On the modelling of mechanical dewatering in papermaking

On the Modelling of Mechanical Dewatering in Papermaking Vinicius Lobosco Doctoral Dissertation Stockholm 2004 Royal Institute of Technology Department of Fibre and Polymer Technology Division of Paper Technology SE-100 44 Stockholm, Sweden Typeset in LATEX. Trita-FPT-Report 2004:1 ISSN 1652-2443 ISRN/KTH/FPT/R-2004/1-SE Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan i Stockholm framlägges till offentlig granskning för avläggande av teknologie doktorsexamen den 27:e februari 2004 klockan 10:00 i STFI-salen, Drottning Kristinas väg 61, Stockholm. Avhandlingen försvaras på engelska. This work is licensed under the Creative Commons Attribution-NonCommercialShareAlike License. To view a copy of this license, visit: http://creativecommons.org/licenses/by-nc-sa/1.0/ Stockholm 2004, Universitetsservice US-AB tryckeri To Hanna & Julia Viagem Literal Se alguém telefonar, diga lhe que tenho saudades, e que o mundo só girou a metade. Num terço ou quarto de volta a gente se encontra. E outra vez gira o céu e a saudade volta a te abraçar. O ciclo nunca pára. As curvas tendem a ser tortas. As almas, vidas vivas, as podres personalidades mortas, são um jogo sem fim da poesia infinita, a obra de Deus que Ele não terminou, o ponto de mutação sempre próximo a nova estação. Se cegarmos os olhos, veremos com o coração: têm frango assado no forno, pêssego em calda na geladeira, e muito som na caixa. Lobosco, Vinicius (2004) On the Modelling of Mechanical Dewatering in Papermaking Abstract Most of the water fed into a paper machine is removed mechanically in the forming and press sections. One of the factor which has an important influence on mechanical dewatering, i.e. in both forming and pressing, is the stress-strain behaviour of the fibre network. The focus of this thesis is on the development of improved mathematical descriptions of the stress-strain behaviour exhibited by fibre networks in the forming and press sections. The first part of the thesis presents a physically based model of the forming and densification of fibre mats in twin-wire formers. The model can calculate the effect of the application of a varied load through the forming section. It was developed from mass and momentum balances of the fibre and liquid phases, the fibre mat stress-porosity relation and an expression for the permeability as a function of the porosity. The fibre-mat stress-porosity relation used is rate-independent and presents hysteresis. Simulations have been conducted to study the effects of roll pressure, blade pulses, wire tension and beating. The effect of sequential blade pressure pulses after the forming roll on the dewatering and the concentration gradients could be characterised. The simulations also exhibited rewetting by expansion when the fibre mats left the forming roll. Increasing wire tension resulted in increased dewatering, but the rate of increase diminished rapidly with increasing tension. The simulation results also indicated that beating has a large influence on dewatering. The second part of the thesis presents two models of the rate-dependent stress-strain behaviour of the fibre network that is observed in wet pressing. The first model was based on the approach pioneered by Perzyna (1966) for strain-rate dependent plasticity and was quite satisfactory for calculating the stress-strain behaviour of the fibre network in single press nips. It was successfully applied for studying densification and dewatering in both normal wet pressing and high temperature wet pressing. However, the first model only includes rate dependence in the compression phase of the compressionexpansion cycle; the expansion phase is treated as being rate independent The second model of the stress-strain behaviour of the fibre network treats both compression and expansion as being rate dependent, according to experimental observations. It is based on the idea that the wet fibre web may v be conceived as a layered network of restricted swelling gels. A swollen fibre is a restricted gel, the inner swelling pressure in a swollen fibre wall being balanced by the stresses in the fibre wall structure. The observed rate dependence of wet webs in both compression and expansion phases was attributed to the flow of water out of and into the fibre walls. The second model gave predictions that are in good agreement with results from uniaxial experiments using pressure pulses of arbitrary shape for both a single pulse and a sequence of pulses. It may therefore be used as a general model for the rheological behaviour of the wet fibre network in wet pressing, provided the model parameters are estimated from experimental data with small experimental error. KEYWORDS: Paper, modelling, dewatering, forming, wet pressing, fibre network stress, rheology, hysteresis, intra-fibre water, compressibility, structural stress, stress-strain, restricted gels, swelling. vi Preface “There is no duty more indispensable than that of returning a kindness. All men distrust one forgetful of a benefit.” Cicero. The mechanical dewatering that takes place in the forming and press sections is of crucial importance for papermaking, not only because most of the water in papermaking is removed mechanically, but also because several of the final paper properties are influenced by these two sections. A factor which plays an important role in mechanical dewatering, in both forming and pressing, is the stress-strain behaviour of the fibre network. The focus of this thesis is on developing improved mathematical descriptions of the stress-strain behaviour exhibited by fibre networks in the forming and press sections. The first part of the thesis presents a model of the forming and densification of fibre mats in twin-wire formers (Paper 1). The model includes hysteresis in the stress-strain relationship of the fibre network. The second part of this thesis deals with the stress-strain behaviour of the fibre network during wet pressing. Since the work of Gustav Carlsson it is generally accepted that the intra-fibre water flow also contributes to the stress-strain behaviour of the fibre network in the press section. Both Carlsson and later researchers have stressed the importance of that phenomenon for wet pressing. Two models of the stress-strain behaviour of the fibre network in wet pressing are presented. The first model follows the approach proposed by Perzyna for rate sensitive plasticity and is here called the Perzyna model. It could describe the rate-dependent stress-strain behaviour of the fibre network in conditions typical for a single press nip (Paper 2). The model could simulate the stress-strain behaviour of different types of pulps and the effect of web temperature (Paper 3). The Perzyna model allowed for rate sensitivity in the compression phase of the nip but not in the expansion phase. A press sections normally has a sequence of press nips and the fibre network expands in the unloaded region in-between the nips. To model the stress-strain behaviour in a sequence of press nips, it is therefore necessary to allow for rate sensitivity in both the compression and expansion phases of the pressure pulse. Hence, a second model, which is here referred to as the vii gel-network model, was developed based on the idea that the fibre web may be conceived as a layered network of restricted swelling gels (Paper 4). It gave predictions which are in good agreement with experimental results for both a single pulse and a sequence of pulses. Since I started this project in September 1997, I have been helped and supported by many people that I would like to thank. First, I would like to greatly thank my supervisor at STFI and co-author in the last three papers, Vikram Kaul, who believed in me, and gave me the opportunity to start working as a postgraduate student at STFI. I would also like to thank him for all the help and advice, and for being so unusually present in the development of my work even after he retired. I would also like to thank the co-authors of the first paper, Bo Norman and Sören Östlund, for their very valuable discussions and comments. Sören Östlund is also thanked for guiding me in – for me – new fields of solid mechanics, and believing in me in critical moments of my work. I feel very much in debt to these three people. I am also in debt with Raul Tempone, for giving me support regarding the numerical methods and with Antônio Ponce de Leon for his help with statistics. Hannes Vomhoff is thanked for making his experimental results available. I am very greatful to Jean-Francis Bloch, Jan-Erik Gustafsson and Eric Schmidt for reading the drafts and I also would like to thank Jonas Funkquist, Inger van Heesch, Richard Holmqvist, Sune Karlsson, Marianne Lockner, Marco Lucisano and Ali Moosavifar for their help and support. Finally, I would like to acknowledge the Joule Project of the European Community, the Center for Chemical Process Design and Control and STFI for providing the funds necessary for this project. I hope I will be able to return their kindness. Stockholm, Autumn 2003 Vinicius Lobosco viii List of Publications This thesis is based on the following four papers: 1. Modelling of Forming and Densification of Fibre Mats in Twin-Wire Formers. Lobosco, V., Norman, B. and Östlund, S., to be submitted for publication to Nordic Pulp and Paper Research Journal. 2. An Elastic/Viscoplastic Model of the Fibre Network Stress in Wet Pressing: Part I Lobosco, V. and Kaul, V. (2001), Nordic Pulp and Paper Research Journal, 16(1):12–17. 3. An Elastic/Viscoplastic Model of the Fibre Network Stress in Wet Pressing: Part 2. Accounting for Pulp Properties and Web Temperature Lobosco, V. and Kaul, V. (2001), Nordic Pulp and Paper Research Journal, 16(4):313–318. 4. The Stress-Strain Relationship of the Fibre Network in Wet Pressing. Lobosco, V. and Kaul, V., to be submitted for publication to Nordic Pulp and Paper Research Journal. ix Other relevant publications not included in the thesis: 1. An Elastic-Viscoplastic Model of the Rheological Behaviour of Wet Fibre Networks in Compression Lobosco, V. and Kaul, V. (1999), International Paper Physics Conference, San Diego, CA, USA, 26-30 Sept. 1999, pp 403-410. 2. A Rheological Model of the Paper Fibre Network in Wet Pressing Lobosco, V. (2000), Licentiate Thesis, Department of Pulp and Paper Chemistry and Technology, Royal Institute of Technology, Stockholm. 3. Densification and Dewatering in High Temperature Wet Pressing Gustafsson, J. E., Kaul, V. and Lobosco, V. (2001), The Science of Papermaking, 12th Fundamental Research Symposium, Oxford, UK, 17-21, Sept. 2001, vol. 1, pp 679-708. 4. A Rheological Model of the Fibre Network in Wet Pressing Lobosco, V. (2002), Nordic Rheology Conference 2002, Gothenburg, Sweden, June 12-14 2002, pp 83-88. 5. High Temperature Pressing of Fibrous Materials Kaul, V., Gustafsson, J.-E., Lobosco V., Bloch, J.-F., Thibault, X., Reverdy, N., Costa, C. A. V., Aguilar, H., Mendes, P., Norman, B., Nilsson, F., Stenström, S., Nilsson, J., Olsson, L., Todorovich, A. (2002), STFI report PUB 7, Stockholm, Sweden: STFI Swedish Pulp and Paper Research Institute, 19pp 6. Modélisation du Pressage du Papier : Consolidation et Expression de l’eau du Gâteau Fibreux Humide Gustafsson, J. E., Kaul, V., Lobosco, V. and Bloch, J.-F. (2003), Rev. ATIP vol. 57, no. 4, Oct. 2003, pp 22, 24, 26-30, 32-35. x Contents 1 Introduction 1 1.1 The Papermaking Process . . . . . . . . . . . . . . . . . . . . 1 1.2 Mechanical Dewatering . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Forming Section . . . . . . . . . . . . . . . . . . . . . . 3 1.2.2 Wet Pressing Section . . . . . . . . . . . . . . . . . . . 5 Fibre/Water Separation Process . . . . . . . . . . . . . . . . . 6 1.3.1 Flow Through Porous Media . . . . . . . . . . . . . . . 6 1.3.2 Fibre Network Compression . . . . . . . . . . . . . . . 9 1.3.3 Flow with Compression . . . . . . . . . . . . . . . . . . 12 1.3 1.4 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4.1 Forming . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4.2 Wet Pressing . . . . . . . . . . . . . . . . . . . . . . . 16 1.5 Summarising . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.6 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2 Modelling of Dewatering in Forming 2.1 19 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.1.1 Constitutive Equations and Physical Properties . . . . 20 2.1.2 Initial and Boundary Conditions . . . . . . . . . . . . . 21 2.1.3 Numerical Solution . . . . . . . . . . . . . . . . . . . . 22 2.2 Results and Discussions . . . . . . . . . . . . . . . . . . . . . 23 2.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . 26 xi 3 The Perzyna Model 3.1 27 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.1.1 Sources of Error . . . . . . . . . . . . . . . . . . . . . . 29 3.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3 Results and Discussions . . . . . . . . . . . . . . . . . . . . . 34 3.4 3.3.1 Parameters Estimation . . . . . . . . . . . . . . . . . . 34 3.3.2 Validation . . . . . . . . . . . . . . . . . . . . . . . . . 36 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . 38 4 The Gel-Network Model 39 4.1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.3 4.4 4.2.1 The Micro- and Macroscopic Perspectives . . . . . . . 40 4.2.2 The Macroscopic Model . . . . . . . . . . . . . . . . . 42 Results and Discussions . . . . . . . . . . . . . . . . . . . . . 44 4.3.1 The Equilibrium Curve . . . . . . . . . . . . . . . . . . 44 4.3.2 Parameters Estimation . . . . . . . . . . . . . . . . . . 44 4.3.3 Validation . . . . . . . . . . . . . . . . . . . . . . . . . 45 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . 47 A Model Derivation 59 A.1 Continuity Equations . . . . . . . . . . . . . . . . . . . . . . . 59 A.2 Momentum Equations . . . . . . . . . . . . . . . . . . . . . . 60 A.3 Momentum Transfer Equations . . . . . . . . . . . . . . . . . 63 B Rate-Independent Fibre-Network Stress 65 C Numerical Solution with FEM 71 xii Chapter 1 Introduction “...rå materialet stötes i stenmortlar tillsammans med vatten. [...]de tidigaste pappersarken formades genom att man förtunnade pappersmassan med vatten tills man hade en slags fibervälling och därefter slog den över en ram med ett utspänt nät.” Bo Rudin about the art of paper production in China at Ts’ai Lun’s time. Ts’ai Lun is often named as the first papermaker. In this chapter, first, the basic principles of the papermaking process are shortly described. Then, the phenomena that are relevant for mechanical dewatering in papermaking are presented together with experimental results, which give support for the assumptions made in the development of the models. 1.1 The Papermaking Process The basic principles of producing paper from fibre suspensions have not changed since the time of Ts’ai Lun in China 105 BC. This is also the principle in the first industrial paper machines (Figure 1.1) as well as in present modern machines. However, production capacity has increased several orders of magnitude, and paper quality has improved dramatically. When the fibre suspension1 enters the paper machine, it has a solids content of approximately 0.5 percent, while the finished paper has a dry material content in excess of 90 percent. Hence, 200 times the weight of the paper produced has to be removed before it reaches the consumer (Figure 1.2). 1 The suspension contains fibres and water. When the fibres are entangled enough to become a structure, they are called fibre mat or wet web, which sometimes also includes air. The fibre network comprehends only the intra-fibre water and the fibres. 1 2 CHAPTER 1. INTRODUCTION Figure 1.1: The paper machine presented by Donkin in Germany 1818 (Pires and Kuan, 1988) Moreover, it is vital for the final paper quality the way water is removed. Therefore, mechanical dewatering plays a central role in paper research. Figure 1.2: A squematic picture of a paper machine showing the dewatering sections, and the approximate fibre concentration at different stages (Pires and Kuan, 1988) Water is removed from the fibre suspension, and later from the fibre web, in three basic parts of a paper machine: in the forming, the press and the drying sections. In the forming and wet pressing sections, water is removed mechanically, whereas it is removed by evaporation in the drying section. In the forming section, a jet with the fibre suspension is distributed onto one or between two wires, where the paper is basically formed. Most of the dewatering takes place in this section by wire tension as well as vacuum, with the fibre web reaching a solids content of approximately 20 percent, when it enters the press section. In the press section, water is also mechanically removed from the web, but now with much higher pressure levels. A mechanical load is applied in a press nip, when the fibre web – covered with one or two felts – passes between two rotating press rolls or between a press roll and a press shoe. Usually, there are three or four press nips in a paper 1.2. MECHANICAL DEWATERING 3 machine, and these lead to an increase in the solids content to approximately 45 percent when the web leaves this section. In the drying section, most of the remaining water is dried by evaporation as it passes over a number of steam-heated cylinders, thus finalising the dewatering process. As may be expected, the effort for removing water increases as the water content decreases, which has consequences for the costs. The drying section has by far the largest costs as water is removed by evaporation. It is, therefore, preferable to remove water, as much as possible, mechanically in the cost-effective forming and pressing sections. Even small improvements in mechanical dewatering could lead to significant cost reductions. Reliable mathematical models of mechanical dewatering may contribute to a better design and the optimisation of dewatering in the forming and pressing sections. Furthermore, various important characteristics of the final paper are determined by mechanical dewatering. Hence, a better understanding of the process, with the help of reliable physically based models, may also lead to improvements in paper quality. 1.2 1.2.1 Mechanical Dewatering Forming Section The first generation of forming sections was the so-called Fourdrinier form- Types of formers ing, where dewatering took place on a horizontal wire by the action of the gravity. In the 1970’s, the second generation, twin-wire formers, started being increasingly used. The main advantages of twin-wire formers are the increased dewatering rate (approximately four times dewatering in a Fourdrinier former), and the lessened two-sidedness. Although the retention is worsened, the advantages, especially of being able to run the paper machine at much faster velocities, exceed the disadvantages, and nowadays most machines have a twin-wire former. Basically, there are two dewatering principles normally used in twin-wire formers, viz. roll and blade formers. In roll formers, the suspension is fed between two wires that wrap a roll. The applied pressure is approximately constant and equal to the wire tension divided by the radius of curvature. In blade formers, the suspension and the wires are deflected through a series of opposing blades. In this case, the suspension experiences a series of small pressure pulses. 4 CHAPTER 1. INTRODUCTION Blade formers generate paper with good formation. There is, however, a trade-off between formation and retention. Usually, to improve formation one worsens retention. The pressure pulses generated by the blades are believed to disrupt the larger flocs in the suspension by stretching, improving large scale formation. The price to pay is the deteriorated retention, which requires the addition of retention aids to keep it at an acceptable level. Roll formers, on the other hand, usually produce paper with bad formation, but with a good retention. Therefore, as pointed out in Nordström and Norman (1994), roll-blade formers endeavour to achieve the best of both giving up as little as possible. The initial mat is formed at a high retention. Later, in the blade former region, the partially formed fibre mats help to enhance retention in the formation-improving phase by acting as filters. Dewatering mechanisms Baines (1967) divided conceptually dewatering in the forming section in a twin-wire former in three zones, viz: the free-jet, the wedge and the press zones2 (Figure 1.3). The free-jet zone is the region the suspension travels after leaving the headbox until it impinges onto the wires. At the impinging point the wedge zone starts, where the fibre mats – separated by the fibre suspension in between them – are built up on the wires until they meet. In this region, drainage occurs by filtration (Hisey, 1956; Parker, 1972). After the contact point, dewatering occurs through compression of the fibre mats. This is the press zone, where dewatering occurs by thickening, i.e. compression of the fibre mat. Comprehensive description of the forming process can be found in Wrist (1962), Norman (1989) and Norman and Söderberg (2001). Figure 1.3: A squematic figure with the different zones in the forming section. The dewatering mechanisms in each zone are indicated. 2 Press zone should not be confused with the press section. Although the dewatering mechanism is the same, they are two distinct things, and occupy different regions in the paper machine. 1.2. MECHANICAL DEWATERING 1.2.2 5 Wet Pressing Section The dewatering principle in wet pressing can be described as an intense thickening. In wet pressing as well as in the press zone in forming, the stress applied on the fibre network is balanced by two components, viz. the hydraulic pressure and the fibre network stress (often called the structural stress). The hydraulic pressure causes water to flow in-between the fibres, whereas the fibre network stress deforms the fibre network. In wet pressing, the difference is that, as the loads applied are much higher, the water flow from the fibre walls may play an important role. Campbell (1947) was the first researcher to make a detailed account of the physics of wet pressing. He introduced Terzaghi’s principle (1944) from soil mechanics into paper science. According to this principle, the total applied stress in the nip, σT , is balanced by the sum of the hydraulic pressure, pw , and the fibre network stress (also called structural stress), σs . Nevertheless, one should note that the Terzaghi’s principle is just a particular case of the of the more general expression (Biot, 1941; Auriault and Sanchez-Palencia, 1977): σT = φeff a pw + σs , Stress decomposition Terzaghi’s principle (1.1) eff when φeff a is equal to the unity. The parameter φa is called the effective areal porosity and have to be determined experimentally (Kataja et al., 1995). Although the total applied stress is approximately constant in the direction of application, the two components vary due to the flow. The hydraulic pressure decreases in the direction of flow and the fibre network stress increases accordingly. Therefore, the layers closest to the surface where the water leaves the medium are the most compressed. This phenomenon is called stratification (MacGregor, 1989). Still based on the stress decomposition in the nip, Wahlström (1969) introduced the concepts of flow-controlled and pressure-controlled dewatering according to the importance of each component in the right-hand side of Equation (1.1). The idea of flow-controlled nips makes it easier to understand the advantage of extended nip presses (ENP)3 , where one roll is replaced by a press shoe. In these cases, the resistance imposed by the liquid flow in-between the fibres is the limiting factor for dewatering. Therefore, the 3 Nevertheless, pressure-controlled nips may also experience dewatering improvements (Wahlström, 1990). Stratification Flow- and pressurecontrolled nips 6 CHAPTER 1. INTRODUCTION increased residence time in ENP enables a longer water flow time from the web, improving dewatering. 1.3 Fibre/Water Separation Process The physical modelling of the fibre/water separation process involves mass and momentum balances of the components and some constitutive equations. As the geometry of the fibres is complex, simplifying assumptions are frequently done, and the different phases are usually treated as continua. A short survey of the of the experiments relevant for mathematical modelling of mechanical dewatering is given below. 1.3.1 Darcy’s law Flow Through Porous Media Momentum exchange between the phases is often postulated to obey Darcy’s law for homogeneous media subject to a pressure difference: v= Forchheimer equation K ∆pw , µ L (1.2) where v is the superficial velocity, pw is the hydraulic pressure, L and K are the medium length and permeability, repectively, and µ is the fluid viscosity. Equation (1.2) neglects inertial forces, which seems to be reasonable for Reynolds number, Re = ρrv/µ, less than unity (Bear, 1972; Scheidegger, 1974; Greenkorn, 1983). r is the average pore size radius4 and ρ is the fluid velocity.Equation (1.2) has been extended in cases where inertial terms are significant into: ∆pw = av + bv 2 , L (1.3) where a and b are constants (Scheidegger, 1974; Dullien, 1979). This equation is usually called the Forchheimer equation, and was suggested in the first decade of the 20th century. It applies for Reynolds numbers larger than unity. 4 Some researchers use the average fibre radius instead. However, some fibre modificating processes, e.g. beating, alter considerably the flow conditions without changing markedly the average fibre radius. 1.3. FIBRE/WATER SEPARATION PROCESS 7 Equation (1.2) was originally presented by Darcy for isotropic, homogeneous5 porous media. However, fibre mats are highly compressible, and many interesting separation processes have non-constant concentration or composition and therefore varying permeability through the medium. Moreover, the flow resistance in the medium may also vary with direction. Equation (1.2) is heuristically extended to heterogeneous, anisotropic media by expressing it in the differential form: v= K gradpw , µ (1.4) where K is the permeability tensor, which has to be described in the whole medium. Thus, Darcy’s law may also be used in compressible anisotropic media. Permeability measurements Several researchers have examined the influence of porous media concentration on the permeability, using thick media in compression-permeability (CP) cells (Collicutt, 1947; Robertson and Mason, 1949; Ingmanson, 1952, 1953; Nilsson and Larsson, 1968; Gren and Ljungkvist, 1983; Ljungkvist, 1983; Carlsson et al., 1983; Rasi et al., 1999). Their experimental results for several different kinds of pulp fit well to Equation (1.2), which suggests that inertial terms are negligible in those experimental conditions. Indeed, Mantar et al. (1995) and Wildfong et al. (2000a) conclude that inertial effects may be present but play only a secondary role in the flow through fibre mats. Different relationships are employed to fit the measured permeability data at different concentrations. Usually, at concentrations typical for wet press- Kozeny-Carman ing, empirical relationships are used for fitting (Nilsson and Larsson, 1968; Carlsson et al., 1983; Bloch, 1995), although considerable advances have been achieved lately in modelling liquid flow through a unstructured medium (Koponen et al., 1998). At concentrations typical for forming, theoretical considerations underpinning the semi-empirical equations are employed to describe the observed data. The theoretical model by Kozeny and Carman is the one most used. It is an extrapolation of the Poiseuille equation to porous medium by using the concept of hydraulic radius (Dullien, 1979). According to this formulation, permeability, K, as a function of the concentration (or porosity) is given by: 5 Homogenity is said of a property that is independent of the position, whereas isotropy is said of a property that varies with the direction. As this text only considers heterogenities and anysotropies in the flow resistance, i.e. with respect to the permeability, this property will be omitted for simplicity. 8 CHAPTER 1. INTRODUCTION K= φ3ext , k0 Sv2 (1 − φext )2 (1.5) where k0 is the Kozeny constant, Sv is the specific surface area – i.e. the external area per unit volume – in contact with the fluid and φext is the external porosity, i.e. the ratio between the volume available for flow and the total volume. According to the measurements by Ingmanson et al. (1959), k0 is approximately constant (= 5.55) for values of effective porosity until 0.8, above which k0 increases dramatically. The external porosity, φext , can be expressed as 1 − ac, where a is the effective volume and c is the concentration. Substituting it into Equation (1.5), one obtains, after some rearrangement: Kc 2
31 =  1 5.55Sw  13 (1 − ac) , (1.6) where Sw = Sv a is the external surface area per weight of fibrous material. Robertson and Mason (1949) examined the permeability of kraft and sulphite pulps beaten to different degrees over a concentration range between five and twelve percent. They calculated the permeability parameters a and Sw based on their measurements for trials with two kraft pulps and five with sulphite pulp. Robertson and Mason’s (1949) and also Ingmanson’s (1952) permeability data are in good agreement with Equation (1.6) for concentrations up to slightly above 15 percent solids content for sulphite pulp, Figure 1.4. Nevertheless, in order to obtain Equation (1.6), it was assumed that the volume not available for flow in the fibre mats is constant, which definitely is not true at high solids content. As the fibre network gets compressed, the specific volume of the fibres diminishes, which will eventually lead to water expression out of the fibre walls. It is also possible that water actually flows through the fibres in conditions typical for pressing (Gustafsson et al., 2001; Lucisano and Martinez, 2001). Both these phenomena would imply somewhat higher permeability values than the Kozeny-Carman equation reckons (indeed, it gives negative permeability for concentrations, c, higher than 1/a); the deviation increasing the higher the solids content is. Therefore, Meyer (1969) suggest that a and Sw were a function of the stress in the fibre network. Wang et al. (2002) give another solution to this problem. They suggest an exponential expression for φext as a function of concentration, in order to avoid that porosity reaches zero. 1.3. FIBRE/WATER SEPARATION PROCESS 9 Figure 1.4: Left: Rectified permeability data as a function of the concentration for a series of beaten samples of sulphite pulp. The freeness numbers are given in the diagram (Robertson and Mason, 1949). Right: Similar results by Ingmanson (1952). 1.3.2 Fibre Network Compression Fibre mats are highly compressible, and the fibre network stress induced by fluid flow through the porous medium has a cumulative effect, resulting in a heterogeneous porous medium. Also, as pointed out above, permeability depends on the porosity. Therefore, in order to quantify the dewatering rate in a heterogeneous medium with the help of Equation (1.4), a mathematical description of the concentration as a function of the fibre network stress is required. Low Loads Several researchers have reported the stress-strain behaviour of the fibre network at low mechanical loads and concentrations (Campbell, 1947; Ingmanson, 1952; Jones, 1963; Han, 1969; Gren and Ljungkvist, 1983; Vomhoff and Schmidt, 1997; Jong, 1998; Boxer and Dodson, 1999). Usually the concentration is expressed as a function of the stress, σ, according to a power law (Campbell, 1947): Stress-strain behaviour 10 CHAPTER 1. INTRODUCTION c=M  σ pr N , (1.7) where c is the concentration, pr is a reference pressure and M and N are fitting constants6 (Gren and Ljungkvist, 1983). In the forming section, fibre networks are at low concentrations (normally below 20 percent). In this range, the fibre network is not compressed to such a high extent that a considerable amount of water from inside the fibre walls is squeezed out. Therefore, one would not expect the stress-strain behaviour of the fibre mats would be rate-dependent in this range. Wilder (1960) studied the influence of creep in the behaviour of a fibre network at low loads. For short compression times (<40 ms) creep is of relatively little importance. Nevertheless, based on the modelling of constant-flow drainage experiments, Sayegh and Gonzalez (1995) obtained good agreement by assuming that the fibre network behaves as a Maxwell element. However, one could suspect that this might also be due to fines distribution in the thickness direction (see section 1.3.3). Hysteresis Jones (1963) observed hysteresis in the compressibility behaviour of the fibre networks (Figure 1.5). It is clear that the strain not only depends on the applied stress, but also on the history of the compression. Therefore, the following notation may be preferred: σ ≡ σ[φ], (1.8) where the square brackets mean that σ at a given time depends on the whole history of φ until that given instant. High Loads In saturated fibre mats, the water located in the interstices between fibres and Intra- and in the fibre lumens is called the extra-fibre water. In contrast, a considerable extra-fibre water amount of water is also held in the porous structure of the fibre wall, the intrafibre water. The intra-fibre water can be estimated by different techniques (Lindström, 1986), amongst the most common is the water retention value (WRV). The first evidence of the importance of the intra-fibre-water flow for pressing comes from experiments showing the expression of water out of fibre walls 6 pr is introduced just to avoid complex units in M and N . 1.3. FIBRE/WATER SEPARATION PROCESS 11 Figure 1.5: Hysteresis behaviour of solids concentration against applied stress for a fibre network made of loblolly pine summerwood (Jones, 1963). Carlsson et al. (1977). Therefore, in the wet-press section, the intra-fibre water flow may play an important role in the dewatering process. Later, the effect of pressing on the intra-fibre water has been shown using different techniques (Laivins and Scallan, 1993b; Maloney et al., 1997; Häggkvist, 1999). Maloney et al.’s (1997) and Häggkvist’s (1999) results show that larger pores are closed first followed by continuously smaller pores, and some pores are permanently closed (hornification). Moreover, some intra-fibre pores can be reopened by dispersion (Maloney et al., 1997). Drying and beating are also shown to influence the amount of intra-fibre water (Laivins and Scallan, 1993a, 1995). Arguably, the water inside the fibre wall is more difficult to be removed than the water in-between the fibres. Therefore, in press nips where a limited impulse is available, the intra-fibre water appears to be the limiting solids content that can be achieved after the press section (Figure 1.6). Hence, it is now widely accepted that the effect of intra-fibre-water flow should be included in the description of the stress-strain behaviour of the fibre network. Accordingly, Wahlström (1990) proposes the further decomposition of the total stress: σT = pw + σs = pw + σs,eq + σs,f l , (1.9) 12 CHAPTER 1. INTRODUCTION Figure 1.6: Influence of the water retention value on the final solids content for several different pulps (Busker and Cronin, 1982). where, the fibre network stress is decomposed in σs,eq and σs,f l , the static compression curve and the resistance to compression due to flow out of the fibre wall, respectively. Expansion Dynamic measurements of the stress-strain behaviour of the fibre network show considerable rate dependence (Ellis, 1981; Szikla and Paulapuro, 1989; Vomhoff, 1998; Lobosco, 2000). Further, during the unloading phase, there is usually little or no expansion of the fibre network before the applied pressure comes to zero (Vomhoff, 1998). However, some slow but significant expansion does occur if the web is unloaded completely. 1.3.3 Specific Resistance Flow with Compression In several experiments, for practical reasons, the fibre mat flow resistance is measured by filtration where a considerable compression of the fibre network takes place due to flow. The fibre mat flow resistance is thus given in terms of flow specific resistance 7 , Rw , independent of the thickness, which is a function of the pressure drop through the medium and is given by: 7 When, for simplicity, the term resistance is used, it is referred to the flow specific resistance. 1.3. FIBRE/WATER SEPARATION PROCESS Rw = ∆pw , vµw 13 (1.10) where v is the superficial velocity and w is the grammage (Ingmanson, 1952). Notwithstanding the fibre mat compression, the medium permeability parameters can be obtained from the specific resistance by taking into account the fibre network deformation with the help of Equation (1.7). Ingmanson (1952, 1953) obtains permeability parameters in good agreement with permeability results from a CP-cell. However, it is usual to just study the specific resistance of fibre mats under different conditions. Herzig and Johnson (1999) have studied the flow resistance due to the wires. Their results fit well to Equation (1.3), i.e. Forchheimer’s. Wildfong et al. (2000b) draw the same conclusion based on results from a laboratorydrainage tester. However, the influence of the resistance to flow due to the wire is acknowledged to be much greater than could be accounted for by simply adding it to the fibre mat resistance (Meyer, 1969; Radvan, 1980). The interaction between fibre mat/wire implies an additional flow resistance, due to the self-healing effect, i.e. the fibres are deposited in apertures in the wires. This phenomenon has been acknowledged as early as in 1962 by Han. His data from an experimental vacuum former show a fast resistance increase until 10–20 g/m2 , of approximately the same order as can be seen in the measurements of Herzig and Johnson (1999) and Jong et al. (1999). Jong et al.’s results show a fast resistance increase until a mat of approximately 20 g/m2 is formed, followed by a monotonic decrease. However, neither of them attribute these phenomena to an interfacial resistance. In fact, by expressing the effective resistance as a sum of the interfacial resistance, Ri , and the mat resistance, Rw , one obtains for grammages above w0 : Reff = Ri w0 + Rw (w − w0 ) , w (1.11) where w0 is the grammage in the interfacial layer (here w0 ∽ 20 g/m2 ) and Reff is the overall (effective) resistance of the mat and the interface. The interfacial resistance, Ri , is given by the average resistance in the interface, w0 (Figure 1.7). Equation (1.11) can be applied to the experimental results by Han (1962) and Jong et al. (1999). For small Ri or large8 w, Reff is approximately Rw . 8 CP-cells usually use high fibre grammages. Wire Interface 14 CHAPTER 1. INTRODUCTION Concluding, flow resistance due to the wires is significant only in the cases of fast-draining pulps and over the initial impingement area (Herzig and Johnson, 1999). Further, the interfacial resistance may play an important role in mechanical dewatering. However, there is not much data on this subject, nor a stablished form of how to model it. Figure 1.7: A possible dependence of the incremental specific resistance, R, on grammage, w, for an unbeaten pulp. Fines Mantar et al. (1995) estimate the fibre-mat resistance to flow using a constant-pressure drainage tester. They show a fibre mat flow resistance increasing with fines content in the pulp. In contrast to Jong et al. (1999), the estimated resistance does not include the contribution from the interfacial resistance, which was accounted together with the wire resistance instead. Therefore, Mantar et al. (1995) do not observe the decaying pattern shown by Jong et al. (1999). Mantar et al.’s results (1995) show an increase in resistance with grammage for pulps where fines are present, and approximately contant resistance for an unbeaten pulp. They attribute this phenomenon to an increase in entrapment of fines with grammage. This is corroborated by the results of Han (1962) and Wildfong et al. (2000a,b), although the supposed interfacial resistance was included in the resistance in their case. In this case, the interfacial resistance must have been small. Han (1962) and Wildfong et al. (2000a) measured the retention as a function of grammage. They observed an increase of retention with grammage, which was attributed to be the reason of the increased flow resistance. Mantar et al. (1995) also studied the influence of the slurry concentration on the fibre-mat resistance, and observed a maximum value at approximately 0.5 percent for bleached softwood kraft pulp. At lower concentrations the 1.4. MODELLING 15 fines concentration decreases, whereas, at higher concentrations, flocculation increases. Both lead to decreasing flow resistance. Concluding, fines and flocculation may play an important role in the resistance to flow. Therefore, a fibre mat with constant concentration may not have a constant permeability due to changes in flocculation condition and the fines content. Accordingly, Sayegh and Gonzalez’s (1995) results could be reinterpreted in a different way than the authors’. They attributed the increase in flow resistance to the visco-elastic compression of the fibre network. However, in opposition to what one would intuitively expect, groundwood pulp presented a viscous parameter that was approximately five times larger than that for unbleached hardwood kraft pulp, which has far more intra-fibre water. Further, the viscous contribution increased with decreasing CSF, which one expects to correlate with the amount of fine material. In contrast to that, the increased flow resistance could be attributed to the increasing fines retention, which enhances flow resistance, as the experimental results by Han (1962), Mantar et al. (1995) and Wildfong et al. (2000a,b) have shown. 1.4 1.4.1 Modelling Forming All the models that take into account the effect of the fibre mat deformation use a power law to describe the relationship between the concentration and the fibre mat stress (Meyer, 1962; Emmons, 1965; Martinez, 1998; Zahrai et al., 1998). There are no models hitherto that take into account the behaviour of the fibre mats during unloading or reloading. Therefore, these models are limited to situations where the applied stress in the fibre mats never decreases, and it is not possible to use these models to study the dewatering of the fibre mats after they leave the forming roll, for instance. Several researchers have modelled dewatering in roll formers (Baines, 1967; Wahren et al., 1975; Hauptmann and Mardon, 1973; Norman, 1979; Miyanishi et al., 1989) and the pressure distribution around blades (Zhao and Kerekes, 1995; Nigam and Bark, 1997; Zahrai et al., 1997; Holmqvist, 2002) without considering the fibre network deformation and effect on permeability. There are, however, a few exceptions. 16 CHAPTER 1. INTRODUCTION Martinez (1998) developed a physically based model of the dewatering rate in twin-wire roll formers taking into account the deformation of the fibre mats. The model disregards the time derivative, which means that the diffusion term balances the convective one completely. As Boxer et al. (2000) pointed out, this assumption implicitly means that the model only can handle constant pressures as a boundary condition on the wires. 1.4.2 Wet Pressing Several models of wet pressing has been proposed. However, most of them show stress-strain relationships of the fibre network, which do not present rate dependence. Some consider the fibre network to be only elastic (Roux and Vincent, 1991; Bloch, 1995), whereas others also consider (rate-independent) plasticity (El-Hosseiny, 1990; Kataja et al., 1992; Riepen et al., 1996; Bezanovic et al., 2002). In contrast to those, Ceckler and Thompson (1982) presented a model of wet pressing, which includes a rate-dependent description of fibre-network compression. This description presents an elastic part, attributed to fibre bending, and a rate-dependent part, attributed to the expulsion of the water from the inside the fibres. However, this mathematical description shows too large an expansion during unloading, which has not been observed in experiments (Vomhoff, 1998). McDonald and Kerekes (1991a,b, 1994) presented an alternative solution to include intra-fibre-water dewatering, the decreasing permeability model. They suggest a model of wet pressing, in which intra- and extra-fibre flow resistance is considered, but the mechanical stiffness of the fibre network is ignored. The model is useful for solids content determination within the operability regions where the parameters were estimated. However, it gives little insight into the real physical phenomena taking place in web consolidation. Concluding, models of the fibre network rheology, which can be used in reliable physically based models of wet pressing are needed, but are not yet available. Indeed, according to Börje Wahlström (1990): “None of the serious attempts of mathematical modeling of water removal by pressing, [...], have taken water in the fiber wall into account and have therefore been unable to properly describe wet pressing.” 1.5. SUMMARISING 1.5 17 Summarising Forming There are good experimental results supporting the modelling of mechanical dewatering by considering the suspesion and the wires as continua. However, the following conclusions can be drawn, which are not taken into account in the modelling of dewatering: ❼ The fibre mats are heterogeneous due to flocculation and the uneven distribution of fines. ❼ The interfacial flow resistance, i.e. due to the first layers of fibres on the wire (10–20 g/m2 ), may play an important role. ❼ The stress-strain relationship may be considered rate independent but presents hysteresis. Wet Pressing In wet pressing, intra-fibre water is pressed out of the fibre walls, which results in a rate dependent stress-strain relationship of the fibre network. Rate dependence is also observed in the slow but significant expansion taking place after the end of the pressure pulse. There is no mathematical model that captures these phenomena. 1.6 Objectives The specific objectives of the work reported in this thesis were: 1. To develop a model of dewatering in twin-wire formers with special focus on the fibre network stress and the deformation gradients in the thickness direction that are generated. Further: ❼ The model should be able to calculate the effects of complex pressure profiles that are generated through the forming section. ❼ The stress-strain relationship used should include hysteresis. 2. To develop a model of the stress-strain behaviour of saturated fibre networks when subjected to pressure pulses that are typical for wet pressing. 18 CHAPTER 1. INTRODUCTION Chapter 2 Modelling of Dewatering in Forming ”I have always found it difficult to read books that cannot be understood without too much meditation. For when following one’s own meditation, one follows a certain natural inclination and gains profit along with pleasure; but one is enormously cramped when having to follow the meditation of others.” Gottfried Wilhelm von Leibniz (1646–1716) In this chapter, a physically based model of the forming and densification of fibre mats in twin-wire formers is presented. The model was used to estimate the dewatering and the concentration gradients in the fibre mats along the forming section. This model can handle time-varying stress as boundary conditions on the wires. Also, the stress-strain behaviour of the fibre mats presents hysteresis1 , which enables to use the model to study dewatering and concentration gradients after the fibre mats leave the forming roll. A number of numerical studies were performed. The effect of sequential blade pressure pulses after the forming roll on the dewatering and the concentration gradients could be characterised. Increasing wire tension resulted in increasing dewatering, but the rate of increase diminished rapidly. The simulations also exhibited an elastic expansion of the fibre mats against the suspension and the wires, sucking back water, when they left the forming roll. The effect of beating showed a large influence on dewatering. 1 This description is rate independent, as intra-fibre water is assumed not to leave the fibre walls. Other sources of rate-dependence were also assumed negligible. 19 20 CHAPTER 2. MODELLING OF DEWATERING IN FORMING 2.1 Model It was assumed that there is no air, that the two wires are symmetrical and that dewatering occurs only in the z-direction (Figure 2.1). Further, inertial terms were neglected. Terzaghi’s principle was considered applicable, and Darcy’s law was assumed to apply. The detailed derivation of the equation used here is presented in the Appendix A. The final convective-diffusive equation obtained for the porosity, φ, obtained was: ∂ ∂φ ∂ + (φvf ) − ∂t ∂z ∂z  K ∂σf 0 µ ∂z  = 0, (2.1) where vf is the fibres velocity, σf 0 is the stress in the fibre network and K and µ are the medium permeability and the viscosity, respectively. Figure 2.1: The system being modelled and the coordinate system used. 2.1.1 Constitutive Equations and Physical Properties Equation (2.1) involves physical properties and constitutive equations, which had to be provided in order for the equations to be solved. Permeability was assumed to obey the Kozeny-Carman relation (Equation (1.5)) and the parameters were taken from Robertson and Mason (1949) presented in Table 2.1. The paramenter k0 was assumed constant and equal to 5.55. Equation (1.5) is a rather simplified description compared to what actually 2.1. MODEL 21 happens in between the wires. It does not take into account the variations in permeability due to the non-homogeneous distribution of fines and flocs in the forming web mats. The mathematical description of the fibre network stress, σf 0 , used here is thoroughly explained in Appendix B. The fluid viscosity, µ, was assumed constant and equal to 10−3 Pa ➲ s (Perry et al., 1984). Table 2.1: The effect of beating in terms of the Canadian Standard Freeness (CSF) on the permeability parameters (Robertson and Mason, 1949). CSF 718 708 598 454 384 2.1.2 kraft 1 a [ cm3 /g] 4.27 4.60 4.60 4.70 4.91 Sw CSF [ cm2 /g] 9100 738 11500 703 28300 541 37300 429 43000 292 kraft 2 a [ cm3 /g] 3.66 4.57 4.70 4.87 4.67 Sw [ cm2 /g] 9500 9700 31600 41600 58400 Initial and Boundary Conditions The mathematical model also requires an initial and two boundary conditions. Initial Condition. The porosity at t = 0 has to be given for every z: φ(t = 0, z) = φ0 (z) = 0.99. (2.2) Boundary Conditions. At the symmetry plane, z = 0, the water velocity was assumed to be equal to the fibre velocity, therefore the hydraulic pressure gradient vanishes at that point for every t. Consequently, the porosity gradient was assumed to vanish: vv (t, z = 0) − vf (t, z = 0) = 0 ⇒ ∂φ ∂z = 0. z=0 (2.3) 22 CHAPTER 2. MODELLING OF DEWATERING IN FORMING For some wires, the superficial open area is large enough to render a negligibly small flow resistance even in comparison to very thin fibre mats (Radvan, 1980). The flow resistance imposed by the wire was thus neglected. However, the fibre/wire interaction may have a significant resistance. This was not, however, taken into account because of lack of reliable data. Nevertheless, the model could be easily extended to also include those effects by attributing the interfacial properties to the first deposited layers of fibres. At the upper boundary, z = h (where h is half of the suspension thickness2 ), the fibre network stress was given, as the hydraulic pressure was assumed zero. Therefore, the fibre network stress was set equal to the total pressure experienced by the suspension and the formed web, i.e. σf 0 (t, z = h) = σT (t) . (2.4) Moreover, with the help of the stress-porosity constitutive equation, the porosity, φ(t, z = h), can be calculated for the given applied stress, σT . 2.1.3 Numerical Solution The final Equation (2.1) together with its initial, Equation (2.2) and boundary conditions, Equations (2.3) and (2.4), the constitutive equations and physical properties were solved with the finite element method. This is described in Appendix C. The model output is porosity, φ (together with φmin and φmax ), as a function of t and z. Based on these results, several different quantities can be calculated, e.g. dewatering rate, fibre and water velocity, structural stress, hydraulic pressure, etc. The results were transformed to solids content, SC, according to the relation: SC = 1 1+ φ ρw 1−φ ρf . The specific dewatering rate, i.e. the Darcian water velocity at z = h, was calculated from: qv (t, z = h) = − Z 0 2 h ∂φ dz. ∂t (2.5) In order to calculate the suspension thickness, the fibre material density, here assumed to be 1550 kg/m2 (Carlsson et al., 1983), and the grammage, which in all the simulations was 100 g/m2 , is required. 2.2. RESULTS AND DISCUSSIONS 2.2 23 Results and Discussions The model was used to simulate the effects of blade pulses, wire tension and beating. Typical simulation results are shown in Figure 2.2, where it is demonstrated how the fibre mats are being formed on the wires, and how the dewatering rate decreases with the amount of deposited fibres3 . The simulations show that the suspension between the fibre mats was being dewatered by filtration. Further, it shows concentration gradients in the fibre mats, indicating they were being dewatered by thickening even when there was a suspension in between them. Figure 2.2: Left: Model predictions for the solids content in a roll-blade former of unbeaten kraft pulp. The dewatering pressure profile was a constant pressure of 8 kPa during 20 ms and two haversine pulses with peak pressures of 15 and 20 kPa and pulse lengths of 10 ms and separated in time by 10 ms. Right: The dewatering rate for this simulation. When the fibre mats left the roll, there was a partial expansion of the mats, which sucked water back into the mat, showing that rewetting may occur at the very early stages in a paper machine. The calculations showed that the fibre mats also expanded into the suspension, contributing to an increase of the amount of deposited fibres, although the average solids content obviously decreased. 3 There was, however, no sharp distinction between the suspension and the fibre mats in the model. The initial concentration of the suspension was already above the sediment concentration, and the fluid-like behaviour of the suspension, which depends on shear forces, was not considered in this study. The suspension region was arbitrarily considered to be determined as the region where less than a 20-percent-concentration change occurred in each simulation. Rewetting 24 CHAPTER 2. MODELLING OF DEWATERING IN FORMING Two blades could be modelled assuming a blade pressure profile. This required the insertion of the pressure profile as a boundary condition but no other modification. The simulations show sharp gradients in solids content at regions close to the wire. Note that this region with large gradients expands after each successive pulse. This was probably a consequence of the compressibility behaviour of the fibre mat, which includes hysteresis. Unfortunately, the model has a limitation with respect to the pressure profile that can be used in the simulations. The “blade pulses” must have a large enough impulse as to deform the elements at least as much as the previous pulse, in order to give a proper description of the fibre network stress during unloading. In order to describe the fibre network stress throughoutly, one needs either more internal variables from the porosity history or a hysteresis operator. Wire tension Model simulations confirmed the phenomenon that dewatering increases nonlinearly with increasing wire tension, T (Figure 2.3). This is related to the rapid decrease of permeability as the fibre mats get compacted. This result has been observed in pilot paper machines but cannot be obtained with models where the compressibility, and consequently the strong reduction in permeability of the fibre mats, are not taken into account. Figure 2.3: Model predictions of the final average solids content as a function of the constant wire generated stress during 20 ms and no stress during 20 ms (pressure profiles analogue to those used in Figure 2.4). Beating The influence of beating was studied by comparing the effect of a beaten (CSF 598) and an unbeaten (CSF 718) kraft pulp on the dewatering and forming of fibre mats in the roll section prior to the first blade (Figure 2.4). 2.2. RESULTS AND DISCUSSIONS 25 A constant pressure pulse of 8 kPa under 20 ms followed by no load during the next 20 ms was applied as boundary condition. Figure 2.4: Model predictions for the solids content in a roll-former at two different beating degrees. Left: unbeaten kraft pulp (CSF 718). Right: beaten kraft pulp (CSF 598). The dewatering pressure was 8 kPa under 20 ms and no pressure under the following 20 ms. There was a large difference in dewatering rate between the two beating levels. In the unbeaten case, approximately 80 percent of the suspension was dewatered, whereas for the beaten pulp only approximately 20 percent of the water was removed in the first 20 ms. This was due to the difference in permeability of the two fibre mats, as the stress-strain behaviour of the fibre network shows not to be dependent on beating (Han, 1969). It is possible that the model is overestimating the effect of beating in dewatering since it has a simplified description of permeability. To get more accurate results it will be necessary to consider the variations in permeability due to the heterogeneous distribution of fines in the forming web mats. The model does not take into account the effect of the wire- and the interfacial wire/fibre-flow resistances. However, wire permeability is important only in the very beginning of the dewatering process, therefore, the results would probably not have changed significantly, if it had been included. The wire/fibre-flow resistance was not taken into account because of lack of data. Its influence should probably increase with increasing number of openings per unit area for modern wires. 26 2.3 CHAPTER 2. MODELLING OF DEWATERING IN FORMING Concluding Remarks This section presents a model of the forming and densification of the fibre mats in twin-wire formers, which endeavours to overcome two limitations in previously published models. First, it allows for time-varying stress in the boundary condition. Second, it can account for hysteresis in the fibre network stress. Therefore, it was possible to simulate the fibre mat recovery at the end of the roll former and pressure pulses in the blade section. Model simulations indicate that increasing wire tension resulted in increasing dewatering, but the rate of increase diminished rapidly. The effect of sequential blade pressure pulses after the forming roll on the dewatering and the concentration gradients could be characterised. Model simulations also indicate the possibility of rewetting when the fibre mats leave the forming roll. The effect of beating showed a large influence on dewatering. However, this influence may be overestimated as the permeability relation employed did not consider the effect of the heterogeneous distribution of fines in the fibre mats. Chapter 3 The Perzyna Model ”The felling of awed wonder that science can give us is one of the highest experiences of which the human psyche is capable. It is a deep aesthetic passion to rank with the finest that music and poetry can deliver. It is truly one of the things that makes life worth living and it does so, if anything, more effectively if it convinces us that the time we have for living is finite.” Richard Dawkins (1941 – ) in Unweaving the Rainbow 1998 This and the next chapters deal with wet pressing. The mathematical modelling of wet pressing, analogously to the modelling of the forming section described in the previous chapter, requires constitutive equations in order to be solved. One of these, is the relation between the fibre network stress1 , σf 0 , and the porosity, usually called stress-strain or rheological behaviour. However, the term rheological is used only to describe stress rate dependence. Also, as an abuse of the terminology, sometimes the model is referred to as a model of the fibre network stress, even though the strain that is modelled. Rheological behaviour As pointed out in the introduction, the stress-strain behaviour of the fibre networks at high loads is rate dependent, and it is, at least partially, dependent on the intra-fibre-water flow. Initially, a model based on the concept of strain-rate dependent plasticity, including a rate-independent elasticity, was developed in order to describe the observed experimental data (Perzyna, 1966). This model is presented in this chapter and is called the Perzyna model. Models of this kind is sometimes referred to in the literature as elastic/viscoplastic models. It showed good agreement with experimental results for the compression phase, therefore it is employed in physically based models of wet pressing (Gustafsson and Kaul, 2001; Gustafsson et al., 2001). However, the Perzyna model gave poorer 1 For simplicity, the subspript will be omitted in this and next chapters. 27 Elastic/ viscoplastic model 28 CHAPTER 3. THE PERZYNA MODEL results, when describing unloading as a rate-independent process. Moreover, it could not describe the experimentally observed behaviour of a sequence of pressure pulses. Therefore the Perzyna model was further developed. This is shown in the next chapter. 3.1 Experiments Vomhoff and Norman (2001) conceived a special measuring cell (Figure 3.1), with which measurements of the rheological behaviour of the fibre networks can be obtained. This cell can be placed in a material testing machine to conduct uniaxial compression tests on saturated fibre networks. Figure 3.1: The compression device conceived by Vomhoff (1998). The inset shows the water flow pattern. Webs placed in the test cell are compressed between a smooth non-porous surface on one side and 1 mm thick bars spaced 1 mm apart on the other. The pressure pulses applied by the testing machine can vary greatly in shape, pulse length, peak pressure and the cell even allow measurements at different temperatures (Vomhoff and Norman, 2001). 3.1. EXPERIMENTS 29 The experimental trials were performed using webs of 20 or 30 g/m2 of thermo-mechanical pulp (TMP) and softwood bleached kraft pulp (SBK) beaten to different levels. The response of the web thickness to the pressure pulse was recorded as a function of time (Figure 3.2). The modified strain was calculated based on web thickness according to: ǫ = ln  h h0  , (3.1) where h0 = w/ρf is the thickness the web would have at zero porosity, and w and ρf are the grammage and the fibre material density. Expressing the experimental results in modified strain reduced greatly the variation between different tests, as it does not require the initial strain, which is difficult to determine experimentally. The results were, however, presented as a function of solids content, SC, according to: SC = ρf , ρf + ρw (hρf /w − 1) (3.2) as it is a better known quantity for the papermaker. The fibre material density, ρf was assumed to be 1550 kg/m3 for bleached sulphate (Carlsson et al., 1983). MODIFIED STRAIN [−] STRESS [MPa] −1 −2 −3 −4 −5 −6 −7 −8 2.6 0 2.4 −1 2.2 STRESS [MPa] 0 2 1.8 1.6 1.4 20 40 60 80 100 120 −3 −4 −5 −6 1.2 −7 1 0 −2 0.8 0 20 TIME [ms] 40 60 TIME [ms] 80 100 120 −8 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 MODIFIED STRAIN [−] Figure 3.2: Left: An arbitrary pulse applied to the wet web. Centre: the response in thickness measured as a function of time. Right: the experimental response in a stress-modified strain graph. 3.1.1 Sources of Error The water from the saturated web flows primarily in the in-plane direction, to the grooves between the bars. Since the design of the test cell ensures 2.6 30 CHAPTER 3. THE PERZYNA MODEL that water flows primarily in the in-plane directions, the compression of the web is assumed to be uniform in the thickness direction, which is one of the requirements to obtain a reliable uniaxial measurements of the stress-strain behaviour of the fibre network. Experimental error Nevertheless, the stress measured is not exactly the fibre network stress. Part of the applied stress is balanced by the hydraulic pressure developed in the extra-fibre space (Vomhoff and Norman, 2001). On the other hand, according to them, the contribution of the hydraulic pressure due to the inplane flow amounts to only seven percent of the total work performed for the fastest of their pulses (20 ms) for a similar pulp with almost the same beating level (23.3 SR in contrast to 25.2 SR used here). For the case for SBK pulps with 16.3, 35.1 and 62.7 SR, no reliable permeability data were available, and the influence of the hydraulic pressure from the flow in between the fibres was not studied. It was probably negligible for the case of 16.3 SR, but could be significant for the cases with 35.1 and 62.7, especially for the shortest pulses. Vomhoff (2000) reports high influence of the hydraulic pressure for the shortest pulses performed with TMP (75 percent of the work for pulses of 25 ms). A second type of experimental error is observed in comparing results from trials with nearly the same pulse profile (Vomhoff and Norman, 2001). They report relative error at minimum achieved modified strain to be less than ten percent. These two contributions are henceforth referred to as the experimental error. In conclusion, although the best experimental equipment available was used, these is, nevertheless, some error in the experimental data obtained. Consequently, the estimated parameter values of the model will carry conModelling error tributions from the experimental error in addition to any eventual modelling error. Initially, the behaviour of TMP was examined, and then the material parameters were estimated. Finally, the constitutive equations were validated. The tests were performed with webs of 20 g/m2 . The experimental data in Table 3.2 were used in the study of the influence of temperature and intrafibre water in the rheological behaviour of the fibre network stress for SBK. The tests were performed with webs of 30 g/m2 made of SBK pulp. Some tests were kindly made available by Vomhoff (1998). 3.1. EXPERIMENTS 31 Table 3.1: Tests performed with webs of 20 g/m2 made of TMP (CSF 235 ml, except the three last rows which have CSF 123 ml). The tests marked with the symbol ‘*’ were conducted by Vomhoff (1998). The quasi-static tests were performed with a pulse length of 130 s made of SBK. Pulse Length Peak Pressure Temperature [ms] [MPa] [o C] (25, 75)* 3, 8, 23 23 250, 2500 3, 8*, 23* 23 27 3, 10, 20 23 270 3 23 25, 75, 250, 2500 8, 23 23 130000 23 23 699000 15 23 699000 15 50 699000 15 80 Pulse Shape — Roll Roll Shoe-press Shoe-press Strain ramp Strain ramp Strain ramp Strain ramp Strain ramp Table 3.2: Tests performed with webs of 30 g/m2 made of SBK pulp. The tests marked with the symbol * were conducted by Vomhoff (1998). The quasi-static tests were performed with a pulse length of 349 or 699 s made of SBK. Pulse Length Peak Pressure WRV [ms] [MPa] [g/g] 20, 200, 2000, 349000 8 1.26 20, 200, 2000, 349000 8 1.43 20, 200, 2000, 349000 8 1.51 20, 200, 2000, 349000 8 1.65 (20, 200, 2000)* 5, 10, 20 — (20, 200, 2000)* 5, 10, 20 — (20, 200, 2000)* 5, 10, 20 — 699000 20 — 699000 20 — 699000 20 — SR Temp. [cl] [o C] 16.3 23 25.2 23 35.1 23 62.7 23 22.3 23 23.7 50 22.3 80 22.3 23 23.7 50 22.3 80 32 CHAPTER 3. THE PERZYNA MODEL Figure 3.3: Two experimentally determined stress-solids content curves obtained from tests with nearly identical applied pulses (8 MPa and 75 ms). The difference between the curves exemplifies the order of magnitude of the highest experimental errors. 3.2 Model A model of the stress-strain behaviour of the fibre network in wet pressing should present rate-dependent strains, as observed in the experimental work of several researchers (Ellis, 1981; Szikla and Paulapuro, 1989; Vomhoff and Norman, 2001). In our first attempts to model the fibre network force we applied the standard approaches to visco-elasticity. However, we were not able to obtain a model which gave good agreement with experimental results. We concluded it would be necessary to apply plasticity theory, and that led us to Perzyna’s approach to visco-plasticity (1966). It has two desirable properties, viz. it allows for time-varying stress and the terms can be nonlinear. Moreover, it could also easily incorporate a term making the ratedependent plasticity also dependent on the strain. Therefore it was employed. In Perzyna’s approach, the total strain rate is decomposed into elastic (recoverable) and plastic (non-recoverable) parts. Using a logarithmic definition of strain, ǫ, the total strain rate is then given by dǫpl dǫel dǫij ij ij = + . dt dt dt (3.3) pl where ǫel ij are the components of the elastic strain, and ǫij the components of the plastic strain tensor. This is a reasonable assumption even for large total 3.2. MODEL 33 strains, provided the elastic strain is small compared to the plastic strain. The two components of the total strain rate were modelled independently, and the plastic part of the strain rate included rate-dependent effects. The proposed model may therefore be classified as elastic/viscoplastic. Non-linear elasticity The submodel proposed for the observed non-linear elasticity was:    σ/3 − pt , ǫ = ln 1 − Dln −pt el (3.4) where σ is the (uniaxial) applied stress, pt and D are parameters. This (rateindependent) expression was used in modelling the elastic part of the total strain. Rate-sensitive plasticity The approach proposed by Perzyna (1966) for modelling rate-sensitive plasticity is followed here. He introduces a static load function, F , in the form: F (σij , ǫpl kl ) = f (σij , ǫpl kl ) κ(ǫpl e ) − 1, (3.5) where the function f is the dynamic load function, κ(ǫpl e ) is a hardening funcpl tion , and ǫkl are the components of the plastic strain tensor (Perzyna, 1966). When the dynamic load function, f , is less or equal to the hardening function, κ(ǫpl e ), then the static load function, F , is equal to the yield criterion in the inviscid plasticity theory. However, in Perzyna’s approach to viscoplasticity, f is allowed to exceed the yield limit, κ(ǫpl e ). The positive difference between pl f and κ(ǫe ) is called the overstress. The function, F , may be interpreted as a normalised overstress. Perzyna proposes that the plastic strain rate be expressed as a function of the normalised overstress, F , according to: dǫpl ∂f ij = γc hΩc (F )i , dt ∂σij (3.6) where γc is a viscosity parameter, and hΩc (F )i is a function that has be chosen to represent the results of tests under dynamic loading2 . A representation of 2 The symbol hai denotes the Macauley brackets having a value of a for a > 0 and vanishing otherwise. 34 CHAPTER 3. THE PERZYNA MODEL the results of the available uniaxial dynamic tests can be given by (Lobosco, 2000): Nc c , Ωc (F ) = Ψm c F (3.7) where  Ψc =  − κ(ǫpl ref ) pl κ(ǫref ) κ(ǫpl e ) !2 mc + 1 . (3.8) ǫpl ref is a reference plastic strain assumed to be 1.9 for any pulp, and mc and Nc are material parameters. As only uniaxial compression was considered here, the dynamic load function, f (σij , ǫpl kl ), was taken to be the applied uniaxial stress, σ, and the hardening function, κ(ǫpl e ), reduced to the static stress-plastic strain relation, pl σy (ǫ ). Thus, the static yield function became: F = σ − 1. σy (ǫpl ) (3.9) Finally, rewriting Equation (3.6) for the uniaxial case, and using Equation (3.7), one obtains: dǫpl = γc Ψmc F Nc , dt (3.10) which was solved numerically with a Runge-Kutta algorithm together with Equation (3.8) in order to obtain dynamic expressions for the modified strain, i.e ǫpl from σ(t) as input variable. 3.3 3.3.1 Results and Discussions Parameters Estimation The elastic parameters of Equation (3.4), pt and D, and the plastic ones from Equation (3.10), γc , mc and Nc , are different for different pulps, and may 3.3. RESULTS AND DISCUSSIONS 35 change with beating and temperature. Obviously, they should not change with peak pressure, nor with pressure pulse shape and nor with pulse length. The parameters were estimated using a small subset of the available experimental data for each experimental setting. The remaining tests were used to validate the model. The elastic parameters were estimated from the tests with the shortest pulse length, viz. 25 ms pulse length for TMP and 20 ms for SBK. They were estimated from the region of the stress-modified strain curve, where the total modified strain starts to recover. The estimated values for TMP were: D = 0.037 and pt = 0.2 MPa. Webs made of TMP showed a significant elastic strain, whereas webs made of SBK showed a much smaller, almost insignificant elastic strain. Therefore, the elastic strain may be disregarded in the case of SBK, without any markedly change in the final results. Having estimated the elastic parameters, the experimental plastic strain was determined by subtracting the calculated elastic part from the modified total strain, according to: ǫpl = ǫ − ǫel . (3.11) The strain decomposition being allowed because the compression is uniaxial. For TMP, the parameters γc , Nc and mc were estimated by varying them in an iterative procedure to minimise the differences between the experimental plastic strain and the calculated plastic strain. The three tests with 25 ms pulses were used in this procedure. The estimated parameter values obtained in this way were Nc = 4.43, γc = -1.2 s−1 and mc = -0.58. Evidently, the estimated parameter values were kept fixed for the prediction of the tests in the same conditions. In the temperature study, the sub-set of tests performed with the highest peak pressure, i.e. 20 MPa with pulse lengths 20, 200 and 2000 ms (see Table 3.2), were used for parameter estimation. In the case of fibre swelling, the two tests with pulse lengths 20 and 2000 ms (for each WRV) were used for parameter estimation. The tests with pulse lengths of 200 ms were used to test the model predictive capability. pl The plastic strain rate, dǫdt , the functions F and Ψm∗ were calculated from the experimental plastic strain, and the parameters were obtained by multiple regression. 36 CHAPTER 3. THE PERZYNA MODEL As m∗c did not show a significant change with temperature nor with fibre swelling – reflected in the WRV –, it was considered independent of both temperature and fibre swelling. The influence of temperature on the first compression could be observed on γc and Nc . The values are presented in Table 3.3. The influence of fibre swelling on the first compression could be observed on γc . The values are presented in Table 3.4. Table 3.3: Estimated values of γc and Nc for webs made of SBK compressed at different temperatures. Temperature γc ➲ 10−3 23 -0.140 50 -0.178 80 -0.255 mc Nc -0.60 1.146 -0.60 1.036 -0.60 0.930 Table 3.4: Estimated values of γc for webs made of SBK beaten at with different WRV. WRV γc ➲ 10−3 1.26 -0.027 1.43 -0.012 1.51 -0.007 1.65 -0.004 3.3.2 mc -0.40 -0.40 -0.40 -0.40 Nc 1.61 1.61 1.61 1.61 Validation The Perzyna model presented above was validated by comparing predictions of web behaviour for all the remaining pressure pulses (i.e. those not used in the parameters estimation, see Tables 3.1 and 3.2) with the corresponding experimental results. In each case, the applied pressure pulse (as recorded in the experiment) was used as the input to the model. The first sampled thickness in the experiment was used as the initial condition. On the left-hand side of Figure 3.4, is shown predictions compared with experimental results for some roll-nip-like pressure pulses with 75 ms in pulse length (Table 3.1). The rate-dependent behaviour of the webs can be seen more clearly on the right-hand side of Figure 3.4 showing the experimental results and the predictions for pulses with 25, 250 and 2500 ms in pulse length (Table 3.1). 3.3. RESULTS AND DISCUSSIONS 37 Figure 3.4: Left: Comparison between test results for 75 ms roll-press type pulses with max peak pressures of 3, 8 and 23 MPa, and predictions. Right: Comparison between test results for roll-press type pulses with length of 25, 250 and 2500 ms and with max peak of 23 MPa and Perzyna model predictions (only loading). The quasi-static curve (pulse length 130 s) is also shown (as a reference). On the left-hand side of Figure 3.5, is presented the predicted and the experimental results for two different degrees of swelling, 1.26 and 1.65 g/g with 200 ms (Table 3.1). In both cases, the pressure pulse was 200 ms in length and achieved 8 MPa in the peak pressure. Figure 3.5: Left: Experimental and predicted results for two different degrees of fibre swelling of SBK pulps. The tests were performed with pulse lengths of 200 ms. Right: Experimental and predicted results for two different web temperatures of SBK pulps. The tests were performed with pulse lengths of 20 ms. 38 CHAPTER 3. THE PERZYNA MODEL On the right-hand side of Figure 3.5, is shown predicted and experimental results for two incoming web temperatures with pressure pulses of 20 ms length and 10 MPa peak pressure (Table 3.1). 3.4 Concluding Remarks A mathematical model of the stress-strain behaviour of the fibre network, here called the Perzyna model was presented and validated. The model predictions were in good agreement with the experimentally observed fibre network rheological behaviour over the whole range of peak pressure, pulse lengths and pulse shapes studied (single pulses). The agreement was good also for different pulps (TMP and SBK), temperatures, and intra-fibre-water content (reflected by pulps WRV). The Perzyna model may therefore be a useful tool to calculate the stress-strain behaviour of fibre network in physically based models of wet pressing for one pulse without relaxation, provided the model parameters are estimated from experimental data with small experimental error. Chapter 4 The Gel-Network Model “We shall never be able to study, by any method, their chemical composition or their mineralogical structure... Our positive knowledge of stars is necessarily limited to their geometric and mechanical phenomena.” August Comte (1798 – 1857) in Cours de philosofie positive 1835 ”Credulity is the man’s weakness, but the child’s strength.” Charles Lamb (1775 – 1834) in Essays of Elia 1823 In order to model the fibre network stress in a sequence of press nips, as are common in press sections, it is necessary to allow for the unloaded expansion between the nips. This requires rate sensivity in both the compression and expansion phases, i.e. the web has to be treasted as a material that shows rate dependence in both the elastic and plastic regions. The determination of the yield conditions for such a material is very difficult and has not yet been done (Perzyna, 1966). Therefore, a new model was developed that utilizes information about the structure of the web. The wet web may concieved as layered network of restricted swelling gels. A fibre in a saturated web is a restricted swollen gel, the inner swelling pressure in a swollen fibre wall being balanced by the stress in the fibre wall structure (Flory, 1953; Christensen and Giertz, 1965). The resistance to deformation of a wet web may therefore to be dependent on the swelling behaviour of the web fibres. The new model is refered to as restricted gel-network model, or just shortly gel-network model. The gel-network model could take into account the experimentally observed rate dependence of saturated webs in both compression and expansion and its predictions were in good agreement with results from uniaxial experiments for both a single nip and a series of consecutive nips, for a wide range of experimental conditions. 39 Compression, expansion and recompression Restricted swelling gels 40 CHAPTER 4. THE GEL-NETWORK MODEL 4.1 Experiments The experimental procedure was the same as in the former chapter. Only some tests were conducted with different shapes in order to explore the effect of unloading. The experimental data in Table 4.1 were employed in the generalisation of the mathematical description for expansion and recompression of fibre sheets. Regarding the hydraulic pressure contribution on the successive pulses, it was assumed that they should be smaller than those observed in the first compression. According to Lindsay and Brady (1993) and Reverdy-Bruas et al. (2001), permeability for previously compressed webs increases at the same porosity. Further, previously dried webs also demonstrate increased permeability (Lindsay, 1994), indicating that the closure of internal pores, leave external pores more open, for a given total porosity. Table 4.1: Tests performed with pulp fibre webs of 30 g/m2 made of SBK. In the cases marked with ‘*’, only one compression pulse was performed. Pulse Length Interval Pulse Shape [ms] [ms] – 20 and 2000 0, 100 and 300 roll 50 and 500 0, 100 and 300 shoe 2000* — see Figure 4.2 20, 200 and 2000* — roll 4.2 4.2.1 SR Peak Pressure [cl] [MPa] 25.2 9 26.2 8 25.2 5 25.2 9 Model The Micro- and Macroscopic Perspectives In this section, we present our view of the relationship between the microscopic concept of a swollen fibre (restricted gel) and the macroscopic variables, viz. applied stress and wet web strain. A swollen cellulose is a restricted swelling gel (Flory, 1953; Christensen and Giertz, 1965) and the fibre web may be regarded as a layered network of restricted gels. From a macroscopic perspective, however, the fibre web may be considered a continuum. Here we were only concerned with situations 4.2. MODEL 41 where the fibre web is immersed in water and the pressure of the inter-fibre water (in the pores between the fibres) is negligible. Therefore, the applied stress perpendicular to the plane of the web was balanced only by the fibre network stress. It is hypothesised that if a stress is applied to the uncompressed web, the hydraulic pressure of the water inside the fibre walls increases, resulting in an outflow of water until static equilibrium is achieved (Laivins and Scallan, 1993b) at a higher swelling pressure. At the macroscopic level, the applied stress on the web in a state of static equilibrium is called the equilibrium stress, σe , which is a function of the history of the web strain1 , ǫ (Figure 4.1). The function σe can be approximated experimentally by uniaxial, quasi-static compression tests. Whenever the applied stress on the web is greater than the equilibrium stress, the web will be compressed, and some intra-fibre water will be pressed out of the fibre walls until equilibrium is attained. If instead the applied stress on the web is less than the equilibrium stress, the web will expand, and, at the microscopic level, water will flow into the fibre walls until equilibrium is attained at a lower swelling pressure, as indicated by the experimental results shown in (Figure 4.2). Figure 4.1: Experimentally determined equilibrium stress under loading and unloading starting from two different initial strains (loading and unloading). 1 ρ The solids content is a function of the strain, ǫ, according to: SC = ρf +ρwf(eǫ −1) , where ρw and ρf are the density of the water and fibre material respectively. Equilibrium stress Reaching equilibrium 42 CHAPTER 4. THE GEL-NETWORK MODEL Figure 4.2: Experimental response to a half-haversine (1000 ms) pulse with sudden change to two different stress levels. The load was kept at a constant level for several minutes. It can be clearly seen how the response approaches the equilibrium curve during the loading (left) and unloading (right) phases. Swelling The stress-strain behaviour of the fibre network was considered to be dependent on the swelling properties of the fibres. The observed rate dependence of wet webs in both compression and expansion phases was attributed to the water flow out of and into the fibre walls. This motivates the use of the concept of overstress – i.e. the difference between the applied stress and the equilibrium stress – under both the compression and the expansion phases. A macroscopic model that is consistent with the hypothesis regarding the swelling and the flow phenomena taking place at the microscopic level is presented below. 4.2.2 Overstress The Macroscopic Model The mathematical description of the rheological behaviour of the fibre network stress, proposed here is based on the hypothesis that compression occurs with positive normalised overstress, i.e. F = σ − σe , σe (4.1) whereas expansion occurs when F is negative. The model represents the rate of fibre network deformation by the following set of expressions: 4.2. MODEL 43 dǫ = γc Ωc (F ) for dt dǫ = 0 for dt dǫ = γe Ωe (F ) for dt F > 0, (4.2) F = 0, (4.3) F < 0. (4.4) The functions Ωc (F ) and Ωe (F ) need to be chosen so that they adequately represent the results of uniaxial tests under dynamic compression/expansion (Figure 4.2). Analysis of the compression parts of the results of such dynamic tests indicates that Ωc (F ) is a function not only of the excess applied structural stress but also of the strain (Lobosco, 2000). The expansion part of the test results showed that Ωe (F ) was also a function of the strain and the maximum strain. A good representation of the test results was given by the functions: Nc c , Ωc = Ψm c F me N e Ωe = Ψe F , (4.5) (4.6) where2 2 σy (ǫ) Ψc (ǫ) = − 1 + 1, σy (ǫref ) Ψe (ǫ, ǫmax ) = (ǫ − 0.9ǫmax ) .  (4.7) (4.8) Equations (4.2) and (4.4) combined with Equations (4.5) and (4.6) respectively, yield: dǫ Nc c , = γc Ψm c F dt dǫ Ne e . = γe Ψm e F dt (4.9) (4.10) The constants γc , γe , Nc , Ne , mc and me are parameters to be estimated from the experimental data. These equations can be solved using a Runge-Kutta algorithm. 2 The factor 0.9 multiplying ǫmax is just to avoid singularities problems when ǫ = ǫmax , as the parameter me obtained was negative. 44 4.3 4.3.1 Hysteresis CHAPTER 4. THE GEL-NETWORK MODEL Results and Discussions The Equilibrium Curve In order to estimate the parameters of the model and to make predictions, one needs to know the equilibrium stress function, σe , approximated by quasistatic measurements of the fibre network. The experimentally determined equilibrium curves observed were quite complex (Figure 4.1) and could be expected to require a hysteresis operator if the responses to arbitrary pulses were to be simulated (e.g. Bergqvist, 1994). However, a much simpler mathematical expression was found to give adequate description of the equilibrium stress functions of the fibre network obtained in this study (Appendix B). The proposed expressions showed good agreement with the experimental results, and were used in the estimation of the dynamic parameters. Figure 4.1 shows a typical example of the equilibrium curves obtained by us. Laivins and Scallan (1993b) have presented experimentally determined curves of the water content of wet webs as functions of the applied compressive pressure which are, in principle, equivalent to the first compression part of the equilibrium curve shown in Figure 4.1. Their results show that the water contents of pads pressed to high pressures (and allowed to achieve equilibrium) correlate linearly with the initial levels of swelling of the fibres (the fibre saturation points). This implies that the solids content at the end of the first compression in our equilibrium curve are linearly correlated with the fibre saturation points of the web fibres. Typically, the curves for successive compressions come closer at higher pressures but never pass through the same point. This can be interpreted as being the result of hornification, i.e. the closing of pores in the fibre walls (Maloney et al., 1997, 1998). 4.3.2 Parameters Estimation Equations (4.9) and (4.10) are intrinsically linear, which means that they can be transformed to linear equations (with respect to the parameters) by a logarithmic transformation, viz.   dǫ ln = ln(γc ) + mc ln(Ψc ) + Nc ln(F ) dt   dǫ ln = ln(γe ) + me ln(Ψe ) + Ne ln(F ). dt (4.11) (4.12) 4.3. RESULTS AND DISCUSSIONS 45 The compression parameters, γc , mc and Nc , and the analogue expansion ones, γe , me and Ne in these two equations could then be straightforwardly estimated by the weighted least-square method (Draper and Smith, 1981). Experimental results for several pulses with a large range of strain rate were used in the estimation, as they provided a larger operation window for the regressed model. Pulses with pulse lengths of 20, 200 and 2000 ms without interval (see Table 4.1), were used for parameter estimation. This multiple regression had to be weighted because the variance of the residuals was clearly not constant. Table 4.2: Material parameters estimated with the weighted least-square method for webs made of SBK. γ m N Compression -0.0095 -0.59 1.68 Expansion 0.0011 -0.38 1.08 The residuals obtained were small for the compression phase, but not for the expansion one. Better suggestions for Ωe will probably improve the residuals. The model parameters were assumed to be the same for all the pulses. 4.3.3 Validation The gel-network model was validated by comparing predictions with all experimental results not used in the parameters estimation (Table 4.1). Parameter estimation was required only once and could be done using any suitable pulse shape. Predictions could then be made for widely different pulse shapes as shown in Figures 4.3 and 4.4. The results shown in Figure 4.4 are consistent with the hypothesis that the webs resistance to deformation is influenced by the swelling behaviour of the component fibres. The left hand side of Figure 4.4 indicates that equilibrium was achieved by static compression and the flow out of the fibre wall, the static applied stress being initially higher than the equilibrium stress. The right hand side of Figure 4.4 shows a situation where the static applied stress was initially lower than the equilibrium stress and equilibrium was achieved by static expansion, partly driven by the swelling of the fibre walls. This expansion is not captured by the Perzyna model, nor, to the knowledge of the authors, by any other published model. Weighted least squares 46 CHAPTER 4. THE GEL-NETWORK MODEL Figure 4.3: Left: The roll press-like pulse with 20 and 2000 ms used to estimate the gel-network model parameters and the model predictions. Right: Comparison between the experimental response and the gel-model prediction for an experiment with a shoe-press-like pulse of 50 ms. Figure 4.4: Comparison between experimental and model responses for half roll-press-like pulses with incomplete unloading to two different load levels. Figure 4.5 compares the gel-network model predictions with the observed response of the web to a series of roll-like press pulses, with and without unloaded periods in-between. Keeping in mind the difficulty of measuring web thickness accurately at near zero load, the agreement is quite satisfactory. Figure 4.6 compares the experimentally observed development of solids content with predictions from the Perzyna and the gel-network models for a sequence of four pressure pulses. The Perzyna model fails to capture the reductions in solids content during the unloaded intervals in-between the pulses, whereas the gel-network model showed good agreement with the experimental data, capturing the general pattern. 4.4. CONCLUDING REMARKS 47 Figure 4.5: Left: Comparison between the experimental and the mathematical prediction for a sequence of roll-press-like pulses without unloaded periods in-between pulses. Right: Similar comparison with unloaded intervals of 100 ms in-between pulses. Figure 4.6: Comparison of the experimental, Perzyna model and gelnetwork model responses for four consecutive roll-press-like pulses with 10 MPa in peak pressure with unloaded periods in-between the pulses 4.4 Concluding Remarks This section presents a new model of the fibre network rheology, the gelnetwork model. It was based on the idea that the wet fibre web may be conceived as a layered network of restricted swelling gels. Further, it gave predictions that agree well with the results of uniaxial experiment. It could simulate the fibre network response to a sequence of pulses of arbitrary shapes with only a single set of estimated parameters. The gel-network model may therefore be used as a general model for the rheological behaviour of the wet 48 CHAPTER 4. THE GEL-NETWORK MODEL fibre network in wet pressing, provided the parameters are estimated from data with small experimental error. Bibliography Auriault, J.-L. and E. 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Norman (1997), ‘An analysis of blade dewatering in a twin-wire paper machine’, Journal of Pulp and Paper Science 23(9), J452–459. Zahrai, S., M. D. Martinez and A. A. Dahlkild (1998), ‘Estimating the thickness of the web during twin-wire forming’, Journal of Pulp and Paper Science 24(2), 67–72. Zhao, R. and R. Kerekes (1995), ‘Pressure distribution between forming fabrics in blade gaps formers: Thin blades’, Journal of Pulp and Paper Science 21(3), 97–103. Appendix A Model Derivation Several researchers have examined and studied the physics of multi-phase flow using a continuum mechanics approach, where averaging techniques have been applied (e.g. Ishii, 1975; Drew, 1983; Soo, 1990; Kaviany, 1995; Enwald et al., 1996). This procedure has the advantage that it shows more clearly how the macroscopic quantities stem from microscopic considerations. Therefore, it was followed here. The system being modelled is shown in Figure 2.1. Accordingly, a volumetric averaging (Kaviany, 1995), i.e. 1 hψk i = ψ˜k φk = V Z ψk dV, (A.1) Vk was performed on the continuity and momentumR equations below for a given quantity, ψk , of phase k (Figure A.1). φk = V1 Vk dV is the volume fraction of phase k = {w, f } (water and fibre)1 . The tilde designates volumetric average whereas the cap designates massic average, viz. R V ψ̂ = Rk A.1 ρk ψk dV ρ dV Vk k . (A.2) Continuity Equations The continuity equation for phase k becomes: 1 Water volume fraction is the porosity, which is the terminology used henceforth. 59 60 APPENDIX A. MODEL DERIVATION Figure A.1: A representative picture of the region being averaged composed of phase k and k. ∂φk ρ˜k + div(φk ρ˜k v̂k ) = 0. ∂t (A.3) No mass flux between the phases was assumed. If both phases are incompressible, Equation (A.3) becomes: ∂φk + div (φk v̂k ) = 0 ∂t A.2 (A.4) Momentum Equations Similarly, the equation for conservation of momentum is obtained by volumetric averaging:  1  ∂ + (φk ρ˜k v̂k ) + div (φk ρ˜k v̂k v̂k ) = div φk T̃k + φk T̃Re k ∂t V Z Sk Tk ➲ nk dS, k (A.5) where nk is the unit vector normal to the surface2 , Sk k , between the phases and T̃k and T̃Re k are the stress tensor in the phase k and the stress due to the (microscopic) velocity variations inside the integration volume, the so-called Reynolds stresses (Bird et al., 1960), respectively. According to Bear (1972) and Kaviany (1995), the deviations observed from Darcy’s law 2 k refers to the adjoining phase to k. A.2. MOMENTUM EQUATIONS 61 at Reynolds numbers higher than unity are attributed to inertial effects, and not to turbulence. Arguably, the Reynolds stresses were supposed to be responsible for this effect. Accordingly, the experiments performed by Wildfong et al. (2000b) give evidences for assuming T̃Re k negligible for the fluid phase. It is obviously negligible for the solid phase too. According to experiments performed by Herzig and Johnson (1999) and Wildfong et al. (2000a), T̃Re k is probably relevant for the water phase flowing through wires at early dewatering stages. Simulations of blade dewatering performed by Roshanzamir et al. (2000) show that this term plays only a modest role in later parts of the forming sections. The macroscopic inertial terms (the two terms on the left-hand side of Equation (A.5)) were neglected as the water velocity variation in the z-direction is very small. The last term on the right hand side accounts for the (diffusive) momentum transfer between phases for a system without phase transition. How this term was taken into account is described in the next section. The influence of a potential (gravitational) field was disregarded for both phases, as this contribution is small in comparison to the stresses applied in twin-wire formers. Equation (A.5) then becomes:  div φk T̃k  1 + V Z Sk Tk ➲ nk dS = 0. (A.6) k On the boundary between phases, the forces that each phase exerts on each other have the same magnitude and opposite direction according to Cauchy’s formula (Landau and Lifchitz, 1987; Chadrasekharaiah and Debnath, 1994), i.e. Tk ➲ nk = −T k ➲ n k , (A.7) where n k = −nk . Thus, adding the momentum equations for the two phases, one obtains:   divTt = div −φw p̃w I + (1 − φw )T̃f = 0, (A.8) where Tt is the total stress and T̃w reduces to the hydraulic pressure, i.e T̃w = −p̃w I (I is the identity matrix ), as the internal shear stresses balance 62 APPENDIX A. MODEL DERIVATION out in the volume averaging (Jacob, 1950; Bear, 1972; Lewis and Scherefler, 1998). The stress in the fibre network, k = f , has two contributions (Biot, 1941; Drew, 1983; Kataja et al., 1995), viz. T̃f = T̃f 0 + T̃f h , (A.9) where T̃f 0 is the stress in the fibre network disregarding the effect of hydraulic pressure, i.e. the fibre network stress, and T̃f h is the stress in the fibre network induced by the hydraulic pressure acting on its surface. Rendulic (p.13 in Terzaghi, 1944) concludes that the mechanical effects of a given state of total stress depend only on T̃f 0 . Kirmanen et al. (1994) conjecture the induced structural pressure to be proportional to the hydraulic pressure, i.e. T̃f h = −C p̃w I (A.10) where the constant C (≤ 1) depends on the material and the contact between particles. For instance, C = 1 for completely immerse solids (Emmons, 1965; Landau and Lifchitz, 1990). Therefore, it depends on the straining of the material. Equation (A.8) can, with the help of Equations (A.9) and (A.10), be rewritten as: Tt = X φk T̃k = hTf 0 i − φeff a p̃w I, (A.11) k where φeff a is called the effective areal porosity (Kataja et al., 1995). The fibre network stress is given by: hTf 0 i = (1 − φw )T̃f 0 , (A.12) and φeff a is a function of C according to: φeff a = C + (1 − C)φw . (A.13) According to measurements by Kataja et al. (1995), φeff a is very close to unity (∽ 0.99) even for an extreme case in the forming section (40 kPa and A.3. MOMENTUM TRANSFER EQUATIONS 63 25 percent solids content ⇒ φeff a = 0.99). Accordingly, Terzaghi’s principle was assumed in the forming section, i.e. Tt = hTf 0 i − p̃w I. A.3 (A.14) Momentum Transfer Equations The momentum transfer term given by the last term in Equation (A.5) is most frequently described by an empirical equation. It includes a dilatational contribution (Garg et al., 1975; Brownell Jr. et al., 1977), i.e. p̃w gradφw , which can be obtained by volumetric averaging in static stress conditions, i.e. when the fluid is under a constant pressure. Expanding, Tk into −p̃w I + Tdk , the momentum transfer term becomes: 1 V 1 Tk ➲ nk dS = p̃w gradφk + V Sk k Z 1 (p̃w − pw ) nk dS + V Sk k Z Z Sk Tdk ➲ nk dS, k (A.15) The second term on the right-hand side of Equation (A.15) comprehend the buoyant and the form effects, whereas the last term accounts for the drag. The last two terms are postulated to have the form (Garg et al., 1975; Brownell Jr. et al., 1977; Lewis and Scherefler, 1998): 1 V 1 (p̃w − pw ) nk dS + V Sk k Z Z Sk Tdk ➲ nk dS = −µφ2w K−1 (v̂k − v̂ k ) , (A.16) k where K is the so-called permeability of the medium, and µ is the viscosity of the fluid. As expressed by Equation (A.7), Equation (A.16) has the same magnitude but opposite signs for the two phases. The final equation was obtained from Equations (A.4), (A.6), (A.14), (A.15) and (A.16) after some rearrangement: ∂φw + div (φw v̂f ) − div ∂t   K divhTf 0 i = 0 µ (A.17) As the porosity gradients are much larger in the thickness direction than those in the MD and the roll radius is large in relation to the suspension 64 APPENDIX A. MODEL DERIVATION thickness, only the uniaxial problem was considered here. Furthermore, only symmetrical dewatering was assumed, i.e. centrifugal forces were neglected, and the blades were considered not to impede the water flow (Figure 2.1). Equation (A.17) then becomes: ∂φw ∂ ∂ + (φw v̂f,z ) − ∂t ∂z ∂z  Kzz ∂hσf o,zz i µ ∂z  = 0, (A.18) where only the vectorial and tensorial components in the z-direction have been considered. The indexes z and zz indicate the z-components of the vectors and tensors in Equation (A.17). By an abuse of notation, they, as well as the “hi” brackets, were omitted and φw was denoted as φ in the following for simplicity. Appendix B Rate-Independent Fibre-Network Stress Here mathematical expressions, which can give good descriptions of the fibre network stress, σf 0 (discussed in Chapter 2), and the fibre network equilibrium stress, σe (discussed in Chapter 4), are suggested. Both stresses are rate-independent. As it was pointed out in the introduction (see Equation (1.8)), the fibre network stress depends on the entire compression history of the fibre mat. The same is valid for the fibre network equilibrium stress. However, here a simplification was made. Those stresses were assumed to be a function of ǫ, ǫmin and ǫmax , instead of taking into account the entire history of ǫ, i.e.1 σ ≡ σ(ǫ, ǫmin , ǫmax ). (B.1) where ǫmin and ǫmax are the onset unloading and the onset reloading modified strain, respectively. Therefore, this description could be divided into three parts, viz. first compression, unloading and reloading. There are some limitations with this approach, but it simplified the solving of the final equations. The numerical solution of the partial differential equations with hysteresis operators is a more complex issue (see e.g. Visintin, 1991; Bergqvist, 1994; Brokate and Sprekels, 1996) than the solution with the proposed Equation (B.1). The expressions had a specific pattern for the three loading situations. Several authors have measured the first compression of wet fibre networks 1 The modified strain can be readily converted to porosity, φ, according to φ = 1 − 65 1 eǫ . 66 APPENDIX B. RATE-INDEPENDENT FIBRE-NETWORK STRESS quasi-statically (Campbell, 1947; Han, 1969; Vomhoff and Schmidt, 1997). According to Campbell (1947), Ingmanson (1952), Han (1969), Gren and Ljungkvist (1983) and Lobosco (2000), beating does not have any significant influence on the compressibility behaviour of fibre networks. Campbell (1947) suggests a mathematical description of this behaviour according to: p σf c = N c/M , pr (B.2) where c is the concentration (c = ρf /eǫ ), σf c is the applied stress, pr is a reference pressure and M and N are fitting constants. This equation was employed in this study as there are several experimental data for different kind of pulps expressed in this form. The unloading behaviour was described with the relation: B− σun (ǫ, ǫmin ) = σf c (ǫmin )  ǫmin −ǫ0 ǫ−ǫ0 B−1 β , (B.3) where B=  ǫmin − ǫ0 ǫf c − ǫ 0 β , (B.4) and ǫf c , and σ are parameters dependent on ǫmin (see below). Equation B.3 is intrinsically non-linear, which requires a non-linear algorithm in order to estimate the parameters. Therefore β, ǫ0 and ǫf c , were estimated for experimental unloading curves using an iterative algorithm (Nelder-Mead simplex method (Lagarias et al., 1998)). From the results presented in Figure B.1, β was assumed constant and equal to 2.3. The two other parameters was considered dependent on ǫmin , viz. β = 2.3 ǫ0 = −0.055 + 0.95ǫmin ǫf c = 0.3 + 1.15ǫmin (B.5) (B.6) (B.7) 67 Figure B.1: Parameters as a function of the modified strain for several unloading experiments (high loads). Figure B.2: Experimental (unloading only) responses to the quasi-static pulses, and the results obtained with the mathematical expression. 68 APPENDIX B. RATE-INDEPENDENT FIBRE-NETWORK STRESS Figure B.3: The experimentally measured fibre network compressibility behaviour at low loads (Vomhoff and Schmidt, 1997). The results obtained with the mathematical expression are also presented for comparison. The fit between the mathematical description suggested for the unloading curves and the experimental results are shown in Figure B.2 and Figure B.3. In order to describe consecutive compressions, it is necessary to know not only the strain at which unloading started, but also the strain at which reloading starts. The following equation was used to describe reloading equilibrium curves starting at zero stress, σ = 0: 1/α ǫ σre,0 (ǫ, ǫmin ) = σf c (ǫmin ) e ǫ−ǫmin min −ǫf c − e−1/α 1 − e−1/α . (B.8) The parameter α, was chosen to be: α(ǫmin ) = −0.02 + 0.12ǫmin for high loads, where as for low loads, α showed to behave according to: α(ǫmin ) = −0.016 + 0.3ǫmin . For reloading curves starting at σ 6= 0, a weighted average of Equations (B.3) and (B.8) was proposed, viz. σre (ǫ, ǫmin , ǫre ) = σre,0 + ǫmin − ǫ (σun (ǫre ) − σre,0 (ǫre )) . ǫmin − ǫre (B.9) SOLIDS CONTENT [%] 69 STRESS [MPa] Figure B.4: Stress-strain behaviour for cyclic compression according to Equation (B.9). The sign of the compression is shown with an arrow. The first compression is closet to the arrow. Equation (B.9) could describe qualitatively the behaviour observed in Jones (1963), see Figure B.4 cf. Figure 1.5. The mathematical description approximated well the quasi-static curves in different experiments (Figure B.5). However, as the error is accumulated, some significant discrepancies was observed at the successive pulses. This accumulated error is not shown, as the experimental ǫmin and ǫmax were used in Equation (B.9). Figure B.5: The experimental response to consecutive completely unloaded quasi-static pulses. 70 APPENDIX B. RATE-INDEPENDENT FIBRE-NETWORK STRESS Appendix C Numerical Solution with FEM Equation (A.18) is markedly more easily solved if the z-coordinate is changed to a convective coordinate, ζ, according to (Jönsson and Jönsson, 1992; Mitchell and Johnson, 2001; Fung and Tong, 2001): dζ = ρ̃f (1 − φ (z, t)) dz w ⇒ ∂ζ ρ̃f = − (1 − φ)v̂f , ∂t w (C.1) where w is the suspension grammage. Applying this coordinate transformation and expanding ∂σf 0 /∂ζ by applying the chain rule, Equation (A.18) becomes: ∂φ (1 − φ)2 ρ̃2f ∂ − ∂t w2 ∂ζ  (1 − φ)K ∂σf 0 [φ] ∂φ µ ∂φ ∂ζ  = 0. (C.2) In order to obtain Equation (C.2) in a dimensionless form, one needs just to multiply by a time constant, for instance, the process time, tf inal , ∂ ∂φ − (1 − φ)2 ∂τ ∂φ  ∂φ 1 De(1 − φ) ∂ζ  = 0. (C.3) where τ = t/tf inal and De is a Deborah number1 : De = µ (w/ρ̃f (1 − φ))2 . tf inal K (∂σf 0 [φ]/∂φ) 1 (C.4) The Deborah number is called in reference to the Biblical prophet Deborah: She said the mountains flow according to the Lord’s time scale and not humans’. 71 72 APPENDIX C. NUMERICAL SOLUTION WITH FEM The process is flow controlled for high De and press controlled, otherwise. The most important advantage with this formulation is that the spatial domain is constant, Ω = [0, 1], yielding a rectangular space-time domain Q = Ω × I, where I = [0, tf inal ]. Equation (C.3) can be rewritten as: 1 ∂φ ∂ + 2 ∂ζ − (1 − φ) ∂τ  ∂σf 0 [φ] ∂φ 1 Ev (1 − φ) ∂φ ∂ζ  = 0, (C.5) by dividing the Deborah number into a viscous stress, Ev = µ(w/ρ̃f (1−φ))2 , tf inal K ∂σ [φ] f0 . The dimensionless forms of the initial and boundary conditions and ∂φ are straightforward. Substitutiong σf 0 [φ] for σf 0 (φ, φmini , φmax ), and then expressing Equation (C.5) in the weak form and integrating the second term by parts, one obtains: 1 1 ∂φ vdζ 2 0 − (1 − φ) ∂τ   Z 1 1 ∂σf 0 ∂φmini ∂σf 0 ∂φmaxi ∂v ∂σf 0 ∂φ + + + dζ = 0, ∂φ ∂ζ ∂φmini ∂ζ ∂φmaxi ∂ζ ∂ζ 0 Ev (1 − φ) (C.6) Z where v is a test function belonging to the infinite space, V , of bounded continuous functions and piecewise continuous derivatives on [0, 1] with v(0) = v(1). Partionining the interval (0, 1) in 0 = ζ0 < ζ1 < · · · < ζN < ζN +1 = 1, we can then introduce the basis functions 2 , ϕi , of the finite-dimensional subspace Vh ∈ V : ϕi =   0, ζ−ζi−1 , ζi −ζi−1   ζ−ζi+1 , ζi −ζi+1 if ζ ∈ / [ζi−1 , ζi+1 ] if ζ ∈ [ζi−1 , ζi ] (C.7) if ζ ∈ [ζi , ζi+1 ]. So, Vh is the space of continuous functions v that are piecewise linear with respect to the partitioning suggested such that v(0) = v(1) = 0. In this subspace, the solution can be written as a linear combination of the basis functions: 2 The are the so-called hat functions. 73 φn φmini,n φmaxi,n PN +1 = i=0 Ai (τ ) ϕi (ζ) PN +1 = Bi (τ ) ϕi (ζ) . Pi=0 N +1 = i=0 Ci (τ ) ϕi (ζ) Substituting Equations (C.8) and v = yields: X ∂Ai Z i ∂τ 0 1 PN +1 j=0 (C.8) ϕj (ζ) in Equation (C.6), it Z 1 1 ∂σf 0 ∂ϕi ∂ϕj 1 dζ 2 ϕi ϕj dζ + Ai − (1 − φ) 0 Ev (1 − φ) ∂φ ∂ζ ∂ζ Z 1 ∂σf 0 ∂ϕi ∂ϕj 1 dζ + Bi 0 Ev (1 − φ) ∂φmini ∂ζ ∂ζ Z 1 1 ∂σf 0 ∂ϕi ∂ϕj + Ci dζ, (C.9) 0 Ev (1 − φ) ∂φmaxi ∂ζ ∂ζ where the derivatives ∂σf 0 /∂φ, ∂σf 0 /∂φmini and ∂σf 0 /∂φmaxi were calculated numerically by finite diferences, and the integral was calculated according to Simpson’s rule. Equations (C.9) were then solved using a Runge-Kutta scheme. For the first compression, fewer elements were required by using the scheme described by Equation (16.26) in Eriksson et al. (1996).