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A numerical method for the evaluation of non-linear transient moisture flow in cellulosic materials

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A numerical method for the evaluation of non-linear transient moisture flow in cellulosic materials
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  INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING  Int. J. Numer. Meth. Engng  2006;  66 :1859–1883Published online 13 January 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1597 A numerical method for the evaluation of non-linear transientmoisture flow in cellulosic materials U. Nyman 1 , ∗ , † , P. J. Gustafsson 1 , B. Johannesson 2 and R. Hägglund 3 1  Division of Structural Mechanics ,  Lund University ,  S-22100 Lund  ,  Sweden 2  Division of Building Materials ,  Lund University ,  Sweden 3 SCA Packaging Research ,  SCA Packaging ,  Sundsvall ,  Sweden SUMMARYA numerical method for the transient moisture flow in porous cellulosic materials like paper and woodis presented. The derivation of the model is based on mass conservation for a mixture containinga vapour phase and an adsorbed water phase embedded in a porous solid material. The principleof virtual moisture concentrations in conjunction with a consistent linearization procedure is used toproduce the iterative finite element equations. A monolithic solution strategy is chosen in order tosolve the coupled non-symmetric equation system.A model for the development of higher order sorption hysteresis is also developed. The modelis capable of describing cyclic hardening as well as cyclic softening of the equilibrium water con-centration. The model is verified by comparison with the measured response to natural variations intemperature and humidity. A close agreement of the simulated results to measured data is found.Copyright    2005 John Wiley & Sons, Ltd. KEY WORDS : mixture model; multiphase model; moisture transport; sorption hysteresis; paper material;virtual moisture concentration 1. INTRODUCTIONMoisture flow in porous materials is encountered in a number of engineering applications andin a wide variety of materials. Examples of porous materials are concrete, soil, wood andfibre networks such as textiles and paper. Traditionally, moisture flow, among other diffusionalprocesses, is modelled by using Fick’s law, where the rate of particle motion is governedby the concentration gradient. This relation constitutes a constitutive dependence between the ∗ Correspondence to: U. Nyman, Division of Structural Mechanics, Lund University, P.O. Box 118, S-22100Lund, Sweden. † E-mail: ulf.nyman@byggmek.lth.seContract / grant sponsor: The Bo Rydin Foundation for Scientific Research  Received 28 April 2004 Revised 31 January 2005 Copyright    2005 John Wiley & Sons, Ltd.  Accepted 20 October 2005  1860  U. NYMAN  ET AL . diffusion velocity and the concentration gradient. By using Fick’s law, a simplified continuumrepresentation of the physical medium is adopted. However, moisture flow in a porous materialmay be a very complex course of events, e.g. phase changes, sorption of vapour in the bulksolid and capillary suction, i.e. flow of a liquid water phase. This leads to the use of amore extensive and detailed modelling of the natural events, such as the use of multiphasemodelling of matter. In a multiphase model, the state variables describing the system areaccounted for on a microscale level of the material. This means that the complete response isachieved for the natural events. Often, this is referred to as mixture models and can involvestate variables such as stress, strain, velocity, solute concentrations and temperature. The keyidea in the modelling of mixtures is that the heterogeneous material is apparently smoothened,so that at a generic point all the constituents are coexisting. The choice of such a detailedmodelling incorporating multiple state variables implies that the response of and interactionbetween different constituents in the mixture are logically achieved, albeit at the expense of the complexity of the model.Multiphase modelling of heat and mass transport problems in materials has been treated byseveral authors. In References  [ 1–3 ]  the fundamental theories of thermodynamics and mixtureproblems are treated in terms of continuum conservation equations. In Reference  [ 4 ]  a com-prehensive presentation of transport processes in concrete is found. Coupled heat and moisturetransfer in building walls is dealt with in Reference  [ 5 ] . In Reference  [ 6 ]  a multiphase transportmodel for drying wood is analysed. Modelling of heat and moisture transfer in multilayer wallconstructions can be found in Reference  [ 7 ] . In Reference  [ 8 ]  the effects of temperature, stressand damage (matrix cracks) are considered in order to form a moisture diffusion model forstress-loaded polymer matrix composites.The moisture concentration level has an extremely large influence on the strength of paper.Moreover, in the manufacturing processes involved in papermaking or in the converting processof corrugated board, the level of moisture concentration is crucial in order to achieve the desiredproperties of the material. In view of this, considerable efforts have been made within the fieldof modelling the moisture transport process in paper. In References  [ 9,10 ]  effective moisturediffusivities were measured for paper. These works presented material parameters relying onFickian transport within the material with concentration independent diffusivity. This is onlya valid approximation in the relative humidity range of about 0–60%, after which a strongnon-linear dependency is observed  [ 11 ] .Five different transport mechanisms may be postulated for paper (porous materials)  [ 9 ] : gasdiffusion (vapour phase in the pores), Knudsen diffusion (vapour phase in pores with a diameterlower than 100Å), surface diffusion (adsorbed phase at the surface of the fibres), bulk–soliddiffusion (adsorbed phase within the fibres) and capillary transport (condensed phase in thepores). At low or medium moisture concentrations, i.e. 0–0.2 weight fraction water per drymaterial, gas diffusion is the predominant mode of transport.A step toward a two phase physical model is suggested in Reference  [ 12 ] , in which themoisture level both in the pores and in the fibres is considered. However, the sorption processin the fibres was considered not to be extended in time, which is the case in Reference  [ 13 ] .The authors of Reference  [ 13 ]  concluded that the transport velocity within the fibre is slowcompared to the velocity in the pores. They developed a physical model, however, where themoisture movement within the fibres is included. Several works based on similar equations arepresented in References  [ 14–16 ] . In Reference  [ 17 ]  an optimization procedure is used in orderto fit the transport model to a number of paper materials. Copyright    2005 John Wiley & Sons, Ltd.  Int. J. Numer. Meth. Engng  2006;  66 :1859–1883  TRANSIENT MOISTURE FLOW IN CELLULOSIC MATERIALS  1861At low levels of relative humidity, the process of sorption in the fibres is addressed tomolecular bonding (hydrogen bonding) of hydrogen atoms to the hydroxyl groups, or OH sites,into the cellulose molecule,  ( C 6 H 10 O 5 ) n . At an increased level of relative humidity, sorptionis addressed to multilayer adsorption, as free water molecules attach to already fixed watermolecules by hydrogen bonds. In addition, capillary condensation might be present at highlevels of relative humidity.This work focuses on moisture transport and sorption of water in paper materials. The primarypurpose is to form a finite element model suitable for one-dimensional modelling based on atwo phase representation of flow in the material. An extension to the multidimensional case isstraightforward. In addition, the sorption hysteresis effect in the fibres is accounted for, and asorption law for higher order processes is developed. The coupled equation system is solved forin a monolithic iteration matrix format rather than a staggered solution scheme. The advantageof this is that no restriction of time step length is produced by staggering steps. Nevertheless,the monolithic iteration format produces a non-symmetric stiffness matrix. In several numericalexamples and by comparison with test results it is shown that modelling of moisture flow andsorption under cyclic variation in ambient moisture levels can be adequately performed.2. MIXTURE BALANCE EQUATIONSIn a Lagrangian frame, the current position,  x i , of a particle  i  is a function of the srcinalposition,  X i , and time x i  =  ( X i ,t)  (1)where    denotes the motion of the particle and  X i  is termed the reference frame.The conservation of mass for an open volume,   , on rate form is given by  t      i  d V   +      i ˙ x i  · n d A −    m i  d V   = 0 , i = 1 ..n  (2)where index  i  refers to the  i th constituent in a mixture of   n  constituents, the last term isthe rate of mass exchange from the other constituents and  ˙ x i  · n  is the normal time derivativeof the particle position  x i . In this case, the mixture is considered to consist of a gas phase(vapour), a liquid phase (water) and a solid phase (fibres). The individual mass densities foreach phase are denoted   g ,   l , and   s , respectively, defined over a unit volume element in   .The corresponding balance equations can be written as  t      g  d V   +      g ˙ x g  · n d A −    m g  d V   = 0  t      l  d V   +      l ˙ x l  · n d A −    m l  d V   = 0  t      s  d V   +      s ˙ x s  · n d A −    m s  d V   = 0(3) Copyright    2005 John Wiley & Sons, Ltd.  Int. J. Numer. Meth. Engng  2006;  66 :1859–1883  1862  U. NYMAN  ET AL . If there is no motion of the solid, or if no mass exchange occurs from other phases to or fromthe solid, the third equation in (4) can be omitted, resulting in  t      g  d V   +      g ˙ x g  · n d A −    m g  d V   = 0  t      l  d V   +      l ˙ x l  · n d A −    m l  d V   = 0(4)By defining the density of the mixture as  m =  g  +  l  +  s  (5)the mass weighted average velocity of the mixture is given by ˙ x m =  g ˙ x g  +  l ˙ x l  +  s ˙ x s  m (6)It might be advantageous to define the velocity of the different constituents in terms of relative velocity. This is justified in order to achieve descriptions of flow of material that isframe indifferent, a restriction which must be followed when constitutive laws are formed forthe material behaviour. The relative velocity of the  i th constituent is v i  = ˙ x i  − ˙ x m  (7)By using (7) and applying the divergence theorem, (4) can be written in the form    ( ˙  g  + div (  g v g ) +  g  div ( ˙ x m ) + ˙ x m  · grad (  g ) − m g ) d V   = 0    ( ˙  l  + div (  l v l ) +  l  div ( ˙ x m ) + ˙ x m  · grad (  l ) − m l ) d V   = 0(8)where the dot on   g  and   l  represents the spatial time derivative. The third and fourth (con-vective) terms in (8) influence the mass flow only when the average velocity in the mixture isconsiderable, e.g. when capillary flow is present. In this case, spatial mass transport is assumedonly to take place in the gas phase by vapour diffusion. Thus, since  ˙ x s  and  ˙ x l  are assumed tobe zero and   g >  m , it can be concluded from (6) that  ˙ x m  is small, which implies that  v g ≈ ˙ x g .Yet, mass exchange is considered to take place in the form of condensation and evaporationof water molecules, which means m g = − m l = 0 (9)Equation (8) now reduces to    ( ˙  g  + div (  g v g ) − m g ) d V   = 0    ( ˙  l  − m l ) d V   = 0(10) Copyright    2005 John Wiley & Sons, Ltd.  Int. J. Numer. Meth. Engng  2006;  66 :1859–1883  TRANSIENT MOISTURE FLOW IN CELLULOSIC MATERIALS  1863In a porous material, i.e. consisting of both bulk material and void spaces, the porosity isdefined as  v =    v  d V      d V  (11)in which   v  is the volume occupied by the voids in   . By using   v , the mass density of thegas phase, for example, can be defined over a void unit volume element instead of over theentire volume element as  v =  g  v (12)and (10) can be written as    (  v  ˙  v  + div (  v  v v g ) − m g ) d V   = 0    ( ˙  l  − m l ) d V   = 0(13)since   v  is independent of time in a non-deformable body. Note that this is an approximationwhich might not be valid if the bulk material undergoes considerable swelling. The reason forrewriting (10) using (12) is that boundary conditions in the form of ambient mass density canbe applied directly on (13). It is convenient to express the mass flow of gas relative to themixture as  j g =  v  v v g  (14)It is noted that any constitutive relation applied on (14) will result in material parametersreferred either to values of    v , or gradient thereof, or   g , depending on the choice of statevariable. The mass concentration of water in the liquid phase, which is assumed to take placesolely within the fibres, can be related to the water mass density by c l =  l  m (15)Assuming that (13) is valid for all parts of    , the strong form of (13) is  v  ˙  v  + div (  j g ) − m g = 0 ,  x g ∈  v  ⊂   , t> 0 ˙  m  c l  − m l = 0 ,  x l , x s   v ,  x l , x s ∈   t> 0  j g  · n −  j   = 0 ,  x g ∈   v  ⊂    , t> 0  v ( x g , 0 ) =  0 ( x g ), c l ( x l , 0 ) = c 0 ( x l ),  x g ∈  v ,  x l   v ,  x l ∈  , t   = 0(16)In (16) the boundary conditions and initial conditions are added to the balance equations. It isnoted that, inherently, for a porous material the solid phase forms the main part of the mixturedensity. Thus, in (16) the rate of change of    m  is very small, i.e.  ˙  m  c l  ≈   m  ˙ c l . In orderto completely describe the system, (16) must also be supplemented by the proper constitutiverelations, which is done in a following section. Further on, the indices denoting vapour phase Copyright    2005 John Wiley & Sons, Ltd.  Int. J. Numer. Meth. Engng  2006;  66 :1859–1883
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