DEVELOPING MIXTURE RULES FOR NON-CONSERVATIVE PROPERTIES FOR PULP SUSPENSIONS by Pin Wen (Wendy) Tsai B.A.Sc., The University of British Columbia, 2003 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (CHEMICAL AND BIOLOGICAL ENGINEERING) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) April 2016 Β© Pin Wen (Wendy) Tsai, 2016 ii Abstract Nowadays new technologies emerge constantly and people continuously strive to meet challenges. The Pulp and Paper industry has been faced with many changes in recent years. One of which is to diversify the fiber baskets to produce a wide range of products. To help papermakers to accommodate this transition from a single pulp component to a multi-component furnish used in their process, this paper first puts effort into developing a sound and effective methodology to characterize mixture rules that predict properties such as tensile strength and pulp freeness. Using an expansion of a higher order Taylor series as the backbone of model development and removing model parameters based on the limitation of the separately refined system and statistical analysis, the tensile strength and pulp freeness models give predictions close to the observed measurements within 10% variance. Furthermore, two methods, one being the minimization approach using least squares, and the other being the one variable approach, when granting more emphasis on one particular mixture parameter than the other is preferred, are established to determine the operating conditions required to satisfy multiple target properties. Lastly, a graphical user interface, built on the defined mixture models, is also constructed to make recommendations of the optimized condition that can be applied to generate a mixture to achieve both target properties at minimum cost. iii Preface This dissertation is submitted in partial fulfillment of the requirements for a Master Degree of Applied Science at the University of British Columbia (UBC). All of the work presented hereafter was conducted under the supervision of Dr. Mark Martinez (UBC) at the laboratory of Canfor Pulp Innovation, located in Burnaby, British Columbia. I was actively engaged in researching, data analyzing and writing this thesis from September 2013 to March 2016. The thesis has been produced solely by the author, Pin Wen Tsai, yet some of the text is based on the research of others, and I have put my best effort forward to provide references to these sources. The research work conducted presents an effective methodology that can be of use to develop mixture rules in predicting mixture properties and means to establish operating conditions required to generate a pulp mixture with specific mixture properties for a separate refining system. iv Table of Contents Abstract .......................................................................................................................... ii Preface .......................................................................................................................... iii Table of Contents ......................................................................................................... iv List of Tables ................................................................................................................ vi List of Figures ............................................................................................................. vii Acknowledgements ...................................................................................................... ix 1 Introduction ............................................................................................................. 1 2 Background ............................................................................................................. 4 2.1 Industrial Motivation ........................................................................................ 4 2.2 Literature Review ............................................................................................. 6 2.2.1 Refining Mechanism .............................................................................. 6 2.2.2 Refining Theory ...................................................................................... 8 2.2.3 The Effects of Refining on Fiber Characteristics ................................ 9 2.2.4 The Effects of Refining on Pulp and Paper Properties ..................... 11 2.2.5 Refining Strategies for Mixtures ......................................................... 13 2.2.6 Mixtures of Different Pulps .................................................................. 14 2.2.7 Model Development ............................................................................. 19 3 Methods and Materials ......................................................................................... 25 3.1 Pulps ............................................................................................................... 25 3.2 Pulp and Paper Testing ................................................................................. 29 4 Tensile Strength Model ........................................................................................ 31 4.1 Case I .............................................................................................................. 31 v 4.2 Case II ............................................................................................................. 37 5 Pulp Freeness Model ............................................................................................ 41 5.1 Case I .............................................................................................................. 41 5.2 Case II ............................................................................................................. 45 6 Applications and Analytical Approach ............................................................... 48 6.1 Case I .............................................................................................................. 49 6.2 Case II ............................................................................................................. 50 7 Methods for Predicting Operating Conditions for Target Properties ............... 52 7.1 Method I: Minimization .................................................................................. 53 7.2 Method II: One Variable Problem .................................................................. 54 7.3 Comparison of Method I and II ...................................................................... 55 7.4 Graphical User Interface (GUI) ...................................................................... 59 8 Conclusions and Recommendations .................................................................. 61 8.1 Conclusions .................................................................................................... 61 8.2 Recommendations ......................................................................................... 62 References ................................................................................................................... 63 Appendix A Derivation of Almin and de Ruvoβs Model......................................... 66 vi List of Tables Table 3.1 Refining conditions for SW1 and HW pulps ..................................................... 27 Table 3.2 Case I mixture experiments ............................................................................. 28 Table 3.3 Case II mixture experiments ............................................................................ 29 Table 4.1 Case I - tensile strength model coefficient elimination ..................................... 32 Table 4.2 Case I - coefficients and model statistics for tensile strength model ............... 33 Table 4.3 Repeatability ratios from the study and TAPPI ................................................ 36 Table 4.4 Case II - tensile strength model coefficient elimination .................................... 38 Table 4.5 Case II - coefficients and model statistics for tensile strength model .............. 38 Table 5.1 Case I - coefficients and model statistics for CSF model ................................ 42 Table 5.2 Case II - coefficients and model statistics for CSF model ............................... 45 vii List of Figures Figure 2.1 Co-refining ........................................................................................................ 5 Figure 2.2 Separate refining .............................................................................................. 5 Figure 2.3 Commercial refiner (GL&V, 2014) .................................................................... 7 Figure 2.4 Photo of refiner plate ........................................................................................ 7 Figure 2.5 Refining mechanism (Hubbe) ........................................................................... 8 Figure 2.6 Swelling of fibers ............................................................................................ 10 Figure 2.7 Fibrillated fiber ................................................................................................ 10 Figure 2.8 Refining curve ................................................................................................ 12 Figure 2.9 Co-refining system ......................................................................................... 13 Figure 2.10 Separate refining system.............................................................................. 13 Figure 2.11 Mixture property prediction ........................................................................... 16 Figure 3.1 Refining Equipment ........................................................................................ 25 Figure 3.2 Refiner plate for SW1 ..................................................................................... 26 Figure 3.3 Refiner plate for HW ....................................................................................... 26 Figure 3.4 Schematics of the refiner loop ........................................................................ 26 Figure 3.5 CSF tester ...................................................................................................... 30 Figure 3.6 L&W tensile strength tester ............................................................................ 30 Figure 4.1 Case I schematics .......................................................................................... 31 Figure 4.2 Case I tensile - residual plot ........................................................................... 34 Figure 4.3 Case I tensile - normality plot ......................................................................... 35 Figure 4.4 Tensile strength prediction with 90% confidence interval ............................... 35 Figure 4.5 Case I - predicted vs. measured tensile strength ........................................... 36 viii Figure 4.6 Case II schematics ......................................................................................... 37 Figure 4.7 Case II tensile β residual plot ......................................................................... 39 Figure 4.8 Case II tensile β normality plot ....................................................................... 39 Figure 4.9 Case II β predicted vs. measured tensile strength ......................................... 40 Figure 5.1 CSF vs. refining energy .................................................................................. 41 Figure 5.2 Case I CSF β residual plot ............................................................................. 43 Figure 5.3 Case I CSF β normality plot ........................................................................... 43 Figure 5.4 CSF prediction with 90% confidence interval ................................................. 44 Figure 5.5 Case I β predicted vs. measured CSF ........................................................... 44 Figure 5.6 Case II CSF β residual plot ............................................................................ 46 Figure 5.7 Case II CSF β normality plot .......................................................................... 46 Figure 5.8 Case II β predicted vs. measured CSF .......................................................... 47 Figure 6.1 Tensile strength vs. refining energy ............................................................... 50 Figure 6.2 Tensile strength vs. refining energy at various SW1 ratios ............................ 51 Figure 7.1 Inverse problem layout ................................................................................... 52 Figure 7.2 Schematic algorithm of Method I .................................................................... 53 Figure 7.3 Schematic algorithm of Method II ................................................................... 55 Figure 7.4 3D volumization plot ....................................................................................... 56 Figure 7.5 SW1 and HW energy combinations at 0.45 SW ratio for Method I ................. 57 Figure 7.6 Method I solutions .......................................................................................... 57 Figure 7.7 SW1 and HW energy combinations at 0.45 SW ratio for Method II ................ 58 Figure 7.8 Method II solutions ......................................................................................... 59 Figure 7.9 GUI display ..................................................................................................... 60 ix Acknowledgements This thesis would not have been possible without the help and support of several individuals who in one way or another contributed to the completion and success of this research work. The author would like to express her deepest gratitude to Dr. Mark Martinez for his mentorship and guidance to help with the framing of this research. Sincere appreciation to Dr. Richard Kerekes, Dr. Frank Meltzer, Dr. Paul Watson and Dr. Paul Bicho for their insightful input and committee members for their valuable feedback to further enhance this paper. She would also like to take the opportunity to gratefully acknowledge Canadian Forest Products for providing financial support for her master study and the University of British Columbia for fostering her academic mindset. Moreover, the author wishes to extend appreciation to all faculties, staff and all of her peers at UBC for their helpfulness. Special thanks to the team at Canfor Pulp Innovation (Patricia McBeath, Ranbir Heer, Alice Obermajer, Daniel Fromm and Onyinyechukwu Ofulue) for their assistance with the experiments. Last but not least, the author is extremely thankful to have her loved ones by her side and their unconditional support and encouragement throughout the years. 1 1 Introduction The problem with mixtures was studied somewhat extensively thirty years ago. Since then, the activity in this area has been rather sparse, due to the lack of advanced knowledge and available resources to conduct comprehensive studies. Moreover, it has become common practice to use a mixed furnish to produce products that cater to a broad spectrum of applications in the Pulp and Paper industry. Hence there existed an urgency to investigate the aforementioned problem. This research work examined a two component mixture problem of which the pulps were separately refined prior to mixing them together in the downstream process to produce a pulp mixture with specific pulp properties. Two cases were considered, one being both pulp components received separate refining treatment at the production plant and the other introduced one of the pulps as pre-developed by the pulp suppliers, no further refining work needed, and was added directly into the process with the other pulp that was refined on-site. A curve fitting exercise was carried out to develop reliable models to predict tensile strength and pulp freeness of the mixture based on the weight fraction of each pulp component and the refining energies used for each pulp. All possible operating conditions required to generate a pulp mixture that could achieve target values of both parameters with acceptable precision were determined in order to deliver optimized solutions to users when dealing with mixtures. 2 With the above considerations in mind, the overall objectives of this research were: 1. To assemble a sound and effective methodology to statistically establish mixture models that predicts the properties of a mixture. 2. To present reliable approaches in the determination of operating conditions for target tensile strength and pulp freeness of a mixture. Upon the successful fulfillment of the set objectives, the proposed research can provide valuable insight into proper refining strategies for the two pulps being evaluated and other pulps that possess similar fiber characteristics as the pulps used in this study. In addition, the establishment of mixture models can bring great significance to the Pulp and Paper industry. With minor modifications in program codes by users, the model can essentially be applied to pulp mixtures that carry similar fiber characteristics as the two pulps examined. The ability to predict mixture properties without allocating an immense amount of resources and manpower to actually conduct full scale experiments can be an appealing benefit to many researchers and pulp users. In the chapters to come, Chapter 2 gives a brief overview of background information relating to the driving force from the industry, the mechanics of refining and the theories behind it, and the effects of refining on fiber structure and paper properties. The latter part of Chapter 2 reviews previous work on the study of mixture rules for single and multiple pulp components and introduces the proposed methodology to develop the mixture models in this research study. Chapter 3 describes the methods and materials used in conducting the experiments. Development of tensile strength and pulp freeness models and results of model predictions with respect to observed measurements are presented in Chapter 4 and 3 5 respectively. While Chapter 6 discusses the applications of the two cases considered and analytical solutions for the established mixture rules, Chapter 7 explains and compares the two approaches (minimization and one variable) applied to determine the operating conditions needed to produce a pulp mixture that meets specific target tensile strength and freeness. Lastly, an itemized summary of the findings and the recommended future work are given in Chapter 8. 4 2 Background 2.1 Industrial Motivation Most commercially produced paper products comprise a mixture of softwood (SW) and hardwood (HW) fibers. Both fibers contribute differently to the properties of the final sheet produced, where SW fibers boost the strength of the sheet, HW fibers provide a smooth and soft finish to the sheet. There is an increasing supply of HW fibers flooding the market place as a result of more hardwood plantation forests and a rapid rise of new production mills in Asia and South America. In recent years, papermakers are challenged to continuously improve existing products and develop new ones in order to maintain their competitive edge. Refining is a critical technical junction between pulp stock preparation and papermaking. It has a huge impact on fiber development and end product properties. However many papermakers often give little attention to it during the production process as they donβt recognize the significance of proper refining. Other than the quality of raw material used, many paper properties are developed based on the refining treatment that the fibers underwent (Baker, 1995). Detailed descriptions of refining mechanism and effects of refining can be found in the subsequent section. Co-refining (Figure 2.1) and separate refining (Figure 2.2) strategies are widely practiced to prepare the blended stock (Campo, et al.,1999). While co-refining both pulps together reduces energy consumption in the refiner, the strength properties can suffer if compared with pure softwood refining. Refining the two pulps separately then combining them together in the downstream process tends to be a more sensible approach as both fibers can be properly developed without being compromised. 5 Figure 2.1 Co-refining Figure 2.2 Separate refining Some pulp producers even pre-refine their pulp during pulp production, so it can be marketed as a ready-to-use pulp. Itβs a time-saving strategy as this enables papermakers to add the purchased pulp straight into their system without putting much effort into refining. However, the price of this convenience may be later reflected in the cost of the purchased pulp. The aim of refining is to improve the bonding ability of fibers. Main effects of refining include internal change in the wall structure, external fibrillation, fiber straightening, fiber shortening and production of fines. The noted different refining effects depends on the refiner and the refining conditions. Optimum refining often leads to increasing of fiber flexibility, creating internal and external fibrillations and minimizing fines generation. The term βrefining energyβ is often used to characterize the degree of refining implemented to the fibers. While 6 dry tensile strength of the resulting paper improves with increasing refining energy, as fibers become more fibrillated and flexible, pulp freeness, a measure of rate of water removal from pulp stock, is compromised at the same time. Paper with poor drainability on the machine can render a slower machine speed and reduce the efficiency of paper production, an undesirable outcome in papermaking. Therefore, the establishment of a sound method to define mixture rules using easily attainable process variables and the ability to choose suitable operating conditions in order to help papermakers provide products that meet specific target property values while keeping cost at a minimum can be quite appealing as this gives the versatility to accommodate a wide range of customersβ demands without making drastic modifications in the production process. 2.2 Literature Review 2.2.1 Refining Mechanism The main goal of refining is to change fiber morphology in order to develop specific paper properties. In the past, the refining work was labor intensive and was carried out by rolling a heavy stone disc over the pulp repeatedly to simulate the beating effect. Nowadays, itβs replaced by a mechanical refiner shown in Figure 2.3. 7 Figure 2.3 Commercial refiner (GL&V, 2014) Inside the refiner housing, there is a set of refiner plates (the stationary stator and the rotor that turns at a constant speed) that are grooved and parallel to each other. An image of a refiner plate can be found in Figure 2.4. During refining, the fiber flows along the grooves between the bars to travel through the refiner. Fiber flocs are collected on the refiner plate bar edges and deformed by shearing and compression force, introduced between the rotor and the stator plates seen in Figure 2.5. Figure 2.4 Photo of refiner plate 8 Figure 2.5 Refining mechanism (Hubbe) 2.2.2 Refining Theory Several theories of refining have been developed over the years in an effort to better understand the process by which energy is applied to the actual fiber and to improve control of the process (Canfor, 2014). Specific Edge Load (SEL) is one of the most commonly practiced refining theories in the industry, which was first introduced by Wultsch and Flucher (1958) and further developed by Brecht and Siewert (1966). SEL or refining intensity (Ws/m or J/m) is calculated according to the formula below: ππΈπΏ = ππππ‘πΆπΈπΏ (Eqn 1) ππππ‘ = ππππ‘ππ β πππ πΏπππ (Eqn 2) where ππππ‘ is the net power (kW), ππππ‘ππ is the total power applied (kW) and πππ πΏπππ is the no load power (kW). No load power refers to the power consumed by the refiner when there is no refining of the fibers. CEL or Cutting Edge Length (km/rev) is the total length of bar edges a fiber can pass over in a revolution and can be described as: πΆπΈπΏ = ππ Γ ππ π‘ Γ π Γ πππππππ π ππππ (Eqn 3) 9 where ππ is the number of rotor bars, ππ π‘ is the number of stator bars and π is bar length (km). While SEL characterizes the amount of effective refining energy applied by the edges of the refining elements to the refining zone (Olejnik, 2013), specific refining energy (SRE) in kWh/t quantifies the amount of energy that is transferred from the refiner to the fibers. ππ πΈ = ππππ‘πΉ Γ πΆ (Eqn 4) where πΉ is the volumetric flow rate (L/min) and πΆ is the pulp stock consistency (%) (Lumiainen, 2000). 2.2.3 The Effects of Refining on Fiber Characteristics The mechanical action of refining can cause considerable changes to the morphology of fibers. Internal fibrillation due to the degradation of the primary wall and the outer layer of the secondary wall (S1), external fibrillation, fiber straightening, fiber shortening and fines generation are reported as the main changes in fiber characteristics due to refining. Internal Fibrillation Fiber flexibility is increased by the mechanical force introduced by the refiner which leads to rupture of the primary wall and delamination of the layers in the fiber wall. When this event takes place in the presence of water, the fibers become swollen because of the breaking of intrafiber hydrogen bonds and replacing the bonds with water molecules (Hartman, 1984; Ebeling, 1980). Figure 2.6 shows a SEM image of fiber swelling. The phenomenon of fiber swelling can be captured by the water retention value test. 10 Figure 2.6 Swelling of fibers External Fibrillation While internal fibrillation increases fiber flexibility, external fibrillation promotes interfiber bonding (Clark, 1969). The action governing the creation of external fibrils is explained as a peeling-off mechanism. When the primary wall and the S1 layer are partially peeled off, the middle layer (S2) becomes exposed and is available for interfiber bonding (Kang, 2007). A microscopic image of a fibrillated fiber is displayed in Figure 2.7. Figure 2.7 Fibrillated fiber 11 Fiber Straightening As the fibers travel through many process units such as pumps and mixers in a pulp mill, they are kinked and crimped which makes them curly. Curly fibers affect drainage on the paper machine (Page, et al.,1985). Tension in the refiner gap can help straighten the fibers, improve the load carrying ability as well as the stress distribution in the fiber network, which can further magnify the strength of the paper (Gharehkhani, et al., 2015). Fiber Shortening and Fines Generation Fiber cutting can be expected when fibers are subjected to a small number of impacts at high refining intensity. Fines formation is the result of cleaving parts of the outer walls of the fiber and the breaking off of the fibrils (Hartman, 1984). Both occurrences are unfavorable in refining as longer fiber usually leads to higher tensile strength in the sheet, however both actions are inevitable. 2.2.4 The Effects of Refining on Pulp and Paper Properties While refining alters fiber structure, it can significantly affect pulp freeness and paper properties. Effect on Pulp Freeness Canadian Standard Freeness (CSF) test measures the drainability of a diluted pulp stock, which can affect machine runnability. A high CSF value means that there is little to no resistance for water to pass through a fiber network. Therefore paper can be fast produced with good drainability. Also freeness is strongly correlated with refining energy. Drainability 12 worsens with the growth of refining energy consumption (Figure 2.8), since fibers become swollen and fines are generated due to the refining treatment. Figure 2.8 Refining curve Effect on Paper Properties Paper density increases with refining due to collapsed fibers and collected fines. Since bulk is inversely proportional to density, bulk of the paper is then naturally reduced with respect to refining. Refining not only densifies the paper, but also yields a less porous sheet, in other words, air permeability decreases. While beating improves the dry tensile strength of paper (Figure 2.8) attributed by the increased bonding surface, it negatively impacts opacity and light scattering for chemical pulps (Lumiainen, 2000). Often excessive pulp refining generates more fines (i.e. slow drainage), the paper machine has to be run at a much slower rate than usual and more energy in the dryer section is required to fully dry the paper. This can significantly affect machine efficiency and paper production rate. 13 2.2.5 Refining Strategies for Mixtures As stated earlier, SW and HW pulps each bring distinct features to the properties of mixture. While SW pulp makes up for the strength of the sheet, HW pulp enhances the printability of the sheet. Co-refining (Figure 2.9) and separate refining (Figure 2.10) are the two common ways to refine different kinds of pulp in the paper industry. Figure 2.9 Co-refining system Figure 2.10 Separate refining system Where π₯π is the mass fraction of pulp π, πππ refers to the ππ‘β property of pulp π and πΈππ denotes the ππ‘β set of refining conditions for pulp π. In the case of co-refining, π₯1 fraction of pulp 1 which has properties of π1π is blended with π₯2 fraction of pulp 2 that possesses π2π 14 properties. The combined stock is then refined under a set of specific refining conditions, πΈπππ₯π, in order to develop multiple properties of the mixture, ππππ₯π. Now consider the scenario of separate refining, where pulp 1 and 2 are refined separately at set refining conditions of πΈ1π and πΈ2π, respectively. Then the two refined pulps are mixed together in the downstream process at a fixed blending ratio. The resultant mixture properties become a combination of properties of pulp 1 (π1π) developed at πΈ1π and properties of pulp 2 (π2π) developed at πΈ2π. Although both refining strategies are widely accepted in the industry, separate refining has gradually become a preferred practice by many papermakers as the benefits of properly developed fibers outweighs the economic savings with a co-refined system (Nutall, et al.,1999; Stevens, 1992). 2.2.6 Mixtures of Different Pulps A rigorous theoretical analysis to establish a mixture model is most desirable; however, it is a formidable task to carry out such an exercise given current limitations on manpower and resources. Similar blending studies have been done with various types of fibers by many researchers in the past (Kibblewhite, 1993; Krkoska, et al.,1989), but limited knowledge has been made available in model development. First, for pure pulp components, Page (1969) shows that fiber properties and relative bonded area can be of use to predict paper strength. While Mohlin (1987) demonstrates the association of a number of pulp and fiber properties with paper quality, GΓΆrres et al. (1989) work on the expansion of an approach, developed by Kallmes and Corte (1960; 1961), to also take the interaction between layers into consideration. 15 When there is more than one component in the mix, the use of linear and nonlinear mixture models to predict mixture properties is considered. The most general approach is the basic linear regression model used to approximate properties of mixtures by summing up the product of the properties possessed by each individual pulp component and the weight fraction of each component in the mixture, as shown in the following equation: ππππ₯π = β π΄ππ₯ππππππ=1 (Eqn 5) β π₯πππ=1= 1 (Eqn 6) where ππππ₯π is the ππ‘β property of the mixture, π΄π is the coefficient, π is the number of components in the mixture and the weight fraction of all pulp components added together should equal to one. Brecht (1963) finds that the mixture properties start to deviate from the linear model when chemical pulps comprise more than 30% in a mechanical and chemical pulp mixture (Figure 2.11). He also makes the claim that if the two pulps differ considerably from one another, deviation from linearity in mixture properties can be expected. 16 Figure 2.11 Mixture property prediction Mohlin and Wennberg (1984) also report a negative deviation from the linear model for tensile strength and bulk, whereas a positive deviation is observed with tearing strength for a similar chemical and mechanical pulp mixture. They speculate that the departure from linearity is caused by a weak interaction between the two types of pulps, due to the high shrinkage and twisting tendency of chemical pulp fibers. Arlov (1963) investigates the properties of mixed furnishes containing two different types of SW and HW pulps at various beating degrees and limited the blends to 1:1 mixtures of the separate pulps. He notes linear mixing for porosity and tensile strength, but shows the greatest drift from the linear additive rule occurs with tearing strength when adding unrefined or slightly beaten HW pulp to a more refined SW pulp, since this can lead to more long fiber pull-out rather than breakage of bonds between short and long fibers. 17 Publications reviewed thus far only examine a narrow range of the pulp properties and do not take into consideration the treatment of individual pulps. Almin and de Ruvo (1967) advocate a second order polynomial to describe the effect of pulp treatment as shown below: ππππ₯π = β π΄ππ₯π + β β π΅πππ₯πππ>ππ₯ππβ1π=1ππ=1 (Eqn 7) π΄π = β π΄πππ‘πβ1ππ=1 (Eqn 8) π΅ππ = β π΅ππππ‘πβ1ππ=1 (Eqn 9) where π is the number of pulp components in the mixture, π is the number of properties (e.g. tensile strength) and π‘ is the degree of treatment (e.g. refining energy) and π β 1 refers to the degree of polynomial. Both π΄π and π΅ππ are coefficeints and defined as functions of the treatment variables. The usefulness of this model is that the degree of treatment, may it be mechanical or chemical, can be easily incorporated into the equation. However the limitation of this method is that the number of experimental observations must exceed the number of coefficients to obtain coefficients in the equation and a wide range of experimental data is desired to ensure the validity of the resulting model. Colley (1973) studies the blends of high and low density eucalyptus pulps and suggests the freeness of the blend can be determined from the proportions by weight of the component freeness values (Eqn 5). As tensile and burst strength of the mixture in his study do not behave in such a predictable manner, both parameters are then estimated 18 using the methodology presented by Almin and de Ruvo (Eqn 7) and equation coefficients are obtained using multiple regression techniques. An article (RISI, 2011) in the Pulp & Paper International magazine showcased the development of a virtual refiner by SΓΆdra, a Scandinavian pulp producer. Itβs a computer based tool to mix different pulps and put together a recipe to help customers create an optimal furnish that meets their specification. The tool was developed by correlating on-line measurements of fiber properties with handsheet properties. However, there is no information on the type of mathematical model used in the paper. A blending study carried out by Chauhan et al. (2013) shows the impact of mixing different proportions of SW and HW pulps on a number of paper properties. Blending models are developed based on either principle of property additivity or regression analysis. However, the pulps used in the study are pre-refined to a fixed energy level. Without examining the effect of fiber treatment, the robustness of the regression model proposed diminishes when using the SW/HW blending ratio as the sole independent variable. In summary, the literature reviewed shows inconsistencies in what properties actually deviate from the linear mixing rule and whether the deviation occurs in a positive or a negative manner. Also, it is unclear if a mixing rule and the respective coefficients are general to a pulp mixture or is it truly dependent on the refining state of each individual pulp component. The models examined do not characterize the specific type of refining treatment outlined in this research study. Therefore, a reliable methodology needs to be established in order to accurately describe mixture rules for the specific refining system detailed in this dissertation. 19 2.2.7 Model Development Taylor series is an infinite power series that can be used to determine the solutions of a given function around a desired point. Itβs known as a useful and effective tool to evaluate defined integrals, understand the asymptotic behavior as well as the growth of the function and solve differential equations. In this paper, Taylor series expansion was employed as the basis for mixture model development assuming the function was continuous. The general theorem of a one-dimensional Taylor series expansion of a real function π(π₯) at a point π₯ = π is given by: π(π₯) = βπ(π)(π)π!(π₯ β π)πβπ=0 (Eqn 10) where π is the degree of polynomial. For multi-dimensional Taylor series expansion, Eqn 10 is expressed in scalar form: π(?Μ ?) = βπ(π)(?Μ ?)π!(?Μ ? β ?Μ ?)πβπ=0 (Eqn 11) where ?Μ ? = π₯1, β¦ , π₯π and ?Μ ? = π1, β¦ , ππ. In the case of a third order Taylor series expansion, it can be written as: π(?Μ ?) = π(?Μ ?) + (?Μ ? β ?Μ ?)ππ·π(?Μ ?) +π·2π(?Μ ?)2!(?Μ ? β ?Μ ?)π(?Μ ? β ?Μ ?) +π·3π(?Μ ?)3!(?Μ ? β ?Μ ?)π(?Μ ? β ?Μ ?)2 (Eqn 12) where π·π(?Μ ?) is the gradient of π evaluated at ?Μ ? = ?Μ ? and π·2π(?Μ ?) is the Hessian matrix. 20 In a multivariable system, Eqn 12 can then be defined as follows: π(?Μ ?) = π(?Μ ?) + βππ(?Μ ?)ππ₯π(π₯π β ππ) +12!β βπ2π(?Μ ?)ππ₯πππ₯π(π₯π β ππ)(π₯π β ππ)ππ=1ππ=1+ππ=1 (Eqn 13) 13!β β βπ3π(?Μ ?)ππ₯πππ₯πππ₯π(π₯π β ππ)(π₯π β ππ)(π₯π β ππ)ππ=1ππ=1ππ=1 Written in expanded form when π (the number of variables) = 4, that is considered in this research, π(?Μ ?) = π(?Μ ?) +ππ(?Μ ?)ππ₯1(π₯1 β π1) +ππ(?Μ ?)ππ₯2(π₯2 β π2) +ππ(?Μ ?)ππ₯3(π₯3 β π3) +ππ(?Μ ?)ππ₯4 (Eqn 14) (π₯4 β π4) +12![π2π(?Μ ?)ππ₯1ππ₯1(π₯1 β π1)2 +π2π(?Μ ?)ππ₯1ππ₯2(π₯1 β π1)(π₯2 β π2) + π2π(?Μ ?)ππ₯1ππ₯3(π₯1 β π1)(π₯3 β π3) +π2π(?Μ ?)ππ₯1ππ₯4(π₯1 β π1)(π₯4 β π4) +π2π(?Μ ?)ππ₯2ππ₯1 (π₯1 β π1)(π₯2 β π2) +π2π(?Μ ?)ππ₯2ππ₯2(π₯2 β π2)2 +π2π(?Μ ?)ππ₯2ππ₯3(π₯2 β π2)(π₯3 β π3) + π2π(?Μ ?)ππ₯2ππ₯4(π₯2 β π2)(π₯4 β π4) +π2π(?Μ ?)ππ₯3ππ₯1(π₯1 β π1)(π₯3 β π3) +π2π(?Μ ?)ππ₯3ππ₯2 (π₯2 β π2)(π₯3 β π3) +π2π(?Μ ?)ππ₯3ππ₯3(π₯3 β π3)2 +π2π(?Μ ?)ππ₯3ππ₯4(π₯3 β π3)(π₯4 β π4) + π2π(?Μ ?)ππ₯4ππ₯1(π₯1 β π1)(π₯4 β π4) +π2π(?Μ ?)ππ₯4ππ₯2(π₯2 β π2)(π₯4 β π4) +π2π(?Μ ?)ππ₯4ππ₯3 (π₯3 β π3)(π₯4 β π4) +π2π(?Μ ?)ππ₯4ππ₯4(π₯4 β π4)2 +13! π3π(?Μ ?)ππ₯1ππ₯1ππ₯1(π₯1 β π1)3 + π3π(?Μ ?)ππ₯1ππ₯1ππ₯2(π₯1 β π1)2(π₯2 β π2) +π3π(?Μ ?)ππ₯1ππ₯1ππ₯3(π₯1 β π1)2(π₯3 β π3) + π3π(?Μ ?)ππ₯1ππ₯1ππ₯4(π₯1 β π1)2(π₯4 β π4) +π3π(?Μ ?)ππ₯1ππ₯2ππ₯1(π₯1 β π1)2(π₯2 β π2) + π3π(?Μ ?)ππ₯1ππ₯2ππ₯2(π₯1 β π1)(π₯2 β π2)2 +π3π(?Μ ?)ππ₯1ππ₯2ππ₯3(π₯1 β π1)(π₯2 β π2) (π₯3 β π3) +π3π(?Μ ?)ππ₯1ππ₯2ππ₯4(π₯1 β π1)(π₯2 β π2)(π₯4 β π4) +π3π(?Μ ?)ππ₯1ππ₯3ππ₯1 21 (π₯1 β π1)2(π₯3 β π3) +π3π(?Μ ?)ππ₯1ππ₯3ππ₯2(π₯1 β π1)(π₯2 β π2)(π₯3 β π3) + π3π(?Μ ?)ππ₯1ππ₯3ππ₯3(π₯1 β π1)(π₯3 β π3)2 +π3π(?Μ ?)ππ₯1ππ₯3ππ₯4(π₯1 β π1)(π₯3 β π3) (π₯4 β π4) +π3π(?Μ ?)ππ₯1ππ₯4ππ₯1(π₯1 β π1)2(π₯4 β π4) +π3π(?Μ ?)ππ₯1ππ₯4ππ₯2(π₯1 β π1) (π₯2 β π2)(π₯4 β π4) +π3π(?Μ ?)ππ₯1ππ₯4ππ₯3(π₯1 β π1)(π₯3 β π3)(π₯4 β π4) + π3π(?Μ ?)ππ₯1ππ₯4ππ₯4(π₯1 β π1)(π₯4 β π4)2 +π3π(?Μ ?)ππ₯2ππ₯1ππ₯1(π₯1 β π1)2(π₯2 β π2) + π3π(?Μ ?)ππ₯2ππ₯1ππ₯2(π₯1 β π1)(π₯2 β π2)2 +π3π(?Μ ?)ππ₯2ππ₯1ππ₯3(π₯1 β π1)(π₯2 β π2) (Eqn 14) (π₯3 β π3) +π3π(?Μ ?)ππ₯2ππ₯1ππ₯4(π₯1 β π1)(π₯2 β π2)(π₯4 β π4) +π3π(?Μ ?)ππ₯2ππ₯2ππ₯1 (π₯1 β π1)(π₯2 β π2)2 +π3π(?Μ ?)ππ₯2ππ₯2ππ₯2(π₯2 β π2)2 +π3π(?Μ ?)ππ₯2ππ₯2ππ₯3(π₯2 β π2)2 (π₯3 β π3) +π3π(?Μ ?)ππ₯2ππ₯2ππ₯4(π₯2 β π2)2(π₯4 β π4) +π3π(?Μ ?)ππ₯2ππ₯3ππ₯1(π₯1 β π1) (π₯2 β π2)(π₯3 β π3) +π3π(?Μ ?)ππ₯2ππ₯3ππ₯2(π₯2 β π2)2(π₯3 β π3) +π3π(?Μ ?)ππ₯2ππ₯3ππ₯3 (π₯2 β π2)(π₯3 β π3)2 +π3π(?Μ ?)ππ₯2ππ₯3ππ₯4(π₯2 β π2)(π₯3 β π3)(π₯4 β π4) + π3π(?Μ ?)ππ₯2ππ₯4ππ₯1(π₯1 β π1)(π₯2 β π2)(π₯4 β π4) +π3π(?Μ ?)ππ₯2ππ₯4ππ₯2(π₯2 β π2)2 (π₯4 β π4) +π3π(?Μ ?)ππ₯2ππ₯4ππ₯3(π₯2 β π2)(π₯3 β π3)(π₯4 β π4) +π3π(?Μ ?)ππ₯2ππ₯4ππ₯4 (π₯2 β π2)(π₯4 β π4)2 +π3π(?Μ ?)ππ₯3ππ₯1ππ₯1(π₯1 β π1)(π₯3 β π3)2 +π3π(?Μ ?)ππ₯3ππ₯1ππ₯2 (π₯1 β π1)(π₯2 β π2)(π₯3 β π3) +π3π(?Μ ?)ππ₯3ππ₯1ππ₯3(π₯1 β π1)(π₯3 β π3)2 + π3π(?Μ ?)ππ₯3ππ₯1ππ₯4(π₯1 β π1)(π₯3 β π3)(π₯4 β π4) +π3π(?Μ ?)ππ₯3ππ₯2ππ₯1(π₯1 β π1) (π₯2 β π2)(π₯3 β π3) +π3π(?Μ ?)ππ₯3ππ₯2ππ₯2(π₯2 β π2)2(π₯3 β π3) +π3π(?Μ ?)ππ₯3ππ₯2ππ₯3 (π₯1 β π1)(π₯2 β π2)(π₯3 β π3) +π3π(?Μ ?)ππ₯3ππ₯2ππ₯4(π₯2 β π2)(π₯3 β π3) 22 (π₯4 β π4) +π3π(?Μ ?)ππ₯3ππ₯3ππ₯1(π₯1 β π1)(π₯3 β π3)2 +π3π(?Μ ?)ππ₯3ππ₯3ππ₯2(π₯2 β π2) (π₯3 β π3)2 +π3π(?Μ ?)ππ₯3ππ₯3ππ₯3(π₯3 β π3)3 +π3π(?Μ ?)ππ₯3ππ₯3ππ₯4(π₯3 β π3)2(π₯4 β π4) + π3π(?Μ ?)ππ₯3ππ₯4ππ₯1(π₯1 β π1)(π₯3 β π3)(π₯4 β π4) +π3π(?Μ ?)ππ₯3ππ₯4ππ₯2(π₯2 β π2) (π₯3 β π3)(π₯4 β π4) +π3π(?Μ ?)ππ₯3ππ₯4ππ₯3(π₯3 β π3)2(π₯4 β π4) +π3π(?Μ ?)ππ₯3ππ₯4ππ₯4 (π₯3 β π3)(π₯4 β π4)2 +π3π(?Μ ?)ππ₯4ππ₯1ππ₯1(π₯1 β π1)2(π₯4 β π4) +π3π(?Μ ?)ππ₯4ππ₯1ππ₯2 (π₯1 β π1)(π₯2 β π2)(π₯4 β π4) +π3π(?Μ ?)ππ₯4ππ₯1ππ₯3(π₯1 β π1)(π₯3 β π3) (Eqn 14) (π₯4 β π4) +π3π(?Μ ?)ππ₯4ππ₯1ππ₯4(π₯1 β π1)(π₯4 β π4)2 +π3π(?Μ ?)ππ₯4ππ₯2ππ₯1(π₯1 β π1) (π₯2 β π2)(π₯4 β π4) +π3π(?Μ ?)ππ₯4ππ₯2ππ₯2(π₯2 β π2)2(π₯4 β π4) +π3π(?Μ ?)ππ₯4ππ₯2ππ₯3 (π₯2 β π2)(π₯3 β π3)(π₯4 β π4) +π3π(?Μ ?)ππ₯4ππ₯2ππ₯4(π₯2 β π2)(π₯4 β π4)2 + π3π(?Μ ?)ππ₯4ππ₯3ππ₯1(π₯1 β π1)(π₯3 β π3)(π₯4 β π4) +π3π(?Μ ?)ππ₯4ππ₯3ππ₯2(π₯2 β π2) (π₯3 β π3)(π₯4 β π4) +π3π(?Μ ?)ππ₯4ππ₯3ππ₯3(π₯3 β π3)2(π₯4 β π4) +π3π(?Μ ?)ππ₯4ππ₯3ππ₯4 (π₯3 β π3)(π₯4 β π4)2 +π3π(?Μ ?)ππ₯4ππ₯4ππ₯1(π₯1 β π1)(π₯4 β π4)2 +π3π(?Μ ?)ππ₯4ππ₯4ππ₯2 (π₯2 β π2)(π₯4 β π4)2 +π3π(?Μ ?)ππ₯4ππ₯4ππ₯3(π₯3 β π3)(π₯4 β π4)2 +π3π(?Μ ?)ππ₯4ππ₯4ππ₯4 (π₯4 β π4)3 23 Evaluating the function in Eqn 14 at ?Μ ? = ?Μ ? and if ?Μ ? = 0, then π(0) = 0, which reduces the problem considerably and it becomes: π(?Μ ?) = π΄1π₯1 + π΄2π₯2 + π΄3π₯3 + π΄4π₯4 + π΄11π₯12 + π΅12π₯1π₯2 + π΅13π₯1π₯3 + π΅14π₯1π₯4 + (Eqn 15) π΅21π₯1π₯2 + π΅22π₯22 + π΅23π₯2π₯3 + π΅24π₯2π₯4 + π΅31π₯1π₯3 + π΅32π₯2π₯3 + π΅33π₯32 + π΅34π₯3π₯4 + π΅41π₯1π₯4 + π΅42π₯2π₯4 + π΅43π₯3π₯4 + π΅44π₯42 + πΆ111π₯13 + πΆ112π₯12π₯2 + πΆ113π₯12π₯3 + πΆ114π₯12π₯4 + πΆ121π₯12π₯2 + πΆ122π₯1π₯22 + πΆ123π₯1π₯2π₯3 + πΆ124π₯1π₯2π₯4 + πΆ131π₯12π₯3 + πΆ132π₯1π₯2π₯3 + πΆ133π₯1π₯32 + πΆ134π₯1π₯3π₯4 + πΆ141π₯12π₯4 + πΆ142π₯1π₯2π₯4 + πΆ143π₯1π₯3π₯4 + πΆ144π₯1π₯42 + πΆ211π₯12π₯2 + πΆ212π₯1π₯22 + πΆ213π₯1π₯2π₯3 + πΆ214π₯1π₯2π₯4 + πΆ221π₯1π₯22 + πΆ222π₯23 + πΆ223π₯22π₯3 + πΆ224π₯22π₯4 + πΆ231π₯1π₯2π₯3 + πΆ232π₯22π₯3 + πΆ233π₯2π₯32 + πΆ234π₯2π₯3π₯4 + πΆ241π₯1π₯2π₯4 + πΆ242π₯22π₯4 + πΆ243π₯2π₯3π₯4 + πΆ244π₯2π₯42 + πΆ311π₯12π₯3 + πΆ312π₯1π₯2π₯3 + πΆ313π₯1π₯32 + πΆ314π₯1π₯3π₯4 + πΆ321π₯1π₯2π₯3 + πΆ322π₯22π₯3 + πΆ323π₯2π₯32 + πΆ324π₯2π₯3π₯4 + πΆ331π₯1π₯32 + πΆ332π₯2π₯32 + πΆ333π₯33 + πΆ334π₯32π₯4 + πΆ341π₯1π₯3π₯4 + πΆ342π₯2π₯3π₯4 + πΆ344π₯3π₯42 + πΆ411π₯12π₯4 + πΆ412π₯1π₯2π₯4 + πΆ413π₯1π₯3π₯4 + πΆ414π₯1π₯42 + πΆ421π₯1π₯2π₯4 + πΆ422π₯22π₯4 + πΆ423π₯2π₯3π₯4 + πΆ424π₯2π₯42 + πΆ431π₯1π₯3π₯4 + πΆ432π₯2π₯3π₯4 + πΆ433π₯32π₯4 + πΆ434π₯3π₯42 + πΆ441π₯1π₯42 + πΆ442π₯2π₯42 + πΆ443π₯3π₯42 + πΆ444π₯43 Eqn 15 can be presented in a more concised form using summation notations: π(?Μ ?) = β π΄ππ₯π4π=1+ β β π΅ππ4π=ππ₯ππ₯π4π=1+ β β β πΆππππ₯ππ₯ππ₯π4π=π4π=π4π=1 (Eqn 16) where π΄π , π΅ππ and πΆπππ are coefficients, π₯1 and π₯2 are mass fraction of pulp 1 and pulp 2, respectively, and π₯3 and π₯4 are refining energy applied to each of the individual pulp component. The polynomial equation proposed by Almin and de Ruvo simply poses the problem as a second order Taylor series expansion characterizing mixture properties as a 24 function of πππ and the weight fraction of ith pulp component. For details, please refer to Appendix A for the derivation of their proposed model using Taylor series expansion. Model coefficients in this research work were determined using the Levenberg-Marquardt method, which is a combination of the Gauss-Newton method and the gradient descent method. The Gauss-Newton method reduces the sum of the squared errors by assuming the least squares function is locally quadratic and finds the minimum of the quadratic. The gradient descent approach updates the parameters in the direction of the greatest reduction of the least squares objects to achieve the minimization of the sum of the squared errors (Gavin, 2011). 25 3 Methods and Materials 3.1 Pulps Case I Chemically bleached SW1 and HW pulps were the two pulps used as raw materials. The SW1 kraft pulp was supplied by Canfor with a furnish basket of white spruce, lodgepole pine and balsam fir. The HW eucalyptus kraft pulp was obtained from UPMβs Uruguay mill. The average fiber lengths for these types of SW1 and HW pulps were approximately 2.40-2.60 and 0.70-0.73mm, respectively. SW1 and HW pulps were refined separately using a 12-inch Andritz pilot scaled single disc-refiner, a batch refining process, located in Burnaby at Canfor Pulp Innovation. Figure 3.1 illustrates the refining equipment and the refiner plates used for SW1 and HW refining are shown in Figures 3.2 and 3.3. Figure 3.1 Refining Equipment 26 Figure 3.2 Refiner plate for SW1 Figure 3.3 Refiner plate for HW A 3.5% consistency pulp stock was transferred from the repulper tank to load up the refiner loop first. Then the pulp slurry was circulated inside the loop at 350L/min and each pass to the refiner was 44 seconds (Figure 3.4). Samples at approximately 0, 30, 60, 100 and 150kWh/t were collected through the sampling valve and multiple runs were conducted to generate sufficient stock for the subsequent blending experiments. The refining conditions of the two pulps are summarized in Table 3.1. Figure 3.4 Schematics of the refiner loop 27 Table 3.1 Refining conditions for SW1 and HW pulps As detailed in Table 3.2, a total of 81 SW1/HW blends were created by first mixing both pulps at various degrees of refining energies with a SW1 content varying from 0-100% in a British disintegrator for five minutes to produce a well-stirred pulp stock for handsheet preparation. Isotropic handsheets with a grammage of 60g/m2 were then made with deionized water, according to TAPPI T205 sp-02, and formed in a conventional sheet former. They were further pressed at a pressure of 345kPa for a total of seven minutes, the blotters that were stacked in between the handsheets were replaced with a new batch of dry blotters at the five minute mark. To prevent fiber shrinkage during drying, the sheets were stacked immediately in rings in a controlled temperature and humidity (CTH) room, where temperature and relative humidity were maintained at 23oC and 50%, respectively. Quintuplet experiments were performed on the HW pulp only to check the significance of the experimental results. SW1 HWStock Consistency (%) 3.5 3.5Refiner Speed (rpm) 1800 1800Refining Intensity (J/m) 1.75 0.2Plate Supplier J&L AFT FinebarPlate ID B12TF102 SWFC121215061320Plate Factor (km/rev) 0.21 1.843Bar Width/Groove Width/ Depth (mm) 6.3 x 3.6 x 5.3 1.0 x 2.0 x 3.2Bar Angle (degrees) 13 1528 Table 3.2 Case I mixture experiments SW1 Ratio HW Ratio SW1 Refining Energy HW Refining Energy0β¦1 1β¦0 0 00β¦1 1β¦0 30 300β¦1 1β¦0 60 600β¦1 1β¦0 100 1000β¦1 1β¦0 150 1500.05, 0.1, 0.2 0.95, 0.9, 0.8 0 300.05, 0.1, 0.2 0.95, 0.9, 0.8 0 1500.05, 0.1, 0.2 0.95, 0.9, 0.8 30 00.3, 0.7 0.7, 0.3 30 600.2 0.8 30 1000.05 0.95 30 1500.05, 0.1, 0.2 0.95, 0.9, 0.8 60 00.05, 0.1, 0.2, 0.6 0.95, 0.9, 0.8, 0.4 60 300.05, 0.1, 0.2, 0.7 0.95, 0.9, 0.8, 0.3 60 1500.05, 0.1, 0.2 0.95, 0.9, 0.8 100 00.05, 0.1, 0.2, 0.6 0.95, 0.9, 0.8, 0.4 100 300.9 0.1 100 00.05, 0.1 0.95, 0.9 100 1500.05, 0.1, 0.2 0.95, 0.9, 0.8 150 00.05, 0.1, 0.2 0.95, 0.9, 0.8 150 3029 Case II A different grade of softwood pulp (SW3) was supplied by Canfor. This particular pulp already received refining treatment to develop superior strength properties prior to mixing with the SW1 pulp obtained from Case I. 51 sets of handsheets were prepared based on the experimental plan shown in Table 3.3. Table 3.3 Case II mixture experiments 3.2 Pulp and Paper Testing All physical testing of handsheets was carried out in the CTH room whereas the freeness of pulp stock was measured in a standard room temperature environment. Canadian Standard Freeness (CSF) In accordance with TAPPI T227 om-99, freeness test provides a means to determine the drainage rate of a diluted suspension that contains three grams of pulp in one liter of deionized water. The uncorrected freeness is the volume of filtrate that flows through the side orifice (Figure 3.5). This volume was corrected to the standard consistency of 0.3% SW1 Ratio SW3 Ratio SW1 Refining Energy1 0 0, 30, 60, 100, 1500 1 -0.1β¦0.9 0.9β¦0.1 00.1β¦0.9 0.9β¦0.1 300.1β¦0.9 0.9β¦0.1 600.1β¦0.9 0.9β¦0.1 1000.1β¦0.9 0.9β¦0.1 15030 and temperature of 20oC using the correction table provided in the aforementioned method and was reported in mL. Ten handsheets were prepared for each blend and five of the sheets were used to determine grammage and tensile strength. Grammage Grammage in g/m2, describes the weight of a sheet per unit area, which is used to index the tensile strength. The detailed method is explained in TAPPI T220 sp-01. Tensile Strength Index Two 15mm wide test strips were cut from the middle of each handsheet and a total of 10 strips per sample were subjected to a constant elongation rate of 25mm/min until breakage was detected (Figure 3.6). The average value was reported in N.m/g as outlined in TAPPI T494 om-01. Figure 3.5 CSF tester Figure 3.6 L&W tensile strength tester 31 4 Tensile Strength Model 4.1 Case I Refining the two pulps separately prior to mixing them together, as shown in Figure 4.1, bestows more power to the papermakers to make a wide range of products. Figure 4.1 Case I schematics Eqn 13 displayed the full model derived for the tensile strength of the pulp mixture for Case I. However when the expression was written out in full (Eqn 15), the regression model contained a large number of coefficients that needed solving. The number of coefficients in the model could be further reduced based on the following reasons, before coefficient estimations. First, there were configurations that were physically impossible to achieve. Since the two pulps were refined separately, there should not be any synergies observed between the weight fraction of SW1 (π₯1) and the HW energy (π₯4) and vice versa for the HW content (π₯2) and the SW1 energy (π₯3). Moreover, no interaction between the SW1 and HW energies should exist for a separate refining system. Therefore the terms with any of these combinations were neglected. Secondly, the model must also describe pure species cases, such that when the furnish only contained one species, the energy terms 32 corresponding to the other missing species should not be included to estimate mixture properties. Thirdly, the full model was overspecified with redundant parameters. Terms that only described pure species could then be condensed into one variable. At the end, a more simplified model could be obtained to characterize the tensile strength of a pulp mixture. A stepwise approach was taken to eliminate the terms from the full model, based on the aforementioned reasoning. Followed by a t-test performed to examine the level of significance for the terms in the model to ensure all of them contribute to the regression model statistically and significantly. As shown in Table 4.1, the number of model coefficients were reduced from 34 to 6 without significantly affecting the values of R2 and the root-mean-square error (RMSE). Table 4.1 Case I - tensile strength model coefficient elimination # of Coeffs R2 RMSE Terms Dropped34 - - -15 0.970 2.6111 0.970 2.6110 0.970 2.599 0.970 2.598 0.970 2.577 0.970 2.566 0.968 2.6321 0.987 1.81π₯1 π₯4 , π₯2π₯3 , π₯3π₯4 , π₯12π₯4, π₯1 π₯2π₯3 , π₯1π₯2π₯4 ,π₯3, π₯32 , π₯33 , π₯4, π₯42 , π₯43π₯12 , π₯13 , π₯22 , π₯23π₯12π₯3π₯1π₯22π₯1π₯2π₯22π₯4 π₯12π₯2π₯1 π₯3π₯4, π₯1 π₯42 , π₯22π₯3 , π₯2π₯32, π₯2π₯3π₯4, π₯32π₯4 , π₯3π₯4233 A R2 of 1 indicates that the regression line fits perfectly with the observed data and a RMSE of 0 means variation in the errors is fully explained by the model (i.e. a perfect fit). With a high R2 value of 0.968 and a low RMSE value of 2.63, the proposed tensile strength model is defined as: ππππ ππππππ₯ = π1π₯1 + π2π₯1π₯3 + π3π₯1π₯32 + π4π₯2 + π5π₯2π₯4 + π6π₯2π₯42 (Eqn 17) From Table 4.2, the p-value of each term is less than the specified significance level (πΌ) of 0.05, indicating these terms are significant to the proposed model. Table 4.2 Case I - coefficients and model statistics for tensile strength model There is no evidence of a clear trend found from the residual plot in Figure 4.2. Therefore itβs safe to assume that the errors in data used for model construction are independent. Most data points in the normality plot align closely to the normal line as shown in Figure 4.3, yet deviations observed at the two ends raise concerns in meeting the normality assumption. With all things considered, the model developed seems fairly robust with the inclusion of those points. Estimate SE t Stat p-Valuea1 33.4161 1.7412 19.1920 0.0000a2 0.4316 0.0499 8.6423 0.0000a3 -0.0009 0.0003 -2.9168 0.0047a4 26.4207 0.6524 40.5000 0.0000a5 0.3498 0.0278 12.5810 0.0000a6 -0.0009 0.0002 -5.2751 0.0000R2RMSE0.968 2.630Total # of Blends8134 The predicted strength values (denoted in red triangles in Figure 4.4) fall within the 90% confidence interval of the measured values. Four out of 81 blends give a prediction that is over 10% but less than 15% error. The data points in the predicted vs. measured (Figure 4.5) reside along the unity line. This demonstrates that the model can provide good tensile strength predictions that are comparable to the experimental results. Figure 4.2 Case I tensile - residual plot 35 Figure 4.3 Case I tensile - normality plot Figure 4.4 Tensile strength prediction with 90% confidence interval 20304050607080901001 11 21 31 41 51 61 71 81Tensile Strength (N.m/g)Blend #Measured Predicted36 Figure 4.5 Case I - predicted vs. measured tensile strength Repeatability test was also carried out to explain how much variation of the observed values from the designed experiments is due to measurement system variation. Based on five repeat testing, the repeatability ratios for both tensile strength and CSF found in this study are close to the ones reported in TAPPI as detailed in Table 4.3. This gives great confidence in results reported in this dissertation. Table 4.3 Repeatability ratios from the study and TAPPI 01020304050607080900 10 20 30 40 50 60 70 80 90Predicted Tensile Strength (N.m/g)Measured Tensile Strength (N.m/g)Lab TAPPI Lab TAPPIRepeatability Ratio % 7 5 6 6Reproducibility Ratio % - 10 - -Tensile Strength Pulp Freeness37 4.2 Case II Case II (Figure 4.6) represents the blending of SW1, which was refined to various refining energies prior to mixing, with a pre-developed pulp (SW3) that was prepared by the pulp producer. Figure 4.6 Case II schematics Since the refining energy of SW3 is held constant, this becomes a three variable scenario. Following the same methodology used in Case I, the full model containing 19 terms was reduced to an easily handled form of only four terms as shown in Table 4.4. The proposed tensile strength model for Case II has the following expression: ππππ ππππππ₯ = π1π₯1 + π2π₯1π₯3 + π3π₯1π₯32 + π4π₯2 (Eqn 18) The estimated parameter coefficients and corresponding statistics are summarized in Table 4.5. 38 Table 4.4 Case II - tensile strength model coefficient elimination The random pattern observed in the residual plot of Figure 4.7 grants strong evidence to the independence of the error term. Moreover, the normality plot in Figure 4.8 and the p-value yielded from the Shapiro-Wilk normality test (p-value = 0.107, if p-value is less than 0.05, a poor fit may be warranted) indicate the data is normally distributed. As expected, with a high R2 of 0.993, the percentage errors in predicted tensile strength for all 51 blends are all within 10%. Therefore, the proposed model provides close predictions to the measured tensile strength as exhibited in Figure 4.9. Table 4.5 Case II - coefficients and model statistics for tensile strength model # of Coeffs R2 RMSE Terms Dropped19 - - -15 0.995 1.4412 0.994 1.488 0.994 1.487 0.994 1.486 0.994 1.465 0.993 1.484 0.993 1.50π₯2 π₯3, π₯1π₯2π₯3, π₯22π₯3 , π₯2 π₯32π₯3, π₯32 , π₯33π₯12 , π₯13 , π₯22 , π₯23π₯1π₯22π₯1π₯2π₯12π₯2π₯12π₯3Estimate SE t Stat p-Valuea1 27.0480 0.7122 37.9780 0.0000a2 0.4511 0.0225 20.0690 0.0000a3 -0.0012 0.0001 -8.6698 0.0000a4 99.1600 0.4391 225.8200 0.0000R2 RMSE0.993 1.500Total # of Blends5139 Figure 4.7 Case II tensile β residual plot Figure 4.8 Case II tensile β normality plot 40 Figure 4.9 Case II β predicted vs. measured tensile strength 0204060801001200 20 40 60 80 100 120Predicted Tensile Strength (N.m/g)Measured Tensile Strength (N.m/g)41 5 Pulp Freeness Model 5.1 Case I CSF of pure pulps are strongly correlated with refining energy, as shown in Figure 5.1. The shaded area describes the interaction between the two pulps when blended together. In addition, based on the curvature observed, a power series can be a suitable representation of the proposed mixture function for CSF. Figure 5.1 CSF vs. refining energy After expanding a fourth order Taylor series and taking the same approach to eliminate terms from the full model, the proposed pulp freeness model is given by: πΆππΉπππ₯ = π1π₯1 + π2π₯1π₯3 + π3π₯1π₯32 + π4π₯12π₯3 + π5π₯2 + π6π₯2π₯4 + π7π₯2π₯42 + π8π₯2π₯43 (Eqn 19) 01002003004005006007000 50 100 150 200 250CSF (mL)Refining Energy (kWh/t)SW1 HW42 Table 5.1 displays the estimated coefficients and basic statistics for the regression model. The current model provides a very good fit to the experimental data and gives five predictions that are over 10% but under 15% error in the total 81 blends. Although the removal of the eighth parameter, of which a third order term exists, can reduce the complexity of the current model, the R2 and RMSE values can change significantly from 0.991 to 0.977 and from 12.8 to 20.5. This can affect the overall goodness of the fit to the experimental data, a consequence that needs to be made aware of, if such action is to be taken. The model assumptions are met as no apparent pattern is found in the residual plot (Figure 5.2) and most data points in the normality plot of Figure 5.3 scatter along the normal line. More importantly, the model possesses a high predictive accuracy as shown in Figure 5.4 and 5.5. Table 5.1 Case I - coefficients and model statistics for CSF model Estimate SE t Stat p-Valueb1 638.1383 7.8835 80.9460 0.0000b2 -2.4199 0.2776 -8.7176 0.0000b3 -0.0028 0.0014 -2.0687 0.0421b4 1.0926 0.1979 5.5216 0.0000b5 409.9300 3.6451 112.4600 0.0000b6 -5.9050 0.2838 -20.8040 0.0000b7 0.0585 0.0047 12.3880 0.0000b8 -0.0002 0.0000 -10.6510 0.0000R2 RMSE0.991 12.800Total # of Blends8143 Figure 5.2 Case I CSF β residual plot Figure 5.3 Case I CSF β normality plot 44 Figure 5.4 CSF prediction with 90% confidence interval Figure 5.5 Case I β predicted vs. measured CSF 01002003004005006007008001 11 21 31 41 51 61 71 81CSF (mL)Blend #Measured Predicted01002003004005006007000 100 200 300 400 500 600 700Predicted CSF (mL)Measured CSF (mL)45 5.2 Case II Similarly, the CSF model of Case I is further simplified by dropping the refining energy of the second pulp component, from Eqn 19, as it is constant for all 51 blends in Case II. Also the term π₯1π₯32 is dropped as it does not contribute to the regression significantly (i.e. p-value = 0.526 > ο‘ = 0.05). Hence, the CSF model of Case II is given below with a R2 value of 0.997 and RMSE value of 8.000 (Table 5.2). πΆππΉπππ₯ = π1π₯1 + π2π₯1π₯3 + π3π₯12π₯3 + π4π₯2 (Eqn 20) Table 5.2 Case II - coefficients and model statistics for CSF model The residual plot in Figure 5.6 displays a bit of a u-shaped pattern suggesting that a nonlinear equation could be a better fit to describe the interactions between SW1 and SW3. Furthermore, the deviations at the two tail ends in the normality plot in Figure 5.7 attributes to a p-value of 0.054 from the Shapiro-Wilk test, posing a danger of not satisfying the assumption that the data is normally distributed. However, based on Figure 5.8, the established model already possesses a high predictive power for a freeness range between 150-670mL and none of the 51 blends has predictions over 10% error. Estimate SE t Stat p-Valueb1 674.3700 3.2183 209.5500 0.0000b2 -2.0589 0.1098 -18.7430 0.0000b3 0.7494 0.1328 5.6427 0.0000b4 142.7900 2.9593 48.2520 0.0000R2 RMSE0.997 8.000Total # of Blends5146 Figure 5.6 Case II CSF β residual plot Figure 5.7 Case II CSF β normality plot 47 Figure 5.8 Case II β predicted vs. measured CSF 01002003004005006007008000 100 200 300 400 500 600 700 800Predicted CSF (mL)Measured CSF (mL) 48 6 Applications and Analytical Approach The system outlined in Case I enables papermakers to have full control of the refining treatment that fibers underwent and to refine at optimum energy and intensity for each pulp component to properly develop both types of fibers for the desired end product properties. For instance, many tissue producers use both softwood and hardwood fibers as their furnish and employ this operation to prevent over-beating hardwood fibers, reducing the softness and density of the product and under-refining softwood fibers, resulting in a fragile sheet. However, for some of them, when itβs not feasible to have two refining systems set up to refine both fibers separately, the layout in Case 2 provides a good alternative to this limitation. One of the pulps can be refined at optimum conditions on-site and then mixed in with the other, pre-developed pulp refined by pulp suppliers, in the downstream process to achieve the same outcome. Since the pulp that has already received pre-treatment of refining at pulp manufacturersβ plants to obtain specific pulp properties that meets tissue makersβ needs, it saves them time and effort not having to refine this pulp on-site prior to mixing the two distinct fibers together. While the numerical method is useful in providing approximate solutions to a complex problem, the analytical approach gives insight into the behavior of a mathematically defined problem. For illustrative purposes, the refining energies used to predict tensile strength of the mixture in both Case I and Case II can be solved analytically as shown below. 49 6.1 Case I Recalling Eqn 17, ππππ ππππππ₯ = π1π₯1 + π2π₯1π₯3 + π3π₯1π₯32 + π4π₯2 + π5π₯2π₯4 + π6π₯2π₯42 (Eqn 17) When there is only SW1 present in the mixture, ππππ ππππππ₯ = π₯1(π1 + π2π₯3 + π3π₯32). Re-arranging the terms to have the following expression: π3π₯32 + π2π₯3 + (π1 βππππ ππππππ₯π₯1) = 0 (Eqn 21) Now solving the quadratic equation for π₯3 (SW1 refining energy) π₯3 =12π3[βπ2 Β± βπ22 β 4π3 (π1 βππππ ππππππ₯π₯1)] (Eqn 22) The same can be done, when only HW is used in the mixture, ππππ ππππππ₯ = π₯2(π4 +π5π₯4 + π6π₯42). π₯4 (HW energy) becomes a function of π₯2 πππ ππππ ππππππ₯ π₯4 =12π6[βπ5 Β± βπ52 β 4π6 (π4 βππππ ππππππ₯π₯2)] (Eqn 23) When the mixture contains both SW1 and HW pulps, π₯3 is defined as: π₯3 =12π3[βπ2 Β± βπ22 β 4π3 (π1 +π4π₯2 + π5π₯2π₯4 + π6π₯2π₯42βππππ ππππππ₯π₯1)] (Eqn 24) where π₯2 = 1 β π₯1, based on the law of conservation of mass. Figure 6.1 shows the response curves of tensile strength development with respect to refining energy for the two pure pulp components. The shaded area describes the interaction of the two pulps when mixed together, which can be characterized by Eqn 24. 50 Figure 6.1 Tensile strength vs. refining energy 6.2 Case II In this case, only the energy of SW1 and the weight fraction of SW1 and SW3 have an impact on the tensile strength of the resultant sheet. As shown in Chapter 4.2, the proposed tensile strength model for Case II has the following expression: ππππ ππππππ₯ = π1π₯1 + π2π₯1π₯3 + π3π₯1π₯32 + π4π₯2 (Eqn 18) Dividing π₯1from both sides of the equation and re-arranging the terms, it becomes π3π₯32 + π2π₯3 + (π1 βππππ ππππππ₯ β π4π₯2π₯1) = 0 (Eqn 25) Then π₯3 is determined by 01020304050607080900 50 100 150 200 250Tensile Strength (N.m/g)Refining Energy (kWh/t)SW HW51 π₯3 =12π3[βπ2 Β± βπ22 β 4π3 (π1 βππππ ππππππ₯ β π4π₯2π₯1)] (Eqn 26) As expected, tensile strength decreases with increasing SW1 pulp content, but the improvement in tensile strength, due to the refining treatment of SW1 pulp, is less pronounced when the mixture contains predominantly SW3 pulp (Figure 6.2). Figure 6.2 Tensile strength vs. refining energy at various SW1 ratios 52 7 Methods for Predicting Operating Conditions for Target Properties Operating conditions required to meet target tensile strength and freeness can be determined using the established mixture rules (Figure 7.1) and either one of two methods (Minimization or One Variable) discussed below. For demonstration purposes, such work developed was based on the Case I system, similar work can be reproduced for the Case II system with minor modifications in program codes. Figure 7.1 Inverse problem layout All possible operating conditions for both properties were then fed to a sorting algorithm. Solutions that had more than one record at the same SW1 ratio were first identified, then within those records, only conditions that had the lowest HW energy for the same level of SW1 energy and SW1 ratio were kept and the rest were removed. The same concepts were applied to the records that contained the same HW energy and SW1 ratio. Both methods employed the same sorting algorithm to reduce the number of solutions in the final reporting in order to deliver useful and meaningful solutions to users. Furthermore, a user friendly graphical interface to display the solutions and give recommendations of the operating conditions needed to achieve target tensile strength and 53 CSF values of a pulp mixture, with other user-defined considerations such as the cost of energy and the price of each pulp component, was created using the MATLAB software. 7.1 Method I: Minimization As illustrated in Figure 7.2, objective functions (β ππ) for both mixture properties with acceptable precision were first defined to describe the normalized differences between the predicted (β π) and the targeted values (β π,ππππππ‘). Then the least squares method was applied to minimize the sum of errors (ππ) of both objective functions in order to obtain solutions that satisfied both target parameters with a set tolerance of 0.05. Figure 7.2 Schematic algorithm of Method I 54 7.2 Method II: One Variable Problem Depending on the end product applications, some mixture properties may be more crucial to users than others. For example, tensile strength of a mixture was assumed to have a higher importance than pulp freeness in this study. Hence, only the objective function for tensile strength (β ππππ πππ,π) was first established, then the value of one was assigned to operating conditions that met the set criteria when the absolute value of the objective function was less than or equal to the set tolerance of 0.05. Conditions that failed to meet the specified criteria were given the value of zero and a matrix (M) of zeros and ones was constructed. Next, by multiplying matrix M with matrix G, which contained freeness values that were generated using all possible combinations of SW1 ratio, HW ratio, SW1 energy and HW energy, matrix N, possessing operating conditions that satisfied target tensile strength and corresponding CSF values, was obtained. Only solutions in matrix N that gave CSF values within 5% variance to the target CSF were reported in order to ensure both target parameters were achieved. 55 Figure 7.3 Schematic algorithm of Method II 7.3 Comparison of Method I and II A graphical representation of data sets, which were defined on a 3D grid using SW1 ratio, SW1 and HW energies as the three axes, was created to show all possible solutions to achieve target tensile strength and freeness of a pulp mixture in Figure 7.4. To compare 56 Method I and Method II, a target tensile strength of 50N.m/g and a CSF of 380mL were considered. Figure 7.4 3D volumization plot As displayed in Figure 7.5, 9 combinations of SW1 and HW energies (red nodes) at 0.45 SW1 ratio are determined to reach both target tensile strength and CSF, where the sum of errors for both objective functions are at a minimum. After the solutions are sorted, Method I gives 24 sets of possible operating conditions that can be used to meet specified target tensile strength and CSF within 10% error as detailed in Figure 7.6. 0.2 57 Figure 7.5 SW1 and HW energy combinations at 0.45 SW ratio for Method I Figure 7.6 Method I solutions 0204060801001201401601802000 20 40 60 80 100 120 140HW Energy (kWh/t)SW1 Energy (kWh/t)SW1 Ratio = 0.45 SW1 Ratio = 0.5 SW1 Ratio = 0.55SW1 Ratio = 0.6 SW1 Ratio = 0.65 SW1 Ratio = 0.758 77 energy combinations are found at 0.45 SW1 ratio to meet target tensile strength using Method II. Only 14 conditions are accepted as they have freeness values within 5% of the set CSF shown in Figure 7.7. Method II reports a total of 30 sets of operating conditions that can help produce a mixture with the specified freeness and tensile strength within acceptable precision (Target ο± 5%) in Figure 7.8. As expected, the second method gives more solutions than the first method as it only considers one variable at a time, in this case, tensile strength first then freeness. While Method I finds solutions close to both targets, but not necessarily matching any, Method II gives the flexibility to grant higher importance to one parameter than the other in order to take account of special circumstances. Nevertheless, both approaches are useful and effective for determining solutions needed to meet multiple target properties. Figure 7.7 SW1 and HW energy combinations at 0.45 SW ratio for Method II 59 Figure 7.8 Method II solutions 7.4 Graphical User Interface (GUI) To take it a step further, a graphical user interface incorporating the two methods and other user-specified criteria was created to deliver a complete package for operators to use. In Figure 7.9, the bottom right corner displays all possible operating conditions that can be manipulated to meet the target tensile strength and freeness specified within a set tolerance for the selected method. The production cost to generate the desired mixture is also computed, based on energy price and price for each pulp component inputted by the user. Based on that, the recommended operating condition that renders the lowest cost is emphasized in red color font. 0501001502002500 20 40 60 80 100 120 140HW Energy (kWh/t)SW1 Energy (kWh/t)SW1 Ratio = 0.45 SW1 Ratio = 0.5 SW1 Ratio = 0.55SW1 Ratio = 0.6 SW1 Ratio = 0.65 SW1 Ratio = 0.7SW1 Ratio = 0.7560 Figure 7.9 GUI display 61 8 Conclusions and Recommendations 8.1 Conclusions Refining is a process in which a pulp undergoes significant changes in fiber morphology, which predominately plays a vital role in the development of end product properties. This treatment becomes even more critical when dealing with mixtures. In the first part of this dissertation, a sound and effective methodology was developed to define mixture rules for mixture properties for a separate refining system. The mixture models established, using easily attainable variables (weight fraction of each pulp component and refining energy for each pulp) and the concept of Taylor series expansion for tensile strength and pulp freeness for both Case I and II, provide predictions within 90% confidence interval of the observed data. The second part of the dissertation focused on the establishment of the two methods in determining the operating conditions required to achieve multiple target properties. Although both offer solutions that can be used to generate pulp mixture with specified target properties within acceptable precisions, Method II is a better option when a weighted parameter is desired. Lastly, a simple but practical graphical user interface was developed, outlining a summarized list of possible operating conditions to achieve specified parameters and operation recommendations to minimize production cost. 62 8.2 Recommendations 1) Relate mixture rules to paper machine application: The work done thus far only simulated the pulp stock in the blend chest. A trial on a commercial paper machine can be beneficial to gain more insight into correlating the properties of the finished product with results found in this study. 2) Develop mixture rules for co-refining system: This dissertation only examined the system of separate refining. It is of interest to see whether the same mixture rules applies to co-refining. 3) It can bring great significance to the pulp and paper industry if users can apply the mixture equations established to predict end product properties and provide corresponding sheet structure to help minimize the breakages in bonding between fibers. 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TAPPI Papermakers Conference Proceedings. Lumiainen, J. J. (2000). Refining of chemical pulp. In H. Paulapuro, Papermaking Part I, Stock Preparation and Wet End (Vol. 4). Fapet Oy. Mohlin, U. B. (1987, November). Pulp quality determines paper function. 14-20. Mohlin, U. B., & Wennberg, K. (1984, January). Some aspect of the interaction between mechanical and chemical pulps. TAPPI Journal, 67(1), 90-93. Nuttall, H. G., Mott, L., Mayhead, G., & Duguid, C. (1999). The influence of key refining variables on energy use. Paper technology, 78-90. Olejnik, K. (2013). Impact of pulp consistency on refining process conducted under constant intensity determined by SEL and SEC factors. BioResource, 8(3), 3213. 65 Page, D. H. (1969, April). A theory for the tensile strength of paper. TAPPI Journal, 52(4), 674-681. Page, D. H., Jordan, R. S., & Barbe, B. D. (1985). Curl, crimps, kinks and microcompressions in pulp fibers - their origin, measurements and significance. In: Papermaking raw materials: their interaction with the production process and their effect on paper properties. Transactions of the 8th fundamental research symposium. 1, pp. 183-227. Oxford: Mechanical Engineering Publications Limited. RISI. (2011, March). Virtual pulp - bringing reality to predicting quality. Pulp & Paper International, pp. 23-27. Stevens, W. V. (1992). Refining. In: Koecurek M. J., Pulp and paper manufacture. Joint Committee of TAPPI and CPPA, 6, pp. 187-219. Atlanta. TAPPI Test Methods T205 sp-02. (2007). Technical Association of the Pulp and Paper Industry. Atlanta: TAPPI Press. Wultsch, V. F., & Flucher, W. (1958). The escher-wyss small refiner as a standard test apparatus for modern stock preparation plants. Das Papier, 12(13/14), 334-342. 66 Appendix A Derivation of Almin and de Ruvoβs Model From this exercise, it can be shown that Eqn 7 used by Almin and de Ruvo (1967) can be derived from the expansion of a second order Taylor series. The development here is confined to a three component case (π = 3). Recalling the expression for the general theorem of a multi-dimensional Taylor series (Eqn 11) and for a second order polynomial, π = 2, the equation becomes π(?Μ ?) = βπ(2)(?Μ ?)2!(?Μ ? β ?Μ ?)2 = π(?Μ ?) + (?Μ ? β ?Μ ?)ππ·π(?Μ ?) +π·2π(?Μ ?)2!(?Μ ? β ?Μ ?)π(?Μ ? β ?Μ ?)βπ=0 Writing out in full, π(?Μ ?) = π(?Μ ?) + βππ(?Μ ?)ππ₯π(π₯π β ππ) +12!β βπ2π(?Μ ?)ππ₯πππ₯π(π₯π β ππ)(π₯π β ππ)3π=13π=13π=1 If ?Μ ? = 0, then π(0) = 0, which further simplifies the problem. A reduced form is obtained. π(?Μ ?) = π΄1π₯1 + π΄2π₯2 + π΄3π₯3 + π΅11π₯12 + π΅12π₯1π₯2 + π΅13π₯1π₯3 + π΅22π₯22 + π΅23π₯2π₯3 + π΅33π₯32 Since π₯12, π₯22 and π₯32 can be described by π₯1, π₯2 and π₯3 respectively. The higher order π₯π2 terms can be neglected. Then, the equation becomes π(?Μ ?) = π΄1π₯1 + π΄2π₯2 + π΄3π₯3 + π΅12π₯1π₯2 + π΅13π₯1π₯3 + π΅23π₯2π₯3 and can be re-written in the expression below, which yields the same form as shown in Eqn 7. π(?Μ ?) = β π΄ππ₯π + β β π΅πππ₯ππ₯πππ>ππβ1π=1ππ=1
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Developing mixture rules for non-conservative properties for pulp suspensions Tsai, Pin Wen (Wendy) 2016
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Title | Developing mixture rules for non-conservative properties for pulp suspensions |
Creator |
Tsai, Pin Wen (Wendy) |
Publisher | University of British Columbia |
Date Issued | 2016 |
Description | Nowadays new technologies emerge constantly and people continuously strive to meet challenges. The Pulp and Paper industry has been faced with many changes in recent years. One of which is to diversify the fiber baskets to produce a wide range of products. To help papermakers to accommodate this transition from a single pulp component to a multi-component furnish used in their process, this paper first puts effort into developing a sound and effective methodology to characterize mixture rules that predict properties such as tensile strength and pulp freeness. Using an expansion of a higher order Taylor series as the backbone of model development and removing model parameters based on the limitation of the separately refined system and statistical analysis, the tensile strength and pulp freeness models give predictions close to the observed measurements within 10% variance. Furthermore, two methods, one being the minimization approach using least squares, and the other being the one variable approach, when granting more emphasis on one particular mixture parameter than the other is preferred, are established to determine the operating conditions required to satisfy multiple target properties. Lastly, a graphical user interface, built on the defined mixture models, is also constructed to make recommendations of the optimized condition that can be applied to generate a mixture to achieve both target properties at minimum cost. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2016-04-26 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0300222 |
URI | http://hdl.handle.net/2429/57834 |
Degree |
Master of Applied Science - MASc |
Program |
Chemical and Biological Engineering |
Affiliation |
Applied Science, Faculty of Chemical and Biological Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2016-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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