en Éléments Finis (GIREF) et le Centre de
Recherche sur le Bois (CRB). Nous soulignons l’important appui financier des partenaires
gouvernementaux et industriels, soient le Conseil de Recherche en Sciences Naturelles et
Génie (CRSNG) du Canada, FPInnovations Québec, Uniboard Canada et Boa-Franc.
La thèse a été réalisée sous la direction d’André Fortin, professeur titulaire au Département
de mathématiques et de statistique de l’Université Laval et directeur du GIREF, et sous la
co-direction d’Alain Cloutier, professeur titulaire au Département des sciences du bois et de
la forêt de l’Université Laval et directeur du CRB. Un apport scientifique considérable,
constant et essentiel à la réalisation des travaux de la thèse a été assuré par Jean Deteix,
professionnel de recherche au GIREF.
Les différentes parties de cette thèse ont été présentées dans plusieurs congrès et
conférences, que ce soit sous forme de présentation orale ou d’affiche.
Présentations orales:
14th European Conference on Composite Materials, 7-10 Juin 2010, Budapest, Hongrie
4th European Conference on Computational Mechanics, 16-21 Mai 2010, Paris, France
4th International Conference on Advanced Engineered Wood and Hybrid Composites, 6-10
Juillet 2008, Bar Harbor, Maine, USA
v
76-ième Congrès de l’ACFAS, 5-9 Mai 2008, Québec, Québec, Canada
9-ième Colloque Panquébécois des étudiants (ISM), 11-13 Mai 2007, Université Concordia,
Montréal, Québec, Canada
8-ième Colloque Panquébécois des étudiants (ISM), 23-25 Mai 2006, Université Laval,
Québec, Québec, Canada
Présentations par affiche :
6-ièmes Journées Montréalaises de Calcul Scientifique, 4-6 Mai 2009, CRM, Université de
Montréal, Montréal, Canada
51st Annual Convention of the Society of Wood Science and Technology, 10-12 Novembre
2008, Concepción, Chile
Prix: 3-ième prix à Student Poster Competition Awards, SWST à Concepción, Chile
4-ièmes Journées Montréalaises de Calcul Scientifique, 16-17 Avril 2007, CRM, Université
de Montréal, Montréal, Canada
vi
Remerciements
Mes premiers remerciements vont à ma famille scientifique du GIREF. Dans la dernière
décénie, le GIREF était ma maison, mon milieu de vie, mon univers. Le GIREF, ce sont des
gens pationnés, pleins d’humour, de joie de vivre et en même temps des gens sérieux,
rigoureux et exigeants sur le plan scientifique. La rigueur et les standards de qualité y sont
très élevés. On y apprend vite qu’une chose qui mérite d’être faite mérite d’être bien faite.
On y enseigne la rigueur et l’honnêteté scientifiques, le travail acharné et le jugement
critique. L’ambiance familiale et fraternelle qui y règne fait du GIREF un milieu de travail
agréable, paisible et rassurant. La diversité de cultures et d’opinions nous enrichit et donne
une nouvelle dimension à notre vision du monde et de la vie. Tous ces gens uniques et cette
ambiance merveilleuse font en sorte qu’une fois qu’on les a connus, on a du mal à partir du
GIREF. Tous ceux qui sont passés par le GIREF savent qu’ils en sont sortis de meilleurs
être humains qu’ils ne l’étaient le jour où ils sont arrivés. Des connaissances, l’homme en
acquiert et en perd en cours de route, mais les qualités humaines qu’il gagne au gré des
rencontres demeurent ancrées en lui à jamais.
J’adresse mes remerciements et mes respects à mes deux directeurs de thèse, professeur
André Fortin (directeur) et professeur Alain Cloutier (co-directeur), qui ont travaillé dur
pour que ce magnifique projet voit le jour et m’en ont confié une partie. Je les remercie
pour leur confiance, l’appui et le soutien inconditionnels et constants durant toutes ces
années. Messieurs, j’ai appris beaucoup avec vous et, surout, j’ai appris beaucoup de vous :
l’honnêteté scientifique, la transparence, le dévouement au travail et l’ouverture d’esprit.
Votre expertise, rigueur et vos encouragements m’ont aidé à persévérer et à mener ce
travail à bien. Votre contribution à mon développement tant sur le plan scientifique que
personnel est très importante. Il y a de ces moments dans la vie où le cœur et la raison
s’affrontent et vous m’avez aidé à voir plus clairement à travers l’écran de fumée qui se
dressait sur le champ de bataille.
En réalité, durant cette thèse, j’avais trois directeurs. Le troisième directeur, celui qui
officiellement était inexistant mais qui, dans les faits, était omniprésent, est Jean Deteix.
Jean était le pilier de ce projet. Jean, ton soutien scientifique et moral au quotidien m’ont
été d’une aide inestimable. C’est simple, je pense que sans toi, je n’y serais pas arrivé. Je
vii
me rappelle encore de ces moments difficiles, ardus, frustrants et durs pour le moral,
surtout au début du projet. Grâce à ta présence quotidienne, ton soutien et ton humour (ah,
ton humour dans les moments noirs d’une longue journée), je m’en suis sorti. Comme je
l’ai répété souvent, durant cette thèse, deux saints veillaient sur moi : saint Jean (Deteix) et
saint Georges (Djoumna). Les gars, vous étiez une source inépuisable de soutien et
d’encouragement. Est-ce qu’un jour je pourrai vous remercier assez ?
Je témoigne également ma reconnaissance aux messieurs Pierre Blanchet et Gilles Brunette
de FPInnovations Québec pour l’appui indéfectible qu’ils ont apporté au projet, et ce dès
les premiers instants.
Cristian Tibirna. Tu as pris soin de moi dès mon premier jour au GIREF. Tu m’as enseigné
à coder de manière propre et structurée et ça m’a forcé à avoir une pensée structurée. Tu
n’hésites pas à me critiquer quand je fais ou dis des niaiseries, et Dieu sait que j’en suis
capable, et tu es toujours là pour me donner une tape sur l’épaule quand je fais les choses
suivant les standards du GIREF. Tu es une inspiration, un exemple vivant de curiosité, de
passion d’apprendre et de découvrir. Quand j’ai besoin de conseils ou que je veux savoir si
je suis sur la bonne piste, je viens te voir et tu as toujours des paroles sages et justes. Il y a
un peu de toi dans chacun de nous et dans chacune des thèses qui se sont faites au GIREF.
Mais tu ne sais pas combien il me ferait plaisir si tu prenais enfin le temps de finir la tienne.
Voilà, je promets de ne plus t’embêter avec mes ennuis de programmation. Il me semble
que c’est une occasion à saisir. 
Mes profonds remerciements vont à Sylvie Lambert que j’appelle affectueusement la reine
du GIREF. En plus de nous garder à l’abri d’innombrables tracasseries administratives, et
elle est d’une efficacité redoutable dans ce domaine, Sylvie veille sur nous et notre bienêtre au quotidien. Sa présence rassure, son sourire, sa gaieté et bonne humeur rendent ces
lieux agréables et paisibles. Un mot d’encouragement de sa part, un conseil, une discussion
sur mille sujets nous donnent de l’énergie et nous propulsent vers de nouvelles réalisations.
C’est à Sylvie qu’on doit beaucoup pour cette harmonie et ambiance familiale qui règnent
au GIREF.
viii
Je me remémore de beaux moments, une ambiance hilarante au labo et de longues
discussions avec Carl Robitaille. Carl et Cristian sont des philosophes dans l’âme et de
redoutables débateurs. Je salue également la contribution d’Éric Chamberland dans mon
développement professionnel. Son point de vue pratique et ses questions en apparence
simples forcent souvent à une réflexion profonde. Ce fut un plaisir de grandir à vos côtés
messieurs. Vous avez réussi à reformater mes méninges à quelques reprises.
Je souligne l’attention, la patience sans borne et une aide soutenue de la part des muses du
Département de mathématiques et de statistique, Sylvie Drolet et Suzanne Talbot. Depuis
que je suis arrivé au Département, et je n’ose même plus compter les années, j’ai été témoin
de leur passion et dévouement au travail. S’il y a une façon de vous éviter des
complications adminstratives et d’accélérer le traitement de vos demandes, vous pouvez
être assuré que ces deux anges vont tout faire pour y arriver. Mes dames, je vous admire et
je vous avoue que, sans vous, le Département n’aurait pas le même charme.
Je ne peux passer sous silence l’aide, support et les conseils reçus de la part du professeur
Roger Pierre. Dans certains moments de doute, que ce soit sur le plan mathématique ou
personnel, sa porte était toujours ouverte. Il prenait le temps de m’écouter, de discuter avec
moi et il était généreux de conseils judicieux. En sortant de son bureau, les doutes s’étaient
dissipés, mes idées étaient plus claires et ma confiance rétablie. Je lui suis également
reconnaissant de m’avoir donné deux occasions d’enseigner. C’étaient des expériences très
enrichissantes qui avaient un effet étonnamment relaxant sur moi.
Lors de mes nombreuses visites au CRB, j’ai toujours été accueilli chaudement et avec un
grand sourire des dames Guylaine Bélanger et Colette Bourcier. L’expertise et l’aide de
David Lagueux ont été précieuses lors de la séance de pressage de panneaux MDF au
laboratoire de pressage du CRB.
Un immense merci à mon frère du destin André Guy Tranquille Temgoua et sa famille pour
leur amour, soutien et encouragements, pour ces pauses-escapades de la monotonie de la
vie d’étudiant et ses tracas, pour ces instants où le temps suspend son vol pour nous
permettre de faire des crêpes avec les enfants et les voir grandir.
ix
Je remercie mes nombreux amis et collègues avec qui j’ai partagé ces belles années de
découvertes, de joies, d’amitié, de déceptions (et oui, il y en avait), d’angoisse à l’approche
d’un futur incertain et d’espoir que tout ça va passer et qu’un jour tout sera pour le mieux
pour nous tous. Nous étions venus d’un peu partout et la vie nous a réunis au GIREF où,
pendant des années, ceux qui étaient plus vieux prenaient soin des plus jeunes. À la
mémoire de la belle gang : Fabien Youbissi, Georges Djoumna, Rym Jedidi, Mériem Saïd,
Étienne Non, Abdoulaye Kane, Bocar Wane, Ngueye et Ndeye Thiam, Khalid Benmoussa,
Aberrahman Elmaliki, Youssef Belhamadia, Benoît Pouliot, Emmanuelle Reny-Nolin,
Mahmood Shabankhah, Babacar Toumbou, Michel Diémé, Yahya Ould Denna, Bassirou
Toumbou.
J’adresse des remerciements particuliers à Richard Bois pour ses nombreuses missions de
sauvetage de mes présentations qui, sans lui, seraient beaucoup moins animées. Un clin
d’œil à un ami de longue date, Rabah Hammoud, qui est une incarnation de la persévérance
et du dur labeur jumelés à un esprit scientifique bouillant. À mon cher ami Nacer Mézouari,
une anthologie des mathématiques appliquées : si seulement j’avais le quart de tes
connaissances et de ta passion, je pense que je serais vraiment bon. Merci pour ces
innombrables discussions et conseils fraternels qui ont grandement contribué à soulever le
voile devant mes yeux.
Comment exprimer ma reconnaissance et ma gratitude envers mes parents et ma sœur pour
leur amour et leur soutien inconditionnels, pour leurs sacrifices, la patience et la tolérance?
A-t-on vraiment à féliciter une colombe parce qu’elle sait voler ? Car, après tout, voler,
c’est dans sa nature. Je ne le sais pas, mais on peut certainement l’admirer et prendre
exemple sur elle, sur ses rêves de paix et de liberté. À vous qui avez su attendre patiemment
et en silence, qui avez mis vos espoirs entre les mains de la foi inébranlable qu’un homme
devrait toujours terminer le travail entamé, puisse cette thèse apporter la juste récompense à
vos attentes et ainsi écourter la liste des rêves trahis.
Aux amis perdus dans le brouillard des années de ce long périple,
À ceux qu’on ne reverra plus
mais dont le souvenir demeurera à jamais présent
car eux aussi font partie de notre présent
tout comme ils ont fait partie de notre passé.
Au nom de tous ces moments de joie intense que le temps ne saurait effacer.
Qu’est-ce que le temps devant l’éternité d’un instant de joie de l’âme ?
Aux plaisirs de la compréhension, à l’euphorie de la découverte,
À cet instant unique où l’intelligence prend conscience qu’entre ses mains
respire l’essence même du phénomène physique qui la hante depuis.
Puisse cette drogue être l’éternelle source de notre persévérance et endurance;
Puisse elle nous amener à nous surpasser.
Sinon, à quoi bon continuer ?
xi
Table des matières
Résumé................................................................................................................................... ii
Abstract................................................................................................................................. iii
Avant-Propos .........................................................................................................................iv
Remerciements.......................................................................................................................vi
Table des matières ............................................................................................................... xii
Liste des figures .................................................................................................................. xiii
Introduction.............................................................................................................................1
Chapitre 1................................................................................................................................9
Étude de sensibilité d’un modèle numérique de transfert de chaleur et de masse durant le
pressage à chaud des panneaux MDF .....................................................................................9
Chapitre 2..............................................................................................................................50
Modélisation numérique du processus de pressage à chaud des panneaux MDF ................50
Modèle couplé de transfert de chaleur et de masse ..............................................................50
Chapitre 3..............................................................................................................................88
Modélisation numérique du processus de pressage à chaud des panneaux MDF ................88
Couplage du modèle mécanique avec le modèle de transfert de chaleur et de masse ..........88
Conclusion ..........................................................................................................................129
Bibliographie ......................................................................................................................132
Liste des figures
Chapitre 1
Figure 1.1. Géométrie 2D
Figure 1.2. Résultats numériques et expérimentaux pour T et P
Figure 1.3. Solutions pour P et M obtenues avec différents modèles de sorption
Figure 1.4. Effets des modèles de sorption sur T, M et P
Figure 1.5. Solutions pour P et M obtenues avec différentes valeurs de Minit
Figure 1.6. Effets de Minit sur T, M et P
Figure 1.7. Effets de KT sur T, M et P
Figure 1.8. Effets de Kp sur T, M et P
Figure 1.9. Effets de hT sur T, M et P
Figure 1.10. Effets de hp sur T, M et P
Figure 1.11. Évolution spatio-temporelle du profil vertical de densité anhydre
p.16
p. 23-24
p. 25-26
p. 27-28
p. 29-30
p. 31-32
p. 33-34
p. 35-36
p. 37-38
p. 39-40
p. 41
Chapitre 2
Figure 2.1. Évolution de l’épaisseur de l’ébauche
p. 56
Figure 2.2. Évolution spatio-temporelle du profil vertical de densité anhydre
p. 57-58
Figure 2.3. Géométrie 2D
p. 68
Figure 2.4. Résultats numériques et expérimentaux pour T et P
p. 76
Figure 2.5. Résultats numériques pour M et h
p. 77
Figure 2.6. Résultats numériques pour Pa et Pv
p. 78
Figure 2.7. Résultats numériques pour le degré de polymérisation de la résine et le taux de
polymérisation de la résine
p. 79
Chapitre 3
Figure 3.1. Géométrie 3D
Figure 3.2. Évolution de l’épaisseur de l’ébauche
Figure 3.3. Prédiction numérique du profil vertical de densité
Figure 3.4. Résultats numériques pour la contrainte verticale
Figure 3.5. Résultats numériques et expérimentaux pour T et P
Figure 3.6. Résultats numériques pour M et h
Figure 3.7. Résultats numériques pour Pa et Pv
Figure 3.8. Résultats numériques 2D versus 3D
Figure 3.9. Résultats maillages raffinés
Figure 3.10. Maillage concentré
Figure 3.11. Résultats maillages concentrés
Figure 3.12. Différentes procédures de fermeture
Figure 3.13. Résultats numériques pour différentes procédures de fermeture
Figure 3.14. Résultats numériques coupes 2D T, M et Pv
p. 98-99
p. 101
p. 105-106
p. 107
p. 108
p. 109
p. 111
p. 112-113
p. 114
p. 115
p. 116
p. 118
p. 119
p. 121-122
xiv
Introduction
Au Canada et tout particulièrement au Québec, l’exploitation des ressources forestières est
l’un des principaux moteurs de l’économie. La demande pour les panneaux MDF (Medium
Density Fiberboard ou panneaux de fibres de densité moyenne) n’a cessé d’augmenter au
cours de la dernière décennie. On s’en sert dans la fabrication de meubles, de portes
d’armoires, de comptoirs de cuisine. Une version à plus haute densité (HDF, high density
fiberboard) entre également dans la composition de planchers stratifiés (planchers
d’ingénierie). Les panneaux MDF sont constitués de fibres de bois légèrement humides
mélangées avec un adhésif thermodurcissable et une faible quantité de cire. Le tout est
pressé jusqu’à une densité moyenne variant entre 500 et 850 kg / m3 à haute température
(environ 200 C ) dans une presse en lot (batch press) ou une presse en continu durant une
période variant entre 3 et 5 minutes. Puisque le temps de pressage affecte la productivité de
la presse, tous les efforts pouvant contribuer à le diminuer sont bienvenus. La durée du
pressage ainsi que le programme de fermeture de la presse sont ajustés en fonction des
propriétés désirées du produit fini. En effet, l’opération de pressage constitue une étape
cruciale dans le processus de fabrication des panneaux composites à base de bois. Les
propriétés mécaniques et physiques du produit fini sont déterminées à sa sortie de la presse,
en particulier le profil de densité, et dépendent grandement des conditions de pressage. Les
expériences menées à l’usine peuvent s’avérer très coûteuses aussi bien en termes de
matériel que de ressources humaines, sans compter la mise hors production d’une presse à
des fins expérimentales. C’est ainsi que le développement de modèles mathématiques
constitue une avenue prometteuse permettant de simuler virtuellement le processus de
pressage. En effet, cela permet de faire des études sur l’ordinateur simulant avec aise
différentes configurations initiales (teneur en humidité, proportion de résine, masse de
fibres) et conditions de pressage (température de la presse, programme de pressage, durée
du pressage). On obtient ainsi une assez bonne idée de l’influence de ces différents
paramètres sur l’évolution et la distribution, entre autres, de la température, de la pression
du gaz, du profil de densité et de la teneur en humidité dans le panneau. De plus, de
nombreux phénomènes physiques, mécaniques et chimiques ont lieu simultanément lors du
pressage de l’ébauche. Leur étendue varie dans le temps et suivant la localisation dans
l’ébauche. La complexité et l’interdépendance de ces processus requièrent une étude
théorique approfondie basée sur les principes fondamentaux de la physique. La
compréhension fondamentale du processus de pressage est donc essentielle pour optimiser
la vitesse de production, les coûts de production, la consommation d’énergie, l’optimisation
et manipulation des propriétés de panneaux finis ainsi que le développement de nouveaux
produits et technologies de production. Depuis des années, de nombreux chercheurs ont
tenté de développer des modèles de plus en plus complets et complexes afin de représenter
le plus fidèlement possible les phénomènes physiques qui interviennent lors du pressage à
chaud des panneaux à base de bois. Ainsi, les chercheurs ont porté un intérêt tout particulier
aux mécanismes de transfert de chaleur et de masse d’eau liée aux fibres de bois. De plus,
des lois de comportement d’une complexité toujours croissante ont été développées afin de
représenter au mieux la rhéologie d’un panneau lors de la compression. De nombreuses
études portant sur les facteurs pouvant influencer ces mécanismes ont été menées dans les
dernières six décennies.
2
Une série d’articles a été publiée par Bolton, Humphrey et Kavvouras en 1988 et 1989
(Bolton et Humphrey 1988; Bolton et al 1989b; Humphrey et Bolton 1989; Kavvouras
1977) faisant le point sur l’état de l’art à l’époque. Une revue de littérature exhaustive y a
été faite et plusieurs facteurs influençant le pressage ont été abordés. Depuis lors, plusieurs
études ont suivi menant à de nouvelles connaissances des phénomènes complexes qui se
produisent durant le pressage. On sait ainsi que la température de la presse, la teneur en
humidité de l’ébauche, la vitesse de fermeture de la presse et le temps de pressage sont des
facteurs très importants. En effet, ils sont à l’origine des phénomènes de transfert de masse
et de chaleur dans l’ébauche et influencent grandement ses propriétés mécaniques et,
ultimement, le développement du profil de densité du panneau (Thömen et Ruf 2008). Une
attention toute particulière doit donc être accordée au processus de transfert de chaleur et de
masse et à son interaction avec les propriétés mécaniques de l’ébauche durant le pressage.
Lors du pressage, la principale source de chaleur est celle fournie par les plaques
chauffantes de la presse. La réaction exothermique de polymérisation de la résine fournit
également de la chaleur mais en quantité nettement inférieure. Il en va de même pour la
chaleur latente libérée lors de la condensation de la vapeur d’eau. Le principal transfert de
chaleur des plaques chauffantes à l’ébauche se fait par contact direct, i.e. la conduction. La
chaleur transférée à l’ébauche par le gaz chaud qui circule dans les espaces vides de
l’ébauche est d’une importance moindre compte tenu de la faible masse de la vapeur d’eau
dans l’ébauche et de sa capacité thermique (deux fois inférieure à celle des fibres).
La cohésion des panneaux composites à base de bois dépend grandement de la force des
liens entre les éléments de bois (fibres, particules, lamelles) créés par la polymérisation de
la résine. La vitesse et l’étendue de la polymérisation de la résine sont donc des éléments
fondamentaux dans la fabrication de panneaux. Les deux types de colle les plus
fréquemment employés dans l’industrie de la fabrication de panneaux sont l’uréeformaldéhyde (UF) et le phénol-formaldéhyde (PF). Le coût plus faible de l’UF fait en
sorte qu’elle est souvent privilégiée dans la pratique. Les deux colles sont des résines
thermodurcissables et leur degré de polymérisation dépend directement de la température à
l’intérieur de l’ébauche. Ainsi, des thermocouples ont communément été employés dès
1959 (Maku et al 1959 ; Strickler 1959) afin de surveiller la montée de la température dans
l’ébauche. Une bonne polymérisation de la colle est nécessaire afin d’éviter la délamination
du panneau. En effet, des zones de haute pression gazeuse se développent au centre du
panneau et, au moment de l’ouverture de la presse, peuvent provoquer la délamination
causant ainsi des pertes indésirables aux fabricants.
Les fibres de bois ne sont jamais parfaitement sèches et ont une teneur en humidité qui
avoisine habituellement 2%. L’adhésif qui est ajouté aux fibres est en solution aqueuse à
pH légèrement basique afin d’empêcher sa polymérisation hâtive. L’ajout de cette eau
augmente la teneur en humidité de l’ébauche de fibres. La teneur en humidité de l’ébauche
se situe typiquement entre 6 et 12% au début du pressage et sa répartition est supposée
uniforme dans l’ébauche. L’énergie fournie par les plateaux de la presse sert également à
évaporer l’eau liée. Ainsi, plus la teneur en humidité de l’ébauche est élevée, plus la montée
en température est retardée dans les zones éloignées de la surface du panneau. La teneur en
humidité influence donc indirectement le processus de polymérisation de la résine. Les
effets de la température et de la teneur en humidité sur l’évolution de la polymérisation ont
3
été étudiés par Humphrey et Ren (1989) et Humphrey (1996) alors qu’Ellis (1995) a fait
une étude sur l’influence du temps de pressage sur la force des liens adhésifs. Ainsi,
l’optimisation du temps de pressage et la qualité de la polymérisation passent par une
meilleure compréhension des phénomènes de transfert de chaleur et de masse d’eau liée et
leur distribution dans l’ébauche durant le processus de pressage (Pizzi 1994; García 2002).
La chaleur fournie par les plateaux de la presse fait en sorte que les couches de surface
atteignent rapidement une température élevée. L’eau liée présente dans ces couches
s’évapore (changement de phase) et la vapeur d’eau est créée. Cela a pour conséquence une
diminution de la teneur en humidité des couches de surface et une augmentation de la
pression de vapeur. Le volume des espaces vides dans l’ébauche au début du pressage est
d’environ 90% et se situe en moyenne aux alentours de 50% à la fin du processus de
pressage. Cette grande proportion de vides facilite le transfert de la vapeur d’eau aussi bien
vers les bords que vers le centre de l’ébauche. En effet, l’évaporation de l’eau liée au
niveau de la surface en contact avec les plaques chauffantes crée une augmentation rapide
de la pression partielle de la vapeur dans cette zone. Il en résulte un gradient de pression
partielle de vapeur entre la surface et le centre de l’ébauche. Cela génère un flux de vapeur
se déplaçant par diffusion moléculaire dans la phase gazeuse (espaces vides dans
l’ébauche). Au début du pressage, ce mouvement se fait principalement vers le plan central,
donc dans la direction perpendiculaire à la surface de l’ébauche. Dans son déplacement, la
vapeur atteint les couches internes dont la température est plus basse ce qui favorise le
processus de condensation. De telle sorte, la teneur en humidité locale près de la surface
diminue et celle des régions plus proches du plan central de l’ébauche s’en trouve
augmentée. Lorsque le front de hautes températures atteindra ces zones, l’eau liée qui y est
présente va s’évaporer progressivement et par le même processus se déplacer vers le plan
central de l’ébauche. Ce déplacement en cascades de l’eau liée se poursuit à travers
l’épaisseur de l’ébauche jusqu’au plan central. C’est ainsi qu’à la fin du pressage on
retrouve les plus grandes concentrations d’humidité dans le plan central du panneau. Une
fois la température de la zone centrale suffisamment élevée, le processus d’évaporation
reprendra et s’accompagnera d’une augmentation substantielle de la pression de gaz dans
l’ébauche et d’une température plateau au centre de l’ébauche. Il en résultera alors un fort
gradient de pression entre le plan central et les bords de l’ébauche provoquant un flux de
gaz qui forcera l’air et la vapeur d’eau à s’échapper par les bords du panneau vers le milieu
ambiant. Cela contribue à la diminution de la teneur en humidité du centre de l’ébauche, à
l’augmentation de la température locale et finalement à la polymérisation de la résine dans
la zone centrale. Une pression excessive, surtout en fin de pressage, peut causer la
délamination du panneau. Afin de surveiller le développement de la pression dans
l’ébauche, des sondes de mesure de pression ont été introduites dès les années 1967
(Denisov et Sosnin 1967 ; Kavvouras 1977, Kamke et Casey 1988 ; Steffen et al 1999). La
précision, l’efficacité et le degré de sophistication des sondes et des thermocouples ont
grandement évolué avec le temps. Le point culminant a été atteint lors de l’introduction du
système de suivi de pressage PressMAN (Alberta Research Council 2003). Chaque sonde
de ce système de pressage peut faire des mesures simultanées de la température et de la
pression du gaz à l’intérieur de l’ébauche au cours du pressage.
Le comportement de l’ébauche (constituée des fibres humides encollées et des espaces
vides) en compression est assez complexe. En effet, en quelques instants, l’ébauche passe
4
d’un amas de fibres sans cohésion et de très faible densité (environ 50 kg / m3 ) à une
ébauche compactée de densité moyenne de 700 kg / m3 (Wang et Winistorfer 2000a, b). La
porosité (la proportion volumique des espaces vides) est très élevée (90%) au début du
pressage et l’ébauche n’offre pratiquement aucune résistance mécanique à la compression.
À mesure que le pressage avance, les espaces vides sont progressivement éliminés et la
porosité diminue pour se stabiliser autour de 50% en moyenne (Bolton et Humphrey 1994).
La répartition de la porosité n’est pas uniforme à travers l’épaisseur. Elle a un
comportement opposé à celui de la densité. Ainsi, dans les couches près des surfaces en
contact avec les plaques de la presse, la densité est très élevée alors que la porosité est
faible. À l’opposé, la zone centrale a la densité la plus faible et la plus grande proportion
d’espaces vides (Bolton et Humphrey 1994). Ces particularités de l’ébauche et les
températures élevées développées durant le pressage posent des défis considérables lorsque
vient le temps de déterminer expérimentalement les paramètres physiques et rhéologiques
de l’ébauche en cours de pressage. Ainsi, dans les simulations numériques, on est souvent
contraint d’utiliser des paramètres obtenus pour le bois ou pour d’autres types de panneaux
(pas nécessairement des panneaux de fibres) à des températures inférieures à celles
rencontrées lors du pressage. Tous ces facteurs combinés influent sur la qualité des résultats
numériques.
Au début, l’ébauche manque de consistance et répond instantanément à la sollicitation
extérieure (la fermeture de la presse). Ce comportement s’apparente à une déformation
élastique. Lorsque l’ébauche est assez compressée et qu’elle a atteint une certaine
consistance, les parois cellulaires des fibres offrent une certaine résistance à la compression
et créent un délai dans la progression de la déformation. La paroi cellulaire est composée de
polymères naturels (lignine, cellulose, hémicellulose) qui sont à l’origine de ce
comportement viscoélastique. La température et la teneur en humidité influencent les
propriétés viscoélastiques de ces polymères (Wolcott et al 1990). Lorsque les contraintes
deviennent trop importantes, les parois cellulaires cèdent. Le comportement non linéaire en
compression des composites à base de bois est attribué à l’effondrement des parois
cellulaires (Wolcott et al 1994). Même si les sollicitations extérieures étaient enlevées à ce
moment, une partie des déformations demeureraient irréversibles (plasticité) à cause de
l’effondrement des parois. Ces phases élastique, viscoélastique et plastique de l’ébauche
sont grandement influencées par la température, la teneur en humidité et le degré de
polymérisation de la résine. Une augmentation de la température et/ou de la teneur en
humidité rend l’ébauche plus souple et donc davantage déformable. À l’opposé, une
diminution aurait un effet durcissant ce qui la rendrait moins sujette à la déformation. Les
variations internes de la température et de la teneur en humidité influencent donc les
propriétés rhéologiques de l’ébauche et le développement du profil vertical de densité qui
poursuit d’ailleurs sa formation même lorsque l’ébauche a atteint son épaisseur finale. La
température élevée, l’humidité et la pression mécanique peuvent orchestrer des
changements dans la structure des polymères de la paroi cellulaire. Ces changements font
en sorte qu’une partie de la déformation visqueuse de la paroi devient irréversible
(déformation plastique) même après le relèvement des contraintes extérieures. En plus de
contribuer aux comportements élastique et viscoélastique, la polymérisation de la résine est
en grande partie à l’origine de la plasticité de l’ébauche en ce sens qu’elle s’oppose à la
récupération de la déformation initiale suite à l’ouverture de la presse. C’est la force des
5
liens mécaniques ou chimiques avec le bois créés lors de la polymérisation de l’adhésif qui
empêche la délamination du panneau.
La formation des liens adhésifs et la distribution de la température et de l’humidité à travers
l’ébauche influencent de manière décisive les propriétés rhéologiques et le comportement
de l’ébauche en compression. Cela a un grand impact sur la densification de différentes
zones de l’ébauche et le développement du profil de densité du panneau. Ces variations
locales de densité ont à leur tour une influence prépondérante sur la porosité locale, la
perméabilité au gaz et la conductivité thermique dans ces régions. C’est ainsi que tous ces
phénomènes physiques sont intimement liés les uns aux autres et s’influencent
mutuellement. Le couplage entre les équations décrivant ces phénomènes est donc
inévitable. De plus, le profil de densité joue un rôle capital dans la détermination des
propriétés mécaniques du produit fini (García 2002). Wang et Winistorfer (2000a, b) et
Wang et al (2001a, b) ont d’ailleurs publié une série d’études où, grâce aux rayons gamma
et trois traceurs placés à des endroits stratégiques dans l’épaisseur de l’ébauche, ils ont pu
observer le développement du profil de densité dans différentes conditions de pressage.
L’appareillage sophistiqué et son coût prohibitif font en sorte que le développement des
modèles mathématiques et des simulations numériques sont des outils incontournables dans
la compréhension des phénomènes physiques qui se déroulent lors du pressage et peuvent
contribuer à améliorer la qualité du produit fini.
Les interactions entre les différents processus physiques qui surviennent pendant le
pressage sont très complexes. Leur modélisation a commencé dans les années 1950 (Kull
1954) et se poursuit aujourd’hui. Les premiers modèles étaient simples et ne tenaient
compte que de certains aspects de nombreux phénomènes physiques impliqués. Le premier
modèle de transfert de chaleur et de masse basé sur les lois de conservation a été présenté
par Humphrey en 1982 dans sa thèse de doctorat (Humphrey 1982). On y tenait compte du
changement de phase, de la convection de vapeur et du transfert de chaleur par conduction
et convection. Le modèle a été développé dans le système de coordonnées cylindriques et
appliqué dans le contexte d’une ébauche circulaire. La discrétisation a été faite par la
méthode des différences finies et les propriétés matérielles ont été mises à jour après
chaque pas de temps. Le développement du profil de densité n’a pas été calculé par le
modèle. On a cependant considéré une fermeture instantanée de la presse et un profil de
densité final prédéfini. Une hypothèse voulant que la phase gazeuse soit uniquement
constituée de la vapeur d’eau a été postulée. Or, cette hypothèse n’est vérifiée que dans la
seconde partie du pressage. Les comparaisons entre les résultats du modèle et les mesures
expérimentales ont été présentées par Bolton et al (1989a). Haselein (1998) a enrichi le
modèle circulaire de Humphrey en y ajoutant une importante composante rhéologique
tenant compte de la compression de l’ébauche et de la relaxation des contraintes. Hubert et
Dai (1998) ont présenté un modèle unidimensionnel pour le pressage de l’OSB (oriented
strandboards, i.e. panneaux de lamelles orientées) qui prenait en compte le transfert de
chaleur par conduction et convection, le changement de phase, la convection de la vapeur,
la polymérisation de la résine et la densification de l’ébauche. Les données empiriques ont
servi au calcul du degré de polymérisation de la résine en fonction de la température et du
temps. Les effets viscoélastiques ont été ignorés alors que l’hypothèse difficilement
réalisable de gradients de pression de vapeur constants dans le plan horizontal a été utilisée
pour estimer la sortie de la vapeur par les bords de l’ébauche.
6
L’importance d’une approche intégrée a été mise en lumière par les travaux des chercheurs
comme Kavvouras (1977) et Bolton et Humphrey (1988). Cette approche suggère de
considérer toutes les variables et leurs interactions simultanément afin de mieux quantifier
l’impact des variations dans les conditions de pressage sur le processus et les propriétés du
produit fini.
Une avancée considérable a été réalisée par Thömen (2000) et Thömen et Humphrey (2003,
2006) qui ont proposé un système d’équations différentielles modélisant le pressage
tridimensionnel aussi bien dans une presse en lot qu’une presse continue. Il s’agit de l’un
des premiers modèles pour le pressage en continu. Le modèle en question incluait le
transfert de chaleur et de masse, les processus de sorption et de changement de phase ainsi
que la densification de l’ébauche et la relaxation des contraintes. Les effets de la
polymérisation de la résine n’ont pas été pris en compte. La discrétisation des équations a
été faite par la méthode des différences finies couplée à un schéma explicite en temps qui
requiert de très petits pas de temps pour assurer la stabilité de la méthode. Les résultats
présentés par Thömen et Humphrey (2006) montrent une bonne ressemblance qualitative
avec les mesures du laboratoire. Du point de vue quantitatif, de la flexibilité et de la
précision des résultats numériques, il reste encore de la place à l’amélioration. Dans sa
thèse de doctorat, Vidal (2006) présente une revue bibliographique assez complète
concernant les modèles de transfert de chaleur et de masse.
Pour le processus de pressage en lot (batch pressing), les travaux de Dai et Yu (2004),
Thömen (2000), Thömen et Humphrey (2003; 2006), Thömen et al (2006), Zombori
(2001), Zombori et al (2003; 2004) constituent le point de départ du travail actuel. Dans le
modèle mathématique, l’ébauche de MDF est considérée comme un mélange homogène de
fibres humides, d’air, de résine et de cire. Notre étude des mécanismes de compression et
de transfert de chaleur et de masse lors du pressage se situe donc au niveau macroscopique.
Les spécificités locales microscopiques telles que la taille des fibres et les dimensions et la
forme des espaces vides dans l’ébauche n’ont pas été considérées. Les propriétés
matérielles telles que la conductivité thermique, la perméabilité aux gaz ou encore la
porosité de l’ébauche ont été exprimées en fonction des valeurs locales de la densité, de la
température, de l’humidité et des pressions du gaz, de l’air et de la vapeur.
Dans cette thèse, nous présentons un modèle tridimensionnel basé sur les principes
fondamentaux de conservation afin de simuler le transfert de chaleur et de masse ainsi que
la densification de panneaux composites à base de bois durant le pressage à chaud. Les
mécanismes considérés sont le transfert d’air et de vapeur dans la phase gazeuse par
convection et diffusion moléculaire, le transfert de chaleur par conduction et convection, les
effets sorptifs ainsi que la polymérisation de la résine, la densification du matériel et le
développement des contraintes internes. Notre modèle de transfert de masse et de chaleur
est composé de trois équations non linéaires fortement couplées. Pour sa part, le modèle
mécanique considère un comportement élastique de l’ébauche en compression et s’exprime
en formulation incrémentale quasi-statique. Pour la discrétisation spatiale du système
d’équations de conservations décrivant les phénomènes physiques durant le pressage, nous
avons employé la méthode des éléments finis. Le code a été implémenté dans le logiciel
MEF++ développé depuis 1995 au GIREF (Groupe Interdisciplinaire de Recherche en
7
Éléments Finis) à l’Université Laval. La polyvalence et la très grande flexibilité du MEF++
ont permis et grandement facilité la stratégie de résolution couplée que nous croyons la
seule à même de représenter fidèlement les interactions intimes entre les mécanismes
rhéologiques et ceux de transfert de masse et de chaleur. À notre avis, cela constitue un pas
de plus vers une meilleure modélisation et compréhension des phénomènes physiques qui
se déroulent dans l’ébauche lors du pressage. Nous avons d’ailleurs comparé nos résultats
numériques aux mesures de température et de pression du gaz prises à l’aide du système
PressMAN lors du pressage en lot de panneaux au laboratoire de pressage du Département
de sciences du bois et de la forêt au Pavillon Gene-H.-Kruger de l’Université Laval. Ces
comparaisons ont montré une concordance très satisfaisante et encourageante entre les
résultats numériques et expérimentaux. Le modèle implémenté permet, en plus de prédire la
température et la pression du gaz, de simuler l’évolution locale de la teneur en humidité,
des pressions partielles de l’air et de la vapeur, de la densité et des contraintes internes.
Cela permet d’observer les mécanismes de transfert de masse et de chaleur ainsi que leur
évolution dans le temps, les changements qui surviennent dans la composition de la phase
gazeuse et le développement du profil de densité.
Les objectifs de cette thèse sont :
1. Développer et tester un modèle de transfert de chaleur et de masse dans le
contexte du pressage à chaud de panneaux MDF. La réalisation de cet
objectif requiert de poser clairement les équations régissant les phénomènes
en jeu en se basant sur les principes physiques de conservation.
2. Coupler le modèle de transfert de chaleur et de masse à un modèle
mécanique de compression de l’ébauche afin de prédire le développement
dynamique du profil de densité.
3. Mieux comprendre l’influence de différents paramètres matériels sur les
phénomènes de transfert de chaleur et de masse. Une étude de sensibilité a
été réalisée en ce sens (Kavazović et al 2010) en se basant sur le modèle
présenté par Thömen et Humphrey 2006.
Ce document est présenté sous forme d’une thèse de publication composée de trois articles
écrits en anglais et répartis dans trois chapitres.
Une étude de sensibilité portant sur le modèle de transfert de chaleur et de masse proposé
par Thömen et Humphrey (2006) est présentée au Chapitre 1. Dans cette étude de
sensibilité, nous avons déterminé l’impact de la variabilité des propriétés matérielles, des
modèles de sorption, des conditions aux limites et de la teneur en humidité initiale sur les
variables d’état et les résultats numériques du modèle mathématique. Le travail a été
présenté dans l’article intitulé “SENSITIVITY STUDY OF A NUMERICAL MODEL OF
HEAT AND MASS TRANSFER INVOLVED DURING THE MEDIUM-DENSITY
FIBERBOARD HOT PRESSING PROCESS” et publié dans Wood and Fiber Science 42(2),
2010, pp.130-149.
Cet article a remporté 2nd place George Marra Award for Excellence in Writing pour
l’année 2010 (Wood and Fiber Science).
Dans le Chapitre 2, nous présentons le développement détaillé d’un modèle mathématique
décrivant le transfert de chaleur et de masse durant le pressage à chaud de panneaux MDF.
8
Il s’agit d’un modèle tridimensionnel couplé et instationnaire qui est basé sur les principes
de conservation de l’énergie, de la masse de l’air et de la masse de la vapeur d’eau. La
validation expérimentale des résultats numériques est également présentée. Ce modèle est
le sujet de l’article intitulé “NUMERICAL MODELING OF THE MEDIUM-DENSITY
FIBERBOARD HOT PRESSING PROCESS. PART 1. COUPLED HEAT AND MASS
TRANSFER MODEL” lequel sera soumis à Wood and Fiber Science.
Le Chapitre 3 contient l’article intitulé “NUMERICAL MODELING OF THE MEDIUMDENSITY FIBERBOARD HOT PRESSING PROCESS. PART 2. COUPLED
MECHANICAL AND HEAT AND MASS TRANSFER MODELS” lequel sera soumis à
Wood and Fiber Science. Nous y présentons le couplage d’un modèle mécanique pour un
matériau élastique vieillissant avec le modèle de transfert de chaleur et de masse développé
au Chapitre 2. Ce modèle global décrit les changements se produisant à l’intérieur du
panneau durant le pressage à chaud. Notamment, le développement du profil de densité qui
est calculé dynamiquement sur une géométrie en mouvement dont la déformation
(compression) est une conséquence de la fermeture de la presse.
Chapitre 1
Étude de sensibilité d’un modèle numérique de transfert
de chaleur et de masse durant le pressage à chaud des
panneaux MDF
Ce chapitre est constitué de l’article intitulé
“SENSITIVITY STUDY OF A NUMERICAL MODEL OF HEAT AND MASS
TRANSFER INVOLVED DURING THE MEDIUM-DENSITY FIBERBOARD HOT
PRESSING PROCESS”
publié en 2010 dans la revue Wood and Fiber Science (Society of Wood Science and
Technology), 42(2), 2010, pp.130-149.
Les auteurs de l’article sont Zanin Kavazović, Jean Deteix, Alain Cloutier et André Fortin.
Cet article a remporté 2nd place George Marra Award for Excellence in Writing pour
l’année 2010 (Wood and Fiber Science).
10
Résumé
L’objectif de ce travail était d’estimer l’impact de la variabilité des propriétés de transfert
de masse et de chaleur de panneaux de fibres de densité moyenne sur les résultats prédits
par un modèle numérique de pressage à chaud. Les trois variables d’état du modèle, soient
la température, la pression de l’air et la pression de la vapeur, dépendent des paramètres qui
caractérisent les propriétés matérielles du panneau qui sont connues avec une précision
limitée. De plus, les différents modèles de sorption et la teneur en humidité initiale ont
également un impact sur les résultats numériques. Dans cette étude de sensibilité, nous
avons déterminé l’impact des variations des propriétés matérielles, des modèles de sorption,
des conditions aux limites et de la teneur initiale en humidité sur les variables d’état. Notre
étude montre d'une part que la conductivité thermique du panneau, le coefficient de
transfert convectif de masse associé à la paroi extérieure ainsi que la perméabilité aux gaz
du panneau ont le plus grand impact sur la température, la pression du gaz et la teneur en
humidité dans le panneau. D'autre part, le coefficient de transfert convectif de chaleur
associé à la paroi extérieure n’a aucun effet sur les variables d’état. Le choix du modèle de
sorption affecte significativement seulement les prédictions de la teneur en humidité dans le
panneau. La teneur en humidité initiale a une très forte influence sur la pression du gaz à
l’intérieur du panneau.
Mots clefs: Étude de sensibilité, pressage à chaud, transfert de masse et de chaleur,
méthode des éléments finis, modèles de sorption, teneur en humidité initiale, propriétés
matériels.
11
Abstract
The objective of this work was to estimate the impact of the variability of the medium
density fiberboard mat heat and moisture transfer properties on the results predicted by a
numerical model of hot pressing. The three state variables of the model, temperature, air
pressure, and vapor pressure, depend on parameters describing the material properties of
the mat known with a limited degree of precision. Moreover, different moisture sorption
models and initial moisture contents also have an impact on the numerically predicted
results. In this sensitivity study, we determined the impact of variations of the mat
properties, sorption models, boundary conditions, and initial MC on the state variables. Our
study shows that mat thermal conductivity, convective mass transfer coefficient of the
external boundary, and gas permeability have the most significant impact on temperature,
gas pressure, and MC within the mat. On the other hand, the convective heat transfer
coefficient of the external boundary has no impact on the state variables. The sorption
model affects significantly mat MC predictions only. The initial MC of the mat has a strong
influence on the internal gas pressure.
Keywords: Sensitivity study, hot pressing, heat and mass transfer, finite element method,
sorption models, initial moisture content, material properties.
12
INTRODUCTION
The hot pressing of medium-density fiberboard (MDF) is a complex process involving
several heat and mass transfer properties of the fiber mat. It has captured the attention of
many researchers over the last few years. A comprehensive literature review can be found
in Bolton and Humphrey (1988). Among the first researchers proposing an integrated
approach were Kavvouras (1977), Humphrey (1982), and Humphrey and Bolton (1989a).
The first multidimensional heat and moisture transfer model was probably proposed and
developed by Humphrey (1982). A series of papers describing the physics involved in the
hot pressing of particleboard and presenting typical predictive results followed (Bolton et al
1989a, 1989b, 1989c; Humphrey and Bolton 1989a). That work is the foundation on which
the comprehensive model proposed by Thömen and Humphrey (2006) was developed.
Different heat and mass transfer models describing the hot pressing process of wood-based
composite panels such as MDF, oriented strandboard, and particleboard have been
proposed (Bolton et al 1989a, 1989b; Humphrey and Bolton 1989a; Carvalho and Costa
1998; Zombori et al 2003; Dai and Yu 2004; Nigro and Storti 2006; Thömen and
Humphrey 2006). Ultimately, all the heat and mass transfer models are based on the mass
conservation of air and water vapor and conservation of heat (Zombori et al 2003; Dai and
Yu 2004; Thömen and Humphrey 2006). To these conservation laws, one can add the cure
kinetics equation of the adhesive system to predict the evolution of resin cure (Loxton et al
2003; Zombori et al 2003). An appropriate model of moisture sorption is also required
(Malmquist 1958; Nelson 1983; Wu 1999; Dai and Yu 2004; Vidal Bastías and Cloutier
2005).
The numerically predicted solutions depend on several heat and mass transfer properties of
the fiber mat. Most of these properties are known to a limited degree of precision,
especially under conditions prevailing during the hot pressing process. Moreover, most of
the material properties are obtained from measurements made on wood or on manufactured
panels (von Haas et al 1998). Furthermore, mats made from fibers of different morphology
will most likely have different properties. We understand that these specifics have an
impact on the precision of the numerical results. To improve the reliability of a
mathematical model as a predictive tool in the development of wood-based composite
products, a better understanding of the influence of the material properties on the
mathematical model results is needed. Therefore, the model sensitivity to the parameters
characterizing heat and mass transfer in the fiber mat must be examined (Zombori et al
2004).
Another important component of every mathematical model of heat and mass transfer
within a composite mat is the moisture sorption model. Several are available in the
literature (Malmquist 1958; Nelson 1983; Wu 1999; Dai and Yu 2004; Vidal Bastías and
Cloutier 2005). Because of their complexity and nonlinearity, it is quite difficult to predict
the impact of the sorption model used on the solution. The initial MC (Minit) of the fiber
mat is also expected to have an influence on the hygrothermal conditions within the mat
during the hot pressing process. Thus, a closer look at those two important components
should also be taken.
13
The objective of this work was to quantify the impact of variations of the fiber mat heat and
mass transfer properties, initial MC, and moisture sorption model on the numerical solution
of the heat and mass transfer model in terms of temperature, gas pressure, and MC. To
achieve this objective, we performed a sensitivity study of the mathematical model to the
mat physical properties and assumed boundary conditions. In this way, we account for the
variability and uncertainty of the material properties and estimate their impact on the
precision of the numerically predicted results. The most influential parameters will thus be
identified. By presenting a deeper and broader insight into the influence of some of the
material properties on the evolution of the internal environment of the fiber mat during the
hot pressing process, this work can be seen as complementary to that of Zombori et al
(2004).
MATERIAL AND METHODS
Material
Refined softwood MDF fibers were obtained from the Uniboard MDF La-Baie plant in
Ville de La-Baie, Quebec, Canada. The fibers were a blend of about 90% black spruce
(Picea mariana) and 10% balsam fir (Abies balsamea). The fibers at 6.5% initial MC were
blended with 12% (fiber oven-dry weight basis) urea–formaldehyde resin and 1% wax in a
laboratory rotary drum blender. The initial MC of the furnish was 12%. A series of six
MDF panels of size 560 mm x 460 mm x 13 mm and target density of 750 kg/m3 at 8%
MC were produced in a Dieffenbacher laboratory batch press equipped with a PressMAN
measurement and control system. The press platens were at 203C. The pressing schedule
of 335 s was divided into five steps. The initial mat thickness of about 182 mm was
reduced to 140% of the final thickness in the first 35 s (Step 1). The press remained in this
position for the next 15 s (Step 2) followed by the second compression lasting 110 s at the
end of which the mat reached its final thickness of 13 mm (Step 3). The hot platens
remained in this position for the next 110 s (Step 4). The final step (Step 5) was the
degassing period (65 s) during which the press was slowly opened and reached 107% of the
final panel thickness at 335s.
Methods
Overall Approach and Assumptions
It was reported by Humphrey and Bolton (1989b) that the size of the board has an effect on
the temperature and gas pressure within the hot pressed mat. In the current study, a single
panel geometry was considered. Therefore, the effect of panel size was not studied. All the
mats were formed using the same raw materials and hot pressed using the same pressing
schedule.
Bound water was assumed to be in equilibrium with water vapor in the lumens and in the
mat voids. Local thermodynamic equilibrium was assumed at every point of the fiber mat
and the relationship among local MC, RH, and temperature was described by the sorption
isotherms considered in this study. Hence, the three state variables of the model are
temperature, air pressure, and vapor pressure. For the numerical study by the finite element
14
method, the physical model used is that proposed by Thömen and Humphrey (2006), and
all of the material properties of the fiber mat were taken from the available literature. None
of the fiber mat material properties was obtained from the panels produced in the
laboratory.
The current study is focused on the heat and moisture transfer phenomena involved in the
hot pressing of the MDF wood fiber mat. The rheology of mat consolidation was not
explicitly considered in this study. Therefore, a predefined time- and space-dependent
oven-dry vertical density profile based on the work of Wang and Winistorfer (2000) (see
APPENDIX 1) was used in the simulations to update the local heat and moisture transfer
properties and porosity of the mat. Consequently, the complex dynamic interactions
between heat and moisture and rheological parameters involved during hot pressing process
were not taken into account. We are aware that this simplification may have an influence on
the model results presented in the current study. A numerical coupling between the
mechanical and the heat and mass transfer models will be presented in future work. The
results presented here should therefore be seen from the perspective of the numerical
methods used and regarded as a numerical study by the finite element method of the
sensitivity of a numerical heat and mass transfer model to some of the key mat properties
and model parameters.
In the present work, we focused on the impact of thermal conductivity, gas permeability,
and convective heat and mass transfer coefficients associated to the boundary conditions on
the solution. Moreover, we examined the impact of the sorption model and the initial MC
of the fiber mat on the results. It is assumed that the initial mat MC is uniform throughout
the thickness. The contribution of resin cure to heat and mass transfer is not taken into
account. All the results were obtained by finite element numerical simulations.
Model of Heat and Mass Transfer in the Fiber Mat
We retained the mathematical model proposed by Thömen and Humphrey (2006). The
model is based on the mass conservation of air and water vapor and conservation of energy.
We restate the original version of this model as follows in terms of the three state variables:
partial air pressure (Pa), partial water vapor pressure (Pv) and temperature (T):
Mass conservation of air





( a )
M
     a K p  a Deff   Pa      a K p  Pv   0
t
RT

 

 

(1)
Mass conservation of water vapor





 ( v  )
M
M
    v K p  Pa       v K p  v Deff   Pv    OD
t
t
RT



 

(2)
15
Energy conservation
 Mat CMat
T
M
 H fg OD
    K T  T   0
t
t
(3)
(see “Nomenclature” and Appendices 1 and 2 for definitions of variables and expressions).
Using the Malmquist’s sorption model (Malmquist 1958; Vidal Bastías and Cloutier 2005),
we can also predict and monitor the evolution of the mat MC with time at any position.
As the moisture content M depends on temperature (T) and partial vapor pressure (Pv), the
M
M M Pv M T
chain rule is applied and the term
is developed as

. This

t
t
Pv t
T t
expression is then substituted into Eqs 2 and 3. The model is thus expressed in terms of the
three state variables: Pa, Pv, and T.
Finite Element Solution Strategy
For each of the three conservation equations (Eqs 1, 2, and 3), a finite element method
discretization is performed in space and the time derivatives are calculated using the Euler
implicit scheme. Each state variable is discretized by Q1 (linear quadrangular) finite
element.
Taking advantage of the symmetry, our computational domain represents a quarter of the
full 2D geometry (see Fig 1.1). Therefore, for the numerical simulation runs, we consider a
rectangular domain in the x–z plane of the following dimensions: 280 mm (half length) by
6.5 mm (half thickness). Figure 1.1 shows details of the 2D geometry and our working
domain. The domain considered for calculation was meshed with a 16 by 16 grid having
256 rectangular elements.
The nonlinear Eqs 1, 2, and 3 are strongly coupled and form a coherent system, which can
be written in the following general form:
 a11
0

 0
0
a 22
a32
 Pa 


  B1 B2
a13   t 
Pv 



a23  
  C1 C2
 t 
 0 0
a33  


T



 t 
0  Pa    Fa 

0    Pv     Fv 
K   T    FT 
(4)
An integrated approach simultaneously considering all important variables during hot
pressing was proposed by Kavvouras (1977), Humphrey (1982), and Bolton and Humphrey
(1988). In the case of a heat and mass transfer model, we achieved it in the following way.
At each time step and for each nonlinear iteration, the three equations forming this system
are solved simultaneously preserving the full coupling between them. At each time step, the
nonlinear system (Eqs 1, 2, and 3) is solved by a fixed point method allowing to predict the
16
evolution of the state variables in space. During each time step, several iterations of a fixed
point method are performed to reach convergence to 10-6 in the residual norm. From one
nonlinear iteration to another, all the local conditions and mat material properties are
updated. This is somewhat different from the approach adopted by Thömen and Humphrey
(2006). Indeed, these authors kept the local conditions and properties constant during a
given time step (Thömen 2000). In our case, we updated values of all parameters for each
nonlinear iteration within each time step.
Given that we use Euler implicit time scheme combined with the finite element method, we
have no constraint on the time step length. However, a too large time step could cause
convergence and precision problems. The results presented in this paper were obtained
using a 0.1-s time step.
a)
b)
Hot platen
Air
Air
Hot platen
6.5 mm
13 mm
Air
Hot platen
z
560 mm
Symmetry axes
280 mm
x
Figure 1.1: a) Full 2D geometry of a fiber mat; b) computational domain in 2D (onequarter of the full geometry).
Boundary Conditions
Appropriate boundary conditions are needed to properly solve Eqs 1, 2, and 3. The
temperature evolution of the surface in contact with the hot platen (Fig 1.1a) was imposed
by a Dirichlet boundary condition based on the data obtained during in situ laboratory
experiments. The surface in contact with the hot platen includes the two end vertices
illustrated by black dots in Fig 1.1b. Moreover, the following fluxes are considered at the
boundaries:
Heat flux : q T =  K T  T
(5)

 M

Air flux : q Pa =   a K p  P    a Deff  Pa 

 
  RT
(6)
17

 M

Vapor flux : q Pv =   v K p  P    v Deff  Pv 

 
  RT
(7)
The hot platen is assumed impervious to gas and therefore q Pa  0 and q Pv  0 . Symmetry
conditions are imposed ( q T  0 , q Pa  0 , q Pv  0 ) on the two symmetry axis illustrated by
dashed lines in Fig 1.1b. On the external edge in contact with the ambient air, the following
convection boundary conditions are imposed for the three state variables: temperature, air
pressure, and vapor pressure, respectively:

q T · n =  h T · ( T  Tamb )



q Pa · n =  h p · a · ( P  Pamb )  105  a · ( Pa  Pa amb )
Pa




q Pv · n =  h p · v · ( P  Pamb )  105  v · ( Pv  Pv amb )

Pv
(8)
(9)
(10)

where n is the outward unit normal vector, and hT and hp are, respectively, the convective
heat and mass transfer coefficients at the edge. In Fig 1.1, the external edge is the righthand side edge of the rectangular domain and is represented by a continuous black line
including the black square (Fig 1.1b). The main mode of mass transfer between the mat and
the environment is the gas bulk flow (Zombori et al 2004) generated by the difference of
total gas pressure within and outside the mat. Diffusion generated by the difference of
partial pressures within and outside the mat plays a secondary role (Zombori et al 2004).
Sensitivity Study
The state variables (Pa, Pv, T) of the heat and mass transfer model depend on many
parameters describing the physical properties of the mat. In our sensitivity study, we
perform “what if” scenarios with regard to variations in the material properties. The
intuitive and simple approach adopted consists of perturbing one parameter, whereas all the
others remain at their reference value. Therefore, the influence of one parameter at a time
on the solution is examined.
The results obtained using reference values of the material properties proposed in the
literature are compared with results obtained with perturbed values of the material
properties. To some extent, the perturbation factors can be seen as uncertainty or
measurement errors on the material property of interest. Hence, a perturbed value of a
material property of interest is obtained by multiplying the reference expression by a given
factor. In this work, the results are presented for the following multiplying factors: 0.5, 0.8,
and 0.9 for a decrease of 50, 20, and 10%, respectively; and 1.1, 1.2, and 1.5 for an increase
of 10, 20, and 50%, respectively. Note that the perturbation coefficients are chosen within a
reasonable range given that the variability of the material parameters and the uncertainty of
the experimental measurements are quite large. Moreover, in their sensitivity study,
Zombori et al (2004) considered constant reference values for parameters of interest.
18
Furthermore, they presented a sensitivity study based on a single perturbation factor: 50%
increase in the parameter reference value.
Comparison of results. The solution obtained with the perturbed value of a parameter is
compared with that obtained using the reference value of the same parameter. The resulting
discrepancy between those solutions can be quantified in different ways. We express it as a
percentage of the maximum relative difference (MaxRelDiff). Therefore, we will monitor
the evolution in time of the maximum relative deviation from the reference solutions. For
instance, Tref refers to the temperature field calculated using the reference values and Tper is
the temperature calculated using a perturbed value of a parameter of interest. Therefore, at
each time step, the following variable is calculated:
MaxRelDiff = 100  sign  Tper  Tref   max

Tper  Tref
Tref
(11)
where MaxRelDiff is the maximum relative difference in percentage depicting the impact
of the variation of a given parameter on the temperature field. After each time step, the
T  Tref
is calculated over the simulation domain  and its maximum
expression per
Tref
value retained. It represents the largest relative discrepancy between the two solutions.
However, it gives no indication on the location of the maximum deviation. The expression
(Tper  Tref) is evaluated at the point corresponding to the largest relative discrepancy
between the two solutions and its sign is taken. The sign of the expression (Tper  Tref)
indicates if the perturbed value of a parameter caused an increase in temperature (when the
sign is positive) or a decrease (when the sign is negative). This generic approach is
systematically used to quantify discrepancies for other variables of interest as well.
Sorption Models
Several sorption models of solid wood are proposed in the literature. We chose some of the
most frequently used sorption models and performed numerical simulations to characterize
the impact of the sorption model on the solution. The following sorption models were
considered.
Malmquist. Vidal Bastías and Cloutier (2005) compared several sorption models and their
study showed that the Malmquist’s sorption model gives the best overall fit to experimental
equilibrium moisture content (EMC) data. Therefore, our reference is Malmquist’s sorption
model (Malmquist 1958; Vidal Bastías and Cloutier 2005) expressing dimensionless
moisture content M as a function of absolute temperature T and dimensionless relative
humidity h:
M Malmquist 
MS
I
 1 3
1  N   1
h 
(12)
19
where MS, N and I are second-order polynomials of the absolute temperature, T, defined as
follows (Vidal Bastías and Cloutier 2005):
MS  0.40221  9.736 105  T  5.8964 107  T 2
N  2.6939  0.018552  T  2.1825 106  T 2
I  2.2885  0.0016742  T  2.0637 106  T 2
(13)
(14)
(15)
Hailwood-Horrobin (HH2). The Hailwood and Horrobin model (Vidal Bastías and
Cloutier 2005) was also considered. We used the two hydrates version of that model
expressing dimensionless moisture content M as a function of absolute temperature T and
dimensionless relative humidity h:
M HH 2 
K1  K  h  2  K1  K 2  K 2  h 2 
18  K  h



M p 1  K  h 1  K1  K  h  K1  K 2  K 2  h 2 
(16)
where the molecular weight of water is 18 g/mol and Mp is the molecular weight of a
polymer unit forming a hydrate expressed in units of g/mol. Polynomial expressions for Mp,
K, K1 and K2 are given as functions of absolute temperature T (Vidal Bastías and Cloutier
2005):
K  0.68405  4.7238 104  T  3.3289 108  T 2
K1  19.641  0.0587818  T  4.05 105  T 2
(17)
(18)
K 2  2.6172  1.6795 103  T  6.414 106  T 2
(19)
4
M p  330.03  2.3468  T  2.8368 10  T
2
(20)
García. García (2002) proposed the following sorption model expressing dimensionless
moisture content M as a function of absolute temperature T and dimensionless relative
humidity h:
M Garcia
1
 B D C
 
     1
 h 



(21)
where B = 1.09603 , C = 2.36069 , D = 1.84447 and
A 
 
 T  A2  4 
  A1 exp  
 
  A3  
with A1 = 0.186575 , A2 = 751.85 , A3 = 1163.31 and A4 = 12.7441.
(22)
20
Nelson. Nelson (1983) proposed a sorption model used later by Wu (1999) and by Dai and
Yu (2004). Dimensionless moisture content M is expressed as a function of absolute
temperature T, dimensionless relative humidity h and two material related parameters
denoted by A and B:
 1 

R T
M Nelson  B 1  ln  2.38846 104 
 ln  h   
Mv
 A 
 
(23)
where Mv is the molar mass of water (0.018015 kg/mol) and R is the universal gas constant
(8.3143 J/mol K). Wu (1999) fitted EMC-RH data for different wood-based products to
Nelson’s sorption model to estimate the two material related parameters, A [dimensionless]
and B [dimensionless]. For MDF, Wu (1999) found that during sorption, B=0.1913 and A
= 4.68, whereas during desorption, B=0.2494 and A =4.94. A difficulty arising in using Eq
23 is the appropriate choice of parameters A and B. Indeed, during hot pressing, we can be
in sorption at a given location within the mat and in desorption at another location.
Therefore, based on values proposed by Wu (1999), we performed simulation runs with
Nelson’s model using mean values for B and A, hence, we used B=0.22035 and A = 4.81.
Initial Moisture Content of the Mat
Because the initial moisture content of the fiber mat (Minit) is expected to have an influence
on the internal conditions of the mat, its impact on the results was studied. We have chosen
several values for Minit to reflect conditions normally encountered in practice. Indeed, tests
were made for the following dimensionless values of Minit: 0.08, 0.10, 0.12, and 0.14. We
selected 0.12 as a reference value for the initial moisture content.
Thermal Conductivity of the Mat (KT)
We used the expression suggested by Thömen and Humphrey (2006) as a reference value
for the thermal conductivity of the fiber mat: KTxy = 1.5·KTz where
and
KTz  KT 030  KT
(24)
KT 030  4.38 10-2  4.63 10-5  OD  4.86 10-8   2OD
(25)
KT  0.49 ·M  1.1·104  4.3·103 ·M  · T  303.15 
(26)
The variables KTz and KTxy represent, respectively, the thermal conductivity in the thickness
and horizontal directions. KT030 is the thermal conductivity measured at 0% M and
30C and KT is the correction term accounting for moisture content and temperature
effects on thermal conductivity. The tensor of thermal conductivity KT is therefore given in
2D by
K
KT   Txy
 0
0 
KTz 
(27)
21
To characterize the impact of variations in thermal conductivity, simulations were
performed with  KT where the perturbation factor  took the values of 0.5, 0.8, 0.9, 1.1,
1.2, and 1.5.
Specific Gas Permeability of the Mat (Kp)
Analytical expressions for the specific gas permeability of MDF mats based on the curve
fitting of experimental data can be found in García and Cloutier (2005) and also in von
Haas et al (1998). The expression proposed by García and Cloutier (2005) is valid for MDF
mats having a density of 400 to 1150 kg/ m3, whereas in von Haas et al (1998), the
permeability of fiber, particle, and strand mats with densities varying from 200 to 1200 kg/
m3 was determined. The samples used by von Haas et al (1998) were prepared from
consolidated panels with an adhesive content of 11%. In our study, the reference expression
and the input data for the specific gas permeability of the MDF mats will be based on
expressions proposed by von Haas et al (1998). Hence, the in-plane permeability (Kpxy) and
the cross-sectional permeability (Kpz) of MDF fiber mats are both described by the
following expression:
1
exp  
 A
(28)
where
A = a + b   Mat +
c
ln( Mat )
(29)
and the coefficients to determine Kpxy are a =  0.041, b = 9.5110-6 , c =  0.015 and
those for Kpz are a =  0.037, b = 1.1 10-5 , c =  0.037.
The tensor Kp of the specific gas permeability of the MDF fiber mat is therefore given in
2D by
 K pxy
Kp  
 0
0 
K pz 
(30)
To examine the influence of variations in specific gas permeability, simulations were run
with  K p where the perturbation factor  took the values previously listed.
Convective Heat Transfer Coefficient on the External Boundary (hT)
The sensitivity of the system’s solution to variations of the convective heat transfer
coefficient associated with the external boundary was also examined. The reference value
for this parameter is hT = 0.35 and is taken from Zombori (2001) and Vidal Bastías (2006).
When simulations are run with a perturbed value of hT, the heat flux at the edge in contact
with the surroundings becomes:
22

q T  n =    h T · ( T  Tamb )
(31)
where the perturbation factor  is taking the same values as previously.
Convective Mass Transfer Coefficient on the External Boundary (hp)
The convective mass transfer coefficient associated with the external boundary represents
the fiber mat boundary gas transport properties and depicts the resistance to gas flow out of
the mat. We examine the impact of this external bulk flow coefficient associated with the
boundary condition imposed on the edge in contact with the ambient air. A reference value
for this coefficient is hp = 10-11, which is somewhat different from that used by Zombori et
al (2004) for flakeboards. Given that the contribution of diffusion to mass transport out of
the mat is not significant (Zombori et al 2004), variations in hp will affect the air and vapor
fluxes at the edge in contact with the surroundings. Therefore, the simulations were run
with the following conditions at the external edge:





q Pa  n =     h p · a ·(P  Pamb )  105  a ·(Pa  Pa amb ) 

Pa


(32)





q Pv  n =     h p · v ·(P  Pamb )  105  v ·(Pv  Pv amb ) 

Pv


(33)
where a perturbation factor  is taking the same values as previously.
23
RESULTS AND DISCUSSION
Temperature and gas pressure were measured during the pressing process at the center of
the panel plane at three points across mat thickness: the core, one-quarter of the thickness,
and the surface. The temperature measurements are presented, together with numerically
predicted results, in Fig 1.2a. The total gas pressure curves are shown in Fig 1.2b. The
model captures major trends and gives results of comparable quality to those of Zombori et
al (2004) and Thömen and Humphrey (2006). It should be kept in mind that the numerical
model used here is based solely on heat and mass transfer mechanisms and that the
influence of the changing moisture content and temperature on rheological mechanisms
was not considered. Moreover, the fiber mat material properties, including thermal
conductivity, gas permeability, and porosity, were taken from the literature and not
determined from the specific material used to make the panels in the laboratory. This can
explain the discrepancies between the model and experimental results shown in Fig 1.2.
a)
200
Temperature ( C )
175
150
125
100
CoreLab
QuarterLab
CoreModel
QuarterModel
SurfaceLab
75
50
25
0
0
50
100
150
200
Time (s)
250
300
24
b)
190
180
CoreModel
170
CoreLab
P (kPa )
160
QuarterLab
150
140
130
120
110
100
0
50
100
150
200
250
300
Time (s)
Figure 1.2: a) Temperature evolution in time: measured and numerically predicted results.
Curve labelled SurfaceLab is the temperature measured in laboratory at the surface in
contact with the hot platen and was imposed as a Dirichlet boundary condition for T at the
surface. On the other hand, curves labelled CoreModel and QuarterModel are obtained by
numerical simulation and represent the temperature at the center and at one quarter of the
thickness, respectively.
b) Total gas pressure evolution in time: measured and numerical results. Curve labelled
CoreModel is obtained by numerical simulation and the other two are measured in
laboratory. 1
Effect of Sorption Models
Figure 1.3 presents the comparison of the results for P at the core, M at the core, and M at
one-quarter of the mat thickness obtained using each one of the four sorption models
considered. Figure 1.4 shows the evolution in time of MaxRelDiff for T, M, and P.
As can be seen in Fig 1.4, the Hailwood-Horrobin two-hydrate sorption model produces
closer results to the reference sorption model (Malmquist). On the other hand, the Nelson
(1983) model with averaged coefficients for MDF (Wu 1999) gives the results that are the
1
Nota: In all figures presented in this document, special symbols like □ , ○,* , ◊, , etc are
used in order to distinguish different curves from each other and they do not represent
experimental data, unless the contrary is explicitly indicated.
25
most different from those obtained using Malmquist’s model. It should not be forgotten that
the MaxRelDiff corresponds to the largest discrepancy in the domain considered (Eq 11).
Temperature and internal gas pressure do not seem to be significantly influenced by the
sorption model used (Figs 1.3 and 1.4). The effect of the sorption model on the moisture
content evolution is more significant. This was expected because different sorption models
describe differently the EMC–RH–T relationship.
a)
191 90
181 80
Total Pressure (kPa)
171 70
161 60
Malmquist
HH2
Garcia
NelsonMean
50
40
141
30
131
20
121
10
111 0
Est
Ouest
Nord
151
101
0
1er 2e trim.3e trim.4e trim.
50
100
150
200
250
trim.
Time (s)
300
350
26
b)
Moisture Content (%)
18
90
80
70
16
60
50
14
40
30
20
12
10
0
Est
Ouest
Nord
Malmquist
HH2
Garcia
NelsonMean
10
0
1er 2e trim.3e trim.4e trim.
50
100
150
200
250
trim.
300
350
Time (s)
c)
Moisture Content (%)
18
90
16
80
70
14
60
12
50
40
10
30
8
20
10
6
0
4
0
Est
Ouest
Nord
Malmquist
HH2
Garcia
NelsonMean
1er 2e trim.3e trim.4e trim.
100
150
200
250
trim.50
Time (s)
300
350
Figure 1.3 : Solutions obtained with different sorption models for: a) total gas pressure P
at the core; b) moisture content M at the core; c) moisture content M at a quarter of the
mat thickness.
27
MaxRelDiff (%)
a)
1
90
0.8
80
0.6
70
0.4
60
0.2
50
0
40
-0.2
300
20
-0.4
-0.6
10
-0.80
-1
50
HH2
100
Garcia
150
200
250
Est
300Ouest
350
Nord
NelsonMean
1er 2e trim.3e trim.4e trim.
Time (s)
trim.
MaxRelDiff (%)
b)
30
90
20
80
10
700
0
-10
60
-20
50
-30
40
-40
-50
30
-60
20
-70
10
-80
-90
0
-100
-110
50
100
150
200
250
300
350
Est
Ouest
Nord
HH2
Garcia
1er 2e trim.3e trim.4e trim.
trim.
Time (s)
NelsonMean
28
c)
MaxRelDiff (%)
4
90
380
70
2
60
50
1
40
030
0
20
-1
10
-2 0
-3
HH2
50
100
150
200
Garcia
250
NelsonMean
Est
Ouest
300 Nord350
1er 2e trim.3e trim.4e trim.
trim.
Time (s)
Figure 1.4 : Effect of sorption models on: a) temperature T; b) moisture content M; c) total
gas pressure P. Graphs present the maximum relative deviation of results obtained for each
sorption model when compared to the reference solutions obtained with the Malmquist’s
sorption model.
Effect of the Initial Moisture Content on the Predicted Results
Figure 1.5 summarizes the results obtained for P and M using each one of the four values of
Minit. Figure 1.6 shows the evolution in time of the MaxRelDiff (Eq 11) for T, M, and P.
From Fig 1.6a, one could conclude that Minit does not seem to have a large impact on the
evolution of temperature within the fiber mat. However, the evolution of moisture content
within the mat is strongly dependent on Minit of the fiber mat (Fig 1.6b), which was
expected and observed experimentally by Zombori et al (2004). Given that the evolution of
the temperature field is very similar for different values of the initial moisture content
considered, the amount of bound water desorbed should therefore be higher for higher
values of Minit. Hence, the internal gas pressure consequently increases within the mat, as
illustrated in Fig 1.5a. This is in agreement with observations made by Zombori et al (2004)
claiming that the total pressure increases with increasing moisture content. Therefore, these
results confirm that lowering the initial moisture content of the mat results in lower gas
pressure within the mat during the hot pressing process.
29
a)
Total Pressure (kPa)
191
90
181
80
171
70
161
60
50
151
40
141
30
131
20
121
10
111
0
Minit=8%
Minit=10%
Minit=12%
Minit=14%
101
0
Est
Ouest
Nord
1er 2e trim.3e trim.4e trim.
100
150
200
250
trim.50
Time (s)
300
350
b)
22
90
80
18
70
16
60
14
50
12
40
10
30
8
20
6
10
4
0
2
Moisture Content (%)
20
0
0
Est
Ouest
Nord
Minit=8%
Minit=10%
Minit=12%
Minit=14%
1er 2e trim.3e trim.4e trim.
50
100
150
200
250
trim.
Time (s)
300
350
30
c)
22
90
80
18
70
16
60
14
50
12
40
10
30
8
20
6
10
4
0
2
Moisture Content (%)
20
0
0
Minit=8%
Minit=10%
Minit=12%
Minit=14%
Est
Ouest
Nord
1er 2e trim.3e trim.4e trim.
trim.
50
100
150
200
250
300
350
Time (s)
Figure 1.5 : Solutions obtained with different values on initial moisture content Minit of the
fiber mat for: a) total gas pressure P at the core; b) moisture content M at the core; c)
moisture content M at a quarter of the mat thickness.
31
a)
MaxRelDiff (%)
2.5
90
80
2
70
1.5
60
1
50
40
0.5
30
0
200
-0.5
10
-1 0
-1.5
Minit=8%
Minit=10%
Minit=14%
Est
Ouest
Nord
50
100
150
200
250
300
350
1er 2e trim.3e trim.4e trim.
trim.
Time (s)
MaxRelDiff (%)
b)
90
30
80
20
70
10
60
0
50
0
40
-10
30
-20
20
-30
10
-400
-50
Minit=8%
50
100
150
Minit=10%
200
250
1er 2e trim.3e trim.4e trim.
trim.
Time (s)
Minit=14%
300
Est 350
Ouest
Nord
32
MaxRelDiff (%)
c)
906
804
702
600 0
-2
50
-4
40
-6
30
-8
20
-10
10
-12
0
-14
-16
50
100
150
200
250
300
350
Est
Ouest
Nord
Minit=8%
Minit=10%
Minit=14%
1er 2e trim.3e trim.4e trim.
trim.
Time (s)
Figure 1.6 : Effect of initial moisture content Minit of the mat on: a) temperature T; b)
moisture content M; c) total gas pressure P. Graphs present the maximum relative
deviation of results obtained for different values of Minit from the reference solutions
obtained with Minit = 12%.
Effect of Thermal Conductivity (KT)
Figure 1.7 presents the impact of variations in thermal conductivity (KT) on temperature,
moisture content, and total pressure. When the thermal conductivity (KT) of the mat is
decreased, heat is conducted more slowly through the mat and its internal temperature
remains lower (Fig 1.7a). Hence, less bound water is desorbed resulting in a lower internal
gas pressure. Of course, local moisture content will decrease more slowly as well.
Conversely, if KT is increased, heat is conducted more quickly through the mat and the local
temperature increases (Fig 1.7a) causing the desorption of more bound water. This lowers
local moisture content and more water vapor is produced, increasing internal gas pressure.
As can be seen in Figs 1.7b and 1.7c, KT has a very significant impact on mass transfer in
the mat during the hot pressing process. Indeed, on average, variations in KT have an effect
on gas pressure in the mat almost five times greater than on temperature (Fig 1.7c). A
similar effect can be observed on moisture content (Fig 1.7b). Moreover, variations in
moisture content seem to be more or less linearly related to variations in thermal
conductivity in the sense that, for instance, a perturbation of 10 or 20% in KT of the mat
will, respectively, induce a 10 or 20% variation in moisture content. Indeed, variations in
33
thermal conductivity seem to affect mainly moisture content and total gas pressure.
Zombori et al (2004) also concluded that variations in KT have the most significant effect
on the system.
a)
6
MaxRelDiff (%)
90
4 80
70
2
60
50
0
0 40
-2 30
20
-4
10
0
-6
-8
50
100
150
200
250
0.5*KT
1er 2e trim.3e trim.4e1.5*KT
trim.
trim.
Time (s)
300
Est
350
Ouest
Nord
0.8*KT
1.2*KT
0.9*KT
1.1*KT
b)
MaxRelDiff (%)
100
90
80 80
70
60
60
40
50
20 40
30
0
020
-20 10
0
-40
-60
0.5*KT
0.8*KT
1.2*KT
1.5*KT
0.9*KT
1.1*KT
Est
Ouest
Nord
50
100
150
200
250
1er 2e trim.3e trim.4e trim.
trim.
Time (s)
300
350
34
c)
MaxRelDiff (%)
30
90
80
20
70
10
60
50
0
400
-10
30
20
-20
10
-30 0
-40
50
100
0.5*KT
0.8*KT
1.2*KT
1.5*KT
150
200
250
300
Est
350
Ouest
Nord
0.9*KT 2e
1.1*KT
1er
trim.3e trim.4e trim.
trim.
Time (s)
Figure 1.7 : Effect of thermal conductivity KT on: a) temperature T; b) moisture content M;
c) total gas pressure P. Graphs present the maximum relative deviation of results obtained
for different values of KT from the reference solutions obtained with the reference value of
KT.
Effect of the Specific Gas Permeability of the Mat (Kp)
Numerical simulations were run with perturbed values of Kp. The results are summarized in
Fig 1.8 as a percentage of the maximum relative difference (Eq 11). As expected, Fig 1.8
suggests that gas permeability is a significant factor affecting mostly mass transfer within
the mat. The sensitivity of the system to Kp and its influence on bulk flow within the mat
was recognized by Zombori et al (2004). Indeed, moisture content and internal gas pressure
seem to be about 10 times more sensitive than temperature to variations of Kp. It can also
be noticed that a decrease in Kp appears to have a more pronounced effect on the solution
than the proportional increase of Kp.
The higher the gas permeability, the easier the gas escapes the mat lowering total internal
gas pressure. Thus, high gas permeability creates conditions that facilitate bound water
desorption, which decreases mat moisture content. Bound water desorption and evaporation
require a certain amount of energy (heat of sorption and latent heat of vaporization) that
will cause a decrease in local temperature. On the other hand, when gas permeability
decreases, the local gas pressure increases. This can result in water vapor condensation and
an increase of the local moisture content. Water vapor condensation and adsorption are
35
exothermic processes that release the latent heat of vaporization and the heat of sorption.
This input of thermal energy increases the local temperature within the mat.
a)
MaxRelDiff (%)
0.6
90
0.580
70
0.4
60
0.3
50
0.240
30
0.1
20
0
10
0
-0.1 0
-0.2
0.5*Kp
0.8*Kp
1.2*Kp
1.5*Kp
0.9*Kp
1.1*Kp
Est
Ouest
Nord
50
100
150
200
250
300
350
1er 2e trim.3e trim.4e trim.
trim.
Time (s)
b)
MaxRelDiff (%)
6
90
5 80
70
4
60
3
50
2 40
30
1
20
0 10
0
0
-1
-2
0.5*Kp
0.8*Kp
1.2*Kp
1.5*Kp
0.9*Kp
1.1*Kp
Est
Ouest
Nord
50
100
150
200
250
1er 2e trim.3e trim.4e trim.
trim.
Time (s)
300
350
36
c)
MaxRelDiff (%)
8
90
7
80
6
70
5
60
4
50
3
40
2
30
1
20
0
10
0
-1
0
-2
-3
0.5*Kp
0.8*Kp
1.2*Kp
1.5*Kp
0.9*Kp
1.1*Kp
Est
Ouest
Nord
50
100
150
200
250
300
350
1er 2e trim.3e trim.4e trim.
trim.
Time (s)
Figure 1.8 : Influence of gas permeability Kp on: a) temperature T; b) moisture content M;
c) total gas pressure P. Graphs present the maximum relative deviation of results obtained
for different values of Kp from the reference solutions obtained with the reference value of
Kp.
37
Effect of the Convective Heat Transfer Coefficient on the External Boundary (hT)
Figure 1.9 depicts the sensitivity of the system’s solution to variations of hT. We concur
with Zombori et al (2004) who found that hT does not have a significant influence on heat
and mass transfer phenomena within the mat. Indeed, the results presented in Fig 1.9
suggest that the convective energy transfer through the interface between the mat and the
ambient air is not a significant factor.
a)
0.3
90
MaxRelDiff (%)
0.25
80
70
0.2
60
0.15
50
0.1
40
0.05
30
0
20
100
-0.05
0
-0.1
-0.15
0.5*hT
0.8*hT
1.2*hT
2*hT
Est
Ouest
Nord
50
100
150
200
250
1er 2e trim.3e trim.4e trim.
trim.
Time (s)
300
350
38
b)
1.5
MaxRelDiff (%)
90
1 80
70
0.5 60
50
0
40
0
-0.5 30
20
-1
10
-1.5 0
-2
50
100
0.5*hT
0.8*hT
1.2*hT
1er
trim.
2e
trim.
2*hT
150
200
250
Est
300
350
Ouest
Nord
3e
4e
trim.
trim.
Time (s)
c)
M axR elD iff (% )
0.05
90
0.04
80
70
0.03
60
0.02
50
40
0.01
30
0.00
200
-0.01
10
-0.020
-0.03
0.5*hT
0.8*hT
1.2*hT
2*hT
Est
Ouest
Nord
50
100
150
200
250
300
350
1er 2e trim.3e trim.4e trim.
trim.
Time (s)
Figure 1.9 : Influence of external heat transfer coefficient hT on: a) temperature T; b)
moisture content M; c) total gas pressure P. Graphs present the maximum relative
deviation of results obtained for different values of hT from the reference solutions
obtained with the reference value hT = 0.35.
39
Effect of the Convective Mass Transfer Coefficient on the External Boundary (hp)
The convective mass transfer coefficient associated with the external boundary depicts the
resistance to gas flow out of the mat. Figure 1.10 summarizes the impact of hp on T, M, and
P. One observes that this coefficient has a very significant impact, especially on mass
transfer within the mat. This is expressed by the significant impact of hp on M and P.
Indeed, these two variables seem to be 10 times more sensitive than temperature to
variations of hp. When the external mass transfer coefficient hp decreases, the resistance to
gas flow out of the mat increases. Hence, the gas remains trapped within the mat,
increasing the internal gas pressure. Water vapor can condense, increasing mat moisture
content. Water vapor condensation and adsorption in the wood fibers releases thermal
energy (latent heat of vaporization and heat of sorption), slightly increasing the local
temperature. If hp increases, the opposite happens. The gas can leave the mat more easily,
decreasing the internal gas pressure and temperature. A lower gas pressure eases the bound
water desorption process that eventually decreases local moisture content. The system
seems to react more strongly to variations of hp than to variations of Kp. Among the
parameters we studied, hp is the second most influential after KT. It should be noticed that in
their study on the influence of external bulk flow coefficient for flakeboard, Zombori et al
(2004) concluded that this parameter does not noticeably influence the results. This
difference may be explained by the higher porosity and gas permeability of MDF compared
with flakeboard. It is therefore plausible that variations in hp have a larger impact for MDF.
a)
MaxRelDiff (%)
2
90
80
1.5
70
60
1
50
0.540
30
020
0
10
-0.5 0
-1
0.5*hp
0.8*hp
1.2*hp
1.5*hp
0.9*hp
1.1*hp
Est
Ouest
Nord
50
100
150
200
250
1er 2e trim.3e trim.4e trim.
trim.
Time (s)
300
350
40
b)
MaxRelDiff (%)
20
90
80
15
70
60
10
50
405
30
0
20 0
10
-5
0
0.5*hp
0.8*hp
1.2*hp
1.5*hp
0.9*hp
1.1*hp
Est
Ouest
Nord
50
100
150
200
250
300
350
1er 2e trim.3e trim.4e trim.
trim.
-10
Time (s)
MaxRelDiff (%)
c)
25
90
80
20
70
15
60
50
10
40
305
200
10
-5
0
-10
0.5*hp
0.8*hp
1.2*hp
1.5*hp
0.9*hp
1.1*hp
Est
Ouest
Nord
0
100
200
300
1er 2e trim.3e trim.4e trim.
trim.
Time (s)
Figure 1.10 : Influence of external convective mass transfer coefficient hp on: a)
temperature T; b) moisture content M; c) total gas pressure P. Graphs present the
maximum relative deviation of results obtained for different values of hp from the
reference solutions obtained with the reference value of hp = 10-11.
41
90
80
70
60
50
40
30
20
10
0
Est
Ouest
Nord
1er 2e trim.3e trim.4e trim.
trim.
Figure 1.11 : Evolution of a space and time dependent predefined oven-dry density profile
used in calculations: density profile in thickness direction at different moments in time.
42
CONCLUSIONS
The coupled nature, complexity, and high nonlinearity of the equations constituting the
model studied here make it difficult to predict the impact of uncertainties of the input
variables on numerical solutions of the model. To gain insight of the influence of different
parameters on the system’s behavior, a sensitivity study of a numerical model of heat and
mass transfer in the MDF mat during hot pressing was performed. Our study suggests that
among the tested material properties, those having the most pronounced effect on heat and
mass transfer within the mat during the hot pressing process are the thermal conductivity of
the mat and the convective mass transfer coefficient associated with the edge in contact
with the ambient air. Given that the latter coefficient plays a very important role in the
quality of the results produced by the model, significant efforts should be made to get
accurate measurements of this coefficient. Indeed, a variation of 20% of the reference value
of hp produces relative discrepancies up to 5% in moisture content and gas pressure results.
The same variation of the reference value of gas permeability creates discrepancies in
moisture content and total gas pressure that are lower than 2%. A variation of 50% of hp
leads to a relative difference of up to 15% in moisture content and up to 22% in gas
pressure relative deviation. A similar perturbation of gas permeability induces variations in
moisture content and gas pressure no greater than 5 and 7%, respectively. Given that the
convective heat transfer coefficient does not seem to be an influential factor, accurate
measurements of thermal conductivity, gas permeability, and convective mass transfer
coefficient associated with the edge in contact with the ambient air are needed to improve
the quality and reliability of model predictions. Also, the choice of an appropriate sorption
model should be addressed with caution because of its impact on the numerically predicted
values of moisture content. Finally, lowering the initial moisture content of fiber mats
contributes to lower internal gas pressure and helps achieve drier conditions within the mat.
43
NOMENCLATURE
t : time [s]
x : length [m]
y : width [m]
z : thickness [m]
T : temperature field [K] ; a state variable calculated by the model
Pa : partial air pressure [Pa] ; a state variable calculated by the model
Pv : partial vapor pressure [Pa] ; a state variable calculated by the model
P : total gas pressure [Pa]
M : moisture content [dimensionless]
h : relative humidity [dimensionless]
PvSAT : saturated vapor pressure [Pa]
Ma : molar mass of air [kg/mol]
Mv : molar mass of water vapor [kg/mol]
R : universal gas constant [J/(mol·K)]
 a : density of air [kg/m3]
v : density of water vapor [kg/m3]
 OD : oven-dry density of the mat [kg/m3] (see APPENDIX 1)
Φ : porosity of the mat [dimensionless]
 Mat : wet density of the mat [kg/m3]
KT : thermal conductivity tensor [J/(m· s·K)]
Kp : tensor of specific (effective) gas permeability of the mat [m3/m]
Deff : tensor of effective diffusion coefficient [m2/s]
Dva : binary molecular diffusion coefficient of the air-vapor gas mixture [m2/s]
kd : obstruction factor [dimensionless]
Hfg : latent heat of vaporization (desorption + evaporation) of bound water [J/kg]
CMat : mass specific heat capacity of the mat at current moisture content [J/(kg·K)]
 : dynamic viscosity of the air-vapor mixture [Pa·s]
a : dynamic viscosity of the air [Pa·s]
v : dynamic viscosity of the water vapor [Pa·s]
hT : convective heat transfer coefficient associated to the external boundary [J/(m2 · s· K)]
hp : convective mass transfer coefficient associated to the external boundary [m]
q T : heat flux [J/(m2· s)]
q Pa : air flux [kg/(m2· s)]
q Pv : water vapor flux [kg/(m2· s)]
EMC : equilibrium moisture content
RH : relative humidity
Minit : initial moisture content of the mat [dimensionless]
Tinit : initial temperature of the mat [K]
hinit : initial value of relative humidity [dimensionless]
44
PvSAT init: initial value of saturated vapor pressure [Pa]
Pv init : initial value of partial vapor pressure [Pa]
Pa init : initial value of partial air pressure [Pa]
Tsurface : temperature at the surface in contact with the hot platen [K]
hamb : relative humidity of ambient gas [dimensionless]
Tamb : temperature of the ambient gas [K]
PvSAT amb: saturated vapor pressure in ambient gas [Pa]
Pamb : ambient gas pressure [Pa]
Pv amb : ambient vapor pressure [Pa]
Pa amb : ambient air pressure [Pa]
45
APPENDIX 1
A predefined mat oven-dry vertical density profile (  OD [kg/m3]) was used for
calculations. This profile is space and time dependent and is based on the results presented
by Wang and Winistorfer (2000). The mathematic representation of the density profile is
given by the expression presented subsequently. For the sake of clarity, this expression is
graphically presented in Fig 1.11 to illustrate the density profile as a function of time and
space (position in the thickness direction). Because the symmetry of the vertical density
profile is assumed, its evolution is only presented for half of the mat thickness. In the
mathematical expression of the profile, “t” represents time and “z” represents position in
the thickness direction. Furthermore, in the thickness (“z”) direction, the density profile is
divided into four sections. In each section, the density profile is expressed by a different
function. Of course, the overall continuity of the density profile is ensured by the way the
four functions are constructed. These functions are defined as follows:
LD = Low-Density Section,
MD = Medium-Density Section,
HD1 = First Part of the High-Density Section,
HD2 = Second Part of the High-Density Section.
OD
 LD

0  z  0.00455


0.00455 < z  0.00585 
 MD


 HD1 0.00585  z  0.006175
 HD 2 0.006175 < z  0.0065 
where
5.736968  t+99.206112
0  t  35 



300
35 < t  50 

LD  

300+(t-50)  (0.909091+99.9001  z ) 50 < t  160 

400.00001+10989.011  z
160 < t 

 BPHD  TPLD 
MD  TPLD   z  0.00455  
 0.00585  0.00455


0  t

0  t  35 
 5.736968  t+99.206112

300
35 < t  50 

TPLD  

231.8181772+1.363636455  t 50 < t  160 

450.00001
160 < t 
46
5.736968  t+99.206112 0  t  35 

300
35 < t  50 

BPHD  

100+4  t
50 < t  160 


740
160 < t 
5.736968  t+99.206112

0  t  35 


300
35 < t  50 

HD1  

300+(t-50)  (-12.36363636+2797.202797  z ) 50 < t  160 

-1060+3.076923077 105  z
160 < t 
5.736968  t+99.206112

0  t  35 


300
35 < t  50 

HD 2  

300+(t-50)  (22.18181827-2797.202797  z ) 50 < t  160 

2740.00001- 3.076923077 105  z
160 < t 
47
APPENDIX 2
Expressions of some parameters used in the calculations.
P = Pa + Pv : Dalton’s law
M : defined at every point in the mat by a sorption model, we use Malmquist’s sorption
model as a reference
MS
M Malmquist 
I
1 
1  N   1 3
h 
where MS, N and I are second order polynomials of absolute temperature T [K] given by
MS  0.40221  9.736 105  T  5.8964 107  T 2
N  2.6939  0.018552  T  2.1825 106  T 2
I  2.2885  0.0016742  T  2.0637 106  T 2
h
Pv
PvSAT
6516.3


PvSAT  exp 53.421 
 4.125  ln T   (Kirchoff’s formula)
T


Ma = 28.951·10-3
Mv = 18.015·10-3
R = 8.314147
M a Pa
(ideal gas law)
RT
M P
v = v v (ideal gas law)
RT
a =
 =1–
OD
(Siau 1984)
1530
 Mat =  OD (1+M)
D
Deff  va I , where I is identity tensor
kd
1.75
 101325   T 
Dva  2.6  105  


 P   298.2 
k  0.334  e A , A  5.08 103  
d
Mat
H fg  2.511 10  2.48 10  T  273.15  1.172 106  e 0.15M 100
6
CMat 
3
1131  4.19  T  273.15  4190  M
1 M
48
Pa
P
 a  v v
P
P
1.37 106  T 1.5
a 
T  85.75
1.12 105  T 1.5
v 
T  2937.85
hT = 0.35
hp = 10-11
Minit = 0.12
Tinit = 298.15
hinit calculated by Malmquist’s formula

3
 1  MSinit

1
 1 
 1 

hinit
 
 N init  M init
I init
where
MSinit  0.40221  9.736 105  Tinit  5.8964 107  Tinit 2
N init  2.6939  0.018552  Tinit  2.1825 106  Tinit 2
I init  2.2885  0.0016742  Tinit  2.0637 106  Tinit 2


6516.3
PvSAT init  exp 53.421 
 4.125  ln Tinit  
Tinit


Pvinit  hinit  PvSAT init
Painit  101325  Pvinit
Tsurface : temperature at the surface in contact with the hot platen; its evolution in time is
imposed by the Dirichlet boundary condition and the values are prescribed by measured
experimental data (see Figure 1.1)
hamb = 0.3
Tamb = 298.15


6516.3
PvSAT abm  exp 53.421 
 4.125  ln Tamb  
Tamb


Pamb = 101325
Pv amb = hamb  PvSAT amb
Pa amb = Pamb  Pv amb
49
ACKNOWLEDGMENTS
The authors wish to thank the Natural Sciences and Engineering Research Council of
Canada (NSERC), FPInnovations – Forintek Division, Uniboard Canada and Boa-Franc for
funding of this project under the NSERC Strategic Grants program.
Chapitre 2
Modélisation numérique du processus de pressage à
chaud des panneaux MDF
Modèle couplé de transfert de chaleur et de masse
Ce chapitre est constitué de l’article intitulé
“NUMERICAL MODELING OF THE MEDIUM-DENSITY FIBERBOARD
HOT PRESSING PROCESS
PART 1. COUPLED HEAT AND MASS TRANSFER MODEL”
Cet article sera soumis à la revue Wood and Fiber Science (Society of Wood Science and
Technology).
Les auteurs de l’article sont Zanin Kavazović, Jean Deteix, Alain Cloutier et André Fortin.
51
Résumé
Nous présentons ici un modèle mathématique décrivant les phénomènes complexes de
transfert de chaleur et de masse qui surviennent durant le pressage à chaud de panneaux de
fibres de densité moyenne. Notre modèle est basé sur trois principes de conservation :
conservation de l’énergie, de la masse de l’air et de la masse de la vapeur d’eau, qui
résultent en un problème instationnaire tridimensionnel dans lequel les variables d’état et
les propriétés matérielles du panneau varient dans le temps et l’espace. Les équations de
conservation sont exprimées comme fonctions des trois variables d’état du modèle, soient
la température, la pression de l’air et la pression de vapeur. Le modèle comprend le
transfert conductif et convectif de chaleur, le changement de phase de l’eau, le transfert
convectif et diffusif de masse. On tient également compte de la polymérisation de la résine
ainsi que de la chaleur latente associée au changement de phase de l’eau. On suppose
l’existence de l’équilibre thermodynamique au niveau local et on emploie le modèle de
sorption de Malmquist pour décrire la dépendance de la teneur en humidité de la
température et de l’humidité relative du panneau. On présente le développement des
équations de conservation ainsi que les relations constitutives caractérisant les propriétés
matérielles du panneau. Le modèle a une formulation mathématique générale en trois
dimensions mais, dans cet article, nous présentons uniquement les résultats
bidimensionnels. La fermeture de la presse et le développement du profil non homogène de
densité sont pris en compte en imposant un profil de densité prédéfini (mais réaliste) qui
varie dans le temps et l’espace. Les calculs sont faits sur une géométrie de référence et les
détails mathématiques décrivant le transfert des équations d’une géométrie réelle à la
géométrie de référence sont également présentés. Les trois équations de conservation
hautement non linéaires sont résolues par la méthode de Newton en tant qu’un système
d’équations couplées. La discrétisation spatiale du système en question est faite par la
méthode des éléments finis et la méthode d’Euler implicite a été utilisée pour la
discrétisation temporelle. Les résultats obtenus par le modèle montrent en général une
bonne concordance avec les mesures expérimentales. Le modèle fournit également de
l’information utile sur les variables d’intérêt telles que la pression du gaz, la température, la
teneur en humidité, l’humidité relative et l’étendue de la polymérisation de la résine.
Mots clefs: Modèle mathématique, pressage à chaud, transfert couplé de chaleur et de
masse, méthode des éléments finis, dynamique de la polymérisation de la résine, domaine
de référence.
52
Abstract
A mathematical model describing complex phenomena of heat and moisture transfer
occurring during the hot pressing of medium-density fiberboard (MDF) mats is presented.
Our model is based on three conservation principles: conservation of energy, air mass and
water vapor mass, resulting in a three-dimensional unsteady-state problem in which the
fiber mat’s properties and state variables vary in time and space. The conservation
equations are expressed as functions of three state variables of the model, namely
temperature, air pressure, and vapor pressure. The model includes conductive and
convective heat transfer, phase change of water, and convective and diffusive mass transfer.
Resin curing kinetics and latent heat associated to phase change of water are also taken into
account. Local thermodynamic equilibrium is assumed and Malmquist’s sorption isotherm
model is used to describe dependence of moisture content of the mat on temperature and
relative humidity. Developments of conservation equations are presented as well as all
constitutive relations describing the material properties of the mat. The model has a general
3D mathematical formulation but, in this paper, only two-dimensional results are presented.
Press closing and development of non homogeneous density profile are taken into account
by imposing a realistic predefined time- and space-dependent density profile. Calculations
are carried out on a reference geometry and mathematical details relevant to the transfer of
equations from real world geometry to reference geometry are presented. The three
nonlinear conservation equations are solved together as a fully coupled system by means of
Newton’s method. Spatial discretization of the system is achieved by finite element method
and the Euler implicit scheme is employed for the time discretization. The model results
exibit very good overall agreement with experimental measurements. The model produces
valuable information on variables of interest such as total gas pressure, temperature,
moisture content, relative humidity, and degree of resin cure in the case of batch pressing.
Keywords: Mathematical model, hot pressing, coupled heat and mass transfer, finite
element method, resin cure dynamics, reference domain.
53
INTRODUCTION
The hot pressing of medium-density fiberboard (MDF) is a complex process involving
several mechanical and heat and mass transfer phenomena in the fiber mat. There is
evidence of numerous efforts that have been deployed by researchers in order to better
understand the heat and mass transfer phenomena occurring in wood-based panel mats
during the hot pressing process. A comprehensive literature review can be found in Bolton
and Humphrey (1988). Among the first researchers proposing an integrated approach were
Kavvouras (1977), Humphrey (1982), and Humphrey and Bolton (1989a). The first multidimensional heat and moisture transfer model was probably developed and proposed by
Humphrey (1982). A series of papers describing the physics involved in the hot pressing of
particleboard and presenting typical predictive results followed (Bolton et al 1989a, 1989b,
1989c; Humphrey and Bolton 1989a). That work is the foundation on which the
comprehensive model proposed by Thömen and Humphrey (2006) was developed.
Different heat and mass transfer models describing the hot pressing process of wood-based
composite panels such as MDF, oriented strandboards (OSB) and particleboards have been
proposed by Carvalho and Costa (1998), Thömen (2000), Nigro and Storti (2001), Zombori
(2001), García (2002), Zombori et al (2003), Dai and Yu (2004), Pereira et al (2006),
Thömen and Humphrey (2006), Vidal Bastías (2006), just to mention a few of them.
Ultimately, all heat and mass transfer models are based on the mass conservation of air and
water vapor and conservation of energy (Zombori et al 2003; Dai and Yu 2004; Thömen
and Humphrey 2006). To these conservation laws, one can add the cure kinetics equation of
the adhesive system to predict the evolution of resin cure (Loxton et al 2003; Zombori et al
2003). Local thermodynamic equilibrium between mat moisture content and water vapor is
assumed and the relationship between the equilibrium moisture content, relative humidity
and temperature (EMC-RH-T) needs to be described. An appropriate moisture sorption
model is required since it is an important component of every mathematical model of heat
and mass transfer within a composite mat. Several of them are available in the literature
(Malmquist 1958; Nelson 1983; Wu 1999; Dai and Yu 2004; Vidal Bastías and Cloutier
2005). It is expected that the choice of a sorption model might have an influence on the
predictions of hygrothermal conditions within the mat when simulating the hot pressing
process. Several complex and nonlinear sorption models were studied by Vidal Bastías and
Cloutier (2005). Their study showed that the Malmquist (1958) sorption model gives the
best overall fit to experimental EMC.
The largest amount of heat is supplied to the mat by the heated press platens. The heat
released by the exothermic reaction of resin polymerization is also taken into account as
well as the energy associated to the phase change of water. The moisture content of the mat
is usually below the fiber saturation point. Thus, only bound water is considered in the
models (Dai and Yu 2004). Hence, in the energy balance equation, the heat of phase change
involves latent heat of desorption and evaporation (bound water to vapor), and latent heat
of condensation and adsorption (water vapor to bound water) (Nigro and Storti 2001;
Zombori et al 2003; Dai and Yu 2004; Thömen and Humphrey 2006). Heat in the fiber mat
is transferred both by conductive heat flux (modeled by Fourier’s law) and convective heat
flux (heat transported by gas flow through the mat).
54
The gas present in the mat is regarded as an ideal gas and assumed to be a mixture of air
and water vapor (Thömen and Humphrey 2006). Gas flow is assumed to be laminar and the
total gas pressure gradient generates a convective gas flow which is modeled by Darcy’s
law. Diffusive fluxes of air and water vapor are both driven by their partial pressure
gradients and are described by Fick’s law.
Furthermore, the complexity and strong coupled nature of the physical processes involved
during heat and mass transfer are widely recognized in the literature (Bolton and Humphrey
1988, Humphrey and Bolton 1989a; Carvalho and Costa 1998; Nigro and Storti 2001;
Zombori et al 2003; Dai and Yu 2004; Thömen and Humphrey 2006). However, a clear
description of how the coupling procedure is incorporated in the numerical resolution
strategy is most often omitted.
The use of a finite element method and an implicit time scheme is not frequent in the
literature. The only work using that approach we found is by Nigro and Storti (2001). Most
authors use finite difference discretizations with explicit and conditionally stable time
integration schemes. That approach implies the use of smaller time steps in order to satisfy
stability conditions (e.g. time step of 0.005 s used by Yu et al 2007).
Despite the literature already available, it is often difficult to reproduce the numerical
results presented. From the finite element simulation point of view, it is important to be
specific about the partial differential equations (PDE) that constitute the model, the
assumed boundary conditions, the material properties and the numerical methods used to
solve the problem. The aim of this work is to present and solve such a model based on a set
of mathematical equations modeling the complex phenomena of heat and moisture transfer
during the hot pressing of the medium-density fiberboard mat. Particular attention is
devoted to the development of the mathematical formulation of conservation equations. At
each time step, the system of coupled equations is solved on a reference domain. The
mathematical details of the transformation from the real world geometry to the reference
one are explained. We also describe in details how the full coupling of conservation
equations is achieved from the modeling and numerical simulation stand points. We use a
flexible and efficient finite element code developed at the GIREF (Laval University,
Quebec) allowing easy-to-incorporate changes of the input data such as: reference
geometry, material properties, predefined density profiles, different time schemes, linear or
quadratic finite element approximation of the state variables, time step, and mesh
adaptation.
MATERIAL AND METHODS
When developing a mathematical model aimed to simulate physical phenomena, it is
important to compare the results produced by the model with laboratory measurements.
Therefore, experimental data were obtained in order to validate our numerical model.
Material
Medium-density fiberboards were produced in the laboratory for model validation
purposes. Temperature and gas pressure were measured in the fiber mat during the pressing
55
process at the center of the vertical panel plane. Temperature was measured at three points
across mat thickness: at the core, at one-quarter of the thickness and at the surface, whereas
total gas pressure was measured at the core and at the surface. Refined softwood MDF
fibers were obtained from the Uniboard MDF La-Baie plant in Ville de La-Baie, Quebec,
Canada.
Methods
The proposed model is based on conservation principles leading to three governing
conservation equations: conservation of energy, air mass and water vapor mass (Zombori et
al 2003; Dai and Yu 2004; Thömen and Humphrey 2006). This results in a threedimensional unsteady-state mathematical-physical model in which the fiber mat’s
properties and state variables vary in time and space. The three state variables of the model
are temperature, air pressure, and vapor pressure. The conservation equations are expressed
as functions of these three main variables. Furthermore, resin cure is predicted by
considering cure kinetics equation of the adhesive system as part of the energy balance
equation (Zombori et al 2003; Loxton et al 2003; Dai and Yu 2004; Yu et al 2007).
Moreover, Malmquist’s sorption isotherm model is used to describe dependence of
moisture content of the mat on temperature and relative humidity.
Useful information related to complex expressions and equations describing material
properties, sorption model and resin cure kinetics are taken from available literature (see
APPENDIX 1). At this stage, press closing and development of non homogeneous density
profile are taken into account by imposing a realistic predefined time- and space-dependent
density profile (see APPENDICES 2 and 3). Calculations were carried out on a reference
geometry and effects of evolving domain geometry were accounted for by transferring of
equations and material properties from evolving real world geometry to reference
geometry.
Panel manufacturing
The fibers were a blend of about 90% black spruce (Picea mariana) and 10% balsam fir
(Abies balsamea). The fibers at 6.5 % initial moisture content were blended with 12 %
(fiber oven-dry weight basis) urea-formaldehyde resin and 1 % wax in a laboratory rotary
drum blender. The initial moisture content of the furnish was 12%. A series of six MDF
panels of size 560 mm x 460 mm x 13 mm and target density of 750 kg/m3 at 8% MC were
produced in a Dieffenbacher laboratory batch press equipped with a PressMAN
measurement and control system. The press platens were at 203C. The pressing schedule
of 335 s was divided in five steps. The initial mat thickness of about 182 mm was reduced
to 140% of the final thickness in the first 35 s (Step 1). The press remained in this position
for the next 15 s (Step 2) followed by the second compression lasting 110 s at the end of
which the mat reached its final thickness of 13 mm (Step 3). The hot platens remained in
this position for the next 110 s (Step 4). The final step (Step 5) was the degassing period
(65 s) during which the press was slowly opened and reached 107% of the final panel
thickness at 335 s. The curve presenting the evolution of mat thickness with time can be
seen in Figure 2.1a.
56
a)
b)
Figure 2.1 : Evolution of : a) mat thickness; b) normalized percentage of mat target
thickness (PTT(t)).
Model development
Overall Approach and Assumptions
It was reported by Humphrey and Bolton (1989b) that the size of the board has an effect on
the temperature and gas pressure within the hot pressed mat. In the current study, a single
panel geometry was considered. Therefore, the effect of panel size was not studied. All the
mats were formed using the same raw materials and pressed using the same pressing
schedule.
Bound water was assumed to be in equilibrium with water vapor in the lumens and in the
mat voids. Local thermodynamic equilibrium was assumed at every point of the fiber mat
and the relationship between local moisture content, relative humidity and temperature was
described by the sorption isotherm. Hence, the three state variables of the model are
temperature, air pressure, and vapor pressure. All of the material properties of the fiber mat
were taken from the available literature (see below and APPENDIX 1). None of the fiber
mat material properties was obtained from the panels produced in the laboratory.
The current study is focused on the heat and moisture transfer phenomena involved in the
hot pressing of the MDF wood fiber mat. The rheology of mat consolidation was not
explicitly considered in this study. Therefore, a predefined time- and space-dependent
oven-dry vertical density profile based on the work of Wang and Winistorfer (2000) (see
APPENDIX 2 and Figure 2.2) was used in the simulations to update the local heat and
moisture transfer properties and porosity of the mat. Consequently, the complex dynamical
interactions between heat and moisture and rheological parameters involved during the hot
pressing process were not taken into account. We are aware that this simplification may
have an influence on the results presented here. A numerical coupling between the
mechanical and the heat and mass transfer models will be presented in Part 2 of this paper.
57
It is assumed that the initial mat moisture content is uniform throughout the thickness. The
contribution of resin cure to heat and mass transfer is also taken into account. All the results
for the coupled three dimensional mathematical model of heat and moisture transfer were
obtained by finite element numerical simulations.
It is also assumed that the mass of oven-dry fiber material in each control region is constant
and a control region of constant volume is considered (Thömen 2000; Thömen and
Humphrey 2006). Our model presents similarities with the models published by Dai and Yu
(2004) and Thömen and Humphrey (2006). However, in our approach, we also take into
account the effect of compression on the global mat geometry. Press closing (Figure 2.1) is
considered and the effect of changing mat thickness on the material properties (thermal
conductivity, gas permeability, and porosity) is also accounted for as will be described
below.
a)
58
b)
c)
Figure 2.2 : a) Evolution of a space- and time-dependent predefined oven-dry density
profile used in calculations: vertical density profile in the center line at different moments
in time. Evolution of: b) predefined oven-dry density profile, values at 4 points in the
vertical center line (BSQ=Middle Point Between Surface and Quarter); c) wet density
profile at 4 points in the vertical center line calculated by  Mat =  OD (1+M).
Total Derivative
The notion of total derivative will frequently appear in the development of conservation
equations. Therefore, a brief reminder will be useful. Suppose that G is a function of an
independent variable t and of three real-valued functions f, g, and h which also depend on t.
Thus, one can write G  G (t , f (t ), g (t ), h(t )) . In this case, the partial derivative of G with
G
respect to t (
) does not give the true rate of change of G with respect to t, because it
t
does not take into account the dependency of f, g and h on t. However, the total derivative
dG
) is taking such dependencies into account. Hence, the true rate of change of G with
(
dt
respect to t is given by
dG G G df G dg G dh




dt
t f dt g dt h dt
(34)
dG
) should be
dt
considered. Hence, all the indirect dependencies (dependencies of other variables on t) are
G
). When developing conservation equations, one should
added to the partial derivative (
t
account for the overall rate of change and the total derivative should be used.
Therefore, to find the overall dependency of G on t, the total derivative (
59
Generic Formulation of a Conservation Principle
When considering a constant arbitrary control volume  , the principle of conservation of a
quantity G contained within the control volume states that the total variation in time of the
quantity G is basically caused by the combined action of internal and external factors: an
internal source producing G, an internal sink consuming G and the net flux of G through
the boundary of the control volume. The net flux represents the result of interactions of G
contained within the control volume with the surroundings. We suppose that the quantity G
is variable in time and space ( G  G (t , x) ), differentiable and its total time derivative
bounded over  . This allows us to put the derivation sign inside of the integral on the left
hand side of the Eq. (35). Hence, we have the following conservation principle for G
d
d

Gd    Gd    S source d    S sink d    qG nd 

dt 
dt




(35)
where qG is the net flux of quantity G through the boundary  of the constant control

volume  and n represents the outward unit normal vector on the boundary  . Of course,
the source term will create a positive variation (increase) of the quantity G contained within
the control volume, whereas the sink term will cause a decrease of concentration of G
(negative sign). When the net flux qG is in the same direction as the outward normal


vector n ( qG n  0 ), then the quantity G is leaving the control volume resulting in a

decrease of concentration of G (negative sign). On the opposite, if qG n  0 , then the net
flux is entering the control volume resulting in an increase of concentration of G

(   qG nd  will then be positive).

Moreover, the divergence theorem
q
G


nd     qG d 
(36)

yields the following expression for the conservation of G
d
d
Gd    Gd    S source d    S sink d     qG d 

dt 
dt




(37)
The previous expression being true for any constant control volume  , we finally have
d
G   qG  S source  S sink
dt
(38)
60
Mass Conservation of Air
It is commonly agreed that air can be considered as an ideal gas and therefore obeys the
ideal gas law. During the hot pressing process of MDF panels, no air is generated or
consumed. It means that there is no source or sink term in the conservation equation.
Therefore, variations of air mass within a control volume of the mat are solely depending
on bulk flow of the gas phase (convection generated by the gradient of total gas pressure
and modeled by Darcy’s law) and molecular diffusion of the air within the gas phase
(diffusion created by the gradient of partial air pressure modeled by Fick’s law). Molecular
diffusion translates the tendency to homogenize the gas phase. However, it is recognized by
many authors (Thömen and Humphrey 2003; Dai and Yu 2004; Thömen and Humphrey
2006) that the bulk flow of the gas phase is the main process causing the air to leave the
mat. The conservation principle applied to the air yields to the following equation
expressing the conservation of the mass of air



d ( a)
M


     a K p   P       a Deff   Pa   0
dt

  RT


 

(39)
Mass Conservation of Water Vapor
Water vapor is also treated as an ideal gas as well as a mixture of air and vapor. Variations
of water vapor mass within a control volume of the mat depend on bulk flow of the gas
phase (modeled by Darcy’s law) and molecular diffusion of the vapor within the gas phase
(modeled by Fick’s law). Given that the mat moisture content is significantly below the
fiber saturation point, only bound water is present in the mat (Dai and Yu 2004). Hence,
during the hot pressing process, bound water present in the MDF mat will partially
evaporate in regions of high temperature and thus produce water vapor. On the other hand,
in cooler regions, condensation of water vapor may occur and increase the bound water
content. This means that the evaporation of bound water can be seen as a source term of
vapor and condensation of water vapor is interpreted as a sink. Bound water evaporation is
equivalent to the loss of bound water due to the moisture content decrease in time (García
2002), and the condensation of vapor is equivalent to the gain of bound water due to the
moisture content increase in time. Hence, the source and the sink terms of water vapor are
dM
dM
both modeled by the following expression:  OD
and the sign of
determines if
dt
dt
dM
dM
evaporation (
 0 ) or condensation (
 0 ) is taking place (García 2002; Thömen
dt
dt
and Humphrey 2006). Therefore, the equation expressing the conservation of the mass of
the water vapor is



d ( v)
dM
M


     v K p   P       v Deff   Pv    OD
dt
dt

  RT


 

(40)
61
Conservation of Energy
The fiber mat is composed of a solid phase (oven-dry fibers, resin and bound water) and a
gaseous phase (air and water vapor). Within a control volume of the mat having a constant
volume and constant mass of oven-dry fiber material and resin, the variation of energy of a
solid phase is represented by
d
d
 ODCMatT   OD CMatT 
dt
dt
(41)
d
 C a T  a   Cv T  v  
dt
(42)
and that of a gaseous phase by
It is assumed that, inside the control volume, the solid phase and the surrounding gaseous
phase have the same temperature. We also assume that the heat generated by friction during
the compression of the mat is negligible. Within the mat, the heat is transferred by
conduction (conductive heat flux) in the solid phase and by convection in the gaseous phase
(bulk flow and molecular diffusion). The energy associated to the phase change
(evaporation and condensation) must also be taken into account. The bound water
evaporation process requires energy (latent heat of desorption and vaporization) and acts as
an energy sink, whereas water vapor condensation is an exothermic process which releases
the same amount of energy (latent heat of condensation and adsorption) and acts as an
energy source. Latent heat associated to phase change depends on local temperature and
moisture content. Therefore, it varies in space and time (Thömen and Humphrey 2003; Dai
and Yu 2004). We use the same expressions for the latent heat as Thömen and Humphrey
(2003) and Dai and Yu (2004), which are based on Humphrey and Bolton (1989a) results.
The impact of the phase change on the energy of the control volume is modeled by the
dM
expression:  H fg   Cbw  Cv  T  OD
(Thömen and Humphrey 2006). The sign of
dt
dM
dM
dM
determines if evaporation (
 0 ) or condensation (
 0 ) is taking place. The
dt
dt
dt
heat generated by the exothermic reaction of resin polymerization is also taken into account
and it acts as an energy source in the control volume. The energy source term Qr related to
the cure kinetics of resin polymerization is considered and a model for the resin curing
process is discussed in the next section.
62
Hence, the conservation of energy of the system is expressed by
d
d
CMatT   CaT a   CvT v       KT  T 
dt
dt


 C M


T

     a Ca  v Cv  K p   P       a a Deff   Pa 
 

 R



OD
(43)
 C M

dM

    v v Deff   Pv   Qr   H fg   Cbw  Cv  T  OD
dt

 R

Resin Cure Kinetics
It is a common belief that the performance of composite panels is strongly related to the
uniformity of resin distribution in the mat and that at a given resin content, a uniform resin
distribution will lead to the best panel properties (Kamke et al 1996; Loxton et al 2003).
Loxton et al (2003) found that “resin distributions changed upon pressing of resinated fiber,
implying that resin was being redistributed during pressing” (Loxton et al 2003). Despite
that observation and for modeling purposes, we still assume a uniform distribution of the
resin throughout the fiber mat.
In the laboratory experiments, we used urea-formaldehyde (UF) thermosetting resin as
adhesive system. Naturally, for a thermosetting resin, a dominant factor controlling resin
cure is the temperature: the higher the temperature, the faster the resin cures. However,
moisture content influences resin cure as well. In regions where moisture content is high,
the temperature rise is delayed by the evaporation process which in turn slows down resin
curing.
To model resin cure kinetics, a phenomenological approach is commonly used ignoring the
chemical details of the reacting system by fitting a mathematical model to experimental
data (Liang and Chandrashekhara 2006). As different resin systems exhibit different curing
behavior, several models of cure kinetics are available in the literature (Harper et al 2001;
Xing et al 2004; Liang and Chandrashekhara 2006). The simplest one is the nth-order
kinetics model which “does not account for any autocatalytic effects and so it predicts
maximum reaction rate at the beginning of the curing” (Liang and Chandrashekhara 2006).
The curing reaction of UF resin is assumed to have a nth-order kinetics (Park et al 2008)
and is therefore modeled by the following ordinary differential equation (ODE)
d
 E 
n
 A  exp   a   1   
dt
R
T
 

(44)
  0  0
where α is a dimensionless variable representing the extent of the resin cure which can be
defined as the ratio of the mass of cured resin to the initial mass of uncured resin.
Therefore, α has values between 0 (no resin has yet been polymerized) and 1 (all the
available resin has been cured) (Liang and Chandrashekhara 2006). The above model states
63
d
) is higher. The function 1   n
dt
describes the decrease of the reaction rate as α increases and the reactants are consumed.
that, at higher temperatures, the reaction rate (
The constant A is the Arrhenius collision frequency factor relating the amount of collisions
that need to occur in a unit time to carry out the reaction (Harper et al 2001), Ea is the
Arrhenius activation energy describing the amount of energy needed to propagate cure
(Harper et al 2001), R is the universal gas constant, T is temperature field in Kelvin and n is
the order of the reaction.
Xing et al (2004) also considered the UF resin curing reaction as nth order reaction and
conducted an experiment using differential scanning calorimetry to quantify the degree of
resin pre-cure after blending. Among other things, they were able to come up with values
for different parameters involved in the equation. We therefore use the following values
from Xing (2003) and Xing et al (2004): A = exp(17) [1/s], Ea = 7* 104 J/mole, n = 1.2 and
also the total heat released during the entire course of the reaction at 105 J/kg (Xing 2003,
Table 4.3).
When solving the ODE for α, the temperature field T is assumed to be known. Given the
initial condition   0   0 , the analytical solution for the ODE describing the evolution of
resin cure is:
when n  1

 Ea  

 R  T 
  t   1  exp   A  t  exp  

(45)
and when n  1
1

 1 n
 E 
  t   1  1  exp  a   A   n  1  t 
 R T 


(46)
The above expressions describe the evolution of the degree of resin cure and we use it in
our heat and mass transfer model. As we consider that the order of reaction is n = 1.2 (Xing
2003; Xing et al 2004), the expression when n  1 is used to predict the evolution of the
degree of resin cure.
Heat Generation Rate
The source term Qr in the energy conservation equation (Eq.(43)) represents the heat
generated by the exothermic reaction of resin polymerization. Of course, it varies in space
and time. The accumulated heat generated by the reaction up to a given moment during the
reaction is given by the following relation
Qr   r  H r 
d
dt
(47)
64
Replacing
d
by its expression given by Eq.(44), one gets
dt
 E 
Qr   r  H r  A  exp   a   1   n
 R T 
(48)
where H r is the total heat released during the polymerization (latent heat of polymerization
estimated at 105 J/kg by Xing 2003) and  r is the resin density expressed as the ratio of
solid resin mass to the total volume. As the UF resin content of the fiber mat is 12 % based
on fiber oven-dry weight basis, the corresponding mass of solid resin is given by 0.12 times
the mass of oven-dry fibers. We also have the following result:  r  0.12 OD .
Sorption Model
Following Dai and Yu (2004), we assume that local isothermal sorptive equilibrium exists
between mat moisture content, gas relative humidity, and temperature. It is also assumed
that the initial mat moisture content is uniform throughout the thickness and its value is set
to 0.12. Using an appropriate sorption model, the evolution of mat moisture content with
time at any position can thus be predicted and monitored. It is still not possible to
experimentally measure directly moisture content and relative humidity during the hot
pressing process. Therefore, predicting space and time evolution of these variables gives
useful information from the practical standpoint.
Several sorption models are proposed in the literature: Malmquist (Malmquist 1958),
Hailwood-Horrobin one and two hydrates (Vidal Bastías and Cloutier 2005), García (2002),
Nelson (1983) and others. Vidal Bastías and Cloutier (2005) compared several of the most
frequently used sorption models and their study showed that the Malmquist’s sorption
model gives the best overall fit to experimental equilibrium moisture content (EMC) data,
especially at high temperatures. This model was therefore used in our numerical
simulations. It expresses dimensionless moisture content M as a function of absolute
temperature T and dimensionless relative humidity h:
M Malmquist 
MS
I
 1 3
1  N   1
h 
(49)
where MS, N and I are second order polynomials of the absolute temperature T defined as
follows (Vidal Bastías and Cloutier 2005):
MS  0.40221  9.736 105  T  5.8964 107  T 2
N  2.6939  0.018552  T  2.1825 106  T 2
I  2.2885  0.0016742  T  2.0637 106  T 2
(50)
(51)
(52)
65
Numerical Model of Heat and Mass Transfer in the Fiber Mat
Our mathematical model can be seen as a generalization of the model proposed by Thömen
and Humphrey (2006). As the model is based on the mass conservation of air and water
vapor and conservation of energy, it is expressed in terms of the three state variables:
partial air pressure (Pa), partial water vapor pressure (Pv) and temperature (T). Therefore, in
each one of the three conservation equations, the chain rule is applied to time derivatives.
Hence, let B represent any of the functions for which the total derivative with respect to
time is considered in the conservation equations. Then, we have
dB B Pa B Pv B T B




dt Pa t Pv t T t t
(53)
B
is the partial time derivative which is different from zero if and only if B depends
t
explicitely on time, i.e. if the time appears explicitly in the expression of B. For instance, as
the moisture content M depends on temperature (T) and partial vapor pressure (Pv) and does
M
M
not explicitly depend on time nor on air pressure, then
 0 and
 0 . When the
t
Pa
dM
dM M Pv M T
is thus developed as
. A
chain rule is applied, the term


dt
dt
Pv t
T t
similar approach is used to perform other time derivatives in the model. Once this is done,
the three conservation equations are written in terms of the three state variables (Pa, Pv, T)
as follows:
where
Mass conservation of air
  (  a  )  Pa   (  a  )  Pv   (  a  )  T
 P  t   P  t   T  t


a
v










M
(  a )
    a K p  a Deff   Pa       a K p   Pv   
RT
t


 

 

(54)
Mass conservation of water vapor
 (  v )
M  Pa   (  v  )
M  Pv   (  v  )
M  T
 P   OD P  t   P  OD P  t   T  OD T  t


a
a 
v
v 








M
(  v )
    v K p   Pa       v K p  v Deff   Pv   
RT
t


 

 

(55)
66
Energy conservation
  CMatT    CaT  a   CvT  v  

M  Pa

  H fg   Cbw  Cv  T  OD

 OD

Pa
Pa
Pa  t


  CMatT    CaT  a   CvT  v  
M  Pv

  H fg   Cbw  Cv  T  OD

 OD





P
P
P
t
v
v
v 


  CMatT    CaT  a   CvT  v  
M  T

  H fg   Cbw  Cv  T  OD

 OD

T
T
T  t




T
CM
     aCa   vCv  K p  a a Deff   Pa 
R




(56)



T
CM
     aCa   vCv  K p  v v Deff   Pv 
R




   KT  T 
 Qr  OD
  CMatT    CaT  a   CvT  v  

t
t
(see NOMENCLATURE and APPENDICES 1 and 2 for definitions of variables and
expressions).
The model is formed by these three highly nonlinear conservation equations which are
strongly coupled and constitute a coherent system.
Finite Element Solution Strategy
An integrated approach considering simultaneously all important variables during hot
pressing was proposed by Kavvouras (1977), Humphrey (1982), and Bolton and Humphrey
(1988). Our solution strategy is quite different from what has traditionally been done.
Indeed, for each of the three conservation equations, a finite element method discretization
is performed in space while the time derivatives are calculated using the stable Euler
implicit scheme. This allows for larger time steps and reduces the calculation burden. Each
state variable is discretized by Q1 (bilinear quadrangles) finite element (Bathe 1982; Reddy
2006). Moreover, at each time step and for each nonlinear iteration, the three nonlinear
equations forming the coupled heat and mass transfer system are solved simultaneously
preserving the full coupling between them.
To address the nonlinearity of this complex system of coupled equations, we proceeded
similarly to Nigro and Storti (2001). Indeed, at each time step, the nonlinear system is
solved by Newton’s method allowing to predict the evolution of the state variables in space.
Unlike the later authors, we explicitly computed the derivatives rather than using a finite
difference approximation of derivatives (Nigro and Storti 2001) to calculate the Jacobian
67
matrix. Unlike Nigro and Storti (2001), no stabilization technique was needed to solve the
system.
It has been a common practice to keep the local conditions and properties constant during a
given time step (Thömen 2000; Thömen and Humphrey 2004). We however adopted a
different approach. Indeed, in our case, within each time step, all the local conditions and
mat material properties are updated from one nonlinear iteration to another. Although more
complicated, we believe that this is a better numerical approach and our code is able to deal
easily with this level of complexity. During each time step, on average 4 iterations of
Newton’s method are performed in order to reach convergence to 10-5 in the residual norm.
Since we are using an Euler implicit time scheme combined with the finite element method,
we have no constraint on the time step length. However, a too large time step could cause
convergence and accuracy problems. The results presented in this paper were obtained
using a time step of 0.1 s.
Computational Domain
It is noteworthy that our mathematical model is three-dimensional (3D). However, at this
stage, in order to reduce the calculation time, it will be applied on a two-dimensional (2D)
geometry. Nevertheless, in Part 2 of this paper, numerical results obtained with a global
model on 3D geometry will be presented.
When studying the hot pressing of multi-layered wood strand composites, Lee et al (2007)
pointed out that a daylight delay (the time necessary for the top platen to touch the mat)
creates a temperature asymmetry which causes asymmetric distribution of internal mat
conditions in the thickness direction. Despite that observation and for modeling reasons, we
will rather follow a common path proposed in the literature and take advantage of the
symmetry (Carvalho and Costa 1998; Carvalho et al 2001; Nigro and Storti 2001; Carvalho
et al 2003; Thömen and Humphrey 2003; Dai and Yu 2004; Pereira et al 2006; Thömen and
Humphrey 2006; Yu et al 2007).
Hence, our computational domain represents a quarter of the full 2D geometry (see Figure
2.3) with the symmetry plans presented in Figure 2.3. Therefore, for the numerical
simulation runs, we consider a rectangular domain in the x-z plane with the following
dimensions: 280 mm (half length) by 6.5 mm (half final thickness). Figure 2.3 shows
details of the 2D geometry and our working domain. The domain considered for calculation
was meshed with a 16 by 16 grid having 256 rectangular elements. More refined grids were
used by Zombori et al (2003) (19 by 19) and Nigro and Storti (2001) (20 by 20). We should
also point out that the mesh used by Nigro and Storti (2001) had a high concentration of
elements towards the edges in contact with hot platens and ambient air, whereas our mesh
is made of homogeneous rectangular elements. Thorough discussion on size of the grids
and examples of its influence on numerical results will be presented in Part 2 of this paper.
In the current study, we worked on a reference domain: 280 mm (half length in the x
direction) by 6.5 mm (half final thickness in the z direction). This thickness corresponds to
the final half thickness of the fiber mat (symmetry is taken into account). It is clear that in
68
different stages of pressing process, the mat has different thickness values. Given that the
total mass of fiber material does not change with the thickness of the mat being
compressed, the material properties of the mat will change during compression. For
instance, in thicker mats, thermal conductivity is lower, whereas gas permeability is higher.
As we work on a reference domain, these changes in material properties must be accounted
for as the pressing process evolves. Thus, the transfer of material properties and the
equations from a real-world evolving domain to the reference domain must be done. In the
next section, we explain the procedure to properly achieve this task.
a)
b)
Hot platen
Air
Air
Hot platen
6.5 mm
13 mm
Air
Hot platen
z
560 mm
Symmetry axes
280 mm
x
Figure 2.3 : a) Full 2D geometry of a fiber mat; b) computational domain in 2D (one
quarter of the full geometry).
Transfer to the Reference Domain
Generic Approach
We first present the basics and then apply them to a simple generic problem in order to
illustrate the transfer of calculations on the reference domain.
Let  be a real-world domain where the coordinates of a given point are  x, y , z  , and let
 be a reference domain (where the calculations are carried out) where the coordinates of



   , G x , y , z   x, y , z 
a given point are x , y , z . The invertible function G : 




transfers a point from the reference domain to a real-world domain. Its inverse
 

 , H  x, y , z   x , y , z transfers a point from the real-world
function H  G 1 :   


domain to the reference domain. If we express the real-world coordinates as a function of
69
the reference domain coordinates, one can write
x  g1 x , y , z ; y  g 2 x , y , z ; z  g 3 x , y , z and we have







G x , y , z   g1 x , y , z , g 2 x , y , z , g3 x , y, z    x, y , z 





 
 

(57)
Similarly, one can express the reference domain coordinates as a function of the real-world
coordinates: x  h1  x, y , z  ; y  h2  x, y , z  ; z  h3  x, y , z  and get

H  x, y , z    h1  x, y , z  , h2  x, y , z  , h3  x, y , z    x , y , z


(58)
Let F  F  x, y , z  be any scalar function expressed on the real-world domain  and


F
 x , y , z the same scalar function expressed on the reference domain 
 . Then, the
F
following derivatives can be calculated and written in matrix notation
 
F
g
   1
  x    x
F
   g

 1

y


   y
  
  F   g1

  z    z
g 2
 x
g 2
 y
g 2
 z
g3   F 

 x   x 
g3   F 

 y   y 
 
g3   F 
 
 z   z 
and
 F   h1
 x   x
  
 F   h1
 y    y
  
 F   h1
 z   z
h2
x
h2
y
h2
z
 
h3    F 

x    x 

h3    F


y    y 

h3    F



z    z 


(59)
or written in a more compact way
 T
F
  
 G  F



and
 T

 T
F
   H T 
 G  F ,
Since F   H  

 

 1

 T  T
 
 


  H  

  G   I i.e.  H   G  .
one
 T
F

F   H  
comes
to
the
(60)
conclusion
that
We will now illustrate on a simple but generic equation the transfer of calculations from the
real-world domain to the reference domain. The equation is
dT
  K T   f
dt
(61)
We used the finite element method. From this approach, the variational formulation of Eq.
(28) on the real-world domain is
70
dT
 dt  d    K T  d    h T  T  dS   h  dS   f  d  ,  V    (62)
EXT
R


N
SR

SN
where  is a test function in a certain functional space V    , S N is the part of the
boundary of  on which a flux is imposed (Neumann boundary condition) and S R is the
part of the boundary of  on which the exchange (Robin or Neumann nonlinear) boundary
condition is imposed, hN and hS are coefficients, T EXT is the ambient temperature, f is the
source term and d   dxdydz . The equivalent expression on the reference domain is
 T
 T
dT  
 dt  Jd    K  H   T  H   Jd 



  hR T  T
S R
EXT

 J SR d SR 

S N

hN  J S N d SN   f  Jd 
 

 V 
(63)


which can also be written as

 T
dT  
 dt  Jd     H  K  H   T  Jd 




EXT
  hR T  T
 J SR d SR 
S R



S N

hN  J S N d SN   f  Jd 
(64)


 1

 G and J and J are the determinants (Jacobians) of
where J  det  H   det 
SN
SR
  d xd
 yd z . We will be more specific about
the transformation on the boundary, and d 
these expressions in the next subsection.
 
It should be noticed that Eqs. (62) and (64) are equivalent and have the same structure.
From the finite element point of view, it means that the transfer is not code invasive.
Indeed, the coefficients in each integral were multiplied by an appropriate Jacobian and the
material properties represented by the tensor K on the real-world domain are now


  H T on the reference domain.
represented by the tensor  H  K


This illustrates the transfer of calculations from the real-world domain to the reference
domain. The same methodology is applied to each of the three conservation equations of
our model (Eqs. (54), (55), (56)).
71
Mat Compression
During the pressing process, the thickness of the mat decreases as a function of the press
closing schedule. The mat target thickness (MTT) is known in advance (MTT = 13 mm in
our case). Of course, when symmetry is assumed, one half of the MTT is used in the
calculations (6.5 mm). The press closing schedule is only time dependent and can be
expressed in terms of percentage of MTT and named PTT  t  . In our simulation runs,
PTT  t  is known (APPENDIX 3 and Figure 2.1b). Therefore, one can express the
evolution of the real mat thickness (RMT) in terms of MTT and PTT  t  as follows
RMT  t  
PTT  t 
MTT
100
(65)
 , its
Since the calculations are performed on an arbitrary but fixed reference domain 
thickness (RDT: reference domain thickness) is also known. Thus, the following useful
expression can be defined
 t  
RMT  t 
RDT
(66)
During the pressing process, the largest variations in mat dimensions occur in the thickness
direction, whereas the variations in the x and y directions can be considered negligible.
Thus, the following relations stand
 
y  y  g  x , y , z 
z    t  z  g  x , y , z 
and
x  x  h1  x, y, z 
(67)
2
and
y  y  h  x, y , z 
2
(68)
3
and
1
z 
z  h3  x, y, z 
 t 
(69)
x  x  g1 x , y, z


F
 x , y , z on
Thus, for scalar fields F  F  x, y , z  on the real-world domain  and F
 , we have
the reference domain 

F
 F 
 

  x  1 0
x 
0


F
 
 F

  0 1
0   
y
  y  
   0 0   t    F 
 
F 

 z 
  z 
and

F
 F  
  
 x  
x 
0  
  1 0


F
 F   0 1


0
 y  
   y 
  
1 

 F  0 0
F



t




 z  
  z 
(70)
72


 1

 G    t  and if a tensor K
 is given
Hence, J  det  H   det 
 
k 13 

k
k
12
 11

 t  




 k 11 k 12 k 13 






k 23 
  H T   k 21
 22
   k 21 k 22 k 23  , then  H  K
.
by K
k

 


 t  






 k 31 k 32 k 33 
 k 31
k 32
k 33 
 t

2
    t   t  
 
We can also have an explicit expression for J S associated with a surface element. As the
treatment of the surface element is similar whether the Neumann or Robin boundary
condition is dealt with, a generic approach is presented. From Eqs. (67), (68) and (69), one
gets dx  d x , dy  d y and dz    t  d z . There are three cases to consider:
 y  d S
dS  dxdy  d xd
and J S  1
dS  dxdz  d x  t  d z    t  d S
and J S    t 
dS  dydz  d y  t  d z    t  d S
and J    t 
S
(71)
(72)
(73)
This strategy was applied to several different reference domains to test the independency of
the results on the reference geometry. All the numerical tests were successful and gave the
same results.
Density Profile
Generally, the vertical density profile of compressed panel is not uniform. It mainly
exhibits a common “M-shaped” profile with higher density close to the surface and lower
density in the core (Carvalho et al 2001, 2003; Wang et al 2001). It is regularly observed
that the transition region from low to high density is rather thin (Carvalho et al 2001, 2003;
Wang et al 2001). This characteristic “M-shaped” density profile is attributed to the
interactions between the heat and mass transfer phenomena and the mechanical
compression of the mat (Dai and Yu 2004).
In the present work, in order to reduce the complexity and calculation time, a realistic non
homogeneous predefined oven-dry density profile of the mat (  OD [kg/m3]) was used
during the simulation runs (see APPENDIX 2 and Figure 2.2; Kavazović et al 2010). This
vertical density profile is time- and space-dependent and is in part inspired by the results
presented by Wang and Winistorfer (2000), Winistorfer et al (2000), and Wang et al (2001,
2004).
Some authors are not very specific about the density profile data when presenting results of
their heat and mass transfer model. However, it is noteworthy that Carvalho and Costa
73
(1998) state explicitly that “it is assumed instantaneous closing of the press. The final
thickness is attained instantaneously, there is no variation of density and the porosity
remains unchanged”.
In our case, the densification process (formation of the vertical density profile) is taken into
account by the time- and space-dependent expression of the oven-dry density profile (see
APPENDIX 2 and Figure 2.2a). Furthermore, the evolution of oven-dry density at four
representative points placed in the vertical central plan of the mat is presented in Figure
2.2b.
The porosity of the mat was calculated by the formula presented by Siau (1984)
 1–
OD
1530
and was thus time and space dependent.
Boundary Conditions
Appropriate boundary conditions are needed to properly solve the system constituted by
Eqs. (54), (55) and (56). The temperature evolution of the surface in contact with the hot
platen (Fig. 2.4a) was imposed by a Dirichlet boundary condition based on the data
obtained during in-situ laboratory experiments. The surface in contact with the hot platen
includes the two end vertices illustrated by black dots in Figure 2.3b. Moreover, the
following fluxes were considered at the boundaries:

 M

Air flux : q Pa =   a K p  P    a Deff  Pa 

 
  RT

 M

Vapor flux : q Pv =   v K p  P    v Deff  Pv 


  RT
 T   a Ca   v Cv 

Heat flux : q T =   KT • T   
K p P 



C M
 C M

  a a Deff  Pa    v v Deff  Pv 
 R
  R

(74)
(75)
(76)
The hot platen is assumed impervious to gas and therefore qPa = 0 and qPv = 0. Symmetry
conditions are imposed (qT = 0, qPa = 0 , qPv = 0) on the two symmetry axis illustrated by
dashed lines in Figure 2.3b. On the external edge in contact with the ambient air, the
following convection boundary conditions are imposed for the three state variables: air
pressure, vapor pressure and temperature, respectively:



q Pa · n =  h p a ( P  Pamb )  105 a ( Pa  Pa amb )
Pa




q Pv · n =  h p v ( P  Pamb )  105 v ( Pv  Pv amb )
Pv

(77)
(78)
74
T   a Ca  v Cv 

q T · n =  h T ( T  Tamb )  h p
( P  Pamb )

CM
CM
105 a a ( Pa  Pa amb )  105 v v ( Pv  Pv amb )
R
R
(79)

where n is the outward unit normal vector, hT and hp are respectively the convective heat
and mass transfer coefficients associated to the external boundary (Zombori 2001; Vidal
Bastías 2006). In Figure 2.3, the external edge is the right hand side edge of the rectangular
domain and is represented by a continuous black line including the black square (Fig. 2.3b).
The main mode of mass transfer between the mat and the environment is the gas bulk flow
(Zombori et al 2004) generated by the difference in total gas pressure within and outside
the mat. Diffusion generated by the difference in partial vapor pressure within and outside
of the mat plays a secondary role (Zombori et al 2004).
Thermal Conductivity of the Mat (KT)
Thermal conductivity increases with the increase of density, temperature and moisture
content of the fiber mat. We used the expression suggested by Thömen and Humphrey
(2006) for the thermal conductivity of the fiber mat: KTxy = 1.5·KTz where
KTz  KT 030  KT
KT 030  4.38  10
and
-2
 4.63 10  OD  4.86 10
-5
(80)
-8

2
OD
KT  0.49 M  1.1104  4.3 103 ·M  · T  303.15 
(81)
(82)
The variables KTz and KTxy represent respectively the thermal conductivity in the thickness
and horizontal directions. KT030 is the thermal conductivity measured at 0% M and
30C and KT is the correction term accounting for moisture content and temperature
effects on thermal conductivity. The tensor of thermal conductivity KT is therefore given by
 KTxy
KT   0

 0
0
KTxy
0
0 
0 

KTz 
(83)
Specific Gas Permeability of the Mat (Kp)
Analytical expressions for the specific gas permeability of MDF mats based on curve fitting
of experimental data can be found in García and Cloutier (2005) and also in von Haas et al
(1998). The expression proposed by García and Cloutier (2005) is valid for MDF mats
having a density between 400 kg/m3 and 1150 kg/m3, whereas in von Hass et al (1998), the
permeability of fiber, particle and strand mats with densities varying from 200 kg/m3 to
75
1200 kg/m3 was determined. The samples used by von Hass et al (1998) were prepared
from consolidated panels with an adhesive content of 11%. In our study, the expression and
the input data for the specific gas permeability of the MDF mats will be based on
expressions proposed by von Haas et al (1998). Hence, the in-plane permeability (Kpxy) and
the cross-sectional permeability (Kpz) of MDF fiber mats are both described by the
following expression
1
exp  
 A
where
A = a + b   Mat +
c
ln( Mat )
(84)
and the coefficients to determine Kpxy are a =  0.041, b = 9.5110-6 , c =  0.015 and
those for Kpz are a =  0.037, b = 1.1 10-5 , c =  0.037.
The tensor Kp of the specific gas permeability of the MDF fiber mat is therefore given by
 K pxy

Kp   0
 0
0
K pxy
0
0 

0 
K pz 
(85)
RESULTS AND DISCUSSION
It should be kept in mind that the numerical model used here is based solely on heat and
mass transfer mechanisms and that the influence of the changing moisture content and
temperature on rheological mechanisms was not considered. The numerically predicted
solutions depend on several heat and mass transfer properties of the fiber mat and most of
these properties are only known to a limited degree of precision, especially under the
conditions prevailing during the hot pressing process. Moreover, the fiber mat material
properties including thermal conductivity, gas permeability and porosity were taken from
the literature and were not determined from the specific material used to make panels. This
can explain some of the discrepancies between the model and the experimental results.
The temperature measurements are presented together with numerically predicted results in
Figure 2.4a where the vertical bars represent standard deviation from the mean value. In
Figure 2.4a, curve labelled SurfaceLab is the temperature measured in the laboratory at the
surface in contact with the hot platen and was imposed as a Dirichlet boundary condition
for T at the surface. Moreover, curves labelled CoreModel and QuarterModel are obtained
by numerical simulation and represent the temperature at the center and at one quarter of
the thickness, respectively. Numerically predicted temperature at the core and at onequarter of thickness (Figure 2.4a) closely follow the evolution of in situ measurements. In
particular, the plateau temperatures and the times when they are reached seem to be quite
similar. The total gas pressure curves (predicted and measured) are shown in Figure 2.4b
76
with standard deviation bars. In the second half of the pressing period, experimental
measurements of the total gas pressure exhibit more dispertion (large standard deviation
bars) whose coefficient of variation (ratio of the standard deviation to the mean) is
approximately 6% (Figure 2.4b). Numerically predicted total gas pressure seems to be
constant through the mat thickness. Hence, the predicted values of gas pressure at the core
and the surface of the mat are equal, and both curves are superimposed (curve labelled
Surface&CoreModel) and are identified by the same symbol in Figure 2.4b. The model
captures the major trends and gives results of good quality and somewhat closer to
experimental results than those presented by Thömen and Humphrey (2006) and by
Zombori et al (2004). When compared with experimental measurements, numerically
predicted results for the temperature and the gas pressure exhibit satisfactory behavior.
However, the absence of the total gas pressure gradient in the vertical center plan should be
underlined. The same phenomenon is observed by most of the investigators presenting
numerically predicted total gas pressure during the hot pressing of wood-based panels
(Carvalho and Costa 1998; Thömen 2000; Carvalho et al 2003; Zombori et al 2003; Pereira
et al 2006; Thömen and Humphrey 2006; Yu et al 2007). Nevertheless, we and all of the
above authors observed the development of a significant horizontal total gas pressure
gradient, especially in the core driving the gas out of the mat.
a)
b)
Figure 2.4: a) Temperature evolution in time: measurements at the surface, the core and
one quarter of the thickness, and numerically predicted results at the core and one quarter
of the thickness.
b) Total gas pressure evolution in time: measurements and numerical results at the surface
and the core. Curve labelled Surface&CoreModel is obtained by numerical simulation and
the other two are measured in the laboratory.
In all figures, special symbols such as □, ○, *, ◊, , etc, are used to distinguish different
curves and do not represent experimental data unless the contrary is explicitly indicated.
77
Figure 2.5 presents numerical predictions of the evolution of moisture content (M) (Fig.
2.5a) and relative humidity (RH) (Fig. 2.5b) in the mat during the hot pressing process. The
results are presented for five equidistant points laying in the vertical center line of the mat:
at the core, at one-quarter of the thickness, at the surface, and at the mid-points between the
core and one-quarter of the thickness (BCQ) as well as between the surface and one-quarter
of the thickness (BSQ). The sorption isotherm model relates moisture content to relative
humidity and temperature. Therefore, it is not surprising to observe very similar behavior of
the curves representing the M evolution and the RH evolution in the mat. Comparable
observations were also made by Yu et al (2007). Furthermore, from the early stages of the
hot pressing process, the temperature of the surface increases rapidly causing the
evaporation of bound water and thus decreasing M and increasing gas pressure at the
surface. This induces vapor flow towards the inner layers. Given that inner layers have
lower temperature, water vapor condenses and thus increases the local moisture content
(Yu et al 2007). A sequence of peaks of local moisture content presented in Figure 2.5a
clearly shows the movement of M from the surface region towards the core layer. Because
of that, the amount of bound water present in the core region of the mat increases with time.
It takes large amount of energy and time to evaporate the accumulated bound water. That
explains the long-lasting temperature plateau in the core (Fig. 2.4a).
a)
b)
Figure 2.5 : Numerical predictions of moisture content and relative humidity at 5
equidistant points in the vertical center line (BSQ=Between Surface and Quarter;
BCQ=Between Center and Quarter). Evolution of : a) moisture content; b) relative
humidity.
Wet mat density is a function of the oven-dry density and the moisture content of the mat.
As can be seen in Figure 2.2, the time and space evolution of wet mat density is mostly
influenced by predefined oven-dry density profile (Figure 2.2a).
78
Figure 2.6 summarizes the results obtained for partial air (Pa) and vapor (Pv) pressures at
five representative locations in the vertical center plan. Figure 2.6a shows that, for the first
30 s of pressing process, the air pressure quickly drops at the surface while it remains
almost stable in the inside layers. At the same time, due to evaporation process which is
taking place close to the hot surface, the vapor pressure exhibits the opposite behavior: it
increases at the surface and remains unchanged elsewhere (Figure 2.6b). This creates
vertical partial air and vapor pressure gradients. However, as the hot pressing process
continues, the air pressure is rapidly decreasing, which indicates that air is leaving the mat
by the edges. As the heat penetrates deeper into the mat, the amount of bound water
evaporated gradually increases causing a noticeable increase of vapor pressure. Hence, the
vapor replaces the air and occupies a very large proportion of the gaseous phase. Higher
temperature (more evaporation) and density (lower permeability) contribute to the gas
pressure build-up, especially in the core. The difference in gas pressure between the core
and the edges results in gas flow in the panel (horizontal) plane. This is in agreement with
observations made by Yu et al (2007).
a)
b)
Figure 2.6 : Numerical predictions of partial air and vapor pressure at 5 equidistant points
in the vertical center line (BSQ=Between Surface and Quarter; BCQ=Between Center and
Quarter). Evolution of : a) partial air pressure; b) partial vapor pressure.
79
Finally, Figure 2.7 summarizes numerical predictions at four representative locations of the
degree of resin cure and the resin curing rate. As expected, at the beginning of the hot
pressing process, the resin curing rate is the highest at the surface (Figure 2.7b) resulting in
the fastest increase in the resin cure degree which rapidly reaches its highest value (Figure
2.7a). Because of the high temperature, all of the available resin at the surface quickly
polymerized. Then, the curing rate in that region quickly vanishes. As the temperature of
the layers closer to the core increases, so does the amount of cured resin. That increases the
degree of resin cure towards its maximum value (Figure 2.7a). Of course, as the reactants
are used up, the rate of resin cure consequently diminishes and tends to zero (Figure 2.7b).
These results are in agreement with those presented by Yu et al (2007).
a)
b)
Figure 2.7 : Numerical predictions of degree of resin cure and resin curing rate at 4 points
in the vertical center line (BSQ=Middle Point Between Surface and Quarter).
Evolution of : a) resin cure degree; b) resin curing rate.
80
CONCLUSIONS
This paper presents a detailed 3D mathematical approach of the development of a physicalmathematical model for heat and mass transfer that occurs in the MDF mat during hot
pressing process. The complex nature and interactions of different physical phenomena are
described by means of three strongly coupled and nonlinear conservation equations. The
conservation equations are expressed as functions of the three state variables of the model,
namely temperature, air pressure, and vapor pressure, and form a coherent system. Those
equations take also into account the curing kinetics of UF resin and Malmquist’s model
describes the sorption isotherm. Physical and heat and mass transfer properties of the fiber
mat are also considered time and position dependent. The fully coupled system of
governing nonlinear equations, discretized by finite element method, is solved by Newton’s
method on a reference domain and mathematical details of the transfer of those equations
from a real-world domain to a reference domain are presented. The model produced good
predictions of the evolution of several variables related to heat and mass transfer.
Numerically predicted results for temperature and gas pressure exhibited a fair
correspondence with experimental data. Predicted evolution of moisture content, relative
humidity, partial air and vapor pressures, and the extent of the resin cure were all in
agreement with previously published results. The model thus provides a good and
reasonably reliable insight in the complex dynamics of heat and mass transfer phenomena.
81
NOMENCLATURE
t : time [s]
x : length [m]
y : width [m]
z : thickness [m]
T : temperature field [K] ; a state variable calculated by the model
Pa : partial air pressure [Pa] ; a state variable calculated by the model
Pv : partial vapor pressure [Pa] ; a state variable calculated by the model
P : total gas pressure [Pa]
M : moisture content [dimensionless]
h : relative humidity [dimensionless]
PvSAT : saturated vapor pressure [Pa]
Ma : molar mass of air [kg/mol]
Mv : molar mass of water vapor [kg/mol]
R : universal gas constant [J/(mol·K)]
 a : density of the air [kg/m3]
v : density of the water vapor [kg/m3]
 OD : oven-dry density of the mat [kg/m3] (see APPENDIX 2)
Φ : porosity of the mat [dimensionless]
 Mat : wet density of the mat [kg/m3]
KT : thermal conductivity tensor [J/(m· s·K)]
Kp : tensor of specific (effective) gas permeability of the mat [m3/m]
Deff : tensor of effective diffusion coefficient [m2/s]
Dva : binary molecular diffusion coefficient of the air-vapor gas mixture [m2/s]
kd : obstruction factor [dimensionless]
Hfg : latent heat of vaporization (desorption + evaporation of bound water; condensation +
adsorption of water vapor) [J/kg]
CMat : mass specific heat capacity of the mat at current moisture content [J/(kg·K)]
Ca : mass specific heat capacity of air [J/(kg·K)]
Cv : mass specific heat capacity of water vapor [J/(kg·K)]
Cbw: mass specific heat capacity of the bound water [J/(kg·K)]
 : dynamic viscosity of the air-vapor mixture [Pa·s]
a : dynamic viscosity of the air [Pa·s]
v : dynamic viscosity of the water vapor [Pa·s]
hT : convective heat transfer coefficient associated to the external boundary [J/(m2 · s· K)]
hp : convective mass transfer coefficient associated to the external boundary [m]
q T : heat flux [J/(m2· s)]
q Pa : air flux [kg/(m2· s)]
q Pv : water vapor flux [kg/(m2· s)]
EMC : equilibrium moisture content [dimensionless]
82
RH : relative humidity [dimensionless]
MTT : mat target thickness [m]
Minit : initial moisture content of the mat [dimensionless]
Tinit : initial temperature of the mat [K]
hinit : initial value of relative humidity [dimensionless]
PvSAT init: initial value of saturated vapor pressure [Pa]
Pv init : initial value of partial vapor pressure [Pa]
Pa init : initial value of partial air pressure [Pa]
Tsurface : temperature at the surface in contact with the hot platen [K]
hamb : relative humidity of ambient gas [dimensionless]
Tamb : temperature of the ambient gas [K]
PvSAT amb: saturated vapor pressure in ambient gas [Pa]
Pamb : ambient gas pressure [Pa]
Pv amb : ambient vapor pressure [Pa]
Pa amb : ambient air pressure [Pa]
83
APPENDIX 1
Expressions of some parameters used in the calculations.
P = Pa + Pv : Dalton’s law
M : defined at every point in the mat by a sorption model, we use Malmquist’s sorption
model as a reference
MS
M Malmquist 
I
1


1  N   1 3
h 
where MS, N and I are second order polynomials of absolute temperature T [K] given by
MS  0.40221  9.736 105  T  5.8964 107  T 2
N  2.6939  0.018552  T  2.1825 106  T 2
I  2.2885  0.0016742  T  2.0637 106  T 2
h
Pv
PvSAT
6516.3


PvSAT  exp 53.421 
 4.125  ln T   (Kirchoff’s formula)
T


Ma = 28.951·10-3
Mv = 18.015·10-3
R = 8.314472
M a Pa
(ideal gas law)
RT
M P
v = v v (ideal gas law)
RT
a =
 =1–
OD
(Siau 1984)
1530
 Mat =  OD (1+M)
D
Deff  va I , where I is identity tensor
kd
1.75
 101325   T 
Dva  2.6  10  


 P   298.2 
k  0.334  e A , A  5.08 103  
5
d
Mat
H fg  2.511 10  2.48 10  T  273.15  1.172 106  e 0.15M 100
6
CMat 
3
1131  4.19  T  273.15  4190  M
Ca = 1003.5
1 M
84
Cv = 1950
Cbw = Cwater = 4190: because of lack of data
P
P
  a  a  v v
P
P
1.37 106  T 1.5
a 
T  85.75
1.12 105  T 1.5
v 
T  2937.85
hT = 0.35
hp = 2*10-11
MTT = 0.013 m
Minit = 0.12
Tinit = 298.15
hinit calculated by Malmquist’s formula
3
 1  MSinit

1
 1 
 1 

hinit
 
 N init  M init
I init
where
MSinit  0.40221  9.736 105  Tinit  5.8964 107  Tinit 2
N init  2.6939  0.018552  Tinit  2.1825 106  Tinit 2
I init  2.2885  0.0016742  Tinit  2.0637 106  Tinit 2


6516.3
 4.125  ln Tinit  
PvSAT init  exp 53.421 
Tinit


Pvinit  hinit  PvSAT init
Painit  101325  Pvinit
Tsurface : temperature at the surface in contact with the hot platen; its evolution in time is
imposed by the Dirichlet boundary condition and the values are prescribed by measured
experimental data (see Figures 2.3 and 2.4a)
hamb = 0.3
Tamb = Tsurface : because the size of the mat is much smaller than the platens of the press, the
temperature of the air surrounding the mat under compression is much higher than 298.15
and is supposed to be equal to the temperature at the surface of the mat


6516.3
PvSAT amb  exp 53.421 
 4.125  ln Tamb  
Tamb


Pamb = 101325
Pv amb = hamb  PvSAT amb
Pa amb = Pamb  Pv amb
85
APPENDIX 2
A predefined mat oven-dry vertical density profile (  OD [kg/m3]) was used for calculations.
This profile is space and time dependent and is based on the results presented by Wang and
Winistorfer (2000). A similar approach was adopted by Kavazović et al (2010). The
mathematic representation of the density profile is given by the expression presented
subsequently. For the sake of clarity, this expression is graphically presented in Figure 2.2
to illustrate the density profile as a function of time and space (position in the thickness
direction). Because the symmetry of the vertical density profile is assumed, its evolution is
only presented for half of the mat thickness. In the mathematical expression of the profile,
“t” represents time and “z” represents position in the thickness direction. Furthermore, in
the thickness (“z”) direction the density profile is divided into four sections. In each
section, the density profile is expressed by a different function. Of course, the overall
continuity of the density profile is ensured by the way the four functions are constructed.
These functions are defined as follows:
LD = Low-Density Section,
MD = Medium-Density Section,
HD1 = First Part of the High-Density Section,
HD2 = Second Part of the High-Density Section.
OD
 LD

0  z  0.00455


0.00455 < z  0.00585 
 MD


 HD1 0.00585  z  0.006175
 HD 2 0.006175 < z  0.0065 
where
36.60305611+13  exp(0.08596321714  t) 0  t  35 

300
35 < t  50 

LD  

50 < t  160 
 300+(t-50)  (0.909091+99.9001  z )

400.00001+10989.011  z
160 < t 

 BPHD  TPLD 
MD  TPLD   z  0.00455  
 0.00585  0.00455


0  t

36.60305611+13  exp(0.08596321714  t) 0  t  35 

300
35 < t  50 

TPLD  

231.8181772+1.363636455  t
50 < t  160 


450.00001
160 < t 
86
36.60305611+13  exp(0.08596321714  t) 0  t  35 

300
35 < t  50 

BPHD  

100+4  t
50 < t  160 


740
160 < t 
0  t  35 
 36.60305611+13  exp(0.08596321714  t)

300
35 < t  50 

HD1  

300+(t-50)  (-12.36363636+2797.202797  z ) 50 < t  160 

-1060+3.076923077  105  z
160 < t 
0  t  35 
 36.60305611+13  exp(0.08596321714  t)

300
35 < t  50 

HD 2  

300+(t-50)  (22.18181827-2797.202797  z ) 50 < t  160 

2740.00001- 3.076923077  105  z
160 < t 
87
APPENDIX 3
The press closing schedule is a time dependent function. It can be expressed in terms of the
mat target thickness (MTT) as a percentage of MTT and named PTT  t  . Figure 2.1
illustrates its evolution in time. For example, when the press is in the position
corresponding to 140% of MTT, PTT=140%. Then, the actual thickness of the mat equals
140% * MTT/100% = 1.4 * MTT. The time evolution of PTT(t)/100% used for our
simulations is given by the following expression:
0  t  35 
 14 - 0.36  t

1.4
35 < t  50 


1
 87

PTT (t ) 
50 < t  160 
t
  55 275

100% 
1
160 < t  270 


7
 461

 650  6500  t 270 < t  335
ACKNOWLEDGMENTS
The authors wish to thank the Natural Sciences and Engineering Research Council of
Canada (NSERC), FPInnovations – Forintek Division, Uniboard Canada and Boa-Franc for
funding of this project under the NSERC Strategic Grants program.
88
Chapitre 3
Modélisation numérique du processus de pressage à
chaud des panneaux MDF
Couplage du modèle mécanique avec le modèle de
transfert de chaleur et de masse
Ce chapitre est constitué de l’article intitulé
“NUMERICAL MODELING OF THE MEDIUM-DENSITY FIBERBOARD
HOT PRESSING PROCESS
PART 2. COUPLED MECHANICAL AND HEAT AND MASS TRANSFER MODELS”
Cet article sera soumis à la revue Wood and Fiber Science (Society of Wood Science and
Technology).
Les auteurs de l’article sont Zanin Kavazović, Jean Deteix, André Fortin et Alain Cloutier.
89
Résumé
Nous présentons le couplage d'un modèle mécanique avec le modèle de transfert de chaleur
et de masse introduit dans la première partie afin de décrire les phénomènes complexes de
compression du panneau et de transfert de masse et de chaleur qui surviennent durant le
pressage à chaud des panneaux de fibres de densité moyenne. Ce modèle global décrivant
les interactions entre les mécanismes rhéologiques et ceux de transfert de masse et de
chaleur est basé sur des principes de conservation non linéaires et fortement couplés. Le
modèle de transfert de chaleur et de masse est constitué des équations de conservation de
l’énergie, de la masse de l’air et de la masse de la vapeur d’eau, résultant en un problème
instationnaire tridimensionnel dans lequel les variables d’état et les propriétés matérielles
du panneau varient dans le temps et l’espace. Ces équations de conservation sont exprimées
comme fonctions des trois variables d’état du modèle (la température, la pression de l’air et
la pression de la vapeur) et sont résolues par la méthode de Newton en tant qu’un système
d’équations couplées. Le couplage entre le modèle mécanique et le modèle de transfert de
masse et de chaleur est pris en compte et le profil non homogène de densité est calculé
dynamiquement durant la simulation. Le comportement d’un matériau élastique vieillissant
est décrit par le modèle mécanique dans un contexte tridimensionnel général. Le processus
de vieillissement est pris en compte et on fait dépendre les propriétés rhéologiques du
temps, de l’espace, de la température, de la teneur en humidité et de la polymérisation de la
résine. De plus, on tient compte des phases de durcissement et d’assouplissement du
matériau qui sont alors représentées par deux lois de comportement distinctes. Le modèle
mécanique s’exprime en formulation incrémentale quasi-statique en termes de champ de
déplacement. Le modèle mécanique ainsi que le modèle de transfert de masse et de chaleur
sont tous les deux discrétisés en espace par la méthode des éléments finis. Le schéma de
Gear (implicite arrière du second ordre) est utilisé pour la discrétisation en temps. Cela
procure davantage de flexibilité dans le choix de la longueur du pas de temps et permet
d’abaisser éventuellement le coût global des calculs. Le modèle comprend le transfert
conductif et convectif de chaleur, le changement de phase de l’eau, le transfert convectif et
diffusif de masse. Les équations du modèle tiennent également compte de la polymérisation
de la résine urée-formaldéhyde ainsi que de la chaleur latente associée au changement de
phase de l’eau. On suppose l’existence de l’équilibre thermodynamique au niveau local et
on emploie le modèle de sorption de Malmquist pour décrire la dépendance de la teneur en
humidité de la température et de l’humidité relative du panneau. La fermeture de la presse
est prise en compte alors que le développement du profil non homogène de densité est
prédit par le modèle mécanique qui est couplé au modèle de transfert de chaleur et de
masse. Les calculs sont faits sur une géométrie en mouvement dont la déformation
(compression) est une conséquence de la fermeture de la presse.
Les résultats obtenus par le modèle montrent en général une bonne concordance avec les
mesures expérimentales. Le modèle fournit également de l’information utile sur les
variables d’intérêt telles que le profil de densité, la pression du gaz, la pression de l’air et de
la vapeur, la température, la teneur en humidité, l’humidité relative et le degré de
polymérisation de la résine.
Mots clefs: Modèle mathématique, pressage à chaud, couplage du modèle mécanique avec
le modèle de transfert de chaleur et de masse, domaine mobile, méthode des éléments finis,
polymérisation de la résine.
90
Abstract
Coupled mechanical and heat and mass transfer mathematical models describing complex
phenomena of mat compression and heat and moisture transfer occurring during the hot
pressing of medium-density fiberboard (MDF) mats are presented. This global model
depicting intimate interactions between rheological and heat and mass transfer mechanisms
is based on coupled and nonlinear conservation principles. The heat and mass transfer
model consists of equations of conservation of energy, air mass and water vapor mass,
resulting in a three-dimensional unsteady problem in which the fiber mat’s properties and
state variables vary in time and space. These conservation equations are expressed as
functions of three state variables (temperature, air pressure, and vapor pressure) and are
solved together as a fully coupled system by means of Newton’s method. The coupling
between mechanical and heat and mass transfer models is taken into account and the non
homogeneous density profile is dynamically calculated during the simulation. Behavior of
the ageing linear elastic material is described by a mechanical model in the general threedimensional context. Ageing process is taken into account and rheological properties of the
mat depend on time, space, temperature, moisture content and resin cure. Moreover, the
hardening and softening phases of the material behavior are accounted for and treated with
separate constitutive laws. The mechanical model is expressed in a quasi-static incremental
formulation as a function of displacement field. Both mechanical and heat and mass
transfer models are discretized in space by the finite element method. The Gear (implicit
second order backward) scheme is employed for time discretization providing more
flexibility in the choice of the time step and eventually lowering the overall computational
cost. Furthermore, the model includes conductive and convective heat transfer, phase
change of water, convective and diffusive mass transfer. UF resin curing kinetics and latent
heat associated to the phase change are also included in the governing equations. Local
thermodynamic equilibrium is assumed and Malmquist’s sorption isotherm model is used
to describe dependence of moisture content of the mat on temperature and relative
humidity. Press closing is taken into account and the development of non homogeneous
density profile is predicted by the mechanical model which is coupled to heat and mass
transfer model. All calculations are carried out on a moving geometry whose deformation
(compression) is a function of press closing schedule.
Model results exibit good overall agreement with experimental measurements from
laboratory batch press. Moreover, under various press closing schedules, the model is able
to produce valuable information on variables of interest such as density profile, total gas
pressure, air and vapor pressure, temperature, moisture content, relative humidity, and
degree of resin cure.
Keywords: Mathematical model, hot pressing, coupled mechanical and heat and mass
transfer models, coupling, moving domain, finite element method, resin cure dynamics, non
homogeneous density profile.
91
INTRODUCTION
When a wood composite mat is hot pressed, mechanical deformation and heat and moisture
transfer processes are intimately coupled and strongly interact with each other (Nigro and
Storti 2001; Zombori et al 2003; Dai and Yu 2004; Thömen and Humphrey 2006; Thömen
and Ruf 2008). Dynamical development of a density profile (outcome of a compression
process and a complex viscoelastic stress-strain relationship) is enhanced by a softening
effect of moisture content and heat (Bolton et al 1989; Thömen and Ruf 2008). On the other
hand, changes in density profile influence the thermal conductivity, gas permeability and
porosity of a composite mat, thus affecting heat and moisture transfer in the mat.
The literature on combined mechanical and heat and mass transfer models of the hot
pressing process rarely presents detailed information about coupling procedure. Among the
first researchers proposing an integrated approach were Kavvouras (1977), Humphrey
(1982), and Humphrey and Bolton (1989). Recently, models have been developed by Dai
(2001), Carvalho et al (2003), Zombori et al (2003), Pereira et al (2006), and Thömen et al
(2006). Meanwhile, Wang and Winistorfer (2000), Wang et al (2001, 2004), Winistorfer et
al (1996, 2000) published a series of papers presenting gamma-ray in situ measurements to
investigate the density profile development during hot pressing. Unfortunately, the
apparatus needed to conduct those experiments is not commonly available. Therefore, to
gain insight into dynamic development of the density profile, an approach based on
numerical simulation appears as a promising avenue (Thömen et al 2006).
The mechanical behavior of wood based composites is influenced by moisture content (M)
and temperature (T) following changes in environmental conditions. This hygrothermal
ageing (time dependence of the mechanical properties) induces dependence of the
rheological parameters upon evolving M and T. Thus, the coupling of the mechanical model
with the heat and mass transfer model becomes necessary and helps describing more
accurately interactions between heat and mass transfer and rheological mechanisms.
Hot pressing is a time dependent mechanical process (Dubois et al 2005). For the
anisotropic case, the literature is mainly focussed on non ageing materials (Zocher et al
1997; Poon et al 1998 and 1999). For the ageing case, Dubois et al (2005) have developed a
1D viscoelastic model conforming to the thermodynamic principles based on a generalized
Kelvin-Voigt model.
The aim of this work was to develop a numerical model for the linear elastic mechanical
behavior of an ageing MDF mat (whose rheologic properties depend on time, temperature,
moisture content and resin cure) and to describe the methodology developed and solution
strategy implemented to simulate MDF hot pressing. To describe the MDF hot pressing
process, we propose a global coupled mechanical and heat and mass transfer numerical
model based on the finite element method. In Part 1 of this paper, equations of conservation
of energy, air mass and water vapor mass were proposed to model heat and mass transfer.
This 3D unsteady mathematical model was expressed as function of three state variables:
temperature, air pressure, and water vapor pressure. The assumed boundary conditions, the
time- and space- dependent material properties of the mat, and the numerical solution
methods and strategy were also presented. Since all relevant details regarding heat and
92
mass transfer model were provided in Part 1, we shall only discuss here the new features
related to the mechanical model and the coupling of those two models. The robustness and
flexibility of the global model were tested under various pressing conditions and the model
was used to perform several tests and case studies.
MATERIAL AND METHODS
The results of our coupled numerical model were validated against the same experimental
data as those presented in Part 1 of this paper. Relevant details regarding materials and
panel manufacturing are also presented in Part 1 and shall not be repeated here.
Methods
We propose and describe an approach to couple mechanical and heat and mass transfer
models describing complex phenomena resulting from interactions between rheological and
heat and moisture transfer mechanisms during the hot pressing of MDF mats. Those
interactions contribute to the development of non homogeneous density profile. Its timeand space-dependent development is predicted by a mechanical model which is combined
with heat and mass transfer model. Since material and rheological properties of the mat
depend on time, density, resin cure, temperature and moisture content (ageing material)
(APPENDIX 1), the coupling of mechanical and heat and mass transfer models seems
necessary. The proposed models have a general 3D mathematical formulation. Numerical
procedure combines the finite element method with a quasi-static incremental formulation.
Linear elastic model for an ageing material was applied and a composite constitutive law
combining both Hooke’s and tangent laws is elaborated in order to meet thermodynamic
principles. Press closing and effect of a changing mat thickness on the material and
rheological properties of the mat are also taken into account. Moreover, the geometry of the
working domain evolves during the pressing process. In Part 1 of this paper, material
(Lagrangian) formulation was adopted and calculations were transferred to reference
domain. In Part 2 however, updated Lagrangian formulation of all equations is used and
calculations are carried out on a dynamically moving domain. The mesh grid moves as well
and is updated after each time step. Updated Lagrangian formulation enables us to capture
the movement of the domain in a natural way. Resolution strategy and coupled mechanical
and heat and mass transfer models were integrated into the finite element code MEF++
developed by the Groupe interdisciplinaire de recherche en éléments finis (GIREF) at Laval
University.
Overall Approach and Assumptions
Expressions and equations describing material properties, sorption model and resin cure
kinetics were obtained from available literature and presented in APPENDIX 1 of Part 1 of
this paper. None of the fiber mat material and rheological properties was obtained from the
panels produced in the laboratory. Expressions of coefficients for the fourth order elasticity
tensor are based on information presented by Thömen et al (2006) for the Burger’s model.
Those coefficients were first obtained by von Haas (1998) (APPENDIX 1) using curve
fitting of experimental data. We slightly modified those expressions to take into account the
93
contribution of resin cure to the change of elastic properties of the fiber mat (APPENDIX
1).
Following the approach proposed by Dubois et al (2005), we imposed that our rheological
model satisfies the second principle of thermodynamics (positive dissipation hypothesis). In
our ageing linear elasticity model, two distinct constitutive laws are proposed to comply
with the thermodynamic requirements: the Hooke’s law for the softening (moistening) and
the tangent law (Bazant 1979) for the hardening (drying) behavior. During hot pressing,
drying and moistening can occur at the same time in different regions of the mat.
It is assumed that the mass of oven-dry fiber material in each grid element is constant
(Thömen 2000; Thömen and Humphrey 2006). Actually, the volume of each element
changes over time as a consequence of mat compression. Thus, the calculated oven-dry
density profile changes over time.
Wang and Winistorfer (2000), Winistorfer et al (2000), Wang et al (2001, 2004), and
Thömen and Ruf (2008) demonstrated the influence of the choice of the pressing schedule
on the development of the vertical density profile. In our numerical study, tests were
conducted for different press closing schedules. However, since only one pressing schedule
was used to perform our laboratory experiments, validation of numerical results was
possible only for that pressing schedule. Press opening (venting period) was not modeled.
The total pressing cycle considered in our numerical study had duration of 268 s.
Moving Domain and Material Derivative
Mathematical concepts introduced in this section are presented in more details in Garrigues
(2007). Since the fiber mat is compressed and changes shape, it can be considered as a
moving domain. Once material particles are within the mat, they always belong to the mat
(no material particles are lost nor added). Therefore, the mass of the mat remains constant
over time. However, since the volume of the mat is changing, its density also changes. To
mimic the compression of the fiber mat and to follow the domain in its movement, all
calculations were performed on a moving geometry. Calculations of the displacement field
over the domain allow keeping track of the movement of each material particle.
To each material particle, p, of a moving domain one can associate different physical
quantities, G, such as scalar functions (temperature, moisture content, pressure, etc),
vectors (displacement, velocity, etc) or tensors (thermal conductivity, strain, etc). Material
(particular) derivative of G is defined as time derivative of G when following a particle p of
the material domain in its movement.
Suppose that G is a function of an independent variable t and of three real-valued functions
f, g, h which are also associated to p and depend on t. Thus, one can write
G ( p, t )  G ( p, t , f ( p, t ), g ( p, t ), h( p, t )) . The chain rule applies and the material derivative
of G associated to an arbitrary but fixed material particle p is given by
94
DG G ( p, t ) G Df ( p, t ) G Dg ( p, t ) G Dh( p, t )




Dt
Dt
Dt
Dt
t
f
g
h
(86)
Since the particle p is followed in its movement, suppose that it occupies a position P1 at
time t1 and a position P2 at time t2 with t2  t1  t . Then, each time derivative on the right
hand side is regarded as a limit (with fixed particle p); for instance,
Df ( p, t )
f ( P2 , t1  t )  f ( P1 , t1 )
 lim

t

0
t
Dt
(87)
From the numerical standpoint, a finite difference scheme is used to discretize and
approximate the time derivatives such as the one presented in Eq.(87). Displacement of a
particle p is calculated at each time step by the mechanical model and its position is
updated. That is how both the shape and position of the working domain evolve in time.
Since no material particle moves in or out of the material domain, the same conservation
principles of physical quantities associated to material particles of a moving domain apply
as those presented in Part 1 of this paper. Therefore, the conservation equations involved in
the heat and mass transfer model presented in Part 1 remain the same and shall not be
repeated here. However, we should make the following remarks regarding the conservation
of mass of oven-dry fiber material in a moving domain. Following the development
proposed by Duvaut (1998), the mass of dry solid media contained within any arbitrary
time evolving subdomain  (t ) of a moving mat (t ) is given by
massOD ( (t )) 


OD d 
for all  (t ) in (t )
(88)
(t )
where  OD is the oven-dry density. The mass of dry solid media does not change over time,
thus the transport theorem (Duvaut 1998) applied to the mass conservation equation gives
D
D
 D

massOD ( (t )) 
OD d    OD  OD div(v) d   0 , for all  (t ) in (t )

Dt
Dt  ( t )

 ( t )  Dt
(89)
where v( x, t ) is a velocity field. Since the last equality is true for any  (t ) in (t ) , it means
D OD
that
 OD div(v)  0 . A consequence of this result (Hughes and Marsden 1976;
Dt
Duvaut 1998) is that, for any arbitrary regular function B( x, t ) , we have
95
 D  OD B 

D
OD Bd   
  OD B  div(v) d  

Dt  (t )
Dt

 (t ) 
DB
DB
 D

B  OD  OD div(v) d   OD
d   OD
d

Dt
Dt
Dt


 (t )
 (t )
 (t )

for all  (t ) in (t )
(90)
0
This result is used when expressing the energy conservation of the system in the heat and
mass transfer model on the moving domain. Therefore, the energy conservation equation is
the same as the one presented in Part 1.
Mechanical Model
The governing equation for the mechanical model is expressed in terms of time- and spacedependent displacement field U  x, t  . Compression of the mat obeys Newton’s second law
 MAT
Dv
 div  U    0
Dt
in   t 
(91)
where  MAT is the mat wet density, v the velocity field, and  the second-order stress
tensor (APPENDIX 1), and   t  represents the evolving computational domain (fiber mat).
Dv
) is considered negligible during the hot pressing
Dt
process and will not be taken into account in further discussions.
In Eq.(91), the inertial term (  MAT
Constitutive Law
To establish a constitutive law relating U to  , the following considerations were taken into
account. The phenomenon of material ageing is considered at the macro-level. Ageing is
defined as the time dependency of the material properties and is expressed as a variation of
mechanical properties as function of time. Following Dubois et al (2005) and based on
Bazant (1979), we suppose that all the components of a rheological model must satisfy the
second principle of thermodynamics (positive dissipation). Bazant (1979) have shown that
two distinct constitutive laws are necessary. For softening material, the classical Hooke’s
law satisfies the thermodynamic condition (Dubois et al 2005). For the hardening material
however, the tangent law (Bazant 1979) is considered to comply with the positive
dissipation condition.
Since wood and wood-based composites are hygroscopic materials, ageing is induced by
variable moisture content and temperature conditions. This is taken into account by making
rheological properties depend upon evolving M and T (Dubois et al 2005); hence material
properties vary in time. An increase in moisture content and temperature softens wooden
material and its stiffness decreases. At the opposite, as moisture content and temperature
96
decrease, the material hardens and becomes stiffer. Therefore, variations in M and T are
directly linked to softening and hardening of the material.
An increase in M and T in solid wood generates swelling, whereas a decrease causes
shrinkage (Hunt and Shelton 1988; Dubois et al 2005). In the current study, as a first
approach, swelling and shrinkage were not taken into account.
During hot pressing, heat and moisture move from the surface in contact with the hot
platens towards the core. There are therefore two opposite phenomena that are
simultaneously taking place within the mat undergoing hot pressing: desorption occurs in
one region and sorption in another. Therefore, to satisfy thermodynamic principles,
Hooke’s law and the tangent law must be simultaneously considered leading to a composite
constitutive law for ageing linear elastic behavior:
D

E:

D 
Dt

D DE
Dt 
:
E:


Dt Dt
DE
 0 (hardening, tangent law)
Dt
DE
 0 (softening, Hooke's law)
Dt
   E :   ( E )  : 
(92)
where  is a second-order Cauchy stress tensor (APPENDIX 1, Eq.(104)), E is a fourthorder elasticity tensor (APPENDIX 1, Eq.(104)), and the second-order strain tensor
1
DE
T
 U   U   U   . For the time derivative of E , E 
, only the negative part of

2
Dt
 E E  0
. The reader should note that the time
each component is retained: ( E )   

0 E  0
derivatives of the tensor E as well as the negativity conditions ( E )  are calculated
component by component. The time dependency of the components of elasticity
tensor E appears through their relation with mat density (  MAT ), local moisture content (M),
temperature (T) and degree of resin cure (APPENDIX 1, Eq. (101)). They are thus
implicitly time-dependent.
97
Incremental Formulation
From Eq.(87), one can see that the time derivatives of  and  can be approximated
(Ghazlan et al 1995; Dubois et al 2005; Beuth et al 2008) by
D 

Dt t
;
D 

t
Dt
(93)
where  and  are instantaneous increment of strain and stress, respectively. With
N  0,1, 2,... and t0  0 , we denote time increment t  t N 1  t N ,  N    t N  ,
EN  E  t N  ,  N    t N  with the time variation of E defined as E  EN 1  EN . The
composite constitutive law for ageing linear elastic material behavior (Eq. (92)) now reads
  EN 1 :    E  :  N

(94)
E E  0
where (E )   
. It is usually assumed that the inertial component is
 0 E  0
Dv
negligible (i.e.  MAT
 0 ) and the problem is regarded as quasi-static (Ghazlan et al
Dt
1995; Beuth et al 2008): at each time step, static equilibrium is assumed (quasi-static
assumption). The time evolution of mat geometry is simulated by imposing successive load
increments upon the mat. The compression is regarded as a step-by-step process evolving
by time increment t . Development of the strain and stresses is then regarded as an
incremental process:
 N 1   N  
;
 N 1   N  
(95)
with  N and  N representing the actual accumulated strain and stress whereas  N 1
and  N 1 represent their values at the end of the next load increment. This allows retaining
the accumulated strain and stress history within storage variables  N and  N .
With these assumptions and Eq. (94), Eq. (91) is written in incremental formulation as
follows
div    div(N )
UN1 UN U

 div EN1 :    E : N   div(N )


T
1
with     U    U     U   

2
i.e.
in N
(96)
System (96) is thus written in terms of unknown displacement increment U and is
discretized by the finite element method. Appropriate boundary conditions will be specified
later on.
98
Computational Domain
It is noteworthy that our mathematical model is written in a general three-dimensional form
and that our code can perform simulations on 2D and 3D geometries (Figure 3.1a).
The effect of a daylight delay (the time necessary for the top platen to touch the mat) (Lee
et al 2007) was ignored and geometric symmetry was assumed (Carvalho and Costa 1998;
Carvalho et al 2001; Carvalho et al 2003; Dai and Yu 2004; Nigro and Storti 2001; Thömen
and Humphrey 2003; Thömen and Humphrey 2006; Pereira et al 2006; Yu et al 2007). In
3D, there are three planes of symmetry (Figure 3.1b): a horizontal mid-plane and two
vertical mid-planes. Therefore, our computational domain represents one eighth of the full
3D geometry or a quarter of the full 2D geometry when calculations are performed in 2D.
The domain considered for calculation was meshed with a non-uniform 24 by 24 by 20 grid
whose hexahedral elements where concentrated towards external planes (surface and
exterior edge). The displacement of elements was generated by a geometric progression
with the common ratio of 0.9. We work on a moving domain (Figure 3.1b and 3.1c): 280
mm (half length in x direction) by 230 mm (half width in y direction) by half mat evolving
thickness (half thickness in z direction starting at 91 mm at the beginning of the pressing
and ending at 6.5 mm). Clearly, mat thickness evolves during the pressing process and
Figure 3.1c shows deformation of computational domain at different moments in time as a
result of press closing. The mat is also free to expand in the x- and y- directions.
a)
b)
Figure 3.1 : a) Full 3D geometry of a fiber mat; b) computational domain in 3D (one
eighth of the full geometry).
99
t=0s
t=5s
t=10s
t=15s
t=35s
t=48s
Figure 3.1 c : Evolving 3D computational domain at different moments in time (one eighth
of the full geometry).
100
Density Profile and Mat Compression
Despite the fact that the total mass of fiber material does not change during compression,
material and rheological properties of the mat, such as local density, porosity, permeability
and thermal conductivity change. For instance, in thicker mats, thermal conductivity is
lower, whereas porosity and gas permeability are higher. When the mat is compressed,
thermal conductivity increases whereas porosity and gas permeability decrease. These and
other changes in material properties are accounted for as the pressing process evolves.
Generally, the vertical density profile of compressed composite panels is not uniform. It
mainly exhibits a characteristic “M-shaped” vertical profile with higher density in the
surface layers and lower density in the core (Carvalho et al 2001, 2003; Wang et al 2001;
Thömen and Ruf 2008). It is regularly observed that the transition region from low to high
density is rather thin (Carvalho et al 2001, 2003; Wang et al 2001). This nonuniform
densification is attributed to variations of T and M, and interactions between the heat and
moisture transfer phenomena and the mechanical compression of the mat (Kamke and
Wolcott 1991; Dai and Yu 2004; Thömen and Ruf 2008). In general, the higher
temperature or moisture content, the softer and more compressive the mat gets (Kamke and
Wolcott 1991; von Haas and Frühwald 2000).
In the present work, the evolution of non homogeneous oven-dry vertical density profile of
the mat (  OD [kg/m3]) during the pressing process was calculated by a mechanical model
for an ageing elastic material. Mat thickness decreases as a function of press closing
schedule (Figures 3.1c and 3.2a) of a Dieffenbacher laboratory batch press. The pressing
schedule of 268 s was divided into five steps. The initial mat thickness of about 182 mm
was reduced to 37 mm in the first 15 s (Step 1). The press remained in this position for the
next 10 s (Step 2) followed by the second compression which reduced the mat thickenss to
19 mm at time of 42 s (Step 3). A slow compression phase lasting 120 s followed at the end
of which the mat reached its final thickness of 13 mm (Step 4) at time of 162 s. The hot
platens remained in this position (Step 5) until the time of 268 s. The curve presenting the
evolution of mat thickness with time can be seen in Figure 3.2a. Venting period was not
modeled and therefore is not presented.
From the numerical simulation’s stand point, the densification process can be described as
follows: at each time step, an increment of the platen displacement is imposed as a
Dirichlet boundary condition at the surface in contact with the top platen. As a reaction to
this solicitation, the mechanical model calculates the corresponding displacement of each
mesh node and the mat geometry is updated accordingly. Since the oven-dry mass of the
material remains constant within each element, the oven-dry density evolves because of the
change in volume of each element.
101
a)
b)
Figure 3.2 : a) Evolution of mat thickness: reading of the distance between the two platens
of Dieffenbacher laboratory batch press. Venting period is not modeled and therefore is not
presented.; b) Evolution of Poisson’s coefficient.
Porosity
Heat and mass transfer properties of the fiber mat such as thermal conductivity and specific
gas permeability were presented in Part 1. Mat porosity is calculated by the following
equation
 

  1.1 1 – OD 
1530 

(97)
This is the equation proposed by Siau (1984) multiplied by a correction factor for MDF
mats proposed by Belley (2009). Since mat porosity is a function of oven-dry density, it is
thus time- and space-dependent.
Initial and Boundary Conditions
Appropriate initial and boundary conditions for the heat and mass transfer model are
described in Part 1 of this paper. We shall now consider initial and boundary conditions for
the mechanical model.
At the beginning of the pressing process ( t  t0  0 ), the mat is assumed at rest and stress
free, and the displacement field is assumed null. This is expressed by the following initial
conditions:

U  0  U0  0
and
  0   0  0
(98)
To mimic press closing (Figure 3.2a), at each time step, an increment of displacement field
( U ) is imposed at the top surface in the z direction by a Dirichlet boundary condition
102
( U Z is deduced from the evolution of the mat thickness, Figure 3.2a). No movement in
the x-y plane is allowed at the top surface ( U X  U Y  0 ). Since we take advantage of
symmetry, the working domain represents one eighth of the mat (Figure 3.1a and 3.1b) and
boundary conditions have to be imposed on the 3 symmetry planes. The symmetry plane
z  0 is not allowed to move in z direction ( U Z  0 ). The symmetry plane x  0 does not
move in the x direction ( U X  0 ) whereas y  0 does not move in the y direction
( U Y  0 ). However, they both compress in the z direction following the movement of the
closing press platen. The exterior faces x  280 mm and y  230 mm follow the press
closing movement in the z direction and are both free to expand in both the x and y
  
directions (zero traction  n  0 , n is outward unit normal).
Numerical Coupling of Mechanical and Heat and Mass Transfer Models
The complexity and strong coupled nature of the physical processes involved during heat
and mass transfer are widely recognized in the literature (Bolton and Humphrey 1988,
Humphrey and Bolton 1989; Carvalho and Costa 1998; Nigro and Storti 2001; Zombori et
al 2003; Dai and Yu 2004; Thömen and Humphrey 2006).
We now describe how coupling is dealt with in our numerical solution strategy. At each
time step, the coupled heat and mass transfer model is solved first. The three nonlinear
conservation equations for heat and mass transfer form a fully coupled system and are
solved together. This system is discretized in space by the finite element method using Q1
finite elements (Bathe 1982; Reddy 2006) and is solved by means of Newton’s method
(Kavazović et al 2010). All material properties are updated at each nonlinear iteration,
except for the oven-dry density profile which remains unchanged at this stage. Once the
convergence criterion is reached, the program provides new values for the three state
variables Pa, Pv, and T from which we calculate M. Those updated variables are then used
as input to the mechanical model. Indeed, as they appear in expressions of the ageing
elasticity tensor coefficients, those new values will update the rheological parameters of the
mat and be used in the calculations of mat compression. The increment of the press platen
position is imposed (Dirichlet boundary condition) at the top surface of the mat. The
displacement vector field is then obtained as a solution of the mechanical model which is
discretized in space by the finite element method using Q2 finite element (Bathe 1982;
Reddy 2006). The position of each grid point is then updated by the corresponding
increment of displacement vector ( U ). As the grid points move, the volume of each
element eventually changes. Consequently, the value of the oven-dry density of each
element changes too. The new oven-dry density field is then used to update mat’s heat and
mass transfer properties. Hence, we are ready to undertake the calculations with the heat
and mass transfer model at the next time step.
103
Gear’s time discretization scheme
After the finite element discretization in space is achieved for the heat and mass transfer
problem, we end up with a very complicated version of a first order initial value differential
equation. In our case, that equation is treated by the Gear’s scheme.
Actually, one can apply numerous discretization formulas to solve the following first order
differential equation having a prescribed initial value ( Y0 ) of the solution Y  t 
dY
 F Y  t  , t 
dt
with Y0  Y  0  given initial value
(99)
Implicit methods are in general more stable than explicit ones, and the precision of a
method increases with its order. For instance, for the same length of a time step, a fourth
order method is more accurate than a second order method which is more accurate than a
first order method. The computational burden of a method increases as the order of a
method is higher. Moreover, implicit methods require solution of a nonlinear system and
are therefore more time consuming per time step than explicit methods. However, for the
sake of stability and precision, we adopted Gear’s implicit second order two step backward
differentiation formula. Since it is a two step formula, it requires solution estimates from
two previous time steps in order to calculate the next approximation. Hence, at the very
beginning of the calculation process, a one step second order formula is needed to calculate
the first time step estimate. Usually, second order implicit Crank-Nicholson’s theta-scheme
is used to produce the approximation at the first time step. Hence, the algorithm for the
Gear’s scheme that we used reads as follows (where N is the total number of iterations)
given the initial value Y0
and
Y1 calculated by second order Crank-Nicholson scheme
(100)
Yn 1 
2t
4
1
F Yn 1 , tn 1   Yn  Yn 1
3
3
3
,
n  1, 2,3,..., N
We applied that algorithm as time discretization scheme for the heat and mass transfer
problem.
104
RESULTS AND DISCUSSION
To perform successful numerical simulations, coefficients and expressions for different
material properties are needed. Some of those expressions have to account for interactions
between several properties. For instance, increasing temperature stimulates resin cure and
creation of bonds which ultimately solidifies the entire mat. Thus, adhesive cure influences
rheological properties of the mat. We proposed a formula to account for effects of the
extent of resin cure on modulus of elasticity (Eq. (105) in APPENDIX 1). The proposed
expression is only the first step in characterization of this complex relationship and
thorough investigation is required to gain a deeper insight. More research is also needed to
better understand dynamics of development of Poisson’s ratio during early stages of
compression. Indeed, at the beginning of pressing process, the mat is a loose material. It
eventually gains more cohesion as the pressing progresses. To reflect this transition, an
appropriate formula for Poisson’s coefficient is needed, especially in early stages of
pressing process. We attempted to address that issue by proposing a sigmoid shape function
allowing for a smooth transition from a loose stage to a more cohesive material (Figure
3.2b and APPENDIX 1, Eq.(103)). Experimental data is needed to validate our hypothesis
and have a better understanding of this phenomenon.
The density profile is one of the most critical properties for MDF (Thömen et al 2006).
Coupling of mechanical and heat and mass transfer 3D models allowed for dynamically
predicting its development as a function of pressing schedule. Figure 3.3a shows numerical
predictions of development of a space- and time-dependent oven-dry vertical density
profile in the panel’s center line (axis connecting upper and lower hot platen and passing
through the core). At the early stage, overall density rapidly increases and the vertical
density seems uniform throughout the thickness. About 15 s after the beginning of the
compression, a steep density gradient develops close to the edges. As press closure
progresses, the overall density of the mat increases whereas its thickness decreases. In
Figure 3.3a, one can notice the development of a “U-shape” profile with a high-density
region near the press platens, a significantly lower density in the core, and a transition
region in between. Panels pressed in laboratory present an “M-shaped” profile mostly
because of the resin pre-cure at the surfaces in contact with the hot platens. Since our
mechanical model does not account for the plastic behavior of the mat (region with the precured resin exhibits plastic behavior), the model was not able to reproduce the “M-shaped”
profile. Some specific locations in one eighth of 3D mat geometry are of particular interest,
especially for validation of numerical results. Figure 3.3b shows a symmetry X-Z (widththickness) plane. Black dots represent locations where thermocouples (Surface, Quarter,
and Core) and pressure probes (Surface, and Core) were installed to monitor temperature
and gas pressure evolution, respectively. Points in the vertical center line named BSQ
(Between Surface and Quarter) and BCQ (Between Core and Quarter) as well as locations
identified by empty circles were used as numerical tracing points for all variables.
Results obained at four representative locations (Figure 3.3b) in the symmetry widththickness midplane of a 3D geometry illustrate numerical predictions of the evolution of the
oven-dry (Figure 3.3c) and wet densities (Figure 3.3d), respectively. As expected, density
of the surface layer increases faster than elsewhere. During the Step 5 (Figure 3.2a), the
press platens remain at the same position. Hence, a zero displacement increment is imposed
105
and the oven-dry density profile remained unchanged (Figure 3.3c). This was expected
since we model a linear elastic behavior. There is a qualitative similarity between our
results and those presented by Wang et al (2001; 2004) for a similar pressing schedule.
Oven-dry Density (kg/m^3)
a)
900
t=5s
800
t=10s
700
t=15s
t=25s
600
t=30s
500
t=35s
400
t=40s
t=50s
300
t=100s
-20 -17 -14 -11
-8
-5
200
t=125s
100
t=150s
0
-2
t=268s
1
4
7
Mat Thickness (mm)
b)
10
13
16
19
106
c)
d)
Figure 3.3 : a) Evolution of space- and time-dependent numerically predicted oven-dry
vertical density profile;
b) symmetry X-Z (width-thickness) plane with equidistant tracing points;
c) predicted oven-dry density profile, values at 4 points in the vertical center line;
d) bulk density profile at 4 points in the vertical center line calculated by
 Mat  OD 1  M  .
In all figures, special symbols such as □, ○, *, ◊, , etc, are used to distinguish different
curves and do not represent experimental data unless the contrary is explicitly indicated.
Figure 3.4 presents the evolution with time of numerically predicted vertical stress
component calculated by our 3D global model. Its development is monitored at five
equidistant points in the vertical center line (Figure 3.3b) as the pressing process
progresses. There is no stress gradient observed in the vertical center line.
Vertical (ZZ) Stress Component (MPa)
107
-0.2 0
-0.4
-0.6
-0.8
-1
-1.2
-1.4
-1.6
-1.8
-2
-2.2
-2.4
-2.6
-2.8
-3
50
100
150
200
250
CoreModel
QuarterModel
SurfaceModel
BSQModel
BCQModel
Time (s)
Figure 3.4 : Evolution in time of numerically predicted vertical stress component at 5
equidistant points in the vertical center line (BSQ=Between Surface and Quarter;
BCQ=Between Center and Quarter).
Numerical results obtained by the 3D coupled mechanical and heat and mass transfer
models exibit good overall agreement with experimental measurements. Figures 3.5a and
3.5b present laboratory temperature and gas pressure measurements, respectively, together
with numerically predicted results. Vertical bars represent standard deviation from the
mean value. In Figure 3.5a, curve labelled SurfaceLab is the mean temperature measured in
the laboratory at the surface in contact with the hot platen. That curve was imposed as a
Dirichlet boundary condition for T at the surface. Moreover, curves labelled CoreModel
and QuarterModel are obtained by numerical simulation and represent the temperature at
the center and at one quarter of the thickness, respectively. As can be seen in Figures 3.5c
and 3.5d, numerically predicted temperature at the core and at one quarter of the mat
thickness in the vertical center plane closely follow the evolution of in situ measurements.
In particular, the plateau temperatures and the time when they are reached are similar.
Numerical results underestimate though mean value curves of temperature. The maximum
discrepancy from the mean of measured core temperatures is approximately 4% (Figure
3.5c) and, at one quarter of the thickness, temperature is underestimated by up to 8%
(Figure 3.5d).
In Figure 3.5b, numerical predictions of the total gas pressure (P) at the core and surface
locations are compared to experimental data. Large standard deviation bars, especially in
the second half of the pressing process, reveal significant variations in the laboratory
measurements of the gas pressure (Figure 3.5b). Maximum value of the coefficient of
variation (ratio of the standard deviation to the mean of measured data) is approximately
6%. Numerical results do not exhibit any vertical gradient in total gas pressure predictions.
108
Thus, curves monitoring numerically predicted evolution of P at the surface and at the core
superimpose (curve labelled Surface&CoreModel) and are identified by the same symbol in
Figure 3.5b. From the qualitative stand point, the time evolution of gas pressure is well
captured by the model. However, numerical results overestimate mean value of gas
pressure experimental measurements by 10% in the case of the pressure at the surface and
by 15% at the core (Figure 3.5b). Nevertheless, when compared to other numerical results
in the literature (Thömen and Humphrey 2006; Zombori et al 2004; Pereira et al 2006), our
results are of similar quality.
a)
b)
c)
d)
Figure 3.5: a) Temperature evolution in time: measurements at the surface, the core and
one quarter of the thickness, and numerically predicted results at the core and one quarter
of the thickness.
b) Total gas pressure evolution in time: measurements and numerical results at the surface
and the core. Curve labelled Surface&CoreModel is obtained by numerical simulation and
the other two are measured in the laboratory.
c) Close-up on the evolution of the temperature field (measured and predicted) at the core.
d) Close-up on the evolution of the temperature field (measured and predicted) at one
quarter of the mat thickness.
109
When examining numerical results, the absence of total gas pressure gradient in the vertical
center plane should be underlined. The same phenomenon is observed in most of the
publications presenting the numerically predicted total gas pressure (Carvalho and Costa
1998; Carvalho et al 2003; Zombori et al 2003; Pereira et al 2006; Thömen and Humphrey
2006; Yu et al 2007). Measurements of cross-sectional gas pressure reported by Thömen
(2000) for MDF mats validate these numerical results. Nevertheless, we and all of the
above authors observed the development of a significant horizontal total gas pressure
gradient, especially in the central plane, which drives the gas out of the mat.
Figure 3.6 displays the evolution of predicted mat moisture content (M) (Figure 3.6a) and
relative humidity (h) (Figure 3.6b) at five equally spaced representative locations across the
thickness (Figure 3.3b): at the core, at one-quarter of the thickness, at the surface, and at the
mid-points between the core and one-quarter of the thickness (BCQ) as well as between the
surface and one-quarter of the thickness (BSQ). As expected, moisture content near the
platen drops rapidly and remains low. The curves in Figure 3.6a (a sequence of peaks of
local moisture content) clearly illustrate the movement of bound water from the hot surface
towards the cooler core region. Accumulation of bound water in the central region of the
mat is characterized by an increasing M at the core. Relative humidity presents similar
patterns (Figure 3.6b) since h and M are linked by the isotherm sorption model. Our
observations are in accordance with Yu et al (2007).
a)
b)
Figure 3.6 : Numerical predictions of moisture content and relative humidity at 5
equidistant points in the vertical center line (BSQ=Between Surface and Quarter;
BCQ=Between Center and Quarter). Evolution of : a) moisture content; b) relative
humidity.
110
Figure 3.7 summarizes the results for partial air (Pa) and vapor (Pv) pressures at the same
five locations mentioned previously. Figure 3.7a shows that, at the beginning of the
pressing process, the air pressure rapidly drops in the surface layers while it remains almost
stable in interior layers. At the same time, due to the evaporation process which is taking
place close to the hot surface (indicated by moisture content decrease), the vapor pressure
exhibits the opposite behavior: it increases at the surface and it remains low elsewhere
(Figure 3.7b). This creates vertical (cross-sectional) partial air and vapor pressure gradients.
Since the total gas pressure remains almost constant during the first half of the pressing
period (Figure 3.5b), it can be concluded that the increase in Pv is proportional to the
decrease of Pa. Hence, water vapor replaces air and becomes the main component of the
gaseous phase. Development of vertical Pa and Pv gradients is clearly visible in Figure 3.7.
Those gradients drive the molecular diffusion of air and water vapor within the gas phase.
Furthermore, since the temperature of the surface increases rapidly, the evaporation process
of bound water is intense in regions close to hot platens. As a result, M at the surface region
decreases and the local vapor pressure increases. As seen in Figure 3.7b, a steep Pv gradient
develops. This facilitates the molecular diffusion of the vapor within the gas phase and
makes the vapor flow towards the inner layers. Given that the inner layers have lower
temperature, water vapor condenses and thus increases the local moisture content (Yu et al
2007). Because of that, the amount of bound water present in the core region of the mat
increases with time (Figure 3.6a). It takes large amounts of energy and time to evaporate
the accumulated bound water. That explains the temperature increase slowdown and the
long-lasting temperature plateau in the core (Figure 3.5a). At time t = 160 s, Figure 3.7
suggests that almost all the air was replaced by water vapor in the gas phase. One also
notices that the vertical partial air and vapor pressure gradients vanish (Figure 3.7). Total
gas pressure starts to increase significantly (Figure 3.5b) and the temperature plateau
establishes at the core (Figure 3.5a). This suggests that intense evaporation process of
bound water in the core layer has begun (Figure 3.6a). Moreover, as densification
continues, the gas permeability of the mat decreases and contributes to the gas pressure
build-up, especially in the core layer. That difference in gas pressure between the core and
the edges results in gas flow in the panel’s horizontal plane which becomes the
predominant direction of the mass transfer. This is in agreement with observations made by
Yu et al (2007).
111
a)
b)
Figure 3.7 : Numerical predictions of partial air and vapor pressure at 5 equidistant points
in the vertical center line (BSQ=Between Surface and Quarter; BCQ=Between Center and
Quarter). Evolution of : a) partial air pressure; b) partial vapor pressure.
Numerical explorations
Our goal was to develop a robust numerical tool able to provide reliable numerical results
under different pressing conditions. After the validations of numerical results presented
above, the model was used to perform several tests and case studies. For each time step, it
was observed that the work load was split as follows: one third of the time was spent to
solve the heat and mass transfer problem (Newton’s method converged in 4 or 5 iterations)
and two thirds of the time were spent solving mechanical model (conjugated gradient
method preconditioned by SOR). In our case, 3D computations were 5 to 10 times more
time consuming than 2D calculations. Since the number of tests to be performed was large,
we decided to run them on 2D geometry.
2D versus 3D predictions
When the calculations are performed on 2D instead of 3D geometry, one could expect some
changes to occur in numerically predicted results. To examine the impact of the passage
from 3D to 2D, calculations were performed on the 3D non-uniform 24 by 24 by 20 grid
presented in Figure 3.1c and the 2D non-uniform mesh (elements concentrated in boundary
regions) having 24 rectangular elements in the width and 20 elements in the thickness
(Figure 3.10 presented later on). In both cases, compression dynamics followed laboratory
closing schedule and 0.5 s time step was used.
Comparisons of 2D and 3D numerical results at the core of the panel are illustrated in
Figure 3.8 for several variables. The 2D and 3D results are very similar during the first half
of the pressing period, but some differences can be observed in the second half, mainly in
the plateau values of the variables. Those differences are mostly caused by small in-plane
dimensions of our laboratory panels (0.56m by 0.46m) which enhanced boundary effects
making it easy for the gas to escape by the edges (better venting). Indeed, while performing
112
2D numerical simulations, it is implicitly assumed that the panel has infinite length.
Therefore, in 2D, there is only one boundary in contact with the ambient air which reduces
venting. Hence, in the second half of pressing when the mat gas permeability has
decreased, we observe that, at the core, total gas pressure in 2D is up to 20% higher than in
3D (Figure 3.8a). As a consequence, partial vapor pressure (Figure 3.8b), temperature
(Figure 3.8c) and moisture content (Figure 3.8e) exhibit higher plateau values in 2D as
well. Plateau value for M is increased by at most 3%, whereas the increase of T is about
5%. On the other hand, oven-dry density predictions (Figure 3.8f) do not seem to be
affected by this transfer from 3D to 2D geometry.
a)
b)
c)
d)
113
e)
f)
Figure 3.8 : Comparison of evolution of 2D and 3D numerical results at the core location
for the fields of: a) total gas pressure; b) partial vapor pressure; c) temperature; d) partial
air pressure; e) moisture content; f) oven-dry density.
Convergence with mesh
We verified that the solutions stabilize when the 2D mesh is refined (convergence with the
mesh). Numerical simulations were run under laboratory closing schedule with a time step
of 0.1 s. Calculations were performed on series of uniform rectangular meshes with
increasing number of elements and their solutions were compared. We worked on
following 2D grids where the first number represents the amount of elements in width (x)
direction and the second number is for the amount of elements in thickness (z): 16 by 16;
32 by 16; 64 by 32; 128 by 64; 256 by 128. Numerical results of time evolution of T and M
are used to illustrate the phenomenon. Convergence with increasing mesh size was
observed very quickly. This is illustrated by the results for T and M at the core (Figure 3.9a
and 3.9c). However, the most important discrepancies were noticed close to the edges
(surface boundary and border in contact with the ambient air) (Figure 3.9b and 3.9d). Those
are regions where important variations occur and more elements are needed to adequately
capture the evolution of different phenomena in those areas. Figures 3.9b and 3.9d illustrate
very well that a 16 by 16 mesh for the computational domain (one quarter of the mat) is not
sufficiently refined close to edges. For instance, Figure 3.9d depicts the evolution of M at
the point laying at half width and ¾ of thickness of the computational domain (close to the
surface in contact with the hot platen). We clearly see two groups of curves: the first one
containing solutions on 16 by 16 and 32 by 16 grids, and the solutions obtained on three
other meshes form the second group. Within the time period between 20s and 70s of
pressing process, a clear difference between those two groups of curves can be noticed
(Figure 3.9d). This supports a lack of precision, in that particular region, of uniform meshes
having low number of elements in thickness. Nevertheless, in the zone close to the core,
very small differences are observed among the solutions (Figure 3.9a and 3.9c) and
convergence with mesh size appears clearly.
114
a)
b)
c)
d)
Figure 3.9 : Comparison of evolution of 2D numerical results of T and M at: a) the core
location for T; b) at the boundary in contact with the ambient air for T; c) at the core
location for M; d) at half width and ¾ of thickness for M. Solutions were calculated on
meshes having increasing number of elements.
115
Concentrated mesh
We have just seen that the largest variations appear close to boundaries (hot platen and
exterior border). Non-uniform mesh with higher concentration of elements in those areas
could help to recover the precision of solution in boundary regions. For that purpose, a nonuniform mesh having 24 rectangular elements in the width and 20 elements in the thickness
was created (Figure 3.10). A geometric progression with the common ratio of 0.9 was used
to concentrate the elements towards the hot platen surface and exterior edge in contact with
the ambient air.
Figure 3.10 : 2D mesh (24 by 20 ) : elements are concentrated in boundary regions.
Results obtained on this grid (Figure 3.10) were compared to those obtained on richer
regular grids: 64 by 32; 128 by 64; 256 by 128. Numerical simulations were run under
laboratory closing schedule with a time step of 0.1 s. Figure 3.11 reveals that, in respective
cases, curves for T and M obtained on those four different grids are extremely close to each
other. For instance, when comparing time evolution of moisture content at the point located
at half width and ¾ of thickness (Figure 3.9d and Figure 3.11c), it appears that the behavior
of M close to the hot platen has been adequately captured by a concentrated 24 by 20 mesh
(Figure 3.11c). On the other hand, at the boundary in contact with the ambient air, results
obtained on concentrated 24 by 20 mesh exhibit very acceptable level of precision (Figure
3.11d). Thus, it seems that even a coarse grid (24 by 20) can capture well the complexity of
physical phenomena under study (Figure 3.11) if the elements of the mesh are concentrated
in appropriate areas of the mat.
116
a)
b)
c)
d)
Figure 3.11 : Comparison of evolution of 2D numerical results of T and M at: a) the core
location for T; b) at the boundary in contact with the ambient air for T; c) at half width and
¾ of thickness for M; d) at the boundary in contact with the ambient air for M. Solutions
were calculated on meshes having increasing number of elements.
Influence of the time step length
Impact on the results of the length of the time step was also examined under laboratory
pressing schedule. Tests were performed on two 2D grids (32 by 16 and 64 by 32) with
several lengths of a time step: 0.025 s; 0.05 s; 0.1 s; 0.2 s; 0.4 s; 0.5 s. We used Gear
(implicit second order backward) scheme for the time discretization. Our study revealed
that the solutions converge with decrease of the time step and that the time step has very
little influence on the solutions. This suggests that the combination of finite element
method and Gear scheme allows one to use larger time steps without losing precision. In
our numerical simulations, time step of 0.5 s was often used in 3D, and time steps of 0.1 s
and 0.5 s in 2D.
117
Cold pressing
The test of cold pressing (platens at T=25C) was performed on 64 by 32 grid under
different pressing schedules with a time step of 0.5 s. Since the temperature of the platens is
equal to the ambient temperature, there is neither the heat nor the mass transfer. However,
the closing press platens induce the mat densification over time. Numerical results for the
cold pressing systematically produced flat density profiles through the thickness; that is, the
mat density was increasing over time but, at each time step, it was homogeneous in space.
That was the expected behavior in these conditions.
Pressing schedules
The robustness and flexibility of the global model were tested in different pressing
situations. Numerical simulations were carried out for seven different pressing schedules on
several grids. However, we will only present some of the results obtained on a 64 by 32
grid under a couple of pressing schedules with a time step of 0.1 s.
Figure 3.12 shows 4 different pressing schedules. The press closing dynamics from our
laboratory experiments was used as a reference for qualitative comparisons with three other
pressing schedules: one, two and four step pressing schedules, respectively (Figure 3.12).
All pressing schedules were simulated over a period of 268 s.
One step closure simulates a rapid compression where the mat reaches the final thickness of
13 mm only 20 s after the beginning of pressing. The press platens remain at the final
position until the end (Figure 3.12 curve identified as “1 Step”). For this one step closure,
the time of first compaction was estimated to 18.6 s. We defined the time of first
compaction as the moment in time when the mat reaches 1.9 times its final thickness. This
parameter is used in our definition for time evolution of Poisson’s ratio (APPENDIX 1,
Eqs. (102) and (103)).
For the two step pressing schedule, the time of first compaction was estimated to 71 s and
mat thickness evolved as follows: 20 s after the beginning of the pressing process, the mat
reached 3 times its final thickness; that same thickness was maintained for the next 40 s; at
time of 60 s, a new compression step started such that the final thickness of the mat was
reached at time of 80 s after the beginning of pressing process; the platens maintained the
final mat thickness until the end (Figure 3.12 curve identified as “2 Steps”).
Four step pressing schedule is a slow compression program where the final thickness was
reached 162 s after the beginning of compression. The pressing program combines a
succession of four rest periods, each lasting for 28 s, and five equal compression efforts,
each having duration of 10 s and the same slope (Figure 3.12 curve identified as “4 Steps”).
In this case, the time of first compaction used in expression for Poisson’s ratio was
estimated to 158.5 s (APPENDIX 1).
118
Figure 3.12 : Evolution of mat thickness as a function of four different pressing schedules.
Typical tendencies at the core for oven-dry density, temperature, moisture content, and total
gas pressure fields are presented in Figure 3.13. As expected, one step closure densifies the
core region the most rapidly (Figure 3.13a), hence increasing thermal conductivity which
results in somewhat faster increase of the core temperature (Figure 3.13b). The higher
temperature stimulates moisture evaporation process in the core region to start earlier
(Figure 3.13c). As a result, in early stages of the pressing, a slightly higher gas pressure is
produced in the core region (Figure 3.13d). Figure 3.13c suggests that the end moisture
content in the core region is lower when a rapid one step press closure is applied.
Nevertheless, the final gas pressure does not seem to be higher than the one obtained when
the laboratory pressing schedule was simulated (Figure 3.13d).
Two step and laboratory press closing programs are the most alike and produce the most
resembling results. Indeed, the evolution of temperature, moisture content and total gas
pressure fields at the core is very similar for those two pressing schedules (Figure 3.13b, c,
d). However, oven-dry densities (Figure 3.13a) exhibit predictably different behavior.
Indeed, since a linear elastic mechanical model is used, the development of oven-dry is
heavily influenced by the press closing dynamics.
Four step closing schedule compresses the mat slowly causing moderate densification of
the core (the lowest core density among all tested pressing scenarios) (Figure 3.13a). This
adversely affects thermal conductivity increase. Therefore, noticeable delays are observed
119
in temperature evolution (Figure 3.13b) when the four steps closing is applied. This
eventually slows down the mass transfer towards the core (delay in moisture content
increase) (Figure 3.13c). As a result, evaporation process in triggered later than in other
pressing scenarios. Moreover, a low core density also results in higher gas permeability.
Thus, when combining all these factors, one observes significant delays in pressure buildup when slow four step schedule is simulated (Figure 3.13d).
a)
b)
c)
d)
Figure 3.13 : Comparison of effects of four different pressing schedules on evolution of 2D
numerical results at the core location for: a) oven-dry density; b) temperature; c) moisture
content; d) total gas pressure.
120
2D results
Figure 3.14 depicts evolution of 2D profiles of T, M, and Pv. Some explanations are needed
to better understand the graphs in Figure 3.14. Results are presented on one half of a 2D
geometry where full thickness and a half width were considered. Top and bottom hot
platens are compressing the mat from the left and right hand sides, respectively. Therefore,
thickness is represented by smaller sides of the rectangle. The small side closer to the
viewer is the boundary where the exchange with the ambient air occurs. Hence, the
symmetry (core) plane is represented by the small boundary located far back on the graph.
All graphs in Figure 3.14 display computational domain meshed by a 256 by 128 element
grid. Domain’s thickness decreases with time as press platens compress the mat following
our laboratory closing schedule. Nevertheless, for the sake of clarity, the width of the
displayed solution surfaces was kept fixed through time.
Time step of 0.1 s was used in calculations and numerically predicted results are presented
at time 30 s, 80 s, 125 s, 175 s, 220 s, and 260 s. When observing graphs of T, M, and Pv
presented in Figure 3.14, one can notice development of vertical (in-thickness) gradients
for the three variables. After 175 s, an interesting transition happens in the Pv field
predictions (image was rotated by 90 degrees to the right hand side to better see the
phenomenon). Indeed, the Pv in-thickness gradient seems to vanish and a horizontal Pv
gradient starts to develop. At t = 220 s and 260 s, in-thickness gradient has completely
disappeared and in-plane horizontal gradient is very well established pushing the gas phase
(at this time mainly composed of water vapor) outside of the mat (higher pressure at the
core than at the external border). At the same time, temperature field still shows main inthickness gradient whereas moisture content exhibits both a very pronounced in-thickness
gradient and a weak in-plane horizontal concentration gradient (slightly higher M values at
the core than close to the external border).
121
Time
30 s
80 s
125 s
175 s
220 s
T
M
Pv
122
260 s
Figure 3.14 : Evolution of 2D profiles for temperature, moisture content and partial vapor
pressure.
Composite constitutive law
Our tests in regard with a composite constitutive law Eq. (92) involving tangent and
Hooke’s laws revealed that the results are the same whether a composite law is used or only
E
 0.001 and often close to 106 .
the Hooke’s law is applied. We found out that the ratio
E
This implies that, from one load increment to another, a relative variation of coefficients of
the elasticity tensor is rather small. This furthermore suggests that the contribution of

terms ( E )  :  in Eq. (92) and respectively  E  :  N in Eq. (94) is not very significant.
Further investigation of this phenomenon would be desirable but is beyond the scope of this
paper.
123
CONCLUSIONS
The main purpose of this paper was to describe the methodology developed and solution
strategy implemented to simulate MDF hot pressing on a moving domain. The proposed
model combines the finite element method with an implicit time scheme providing more
flexibility in the choice of the time step and potentially lowering the overall computational
cost. A composite constitutive law (involving tangent and Hooke’s laws) was successfully
applied respecting thermodynamic principles. Our preliminary test suggested that using
Hooke’s law alone would lead to the same numerical predictions. Further investigation is
needed to examine this affirmation.
Results predicted by the global model for T and P exhibit good overall agreement with
laboratory batch press experimental measurements. Temperature evolution and formation
of characteristic temperature plateau are well captured. Gas pressure gradient develops in
the horizontal plane whereas numerical results reveal the absence of the vertical gas
pressure gradient in the center plane. Model also produces valuable predictions for
variables of interest that are difficult to measure in laboratory such as evolution of density
profile, partial air and vapor pressures, stress, moisture content, relative humidity, and
degree of resin cure. During the first half of the pressing process, partial air and vapor
pressure gradients are well developed in the cross-sectional direction. They vanish though
in the second half of pressing and horizontal vapor pressure gradient develops.
The cross-sectional density profile plays an important role in evolution of rheological
properties and inner conditions of the mat. In situ measurements inside the mat are
commonly made for temperature and gas pressure, but not for moisture content and
adhesive cure, nor for the development of the density profile. Despite the fact that some
researchers succeeded in measuring density development during hot pressing at three crosssectional positions, representation of the complete evolution of density profile is still a
challenge. Even though the predictions in our study were based on a simplified mechanical
model, they are in good agreement with the experimental results and can be used for
understanding of the development of the vertical density profile during the compression of
the mat. Development of density profile was reasonably well predicted by this simple
mechanical model. Indeed, numerical results showed evolution of high density towards the
surfaces and significantly lower density at the mat center. The improvement of the model
requires a better knowledge of mat rheological properties and their characterization at low
density values (beginning of pressing). Also, in order to include visco-plastic behavior and
the venting phase of pressing cycle into the model, the corresponding rheological
parameters are required. However, they are rare in the literature and are not easy to obtain
experimentally.
Characterization of rheological properties of the mat is of extreme importance for
numerical simulations. The local rheological mat conditions change in space and over time
and are functions of mat density, temperature, moisture content, and state of adhesive cure.
Von Haas (1998) described the dependence of rheological properties on the first three
variables. We proposed a formula to account for effects of adhesive cure on the rheological
mat characteristics. This newly proposed expression is only the first step in characterization
124
of this complex relationship. More effort is needed to gain a deeper insight. Sensitivity
study could be an avenue in investigating the relative importance and influence of resin
cure on the rheological properties and numerical results.
Furthermore, evolution of mat’s Poisson ratio during early stages of pressing definitely
requires more investigation. We proposed an expression allowing for a smooth transition
and increase of Poisson ratio as a function of mat thickness. However, it would be
interesting to further explore the correlation between Poisson’s ratio and local density. This
work is still to come.
The present model provides a reasonably reliable insight in complex dynamics of rheologic
and heat and mass transfer phenomena occurring during the hot pressing of the fiber mats.
It was tested under various pressing scenarios and numerical results systematically showed
good and reasonable tendencies. Our model and finite element code prove to be robust tools
to conduct further case studies.
125
NOMENCLATURE
Partial nomenclature is presented here, whereas complementary information can be found
in NOMENCLATURE section in Part 1 of this paper.
t : time [s]
x : length [m]
y : width [m]
z : thickness [m]
T : temperature field [K] ; a state variable calculated by the model
Pa : partial air pressure [Pa] ; a state variable calculated by the model
Pv : partial vapor pressure [Pa] ; a state variable calculated by the model
U : displacement field [m] ; a state variable calculated by the model
P : total gas pressure [Pa]
M : moisture content [dimensionless]
h : relative humidity [dimensionless]
 OD : oven-dry density of the mat [kg/m3]
Φ : porosity of the mat [dimensionless]
 Mat : wet density of the mat [kg/m3]
 = resin cure degree [dimensionless]
Eel = coefficient of modulus of elasticity [Pa]
E X = modulus of elasticity in the x direction [Pa]
EY = modulus of elasticity in the y direction [Pa]
EZ = modulus of elasticity in the z direction [Pa]
GXY = modulus of shear stress in the x-y plane [Pa]
GYZ = modulus of shear stress in the y-z plane [Pa]
GXZ = modulus of shear stress in the x-z plane [Pa]
Poisson = Poisson’s coefficient of compression [dimensionless]
126
APPENDIX 1
Here are presented parameters describing mechanical properties of the mat (coefficients of
the forth-order elasticity tensor). Constants a i ,bi ,ci (i  1, 2) and expressions A, B, and C
can be found in Thömen et al (2006) and von Haas (1998). Parameters related to heat and
mass transfer model are presented in Part 1 of this paper.
a1 =4.22102 b1 =  2.74102 c1 = 3.25
a 2 =  1.86102 b 2 =3.24103 c2 =  5.10
A  a1 M 100  b1  T  273.15   c1 ;
B  a 2 M 100  b 2  T  273.15   c 2



C  exp A   MAT exp  B   ln  MAT  
 198.3  

C106
E el 
66
1 

F1  2.13 1 

b

(101)
(102)
t

a  27   1  1
 t fc 
with t fc  time of the first compaction
where b  1  7exp  a 
and
We define the time of the first compaction t fc as the moment in time when the mat reaches
1.9 times its final thickness. That is the moment when we estimate that the mat gained
sufficient cohesion level and set its Poisson’s ratio to a nominal value (which is in our case
0.25).
0.25 F1 F1  1
(103)
Poisson  
F1  1
 0.25
In the case of our laboratory pressing schedule, the time of the first compaction was
estimated at 35 s. The evolution in time of the resulting Poisson’s coefficient can be seen in
Figure 3.2b.
The fourth order elasticity tensor E can be written as a 6 by 6 symmetric matrix and a
relation between stress and strain can be expressed as follows
127
 1  v23v32
 EES
2 3
 1  

  v21  v23v31

2

  EES
2 3
 3  

   v31  v21v32
  23   E E S
2 3
  
13



  12  



with
S
v21  v23v31
E2 E3 S
v31  v21v32
E2 E3 S
1  v31v13
E1 E3 S
v23  v21v13
E1 E2 S
v23  v21v13
E1 E2 S
1  v21v12
E1 E2 S
0 0 0
0 0 0
0 0 0
0 0
0 0
0 0
G23
0
0
0
G13
0


0   1 

0  2 
 
0   3 
 
   23 
 
0   13 
 
0   12 
G12 
(104)
1
1  2v21v32v13  v13v31  v23v32  v12v21  .
E1 E2 E3
Assuming MDF as plane isotropic material, and directions X and Y (1 and 2) to be the
E1
plane of isotropy, we further have E1  E2 , G23  G13 , G12 
, v12  v21 ,
2 1  v12 
v23  v32  v13  v31 (Ganev et al. 2005).
In our case, we moreover assumed
E1  E2  Eel 1  32 
E3  Eel
v12  v23  Poisson
G12 
E3
E1
; G13  G23 
21  Poisson 
21  Poisson 
(105)
(106)
(107)
ACKNOWLEDGMENTS
The authors wish to thank the Natural Sciences and Engineering Research Council of
Canada (NSERC), FPInnovations – Forintek Division, Uniboard Canada and Boa-Franc for
funding of this project under the NSERC Strategic Grants program.
128
Conclusion
Un modèle couplé basé sur les lois de conservation et résultant en un système d’équations
non linéaires et fortement couplées a été développé dans le but de modéliser et mieux
comprendre les phénomènes physiques impliqués lors du pressage à chaud de panneaux
MDF. Le développement des équations de conservation appuyé sur les principes physiques
fondamentaux a été détaillé ainsi que l’interconnexion entre ces différentes relations de
conservation. La complexité des couplages a ainsi été mise en évidence. Le couplage entre
les modèles mécanique et celui de transfert de chaleur et de masse a été réalisé et décrit en
détails. Le développement dynamique du profil de densité est ainsi prédit par le modèle et
son influence sur les propriétés mécaniques et matérielles est prise en compte de manière
dynamique. L’influence de la température et de la teneur en humidité sur les paramètres
rhéologiques a été assurée par une mise à jour des paramètres rhéologiques à chaque pas de
temps. De plus, l’étendue de la polymérisation de la résine est également calculée par le
modèle et son influence sur les paramètres rhéologiques de l’ébauche est considérée. Nous
avons aussi proposé une première façon de prendre en compte le développement
dynamique du coefficient de compressibilité de Poisson dans une ébauche. Cela joue un
rôle important surtout au début du pressage.
Le système d’équations de conservation a été discrétisé par la méthode des éléments finis.
La résolution du système d’équations complexes a été réalisée grâce au logiciel MEF++
développé au sein du GIREF à l’Université Laval. Le modèle de transfert de chaleur et de
masse a été exprimé en termes de trois variables d’état, soient la température, les pressions
partielles de l’air et de la vapeur, respectivement. Toutes les autres variables et les
propriétés physiques de l’ébauche sont exprimées en fonction de ces trois variables d’état.
Le système non linéaire a été résolu par la méthode de Newton. La discrétisation temporelle
est faite par un schéma implicite d’ordre deux (Gear). Cela nous a permis d’employer des
pas de temps de longueur de 0.5 s. Cette longueur est 100 fois plus élevée que celle
communément rencontrée dans la littérature. La formulation incrémentale quasi-statique a
été adoptée pour résoudre les équations du modèle mécanique et la variable d’état était
l’incrément de déplacement tridimensionnel. Au début de la simulation, le maillage initial
est homogène mais au fur et à mesure que le pressage progresse et l’ébauche se déforme
sous l’action de la presse, la taille des éléments diminue et leur forme change. Ainsi, le
maillage et la géométrie du panneau se déforment dynamiquement au cours du pressage.
La validation des résultats numériques a été accomplie par la comparaison aux mesures
expérimentales de température et de pression obtenues au laboratoire de pressage du
Département de sciences du bois et de la forêt à l’Université Laval. Des panneaux MDF à
base de fibres d’épinette noire et de sapin baumier ont été fabriqués et des mesures de
température et de pression ont été prises lors du pressage dans le plan vertical au centre de
l’ébauche. Ces mesures ont été effectuées grâce aux sondes PressMAN qui ont été
installées à la surface et au centre de l’ébauche. Un thermocouple additionnel a été inséré à
un quart d’épaisseur de l’ébauche afin de prendre d’autres mesures de température.
Les résultats numériques obtenus montrent une bonne concordance avec les mesures
expérimentales. Toutefois, les prédictions numériques de la pression du gaz surpassent
130
quelque peu les mesures du laboratoire. Une solution d’appoint serait d’augmenter le
coefficient d’échange gazeux au bord de l’ébauche à la frontière avec le milieu ambiant.
Cependant, une autre avenue pourrait être envisagée. En effet, dans notre modèle
mécanique, nous avons fait appel à l’hypothèse de petites perturbations et petites
déformations communément employée dans le domaine. Or, il serait intéressant d’explorer
le cas des grandes déformations dans le cadre du pressage. Cela pourrait peut-être fournir
une description plus adéquate dans l’évolution de la contrainte (stress) et éventuellement
influencer les résultats numériques. Toutefois, cette étude reste à faire.
Nonobstant les considérations physiques dans le développement du modèle mécanique, une
meilleure connaissance des propriétés rhéologiques de l’ébauche lors du pressage, et tout
particulièrement à de faibles valeurs de densité (ce qui correspond au début du processus de
pressage), serait souhaitable. Pour obtenir de meilleurs résultats, une meilleure
connaissance des valeurs des paramètres physiques en cause dans toute la gamme de
densités, de températures et de teneurs en humidité est nécessaire. Au cours de nos travaux,
nous avons noté des lacunes concernant ces valeurs qui sont fondamentales pour la
simulation numérique des processus de pressage et de transfert de chaleur et de masse. En
effet, nous avons démontré que ces propriétés influencent de manière déterminante les
résultats numériques. Ainsi, la connaissance plus précise des paramètres est primordiale et
les efforts supplémentaires consentis à leur détermination seront salutaires puisqu’ils se
traduiront par une simulation plus précise du phénomène à l’étude. En attendant ces
importantes avancées, nous avons utilisé les valeurs proposées dans la littérature. Elles ont
rarement été déterminées pour les panneaux de fibres ce qui aurait possiblement une
influence sur les résultats numériques. Les propriétés importantes de l’ébauche de faible
densité telles que la conductivité thermique, la perméabilité au gaz ou encore les propriétés
rhéologiques (module de Young, coefficient de Poisson) restent encore à être déterminées
par des études futures. L’importance et l’influence du coefficient de transfert de masse aux
bords (des conditions aux limites du système) sur les résultats ont été démontrées par notre
étude de sensibilité. En effet, ce coefficient détermine l’aisance avec laquelle le gaz quitte
l’ébauche vers le milieu ambiant. Il influence donc grandement les conditions de pression à
l’intérieur de l’ébauche. En se fiant à notre étude de sensibilité, le développement d’un
protocole expérimental permettant de le mesurer avec précision dans différents cas de
figures serait un défi à relever dans le futur qui contribuerait à une meilleure simulation des
phénomènes physiques lors du pressage.
En ce qui nous concerne, pour la suite des travaux de modélisation, le modèle mécanique
devrait être enrichi par l’ajout d’aspects viscoélastiques et plastiques (force des liens
adhésifs) au comportement de l’ébauche. Cela permettra de considérer l’ouverture de la
presse ainsi que des programmes de fermeture de la presse comportant des périodes de
surcompression de l’ébauche où, pendant un laps de temps, l’épaisseur de l’ébauche est
plus petite que l’épaisseur finale désirée. De nouveau, la qualité des prédictions numériques
dépendra de la précision et de la pertinence des mesures des coefficients viscoélastiques et
plastiques.
Pour les variables qui représentent un grand intérêt telles que la densité, la teneur en
humidité, les pressions partielles de vapeur et d’air, les mesures expérimentales sont
difficiles sinon impossibles. Dans ces cas, la modélisation numérique demeure la seule
131
alternative permettant d’obtenir de l’information fiable quant à leur évolution durant le
pressage. Le numérique aide également à mieux comprendre l’importance de certains
processus. Ainsi, les prédictions numériques de la pression partielle de vapeur permettent
de voir que le transfert par diffusion moléculaire de la vapeur d’eau dans la phase gazeuse
joue un rôle déterminant dans le transfert de la masse d’eau dans l’ébauche de MDF. En
effet, le gradient vertical de pression totale du gaz étant faible sinon nul, le transfert par
diffusion moléculaire s’impose en tant que principal moyen de transfert de la masse d’eau
dans la direction verticale (en épaisseur). D’autre part, le gradient horizontal de pression du
gaz assure l’évacuation de la masse d’eau (sous forme de vapeur) à l’extérieur de l’ébauche
et demeure présent tout au long du pressage. De plus, nous avons employé le modèle
couplé afin de simuler différents programmes de fermeture de la presse. Les résultats
numériques obtenus montrent les bonnes tendances et notre modèle pourrait désormais
servir en tant qu’outil d’analyse de procédés.
Il importe de souligner que les modèles mathématiques proposés dans ce travail sont basés
sur les principes physiques fondamentaux et demeurent valides même si la précision de
certains paramètres utilisés dans les simulations numériques reste à être améliorée. La
stratégie couplée de résolution numérique combinée à la méthode des éléments finis est
efficace et donne des résultats probants que l’enrichissement des modèles ne pourrait
qu’améliorer.
Enfin, il convient de rappeler que, peu importe le niveau de complexité d’un modèle donné,
sa capacité à prédire adéquatement les phénomènes physiques sera tributaire de la
connaissance et de l’exactitude des paramètres mécaniques et rhéologiques qui caractérisent
le matériau à l’étude. La qualité des prédictions d’un modèle et son applicabilité dans le
milieu industriel dépend donc grandement de la qualité des données fournies au modèle.
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