### 5.1.1 Application à la variation de la pression totale P₁ en fonction du temps

<table><caption>Tableau 5.2</caption><thead><tr><th></th><th>T [°C]</th><th>1/T[1/°K]</th><th>C<br>ln(dP1/dt/T)</th><th>slope</th><th>Activation<br>energy<br>E [J/mole]</th></tr></thead><tbody><tr><td>C1</td><td>45</td><td>0.0031</td><td>3.64</td><td>-3288.03</td><td>2.73E+04</td></tr><tr><td>C2</td><td>45</td><td>0.0031</td><td>3.86</td><td></td><td></td></tr><tr><td>C4</td><td>26</td><td>0.0033</td><td>2.92</td><td></td><td></td></tr><tr><td>C5</td><td>26</td><td>0.0033</td><td>2.36</td><td></td><td></td></tr><tr><td>C6</td><td>11</td><td>0.0035</td><td>2.27</td><td></td><td></td></tr><tr><td>C7</td><td>11</td><td>0.0035</td><td>2.80</td><td></td><td></td></tr></tbody></table>

Le tableau 5.2 et la figure 5.4 illustrent la validité de la loi d'Arrhenius dans le cas de la variation de la pression totale P₁ en fonction du temps.

<figure><img src="image_179.png" /><figcaption>Figure 5.4 : Application de la loi d'Arrhenius à la variation de la pression totale P₁ en fonction du temps et de la température</figcaption></figure>

Il convient alors de vérifier la validité de cette approche pour d'autres paramètres concernant les mêmes expériences.
5.1.2. Application à la variation du coefficient de compression, C<sub>v</sub>, en fonction du temps

<table><caption>Tableau 5.3</caption><thead><tr><th></th><th>T [°C]</th><th>1/T [1/°K]</th><th>C ln(dCv/dt/T)</th><th>slope</th><th>Activation energy E [J/mole]</th></tr></thead><tbody><tr><td>C1</td><td>45</td><td>0.0031</td><td>-21.05</td><td>-2991.16</td><td>2.49E+04</td></tr><tr><td>C2</td><td>45</td><td>0.0031</td><td>-17.78</td><td></td><td></td></tr><tr><td>C4</td><td>26</td><td>0.0033</td><td>-20.95</td><td></td><td></td></tr><tr><td>C5</td><td>26</td><td>0.0033</td><td>-21.30</td><td></td><td></td></tr><tr><td>C6</td><td>11</td><td>0.0035</td><td>-21.70</td><td></td><td></td></tr><tr><td>C7</td><td>11</td><td>0.0035</td><td>-19.29</td><td></td><td></td></tr></tbody></table>

Le tableau 5.3 et la figure 5.5 illustrent la validité de la loi d'Arrhenius dans le cas de la variation du coefficient de compression, C<sub>v</sub>, en fonction du temps.

<figure><img src="image_180.png" /><figcaption>Figure 5.5 : Application de la loi d'Arrhenius à la variation du coefficient de compression, C<sub>v</sub>, en fonction du temps et de la température</figcaption></figure>

On peut également appliquer cette approche à la variation du coefficient de compression, C<sub>e</sub>, en fonction du temps.
### 5.1.3. Application à la variation du coefficient de compression, C<sub>e</sub>, en fonction du temps

<table><caption>Tableau 5.4</caption><thead><tr><th></th><th>T [°C]</th><th>1/T[1/°K]</th><th>C<br>ln(dCe/dt/T)</th><th>slope</th><th>Activation<br>energy<br>E [J/mole]</th></tr></thead><tbody><tr><td>C1</td><td>45</td><td>0.0031</td><td>-3.38</td><td>-2855.51</td><td>2.37E+04</td></tr><tr><td>C2</td><td>45</td><td>0.0031</td><td>-1.55</td><td></td><td></td></tr><tr><td>C4</td><td>26</td><td>0.0033</td><td>-2.77</td><td></td><td></td></tr><tr><td>C5</td><td>26</td><td>0.0033</td><td>-3.83</td><td></td><td></td></tr><tr><td>C6</td><td>11</td><td>0.0035</td><td>-4.18</td><td></td><td></td></tr><tr><td>C7</td><td>11</td><td>0.0035</td><td>-2.87</td><td></td><td></td></tr></tbody></table>

Le tableau 5.4 et la figure 5.6 illustrent la validité de la loi d'Arrhenius dans le cas de la variation du coefficient de compression, C<sub>e</sub>, en fonction du temps.

<figure><img src="image_181.png" /><figcaption>Figure 5.6 : Application de la loi d'Arrhenius à la variation du coefficient de compression, C<sub>e</sub>, en fonction du temps et de la température</figcaption></figure>

### 5.1.4. Conclusion

Le tableau 5.5 résume les valeurs des énergies d'activation associées à la variation en fonction du temps des trois paramètres de compression étudiés. Il est remarquable de constater que la valeur de l'énergie d'activation ne varie pas en fonction du paramètre étudié ; ceci confirme de façon très nette le fait que la compression du produit C est contrôlée par la température.
<table><thead><tr><th>Function studied</th><th>Activation Energy E [kJ/mole]</th></tr></thead><tbody><tr><td>P [N/m²]</td><td>27.3</td></tr><tr><td>C<sub>v</sub> [m²/s]</td><td>24.9</td></tr><tr><td>C<sub>e</sub> [kg²/s·m⁴]</td><td>23.7</td></tr></tbody></table>

Tableau 5.5

EYRING (1936) a mesuré et a calculé les énergies d'activation correspondant à des tests de fluage et donne les valeurs suivantes concernant l'eau et les matières plastiques :

$$ \text{Eau} \quad E = 16 - 21 \ [kJ/mole] $$

$$ \text{Plastiques} \quad E = 30 - 60 \ [kJ/mole] $$

Les valeurs du tableau 5.5 et la référence ci-dessus confirment la nature rhéologique du type de liquide à déplacer lors de la compression de C.

Sans prétendre donner une valeur statistique à ce qui suit, nous avons aussi exploité deux expériences conduites sur le même produit lors des tests de pressage à vis (Tableau 5.6). Ce procédé est bien différent de la compression en cellule piston sur laquelle reposent les données précédentes. De plus, deux points pour décrire une droite c'est « prêter le flanc » ! Toutefois la comparaison est édifiante, comme le montrent le tableau 5.7 et la figure. 5.7.

<table><thead><tr><th>Reference</th><th>Material</th><th>temperature [°C]</th><th>Screw</th><th>rotation [rpm]</th><th>P feed [kPa]</th><th>P out [kPa]</th><th>Out mass Flow QI [g/s]</th><th>Out mass Flow Qs [g/s]</th><th>Dry content before process [%]</th><th>Dry content after process [%]</th></tr></thead><tbody><tr><td>Cc6-T38</td><td>Cc</td><td>38</td><td>HCR</td><td>6</td><td>48.3</td><td>132.6</td><td>1</td><td>1.2</td><td>33.5</td><td>51.1</td></tr><tr><td>Cc6-T11</td><td>Cc</td><td>11</td><td>HCR</td><td>6</td><td>48.3</td><td>119.4</td><td>1.8</td><td>4.3</td><td>33.5</td><td>41.7</td></tr></tbody></table>

Tableau 5.6

<table><thead><tr><th></th><th>T [°C]</th><th>1/T[1/°K]</th><th>C ln(dCe/dt/T)</th><th>slope</th><th>Activation energy E [J/mole]</th></tr></thead><tbody><tr><td>Cc6-T38</td><td>38</td><td>0.0032</td><td>-10.28</td><td>3691.34</td><td>3.07E+04</td></tr><tr><td>Cc6-T11</td><td>11</td><td>0.0035</td><td>-9.15</td><td></td><td></td></tr></tbody></table>

Tableau 5.7
<figure><img src="image_182.png" /><figcaption>Figure 5.7 : Application de la loi d'Arrhenius à la variation du coefficient de compression, C<sub>e</sub>, en fonction du temps, tests TSPU©</figcaption></figure>

La valeur de l'énergie d'activation de 30.7 kJ/mole reste en accord avec tout ce qui précède. MAFART (1981) cite pour une eau fortement liée à une pâte d'amidon une valeur de 160 Kcal/kg<sup>-1</sup>. Ceci reste dans le domaine d'énergie décrit ci-dessus pour l'eau.

La corrélation entre les données expérimentales et la loi d'Arrhenius suggère la poursuite de cet axe de recherche, à savoir l'application de la thermodynamique statistique à la compression. A cette évolution, on pourrait ajouter l'influence des tensions de surfaces qui devraient aussi corroborer les énergies mesurées et calculées.

Sur le plan de l'ingénierie, d'autres approches sont envisageables. Une nouvelle fois sans aucune prétention tant sur le plan prédictif que statistique, nous proposons une autre voie d'accès à des paramètres énergétiques caractéristiques.

## 5.2 D'AUTRES ORIENTATIONS PROMETTEUSES

Nous proposons un critère de calcul de l'énergie de compression fondé sur la grandeur suivante :
QEV (noté I) = la quantité d'énergie nécessaire à la déformation d'une unité élémentaire de SPM (saturated porous material) par unité de volume de SPM en cours de déformation.

Définissons succinctement les équations de base.

Rappelons tout d'abord :

$$P(\varepsilon_E) = \frac{\bar{F}}{A} = \frac{\text{Force Compressive}}{\text{Surface Drainage}} : [N \cdot m^{-2}]$$

et

$$\varepsilon : \frac{\Delta Z_0}{Z} = \frac{\text{Déplacement}}{\text{Hauteur SPM}} = [m/m]$$

$$\text{Dans ce cas : } I = \int_{0}^{\varepsilon} P(\varepsilon) d\varepsilon = [N \cdot m^{-2}] \cdot [m \cdot m^{-1}] = \frac{[N \cdot m]}{[m^3]} = \frac{[J]}{[m^3]} = QEV \quad \text{Eq. 5.9}$$

On peut assez aisément et pour des raisons associées à la nature même des matériaux testés – étude rhéologique qui dépasse le cadre de ce mémoire – établir, comme le montre la figure 5.8, une relation de comportement des matériaux testés sur la base de la relation suivante :

$$P(t) = A(t - t_0)^p + B \quad \text{et} \quad P(\varepsilon) = A(\varepsilon - \varepsilon_0)^p + B \qquad \text{Eq. 5.10}$$

<figure><img src="image_183.png" /><figcaption>Figure 5.8 : Modélisation des résultats expérimentaux à partir de l'Eq. 5.10</figcaption></figure>

A partir de l'Eq. 5.10, l'intégration de l'Eq. 5.9 se trouve facilitée :
$$I = \int_{0}^{\varepsilon} P(\varepsilon) d\varepsilon = \frac{A}{p+1} (\varepsilon - \varepsilon_{0})^{p+1} + B \cdot \varepsilon + Cte \equiv QEV \quad \text{Eq. 5.11}$$

A titre d'illustration, voici une série de figures correspondant aux trois produits principalement étudiés en prenant en compte deux variables opératoires, à savoir le taux de déformation, ou plus simplement la vitesse de descente du piston, et la température.

<figure><img src="image_184.png" /><figcaption>Figure 5.9 : QEV (Eq. 5.11) = f(ε)</figcaption></figure>

<figure><img src="image_185.png" /><figcaption>Figure 5.10 : QEV = f(1/Cv)
1/Cv [min/cm²]</figcaption></figure>
La figure 5.10 souligne qu'une relation quasi-linéaire existe entre QEV et le rapport 1/Cv, qui permet la classification des produits testés selon la figure 5.11.

<figure><img src="image_186.png" /><figcaption>Figure 5.11 : La classification des produits testés à partir de la pente de la relation QEV = f(1/Cv)</figcaption></figure>

La matière la plus élastique, à savoir la pulpe de betterave, F, requière le moins d'énergie, QEV. Ceci fait penser à une contribution de relaxation élastique (elastic recovery) favorable à la surpression interne. Le produit semi-solide, C, protéines lactiques précipitées, quant à lui requière le plus d'énergie, QEV, tandis que le produit semi-liquide, Cr, airelles broyées, se place en position intermédiaire. C se caractérise par une forte résistance à la déformation structurelle de la matrice solide tandis que Cr aurait tendance à présenter une perméabilité plus faible, d'où la classification observée.

On peut également selon la même méthode, examiner l'influence de la vitesse de compression sur l'énergie nécessaire QEV. Pour cela nous avons choisi le SPM, F, qui présente des propriétés viscoélastiques non-linéaires. Le taux de déformation ou plus exactement la vitesse de déplacement du piston, varie dans un rapport de 1 à 30.

Les figures 5.12 et 5.13 illustrent la validité de l'approche proposée. Les relations linéaires qui existent entre QEV et 1/Cv (Fig. 5.12) et la pente de ces dernières en fonction du taux de déformation (Fig. 5.13), rendent l'utilisation de cette approche extrêmement convenable pour le calcul anticipatoire des énergies mises en jeu et de leur classement relatif.
<figure><img src="image_187.png" /><figcaption>Figure 5.12 : La variation de QEV = f(1/Cv) avec dZ/dt → pour F7 = 7.62 (cm/min), F5 = 2.54, F4 : 0.762 et F2 : 0.254 (cm/min)</figcaption></figure>

<figure><img src="image_188.png" /><figcaption>Figure 5.13 : La variation de la pente de la relation QEV = f(1/Cv) en fonction du taux de déformation</figcaption></figure>
Enfin il serait intéressant de pouvoir corréler les observations faites en utilisant l'approche fondée sur l'usage de la loi d'Arrhenius et celles que l'on peut faire à partir de QEV. Pour cela les mêmes produits sont analysés.

<figure><img src="image_189.png" /></figure>

<figure><img src="image_190.png" /></figure>
Comme le montrent les figures 5.14 et 5.15, une relation existe de façon nette entre la température du SPM, C et l'énergie requise lors de la compression. Ceci renforce l'idée que les deux approches proposées dans ce chapitre constituent des axes de recherche future attrayants.

En résumé, les quatre grands axes de recherche que nous préconisons, suite à ces travaux, sont :

1. l'usage de la thermodynamique statistique pour appréhender les phénomènes sensibles à l'échelle moléculaire, au niveau des agrégats et des couches lors de la compression ; ceci quel que soit le mode de déformation envisagé.

2. l'approfondissement de la valeur à l'échelle de l'ingénierie de la quantité d'énergie nécessaire pour provoquer une déformation unitaire rapportée au volume de matière en cours de déformation, QEV. Cette grandeur présente des caractéristiques propres à la matrice déformée mais aussi une sensibilité apparemment significative vis-à-vis des conditions opératoires du procédé sélectionné.

3. la relaxation suivant la compression est aisément « linéarisable » comme nous l'avons montré dans nos travaux P8 et P10 au chapitre 3. Ce phénomène de retour à un équilibre reflète des propriétés intrinsèques à la matière dans son nouvel état déformé. Il serait judicieux d'exploiter les travaux de PELEG pour entrevoir les conséquences associées au point 1 et 2 qui précédent.

4. Etant donné la dépendance que nous avons démontrée entre les caractéristiques de compression d'un SPM et le rapport des pressions hydraulique et totale ou solide, il devient intéressant de considérer la compression à gradient de pression contrôlé. Cet axe de recherche offre la possibilité de mieux prendre en compte les phénomènes d'interface, comme l'extrusion à travers le médium filtrant, la ségrégation... D'un point de vue thermodynamique l'étude des composantes des contraintes imposées et induites, ainsi que de leurs rapports, pourrait mettre également en évidence le rôle de la tension de surface solide liquide.
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Conclusion

Environnement, énergie et développement durable sont intrinsèquement liés. C'est dans ce contexte que se situent les travaux techniques et scientifiques rapportés dans ce mémoire. Un potentiel important de réduction de la consommation d'énergie existe encore dans de nombreux secteurs pouvant faire usage des modes mécaniques de séparation de phases.

Ce sont par exemple les industries minières, agroalimentaires, papetières, électroniques, pharmaceutiques, biotechnologiques. Elles ont toutes à traiter de séparations solides-liquides au combien pesantes, non seulement dans leur bilan économique et énergétique global, mais également au niveau de leur « transférabilité ».

Le type de séparation solide-liquide qui nous intéresse ici prend en compte des produits et dérivés provenant de ces six secteurs industriels. Attachés à accroître et à mettre à disposition la compréhension fondamentale et sa composante pratique de la séparation de phases par déformation mécanique multidimensionnelle de milieux poreux saturés, quatre modes de compression sont présentés sur le plan théorique et sont illustrés.

Pour cela plusieurs cellules de compression multi-mode (MMCC©) et une unité pilote de pressage à vis (TSPU©) ont été conçues et favorablement testées vis-à-vis de leur compatibilité et de leur potentiel quant à la prédiction de grandeurs caractéristiques de la compression multidimensionnelle.

Des travaux conduits sur la compression à pression constante, CPC, nous dégageons les conclusions suivantes :

1. Les modèles décrits précédemment suivent bien le comportement des milieux poreux saturés, SPM, d'origine minérale.

2. Ces modèles sont d'usage plus limité lorsqu'il s'agit de traiter des matériaux biologiques.

3. La durée des expériences n'est pas toujours représentative des pratiques.

4. Cet aspect rend quelque peu superflue l'analyse de la phase de compression secondaire. Il faudrait peut-être substituer ces expériences longues par des analyses en relaxation comme nous l'avons illustré dans nos publications jointes.

5. L'effet choc que peut exercer sur le produit l'application d'une pression soudaine est aussi un facteur à surveiller.

6. Il est toujours possible, à partir d'un nombre plus ou moins grand d'éléments placés en série ou en parallèle, de faire coïncider une série de points
expérimentaux à une courbe dite théorique. L'usage des modèles rhéologiques classiques est donc à pratiquer avec prudence.

D'une façon plus générale, la compression à déformation constante, CDC nous a permis, sans hypothèse excessivement restrictive concernant les deux paramètres essentiels et caractéristiques de l'écoulement et de la résistance mécanique de la matrice, de mettre à disposition trois approches théoriques adaptables à une large gamme de situations.

Ces trois approches ont des points communs mais aussi des différences notables quant à l'exploitation des résultats et la rapidité d'extraction d'informations essentielles.

Un des apports extrêmement important de la CDC réside dans la rapidité d'exécution des tests, de l'ordre de plusieurs minutes et non pas de plusieurs heures comme dans le cas de la CPC. Ceci est d'autant plus important que les processus industriels de séparations solide-liquide s'insèrent dans des procédés de plus en plus complexes et rapides.

Nous avons pu souligner, dans nos travaux rapportés dans les publications jointes, que le rapport de la pression hydraulique à la pression totale au cours de la compression est un critère important. A ce sujet il est tout à fait concevable de travailler en contrôlant ce rapport. Il s'agirait d'une méthode nouvelle dans le domaine du génie des procédés. La connaissance des états d'équilibres intermédiaires et de réponse réelle du SPM à la compression en serait élargie.

Il est important de noter que le choc exercé sur la matière lorsque l'on travaille à pression constante, et qu'il n'y a pas de phase de filtration préliminaire, a des conséquences qui ne sont prévisibles ou mesurables que par l'approche CDC.

Ceci nous amène aux études de la relaxation qui suit la compression CDC. Ces dernières, rapportées dans les publications jointes, montrent l'incidence du taux de déformation initial sur les caractéristiques en compression et en relaxation du produit. Ceci ouvre une nouvelle voie de recherche qui pourrait faire usage des relations simples que nous avons développées concernant la relaxation.

En matière de compression avec écoulement multi-axial nous développons tout d'abord la compression axiale avec écoulement radial central ou périphérique, puis nous l'utilisons pour aborder le cas très général de la compression axiale avec écoulement tridimensionnel que nous réduisons enfin au cas du pressage à vis. Notre démarche est cohérente avec l'idée que ce mémoire puisse fournir des bases théoriques utiles au-delà des applications sélectionnées. Pour le pressage à vis, la connaissance du temps de séjour et du taux de compression au sein de la vis, suffisent pour calculer le coefficient de compression puis éventuellement le coefficient de perméabilité (pour un gradient hydraulique interne estimé donné). Deux méthodes sont envisagées pour le calcul du temps de séjour à partir des données expérimentales (TSPU©) qui concourent à valider l'approche théorique.

Nous avons développé un équipement de pressage à vis, TSPU©, qui paraît tout à fait représentatif des conditions rencontrées au laboratoire lorsque la cellule MMCC©, que nous avons créée et mise au point, est utilisée. On note aussi un bon degré de prévisibilité des performances du pilote à partir de la cellule de laboratoire.

Les facteurs géométriques, rapport du diamètre de cellule et hauteur des couches comprimées, les mesures des pressions hydrauliques et totales, et les frictions ne doivent pas être négligés pour obtenir une bonne cohérence. La préparation des
échantillons et en particulier l'homogénéisation de ceux-ci, nécessite un soin particulier car nous avons à faire à des produits biologiques variables.

D'un point de vue théorique on peut raffiner l'approche, encore faut-il rester attachés à des paramètres, comme Ce, qui représentent le mieux et le plus simplement possible ce double phénomène de résistance à l'écoulement et de résistance de la matrice. Les coefficients Ce ou Cv se prêtent bien à ce genre d'exploitation, utile sur le plan exploratoire et sans doute suffisante pour l'ingénierie de base lors du choix de certains équipements et de leur dimensionnement.

Les trois produits principalement étudiés, viscoélastique non linéaire pour les pulpes de betteraves, F, semi-solide pour les protéines laitières coagulées, C et semi-liquide pour les airelles, Cr, se distinguent parfaitement dans leur comportement vis-à-vis de cette double résistance. Ceci parfois de façon compensatoire car une grande résistance à l'écoulement associée à une faible résistance mécanique de la matrice peut correspondre en apparence à une faible résistance à l'écoulement et une grande résistance mécanique. Il convient de méditer sur le fait que le Kaolin floculé testé dans la première partie de nos travaux présente, à volume spécifique équivalent, un coefficient de compression Ce proche de celui de la pulpe F.

Le raffinement de l'exploitation des résultats qui précèdent aboutirait à entrevoir d'autres aspects plus spécifiques de la matière SPM traitée et de sa rhéologie.

Enfin nous proposons une application de la thermodynamique statistique de BOLTZMANN et le développement du calcul de l'énergie consommée par unité de volume de matière comprimée. Nous avons pu à partir de 3 grandeurs différentes obtenir une valeur de l'énergie d'activation, associée à la compression de milieux poreux saturés étudiés, presque identique ce qui renforce l'à propos de ce concept.

En résumé, les quatre grands axes de recherche que nous préconisons, suite à ces travaux, sont :

1. L'usage de la thermodynamique statistique pour appréhender les phénomènes sensibles à l'échelle moléculaire, au niveau des agrégats et des couches lors de la compression ; ceci quel que soit le mode de déformation envisagé.

2. L'approfondissement de la valeur, à l'échelle de l'ingénierie, de la quantité d'énergie nécessaire pour provoquer une déformation unitaire rapportée au volume de matière en cours de déformation, QEV. Cette grandeur présente des caractéristiques propres à la matrice déformée mais aussi une sensibilité apparemment significative vis-à-vis des conditions opératoires du procédé sélectionné.

3. La relaxation suivant la compression est aisément « linéarisable » comme nous l'avons montré dans nos travaux. Ce phénomène de retour à un équilibre reflète des propriétés intrinsèques de la matière dans son nouvel état déformé. Il serait judicieux d'exploiter les travaux de PELEG pour entrevoir les conséquences associées au point 1 et 2 qui précèdent.

4. Etant donné la dépendance que nous avons démontrée entre les caractéristiques de compression d'un SPM et le rapport des pressions hydraulique et totale, il devient intéressant de considérer la compression à gradient de pression contrôlé. Cet axe de recherche offre la possibilité de mieux prendre en compte les phénomènes d'interface, comme l'extrusion à travers le médium filtrant, la ségrégation...
5. D'un point de vue thermodynamique l'étude des composantes des contraintes imposées et induites, ainsi que leurs rapports, pourrait mettre également en évidence le rôle de la tension de surface solide liquide.
# Conclusion Générale et Synthèse Prospective

1. CPC : pour l'Orientation

2. CDC : pour la Compréhension

3. CAR : pour l'Application

4. Et au-delà :
- l'approche thermodynamique
- l'aspect énergétique QEV
- la relaxation « réversible »
- le rapport P₁/Pf₁
- l'exploitation MMCC® radiale

L'auteur,

<figure><img src="image_191.png" /></figure>

, vous remercie.

<figure><img src="image_192.png" /></figure>
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Nomenclature

a = distance à partir du piston [m]

a₀ = distance entre le piston et la surface de drainage [m]

a_v = coefficient de compressibilité [m²·N⁻¹]

A = aire de la section du SPM ≈ aire de la surface filtrante [m²]

A = (chap.2 et A2 module de compression du ressort 'a' [Pa]

A₀ = distance réduite A₀ = a₀/Z₀ [-]

A₁ = facteur préexponentiel (loi d'Arrhenius) dimension de k₁ […]

A₂ = facteur préexponentiel (loi d'Arrhenius) dimension de k₂ […]

b = rapport des coefficients de perméabilité : b = Kₕ₀/Kₕ₀ [-]

b² = constante de séparation [-]

B = (chap.2 et A2 module de compression du ressort 'b' [Pa]

c = coefficient intermédiaire c = K(1+e) / γf et (c = C_v · a_v) [m³s·kg⁻¹]

c_c = pente de la relation entre e et log₁₀ P_s [m²·N⁻¹]

c_k = pente de la relation entre e et log₁₀ K [s·m⁻¹]

Δe/Δlog P_s = C_C […]

C_e = C_v/v² = coefficient de compression modifié [kg²m⁻⁴s⁻¹]

C_em = coefficient de compression(modèle de Shirato) [m²s⁻¹]

C_r = coefficient de compression(radial) [m²s⁻¹]

C_v = coefficient de compression C_v = K/γf·m_v [m²s⁻¹]

C_v = coefficient de compression apparent [m²s⁻¹]

C_vr = coefficient de compression (vertical et radial) [m²s⁻¹]

C_C = C_vr = indice de compressibilité rapporté à V_s/V_T [m³·kg⁻¹]

d = distance séparant des niveaux d'énergie minimale [m]

d_e = diamètre externe de l'échantillon d_e = 2·r_e [m]

d_e = diamètre externe de l'échantillon d_e = 2·r_e [m]

d_e = diamètre externe de l'échantillon d_e = 2·r_e [m]

d_e = diamètre externe de l'échantillon d_e = 2·r_e [m]

d_e = diamètre externe de l'échantillon d_e = 2·r_e [m]
$d_c$ = diamètre hydraulique [-]

$e = \frac{V_v}{V_s}$ = rapport de vide local [-]

$\bar{e}$ = rapport de vide moyen $\bar{e} = \frac{1}{Z_0} \int_{0}^{Z_0} e \, dz$ [-]

$e_B$ = rapport de vide à la base du gâteau [-]

$e_T$ = rapport de vide au sommet du gâteau [-]

$e_0$ = rapport de vide initial, t=0 [-]

$E$ = énergie d'activation $[J \cdot mole^{-1}]$

$f(d_c)$ = fonction de répartition [-]

$\bar{F}$ = force de compression induisant un cisaillement $[N]$

$g$ = accélération de la pesanteur $(9.81 \ [m \cdot s^{-2}])$ $[m \cdot s^{-2}]$

$h(cte)$ = constante de Planck $6.6262 \cdot 10^{-34} [J \cdot s]$ $[J \cdot s]$

$h$ = rapport de contraintes effectives: $h = P_{s\infty}/P_{s0}$ [-]

$h_i$ = épaisseur d'une couche SPM (multicouche) $[m]$

$i$ = gradient hydraulique $i = \partial h/\partial z$ [-]

$I$ (QEV) = énergie par unité de volume $[J \cdot m^{-3}]$

$j$ = nombre de surfaces de drainage [-]

$J_n(r)$ = fonctions de Bessel de première espèce d'ordre n [-]

$k_b$ = constante de Boltzmann, $1.38 \cdot 10^{-23} [J \cdot K^{-1}]$ $[J \cdot K^{-1}]$

$k_1, k_2$ = constante de vitesse (réaction chimique) $[\dots]$

(dimension variable suivant l'ordre de la réaction)

$K$ = coefficient de perméabilité $[m \cdot s^{-1}]$

$K_h$ = coefficient de perméabilité horizontale $[m \cdot s^{-1}]$

$K_{hf}$ = coefficient de perméabilité horizontale en fin de test $[m \cdot s^{-1}]$

$K_{h0}$ = coefficient de perméabilité horizontale à t=0 $[m \cdot s^{-1}]$

$K_{h\infty}$ = coefficient de perméabilité horizontale à t → ∞ $[m \cdot s^{-1}]$

$K_i$ = coefficient de perméabilité intrinsèque $[m^2]$

$K_p$ = constante d'équilibre thermodynamique $[\dots]$

(dimension variable suivant l'ordre de la réaction)

$K_v = K_z$ = coefficient de perméabilité verticale $[m \cdot s^{-1}]$

$l$ = $l = \frac{T_v}{T_c} \cdot P_c$ charge de compression proportionnelle $[N \cdot m^{-2}]$

$m$ = entier positif $m \in [0,1,2,...,\infty[$ [-]

$m_v$ = coefficient de compressibilité volumique $m_v = \frac{a_v}{1+e}$ $[m^2 \cdot N^{-1}]$

$\frac{M_v + M_s}{M_s} = m_o$ = rapport initial : masse du gâteau / matière sèche [-]

$M$ = masse initiale totale de l'échantillon $[kg]$
<table>
  <tbody>
    <tr>
      <td>M<sub>i</sub></td>
      <td>= M<sub>i</sub> = (2m + 1)π/2</td>
      <td>[-]</td>
    </tr>
    <tr>
      <td>M<sub>s</sub></td>
      <td>= masse initiale de matière sèche au sein de l'échantillon</td>
      <td>[kg]</td>
    </tr>
    <tr>
      <td>M<sub>v</sub></td>
      <td>= masse initiale de fluide au sein de l'échantillon</td>
      <td>[kg]</td>
    </tr>
    <tr>
      <td>n = V<sub>v</sub>/V<sub>T</sub></td>
      <td>= porosité</td>
      <td>[-]</td>
    </tr>
    <tr>
      <td>n</td>
      <td>= (chap. 4)rapport des rayons externe/interne n = r<sub>e</sub>/r<sub>w</sub></td>
      <td>[-]</td>
    </tr>
    <tr>
      <td>n</td>
      <td>= ordre d'un polynôme de lissage</td>
      <td>[-]</td>
    </tr>
    <tr>
      <td>n</td>
      <td>= entier n ∈ [1,3,5,...,∞]</td>
      <td>[-]</td>
    </tr>
    <tr>
      <td>N</td>
      <td>= nombre d'Avogadro, 6.022·10<sup>23</sup> [mole<sup>-1</sup>]</td>
      <td>[mole<sup>-1</sup>]</td>
    </tr>
    <tr>
      <td>p</td>
      <td>= exposant fonction puissance (lissage de courbes)</td>
      <td>[-]</td>
    </tr>
    <tr>
      <td>p(ΔE)</td>
      <td>= probabilité (distribution de Boltzmann)</td>
      <td>[%]</td>
    </tr>
    <tr>
      <td>P</td>
      <td>= contrainte totale exercée sur le piston</td>
      <td>[N·m<sup>-2</sup>]</td>
    </tr>
    <tr>
      <td>P - P<sub>fr</sub> → P<sub>n</sub></td>
      <td>= contrainte verticale nette (sans frottement latéral P<sub>fr</sub>)</td>
      <td>[N·m<sup>-2</sup>]</td>
    </tr>
    <tr>
      <td>P<sub>c</sub></td>
      <td>= pression capillaire (milieux poreux insaturés)</td>
      <td>[N·m<sup>-2</sup>]</td>
    </tr>
    <tr>
      <td>P<sub>c</sub></td>
      <td>= pression instantanée du fluide (CPC)</td>
      <td>[N·m<sup>-2</sup>]</td>
    </tr>
    <tr>
      <td>P<sub>f</sub></td>
      <td>= pression du liquide (≈ fluide)</td>
      <td>[N·m<sup>-2</sup>]</td>
    </tr>
    <tr>
      <td>P&#773;<sub>f</sub></td>
      <td>= pression moyenne du liquide pour l'ensemble du gâteau SPM</td>
      <td>[N·m<sup>-2</sup>]</td>
    </tr>
    <tr>
      <td>P&#773;<sub>f</sub></td>
      <td>= pression réduite du liquide (solution numérique)</td>
      <td>[-]</td>
    </tr>
    <tr>
      <td>P<sub>fl</sub></td>
      <td>= pression hydrostatique</td>
      <td>[N·m<sup>-2</sup>]</td>
    </tr>
    <tr>
      <td>P<sub>fl</sub></td>
      <td>= pression initiale du fluide (≈ liquide)</td>
      <td>[N·m<sup>-2</sup>]</td>
    </tr>
    <tr>
      <td>P<sub>f0</sub></td>
      <td>= pression du fluide au niveau de la plaque de drainage</td>
      <td>[N·m<sup>-2</sup>]</td>
    </tr>
    <tr>
      <td>P<sub>fR</sub></td>
      <td>= pression liquide de référence (solution numérique)</td>
      <td>[N·m<sup>-2</sup>]</td>
    </tr>
    <tr>
      <td>P<sub>f</sub></td>
      <td>= pression totale interstitielle</td>
      <td>[N·m<sup>-2</sup>]</td>
    </tr>
    <tr>
      <td>P<sub>f0</sub></td>
      <td>= pression initiale du fluide ≈ constante sur toute l'épaisseur SPM</td>
      <td>[N·m<sup>-2</sup>]</td>
    </tr>
    <tr>
      <td>P<sub>f1</sub></td>
      <td>= pression du fluide au niveau du piston (surface non drainante)</td>
      <td>[N·m<sup>-2</sup>]</td>
    </tr>
    <tr>
      <td>P<sub>f1, t, l</sub></td>
      <td>= pression du fluide à l'interface de deux SPM</td>
      <td>[N·m<sup>-2</sup>]</td>
    </tr>
    <tr>
      <td>P<sub>s</sub></td>
      <td>= contrainte effective locale ou pression locale du solide</td>
      <td>[N·m<sup>-2</sup>]</td>
    </tr>
    <tr>
      <td>P&#773;<sub>s</sub></td>
      <td>= moyenne de la contrainte effective sur toute l'épaisseur</td>
      <td>[N·m<sup>-2</sup>]</td>
    </tr>
    <tr>
      <td>P<sub>sf</sub></td>
      <td>= contrainte effective finale (en fin de compression)</td>
      <td>[N·m<sup>-2</sup>]</td>
    </tr>
    <tr>
      <td>P<sub>sp</sub></td>
      <td>= précontrainte de compaction</td>
      <td>[N·m<sup>-2</sup>]</td>
    </tr>
    <tr>
      <td>P<sub>s0</sub></td>
      <td>= valeur de P<sub>s</sub> au niveau de la surface drainante z = 0</td>
      <td>[N·m<sup>-2</sup>]</td>
    </tr>
    <tr>
      <td>P<sub>s1</sub></td>
      <td>= valeur de P<sub>s</sub> au niveau du piston (non-drainant) z = Z<sub>0</sub></td>
      <td>[N·m<sup>-2</sup>]</td>
    </tr>
    <tr>
      <td>P<sub>s∞</sub></td>
      <td>= contrainte effective lorsque t tend vers l'infini</td>
      <td>[N·m<sup>-2</sup>]</td>
    </tr>
    <tr>
      <td>P&#773;<sub>os</sub></td>
      <td>= moyenne de la contrainte effective initiale</td>
      <td>[N·m<sup>-2</sup>]</td>
    </tr>
    <tr>
      <td>P<sub>1</sub></td>
      <td>= contrainte extérieure totale exercée sur le piston</td>
      <td>[N·m<sup>-2</sup>]</td>
    </tr>
  </tbody>
</table>
$$q = \text{rapport: } q = c_c/c_k \qquad [m^3 \cdot N^{-1}s^{-1}]$$

$$q_x = \text{débit entrant dans la direction } x \qquad [m^3 \cdot s^{-1}]$$

$$q_y = \text{débit entrant dans la direction } y \qquad [m^3 \cdot s^{-1}]$$

$$q_z = \text{débit entrant dans la direction } z \qquad [m^3 \cdot s^{-1}]$$

$$r = \text{rayon} \qquad [m]$$

$$\bar{r} = \text{rayon réduit (solution numérique)} \qquad [-]$$

$$R(\text{cte}) = \text{constant des gaz parfaits, } 8.314 [J \cdot K^{-1} \cdot \text{mole}^{-1}] \qquad [J \cdot K^{-1} \cdot \text{mole}^{-1}]$$

$$R \rightarrow dZ/dt = \text{vitesse de descente du piston} \approx \text{vitesse de compaction} \qquad [m \cdot s^{-1}]$$

$$r_c = r_c = \frac{d\bar{e}}{dt} = Cte, \text{ CDC test} \qquad [s^{-1}]$$

$$r_e = \text{rayon externe de l'échantillon} \qquad [m]$$

$$r_R = \text{référence initiale, pour le rayon externe (solution numérique)} \qquad [m]$$

$$R_s = \text{rapport adimensionnel } R_s = T_v/T_s \qquad [-]$$

$$r_w = \text{rayon interne de l'échantillon} \qquad [m]$$

$$s = \text{épaisseur de la couche SPM} \qquad [m]$$

$$s(\infty) = \text{épaisseur à } t \rightarrow \infty \qquad [m]$$

$$\text{SPM (MPS)} = \text{milieu poreux saturé (saturated porous material)}$$

$$SR = \text{rapport des déformations aux extrémités} = \frac{\bar{\varepsilon} \text{ at } A_0 = 1}{\bar{\varepsilon} \text{ at } A_0 = 0} \qquad [-]$$

$$SR(T_v) = SR(T_v) = \frac{\varepsilon_B}{\varepsilon_T} \text{ rapport de déformation} \qquad [-]$$

$$t = \text{temps} \qquad [s]$$

$$T = \text{température en Kelvin} \qquad [K^\circ]$$

$$\bar{t} = \text{temps réduit (solution numérique)} \qquad [-]$$

$$t_a = \text{durée d'application d'une charge} \qquad [s]$$

$$t_c = \text{durée d'application d'une charge constante} \qquad [s]$$

$$t_R = \text{référence arbitraire temps (solution numérique)} \qquad [s]$$

$$T_c = \text{temps réduit } T_c = \frac{C_v t_c}{Z_0^2} \qquad [-]$$

$$T_r = \text{temps réduit } T_r = \frac{C_r t}{(2 \cdot r_e)^2} \qquad [-]$$

$$T_s = \text{temps réduit} \qquad [-]$$

$$T_v = \text{temps réduit } T_v = \frac{C_v t}{Z_0^2} \qquad [-]$$

$$T' = \text{temps réduit } T' = Rt/Z_0 \text{ (CDC test)} \qquad [-]$$

$$U_{av} = \text{taux de compression moyen de l'ensemble de la couche } Z_t \qquad [-]$$
$$U_{av} = \frac{(1/Z_t) \int_0^{Z_t} P_{fi} dz - (1/Z_t) \int_0^{Z_t} P_{fi} dz}{(1/Z_t) \int_0^{Z_t} P_{fi} dz}$$

$U_r$ = taux de compression moyen, cas de l'écoulement radial [-]

$U_v$ = taux de compression moyen, cas de l'écoulement vertical [-]

$U_{vr}$ = taux de compression moyen, écoulement radial et vertical [-]

$$U_{av} = 1 - (1 - U_v)(1 - U_r)$$

$U_z$ = taux de compression local à la cote z [-]

$$U_z = \frac{P_{fi} - P_f}{P_{fi}} = \frac{\text{fluid pressure dissipated}}{\text{initial fluid pressure}}$$

= taux de compression au temps $t = (e_0 - e)/(e_0 - e_f)$ [-]

= volume spécifique local = volume du gâteau par unité de masse de solide $[m^3 \cdot kg^{-1}]$

$\bar{v}$ = valeur moyenne de $v$ pour l'ensemble du gâteau SPM $[m^3 \cdot kg^{-1}]$

$\vec{v}$ = vitesse du fluide (notation vectorielle) $[m \cdot s^{-1}]$

$v_f$ = vitesse apparente de la phase liquide $[m \cdot s^{-1}]$

$\tilde{v}_f$ = vitesse réelle de la phase liquide $[m \cdot s^{-1}]$

$v_F$ = valeur finale de $v$ $[m^3 \cdot kg^{-1}]$

$v_0$ = valeur initiale de $v$ $[m^3 \cdot kg^{-1}]$

$V_f$ ou $V_v$ = volume du liquide (des pores) au sein du SPM $[m^3]$

$v_s$ = vitesse apparente de la phase solide $[m \cdot s^{-1}]$

$\tilde{v}_s$ = vitesse réelle de la phase solide $[m \cdot s^{-1}]$

$V_s$ = volume de solide $[m^3]$

$V_T$ = volume total $V_T = V_s + V_v$ $[m^3]$

$V_v$ ou $V_f$ = volume des pores (du liquide) au sein du SPM $[m^3]$

$w = \frac{V_{s(z=z)}}{A}$ = volume de solide par unité de surface filtrante à la distance $z$ de la surface filtrante (Shirato) $[m^3 \cdot m^{-2}]$

$W = \frac{V_s}{A}$ = rapport : volume de solide total / surface filtrante $[m^3 \cdot m^{-2}]$

$X$ = Taux de compactage $1 + e_0 / (1 + e) = V_T / V_{T \text{ initial}}$ [-]

$Y_n(r)$ = fonctions de Bessel de deuxième espèce d'ordre n [-]

$z$ = distance à partir de la surface de drainage $[m]$

$z(a)$ = coordonnée lagrangienne ou spatiale $z(a) = \int_0^a \frac{1}{1+e(a',0)} da'$ $[m^3/m^2]$

$\bar{z}$ = hauteur SPM réduite (solution numérique) [-]

$z_R$ = hauteur initiale de référence du gâteau SPM (solution numérique) $[m]$
$Z$ = épaisseur totale du gâteau [m]

$Z_t$ = hauteur initiale du gâteau (2 surfaces de drainage) [m]

$Z_o$ = hauteur initiale du gâteau [m]

$Z'_o$ = hauteur fictive initiale du gâteau (cas des multicouches) [m]

## Lettres grecques

$\alpha(\alpha\alpha\alpha)$ = résistance spécifique locale du gâteau [m·kg⁻¹]

$\alpha'$ = rapport $T_v/T_c$ [-]

$\gamma_f(\gamma\gamma\gamma)$ = poids volumique du liquide: pour l'eau 9.81 [kN/m³] [$N·m⁻³$]

$\gamma_s$ = poids volumique du solide [$N·m⁻³$]

$\Gamma(n)$ = fonction Gamma $\Gamma(n) = \int_{0}^{\infty} r^{n-1}e^{-r}dr$ [-]

$\partial s(\delta\delta\delta)$ = variation de l'épaisseur au temps $t$ [m]

$\Delta A_0$ = intervalle, distance (adimensionnel) [-]

$\Delta H$ = enthalpie $\Delta H = \Delta E$ [$J·mole⁻¹$]

$\Delta \bar{r}$ = différence de rayon réduit (solution numérique) [-]

$\Delta \bar{t}$ = intervalle de temps réduit (solution numérique) [-]

$\Delta T'$ = intervalle de temps (CDC) [-]

$\Delta \bar{z}$ = différence de hauteur réduite (solution numérique) [-]

$\varepsilon(\epsilon\epsilon\epsilon\epsilon)$ = déformation vraie [-]

$\varepsilon$ = déformation axiale $\partial\varepsilon = \frac{\partial Z_0}{Z_0}$, [-]

$\bar{\varepsilon}$ = déformation apparente [-]

$\varepsilon_B$ = déformation à la base du gâteau SPM [-]

$\varepsilon_E$ = déformation eulerienne $\varepsilon_E = \varepsilon_L/(1-\varepsilon_L) = (Rt/(Z_0 - Rt))$

$\varepsilon_H$ = déformation (Hooke) [-]

$\varepsilon_K$ = déformation (Kelvin) [-]

$\varepsilon_L$ = déformation lagrangienne $\varepsilon_L = (Rt/Z_0)$

$\varepsilon_T$ = déformation au sommet du gâteau SPM [-]

$\varepsilon_\infty$ = déformation : $t \to \infty$ [-]

$\lambda(\lambda\lambda\lambda)$ = distance séparant des positions successives d'équilibre [m]

$\lambda_k$ = racine fonction de Bessel d'ordre n [-]
$\mu(\mu\mu)$ = viscosité du fluide exprimé $[N \cdot m^{-2} \cdot s]$

$\nu(\nu\mu)$ = fréquence d'activation $[s^{-1}]$

$\nu(+)$ = accroissement de la fréquence d'activation $[s^{-1}]$

$\nu(-)$ = décroissement de la fréquence d'activation $[s^{-1}]$

$\xi(\kappa\kappa\mu)$ = coordonnée eulerienne ou convective $[m]$

$\rho_f(\rho\rho\mu)$ = masse volumique du fluide $[kg \cdot m^{-3}]$

$\rho_s$ = masse volumique du solide $[kg \cdot m^{-3}]$

$\tau(\tau\tau\mu)$ = contrainte non linéaire $[N \cdot m^{-2}]$

$\Phi_s(\vec{v})(\phi\phi)$ = $\Phi_s(\vec{v}) = \iint_S \vec{v} \cdot \vec{N}_u dS$ flux total à travers la surface S $[m \cdot s^{-1} \cdot m^2]$
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Références

Parmi les 700 références répertoriées dans ce mémoire certaines ont été directement fournies par les auteurs au cours de diverses rencontres, il convient dans ce cas de les contacter directement pour examiner la disponibilité de copies éventuelles.

Références les plus utilisées

BARRON R.A. Consolidation of fine-grained soils by drain wells. American Society of Civil Engineers, 811-835, (1947).

CRANK J. The Mathematics of Diffusion. Oxford Science Publications, (1973).

EYRING H. Viscosity, plasticity and diffusion as examples of absolute reaction rates. J. Chem. Phys. (4) April (1936), 283-291.

HARPER J.M. Extrusion of Foods. CRC (I), (1987).

HILDEBRAND F.B. Advanced Calculus for Applications. Prentice-Hall, (1975).

PELEG M. Physical characteristics of food powders. AVI Publishing Co, Westport
CT, (1983).

SPIEGEL M. Formules et Tables de Mathématiques. Schaum, (1974).

SCHEIDEIGER A.E. The Physics of Flow through Porous Media. University of Toronto Press, (1974).

VILLERMAUX J. Génie de la réaction chimique : Conception et fonctionnement de réacteurs. Tec & Doc (1982).
<figure><img src="image_193.png" /></figure>
Références classées

COMPRESSION-RELAXATION : APPROCHE TRADITIONNELLE PRINCIPALEMENT BASEE SUR LA MECANIQUE DES FLUIDES APPLIQUEE A L'ECOULEMENT DANS LES MILIEUX POREUX.

P. 208

COMPRESSION - RELAXATION : APPROCHE RHEOLOGIQUE BASEE SUR LA DEFORMATION DES MATERIAUX ANISOTROPES.

P. 216

CONSOLIDATION : APPROCHE BASEE SUR LA MECANIQUE DES SOLS.

P. 221

COMPRESSION - DESHYDRATATION APPLIQUEE A L'INDUSTRIE SUCRIERE ET AU RECYCLAGE DES PULPES INCLUANT EGALEMENT LA DESHYDRATATION MECANIQUE DES FOURRAGES ET DES BOUES RESIDUIRES.

P. 225

COMPRESSION - DESHYDRATATION APPLIQUEE A L'INDUSTRIE DU PAPIER.

P. 227

COMPRESSION - DESHYDRATATION APPLIQUEE A LA PRODUCTION DE PRODUITS ALIMENTAIRES EN PARTICULIER DES JUS DE FRUIT.

P. 229

COMPRESSION - FILTRATION : PROCEDES ET EQUIPEMENTS.

P. 230

FILTRATION : APPROCHE TRADITIONNELLE.

P. 235

FILTRATION : SEDIMENTATION - DRAINAGE ET ASPECTS CARACTERISATION DES PARTICULES.

P. 241

PRESSAGE A VIS ET EXTRUSION.

P. 243

DESHYDRATATION : ASPECTS ECONOMIQUES ET ENERGETIQUES.

P. 245
COMPRESSION-RELAXATION : APPROCHE TRADITIONNELLE PRINCIPALEMENT BASEE SUR LA MECANIQUE DES FLUIDES APPLIQUEE A L'ECOULEMENT DANS LES MILIEUX POREUX.

ABBOTT G.M. & NASON D. Role of press wall friction in the high-density compression of wool bales. Textile Research Journal 56, N° 12, 715-721, (1986).

BALUAIS G., LECLERC D., MOLL M. & LENOEL M. Washing and compression dewatering of brewing mash. European Brewery Convention Symposium on wort production, Maffliers, April 1986, Monograph XI, (1986).

BALUAIS G., LECLERC D. & REBOUILLAT S. Evaluation de la résistance spécifique des boues floculées lors de la déshydratation par compression mécanique. Filtra 84, Paris, (1984).

BALUAIS G., REBOUILLAT S., LECLERC D. & DODDS J. Modelisation of compression dewatering and the application to filter design. Solids/Liquids separation practice. Leeds (England), 32-41, (1984).

BANKS P.J. Theory of constant-rate expression and subsequent relaxation. 12th Australian Chemical Engineering Conference, Melbourne, (1984).

BEAR J. Dynamics of fluid in porous media. Elsevier, New York, (1972).

BEAVERS C., SPARROW A., HAJJI A. & WITTENBERG M. Fluid flow through a class of highly deformable porous media. Journal of Fluids Engineering ASME 103 (9), 432-460, (1981).

BLONDEL J.M. Modélisation informatique de l'opération de filtration-compression. D.E.A., Génie des Procédés, INPL Nancy, (1982).

BORODULYA V.A., TEPLITSKY Y.S., YEPANOV Y.G. & MARKEVICH I.I. Heat transfer between a surface and an infiltrated bed of solid particles. Int. J. Heat Mass Transfer 32 (9), 1595-1604, (1989).

CARLETON A.J. Optimization of dewatering filters. Journal of Separation Proc. Technology 3 (4), 14-20, (1982).

CHARPENTIER J.C. Hydrodynamics of two phase flow through porous media. Journal of Powder Bulk Solids Technology 2, 53-60, (1978).

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STEINER P., GALLING G., BUTTERSACK C. & BUCHHOLZ K. Die Ultrastruktur der Zuckerrübe während der industriellen Zuckergewinnung. (The ultrastructure of sugar beet during sugar manufacture.), Zuckerindustrie 115 (10), 840-851, (1990).

STEINER P., GALLING G. & BUTTERSACK C. Die Zellwand im Speichergewebe der Zuckerrübenwurzel. (The cell wall in the storage tissue of the sugarbeet root.), Zuckerindustrie 115 (2), 95-102, (1990).

WUCHERPFENNIG K. & POSSMANN Ph. Zur Entwicklung der Entsaftungsverfahren. Flüssiges Obst 8, 282-289, (1979).

ZAMA F. Considerazioni sulle possibilità di riduzione dei consumi energetici in zuccherificio. L'Industria Saccarifera Italiana 2, 27-32, (1977).

## COMPRESSION - DESHYDRATATION APPLIQUEE A L'INDUSTRIE DU PAPIER.

AURIAULT J.L. Non-saturated deformable porous media : quasistatics. Transp. In Prorous Media. 2(3), 45-64, (1987).

ASKLÖF C.A., LARSSON K.O. & WAHLSTROM P.B. Studies of compressibility and drainage resistance of wet paper webs. Pulp & Paper Magazine of Canada, 339-347, Aug. (1964).

ASKLÖF C.A., LARSSON K.O., LINDEROTH J. & WAHLSTROM P.B. Flow conditions in a felt in a plain press nip. Pulp & Paper Magazine of Canada, 246-250, June (1964).

BLOCH J.F. & ROUX J.C. A physical modelling of hot pressing of paper. Second European Congress on Chemical Engineering. Montpellier (1999).

BLOCH J.F. Transferts de masse et de chaleur dans les milieux poreux déformables non saturés : application au pressage du papier. Thèse de Doctorat de l'INP de Grenoble (1995).

BLOCH J.F., SILVY J., ROUX J.C. & AURIAULT J.L. Wet pressing of paper and filtration laws, 7th World Filtration Congress, May 20-23, Budapest, (1996).

BRITT K.W., UNBEHEND J.E. & SHRIDHARAN R. Observations on water removal in papermaking. Tappi Journal, 76-79, July (1986).

CHANG N.L. Dynamic compression of handsheets. Tappi Journal, 93-106, (1978).
CONSIDINE J.M, THELIN P. & FELLERS C. Compressive creep behaviour of paperboard in a cyclic humidity environment - exploratory experiment. Tappi Journal, 131-136, Nov. (1989).

CREEK J. Application and operation of the double belt filter for sludge dewatering in the North American pulp and paper industry. Technical Bulletin.

CRONEWITZ Th. Properties of beet pulp fibers in the mechanical dewatering and subsequently resulting possibilities for further development of beet pulp presses. 18th Assemblée Générale Commission Int. Tech. Sucrière, 153-185, (1982).

DAVIS E.J., STRATTON R.A. & CHANG N.L. Water removal studies with the wet press simulator. Journal of Pulp and Paper Science 68-73, July (1983).

GUNDERSON D.E., CONSIDINE J.M. & SCOTT C.T. The compressive load-strain curve of paperboard: rate of load and humidity effects. Journal of Pulp & Paper Science 14 (2), 37-41, (1988).

HOERING J.F., ELLIS E.R., JEWETT K.B., CECKLER W.H. & THOMPSON E.V. Dynamic compression of paper. II. Compression behaviour of saturated cellulose mats. Journal of Pulp and Paper Science 140-145, Nov. (1983).

JEWETT K.B., CECKLER W.H., BUSKER L.H. & CO A Computer model of a transversal flow press nip. AIChE Symposium Series 76 (200), 59-70, (1980).

KUFFERATH W. Dewatering on the paper machine. Tappi Seminar Notes, 195-239, (1984).

MCDONALD J.D. & PIKULIK I.I. Evaluating the press dewatering of pulps. Tappi Journal, 95-98, Oct. (1989).

MUKHOPADHYAY A.K. & KINGSBURY H.B. A study of two dimensional flow in the press nip. Dep. Mechanical & Aerospace Engineering, University of Delaware, Newark, DE 19711.

ÖSTERBERG L.B. Compression of press felts. Nordic Pulp and Paper Research Journal, 36-39, (1988).

RAYMOND D.R. & WILLIAMS D.G. Water removal on the Fourdrinier - Prediction of top sheet drainage requirements for linerboard. AIChE Symposium Series 207 (77), 26-31, (1981).

ROUX J.C. Modélisation et optimisation du fonctionnement d'une section de presse de machine à papier. Thèse de Doctorat de l'INP de Grenoble (1986).

SCALFAROTTO R.E. & TARVIN R.F. Efficiency of polymeric additives in drainage and dewatering. Tappi Journal 67 (4), 80-83, (1984).

WAHLSTROM P.B. Our present understanding of the fundamentals of pressing. Pulp & Paper Magazine of Canada, 76-96, Oct. (1969).

WAHLSTROM P.B. A long term study of water removal and moisture distribution on a newsprint machine press section. Part I. Pulp & Paper Magazine of Canada, 379-401, Aug. (1960).

WAHLSTROM P.B. A long term study of water removal and moisture distribution on a newsprint machine press section. Part II. Pulp & Paper Magazine of Canada, 418-442, Sept. (1960).

WRIST P.E. The present state of our knowledge of the fundamentals of wet pressing. Pulp & Paper Magazine of Canada, 284-296, July (1964).

## COMPRESSION - DESHYDRATATION APPLIQUEE A LA PRODUCTION DE PRODUITS ALIMENTAIRES EN PARTICULIER DES JUS DE FRUIT.
ANTHONEY K. ENNIS D. & COOK P. Lemon juice color evaluation: sensory and instrumental studies. Journal of Food Science 49, 1435-1437, (1984).

BARTSCH J.A. & BLANPIED G.D. Cooling rates of apples in tight and spaced stacking patterns. Staff report N° 84-3, Dept. of Agricultural Eng., New York State College of Agriculture & Life Sciences, Ithaca, NY 14853.

BATES R.P., MILLS D., MORTENSEN J.A. & CORNELL J.A. Prefermentation treatments affecting the quality of muscadine grape wines. Am. Jour. Enol. Vitic. 31 (2), 136-143, (1980).

BLANPIED G.D. & SMOCK R.M. Storage of fresh market apples. Information Bulletin 191, Cooperative Extension Cornell University, Ithaca NY 14850.

BRUHN H.D., STRAUB R.J. & KOEGEL R.G. A systems approach to the production of plant juice protein concentrate. International Grain & Forage Harvesting Conference Proceedings, (1978).

CHENCHIN K., YUGAWA A. & YAMAMOTO H.Y. Enzymic degumming of pineapple and pineapple mill juice. Journal of Food Science 49, 1327-1329, (1984).

CLARK R.C. & RAO V.N.M. Dynamic testing of fresh peach texture. Transactions of the ASAE, 777-781, (1978).

COFFELT R.J. & GIANNINI G.R. The serpentine fruit press. Transactions of the ASAE, 869-871, (1973).

DUAN L. & HEILSCHER K. Berechnung des Gaseinschlusses in Früchten. Lebensmittelindustrie 25, 223-224, (1978).

ECCHER T., ECCHER ZERBINI P. & GORINI F.L. La determinazione degli indici di qualità delle mele. Istituto Sperimentale per la Valorizzazione Tecnologica dei Prodotti Agricoli, Annali I.V.T.P.A. Vol. VII, 151-165, (1976).

ECCHER ZERBINI P., ECCHER T. & GORINI F.L. Parametri di qualità di mele Golden Delicious provenienti da frutteti a diversa altitudine. Istituto Sperimentale per la Valorizzazione Tecnologica dei Prodotti Agricoli, ATTI I.V.T.P.A. Vol. I, 45-70, (1978).

EMCH F. Extraktion und kontinuierliche Schönung. Flüssiges Obst 46 (1), 12-14, (1979).

EMCH F. Extraktion und kontinuierliche Schönung. Flüssiges Obst 46 (6), 216-229, (1979).

HAUSHOFER H. & MEIER W. Ergebnisse von Grossversuchen mit zwei kontinuierlichen Schneckenpressen im Vergleich zu horizontalen Spindelpressen bei Traubenmaischen. (Résultats d'essais entrepris à grande échelle dans le but de comparer la performance de deux pressoirs hélicoïdaux continus et celle de pressoirs à vis horizontaux pour le pressurage de raisins foulés.) Technical Bulletin.

HURLER A. & WEY R. Technique of grinding and making the mash. Confructa Studien 28 (11), 100-169, (1984).

KINZER G. & SCHREIER P. Influence of different pressing systems on the composition of volatile constituents in unfermented grape musts and wines. Am. Jour. Enol. Vitic. 31 (1), 7-13, (1980).

LAFON-LAFOURCADE S., DUBOURDIEU D., HADJINICOLAOU D. & RIBEREAU-GAYON P. Incidence des conditions de travail des vendanges blanches sur la clarification et la fermentation des moûts. Connaissance de la Vigne et du Vin 14 (2), 127-138, (1980).

LEMPERLE E. & KERNER E. Untersuchungen zur Qualitätsbeeinflussung der Moste durch Traubenpressen. Weinberg & Keller 26, 26-39, (1979).
LOZANO J.E., ELUSTONDO M.P. & ROMAGNOLI J.A. Control studies in an industrial apple juice evaporator. Journal of Food Science 49, 1423-1427, (1984).

O'BEIRNE D., VAN BUREN P. & MATTICK L.R. Two distinct fractions from senescent Idared apples extracted using non-degradative methods. Journal of Food Science 47, 173-176, (1981).

PERA J. & PERA D. Pressoir à vis. Demande de brevet d'invention N° 82 03408, (1982).

PRIEST K.L & LOUGHEED E.C. Evaluating apple maturity using the starch-iodine test. Ministry of Agriculture and Food, Ontari®, No 81-025, (1981).

SMITH R.B. & CLINE R.A. A laboratory and field study of the relationship between calcium sources and browning in Apple juice. Journal of Food Science 49, 1419-1421, (1984).

STRAUB R.J., BRUHN H.D. & BARRINGTON G.P. Plant juice protein separation and recovery. Transactions of the ASAE 27 (2), 337-340 & 347, (1984).

STRAUB R.J., BARRINGTON G.P., KOEGEL R.G. & BRUHN H.D. Heating effects on plant juice protein recovery. Transactions of the ASAE 26 (3), 673-677, (1983).

## COMPRESSION - FILTRATION : PROCEDES ET EQUIPEMENTS.

ALLENSON S.J., BRADEN M.L. & BANIK J.A. Method for dewatering a slurry using a twin belt press with cationic amine salts. United States Patent N° 4,792,406, (1988).

ARUS-ANDRITZ. Demonstration testings, Arus-Andritz, Mansfield, Texas 76063.

AUSTIN E.P. The filter belt press. Application and design. Filtration & Separation July/August 1978, 320-330.

BABBINI Presse à pulpes pour betteraves à sucre, à deux hélices, type Babbini P 18 N, à haute teneur en matière sèche. S.A.S. Fratelli Babbini 47010 Galeata Italie. (1983).

BAGDASARIAN A. & TILLER F.M. Operational features of staged, high pressure, thin cake filters. Filtration & Separation Dec. 1978, 594-598.

BARBULESCU A., BOULET R. & LESSARD D. System and method for extracting liquid from a humid mass by compression. United States Patent N° 4,534,868, (1985).

BASKERVILLE R.C., BRUCE A.M. & DAY M.C. Laboratory techniques for predicting and evaluating the performance of a filterbelt press. Filtration & Separation Sept./Oct. 1978, 445-450.

BASNETT D. Filter-pressing: an investigation into the causes of moisture-content variation in press-cakes. British Ceramic Trans. Journal 88 (3), 97-106, (1989).

BREDESON D.K. Mechanical oil extraction. Journal of the American Oil Chemists' Society 60 (2), 211-213, (1983).

BREDESON D.K. Mechanical extraction. Journal of the American Oil Chemists' Society 55 (11), 762-764, (1978).

BROWN A. The tube press. Filtration & Separation Sept/Oct. 1979, 468-474.

CARLETON A.J. Choosing a compression filter. The Chemical Engineer April 1985, 20-23.

CONDOLIOS M.E. Le compactage et le stockage des boues dans les industries extractives. Le compactage des boues par procédé Tasster. 53-56, bulletin technique.
COVINGTON R.A. & EKINER O.M. Mechanical dewatering process for elastomer slurries. United States patent N° 4,132,845, (1979).

CREEK J. & BUSCH R.A. The endless double belt filter. The economical answer to the mining industry's refuse dewatering and recovery problems. 108th Annual Meeting of the American Institute of Mining, New Orleans, (1984).

CRUZ I.E. A study and evaluation of a small-capacity oil expeller. Phil. J. Coco. Stud. 5 (2), (1980).

DAHL H.K. Anwendung der Theorie des Naßpressens für heutige Pressen. (Application of the theory of wet pressing to modern presses). Wochenblatt für papierfabrikation 3, 91-101, (1989).

DERAMMELAERE R.H. & WASP E.J. Mechanical dewatering, thermal drying and dust collection : an integrated approach. Provided by the authors.

ENGLISH R.R. & RADFORD B.C. Filter presses in coal preparation. Filtration Society meeting "Filter presses ; users' experiences", Stoke on Trent, (1977).

FERRIANI Presse à pulpes type 2F 1500 SPL. Brochure technique. Ferriani S.p.a. 44047 S. Agostino, Italie, (1982).

FERRIANI Presse à pulpes type 2F 2000 SPL. Brochure technique. Ferriani S.p.a. 44047 S. Agostino, Italie, (1982).

FINNEY J.W. Jr. Dewatering of alum sludge at Somerset, Kentucky. Kentucky-Tennessee AWWA annual meeting, Owensboro, (1979).

FISHER J., BUCHER Fa. & KLETTGAU B. Einsatz von Horizontalpressen in mittelständischen Betrieben. Flüssiges Obst 8, 255-257, (1981).

FISCHER M.C. & SCHILL M.G. The dewatering of fine coal refuse with a continuous high performance belt filter press. Society of Mining Engineers of AIME, fall meeting, Tucson, Arizona, Oct. (1979).

FOORD B.A. Computer applications in filtration analysis. The second World Filtration Congress, London, (1979).

GAUDFRIN G. Nouveaux développements du filtre à bande. Informations Chimie 289, 97-108, (1987).

GÜRTLER H.G. Filter media for the filtration of liquids, their properties and applications. Provided by the author.

GWILLIAM R.D. The E.E.C. tube filter press. Filtration & Separation March/April 1971, 173-180.

HARKER H. Getting the best out of batch filtration. Processing Feb. 1979, 34-35.

HAWORTH P.J. & ROBERTS K. Development and application of filterbelt presses. Filtration & Separation Nov./Dec. 1975, 572-576.

HEATON H.M. The practical application of the membrane filter plate to increase filter press productivity and overall economics. Filtration & Separation 15 (3) May/June 1978, 232-238.

HENRICSON K, PIKKA O. & MÄKELÄ M. Hydraulic pressurized dewatering. Paperi ja Puu - Paper and Timber 71 (3), 249-251, (1989).

HERATH B., ALBANO C., ANTILLA A. & FLYKT B. Empirical modelling of a dewatering process using multivariate data analysis. Dewatering Technology & Practice, Brisbane Australia, 129-136, (1989).
HOMANN V.T., KNUTH M., MIKSCHE K.D. & STEIN W. Die mechanische Fest-Flüssig-Trennung mittels Seiherschneckenpressen in der Speiseölindustrie. Fette. Seifen. Anstrichmittel 80 (4), 146-149, (1978).

IDE T. & KATAOKA K. A technical innovation in sludge blanket clarifiers. The second World Filtration Congress, London, 377-384, (1979).

JACKSON M.E., VINCENT M.E. & VADNAY A. Dispositif de filtration à jeter après usage. Demande de Brevet d'Invention N° 2 525 118, (1983).

JAHREIS C.A. Recent developments in pressure filtration. Chemical Engineering, Feb. 1976, 80-94.

KAWASAKI K., MATSUDA A. & MIZUKAWA Y. Compression characteristics of excess activated sludges treated by freezing-and-thawing process. Journal of Chemical Engineering of Japan 24 (6), 743-748, (1991).

KERR R.M. Development of a band filter for dewatering distillery spent wash. Filtration & Separation. Provided by the author.

KOCH K. Einrichtung zum mechanischen Trennen von Flüssigkeiten aus Flüssigkeitsfeststoffgemischen in einer Schneckenpresse. Bundesrepublik Deutschland Offenlegungsschrift DE 30 43 194 A1, (1980).

KREYE K., RICHTER H-J., RADOBERNTZKY G. & ROCH R. Verfahren zur Herstellung und Dosierung eines Preßhilfsmittels. Deutsche Demokratische Republik, Patentschrift DD 265 430 A1, (1987).

KURITA K. Filtration behaviour of filter presses under constant pressure. Filtration & Separation Jan./Feb. 1969, 31-39.

KURITA T. & SUWA S. The fully automatic filter press used for coal preparation waste water filtration. Filtration & Separation Sept./Oct. 1982, 384-386.

KURITA T. & SUWA S. Filtration and expression of municipal sludges by the automatic press. Filtration & Separation May/June 1980, 242-247.

KURITA T. & SUWA S. How the fully automatic filter press has developed. Filtration & Separation March/April 1979, 109-112.

KWADE A., SCHULZE D. & SCHWEDES J. Determination of the stress ratio in uniaxial compression tests. Powder Handling & Processing 6 (1), 61-65, (1994).

MATTESON M.J. & ORR C. Filtration: Principles and Practices. Marcel Dekker Inc., (1987).

MEIER W. & BAYER E. Prüfung der pneumatischen Willmespresse UP 600. Mitt. Klosterneuburg 32, 111-116, (1982).

MILLER P.A. & BEATON N.C. The vacuband filter - A new approach to liquid filtration. Filtration Society's Filtech /81 Conference, London, (1981).

MININNI G., SPINOSA L. & MISITI A. Evaluation of filter press performance for sludge dewatering. Journal WPCF 56 (4), 331-336, (1984).

MOIR D.N. Selecting batch pressure filters. Provided by the author.

MORLAND C.D. & GECOSALA S.R. Predicting filtration efficiency performance of fibrous media. World Filtration III, 409-419.

MORRIS K. Microcomputers and microprocessors in filter design and control. Filtration & Separation March/April 1980, 109-112.
MUNRO P.A. & STEVENSON S.R. Evaluation of a commercial belt press for the mechanical dewatering of lactic casein curd. New Zealand Journal of Dairy Science and Technology 20, 11-18, (1985).

MUNRO P.A. & VU J.T. Optimal operation of a roller press for the mechanical dewatering of acid casein curd: pilot-scale and industrial-scale experiments. New Zealand Journal of Dairy Science and Technology 18, 93-100, (1983).

MUNRO P.A., VU J.T. & MOCKETT R.B. Comparison of three pilot-scale machines for the mechanical dewatering of casein curd. New Zealand Journal of Dairy Science & Technology 18, 35-46, (1983).

NELSON J.L. & DUNWODY R.B. Selection of expression equipment for food related operations. AIChE National Meeting, San Francisco CA, Nov. (1984).

NELSON J.L. & DUNWODY R.B. Dewatering of food processing waste streams. AIChE Summer National Meeting, Philadelphia PA, Aug. (1984).

NELSON F.W., BARRINGTON G.P., STRAUB R.J. & BRUHN H.D. Design parameters for rotary extrusion macerators. Transactions of the ASAE 26 (4), 1011-1015, (1983).

NELSON F.W., BARRINGTON G.P. & BRUHN H.D. Throughput variables for rotary extrusion macerators. Transactions of the ASAE 24 (5), 1146-1148, (1981).

NELSON F.W., BRUHN H.D., KOEGEL R.G. & STRAUB R.J. Rotary extrusion devices. Transactions of the ASAE 23 (6), 1596-1599, (1980).

OLDHAM D.J. Filter presses - It's surprising what you save. Filtration Society's meeting "Filter presses users' experiences, Stoke-on-Trent, (1977).

PITT R.E, STRAUB R.J., KOEGEL R.G. & BRUHN H.D. Design criteria for a forage plant fractionation press. Transactions of the ASAE 25 (1), 18-22, (1982).

PRINSSEN A.A.J.A. Enclosed belt filters and the Pannevis Pressbelt device. Filtration & Separation Sept./Oct. 1982, 396-398 & 414.

PRINSSEN A.A.J.A. Vacuum belt filters. Filtration & Separation March/April 1979, 176-180.

PURCHAS D.B. Solid / liquid separation technology. Uplands Press, (1981).

REGAN J.M. Filter presses - Still alive and kicking! Filtration Society's meeting "Filter presses Users' experiences, Stoke-on-Trent.

RENCKEN G.E., BUCHAN M.I., TREFFY-GOATLEY K., BUCKLEY C.A. & WATER U. The tubular filter press: A locally developed alternative for sludge dewatering. ChemSA August 1989.

RICCI L. Separation techniques, II. Gas / liquid / solid systems. Chemical Engineering Magazine, Mc Graw Hill Publications, (1980).

RIVET P. Guide de la séparation liquide-solide. Société française de filtration, Idexpo, (1981).

ROBERTS K. Theory and practice of filterbelt pressing. Second World Filtration Congress, London, 495-505, (1979).

ROPERTZ G. The successful application of HP filter presses in kaolin processing. Aufbereitungs - Technik 4, 226-232, (1989).

ROSENAU J.R. Non-rennet / non-fermentative methods for process cheese manufacture. Ed. Peleg M. & Bagley E.B. AVI Publishing Company Inc. (1983).
RUSUM G., ROJANSKI V., IANCULESCU S. & FAKEKAS I. Criteria for the filters selection optimization in the waters and waste waters treatment techniques. Provided by the authors.

SATA H., ETO S. & SUZUKI H. Relationship between amount of bound water and dewatering characteristics of alum sludges. World Filtration Congress III proceedings (1986).

SCHWALBACH W. The pressbelt drum filter - A continuous filter system of combined vacuum and pressure filtration. Aufbereitungs-Technik 30 (2), 91-97, (1989).

SCHWEIGLER N. & STAHL W. High performance disc filter for dewatering mineral slurries. Filtration & Separation Jan./Feb. 1990, 38-41.

SHIRATO M., MURASE T. & ATSUMI K. Industrial expression theories and optimization of membrane-compression type filter-press operation. 2nd World Congress of Chemical Engineering Montreal Canada, Proceedings IV, 107-110, (1981).

SHIRATO M., MURASE T. & ATSUMI K. Optimization of filter-press operation with membrane-compression mechanism. Kagaku Kogaku Ronbunshu 6 (6), 643-650, (1980).

SHIRATO M., MURASE T., MORI H., OKADA A., MIZUNO Y. & NIWA H. Air blow drainage of granular beds. Kagaku Kogaku Ronbunshu 6 (4), 410-416, (1980).

SHOEMAKER W. The industrial filtration market for non-wovens. Filtration & Separation Jan./Feb. 1975.

SHOEMAKER W. The spectrum of filter media. Filtration & Separation Jan./Feb. 1975.

SINGER M. Theoritische Erkenntnisse und ihre praktischen Anwendungen bei der Extraktion von Ölsaaten. Seifen - Öle - Fette - Wachse 108 (19), 605-607, (1982).

SINGER M. Theorie und Praxis der Ölgewinnung durch Pressung. Seifen - Öle - Fette - Wachse 107 (19), 585-587, (1981).

SIVIK B. Dehydration of by-products from food industries and catering. University of Lund Sweden, Division of Food Engineering, Dissertation, (1979).

STEGEMAN G. Benefit from dewatering sludge wastes. Power April (1980).

STRAUB R.J. & KOEGEL R.G. Evaluation of a cone press for forage fractionation. Transactions of the ASAE 26 (4), 1016-1021, (1983).

SWIENTEK J. Expression-type belt press delivers high juice yields. Food Processing, 110-111, April (1985).

SWIENTEK J. Automated apple press system offers high juice recovery with low solids content. Food Processing, 134-135, April (1984).

TAKAMA K, HATANO M. & ZAMA K. Developments in the protein utilization of abundantly caught fatty fish. V. Effects of pressure treatments on eliminated lipids and recovered proteins of fish. J. of Food Sci. 30(1), 74-83, (1979).

TESCHNER F., ILLYES K. & ILLYES A. Über das Auspreßverhalten von Rübenmark. Die Lebensmittel Industrie 23, 551-553, (1976).

THOMAS C.M. Filter media for filter pressing applications. Conference on Fibrous Materials for the Filtration of Liquids at the Shirley Institute, Manchester, (1975).

TILLER F.M. Bench-scale design of SLS systems. Chemical Engineering April 1974, 117-119.
TODOROV T. & MATCHEV Ts. Appareil à presser les saucissons secs. 130-144. Provided by the author.

TOKUNAGA K., YAMAGUCHI N., NAKANO I., MIYAMOTO H. & MORIMINE R. Research on various features of filter cloth of full automatic filter press of the longest endless filter cloth travelling type and single-plate, double side filtration type with compressive mechanism. Provided by the authors.

TRAVINSKI H. Hydrocyclones: Applications and practical operation. Filtration Society's Conference Filtech / 77, London, (1977).

TRUMPLER A.L. & GOSS J.W. Sludge dewatering: current industrial practices in the U.S.A. Filtration & Separation Nov./Dec. 1982, 473-482.

USHER S.P., DE KRESTER R.G. & SCALES P.J. Validation of a new filtration technique for dewaterability characterization. AIChE J., 47 (7), 1561-1570, (2001).

VAN DEELEN W. & STEINBUCH E. Anwendungsbericht über die Ensink-Bandpresse zur Saftherstellung aus Lageräpfein. Flüssiges Obst 9, 394-397, (1983).

VAN DEELEN W. & STEINBUCH E. Juice making. Application of the ENSINK-belt-press for juice making of stored apples. Flüssiges Obst 9, 423-424, (1983).

VILLIERS R.V. & FARRELL J.B. A look at newer methods for dewatering sewage sludges. Civil Engineering - ASCE Dec. (1977).

VON WEINBAU I. & MAUL D. Arbeitsaufwand und Kosten verschiedener Keltersysteme. Die Weinwirtschaft 12 (6), 311-315, (1982).

VON WEINBAU I. & MAUL D. Arbeitsaufwand und Kosten verschiedener Keltersysteme. Die Weinwirtschaft 12 (3), 316-322, (1981).

WAKEMAN R.J. On the use of compressibility coefficients to model cake filtration processes. 2nd World Filtration Congress, London, 57-65, (1979).

WAYNE L. & OBERLIN D.W. Wastewater sludge dewatering & coolant filtration. Filtration & Separation Jan./Feb. 1982, 58-62.

## FILTRATION : APPROCHE TRADITIONNELLE.

ALT C. Centrifugal separation. NATO Advanced Study Institute, Mathematical models and design methods in solid-liquid separation, Lagos Portugal, (1982).

ARAGAKI T., HIRAOKA S., YAMADA I., MURASE T., IRITANI E. & SHIRATO M. Generalized relation between liquid pressure and solid compressive pressure in non-unidimensional filter cakes. Journal of Chemical Engineering of Japan 22 (4), 358-363, (1989).

ATSUMI K. & AKIYAMA T. A study of cake filtration - Formulation as a stefan problem. Journal of Chemical Engineering of Japan 8 (6), 487-492, (1975).

BENDER W. Filtering, washing and deliquoring of finely dispersed solids. International Chemical Engineering 18 (1), 48-59, (1978).

CHAO C.C-M. & CHANG W.V. Pressure drop across compressible random-fiber porous medium: a unified filtration theory. World Filtration Congress III, 610-617.

CHASE G.G., WILLIS M.S. & DESAI F. Compressive cake filtration. Chem. Eng. Sci., 47 (6), 1373-1381, (1992).
CHEMTOB M. Filtration en pression. Caractéristiques et domaines d'emploi. Informations Chimie 275, 193-208, (1989).

CHEN N.H. Liquid-solid filtration: Generalized design and optimization equations. Chemical Engineering July 1978, 97-101.

CINTRE M., DODDS J. & LECLERC D. Efficiency of particle capture in colloid filtration. Filtration & Separation, 259-263, (1987).

CINTRE M., CAMBON S., LECLERC D. & DODDS J. Sizing synthetic mixtures of latex and various colloidal suspensions by photon correlation spectrometry. Anal. Chem. 58, 86-90, (1986).

DARCY A. Les Fontaines Publiques de la Ville de Dijon. Dalmont, Paris, (1856).

DE MOOR A.E.L. & GREGORY J. Automated method for filtrability determination. World Filtration Congress III, 253-261.

FATHI-NAJAFI M. & THELANDER H. Determination of local filtration properties at constant pressure. Sep. Tech. 5, 165-178, (1995).

GRANGER J., DODDS J., LECLERC D. & MIDOUX N. Flow and diffusion of particles in a channel with one porous wall : polarization chromatography. Chemical Engineering Science 41 (12), 3119-3128, (1986).

GRANGER J., DODDS J.A., LECLERC D. & MOLL M. Filtration de la bière et de suspensions de latex de faibles concentrations sur membranes cellulosiques. Bios 16 (1-2), 29-34, (1985).

GRANGER J., DODDS J. & LECLERC D. Filtration of low concentrations of latex particles on membrane filters. Filtration & Separation, 58-60, (1985).

HARVEY M.A., BRIDGER K. & TILLER F.M. Apparatus for studying incompressible and moderately compressible cake filtration. Filtration & Separation Jan./Feb. 1988, 21-29.

HERMIA J. Constant pressure blocking filtration laws - Application to power-law non-newtonian fluids. Institut de Génie Chimique UCL, Louvain-la-Neuve Belgique. Provided by the author.

HERMIA J. Filter cake washing. Institut de Génie Chimique UCL, Louvain-la-Neuve Belgique. Provided by the author.

HERMIA J. Constant pressure blocking filtration laws. Trans. Inst. Chem. E.V. 60, 183-187, (1982).

HERZIG J.P., LECLERC D. & LE GOFF P. Flow of suspensions through porous media. Application to deep filtration. Ind. & Eng. Chem. 62 (5), 9-35, (1970).

LECLERC D. Vingt-cinq ans de contribution à l'étude de la filtration. Thèse de Docteur es Sciences de L'INPL de Nancy, (1988).

LECLERC D., DODDS J. & LEITZELEMENT M. La filtration en profondeur. Annales des Mines, 109-112, mai-juin (1984).

LECLERC D. & LE LEC P. Filtration : théorie. Techniques de l'Ingénieur A 5500, 1-19 & A 5501, 1-7, (1981).

LECLERC D., LE LEC P. & LE GOFF P. La filtration sur support. Techniques de l'Ingénieur A 5500, 1-19, (1980).

LECLERC D. Characteristics of unconsolidated filter media. The scientific basis of filtration. Noordhoff, Leyden, (1975).
LECLERC D. Permeability of filter media. The scientific basis of filtration. NATO ASI E 2, Noordhoff, Leyden, (1975).

LECLERC D., CHARPENTIER J.C. & LE GOFF P. Le désaérage des liquides dans les milieux filtrants. 1er Congrès Mondial de la Filtration, Technique et Documentation, Paris, (1974).

LE GOFF P., LECLERC D. & DODDS J. The structure of packed beds : continuity of research in Nancy and some new results. Powder Technology 42, 47-53, (1985).

LEITZELEMENT M., DODDS J. & LECLERC D. Measurement of the particle size of submicron mineral particles, emulsions and biological material by hydrodynamic chromatography and photon correlation spectroscopy. Particle size analysis 1985, 619-631, Wiley, Chichester, (1985).

LLOYD P.J. Particle characterization. Chemical Engineering, 120-122, April (1974).

LU W-M., TILLER F.M., CHENG F-B. & CHIEN C-T. A new method to determine local porosity and filtration resistance of filter cakes. Journal of Chinese Institute of Chemical Engineers 1, 45-53, (1970).

MAHGOUB I., PROST C. & DODDS A. Experimental relative permeability curves compared with a new cutting and joining model of porous media with a distribution of pore sizes. Powder Technology 31, 143-151, (1982).

McTAGGART J.A. Growth in filtration and separation products over the last seven years in the U.S. World Filtration Congress III, 703-706.

MERIGUET G. Filtration. II. Technologie. Techniques de l'Ingénieur J 1542, (1976).

MULDOON J. Screens and screening. Chemical Engineering, Feb. 1984, 90-94.

MURASE T., KOBAYASHI K., IRITANI E., ITO K. & SHIRATO M. Constant-pressure filtration on cylindrical surfaces of power-law non-newtonian fluids. Journal of Chemical Engineering of Japan 18 (3), 230-236, (1985).

MURASE T., IRITANI E., CHO J.H. & SHIRATO M. Determination of filtration characteristics based upon filtration tests under step-up pressure conditions. Journal of Chemical Engineering of Japan 22 (4), 373-378, (1989).

NELSON R.W. Approximate theories of filtration and retention. Tappi Journal 47 (12), 752-764, (1964).

OKAMURA S. & SHIRATO M. Liquid pressure distribution within cakes in the constant pressure filtration. Kagaku Kogaku 19 (3), 104-110, (1955).

OLSON W.P., ERAS M.H. & PARKS R.G. Problems of industrial filtration and membrane filter testing. World filtration Congress III, 484-490.

PURCHAS D.B. A practical view of filtration theory. Filtration & Separation March/April 1980, 147-151.

RUSHTON A., HOSSEINI M. & HASSAN I. The effects of velocity and concentration on filter cake resistance. J. Separ. Proc. Technol. 1 (3), 35-41, (1980).

RUSHTON A. Centrifugal filtration, dewatering and washing. Chemical Engineering Department UMIST Manchester. Provided by the author.

SATOH T. & ATSUMI K. Forced expression of liquid from pastry mixture of liquid and fine solid particles and its numerical solution by method of moments. Journal of Chemical Engineering of Japan 3 (1), 92-98, (1970).
SECO TORRECILAS A., AUCEJO PEREZ A. & SANCHO-TELLO DE CARRANZA M. Filtration a presion de pastas ceramicals. II Estudio de la influencia de la presion y la granulometria sobre la resistencia especifica de la torta. Bol. Soc. Ceram. Vidr. 20 (4), 235-240,

SHIRATO M. & IRITANI E. Filtration principles. Journal of Japan Society of Lubrication Engineers 29 (12), 12-15, (1984).

SHIRATO M. & IRITANI E. Filtration theory for non-newtonian fluids. 3rd Pacific Chemical Engineering Congress, Seoul, (1983).

SHIRATO M., IRITANI E., HAYASHI N. & ITO K. Experimental study on filter cake dewatering by permeation of non-newtonian fluids. Journal of Chemical Engineering of Japan 16 (2), 159-161, (1983).

SHIRATO M., ARAGAKI T., IRITANI E. & FUNAHASHI T. Constant rate and variable pressure-variable rate filtration of power-law non-newtonian fluids. Journal of Chemical Engineering of Japan 13(6), 473-478, (1980).

SHIRATO M., ARAGAKI T. & IRITANI E. Analysis of constant pressure filtration of power-law non-newtonian fluids. Journal of Chemical Engineering of Japan 13 (1), 61-66, (1980).

SHIRATO M., ARAGAKI T. & IRITANI E. Blocking filtration laws for filtration of power-law non-newtonian fluids. Journal of Chemical Engineering of Japan 12 (2), 162-164, (1979).

SHIRATO M., ARAGAKI T., IRITANI E., WAKIMOTO M., FUJIYOSHHI S. & NANDA S. Constant pressure filtration of power-law non-newtonian fluids. Journal of Chemical Engineering of Japan 10 (1), 54-60, (1977).

SHIRATO M., MURASE T., SAMBUICHI M. & ARAGAKI T. Industrial filtration - Theory and practice. Kagaku Kogaku 40 (11), 606-612, (1976).

SHIRATO M., MURASE T. & ARAGAKI T. Study on industrial filtration of non-newtonian fluid. Asahi Garasu Kogyo Gijutsu Shorei-kai Kenkyu Hokoku 28, 269-286, (1976).

SHIRATO M., KOBAYASHI K. & TANIMURA M. Analysis of constant pressure filtration of compressible cakes on cylindrical surface. Kagaku Kogaku 37 (1), 76-82, (1973).

SHIRATO M., DOI H. & SHIMAOKA H. Analysis of filtration processes in plate-and frame press. Kagaku Kogaku 36 (7), 781-787, (1972).

SHIRATO M. & ARAGAKI T. Verification of internal flow mechanism theory of cake filtration. Filtration & Separation May/June 1972.

SHIRATO M., ARAGAKI T., ICHIMURA K. & OOTSUJI N. Porosity variation in filter cake under constant-pressure filtration. Journal of Chemical Engineering of Japan 4 (2), 172-177, (1971).

SHIRATO M., SAMBUICHI M., KATO H. & ARAGAKI T. Internal flow mechanism in filter cakes. AIChE Journal 15 (3), 405-409, (1969).

SHIRATO M., ARAGAKI T., MORI R. & IMAI K. Study on variable-pressure variable-rate filtration. Kagaku Kogaku 33(6), 576-581, (1969).

SHIRATO M. & ARAGAKI T. The relations between hydraulic and compressive pressure in non uni-dimensional filter cakes. Kagaku Kogaku 33 (2), 205-207, (1969).

SHIRATO M., ARAGAKI T., MORI R. & SAWAMOTO K. Predictions of constant pressure and constant rate filtrations based upon an approximate correction for side wall friction in compression permeability cell data. Journal of Chemical Engineering of Japan 1 (1), 86-90, (1968).
SHIRATO M., MURASE T. & KOBAYASHI K. The method of calculation for non uni-dimensional filtration. Filtration & Separation May/June (1968).

SHIRATO M. & KOBAYASHI K. Studies in non uni-dimensional filtration. Filtration on cylindrical spherical and square surfaces. Memoire of the Faculty of Engineering, Nagoya University 19 (2), (1967).

SHIRATO M., SHIMIZU S. & KAKIZOE M. Measurement of pore size distribution of filter medium. Kagaku Kogaku 31 (1), 60-66, (1967).

SHIRATO M., MURASE T., HIRATE H. & MIURA M. Study on non uni-dimensional filtration - Definition of effective coefficient of filter area. Kagaku Kogaku 29 (12), 1007-1013, (1965).

SHIRATO M., SAMBUICHI M. & MURASE T. Hydraulic pressure distribution in filter-cakes under constant pressure filtration. Memoires Faculty of Engineering Nagoya Univ. 16, 68-74, (1964).

SHIRATO M., SAMBUICHI M. & OKAMURA S. Filtration behavior of a mixture of two slurries. AIChE Journal 9 (5), 599-605, (1963).

SHIRATO M. Recent theory and technique of filtration. Kagaku Kogaku 23 (12), 823-830, (1959).

SHIRATO M. & OKAMURA S. Behaviours of slurry mixtures at constant-pressure filtration. Kagaku Kogaku 23 (4), 226-234, (1959).

SHIRATO M. & OKAMURA S. Behaviours of various kaolin slurries in constant pressure filtration. Kagaku Kogaku 23 (1), 11-17, (1959).

SILVERBLATT C.E., RISBUD H. & TILLER F.M. Batch, continuous processes for cake filtration. Chemical Engineering, April 1974, 127-136.

SMILES D.E., RAATS P.A.C. & KNIGHT J.H. Constant pressure filtration: The effect of a filter membrane. Chemical Engineering Science 37 (5), 707-714, (1982).

TETTAMANTI B. Modelling and scaling of filtration in industry. International Chemical Engineering 22(3), 561-571, (1982).

TILLER F.M., CRUMP J.R. & VILLE F. A revised approach to the theory of cake filtration. Fine particles processing Vol.2, 1949-1980, Amer. Inst. Min. Met. Pet. Eng. (1980).

TILLER F.M., CRUMP J.R. & VILLE F. Filtration theory in its historical perspective, a revised approach with surprises. The second World Filtration Congress, London, (1979).

TILLER F.M. & ANDERSON M.D. Solid-liquid separation : an overview with emphasis on compressible cake filtration. ASEE Summer School, (1977).

TILLER F.M. Bench-scale design of SLS system. Chemical Engineering, April 1974, 117-119, (1974).

TILLER F.M., WILENSKY J. & FARRELL P.J. Pretreatment of slurries. Chemical Engineering, April 1974, 123-126.

TILLER F.M. & RISBUD H. Hydraulic deliquoring of filter cakes. PACHEC 72, 80-87.

TILLER F.M. & LU W-M. The role of porosity in filtration. VIII: cake nonuniformity in compression-permeability cells. AIChE Journal 18 (3), 569-572, (1972).

TILLER F.M., HAYNES S. & LU W-M. The role of porosity in filtration. VII: effect of side-wall friction in compression-permeability cells. AIChE Journal 18 (1), 13-20, (1972).

TILLER F.M. & SHIRATO M. The role of porosity in filtration. VI: New definition of filtration resistance. AIChE Journal 10 (1), 61-67, (1964).
TILLER F.M. & COOPER H. The role of porosity in filtration. V: porosity variation in filter cakes. AIChE Journal 8 (4), 445-449, (1962).

TILLER F.M. & COOPER H. The role of porosity in filtration. IV: constant pressure filtration. AIChE Journal 6 (4), 595-601, (1960).

TILLER F.M. The role of porosity in filtration. III: variable-pressure, variable rate filtration. AIChE Journal 4 (2), 170-174, (1958).

TILLER F.M. The role of porosity in filtration. Numerical methods for constant rate and constant pressure filtration based on Kozeny's law. Chemical Engineering Progress 49 (9), 467-479, (1953).

TOSUN I. & WILLIS M.S. Cake filtration theory : an overview. 4th World Filtration Congress, Anvers, (1986).

VAN BRAKEL J., KLEIZEN H.H., WYPKEMA. & DOSOUDIL M. Continuous pressure filtration of coal and prediction of cake permeability. Filtration & Separation July/August 1987, 268-271.

VAN DEN BERG G.B. & SMOLDERS C.A. Flux decline in membrane processes. Filtration & Separation March/April 1988, 115-121.

VAN DOOREN K.G.J. Les tissus filtrants en fibre fluorocarbonée Teflon® dans le dépoussiérage intensif des centrales au charbon, Du Pont de Nemours International S.A. CH1211 Genève.

WAKEMAN R.J. Thickening and filtration: a review and evaluation of recent research. Trans. IChemE 59, 147-160, (1981).

WAKEMAN R.J. & RUSHTON A. The removal of filtrate and soluble material from compressible filter cakes. Filtration & Separation Sept./Oct. 1976, 450-454.

WILLIS M.S. & TOSUN I. A rigorous cake filtration theory. Chemical Engineering Science 35, 2427-2438, (1980).

WILLIS M.S., TOSUN I. & COLLINS R.M. A rigorous filtration theory and a new characterization of filter cakes. Second World Filtration Congress, 629-636, (1979).

YOSHIOKA N., UEDA K. & MIYOSHI K. Liquid pressure distribution for filtration through curved leaves. Journal of Chemical Engineering of Japan 5 (3), 77-82, (1972).

## FILTRATION : SEDIMENTATION - DRAINAGE ET ASPECTS CARACTERISATION DES PARTICULES.

AKERS R.J. The compression of flocculated sediments. Filtration & Separation July/Aug. 1980, 354-356.

ALEXANDER K.S., AZIZI J., DOLLIMORE D. & PATEL F.A. An interpretation of the sedimentation behavior of pharmaceutical kaolin and other kaolin preparations in aqueous environments. Drug development and industrial pharmacy 15 (14-16), 2559-2582, (1989).

BERSA L., SENGUPTA D.K. & ROY S.K. Particle characteristics and their influence on dewatering of kaolin, calcite and quartz suspensions. Int. J. Mineral Proc. 59, 89-112, (2000).

BOCKSTAL F., FOUARGE L., HERMIA J. & RAHIER G. Constant pressure cake filtration with simultaneous sedimentation. Filtration & Separation July/Aug. 1985, 255-257.

CHRISTENSEN G.L. & DICK R.I. Specific resistance measurements. Environmental Engineering, 111, (3), 265-270, (1985).
CHRISTENSEN G.L. & DICK R.I. Specific resistance measurements: non parabolic data. Journal of Environmental Engineering 111 (3), 243-257, (1985).

DOLLIMORE D., HORRIDGE T.A. & ROBINSON R. The effect of the hydration of the exchangeable cation on the flocculation behaviour of china clay-polyacrylamide suspensions, Powder Technology 17, 207-212, (1977).

DOLLIMORE D. & HORRIDGE T.A. The filtration of beds of china clay flocculated by polyacrylamide. Powder Technology 5, 111-114, (1971-1972).

EL ATTAR SOFI Y., CARQUILLE C., LAMARCHE J.M. & FOISSY A. Influence of calcium and phosphate ions on the adsorption of partially hydrolyzed polyacrylamides on TiO2 and CaCo3. Prog. Colloid. Polym. Sci. 82, 43-51, (1990).

FLOERGER Floculants - essais de laboratoire. Brochure technique.

FOISSY A., PERSELLO J., LAMARCHE J.M. & ROBERT G. Stability measurement and dispersion process of CaCO3. Suspensions in sodium polyacrylate aqueous solutions. J. Dispers. Sci. Technol. 3, 105, (1982).

FOISSY A., EL ATTAR A. & LAMARCHE J.M. Adsorption of polyacrylic acid on titanium dioxide. Journal of Colloid and Interface Science 96 (1), 275-287, (1983).

FRANCOIS R.J., VAN HAUTE A.A. & WINDERICKX G.C. Influence of process parameters of flocculation and floc characteristics. Brochure technique.

GOTOH K., SAKAI T., KATO H. & SHIRATO M. Sedimentation of flocculated slurry. Kagaku Kogaku 32 (11), 1105-1110, (1968).

GY J. Determination of the filtration characteristics with the aid of the compression permeability measuring technique. Acta Technica 89 (1-2), 127-144, (1979).

HUNTER R.J. & DELL C.C. Development of a fluxmeter for studying the dewatering of flocculated clay. Provided by the authors.

KAVANAGH B.V. The dewatering of activated sludge: measurement of specific resistance on filtration and capillary suction time. Water Pollution Control, 388-398, (1980).

KOS P. Theory of gravity thickening of flocculent suspensions and a new method of thickener sizing. Second World Filtration Congress London, 595-603, (1979).

LAMARCHE J.M., PERSELLO J. & FOISSY A. Influence of molecular weight of sodium polyacrylate in calcium carbonate aqueous dispersions. I & EC Product Research & Development 22, 123-126, (1983).

LAMARCHE J.M., FOISSY A., ROBERT G., REGGIANI J.C. & BERNARD J. Adsorption d'un polymère de l'acide acrylique sur le dioxide de titane. Fournie par les auteurs.

LLOYD P.J. Particle characterization. Chemical Engineering April 1974, 120-123.

MURASE T., IWATA M., KATO I., LEE W.D. & SHIRATO M. Determination of specific resistance of highly compressed cake by use of expression data. Journal of Chemical Engineering of Japan 21 (2), 204-206, (1988).

NONAKA M. Micro-processes in the sedimentation and thickening processes. World Filtration Congress III proceedings, 380-389, (1982)

PEARSE M.J. & BARNETT J. Chemical treatments for thickening and filtration. Filtration & Separation Sept./Oct. 1980, 460-470.
PIERRE A., LAMARCHE J.M., MERCIER R., FOISSY A. & PERSELLO J. Calcium as potential determining ion in aqueous calcite suspensions. J. of Disp. Sci. & Technol. (1990).

PIERRE A. ET AL. The adsorption of cement superplasticizers on to mineral dispersions. Adsorption Science & Technology 6 (4), 219-231, (1989).

PIERRE A., LAMARCHE J.M., MERCIER R. & FOISSY A. Adsorption d'un fluidifiant du ciment sur le carbonate de calcium. Cement and Concrete Research 19, 692-702, (1989).

POPPING B., DERATANI A., SEBILLE B., DESBOIS N., LAMARCHE J.M. & FOISSY A. The effects of electrical charge on the adsorption of a weak cationic polyelectrolyte onto silica, silicon carbide and calcium fluoride. Colloids and Surfaces 64, 125-133, (1992).

READ A.D. The use of high molecular weight polyacrylamides in the selective flocculation separation of a mineral mixture. Br. Polym. J. 4, 253-264, (1972).

REBOUILLAT S. & SCHWARTZBERG H.G. Dynamic and static filtration resistances during mechanical expressions involving foods. LE MAGUER M. & JELEN P. Ed., Elsevier, (2), 281-292, (1986).

RUDZINSKI W., CHARMAS R., PARTIKA S. & FOISSY A. A calorimetric - thermodynamic study of ion adsorption at the water/anatase interface, based on the surface complexation model. New Journal of Chemistry 15 (5), (1991).

SHIRATO M., MURASE T., MORI H., MIZUNO Y., OGURA T. & NAKAI E. Gravitational drainage of a granular packed bed. International Chemical Engineering 21 (2), 294-302, (1981).

SHIRATO M., MURASE T., MORI H., OKADA A., MIZUNO Y. & NIWA H. Air blow drainage of granular beds. Kagaku Kogaku Ronbunshu 6 (40), 410-416, (1980).

SYNDICAT D'ETUDE DE L'EAU S.E.D.E. The practical theory of flocculation processes. Proceedings of S.E.D.E., K.U. Leuven, 1-20, (1981).

TAKAI R., ABE H., WATANABE H., HASEGAWA H. & SAKAI Y. Average specific cake resistance determined in the presence of sedimentation in filtration of starch slurry under constant pressure. Journal of Food Process Engineering 9, 265-275, (1987).

WAKEMAN R.J. & HOLDICH R.G. Theoretical and experimental modelling of solids and liquid pressures in batch sedimentation. World Filtration Congress III, 346-353, (1982).

WARD A.S. Pretreatment processes. Department of Chemical Engineering, University of Technology Loughborough, Leics. U.K., (1982).

WARDEN J.H Continuous thickening of flocculent sludge. Provided by the author.

ZEITSCH K. Eine neue Theorie für die Permeabilität eines durch Sedimentation entstandenen Filterkuchens. Tech. Mitt. Krupp. Forsch. Ber. 42 (2), 71-76, (1984).

ZERROUK R., FOISSY A., MERCIER R., CHEVALLIER Y. & MORAWSKI J-C. Study of Ca2+ - Inducéd silica coagulation by small angle scattering. Journal of Colloid and Interface Science 139 (1), 20-29, (1990).

## PRESSAGE A VIS ET EXTRUSION.

BONAMICI F. Filtre-presse à vis à étages multiples pour extraire et filtrer les matières solubles contenues dans les matières végétales. Demande de Brevet d'Invention N° 82 03766, (1982).
BOSTANDZHIYAN S.A., BOYARCHENKO V.I. & KARGAPOLOVA G.N. The non-isothermal extrusion of anomalously viscous fluids under complex shear conditions. International Chemical Engineering 12 (3), 379-384, (1972).

BREDESON D.K. The role of the screw press in a modern vegetable oil plant. AIChE Summer National Meeting, Denver, (1983).

BRETSCHNEIDER R., DYNTAR J. & STRANSKY J. Press TL-1000. Listy Cukrovarnick 96, 8-16, (1980).

DYER D.F. A numerical solution for the single-screw extrusion of a polymer melt. AIChE Journal 15 (6), 823-828, (1969).

EDWARDS R.H. & KOHLER G.O. Wet fractionation of forage to reduce energy requirements of dehydration. U.S. Department of Energy Report, DE-AI01-78CS40090, (1982).

FARMER J. The theory, design and operation of multiple barrelled screw presses for bagasse. Factory Engineering. Provided by the author.

ITO S. & KAJIUCHI T. Mixing effects in a single-screw extruder. Kagaku Kogaku Ronbunshu 1 (2), 131-136, (1975).

ITO S. & KAJIUCHI T. On power characteristics of single-screw extruder. Kagaku Kogaku 38 (2), 138-144, (1974).

ITO S. & KAJIUCHI T. On flow characteristics in single-screw extruder. Kagaku Kogaku 37 (6), 581-587, (1973).

KEMBLOWSKI Z. & SEK J. Residence time distribution in a real single screw extruder. Polymer Engineering and Science 21 (18), 1194-1202, (1981).

LEE M.C. Surimi process technology. Food Technology, 69-80, Nov. 1984.

LEVINE L. & ROCKWOOD J. Simplified Models for estimating operating characteristics of food extruders. AIChE National Meeting, San Francisco, (1984).

LÜTHI H.R. & GLUNK U. Extraktion und kontinuierliche Schönung. 3. Entwicklung der Extraktion von Appfelsaft und Betriebserfahrungen. Flüssiges Obst 46, 111-120, (1979).

MATON A.E. Screw feeder design. Powder Handling & Processing 6 (1), 87-89, (1994).

SCHÜLE H., MEDER S. & MÜLLER C. Determining the residence time behavior in extruders. Kunststoffe German Plastics 78 (1), 78-83, (1988).

SCHÜLE H., MEDER S. & MÜLLER C. Grundlagen des Verweilzeitverhaltens in Extrudern. (Fundamentals of the residence time behaviour in extruders.) Kunststoffe German Plastics 77 (12), 1269-1272, (1987).

SHIRATO M., MURASE T., IWATA M., HAYASHI N. & OGAWA Y. Continuous expression of slurry in a screw press. International Chemical Engineering 25 (1), 88-96, (1985).

SHIRATO M., MURASE T. & IWATA M. Pressure profile of a Bingham fluid in constant-pitch, straight-taper and decreasing-pitch screw extruders. International Chemical Engineering 23 (4), 689-698, (1983).

SHIRATO M., MURASE T., HAYASHI N. & IWATA M. Basic concept and its application of continuous screw expression. 3rd Pacific Chemical Engineering Congress, Seoul, (1983).
SHIRATO M., MURASE T. & IWATA M. Pressure profile in a power-law fluid in constant-pitch, straight-taper and decreasing-pitch screw extruders. International Chemical Engineering 23 (2), 323-332, (1983).

SHIRATO M., MURASE T., IWATA M., HAYASHI N. & OGAWA H. Studies in screw expression of slurries. Kagaku Kogaku Ronbunshu 9 (3), 302-308, (1983).

SHIRATO M., MURASE T. & IWATA M. Pressure profiles in constant-pitch, straight-taper and decreasing-pitch screw extruders. International Chemical Engineering, 22(3), 470-478, (1982).

SHIRATO M., MURASE T. & IWATA M. Pressure profile of power law non-Newtonian fluids in screw extruders of constant pitch-straight taper type and varying pitch type. Kagaku Kogaku Ronbunshu 7 (6), 556-563, (1981).

SHIRATO M., MURASE T. & IWATA M. Pressure profile in screw extruders of constant pitch-straight taper type and varying pitch type. Kagaku Kogaku Ronbunshu 6 (5), 461-467, (1980).

SHIRATO M., MURASE T., HAYASHI N., MIKI K., FUKUSHIMA T., SUZUKI T., SAKAKIBARA N. & TAZIMA T. Fundamental studies on continuous extrusion using a screw press. International Chemical Engineering 18 (4), 680-688, (1978).

TINDALE L.H. & HILL-HAAS S.R. Current equipment for mechanical oil extraction. Am. Oil Chemists Soc. 53, 265-264, (1976).

UNDERWOOD W.M. Experimental method for designing extrusion screws. Provided by the author.

UTVIK A. Liquid/solid separation by twin screw pressing. AIChE Meeting, Denver, (1983).

WARD J.A. Processing high oil content seeds in continuous screw presses. J. Am. Oil Chemists Soc. 53, 261-264, (1976).

## DESHYDRATATION : ASPECTS ECONOMIQUES ET ENERGETIQUES.

ASSOCIATION POUR LA PROMOTION INDUSTRIE AGRICULTURE. Le séchage dans les industries agroalimentaires. Aspects énergétiques. 2ème Salon International du Génie Industriel Alimentaire. (1978).

BAUSIER A. Le séchage des pulpes et les économies éventuelles de combustibles. La Sucrerie Belge 93 (7), 293-307, (1974).

CARLETON A.J. Dewatering filter cakes by compression. I.CHEM.E. Symposium Series 91, 175-186. Provided by the author.

CONFEDERATION GENERALE DES PLANTEURS DE BETTERAVES L'économie betteravière. Editions SEDA, Paris, (1995).

DEMAUX M. Pressage des pulpes de betteraves. Essai d'optimisation technico-économique. Ind. Agr. et Ali. 95 (7-8), 811-820, (1978).

DEMAUX M. La pulpe sèche et les économies d'énergie. Ind. Agr. et Ali. 94 (7-8), 737-747, (1977).

FRERE G. Economie d'énergie en déshydratation de produits agricoles. Ind. Ali. et Agr., 491-497, (1981).

GAUTHIER A. & BIMBENET J-J. Simulation et optimisation technico-économique du séchage des fourrages dans les séchoirs à tambour rotatif. Ind. Ali. et Agr., 371-380, (1977).

HAUBRY A. Technical and economical impact of the squeezor in the sludge filtration field. Symposium Société Belge de Filtration Louvain-la-Neuve, (1981).
KOCIENCKI H.L. & ROE R.A. Waste disposal costs reduced 90% by dewatering. Chemical Processing, Sept. (1980).

LECLERC D. & REBOUILLAT S. Economic Optimization Strategy in Solid/liquid Separation Processes: Mechanical Dewatering of Flocculated Sludges – Theoretical evaluation of energetic and economic optima. Int. Symp., Louvain-la-Neuve, Belgium, SBF Ed., 283-313, Nov. (1985).

LECLERC D. & REBOUILLAT S. Déshydratation des suspensions solide-liquide par filtration-séchage. Energétique industrielle, Tome 3, Technique et Documentation, Lavoisier, Paris, (1982).

LECLERC D. & REBOUILLAT S. Déshydratation des suspensions solide-liquide par filtration-séchage. Energétique industrielle, Tome 3, Technique et Documentation, Lavoisier, Paris, (1982).

LE GOFF P. Energétique industrielle. Tome 2, Chap. 9 & 10. Tome 3, Chap. 11. & 15, Technique & Documentation, Lavoisier, Paris, (1980).

LE GOFF P. Optimisation énergétique ou optimisation économique des procédés industriels. Application à la séparation des mélanges. Fournie par l'auteur.

LUCAS M., MORIN M. & LEDRU M. Economies d'énergie dans la déshydratation des pulpes et des luzernes par séchage basse température. Ind. Ali. et Agr., 739-742, (1980).

MAFART P. Coût Energétique des Opérations d'Elimination d'Eau. Tech et Doc Ed., Paris, (1980).

MARC M. La filtration en pleine évolution. Alimentation 91, 77-84, (1981).

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<figure><img src="image_194.png" /></figure>
Annexes

ANNEXE A1

P. 244

La compression unidirectionnelle à pression constante (CPC)
Théorie Traditionnelle

ANNEXE A2

P. 261

L'usage des modèles rhéologiques pour l'étude théorique de la CPC

ANNEXE A3

P. 269

La compression unidirectionnelle à déformation constante (CDC)
Conditions opératoires / Aspects théoriques de la CDC1 et de la CDC2

ANNEXE A4

P. 285

La compression axiale avec écoulement radial (CAR) et le pressage à vis

ANNEXE A5

P. 290

Publication P1

ANNEXE A6

P. 326

Publication P5

ANNEXE A7

P. 334

Publication P7

ANNEXE A8

P. 337

Publication P8

ANNEXE A9

P. 341

Eléments de réponse aux QFP
<figure><img src="image_195.png" /></figure>
# ANNEXE A1

## LA COMPRESSION UNIDIRECTIONNELLE A PRESSION CONSTANTE (CPC)

### ASPECT THEORIQUE

## THE THEORY OF ONE-DIMENSIONAL COMPRESSION

### THE STUDY OF DIFFERENT CASES REGARDING $P_{fj}$ VARIATION :

1. Constant $P_{fj}$ with cake height

2. Linear variation of $P_{fj}$ with cake height

3. Sinusoidal variation of $P_{fj}$ with cake height
## The Theory of one-dimensional Compression

1. The saturated porous material (SPM) is homogeneous.

2. The SPM is saturated.

3. Both the liquid and solid constituents are perfectly incompressible compared to the compressibility of the SPM.

4. The compression of the SPM layer is due to the volume change only, which is due to the expression of liquid from the voids.

5. Darcy's law is applicable.

6. Deformation of SPM takes place in the direction of the load application.

7. The coefficient of compression $C_v$ is constant during the compression.

In the case of one-dimensional compression the flow of liquid is in one direction (z direction), so the flows $q_x$, $q_y$ and $dq_x$, $dq_y$ are equal to zero (Fig. A1.1) and (Fig. A1.2).

<figure><img src="image_196.png" /><figcaption>Figure A1.1 : One-dimensional compression</figcaption></figure>

<figure><img src="image_197.png" /><figcaption>Figure A1.2 : Flow pattern from point A</figcaption></figure>

To calculate the flow $q_z$ entering the SPM prism at point A in z direction, we use Darcy's law:

$$q_z = K_z \frac{\partial h}{\partial z} dx dy \qquad \text{Eq. A1.1}$$

$$\frac{\partial h}{\partial z} = \text{hydraulic gradient [-]}$$

$q_z$ = flow entering in direction z [$m^3 s^{-1}$]

$K_z$ = coefficient of permeability in direction z [$m s^{-1}$]

$h$ = hydraulic head at point A [m]

$dx dy$ = surface in direction z [$m^2$]

We can express $K_z$ by the relation:

$$K_z = \frac{K_i \cdot \rho \cdot g}{\mu} \qquad \text{Eq. A1.1b}$$

$K_i$ : intrinsic coefficient of permeability [$m^2$]

$\mu$ : fluid viscosity [$kg/(m \cdot s)$] [$Pa \cdot s$]
ρ : mass density of the fluid [kg/m³]

g : Acceleration due to gravity [m/s²]

The flow leaving the prism in z direction ($q_z + dq_z$)

with $dq_z = K_z \frac{\partial^2 h}{\partial z^2} dx dy$

Hence, $(q_z + dq_z) = K_z \left( \frac{\partial h}{\partial z} + \frac{\partial^2 h}{\partial z^2} \right) dz$ $dx dy$ Eq. A1.2

Rate of change of volume of the SPM element :

$\frac{\partial V_T}{\partial t} = (q_z + dq_z) - q_z$ Eq. A1.3

with $V_T = dx dy dz$ Eq. A1.4

Substituting Eqs. A1.1 and A1.2 into Eq. A1.3, we obtain Eq. A1.5.

$(q_z + dq_z) - q_z = K_z \left( \frac{\partial h}{\partial z} + \frac{\partial^2 h}{\partial z^2} \right) dz$ $dx dy - K_z \frac{\partial h}{\partial z} dx dy = \frac{\partial V_T}{\partial t}$

$K \frac{\partial^2 h}{\partial z^2} dx dy dz = \frac{\partial V_T}{\partial t}$ Eq. A1.5

K : Coefficient of permeability [m·s⁻¹]

However, $h = \frac{P_f}{\gamma_f}$ [m] Eq. A1.6

$P_f$: Excess pore fluid pressure [N/m²]

$\gamma_f$: Unit weight of fluid for water: $\gamma_f = 9.81[kN/m^3]$ or $[kg/m^3 \cdot m/s^2]$

Substitution of Eq. A1.6 into Eq. A1.5

$$ \frac{K}{\gamma_f} \frac{\partial^2 P_f}{\partial z^2} = \frac{1}{dxdydz} \frac{\partial V_T}{\partial t} $$ Eq. A1.7

We assume that during compression the rate of change of volume is equal to the rate of change of void volume (liquid), the cross section is constant. So,

$\frac{\partial V_T}{\partial t} = \frac{\partial V_f}{\partial t}$ Eq. A1.8

$V_f$ = Volume of voids in the SPM element (volume of liquid), and

$e$: Void ratio, $e = \frac{V_f}{V_s}$ [-] Eq. A1.9

$V_s$ = volume of solids in the SPM element (constant), so

$$ \frac{\partial V_T}{\partial t} = V_s \frac{\partial e}{\partial t} = V_T \frac{V_s}{V_T} \frac{\partial e}{\partial t} = V_T \frac{V_s}{V_s + V_f} \frac{\partial e}{\partial t} = V_T \frac{V_s}{V_s + V_f} \frac{\partial e}{\partial t} = V_T \frac{1}{1+e} \frac{\partial e}{\partial t} $$

$$ \frac{\partial V_T}{\partial t} = \frac{dxdydz}{1+e} \frac{\partial e}{\partial t} $$ Eq. A1.10
Substituting the above relation into Eq. A1.7, we get Eq. A1.11.

$$ \frac{K}{\gamma_f} \frac{\partial^2 P_f}{\partial z^2} = \frac{1}{\text{dxdydz}} \frac{\text{dxdydz}}{1+e} \frac{\partial e}{\partial t} = \frac{1}{1+e} \frac{\partial e}{\partial t} \qquad \text{Eq. A1.11} $$

The change in void ratio, $\partial e$, is due to the increase of effective stress, assuming that they are linearly related, then

$$ \partial e = -a_v \partial(P_s) \qquad \text{Eq. A1.12} $$

$a_v$ = coefficient of compressibility [$m^2/N$]

The increase of effective stress is due to the decrease of excess pore fluid pressure, $\partial P_f$.

In other words:
$$ \partial P_s + \partial P_f = \partial P $$
$$ \uparrow \text{ solid} + \downarrow \text{ liquid} = \text{Cte} $$

Hence,

$$ \partial e = a_v \partial P_f \qquad \text{Eq. A1.13} $$

$\partial P = Cte = 0$ in the case of CPC.

Combining Eq. A1.11 with Eq. A1.13, we obtain Eq. A1.14

$$ \frac{K}{\gamma_f} \frac{\partial^2 P_f}{\partial z^2} = \frac{a_v}{1+e} \frac{\partial P_f}{\partial t} = m_v \frac{\partial P_f}{\partial t} \qquad \text{Eq. A1.14} $$

$m_v$: Coefficient of volume compressibility [$m^2 N^{-1}$], $m_v = \frac{a_v}{1+e}$ \qquad Eq. A1.15

or

$$ \frac{\partial P_f}{\partial t} = \frac{K}{\gamma_f \cdot m_v} \cdot \frac{\partial^2 P_f}{\partial z^2} = C_v \cdot \frac{\partial^2 P_f}{\partial z^2} \qquad \text{Eq. A1.16} $$

We can use at this stage the mathematics of diffusion by CRANK (1973).

$C_v$: Coefficient of compression [$m^2 s^{-1}$], $C_v = \frac{K}{\gamma_f \cdot m_v}$ \qquad Eq. A1.17

Eq. A1.16 can also directly be obtained from Eq. 1.33. Eq. A1.16 is more or less one form of the basic differential equation of TERZAGHI's consolidation theory.

A standard method to obtain a solution of a partial differential equation is to assume that the variables are separable, CRANK (1973). We therefore assume that $P_f$ is the product of 2 functions,

$$ P_f = F(z)G(t) \qquad \text{Eq. A1.18} $$

$$ \text{So, } \frac{\partial P_f}{\partial t} = F(z) \frac{\partial}{\partial t} G(t) = F(z) G'(t) \qquad \text{Eq. A1.19} $$

$$ \text{And } \frac{\partial^2 P_f}{\partial z^2} = \frac{\partial^2}{\partial z^2} F(z)G(t) = F''(z)G(t) \qquad \text{Eq. A1.20} $$

From Eqs. A1.16, A1.19 and A1.20,

$\frac{\partial P_f}{\partial t} = C_v \cdot \frac{\partial^2 P_f}{\partial z^2}$ with proper substitution we obtain:

$F(z)G'(t) = C_v F''(z)G(t)$ (The variables are separated)

$$ \text{Or } \frac{F''(z)}{F(z)} = \frac{G'(t)}{C_v G(t)} \qquad \text{Eq. A1.21} $$
The right part of Eq A1.21 is a function of z only and independent of t. The left part is a function of t and is independent of z. The ratio must therefore be equal to a constant, -b², this constant is generally called the separation constant, (Advanced Calculus for Applications, second edition, F. B. HILDEBRAND (1975).

$$ \frac{F''(z)}{F(z)} = -b^2, \text{ So,} $$

$$ F''(z) = -b^2 F(z) \qquad \text{Eq. A1.22} $$

A solution to Eq. A1.22 (Equation of 2nd order, linear and homogenous) can be given by

$$ F''(z) = -b^2 F(z) $$

$$ F''(z) + b^2 F(z) = 0 $$

$$ m^2 + b^2 = 0, \quad m = 0 \pm i\sqrt{b^2}, \quad \text{hence} $$

$$ F(z) = e^{0z}(A_1 \cos bz + A_2 \sin bz) $$

$$ F(z) = (A_1 \cos bz + A_2 \sin bz) \qquad \text{Eq. A1.23} $$

where $A_1$ and $A_2$ are constants.

The right-hand side of Eq. A1.21 may be written as

$$ \frac{F''(z)}{F(z)} = \frac{G'(t)}{C_v G(t)} $$

$$ G'(t) = -b^2 C_v G(t) \qquad \text{Eq. A1.24} $$

A solution to Eq. A1.24 (First order linear equation) can be given by:

$$ \frac{G'(t)}{G(t)} = -b^2 C_v $$

$$ \rightarrow \text{Integration : } \ln|G(t)| = -b^2 C_v t $$

$$ |G(t)| = e^{-b^2 C_v t + Cte} \quad \forall -b^2 C_v t + Cte, \quad e^{-b^2 C_v t + Cte} > 0, \text{ then } |G(t)| = G(t) $$

$$ G(t) = e^{-b^2 C_v t} \cdot e^{Cte} \quad \text{with} \quad e^{Cte} = A_3 $$

$$ G(t) = A_3 \cdot e^{-b^2 C_v t} \qquad \text{Eq. A1.25} $$

where $A_3$ is a constant. Combining Eqs. A1.18, A1.23, A1.25, ($P_f = F(z)G(t)$)

$$ P_f = (A_1 \cos bz + A_2 \sin bz) \cdot A_3 \cdot e^{-b^2 C_v t} \qquad \text{Eq. A1.26} $$

$$ P_f = (A_4 \cos bz + A_5 \sin bz) \cdot e^{-b^2 C_v t} $$

$$ A_4 = A_1 A_3 \text{ and } A_5 = A_2 A_3 $$

The constants in Eq. A1.26 can be calculated using the boundary conditions, which are:

1. At time $t=0$, $P_f = P_{f1}$

2. $P_f = 0$ at $z = 0$

3. $P_f = 0$ at $z = Z_t = 2Z_0$
$Z_0$ is the length of the longest path. In this case, with a two-way drainage system, $Z_0$ equal to one half of the total thickness of the SPM layer, $Z_t = 2Z_0$.

The second boundary condition ($P_f = 0$ at $z=0$) dictates that $A_4 = 0$. Therefore, Eq. A1.26 can be simplified to obtain an expression that satisfies the differential Eq. A1.16.

$$P_f(z,t) = (A_5 \sin bz) \cdot e^{-b^2 C_v t}$$

And from the third boundary condition ($P_f = 0$ at $z = Z_t = 2Z_0$) one obtains:

$$A_5 \sin(2bZ_0) = 0 \text{ and } A_5 \neq 0$$

Or $2bZ_0 = n\pi$ with $n \in [1, \infty]$, and $b = \frac{n\pi}{2Z_0}$

where $n$ is an **integer**. Consequently $P_f(z,t)$ is the first particular solution of Eq. A1.16 Further particular solutions have the form given by the Fourier calculation,

$$P_f = \sum_{n=1}^{\infty} (A_n) \cdot \sin \frac{n\pi z}{2Z_0} \cdot \exp \left( \frac{-n^2 \pi^2 T_v}{4} \right) \qquad \text{Eq. A1.27}$$

$T_v$ is the nondimensional time given by: $T_v = \frac{C_v t}{Z_0^2}$ [-]

The first boundary condition ($t=0, P_f = P_{f1}$) is satisfied when we have the coefficient of $A_n$ in the form as follows,

$$P_{f1} = \sum_{n=1}^{\infty} (A_n) \cdot \sin \frac{n\pi z}{2Z_0} \qquad \text{Eq. A1.28}$$

$P_{f1}$ is an uneven periodic function, $P_{f1}(-z) = -P_{f1}(z)$, then we can limit the study the Fourier sine series only on the interval $[0, 2Z_0]$.

Equation A1.28 being a Fourier sine series, $A_n$ can be given by

$$A_n = 2 \left[ \frac{1}{2Z_0} \int_0^{2Z_0} P_{f1} \cdot \sin \frac{n\pi z}{2Z_0} dz \right] \\ A_n = \frac{1}{Z_0} \int_0^{2Z_0} P_{f1} \cdot \sin \frac{n\pi z}{2Z_0} dz \qquad \text{Eq. A1.29}$$

Combining Eqs. A1.27 and A1.29, leads to the general solution Eq. A1.30.

$$P_f = \sum_{n=1}^{\infty} \left( \frac{1}{Z_0} \int_0^{2Z_0} P_{f1} \cdot \sin \frac{n\pi z}{2Z_0} dz \right) \cdot \sin \frac{n\pi z}{2Z_0} \cdot \exp \left( \frac{-n^2 \pi^2 T_v}{4} \right) \qquad \text{Eq. A1.30}$$
The Study of different cases regarding P<sub>f</sub> variation:

1. P<sub>f</sub> Constant with SPM height:

Referred to Fig. A1.3

P<sub>f</sub> = P<sub>f0</sub> So, referring to Eq. A1.30

$$
A_n = \frac{1}{Z_0} \int_{0}^{2Z_0} P_{f0} \cdot \sin \frac{n\pi z}{2Z_0} dz = \frac{2P_{f0}}{n\pi} (1 - \cos n\pi)
$$

<figure><img src="image_198.png" /><figcaption>Figure A1.3 : P<sub>n</sub> constant with SPM height</figcaption></figure>

$$
\text{so, } P_f = \sum_{n=1}^{n=\infty} \left( \frac{2P_{f0}}{n\pi} (1 - \cos n\pi) \right) \cdot \sin \frac{n\pi z}{2Z_0} \cdot \exp\left(\frac{-n^2\pi^2 T_v}{4}\right) \quad \text{Eq. A1.31}
$$

(1 - cos nπ) is zero for cases where n is even; therefore, P<sub>f</sub> equal to zero as well. For
the non-zero situations, it is useful to take n = 2m + 1, (n ∈ [1,+∞[, then m ∈ [0,+∞[),
m is an **integer**. Eq. A1.31 becomes:

$$
P_f = \sum_{m=0}^{m=\infty} \left( \frac{2P_{f0}}{(2m+1)\pi} (1 - \cos(2m+1)\pi) \right) \cdot \sin \frac{(2m+1)\pi z}{2Z_0} \cdot \exp\left(\frac{-(2m+1)^2 \pi^2 T_v}{4}\right) \quad \text{Eq. A1.31b}
$$

The term $(1 - \cos(2m + 1)\pi)$ always equals 2, where m is defined in the interval
$[0, \infty[$, then:

$$
\begin{align*}
P_f &= \sum_{m=0}^{m=\infty} \left( 2 \cdot \frac{2P_{f0}}{(2m+1)\pi} \right) \cdot \sin \frac{(2m+1)\pi z}{2Z_0} \cdot \exp\left( \frac{-(2m+1)^2 \pi^2 T_v}{4} \right) \\
P_f &= \sum_{m=0}^{m=\infty} \frac{2P_{f0}}{M_i} \cdot \sin \frac{M_i z}{Z_0} \cdot \exp(-M_i^2 T_v) \tag*{Eq. A1.32}
\end{align*}
$$

$$
\text{where } M_i = (2m + 1) \cdot \frac{\pi}{2}
$$

Simulation, Fig. A1.4

<figure><img src="image_199.png" /><figcaption>Figure A1.4 : Eq. A1.32 graphical representation, P<sub>f</sub> = f(Tv, z)</figcaption></figure>
The degree of compression is defined as a function of z :

$$ U_z = \frac{P_{fi} - P_f}{P_{fi}} = 1 - \frac{P_f}{P_{fi}} = \frac{P_s}{P_{fi}} = \frac{P_s}{P_{f0}} = \frac{\text{dissipated fluid pressure}}{\text{initial fluid pressure}} \qquad \text{Eq. A1.33} $$

$P_s$ is the local effective stress z provoked by the compression.

Combining Eqs. A1.32 and A1.33

$$ \begin{gathered} U_z = \frac{P_{fi} - P_f}{P_{fi}} = 1 - \frac{P_f}{P_{fi}} = 1 - \frac{\sum_{m=0}^{m=\infty} \frac{2P_{f0}}{M_i} \cdot \sin \frac{M_i z}{Z_0} \cdot \exp(-M_i^2 T_v)}{P_{fi}} \\ U_z = 1 - \frac{\sum_{m=0}^{m=\infty} \frac{2P_{f0}}{M_i} \cdot \sin \frac{M_i z}{Z_0} \cdot \exp(-M_i^2 T_v)}{P_{f0}} \end{gathered} $$

Hence,

$$ U_z = 1 - \sum_{m=0}^{m=\infty} \frac{2}{M_i} \cdot \sin \frac{M_i z}{Z_0} \cdot \exp(-M_i^2 T_v) \qquad \text{Eq. A1.34} $$

The simulation, Fig. A1.5

<figure><img src="image_200.png" /><figcaption>Figure A1.5 : Eq. A1.34 graphical representation, $U_z = f(T_v, z)$</figcaption></figure>

The average degree of compression for the all SPM cake is given by:

$$ U_{av} = \frac{\int_{0}^{Z_t} (1/Z_t) \int_{0}^{Z_t} P_{fi} dz - \int_{0}^{Z_t} (1/Z_t) \int_{0}^{Z_t} P_{fi} dz}{\int_{0}^{Z_t} (1/Z_t) \int_{0}^{Z_t} P_{fi} dz} = \frac{\text{compression at any time}}{\text{maximum compression}} \qquad \text{Eq. A1.35} $$

For $Z_t = 2Z_0$, and $P_{fi} = P_{f0}$

$$ U_{av} = \frac{\int_{0}^{Z_t} (1/Z_t) \int_{0}^{Z_t} P_{fi} dz - \int_{0}^{Z_t} (1/Z_t) \int_{0}^{Z_t} P_{fi} dz}{\int_{0}^{Z_t} (1/Z_t) \int_{0}^{Z_t} P_{fi} dz} = 1 - \frac{\int_{0}^{Z_t} (1/Z_t) \int_{0}^{Z_t} P_{fi} dz}{\int_{0}^{Z_t} (1/Z_t) \int_{0}^{Z_t} P_{fi} dz} $$
$$U_{av} = \frac{(1/Z_t) \int_0^{Z_t} P_{f,t} dz - (1/Z_t) \int_0^{Z_t} P_{f,t} dz}{(1/Z_t) \int_0^{Z_t} P_{f,t} dz} = 1 - \frac{(1/Z_t) \int_0^{Z_t} P_{f,t} dz}{(1/Z_t) \int_0^{Z_t} P_{f,t} dz}$$

$$U_{av} = 1 - \frac{(1/Z_t) \int_0^{Z_t} P_{f,t} dz}{(1/Z_t) \int_0^{Z_t} P_{f,t} dz} = 1 - \frac{(1/Z_t) \sum_{m=0}^{Z_t} \sum_{m=0}^{m=\infty} \frac{2P_{fo}}{M_i} \sin \frac{M_i z}{Z_0} \exp(-M_i^2 T_v) dz}{(1/Z_t) \int_0^{Z_t} P_{f,t} dz}$$

$$U_{av} = 1 - \frac{(1/Z_t) \sum_{m=0}^{m=\infty} \frac{2P_{fo}}{M_i} \exp(-M_i^2 T_v) \int_0^{Z_t} \sin \frac{M_i z}{Z_0} dz}{(1/Z_t) \int_0^{Z_t} P_{f,t} dz}$$

After integration,

$$U_{av} = 1 - \frac{(1/Z_t) \sum_{m=0}^{m=\infty} \frac{2P_{fo}}{M_i} \exp(-M_i^2 T_v) \left( -\frac{Z_0}{M_i} \cos \frac{M_i z}{Z_0} \right)_0^{Z_t}}{(1/Z_t) (P_{f0} Z_t)_0^{Z_t}}$$

$$U_{av} = 1 - \frac{(1/Z_t) \sum_{m=0}^{m=\infty} \frac{2P_{fo}}{M_i} \exp(-M_i^2 T_v) \left( -\frac{Z_0}{M_i} (\cos 2M_i - 1) \right)}{(1/Z_t) (P_{f0} Z_t)}$$

$$U_{av} = 1 - \frac{(1/Z_t) \sum_{m=0}^{m=\infty} \frac{2P_{fo}}{M_i} \exp(-M_i^2 T_v) \left( -\frac{Z_0}{M_i} (-2) \right)}{(1/Z_t) (P_{f0} Z_t)} = 1 - \frac{\sum_{m=0}^{m=\infty} \frac{2P_{fo}}{M_i} \exp(-M_i^2 T_v) \left( \frac{Z_t}{M_i} \right)}{(P_{f0} Z_t)}$$

Further simplifying,

$$U_{av} = 1 - \sum_{m=0}^{m=\infty} \frac{2}{M_i^2} \exp(-M_i^2 T_v)$$

Eq. A1.36

Simulated on Fig. A1.6:

In addition the empirical model from SIVARAM and SWAMEE (1977) is superimposed to the simulation of Eq. A1.36.
<figure><img src="image_201.png" /><figcaption>Figure A1.6 : SIVARAM and SWAMEE (1977)'s model is superimposed to the curve of Eq. A1.36</figcaption></figure>

SIVARAM and SWAMEE (1977) gave the following equation for $U_{av}$ varying from 0 to 100%:

$$ \frac{U_{av}\%}{100} = \frac{\left(\frac{4T_v}{\pi}\right)^{0.5}}{\left[1 + \left(\frac{4T_v}{\pi}\right)^{2.8}\right]^{0.179}} \qquad \text{Eq. A1.37} $$

Or

$$ T_v = \frac{\left(\frac{\pi}{4}\right) \cdot \left(\frac{U_{av}\%}{100}\right)^2}{\left[1 - \left(\frac{U_{av}\%}{100}\right)^{5.6}\right]^{0.357}} \qquad \text{Eq. A1.38} $$

## 2. Linear variation of $P_{f1}$ across the SPM cake:

Referring to Fig. A1.7

$$ P_{f1} = P_{f1} - P_{f2} \cdot \frac{Z_0 - z}{Z_0} $$

<figure><img src="image_202.png" /><figcaption>Figure A1.7 : Linear variation of $P_{f1}$ across the SPM cake</figcaption></figure>

With:

$P_{f1}$ = initial excess pore fluid pressure at $z = Z_0$

$P_{f2}$ = initial excess pore fluid pressure - $P_{f1}$ at $z = 2Z_0$

so $P_{f1} = P_{f1} + P_{f2}$ at $z = 2Z_0$
$P_f$ being integrated into Eq. A1.30 yields

$$P_f = \sum_{n=1}^{\infty} \left( \frac{1}{Z_0} \int_0^{2Z_0} \left( P_{f1} - P_{f2} \cdot \frac{Z_0 - z}{Z_0} \right) \sin \frac{n\pi z}{2Z_0} dz \right) \sin \frac{n\pi z}{2Z_0} \exp \left( \frac{-n^2 \pi^2 T_v}{4} \right) \quad \text{Eq. A1.39}$$

Simulation, Fig. A1.8

<figure><img src="image_203.png" /><figcaption>Figure A1.8 : Graphical representation of Eq. A1.39</figcaption></figure>

The average degree of compression: we referred to the definition Eq. A1.35

The average degree of compression at any time is presented as follows, for the situation where $Z_t = 2Z_0$ and $P_f = P_{f1} - P_{f2} \cdot \frac{Z_0 - z}{Z_0}$

$$U_{av} = \frac{\int_{0}^{Z_t} P_f dz - \int_{0}^{Z_t} P_f dz}{\int_{0}^{Z_t} P_f dz} = 1 - \frac{\int_{0}^{Z_t} P_f dz}{\int_{0}^{Z_t} P_f dz}$$

$$U_{av} = 1 - \frac{\int_{0}^{Z_t} P_f dz}{\int_{0}^{Z_t} (P_{f1} - P_{f2} \cdot \frac{Z_0 - z}{Z_0}) dz}$$

$$= 1 - \frac{\int_{0}^{Z_t} \sum_{n=1}^{\infty} \left( \frac{1}{Z_0} \int_{0}^{2Z_0} \left( P_{f1} - P_{f2} \cdot \frac{Z_0 - z}{Z_0} \right) \sin \frac{n\pi z}{2Z_0} dz \right) \sin \frac{n\pi z}{2Z_0} \exp \left( \frac{-n^2 \pi^2 T_v}{4} \right) dz}{\int_{0}^{Z_t} (P_{f1} - P_{f2} \cdot \frac{Z_0 - z}{Z_0}) dz}$$
After integration,

$$ \int_0^{2Z_0} \left( P_{f1} - P_{f2} \cdot \frac{Z_0 - z}{Z_0} \right) \cdot dz = \left( P_{f1}z - P_{f2}z + P_{f2} \frac{z^2}{2Z_0} \right) \Big|_0^{2Z_0} = 2P_{f1}Z_0 $$

$$ \frac{1}{Z_0} \int_0^{2Z_0} \left( P_{f1} - P_{f2} \cdot \frac{Z_0 - z}{Z_0} \right) \cdot \sin \frac{n\pi z}{2Z_0} dz = $$

$$ \frac{1}{Z_0} \int_0^{2Z_0} \left( P_{f1} \sin \frac{n\pi z}{2Z_0} - P_{f2} \sin \frac{n\pi z}{2Z_0} + \frac{P_{f2}z}{Z_0} \sin \frac{n\pi z}{2Z_0} \right) \cdot dz $$

$$ \frac{1}{Z_0} \int_0^{2Z_0} \left( P_{f1} - P_{f2} \cdot \frac{Z_0 - z}{Z_0} \right) \cdot \sin \frac{n\pi z}{2Z_0} dz = $$

$$ -2 \frac{P_{f1} \cos(n\pi) \cdot n\pi + P_{f2} \cos(n\pi) \cdot n\pi - 2P_{f2} \sin(n\pi) - P_{f1}(n\pi) + P_{f2}(n\pi)}{n^2 \pi^2} $$

We used the integration by parts for the third term.

With proper substitution we get:

$$ U_{av} = $$

$$ 1 - \frac{\int_{0}^{Z_0} \sum_{n=1}^{\infty} \left( -2 \frac{P_{f1} \cos(n\pi) \cdot n\pi + P_{f2} \cos(n\pi) \cdot n\pi - 2P_{f2} \sin(n\pi) - P_{f1}(n\pi) + P_{f2}(n\pi)}{n^2 \pi^2} \right) \cdot \sin \frac{n\pi z}{2Z_0} \cdot \exp \left( \frac{-n^2 \pi^2 T_v}{4} \right) dz}{2P_{f1}Z_0} $$

$$ \text{with } n = (2m+1) $$

$$ U_{av} = $$

$$ 1 - \frac{\int_{0}^{Z_0} \sum_{n=1}^{\infty} \left( -2 \frac{P_{f1}(-1) \cdot (2m+1)\pi + P_{f2}(-1) \cdot (2m+1)\pi - 2P_{f2} \cdot (0) - P_{f1}((2m+1)\pi) + P_{f2}((2m+1)\pi)}{(2m+1)^2 \pi^2} \right)}{2P_{f1}Z_0} $$

$$ \times \frac{\sin \frac{(2m+1)\pi z}{2Z_0} \cdot \exp \left( \frac{-(2m+1)^2 \pi^2 T_v}{4} \right) dz}{1} $$

$$ U_{av} = 1 - \frac{\int_{0}^{Z_0} \sum_{n=1}^{\infty} \left( -2 \frac{-P_{f1} \cdot \pi - P_{f2} \cdot \pi - P_{f1}(\pi) + P_{f2}(\pi)}{(2m+1)^2 \pi^2} \right)}{2P_{f1}Z_0} \times \frac{\sin \frac{(2m+1)\pi z}{2Z_0} \cdot \exp \left( \frac{-(2m+1)^2 \pi^2 T_v}{4} \right) dz}{1} $$

$$ U_{av} = 1 - \frac{\int_{0}^{Z_0} \sum_{n=1}^{\infty} \left( -2 \frac{-P_{f1} - P_{f2} - P_{f1} + P_{f2}}{(2m+1)\pi} \right)}{2P_{f1}Z_0} \times \frac{\sin \frac{(2m+1)\pi z}{2Z_0} \cdot \exp \left( \frac{-(2m+1)^2 \pi^2 T_v}{4} \right) dz}{1} $$
$$
\begin{align*}
U_{av} &= 1 - \frac{\int_{0}^{z_t} \sum_{n=1}^{\eta=\infty} \left( -2 \frac{P_{f1} - P_{f2} - P_{f1} + P_{f2}}{(2m+1)\pi} \right) \sin\left(\frac{(2m+1)\pi z}{2Z_0}\right) \exp\left(\frac{-(2m+1)^2 \pi^2 T_v}{4}\right) dz}{2P_{f1}Z_0} \times \frac{1}{1} \\
&= 1 - \frac{\int_{0}^{z_t} \sum_{n=1}^{\eta=\infty} \left( -2 \frac{2P_{f1}}{(2m+1)\pi} \right) \sin\left(\frac{(2m+1)\pi z}{2Z_0}\right) \exp\left(\frac{-(2m+1)^2 \pi^2 T_v}{4}\right) dz}{2P_{f1}Z_0} \\
&= 1 - \frac{\int_{0}^{z_t} \sum_{n=1}^{\eta=\infty} \left( 2 \frac{1}{(2m+1)\pi} \right) \sin\left(\frac{(2m+1)\pi z}{2Z_0}\right) \exp\left(\frac{-(2m+1)^2 \pi^2 T_v}{4}\right) dz}{Z_0} \\
&\quad \text{or } M_i = (2m+1) \frac{\pi}{2} \\
U_{av} &= 1 - \frac{\sum_{m=0}^{\eta=\infty} \left( \frac{1}{M_i} \int_{0}^{Z_t} \sin\left(\frac{M_i z}{Z_0}\right) \exp(-M_i^2 T_v) dz \right)}{Z_0} \\
&= 1 - \frac{\sum_{m=0}^{\eta=\infty} \left( \frac{1}{M_i} \exp(-M_i^2 T_v) \int_{0}^{Z_t} \sin\left(\frac{M_i z}{Z_0}\right) dz \right)}{Z_0} \\
U_{av} &= 1 - \frac{\sum_{m=0}^{\eta=\infty} \left( \frac{1}{M_i} \exp(-M_i^2 T_v) \frac{Z_0}{M_i} (1 - \cos(2M_i)) \right)}{Z_0} \\
&= 1 - \frac{\sum_{m=0}^{\eta=\infty} \left( \frac{1}{M_i} \exp(-M_i^2 T_v) \frac{Z_0}{M_i} \cdot (2) \right)}{Z_0}
\end{align*}
$$

after simplifying

$$U_{av} = 1 - \sum_{m=0}^{m=\infty} \frac{2}{M_i^2} \exp(-M_i^2 T_v)$$

Finally we obtain the same equation than Eq. A1.36 (excess pore fluid pressure is constant with depth)

$$U_{av} = 1 - \sum_{m=0}^{m=\infty} \frac{2}{M_i^2} \exp(-M_i^2 T_v) \qquad \text{Eq. A1.40}$$

### 3. Sinusoidal variation of $P_{f1}$ with height:

Referring to Fig. A1.9

<figure><img src="image_204.png" /><figcaption>Figure A1.9 : Sinusoidal variation of $P_{f1}$ with height</figcaption></figure>

$$P_{f1} = P_{f3} \cdot \sin\left(\frac{\pi z}{2Z_0}\right) \qquad \text{Eq. A1.41}$$

Substitution of the above equation into Eq. A1.30 yields

$$P_f = \sum_{n=1}^{\eta=\infty} \left( \frac{1}{Z_0} \int_{0}^{2Z_0} P_{f3} \cdot \sin\left(\frac{\pi z}{2Z_0}\right) \cdot \sin\left(\frac{n\pi z}{2Z_0}\right) dz \right) \cdot \sin\left(\frac{n\pi z}{2Z_0}\right) \exp\left(-\frac{n^2\pi^2 T_v}{4}\right) \quad \text{Eq. A1.42}$$
and the simulation, Fig. A1.10

<figure><img src="image_205.png" /><figcaption>Figure A1.10 : Graphical representation of Eq. A1.42</figcaption></figure>

For the average degree of compression: we refer again to the Eq. A1.35 with $Z_t = 2Z_0$:

$$U_{av} = \frac{(1/Z_t) \int_0^{Z_t} P_{f1} dz - (1/Z_t) \int_0^{Z_t} P_f dz}{(1/Z_t) \int_0^{Z_t} P_f dz}$$

$$U_{av} = \frac{(1/Z_t) \int_0^{Z_t} P_{f1} dz - (1/Z_t) \int_0^{Z_t} P_f dz}{(1/Z_t) \int_0^{Z_t} P_f dz} = 1 - \frac{(1/Z_t) \int_0^{Z_t} P_f dz}{(1/Z_t) \int_0^{Z_t} P_f dz}$$

Using Eq. A1.41:

$$\begin{align*} U_{av} &= 1 - \frac{(1/Z_t) \int_0^{Z_t} P_f dz}{(1/Z_t) \int_0^{Z_t} P_{f3} \cdot \sin\left(\frac{\pi z}{2Z_0}\right) dz} \\ &= 1 - \frac{\left( (1/Z_t) \sum_{n=1}^{Z_t} \left( \frac{1}{Z_0} \int_0^{2Z_0} P_{f3} \cdot \sin\left(\frac{\pi z}{2Z_0}\right) \cdot \sin\left(\frac{n\pi z}{2Z_0}\right) dz \right) \cdot \sin\left(\frac{n\pi z}{2Z_0}\right) \exp\left(\frac{-n^2\pi^2 T_v}{4}\right) dz}{(1/Z_t) \int_0^{Z_t} P_{f3} \cdot \sin\left(\frac{\pi z}{2Z_0}\right) dz} \end{align*}$$

Integrating,

$$\int_{0}^{Z_t} P_{f3} \cdot \sin\left(\frac{\pi z}{2Z_0}\right) dz = -2 \frac{P_{f3} \cdot Z_0 \cdot (\cos(\pi) - 1)}{\pi} = 4 \frac{P_{f3} \cdot Z_0}{\pi}$$
$$
\begin{align*}
\frac{1}{Z_0} \int_0^{2Z_0} \left( P_{f3} \cdot \sin\left(\frac{\pi z}{2Z_0}\right) \right) \cdot \sin\frac{n\pi z}{2Z_0} dz &= -2 \frac{P_{f3}(-\cos(\pi) \cdot \sin(n\pi) + n \sin(\pi) \cdot \cos(n\pi))}{\pi \cdot (n-1)(n+1)} \\
&= -2 \frac{P_{f3}(\sin(n\pi) + n \sin(\pi) \cdot \cos(n\pi))}{\pi \cdot (n-1)(n+1)}
\end{align*}
$$

To solve this integration we make use of the transformation:

$$
(\sin(A) \sin(B) = \frac{\cos(A - B) - \cos(A + B)}{2})
$$

We then substitute:

$$
U_{av} = 1 - \frac{\left(\frac{1}{Z_t}\right) \int_0^{Z_t} \sum_{n=1}^{n=\infty} \left(-2 \frac{P_{f3}(\sin(n\pi) + n \sin(\pi) \cdot \cos(n\pi))}{\pi \cdot (n-1)(n+1)}\right) \sin \frac{n\pi z}{2Z_0} \exp\left(\frac{-n^2 \pi^2 T_v}{4}\right) dz}{\left(\frac{1}{Z_t}\right) \cdot 4 \frac{P_{f3} \cdot Z_0}{\pi}}
$$

with $n = (2m + 1)$

$$
= 1 - \frac{\left(\frac{1}{Z_t}\right) \sum_{m=0}^{Z_t} \left(-2 \frac{P_{f3}(\sin((2m+1)\pi) + (2m+1)\sin(\pi)\cos((2m+1)\pi))}{\pi \cdot ((2m+1)-1)((2m+1)+1)}\right) \sin\frac{(2m+1)\pi z}{2Z_0} \exp\left(-\frac{(2m+1)^2 \pi^2 T_v}{4}\right) dz}{\left(\frac{1}{Z_t}\right) \cdot 4 \frac{P_{f3} \cdot Z_0}{\pi}}
$$

$$
= 1 - \frac{\left(\frac{1}{Z_t}\right) \sum_{m=0}^{Z_t} \left(-2 \frac{P_{f3}(\sin((2m+1)\pi) + (2m+1)\sin(\pi)\cos((2m+1)\pi))}{\pi \cdot (2m)(2m+2)}\right) \sin\frac{(2m+1)\pi z}{2Z_0} \exp\left(-\frac{(2m+1)^2 \pi^2 T_v}{4}\right) dz}{\left(\frac{1}{Z_t}\right) \cdot 4 \frac{P_{f3} \cdot Z_0}{\pi}}
$$

With integration, and $M_i = (2m+1) \cdot \frac{\pi}{2}$:

$$
U_{av} = 1 - \frac{\sum_{m=0}^{m=\infty} \left( - \frac{P_{f3}(\sin((2m+1)\pi) + (2m+1)\sin(\pi)\cdot-1)}{\pi \cdot (2m)(2m+2)} \right) \cdot \frac{Z_0}{M_i} (1 - \cos(2M_i)) \cdot \exp(-M_i^2 T_v)}{2 \frac{P_{f3} \cdot Z_0}{\pi}}
$$

$$
U_{av} = 1 - \frac{\sum_{m=0}^{m=\infty} \left( - \frac{P_{f3}(\sin((2m+1)\pi))}{\pi \cdot (2m)(2m+2)} \right) \cdot \frac{Z_0}{M_i} (1 - \cos(2M_i)) \cdot \exp(-M_i^2 T_v)}{2 \frac{P_{f3} \cdot Z_0}{\pi}}
$$

$$
U_{av} = 1 - \sum_{m=0}^{m=\infty} \left( - \frac{P_{f3}(\sin((2m+1)\pi))}{\pi \cdot (2m)(2m+2)} \right) \cdot \frac{Z_0}{M_i} \cdot 2 \cdot \exp(-M_i^2 T_v) \cdot \frac{\pi}{2P_{f3}Z_0}
$$

Now simplify

$$
U_{av} = 1 - \sum_{m=0}^{m=\infty} \left( - \frac{\sin((2m+1)\pi)}{(2m)(2m+2)} \right) \cdot \frac{1}{M_i} \exp(-M_i^2 T_v)
$$

with $M_i = (2m+1) \cdot \frac{\pi}{2}$
$$
\begin{align*}
U_{av} &= 1 - \sum_{m=0}^{m=\infty} \left( - \frac{\sin((2m+1)\pi)}{(2m)(2m+2)} \right) \cdot \frac{2}{\pi(2m+1)} \cdot \exp(-M_l^2 T_v) \\
&= 1 - \sum_{m=0}^{m=\infty} \left( - \frac{\sin((2m+1)\pi)}{\pi(2m+1)} \right) \cdot \frac{1}{(m)(2m+2)} \cdot \exp\left( - \frac{(2m+1)^2 \pi^2 T_v}{4} \right) \\
&= 1 - \sum_{m=0}^{m=\infty} \frac{0}{0} \quad \text{IF}
\end{align*}
$$

This is an indeterminate form (IF), for $m = 0$, so we should remove this IF by using the L'Hospital-Bernoulli theorem:

$$
\lim_{m \to a} \frac{f(m)}{g(m)} = \lim_{m \to a} \frac{f'(m)}{g'(m)} \text{ if the limit ratio exists.}
$$

In our case we define two functions,

$$
\begin{align*}
f(m) &= -\sin((2m+1)\pi) \cdot \exp\left(\frac{-(2m+1)^2 \pi^2 T_v}{4}\right) \\
g(m) &= (2m+1)\pi \cdot (m)(2m+2)
\end{align*}
$$

$$
\text{We have: } \lim_{m \to a} \frac{f(m)}{g(m)} = \frac{0}{0}
$$

Hence, calculus of derivatives

$$
\begin{align*}
f'(m) &= -2 \cos((2m+1)\pi)\pi \cdot \exp\left(\frac{-(2m+1)^2 \pi^2 T_v}{4}\right) + \frac{\sin((2m+1)\pi)\pi^2 T_v \cdot \exp\left(\frac{-(2m+1)^2 \pi^2 T_v}{4}\right)}{2} \\
g'(m) &= 2\pi \cdot m(2m+2) + \pi \cdot (2m+1)(2m+2) + 2\pi \cdot (2m+1)m
\end{align*}
$$

Now we study the limit of the derivative ratio.

$$
\lim_{m \to a} \frac{f'(m)}{g'(m)} = \lim_{m \to 0} \frac{-2 \cos((2m+1)\pi)\pi \cdot \exp\left(\frac{-(2m+1)^2 \pi^2 T_v}{4}\right) + \frac{\sin((2m+1)\pi)\pi^2 T_v \cdot \exp\left(\frac{-(2m+1)^2 \pi^2 T_v}{4}\right)}{2}}{2\pi \cdot m(2m+2) + \pi \cdot (2m+1)(2m+2) + 2\pi \cdot (2m+1)m}
$$

After simplification, we have,

$$
\begin{align*}
\lim_{m \to a} \frac{f'(m)}{g'(m)} &= \frac{1}{4} \cdot \exp\left(\frac{-\pi^2 T_v}{4}\right) \cdot (-4 \cdot \cos(\pi) + \sin(\pi) \pi T_v) \\
\lim_{m \to a} \frac{f'(m)}{g'(m)} &= \frac{1}{4} \cdot \exp\left(\frac{-\pi^2 T_v}{4}\right) \cdot (-4 \cdot (-1) + 0) = \exp\left(\frac{-\pi^2 T_v}{4}\right)
\end{align*}
$$
Finally we obtain:

$$
\begin{align*}
U_{av} &= 1 - \sum_{m=0}^{m=\infty} \left( - \frac{\sin\left(\frac{(2m+1)\pi}{2m}(2m+2)\right)}{(2m)(2m+2)} \right) \cdot \frac{2}{\pi(2m+1)} \cdot \exp(-M_i^2 T_v) \\
&= 1 - \lim_{m \to 0} \frac{f'(m)}{g'(m)} = 1 - \exp\left(-\frac{\pi^2 T_v}{4}\right)
\end{align*}
$$

This last transition is possible because for $m \in [1, \infty[$, the term that contains the sum is
always null $(\sum_{m=1}^{m=\infty} \frac{\sin((2m+1)\pi)}{(2m)(2m+2)}) \cdot \frac{2}{\pi(2m+1)} \cdot \exp(-M_i^2 T_v) = \frac{0}{Cte})$

Then the average degree of compression:

$$
U_{av} = 1 - \exp\left(-\frac{\pi^2 T_v}{4}\right) \qquad \text{Eq.A1.43}
$$

and its representation on Fig. A1.11:

<figure><img src="image_206.png" /><figcaption>Figure A1.11 : A graphical representation of Eq. A1.43</figcaption></figure>
# ANNEXE A2

## LA COMPRESSION UNIDIRECTIONNELLE A PRESSION CONSTANTE (CPC)

### ASPECT THEORIQUE

#### Analysis of saturated porous materials based on linear rheological models

##### Saturated porous materials (SPM) as a Hookean Body in series with a Kelvin Body

<figure><img src="image_207.png" /><figcaption>Figure A2.1 : Saturated porous materials (SPM) as a Hookean Body in series with a Kelvin Body</figcaption></figure>

The total strain $\epsilon$ is the sum of the strain $\epsilon_H$ of the Hookean body and the strain $\epsilon_K$ of the Kelvin element:

$$ \epsilon = \epsilon_H + \epsilon_K \qquad \text{Eq. A2.1} $$

The modulus of linear compression of the spring 'a' may be denoted by $A$.

The modulus of the spring 'b' may be denoted by $B$.

The coefficient of viscosity of the dashpot 'λ' may be denoted by $\lambda$.

$$ \epsilon_H = m_{vs} P_s(t) \text{ Or } \epsilon_H = \frac{1}{A} P_s(t) \text{ Hooke law} $$

For Kelvin element (parallel association)

$$ P_s = P_{s1} + P_{s2} $$

$$ \epsilon = \epsilon_1 = \epsilon_2 $$
Then we have the differential equation,

$$ \dot{\eta} \frac{d\varepsilon(t)}{dt} + B\varepsilon(t) = P_s \quad \text{or} $$

$$ \frac{\eta}{B} \frac{d\varepsilon(t)}{dt} + \varepsilon(t) = \frac{P_s}{B} $$

Using separate variable method,

$$ \frac{d\varepsilon(t)}{dt} = \frac{B}{\eta}\left(\frac{P_s}{B} - \varepsilon(t)\right) \Rightarrow \frac{\eta}{B}\frac{d\varepsilon(t)}{\left(\frac{P_s}{B} - \varepsilon(t)\right)} = dt $$

By integration and assuming $\varepsilon = 0$, at time $t = 0$

$$ - \int_{0}^{\varepsilon_{K}} \frac{\eta}{B} \frac{d\left(-\varepsilon(t) + \frac{P_s}{B}\right)}{\left(\frac{P_s}{B} - \varepsilon(t)\right)} = \int_{0}^{t} dt, \text{ then } -\frac{\eta}{B} \ln\left(\frac{P_s}{B} - \varepsilon(t)\right) \Big|_{0}^{\varepsilon_{K}} = t \Rightarrow -\frac{\eta}{B} \left[ \ln\left(\frac{P_s}{B} - \varepsilon_{K}\right) - \ln\left(\frac{P_s}{B}\right) \right] = t $$

$$ \ln\left(\frac{P_s}{B} - \varepsilon_K\right) = -\frac{B}{\eta}t, \text{ then } \left(\frac{P_s}{B} - \varepsilon_K\right) = \exp\left(-\frac{B}{\eta}t\right) \Rightarrow \left(1 - \frac{B}{P_s}\varepsilon_K\right) = \exp\left(-\frac{B}{\eta}t\right) $$

$$ \varepsilon_K = \frac{P_s}{B} \left( 1 - \exp\left(-\frac{B}{\eta}t\right) \right) $$

In integral notation we have the expression for the Kelvin body

$$ \varepsilon_K = \frac{1}{\eta} \int_{0}^{t} P_s(\tau) \cdot \exp\left(\frac{B}{\eta}(\tau - t)\right) d\tau $$

Then the combination of spring and the Kelvin body results to Eq. A2.2.

$$ \varepsilon = \varepsilon_H + \varepsilon_K = \frac{1}{A} P_s(t) + \frac{1}{\eta} \int_{0}^{t} P_s(\tau) \cdot \exp\left(\frac{B}{\eta}(\tau - t)\right) d\tau \quad \text{Eq. A2.2} $$

The pore pressure dissipation is assumed to be governed by the fundamental differential equation of one-dimensional compression. By substituting

$$ \frac{-de}{1+e} = d\varepsilon \qquad \text{Eq. A2.3} $$

and assuming the permeability $K = Cte$, and the initial hydraulic gradient $i_0 = 0$, then

$$ \frac{K}{\gamma_f} \frac{\partial^2 P_f}{\partial z^2} = \frac{1}{1+e} \frac{\partial e}{\partial t} = -\frac{\partial \varepsilon}{\partial t} \qquad \text{Eq. A2.3} $$

with

$$ dP_f = -dP_s \qquad \text{Eq. A2.4} $$

Combining Eq. A2.2 and Eq. A2.3 :
$$ \frac{K}{\gamma_f} \frac{\partial^2 P_s}{\partial z^2} = \frac{\partial \epsilon}{\partial t} $$

$$ \frac{K}{\gamma_f} \frac{\partial^2 P_s}{\partial z^2} = \frac{\partial \left( \frac{1}{A} P_s(t) + \frac{1}{\eta} \int_0^t P_s(\tau) \cdot \exp \left( \frac{B}{\eta}(\tau-t) \right) d\tau \right)}{\partial t} $$

Eq. A2.5

$$ \frac{K}{\gamma_f} \frac{\partial^2 P_s}{\partial z^2} = \frac{1}{A} \frac{\partial P_s(t)}{\partial t} + \frac{P_s(t)}{\eta} - \frac{B}{\eta^2} \int_0^t P_s(\tau) \cdot \exp\left(\frac{B}{\eta}(\tau-t)\right) d\tau $$

Eq. A2.5 is the general differential equation, with the boundary conditions defined for $0 \le t \le \infty$ (see Fig. A2.2).

1. $\frac{\partial P_s}{\partial z} = 0$, at $z = Z_0$

Eq. A2.6

2. $P_s = P(t)$, at $z = 0$

<figure><img src="image_208.png" /><figcaption>Figure A2.2 : The boundary conditions</figcaption></figure>

Applying a Laplace transform to both sides of Eq. A2.5, and using the convolution theorem for the last term on the right hand side, one can derived equations expressing the compression ratio and the fluid pressure for most types of $P(t)$ profiles.

With, $C_v = A \frac{K}{\gamma_f} [cm^2/s]$ coefficient of compression and $\beta = \frac{B}{\eta}$, we have

$$ C_v \frac{\partial^2 P_s}{\partial z^2} = \frac{\partial P_s(t)}{\partial t} + A \frac{P_s(t)}{\eta} - \frac{A \cdot \beta}{\eta} \int_0^t P_s(\tau) \cdot \exp(\beta(\tau-t)) d\tau $$

If $y = P_s(t, z)$, then

$$ C_v \frac{\partial^2 y}{\partial z^2} = \frac{\partial y}{\partial t} + A \frac{y}{\eta} - \frac{A \cdot \beta}{\eta} \int_0^t y(\tau) \cdot \exp(\beta(\tau-t)) d\tau $$

Eq. A2.7

Using the convolution theorem (HILDEBRAND 1975),

$$ \int_0^t y(\tau) \cdot \exp(\beta(\tau-t)) d\tau = \int_0^t f(\tau) g(t-\tau) d\tau = f(t) * g(t) $$

We can find the Laplace transform, by identification

$$ f(\tau) = y(\tau), \text{ and } g(t-\tau) = \exp(-\beta(t-\tau)) $$
Then $f(t) * g(t) = \int_{0}^{t} y(\tau)\exp(-\beta(t-\tau))d\tau$, with $\tau = u$ (new variable) we have:

$$ 
\begin{aligned}
f(t) * g(t) &= \int_{0}^{t} y(u)\exp(-\beta(t-u))du = \int_{0}^{t} y(u)e^{-(\beta(t-u))}du \\
f(t) * g(t) &= y(t) * e^{-\beta t} 
\end{aligned}
\qquad \text{Eq. A2.8}
$$

So we substitute Eq. A2.8 into Eq. A2.7, we get

$$ C_v \frac{\partial^2 y}{\partial z^2} = \frac{\partial y}{\partial t} + A \frac{y}{\eta} - \frac{A \cdot \beta}{\eta} [y(t) * e^{-\beta t}] \qquad \text{Eq. A2.9} $$

for the last term using Laplace transform, then

$$ LC_v \frac{\partial^2 y}{\partial z^2} = pY(z, p) - y(z, 0) + \frac{A}{\eta}Y(z, p) - \frac{A \cdot \beta}{\eta} \left[ Y(z, p) \cdot \frac{1}{p + \beta} \right] \qquad \text{Eq. A2.10} $$

rearranging

$$ 
\begin{aligned}
C_v \frac{\partial^2 Y(z, p)}{\partial z^2} &= Y(z, p) \left[ p + \frac{A}{\eta} - \frac{A \cdot \beta}{\eta} \left[ \frac{1}{p + \beta} \right] \right] - y(z, 0) \\
C_v \frac{\partial^2 Y(z, p)}{\partial z^2} &= Y(z, p) \left[ p + \frac{A}{\eta} \left\{ 1 - \left[ \frac{\beta}{p + \beta} \right] \right\} \right] - y(z, 0),
\end{aligned}
\qquad \text{Eq. A2.11}
$$

$$ \text{and } y(0, t) = P \qquad \text{Eq. A2.12} $$

$$ C_v \frac{\partial^2 Y(z, p)}{\partial z^2} = Y(z, p) \left[ p + \frac{A}{\eta} \left\{ 1 - \left[ \frac{\beta}{p + \beta} \right] \right\} \right] - P_0 = Y(z, p) \frac{p(p + \lambda(A + B))}{(p + \beta)} \qquad \text{Eq. A2.13} $$

$$ \text{Eq. A2.13 can be written, } C_v \frac{\partial^2 Y(z, p)}{\partial z^2} = Y(z, p) \cdot V(p) - P_0 \qquad \text{Eq. A2.15} $$

We recognise a second order linear homogenous equation, then

$$ \frac{\partial^2 Y(z, p)}{\partial z^2} - Y(z, p) \cdot \frac{V(p)}{C_v} = - \frac{P_0}{C_v} \qquad \text{Eq. A2.16} $$

The general solution is defined as:

$$ Y(z, p) = C_1 e^{\alpha_1 z} + C_2 e^{\alpha_2 z} + \frac{P_0}{V(p)} \qquad \text{Eq. A2.17} $$

$$ \text{where } \alpha_1 = \sqrt{\frac{V(p)}{C_v}}, \text{ and } \alpha_2 = -\sqrt{\frac{V(p)}{C_v}} \qquad \text{Eq. A2.18} $$

referring to Eq. A2.12, we have $Y(0, p) = P \cdot \frac{1}{p}$

$$ \left. \frac{\partial y(z, t)}{\partial z} \right|_{z=z_0} = 0 \qquad \text{Eq. A2.19} $$

With the Laplace transformation, it comes,
$$p \cdot Y(z, p)|_{z=z_0} - y(0,t) = 0 \qquad \text{Eq. A2.20}$$

$y(0,t) = P$ and $p \cdot Y(Z_0, p) - P = 0$ then

$$Y(Z_0, p) = \frac{P}{p}$$

The Eq. A2.17 can be expressed in the form:

$$Y(Z_0, p) = C_1 e^{\alpha_1 Z_0} + C_2 e^{\alpha_2 Z_0} + \frac{P_0}{V(p)} \qquad \text{Eq. A2.21}$$

From above and Eq. A2.18:

$$C_1 + C_2 = \frac{P}{p} - \frac{P_0}{V(p)} \qquad \text{Eq. A2.22}$$

$$C_1 e^{\alpha_1 Z_0} + C_2 e^{\alpha_2 Z_0} = \frac{P}{p} - \frac{P_0}{V(p)}$$

Thus, using Cramer method we have access to $C_1$ and $C_2$

$$a_1 C_1 + b_1 C_2 = c_1$$

$$a_2 C_1 + b_2 C_2 = c_2$$

$$C_1 = \frac{\det\begin{pmatrix} \vec{r} \\ c; \vec{b} \end{pmatrix}}{\det\begin{pmatrix} \vec{r} \\ a; \vec{b} \end{pmatrix}} = \frac{\left(\frac{P}{p} - \frac{P_0}{V(p)}\right)(e^{\alpha_2 Z_0} - 1)}{e^{\alpha_2 Z_0} - e^{\alpha_1 Z_0}} \qquad \text{Eq. A2.23}$$

$$C_2 = \frac{\det\begin{pmatrix} \vec{r} \\ a; c \end{pmatrix}}{\det\begin{pmatrix} \vec{r} \\ a; \vec{b} \end{pmatrix}} = \frac{\left(\frac{P}{p} - \frac{P_0}{V(p)}\right)(-e^{\alpha_1 Z_0} + 1)}{e^{\alpha_2 Z_0} - e^{\alpha_1 Z_0}} \qquad \text{Eq. A2.24}$$

$C_1$ and $C_2$ into Eq. A2.17 give,

$$Y(z, p) = \left[ \frac{\left( \frac{P}{p} - \frac{P_0}{V(p)} \right) (e^{\alpha_2 Z_0} - 1)}{e^{\alpha_2 Z_0} - e^{\alpha_1 Z_0}} \right] \cdot e^{\alpha_1 z} + \left[ \frac{\left( \frac{P}{p} - \frac{P_0}{V(p)} \right) (-e^{\alpha_1 Z_0} + 1)}{e^{\alpha_2 Z_0} - e^{\alpha_1 Z_0}} \right] \cdot e^{\alpha_2 z} + \frac{P_0}{V(p)} \qquad \text{Eq. A2.25}$$

considering the term

$$\frac{P_0}{V(p)} = \left[ \frac{P_0}{p + \frac{A}{\eta}\left\{1 - \left[\frac{\beta}{p+\beta}\right]\right\}} \right] \qquad \text{Eq. A2.26}$$

$$\frac{P_0}{V(p)} = \frac{P_0}{\left[ p + \frac{A}{\eta} - \frac{A}{\eta} \left[ \frac{\beta}{p+\beta} \right] \right]} = \frac{P_0}{\left[ \frac{p(p+\beta)}{p+\beta} + \frac{(p+\beta)A}{p+\beta\eta} - \frac{A}{\eta} \left[ \frac{\beta}{p+\beta} \right] \right]}$$
$$ \frac{P_0}{V(p)} = \frac{P_0}{\frac{p(p+\beta)+\frac{A}{\eta}(p+\beta)-\frac{A}{\eta}\beta}{p+\beta}} = \frac{P_0(p+\beta)}{p(p+\beta)+\frac{A}{\eta}(p+\beta)-\frac{A}{\eta}\beta} $$ $$ \frac{P_0}{V(p)} = \frac{P_0(p+\beta)}{p(p+\beta)+\frac{A}{\eta}p+\frac{A}{\eta}\beta-\frac{A}{\eta}\beta} = \frac{P_0(p+\beta)}{p(p+\beta)+\frac{A}{\eta}p} = \frac{P_0(p+\beta)}{p[(p+\beta)+\frac{A}{\eta}]} \quad \text{Eq. A2.27} $$

splitting the denominator

$$ \frac{P_0}{V(p)} = \frac{P_0(p+\beta)}{p\left[(p+\beta)+\frac{A}{\eta}\right]} = \frac{P_0\beta}{\beta+\frac{A}{\eta}p} + \frac{P_0\left(-\left(\beta+\frac{A}{\eta}\right)+\beta\right)}{-\left(\beta+\frac{A}{\eta}\right)\left[p\left(\beta+\frac{A}{\eta}\right)+\frac{A}{\eta}\right]} = 1 $$ $$ \frac{P_0}{V(p)} = \frac{P_0\beta}{\left(\beta+\frac{A}{\eta}\right)p} + \frac{P_0\frac{A}{\eta}}{\left(\beta+\frac{A}{\eta}\right)\left[p+\beta+\frac{A}{\eta}\right]} = 1 \quad \text{Eq. A2.28} $$

substituting Eq. A.2.28 into Eq. A2.25, we obtain:

$$ Y(z,p) = \left[ \frac{\left( \frac{P}{p} - \frac{P_0 \beta}{\left(\beta + \frac{A}{\eta}\right)p} + \frac{P_0 \frac{A}{\eta}}{\left(\beta + \frac{A}{\eta}\right)\left[p + \beta + \frac{A}{\eta}\right]} \right) (e^{\alpha_1 z_0} - 1)}{e^{\alpha_2 z_0} - e^{\alpha_1 z_0}} \right] \cdot e^{\alpha_2 z} + $$ $$ \left[ \frac{\left( \frac{P}{p} - \frac{P_0 \beta}{\left(\beta + \frac{A}{\eta}\right)p} + \frac{P_0 \frac{A}{\eta}}{\left(\beta + \frac{A}{\eta}\right)\left[p + \beta + \frac{A}{\eta}\right]} \right) (-e^{\alpha_1 z_0} + 1)}{e^{\alpha_2 z_0} - e^{\alpha_1 z_0}} \right] \cdot e^{\alpha_2 z} + \frac{P_0 \beta}{\left(\beta + \frac{A}{\eta}\right)p} + \frac{P_0 \frac{A}{\eta}}{\left(\beta + \frac{A}{\eta}\right)\left[p + \beta + \frac{A}{\eta}\right]} $$

With $\alpha_1 = \sqrt{\frac{V(p)}{C_v}}$, and $\alpha_2 = -\sqrt{\frac{V(p)}{C_v}}$, then Eq. A2.30
$$Y(z,p) = \frac{\left[ \left( \frac{P}{p} - \frac{P_0 \beta}{\left(\beta + \frac{A}{\eta}\right)p} + \frac{P_0 \frac{A}{\eta}}{\left(\beta + \frac{A}{\eta}\right)\left[p + \beta + \frac{A}{\eta}\right]} \right) \left( e^{-\sqrt{\frac{V(p)}{c_v}} z_0} - 1 \right) \right]}{e^{-\sqrt{\frac{V(p)}{c_v}} z_0} - e^{-\sqrt{\frac{V(p)}{c_v}} z_0}} \cdot e^{\sqrt{\frac{V(p)}{c_v}} z} +$$

$$\frac{\left[ \left( \frac{P}{p} - \frac{P_0 \beta}{\left(\beta + \frac{A}{\eta}\right)p} + \frac{P_0 \frac{A}{\eta}}{\left(\beta + \frac{A}{\eta}\right)\left[p + \beta + \frac{A}{\eta}\right]} \right) \left( -e^{\sqrt{\frac{V(p)}{c_v}} z_0} + 1 \right) \right]}{e^{-\sqrt{\frac{V(p)}{c_v}} z_0} - e^{-\sqrt{\frac{V(p)}{c_v}} z_0}} \cdot e^{-\sqrt{\frac{V(p)}{c_v}} z} + \frac{P_0 \beta}{\left(\beta + \frac{A}{\eta}\right)p} \frac{P_0 \frac{A}{\eta}}{\left(\beta + \frac{A}{\eta}\right)\left[p + \beta + \frac{A}{\eta}\right]}$$

and replace $V(p)$ into Eq. A2.30,
and we define the total load equal as $P = P_0$, then

Eq. A2.31

$$Y(z,p) = \frac{\left[ \left( \frac{P_0}{p} - \frac{P_0 \beta}{\left(\beta + \frac{A}{\eta}\right)p} + \frac{P_0 \frac{A}{\eta}}{\left(\beta + \frac{A}{\eta}\right)\left[p + \beta + \frac{A}{\eta}\right]} \right) \left( e^{-\sqrt{\frac{P_0 \frac{A}{\eta}}{c_v}} \left\{ 1 - \left[ \frac{\beta}{p+\beta} \right] \right\}} z_0 - 1 \right) \right]}{e^{-\sqrt{\frac{P_0 \frac{A}{\eta}}{c_v}} \left\{ 1 - \left[ \frac{\beta}{p+\beta} \right] \right\}} z_0} \cdot e^{\sqrt{\frac{P_0 \frac{A}{\eta}}{c_v}} \left[ \left( 1 - \left[ \frac{\beta}{p+\beta} \right] \right) z_0 + 1 \right]} + \\ \left[ \left( \frac{P_0}{p} - \frac{P_0 \beta}{\left(\beta + \frac{A}{\eta}\right)p} + \frac{P_0 \frac{A}{\eta}}{\left(\beta + \frac{A}{\eta}\right)\left[p + \beta + \frac{A}{\eta}\right]} \right) \left( -e^{\sqrt{\frac{P_0 \frac{A}{\eta}}{c_v}} \left\{ 1 - \left[ \frac{\beta}{p+\beta} \right] \right\}} z_0 + 1 \right) \right]}{e^{-\sqrt{\frac{P_0 \frac{A}{\eta}}{c_v}} \left\{ 1 - \left[ \frac{\beta}{p+\beta} \right] \right\}} z_0} \cdot e^{-\sqrt{\frac{P_0 \frac{A}{\eta}}{c_v}} \left[ \left( 1 - \left[ \frac{\beta}{p+\beta} \right] \right) z_0 + 1 \right]}$$

$$+ \frac{P_0 \beta}{\left(\beta + \frac{A}{\eta}\right)p} + \frac{P_0 \frac{A}{\eta}}{\left(\beta + \frac{A}{\eta}\right)\left[p + \beta + \frac{A}{\eta}\right]}$$

At this level it is far more efficient to use a specialized computerized mathematical tool, such as Maple® to obtain the inverse Laplace transform, nonetheless the result shall be identical to the analytical procedure proposed hereafter.
The inverse Laplace transform then provides the effective pressure ($P_s(z,t)$) as a function of position and time and finally the excess fluid pressure ($P_f(z,t)$).

The analytical route can be outlined as follows:

$$ \text{From Eq. A2.16, } \frac{\partial^2 Y(z, p)}{\partial z^2} - Y(z, p) \cdot \frac{V(p)}{C_v} = -\frac{P_0}{C_v} $$

Supposing that at time $t = 0$, $P = P_0 = 0$, then Eq. A2.16 can be rewritten:

$$ \frac{\partial^2 Y(z, p)}{\partial z^2} - Y(z, p) \cdot \frac{V(p)}{C_v} = 0 \qquad \text{Eq. A2.32} $$

$$ \frac{\partial^2 Y}{\partial z^2} - Y \cdot r^2 = 0 $$ 
with $r^2 = \frac{p(p + \lambda(A + B))}{C_v(p + \beta)}$ Eq. A2.33

We recognise that Eq. A2.33 satisfies the general solution:

$$ Y(z, p) = C_1 \frac{e^{(rz)} - e^{(-rz)}}{2} + C_2 \frac{e^{(rz)} + e^{(-rz)}}{2} = C_1 \sinh(rz) + C_2 \cosh(rz) \quad \text{Eq. A2.34} $$

with the boundary conditions

$$ \frac{\partial Y}{\partial z} = 0 \text{ at } z = Z_0, Y = P(p), \text{ at } z = 0 $$

it follows that

$$ C_1 = -P(p) \cdot \tanh(rZ_0) = -\frac{P}{p} \cdot \tanh(rZ_0) $$

$$ C_2 = P(p) = \frac{P}{p} $$

hence $Y(z,p) = -P(p) \cdot \tanh(rZ_0) \sinh(rz) + P(p) \cosh(rz)$

$$ Y(z, p) = -P(p) \cdot \frac{\sinh(rZ_0)\sinh(rz)}{\cosh(rZ_0)} + P(p) \cdot \frac{\cosh(rZ_0)\cosh(rz)}{\cosh(rZ_0)} $$

$$ Y(z, p) = P(p) \cdot \frac{\sinh(rZ_0)\sinh(-rz)}{\cosh(rZ_0)} + P(p) \cdot \frac{\cosh(rZ_0)\cosh(-rz)}{\cosh(rZ_0)} \quad \text{then} $$ 
$$ Y(z, p) = P(p) \frac{\cosh[r(Z_0 - z)]}{\cosh(rZ_0)} = \frac{P}{p} \frac{\cosh[r(Z_0 - z)]}{\cosh(rZ_0)} \quad \text{Eq. A2.35} $$

then the effective pressure is equal to

$$ L^{-1} Y(z, p) = P_s = \frac{1}{2\pi i} \int_{-i\infty}^{i\infty} \frac{P \cosh[r(Z_0 - z)]}{p \cosh(rZ_0)} e^{pt} dp, \quad (\text{HILDEBRAND 1975}) $$

and finally the excess fluid pressure

$$ P_f(z,t) = P(t) - P_s = P(t) - \frac{1}{2\pi i} \int_{-i\infty}^{i\infty} \frac{P \cosh[r(Z_0 - z)]}{p \cosh(rZ_0)} e^{pt} dp \quad \text{Eq. A2.36} $$

This is the complete mathematical development of the equations utilized to illustrate the use of rheological models in chapter 2.