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Water purification using polyelectrolyte modified cellulose fibers and filters to adsorb bacteriaOttenhall, Anna January 2017 (has links)
Clean water is necessary for human survival and there is a need for development of cheap and easy water purification techniques to use in emergency situations when there is no access to safe drinking water. Bacteria contaminated water can cause lethal diarrheal diseases and is globally the second most common cause of death among children less than five years of age. Bacteria adsorbing filter paper made from cellulose could be an environmentally and economically sustainable alternative for disposable water purification filters. This thesis investigates the possibility to use polyelectrolyte multilayer modified cellulose pulp fibers and filter papers to adsorb and remove bacteria from water. The bacterial removal efficiency of the modified materials has been tested both in suspension and through filtration. The surface modification provides the cellulose fibers with a positively charged surface that can attract and bind the negatively charged bacteria. The bacterial adsorption through electrostatic interactions makes it possible to remove bacteria, even when the pore size of the cellulose filters is larger than bacteria. Bacterial reduction tests shows that it is possible to remove over 99.9 % of the bacteria when filtering water through the modified materials. An increased amount of adsorbed cationic polyelectrolyte, polyvinylamine, resulted in an increased bacterial removal capacity. It has also been shown that the bacterial removal efficiency increases with an increased the amount of bacteria adsorbing materials in the filter. The modified materials have been compared with a commercial product and the filtration efficiency has shown to be greater for the polyelectrolyte-modified materials, under the test conditions used in this thesis. Tests with natural water samples shows that it is important to use a filtration mode to remove particles from the water in combination with the bacterial adsorption, as the particles interfere with the bacterial adsorption. / Säkert dricksvatten är nödvändigt för överlevnad och det finns ett stort behov av att utveckla nya billiga och enkla tekniker för att rena vatten i nödsituationer där det inte finns tillgång till detta. Dricksvatten förorenat av bakterier kan orsaka dödliga diarrésjukdomar och är globalt den näst vanligaste dödsorsaken bland barn under fem år. Denna avhandling undersöker möjligheten att använda cellulosafibrer och filterpapper, ytmodifierade med multilager av katjoniska polyelektrolyter, för att adsorbera och avlägsna bakterier från vatten. Bakterieradsorberande filterpapper tillverkat av cellulosa kan vara ett miljövänligt och ekonomiskt hållbart alternativ för vattenreningsfilter för engångsbruk. De modifierade materialens förmåga att adsorbera bakterier har testats både i suspension och via filtrering. Ytmodifieringen ger cellulosafibrerna en positivt laddad yta som kan attrahera och binda de negativt laddade bakterierna. Avlägsnandet av bakterier genom elektrostatisk interaktion gör det möjligt att ta bort bakterier, även när filtret har en porstorleken som är större än bakterierna. Bakteriereduktionstesterna visar att det är möjligt att avlägsna mer än 99,9 % av bakterierna vid filtrering genom de modifierade materialen. En ökad mängd adsorberad katjonisk polyelektrolyt, polyvinylamin, resulterade i en ökad bakterieavlägsningskapacitet. Det har också visat sig att effektivitet ökar väsentligt med ökad mängd bakterieadsorberande material i vattenfiltren. De ytmodifierade materialen har jämförts med en kommersiell produkt för vattenrening med goda resultat. Filtreringstesterna utförda i den här avhandlingen visar att de modifierade materialen tar bort mer bakterier än vad det kommersiella filtret inaktiverar. Tester med naturliga vattenprov visar att det är viktigt att använda filtrering för att avlägsna partiklar från vattnet för att uppnå en önskad bakterieadsorption, eftersom partiklarna påverkar bakterieadsorptionen och minskar effektiviteten hos de bakterieadsorberande materialen / <p>QC 20170328</p>
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Testing and Evaluation of Interfibre Joint Strength under Mixed-Mode LoadingMagnusson, Mikael S. January 2013 (has links)
The failure properties of interfibre bonds are the key for the build-up of strength in fibrous materials such as paper and paperboard. In order to tailor the properties of such materials by chemical or mechanical treatments and to learn how such modifications influence the properties at a microscopic level, direct measurement of individual fibre--fibre crosses are typically performed. However, the state of loading in the interfibre joint, in testing of individual fibre--fibre crosses, is in general very complex and a greater understanding for how to evaluate the mechanical properties of interfibre joints is desirable. In Paper A, a method for manufacturing multiple fibre--fibre cross specimens and a procedure for testing interfibre joints at different modes of loading is presented. The method is applied to investigate the strength of fibre-fibre crosses with different geometry and at two principally different modes of loading. Also, an investigation on the influence of drying pressure, the drying method as well as a comparison of pulp fibres from two different degrees of refining is presented. The force at rupture is scaled in terms of different geometric parameters; nominal overlap area, length and width of the joint region. It is shown that neither of the methods of scaling unambiguously reduced the coefficient of variation of the mean strength and that the force at rupture in a peeling type of loading was about 20% of the ones tested in the conventional shearing type of loading. In Paper B, a procedure for evaluating interfibre joint strength measurements in terms of resultant forces and moments at rupture is presented. The method is applied to investigate the state of loading in fibre-fibre crosses tested in two principally different modes of loading. It is shown that for a typical interfibre joint test, the modes of loading other than pure shear, cannot in general be neglected and is strongly dependent on the structural geometry of the fibre-fibre crosses. Also, the stress state in the interface centroid was estimated in order to quantify how the mode of loading influence the amount of normal stresses that develop in relation to the amount of shear stresses in the interfibre joint. / De brottmekaniska egenskaperna hos fiberfogar är nyckeln för uppbyggnaden av styrka hos fibrösa material såsom papper och kartong. För att effektivt skräddarsy sådana materials egenskaper genom kemisk eller mekanisk behandling och för att förstå hur sådana modifieringar påverkar egenskaperna på en mikroskopisk nivå är provning av individuella fiber-fiber-kors en allmänt använd metod. Belastningen i en fiberfog vid provning av individuella fiber-fiber kors är dock generellt mycket komplicerad och ytterligare kunskap om hur fiberfogars mekaniska egenskaper skall utvärderas är önskvärd. I Artikel A, presenteras en metod för samtidig tillverkning av flera fiber-fiber kors samt en metod för mekanisk provning av dessa med olika typer av belastning. Metoden tillämpades för att studera styrkan av fiber-fiber-kors med olika geometri och vid två olika lastfall. En undersökning av hur torktrycket, torkmetoden samt graden av malning inverkar på fogstyrkan presenteras. De uppmätta brottlasterna skalades med olika karakteristiska längder för fogen; nominell överlapparea samt fogens längd och bredd. Resultaten visade att ingendera av normaliseringsmetoderna reducerade variationskoefficienten (av medelvärdet av styrkan) samt att brottlasten för en globalt fläkande belastning var omkring 20 % av brottlasten för prov utförda med den konventionella skjuvande belastningen. I Artikel B, presenteras en metod för utvärdering av mätningar av styrkan hos fiberfogar med hänseende på kraft- och moment- resultanterna i gränsytan mellan fibrerna. Metoden används för att studera belastningsmoden hos fiber-fiber--kors provade i två principiellt olika lastfall. Resultaten visar att för ett typiskt fiberfogsprov av isolerade fiber-fiber-kors med långa fria fibersegment, så kan inte belastningsmoderna vid sidan av skjuvning försummas och att de är starkt beroende av fiber-fiber-korsets geometri. För att kunna jämföra fiberfogar av olika storlek och kvantifiera förhållandet mellan normal- och skjuvbelastningen i fogen skalades de resulterande krafterna och momenten med tvärsnittsstorheter baserade på en approximation av fogareans utformning. / <p>QC 20130125</p> / BiMaC Innovation
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The Influence of Xylan on Precipitation and Filtration Properties of Lignin : A Study in the Context of the LignoBoost Process / Inverkan av xylan på utfällning och filtrering av sulfatligninSchneider, Helen, Schneider, Lynn January 2016 (has links)
The LignoBoost process is a valuable supplement to the Kraft process. It can increase the pulp production rate of a Kraft mill and it enables lignin separation from black liquor with a high degree of purity. However, residual xylan in black liquor has been observed to increase filtration resistance of lignin during the LignoBoost process. In order to uncover underlying mechanisms, this thesis investigates the potential influence of xylan during lignin precipitation and filtration, which are the two main steps of the LignoBoost process. For this purpose experiments based on a model system were designed. Model liquors consisted of lignin and xylan as the only organic compounds and contained lower salt concentrations (4.2-5.9 wt%) compared to black liquor. Furthermore, reference liquors were prepared without xylan addition. Precipitation mechanisms were studied in the onset precipitation region (i.e. alkaline regime) by in-situ focused beam reflectance measurements (FBRM) during step-by-step acidic precipitation of the model liquor. It was found that the onset precipitation pH does not change with the presence of xylan as all liquors started precipitation around pH 9.15. The filtration process was investigated on model liquors that had been precipitated by fast acidification to acidic regimes (pH 6.5-2.87). The use of FBRM during acid precipitation of model liquors suggested that temperature had a significant influence on the chord length distribution (CLD) of the particles. In all filtration experiments, a decrease in CLD was observed when the temperature was changed from 80 °C to 25 °C. Moreover, this thermal instability of particles seemed to be higher when added xylan was present in the liquor. The investigation of the resulting filer cakes with HPLC showed that xylan was evenly distributed through the cake. Further findings on the influence of xylan were impeded due to variations in ionic strength in the model liquors. It was found that the effect of ionic strength on filtration properties and particle sizes overshadows the effect of xylan. Higher ionic strength was observed to yield a lower filtration resistance, a higher solidosty, larger particles and lower solid surface area, as investigated by filtration measurements, laser diffraction and BET analysis. Finally, xylan was fluorescently tagged (i.e. dyed) with Remazol Brilliant Blue R to investigate xylan position in the ligninxylan filer cake, using a confocal fluorescence microscope. However, due to the autofluorescence of lignin as well as low emission intensity of the synthesized dyed xylan, xylan could not been tracked within the lignin particle. Nevertheless, valuable insight was gained into the preparation of dyed xylan and the bond stability.
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Dissolution of cellulose in cold alkali followed by precipitation / Upplösning av cellulosa i kall alkali följt av fällningCurman, Johan January 2024 (has links)
Den vanligaste biopolymeren på jorden är cellulosa och det är ett viktigt material i utvecklingen mot ett hållbart samhälle. Cellulosan och dess egenskaper kan modifieras genom kemiska reaktioner. Reaktiviteten hos cellulosans OH-grupper är låg på grund av en tät kristallstruktur, men kan förbättras genom t.ex. svällning eller upplösning följt av utfällning. I det här projeket kommer cellulosa från tre källor, microkristallin cellulose (Avicel), lövedsmassa (björk och asp) och barrvedsmassa (tall och gran) att lösas upp i natriumhydroxid och sedan fällas ut med två olika syror och ett organiskt lösningsmedel. Syrorna som användes var ättiksyra och svavelsyra och det organiska lösningsmedlet var etanol. Därefter jämfördes utbytet av den utfällda cellulosan. Upplösning av Avicel 2.5 viktprocent gjordes i natriumhydroxid-lösning (10%) vid 5 oC och fälldes ut med de tre fällningsmedlen för att bestämma vilket som gav högst utbyte. Ättiksyra gav de bästa resultaten och användes i följande experiment på massa. Massornas löslighet var låg och den olösta massan var tvungen att separeras genom centrifugering innan utfällning av lösningen kunde göras. Hydrolys av massorna utfördes för att öka lösligheten och utfällningsutbytet. Upplösning av Avicel i kall natriumhydroxid och utfällning var framgångsrik. Utbytet vid utfällning med syror var högt. Det fanns ingen topp för maximalt utbyte när syrorna användes som antilösningsmedel, men etanol hade ett maximalt utbyte men lägre än för syrorna. Resultaten med massa var inte lika bra som för Avicel. De separerade olösta delarna var stora för båda massorna, men HW hade större del olöst material. Utbytet av fällning var lågt eftersom det fanns mindre cellulosa upplöst som kunde fällas ut. Hydrolysen av HW och SW var framgångsrik. Detta gjorde det möjligt att lösa upp och fälla ut mer cellulosa från massan. / Cellulose is the most abundant biopolymer on earth and is an important material in developing a sustainable society. The properties of cellulose can be modified by chemical treatment or reactions. The reactivity is fairly low in crystalline cellulose but can be increased if the dense crystal structure is disrupted by e.g. swelling or dissolution followed by precipitation. Dissolution of cellulose is difficult and this project aims to dissolve cellulose from three sources (Avicel and two sulfate pulps). One of the pulps had a tiny fraction of softwood and mostly birch and aspen, hardwood pulp (HW). The other of the pulps was made of pine and spruce, softwood pulp (SW). The raw materials were dissolved in sodium hydroxide (NaOH) and then precipitated with two acids and an anti-solvent (acetic acid (HOAc), sulfuric acid (H2SO4), or ethanol (EtOH)). The yields of the precipitated products were compared. Avicel 2.5 wt% was dissolved in NaOH solution (10%) at 5 oC and precipitated with the two acids and EtOH to determine the highest yield. HOAc gave the best results and was used in the following experiments on pulp. The solubility of pulps was low and the undissolved pulp had to be separated by centrifugation before precipitation of the solution. Hydrolyzation of the pulps was performed to increase the solubility and the precipitation yield. Dissolution of Avicel in cold NaOH and precipitation was successful. The yield at precipitation with acids was high. There was no peak of maximal yield when the acids were used to precipitate, but EtOH had a maximal yield but lower than for the acids. The results with pulp were not as good as for Avicel. The separated undissolved parts were big for both pulps, but HW was the worst. The yield of precipitation was low as there was less cellulose dissolved that could be precipitated. The hydrolysis of HW and SW was successful. This made it possible to dissolve and precipitate more cellulose from the pulp.
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Modification of Paper into Conductive Substrate for Electronic Functions : Deposition, Characterization and DemonstrationMontibon, Elson January 2011 (has links)
The thesis investigates the modification of paper into an ion- and electron-conductive material, and as a renewable material for electronic device. The study stretches from investigating the interaction between the cellulosic materials and the conducting polymer to demonstrating the performance of the conductive paper by printing the electronic structure on the surface of the conductive paper. Conducting materials such as conducting polymer, ionic liquids, and multi-wall carbon nanotubes were deposited into the fiber networks. In order to investigate the interaction between the conducting polymer and cellulosic material, the adsorption of the conducting polymer poly(3,4-ethylenedioxythiophene): poly(4-styrene sulfonate) (PEDOT:PSS) onto microcrystalline cellulose (MCC) was performed. Electroconductive papers were produced via dip coating and rod coating, and characterized. The Scanning Electron Microscopy (SEM) / Energy Dispersive Spectroscopy (EDS) images showed that the conducting polymer was deposited in the fiber and in fiber-fiber contact areas. The X-ray Photoelectron Spectroscopy (XPS) analysis of dip-coated paper samples showed PEDOT enrichment on the surface. The effects of fiber beating and paper formation, addition of organic solvents and pigments (TiO2, MWCNT), and calendering were investigated. Ionic paper was produced by depositing an ionic liquid into the commercial base paper. The dependence to temperature and relative humidity of the ionic conductivity was also investigated. In order to reduce the roughness and improve its printability, the ionic paper was surface-sized using different coating rods. The bulk resistance increased with increasing surface sizing. The electrochemical performance of the ionic paper was confirmed by printing PEDOT:PSS on the surface. There was change in color of the polymer when a voltage was applied. It was demonstrated that the ionic paper is a good ionic conductor that can be used as component for a more compact electronic device construction. Conductive paper has a great potential to be a flexible substrate on which an electronic structure can be constructed. The conduction process in the modified paper is due to the density of charge carriers (ions and electrons), and their short range mobility in the material. The charge carrying is believed to be heterogeneous, involving many charged species as the paper material is chemically heterogeneous. / <p>Fel ordningsnummer (2010:28) är angivet på omslaget av fulltextfilen.</p> / Printed Polymer Electronics on Paper
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Correlations between fibre properties and paper properties / Inverkan av fiberegenskaper på pappersegenskaperJohansson, Anna January 2011 (has links)
The understanding of what properties the fibre should have in order to give the right end-product paper properties, along with the type of processing, is a subject for a lot of research and development. Today the ability to measure fibre properties on-line is widely used for pulps. It is often necessary to measure many properties and variables in a process. The data collected is therefore almost always multivariate. It is very hard to analyse process data due to a lot of noises. Correlations between fibre and paper properties are hard to find, but this does not mean that correlations do not exist. Fibre properties, measured by the pulp analyser PulpEye, were investigated and correlations to paper properties were studied. The work was divided into three different studies. Study 1 was an investigation of historical process data, in study 2 pulp samples from the production was analysed and study 3 was a refiner setting trial, were different refiner segments, flows through refiners and intensities were studied. Both the group-plots and MVDA’s based on the historical process data in study 1, showed that the Scott Bond was increased with increased amount of kinks and curl for the unbleached pine pulp (softwood pulp). Coarseness measurements, made in the study of historical data, indicated that the coarseness was varying in such a large extent that it was believable that it had effects in the papermaking process. Another interesting fibre property, investigated in the refiner setting trial, was crill. The amount of crill is said to have strong correlation to paper strength. The analysis showed that the incoming pulp had different amount of crill and that the amount of crill after the refiners also was varying for the different samples. The development of crill at different kappa numbers and for pulps refined with different segments and refiner strategies should be further investigated. In this work it has been difficult to find correlations between fibre properties and paper properties in the refiner setting trial. This could have been due to small variations of the different parameters. This work showed that the normal production can be handled very well and variations are rather small. It can be seen though, that problems do appear when parameters are deviating from the normal case. An efficient way to work is to do measurements when the incoming pulp parameters are deviating. It should also be more investigated how the most common deviating pulp parameters should be handled in the refining process and at the board machine. The communication between the pulp production and the board machines is recommended to be further developed, especially when the pulp production have disturbances that can be affecting the refining and further the board production.
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Experimentell studie av kinetiken vid peroxidblekning av pappersmassa / Experimental study of the kinetics of peroxide bleaching of pulpAlberth, Lena January 2011 (has links)
Vid TCF-blekning av pappersmassa är väteperoxid en av huvudkemikalierna. I dagsläget är det svårt att styra blekstegen. Operatörerna måste ta hänsyn till produktionstakt, blekbarheten på massan varierar och fördröjning i trender. PO-bleksteget på Södra Cell Värö har en ungefärlig uppehållstid på 4 timmar och det är först efter så lång tid som man kan se hur en förändring verkligen slår igenom. Överblekning kostar väldigt mycket i kemikalieåtgång och det är därför efterfrågat ett sätt att kunna styra och optimera peroxidblekningssteget. Ett delsteg för detta var att simulera steget experimentellt och från dessa data ta fram en matematisk modell som förutsäger ljusheten. Massa togs ut från fabriken och blektes på labb under kontrollerade förhållanden med satsningar liknande verkliga i fabriken. De parametrar som studerades på labb var temperatur, väteperoxidsats och alkalisats då de påverkar ljusheten mest. En modell för ljushetsökning anpassades sedan till framtagna blekdata. Ekvationer för förbrukning av väteperoxid och hydroxidjoner togs också fram från analyser på blekfiltraten. Med ingående parametrar som startkoncentrationer av kemikalier, temperatur och massakoncentration förutsäger modellen ljusheten väl vid verifiering jämfört med fabrik och verifieringsblekning på labb. Modellen ligger något högre jämfört med fabrik men det var förväntat då förhållanden på labb ger ett renare system. För de parametrar som studerats anpassade sig modellen bra till förändringar och vid verifiering på labb syntes att modellen även svarade med en korrekt ljushet vid förändring i massakoncentration och ingående ljushet i steget. / During TCF-bleaching of pulp, hydrogen peroxide is one of the main chemicals. Today it is difficult to control the bleaching stages. The process operators must consider the production rate, changes in the bleach ability of the pulp and delays of trends in the control system. The PO-bleaching stage at Södra Cell Värö has approximately a retention time of four(4) hours and it is first after that, a result of a change in the process can be seen. Overbleaching is expensive due to high chemical need and therefore it is requested to find a way to control and optimize the peroxide bleaching stage. One way to do this was too simulate the stage experimentally and from those kinetic data accomplish a mathematical model that predicts the brightness increase of the pulp. Pulp from the mill was bleached at the laboratory under controlled conditions and with charges similar to what are used in the plant. The parameters that were studied at lab were temperature, hydrogen peroxide charge and alkali charge as they affect the brightness mostly. A model for brightness increase was adjusted to the achieved bleaching data. Equations for consumption of hydrogen peroxide and hydroxide anions were also developed from analyses of the bleaching filtrates. From the start parameters, as concentration of the chemicals, temperature and pulp concentration, the model predicts the brightness well according to verification of data from mill and verification bleaching at laboratory. The value from the model is somewhat higher compared to that of the plant but that was expected due to the fact that the conditions at lab give a cleaner system. For those parameters that were studied, the model did adjust well to changes made and according to the verification at lab it was seen that the model also worked for changes in pulp concentration and different brightness at the beginning of the stage.
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Predictions of Pulp and Paper Properties Based on Fiber Morphology / Prediktering av massa- och pappersegenskaper baserat på fibermorfologiSundblad, Sara January 2015 (has links)
The aim is to investigate models that predicts the potential of pulp and evaluate the relevance of the zero-span tensile index within these. Two chemical pulps made from softwood and eucalyptus were refined in a Voith-beater with different energy input in order to study the change of fiber morphology signals and other pulp and paper properties. Chemical, THP pulp from Södra Värö is also used as an initial analysis for morphological connections to Zero-span tensile index. The L&W Fiber Tester Plus is used in order to study the pulps fiber morphology and Pulmac 2000 for zero span. Handsheets are made for mechanical tests such as tensile properties, ZD-strength and optical properties. Many of the given signals change according to clear patterns with increasing refining energy. Using least square methods, formulas describing the development with high adaptation could be formulated. Many of the measured aspects changes over already known patterns. These are then applied in the models. Three possible models is tested: linear regression, Shear-Lag and Page. Of the three, only the two first ones where able to produce reliable models, whereas the third required data that was difficult to acquire at the same time as the adaptation was very low. The only model that use exclusively morphology data is linear regression.
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Pyrolysis and thermogravimetric analysis of wood and its components / Pyrolys och termogravimetrisk analys av trä och dess komponenterPascoa Dos Santos, Magaia January 2014 (has links)
The present study investigates the thermochemical conversion of spruce wood and its extracted components by thermogravimetric analysis. The extracted components are two pulps, three xylan-lignin samples and one lignin sample; they were produced by the kraft cooking method with different cooking times. The study involves characterization of the biomass through proximate analysis and pyrolysis. A qualitative comparison between the thermal behaviours of the extracted components and wood is also performed. The study showed that the thermal behaviour of the biomass was highly influenced by the content of cellulose and lignin in the samples. Compounds rich in cellulose produced large quantities of volatiles and had a higher rate of pyrolysis compared to compounds rich in lignin, which produced more char and had a slower rate of pyrolysis. It was also shown that, the amount of char is not solely depending on the amount of the lignin; the structure of the compound also plays a role. On the other hand, the original wood sample showed some deviations regarding the trends in volatile and char production and these deviations were attributed to component interactions. Both cellulose and lignin rich compounds had an increase in thermal stability with increasing cooking time. For the pulps the increase in thermal stability is believed to be caused by increase in crystallinity, while for the lignin rich samples is believed to be caused by the increase in lignin content and structural changes in the compounds. The results also show that although changes are introduced in the cooking process, the extracted component still retain properties exhibited by the source biomass.
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Modelling of heat and moisture transport in a corrugated board stack / Modellering av värme- och fukttransport i en wellpappstackXynou, Marianna January 2014 (has links)
The corrugated board is considered as the second most used packaging material and the world’s environmentally acceptable solution for packaging, with wide range of applications. After the manufacturing process, the corrugated board is cut into sheets and stored in a stack until optimum moisture content has been reached in order to avoid undesired properties. However, due to complex and various structures, it is difficult to estimate the appropriate time so to achieve the acceptable moisture level of the corrugated board stack. So a homogenized model of the stack has to be created which will have the same average properties as the real stack. In order to achieve this goal the behavior of a smaller part of the stack, the unit cell, is investigated. In the second step a homogenized model is created with the average transport of mass and heat. At the end, the unit cell is scaled up. In this master thesis, only the first and the second steps were simulated. This was achieved by creating a 3-D mathematical model using finite element method and simulating its properties in COMSOL Multiphysics®. Four mathematical models were used in the description of the 3-D model: the heat transfer, the moisture transfer, the vapour concentration and the gas pressure. Moreover, by applying the gradient in one direction in each case, the behavior of the detailed unit cell was investigated. Finally different simplified geometries were created and investigated so to approach a homogenized model which described better the average properties of the detailed model. By comparing the results of the models, it was concluded that the homogenized models 2 and 3 approached the values of the second detailed model but only inside of the unit cell. However, the deviation was not negligible and further investigation is required in order to find a new homogenized method.
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