Applied Science, Faculty of
DSpace
UBCV
McIntosh, Nicholas Joseph
Master of Applied Science - MASc
University of British Columbia
Compression of wood chips coupled with chemical treatment in a liquid bath has been shown to be an effective method to reduce the energy required to mechanically produce wood pulp. Though the utility of such compression/chip impregnation processes has become well known through implementation at pilot and mill scales, a fundamental understanding of the operational mechanism of these processes remains to be obtained. Towards this end, an experimental and statistical study has been conducted. In this work, better understanding the mechanism of action and measuring the effectiveness of a given compression/impregnation process has come about via developing two complementary methods to characterize arbitrarily compressed wood chips. The first method, applied to chips compressed at mill and pilot-scale, involves extracting a pore size distribution from a wood sample by use of a combination of microscope imaging, digital image analysis, and model fitting to the data obtained from image analysis. The second method, applied to wooden blocks compressed at lab-scale, involves directly measuring liquid uptake of a wooden block from a reservoir. Results from image analysis suggest that compression changes the structure of wood in a way which hinders liquid uptake. This observation, taken with existing heuristic relationships between compression and liquid uptake ability of wood chips, implies that the effectiveness of a wood chip compression/impregnation process relies on the alteration of chip properties on a length scale larger than that captured by the image analysis (i.e., that of individual fibres). Direct measurement of liquid uptake of compressed wood blocks at lab-scale resulted in the emergence of a stochastic relationship between compression conditions (compression rate and total strain applied to a block) and the changes imparted to wood’s micro-structure. Taken in combination, the results of the experimental program conducted suggest that the effectiveness of wood chip compression/impregnation processes lies within the ability of compression to modify the structure of an aggregate of wood chips rather than individual wood chips on a fibre scale.
Measurement of the Changes to the LiquidTransport Properties of Wood Due toCompressionbyNicholas Joseph McIntoshB.Sc. Chemical Engineering, University of Calgary, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Chemical and Biological Engineering)the university of british columbia(Vancouver)December, 2016© Nicholas Joseph McIntosh 2016AbstractCompression of wood chips coupled with chemical treatment in a liquid bath has beenshown to be an eective method to reduce the energy required to mechanically producewood pulp. Though the utility of such compression/chip impregnation processes hasbecome well known through implementation at pilot and mill scales, a fundamentalunderstanding of the operational mechanism of these processes remains to be obtained.Towards this end, an experimental and statistical study has been conducted.In this work, better understanding the mechanism of action and measuring theeectiveness of a given compression/impregnation process has come about via de-veloping two complementary methods to characterize arbitrarily compressed woodchips. The first method, applied to chips compressed at mill and pilot-scale, involvesextracting a pore size distribution from a wood sample by use of a combination ofmicroscope imaging, digital image analysis, and model fitting to the data obtained fromimage analysis. The second method, applied to wooden blocks compressed at lab-scale,involves directly measuring liquid uptake of a wooden block from a reservoir.Results from image analysis suggest that compression changes the structure ofwood in a way which hinders liquid uptake. This observation, taken with existingheuristic relationships between compression and liquid uptake ability of wood chips,implies that the eectiveness of a wood chip compression/impregnation process relieson the alteration of chip properties on a length scale larger than that captured by theimage analysis (i.e., that of individual fibres). Direct measurement of liquid uptakeof compressed wood blocks at lab-scale resulted in the emergence of a stochasticrelationship between compression conditions (compression rate and total strain appliedto a block) and the changes imparted to wood’s micro-structure.Taken in combination, the results of the experimental program conducted suggestthat the eectiveness of wood chip compression/impregnation processes lies within theability of compression to modify the structure of an aggregate of wood chips ratherthan individual wood chips on a fibre scale.iiPrefaceThis thesis is based on the experimental and analytical work of Nicholas JosephMcIntosh, supervised by Drs. James Olson and Mark Martinez.The production and collection of compressed chips from Andritz’s pilot-scalerefining/R&D facility in Springfield, Ohio, USA (discussed in §3.3.1) was completed bythe operational and engineering sta of the facility. Sectioning, imaging, and modelfitting work done subsequently to the collected chipettes was the sole work of theauthor.The production and collection of compressed chips from Millar Western’s mill inWhitecourt, Alberta, Canada (discussed in §3.3.2) was completed by the operationaland engineering sta at the mill. Sectioning, imaging, and model fitting work donesubsequently to the collected chipettes was the sole work of the author.The hydraulic piston setup used to compress wood blocks (discussed in Chap-ter §4.1.1) was modified and instrumented by Daniel Patterson. The setup and executionof several of the compression tests was completed as a joint eort between him andthe author.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Listings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Energy and Refining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Study Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Thesis Overview and Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 The Structure of Wood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.1 Overall Organization and Growth . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Fibre Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Screw Press Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.1 Eects on Energy Usage and Pulp Properties . . . . . . . . . . . . . . . 82.2.2 Eects on Wood (chip and fibre) Morphology . . . . . . . . . . . . . . . 102.3 Capillary Driven Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.1 Basic Descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.2 Bundle of Capillary Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14ivTABLE OF CONTENTS2.3.3 Non-cylindrical Capillaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.4 Initially Fast Imbibition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4 Research Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.5 Overview of Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Image Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.1 Imaging of Wood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2 Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2.1 Obtaining Pore Sizes from Images . . . . . . . . . . . . . . . . . . . . . . . . 253.2.2 Pore Population Modeling via Gaussian Mixture Models . . . . . . 273.2.3 Determining Adequate Pore Size Sample Size . . . . . . . . . . . . . . . 293.3 Data and Analysis of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3.1 Pilot-scale Chipettes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3.2 Multiple Compression Stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 Liquid Uptake Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.1 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.1.1 Wood Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.1.2 Liquid Uptake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.1.3 Power Curve Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.2 Analysis of Measured Liquid Uptake Data . . . . . . . . . . . . . . . . . . . . . . . . . 454.2.1 Overall Trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2.2 Statistical Perspective on Power Curves . . . . . . . . . . . . . . . . . . . . 495 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.1 On Wood (chip and fibre) Morphological Changes Due to Compression 525.2 On Liquid Uptake Ability of Chipettes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Appendix A Imaging/Image Analysis Details . . . . . . . . . . . . . . . . . . . . . . . . 61A.1 Image Analysis Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61A.2 Camera Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Appendix B Liquid Uptake Time Series Acquisition . . . . . . . . . . . . . . . . . . 63Appendix C Density Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67C.1 Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67C.2 Kernel Density Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68C.3 (Gaussian) Mixture Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69vTABLE OF CONTENTSC.3.1 Model Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Appendix D Non-uniqueness of Tube Bundles . . . . . . . . . . . . . . . . . . . . . . 75viList of Tables3.1 Pilot-scale screw press trial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 30A.1 Camera parameters used for imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62viiList of Figures1.1 Simplified process flow diagram for a conventional (C)TMP pulpingprocess with a screw press added prior to refining . . . . . . . . . . . . . . . . . 11.2 Simplified schematic of screw press internals . . . . . . . . . . . . . . . . . . . . . . 22.1 Schematic of wood’s macro- and micro-structure . . . . . . . . . . . . . . . . . . . 62.2 Schematic of fibre wall layers and matrix structure . . . . . . . . . . . . . . . . . 92.3 Generic cylindrical capillary tube geometry . . . . . . . . . . . . . . . . . . . . . . . 132.4 Generic non-cylindrical capillary tube geometry . . . . . . . . . . . . . . . . . . . 162.5 Initially fast capillary flow time evolution . . . . . . . . . . . . . . . . . . . . . . . . . 213.1 Illustration of wood sample preparation process for microscope imaging 243.2 Progression from raw image to found contours . . . . . . . . . . . . . . . . . . . . 283.3 Sample images with contours from pilot-scale produced chipettes . . . . . 323.4 Pore size distributions obtained from image analysis of chipettes pro-duced at pilot-scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.5 Simplified multi-stage compression process flow diagram . . . . . . . . . . . . 363.6 Sample images with contours from mill-scale, multi-stage compressionproduced chipettes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.7 Pore size distributions obtained from image analysis of multi-stagecompression chipettes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.8 Capillary driven liquid uptakes predicted from the pore size distribu-tions obtained from image analysis of multi-stage compression chipettes 404.1 Labeled photograph of the hydraulic piston and bearing/plate assemblyused to simulate wood chip compression at lab-scale . . . . . . . . . . . . . . . 434.2 Schematic of the liquid uptake measurement laboratory setup . . . . . . . . 434.3 All measured uptake curves grouped by compression rate used duringcompression of the sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.4 Uptake time series comparison chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.5 All measured uptake curves grouped by total amount of compressionsample was subjected to . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47viiiLIST OF FIGURES4.6 Power curve parameters a and b grouped by their associated blockette’scompression rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.7 Power curve parameters a and b grouped by the total strain applied totheir associated blockettes during compression . . . . . . . . . . . . . . . . . . . . 49ixList of ListingsA.1 find_pores() function definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61B.1 run_experiment() function definition . . . . . . . . . . . . . . . . . . . . . . . . . . 63B.2 SartScale object class definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64xAcknowledgementsThis project simply would not have been possible if not for the support and guidanceof a number of individuals and organizations.Foremost, the seemingly endless supply of support and guidance (technical orotherwise) I have received from my supervisors Drs. James Olson and Mark Martinez isvery much so deserving of recognition. My graduate school pursuits would not havebeen possible without them.As research engineers responsible for leading the Energy Reduction in MechanicalPulping project this work is part of, Francisco Fernandez and Meaghan Miller haveproven themselves invaluable in all matters related to project logistics. Their help andsupport throughout the process has alleviated much trouble and is greatly appreciated.Around the Pulp and Paper Centre, both George Soong and Anna Jamroz have beena pleasure to work with and have provided much appreciated support during both laband o ce work.Dr. Nuwan Sella Kapu was particularly helpful and instructive in the early stagesof the image analysis work. His kindness and willingness to help is impossible not torecognize and is very much appreciated.In setting up the compression tests, Daniel Patterson has shown himself to simulta-neously be a hard worker, talented experimentalist, and kind person. His help in thelab is very much so valued.The support and encouragement of Andrew Hawryluk, Dr. Josephine Hill, and EltonLawes during the early stages of my graduate school pursuits is very much appreciated.I thank Jorge Rubiano for not only oering fruitful discussion and help with respectto technical matters, but also his friendship during my time in graduate school.The consistent and ceaseless love, encouragement, and support of my family,friends, and partner has provided me with inspiration and drive to do well in all I do;for that I am forever indebted to all of them.This work has been completed as part of the Energy Reduction in MechanicalPulping project and the financial or in-kind support of all of the project’s partners isgreatly appreciated.xiDedicationFor my parents, Joe and Lynn.Any success I may attain is due, principally, to their relentless love and support.xii1 Introduction1.1 Energy and RefiningMechanically dismantling wood chips into fibres suitable for paper making applicationsaccounts for 11% of BC’s annual electrical energy production [62]. Economic, environ-mental, and process e ciency concerns make reducing the energy consumption ofthe process (broadly referred to as mechanical pulping or refining ) a desirable end,while retaining final pulp properties and quality is generally a requirement. Earlyapproaches to energy reduction include steaming wood chips prior to refining [55](either at or above atmospheric pressure) which acts to soften both the fibre andlignin components of the wood chips and subsequently reduce the energy required toseparate fibres while helping to preserve native fibre length. Pulp produced in such aprocess is known as thermo-mechanical pulp, or TMP. Another class of approachesinvolve the application of softening chemicals (commonly sodium sulphite [4] or alkalinehydroxide [11]) to soften the wood matrix and partially break down the lignin-richmiddle lamella holding fibres together to ease the dismantling of wood chips to woodfibres; when combined with a steaming process, pulps produced in this way are knownas chemi-thermo-mechanical pulps, or CTMP. The eectiveness of both the steamingstep and the chemical applications stems in a CTMP process are both dependent onthe way in which the chips interact with the steam and liquid chemical bath. Improvingboth of these subprocesses as well as achieving mild chip defibration and extractiveremoval [56] has been shown to be achievable by compression of the chips prior torefining and liquid chemical bath treatment [14, 22].ChipsScreens/Cyclonesto papermachineLCRLCRHCRScrew PressFigure 1.1: Simplified process flow diagram for a generic, conventional (C)TMP pulping processwith a screw press added prior to high consistency (HCR) and low consistency (LCR) refining,pulp cleaning/stock prep, and paper making. A conventional process would be identical to thatdepicted above if the screw press and uptake tank in the dashed box were removed.1CHAPTER 1. INTRODUCTIONChip compression is commonly achieved in a screw press [21, 22, 35, 56]. Inside ofa screw press, compression/chemical contacting is achieved by conveying chips throughan increasingly narrowing annular space around a central screw axle before exiting intoa liquid chemical bath external to the screw body (see Figure 1.2 for a simplified screwpress schematic). The products that emerge from the screw press are distinct fromthe chips that were used as feed with respect to appearance, microstructure, and bulkproperties and will thus be referred to in this work as chipettes to distinguish them fromtheir precompression counterparts. It is suggested in [22] and [35] that the productionof pulp from wood chips be viewed as the product of two distinct steps: defibration ofchips into fibres and subsequent development of fibre properties to make them moreamenable to paper making. In a classical (C)TMP process, defibration occurs when thechips first enter the refiner and are cyclically sheared/compressed within the ‘breakerbar’ section of a high consistency refiner. Salmén and colleagues [47] suggest that alower compression loading frequency than that present in the breaker bar section of anormally operated refiner would result in a greater wood destructuring eectiveness(in terms of energy input per wood matrix area liberated). This study provides atheoretical justification for the (relatively) slow compression which is present in ascrew compression process: by compressing chips before refining (i.e., to some extentdecoupling wood defibration and fibre fibrillation) the energy imparted to the chipwithin the refiner may go towards the more targeted development of fibre propertiesrather than (ine ciently) achieving simultaneous defibration.Figure 1.2: Cutaway of screw compression device. Chips are fed to the device around theleftmost part of the screw and are conveyed left to right through a narrowing annular space.Compressed chips then exit into a chemical bath where they imbibe liquid. Note the perforatedscrew housing which allows for water/extractive removal along the length of the screw throughthe compression section. (Image from Christer Sandberg, 2014)Though a fair amount of work has been done to understand the eectivenessof a screw press/chemical treatment process’s inclusion prior to the conventionalmechanical pulping process in terms of both energy consumption and pulp qualitychanges [21, 22, 31, 35, 56], a gap in the knowledge of how chip morphology changeswith compression as well as how chip (micro- and macro-) geometry eects liquiduptake remains. This work aims to, at least partially, fill that gap.2CHAPTER 1. INTRODUCTION1.2 Study ObjectivesThe overall objective of this work centers on better understanding the fundamentaloperational mechanism of screw press machines with a focus, specifically, being on:Objective i. understanding the structural changes incurred by wood chips whichundergo a compression process, andObjective ii. understanding how the structure of wood chips relates to their ability toabsorb liquid.1.3 Thesis Overview and OrganizationThis thesis is organized as follows:Chapter 2 provides the necessary background information to comfortably under-stand this work’s motivation and content. The anatomy, organization, and structureof wood as a material is presented and the nomenclature is clarified. The currentstate of knowledge surrounding screw press mechanism of action (specifically, howchip and fibre morphology has been observed to change with compression) as well asa summary of the observed advantages to screw press inclusion in a CTMP processis discussed. A formal description of capillary flows for various tube geometries andcombinations as well as an analysis of an idealized flow into a rapidly expandingcapillary (this analysis narrows the scope of the liquid uptake objective 1.2.ii to that ofa static problem) is also given. A brief outline of the approach and methods used inachieving the objectives stated in §1.2 is also given for narrative benefit.Chapter 3 details the image analysis work which was done to characterize woodchips which had been treated by a pilot-scale screw compression device as wellas samples obtained from an mill-scale multi-stage compression process. Pore sizedistributions within the treated chips as well as predicted liquid uptake capacities of thetreated chips are found and the changes to both tracked with the various compressionconditions the samples underwent.Chapter 4 details the lab-scale work which was done to characterize single, com-pressed wood blocks by directly measuring their liquid uptake ability. The experimentsdiscussed in this chapter remove any eects of chip-chip interaction present at largerscales and examine the simplest possible idealized compression of wood in an attemptto observe the fundamental mechanism of chip destructuring. Lab methods areoutlined and the results of the experiments are analyzed using statistical tests whichexamine the stochastic nature of wood structure’s response to compression.Chapter 5 acts to summarize the findings of the overall lab program carried out aswell as outlines the conclusions arrived at based on these results. A dierence betweenthe results found from the single block experiments and the observed eectiveness of3CHAPTER 1. INTRODUCTIONindustrial-scale screw press processes is noted. Several recommendations for futurework which may reconcile these dierences are suggested.Appendix A provides a commented copy of the image analysis code used to obtainpore size distributions from thin sections of treated chips as well as a summary of thephotographic setup used to obtain images of the aforementioned chips.Appendix B provides a commented copy of the code used to collect a time seriesof mass readings from a lab balance for the lab-scale liquid uptake experiments. Thiscode relies on the use of a serial port emulation interface to retrieve data from thebalance and so a commented version of the code used for this purpose is also provided.Appendix C provides a formal exposition of the density estimation methods usedto model the pore size distribution of wood chips.Appendix D provides a brief proof of the non-uniqueness of a tube bundle whenconsidering a bundle’s imbibition dynamics. This fact has implications within thediscussion of an ‘optimal’ pore size distribution being achieved by compression and sois included in this thesis for completeness.42 Background and MotivationThis chapter acts, eectively, as a preface to the remainder of the work presented andprovides the technical background required in taking the research approach prescribedin §2.4. Firstly, an overview of wood as a material is presented with an emphasis onstructure, function, and organization. Previous studies concerning screw press/liquiduptake processes are summarized and the gaps in the knowledge base are highlighted.Formalism surrounding capillary driven flows in various capillary geometries is workedthrough and the relative importance of the driving forces/losses present during initiallyrapid imbibition is evaluated for an idealized geometry. The results of this sub-studysuggest that the imbibition of liquid into a wood chip may be regarded as a static (withrespect to chip geometry) process and set forth a precedent for the approach taken inachieving the objectives stated in §1.2. An outline of these approaches is given and themethods used in applying them are briefly described.2.1 The Structure of Wood2.1.1 Overall Organization and GrowthAlthough on a per-species basis properties can vary quite markedly, the general organi-zation, chemical makeup, and structure of wood is quite uniform across both speciesand biological groups. Mechanically, wood is a complex, anisotropic cellular/porousmaterial [20, 55], though an overall organization of this complex structure exists and isdue to the manner in which wood cells (i.e., fibres or tracheids) grow and multiply.Though often the most visually striking feature of a tree, the crown (leaves andbranches) does not directly participate in wood production. Instead, the trunk ofa tree acts as physical support, vascular network, and fibre production center. Thephotochemical reaction of photosynthesis which occurs in the leaves or needles ofa tree provides energy for cell replication and growth which occurs only in a thinradial cylinder between the outermost trunk wood and bark known as the cambium(see Figure 2.1 for cutaway of the trunk structure). It is within the cambium that celldivision occurs with new fibres being deposited ‘inside’ the tree as the tree growsradially outward. The rate of fibre growth in the cambium is heavily dependent5CHAPTER2.BACKGROUNDANDMOTIVATIONFigure 2.1: Illustration of the main structural sections of (soft) wood as it grows within a tree (trunk cutaway shown) as well as the structureand organization of wood at the fibre-scale. Though all of the wood within a tree is structurally similar, the activity and specific role of aregion at any given time is dependent on the time within growth season and relative location within the tree. Dierent functions and activityis explained further in §2.1.1. Note the size variation between early wood and late wood tracheid’s walls and lumens. (Illustrations sourced fromSmook, 1992 [55])6CHAPTER 2. BACKGROUND AND MOTIVATIONon environmental conditions with the largest amount of variability due to seasonalfluctuations; in the relatively biologically prosperous spring and fall it is large, thin-walled fibres which develop (i.e., early wood tracheids) while cooler fall weather resultsin the development of thick-walled, small diameter fibres (i.e., late wood tracheids). Asthere is often no clear transition from early to late wood, tracheids which are moderatein their size and wall thickness are often referred to as transition wood. It is thisannual size variability in the fibres which are produced that results in the familiar ringpattern in a trunk’s cross section (i.e., annual or growth rings).The layer of bark at the outermost radial extent of the trunk serves two mainpurposes: the inner bark acts to transport sap axially within the tree while the outerbark acts as a protective outer jacket which shields inner structures from environmentalstresses (e.g., insects and pests, fire, temperature, precipitation). The fluid transportsystem within the inner bark (predominantly allowing for axial conveyance of fluid) isconnected to that within the rest of the tree via radially oriented tracheids known asrays. Just within the cambium layer of a trunk is the light-colored sapwood. Sapwoodfunctions as an area for sugar storage, acts as structural support for the crown,and is part of the water conduction system from roots to crown. The innermostcolumn of fibre in a tree is known as heartwood and only fills the role of structuralsupport for the tree. The soft central portion of the heartwood is known as the pith.Heartwood is relatively rich in organic compounds which make it often darker, denser,and mechanically tougher than sapwood. Small, metabolically active cells knownas parenchyma are also scattered throughout the wood matrix within trees. Thesesmall cells are arranged together in axial or radial (often ray-like) structures and areresponsible for nutrient and water storage and conduction.All of the information stated above as to the general structure and function ofwoods constituent parts is equally valid for all types of wood, however two mainsub-groups exist within unique structural elements being present in each. Softwood(i.e., gymnosperm or conifer) species are almost entirely composed of simple tracheidswith additional resin canals and rays to aid in fluid transport within the tree. Hardwood(i.e., angiosperm) species are, in general, much more structurally complex. Tracheidfibres in a hardwood are, in general, much shorter than those in a softwood speciesand are accompanied by much shorter, wider fibres called vessels. Vessels make thelarge, rapid growth which occurs in hard wood species during the spring possible byfacilitating increased fluid transport.2.1.2 Fibre StructureJust as the larger-scale fibre network within a tree does, the physical construction ofindividual fibres within the wood matrix also play a large role in the overall biologicaland structural function of a tree. Each fibre within a tree is a hollow, tapered structurethat is closed at either end. Actual fibre length varies with species and biological group7CHAPTER 2. BACKGROUND AND MOTIVATIONbut, on average, softwood fibres are longer than their hardwood counterparts. Fibrelengths of 3-4mm are not uncommon for softwoods and 1-2mm for hardwoods [10, 55].Fibres also have relatively large aspect ratios with outer diameters of 30-40µm, thoughlarger lengths and diameters are not unheard of for particular wood species. Forexample, Redwood species (those belonging to the Sequoioideae sub-family) may havefibre lengths reaching 6.1mm with diameters of 50-65µm [55]. Wall thickness of fibresmy also vary substantially from species to species, though are generally ∼1µm. Forexample, Norway spruce early wood fibres have 1-3µm thick walls while the smallerdiameter, thicker walled late wood fibres have 2-7µm thick walls [13].Individual fibres are held together within the wood matrix by an adhesive layer(i.e., the middle lamella) made of the class of amorphous, phenolic polymers knowncollectively as lignin [8]. The layer of the fibre wall in direct contact with the middlelamella (primary wall layer, P1) is relatively thin (∼0.05µm [55]) and precedes threesecondary wall layers (S1, S2, S3) which dier from one another in the winding angleand alignment of their constituent cellulose strands known as microfibrils. Togethercellulose and lignin make up ∼75% of the mass of a tree (on a dry basis) [8], with theremainder being a mixture of light1 sugar polymers known as hemicelluloses. The voidspace within the S3 layer is known as the fibre’s lumen. See Figure 2.2 for a schematicof each layer.Fibre-fibre fluid transport is made possible by small holes through the walls ofadjacent fibres called pits or simply pores. These pits give adjacent fibres the ability toexchange fluids between lumens and represent the lowest length scale features of thetree’s vascular network. Pits which border rays allow for exchange and fluid transportradially within the tree all the way to the cambium and inner bark, while ordinaryfibre-fibre pits allow for axial fluid transport.2.2 Screw Press Operation2.2.1 Eects on Energy Usage and Pulp PropertiesThe treatment of wood chips with screw press machines and softening chemicals priorto refining have been shown at mill and pilot scale [21, 22, 31, 35, 39, 56] to both reducespecific energy during pulp production and improve select pulp properties. Morespecifically:In [35], chip treatment with a screw compression device prior to refining was shownto reduce specific energy consumption for overall pulp production by up to 6%. In thesame work, the use of a sulphite chip treatment in addition to chip compression was1Hemicelluloses are both lighter and more chemically diverse than cellulose: while cellulose is strictlya glucose polymer with 7000-15000 monomer units, hemicelluloses generally only contain 500-3000monomer units which can include xyloglucans, xylans, glucomannans and galactoglucomannans [19].8CHAPTER 2. BACKGROUND AND MOTIVATIONFigure 2.2: Schematic of fibre wall layers and matrix structure. Note that the lumen region,marked ‘L’, is void space within the fibre. The concentration of lignin/cellulose/hemicellulosesvary from ∼(75,25,0)% in the middle lamella (ML) to ∼(15,40,45)% in the S3 layer [38]. Similarly,the physical structure of each of the fibre wall layers varies in cellulose strand (microfibril)orientation and organization. As such, each of the wall layers behaves dierently both chemicallyand mechanically. (Illustration sourced from Smook, 1992 [55])observed to further reduce the specific energy requirements by 12%. Additionally, theuse of sodium sulphite in addition to compression was observed to reduce shive contentin pulps relative to those produced at equal specific energy without sulphite treatment.Kure et al. in [31] observe similar results: shive content was reduced while tensile andtear strength increased for pulps produced with compressive pre-treatment relative tothose produced without (at equal specific energy). Interestingly, in the same study, anincrease in refining intensity was not observed to reduce fibre length when coupledwith chip compression prior to refining. When a compressive wood chip treatmentwas combined with a fiberizer (a ‘moderately pressurized refiner where defibrationof pre-treated chips into fibre bundles takes place’ [21]) treatment, high intensity firststage refining, and chemical treatment (magnesium hydroxide and hydrogen peroxide)following chip compression specific energy was observed to fall by 42% for lobollypine pulps produced to a given tensile index [21]. This energy value is relative to theenergy consumed by a process similar to that described above but which excludes chipcompression, chemical treatment, and fiberizer treatment. A reduction in fines contentwas simultaneously observed for this treatment recipe.Overall, the energy saving merit of compressive pretreatment processes seems tolie in the ability of such processes to decouple (to some extent) the subprocessesof wood defibration and fibre property development/fibrillation. This idea is statedexplicitly in [31]: the energy expended during pulping (‘fibre liberation’) is groupedinto three categories: that which goes to the creation of new fibre surfaces, that9CHAPTER 2. BACKGROUND AND MOTIVATIONwhich goes to plastic fibre deformation, and that which goes to fibre breakage. Koran,in [30], states that the energy reduction capacity of a given process is maximized if theenergy expended on plastic fibre deformation and breakage is minimized. Compressivepretreatment of wood chips seems to be a a good fit for meeting this criteria, at least atthe less intense end of possible operational circumstances (i.e., low, slow compression).For example: in [39], Parkinson et al. found that fibre cutting occurred at high (5:1)volumetric compression ratios but not at lower (3:1) ratios.In [36, 56] it is noted that the extractive removal accomplished in compressive chippretreatments (up to 40% of the extractives in raw wood chips were observed to beremoved in the single stage of screw compression studied) may contribute to increasedpaper machine e ciencies and improve optical properties of the final pulp.2.2.2 Eects on Wood (chip and fibre) MorphologyChanges to chip structure have been noted in many trials carried out on screwcompression equipment [21, 35, 39, 42]. Both smaller average chip size and a smallerchip size variation is noted in the products of compression ([21, 35, 42]); this resultsin a more even moisture content [36, 39] of the bed, a reduction in the variability ofbed bulk density [39], and increasing uptake of liquid chemicals[39, 56]. Consideringmultiple compression stages: chip bed bulk density and moisture content was observedto decrease and increase, respectively, markedly only after the first stage of compressionand minimally thereafter, while chemical uptake increased uniformly after successivecompression stages [39].With respect to smaller length scale features, fibre/middle lamella delaminationhas been observed to occur predominantly at the S1/middle lamella interfaces of thewood matrix ([21, 35]. This results in cracks forming in a direction parallel to the fibreaxis. With regard to the volumetric compression ratio which many screw devices havenominally attributed to them, a threshold over which fibre breakage occurs seems toexist and has been observed by Parkinson et al. in [39]. Fibre cutting was observed at acompression ratio of 5:1, but not at 3:1 indicating that there exists a maximum usefulamount of compression before defibration eects are overshadowed by fibre breakageand fines generation. However, the fines generation associated with compressive chippre-treatments has been reported in [21] as advantageous to paper properties. Thefines generated act to increase fibre bonding (by way of both sheet densification andenhancing fibre-fibre bond strength) as well as paper surface properties [21].The changes in pore size distributions within the wood matrix of compressedScandinavian spruce chips were quantitatively studied by Peng and Granfeldt in [42] viamercury porosimetry. They found that a majority of the pore volume created by thecompression and defibration of spruce in a screw compression device was found to beattributable to large pores of the order of 20µm while an increase in the surface areaavailable to fluid flow in the wood matrix was mainly attributable to small pores on the10CHAPTER 2. BACKGROUND AND MOTIVATIONorder of 0.5µm. The method of mercury porosimetry employed in this study, however,is questioned as potentially altering the measured pore spaces; the high pressuresrequired for mercury invasion of the pore spaces may have contributed to pore collapseand thus eected the measured values [42].The eect of multiple compressive loading cycles and loading frequency on woodblocks has been studied in [57]. It was found that the relative modulus of a wood samplefalls with subsequent loading cycles. It is suggested that this indicates various stagesof wood matrix damage: first early wood fibre collapse, then late wood fibre collapseand matrix delamination. In the same work it was also observed that variation in thestrain rate (i.e., loading frequency) imposed on the wood samples resulted in varyingdegrees of structural damage; higher frequency loading resulted in a greater drop ofcompressive modulus and a greater degree of structural damage to the macroscopicwood matrix, however lower frequency loading resulted in more widespread damage tothe early wood fibres in the samples. The rate dependence of structural damage (bothwith respect to amount and type of damage) incurred suggests a visco-elasto-plasticresponse from wood even at the relatively low strain rates studied in [42] and presentin a screw press device. The non-linear, rate dependent mechanical properties of woodare confirmed by Placet et al. in [43] with single wood block, lab-scale mechanical tests.As highlighted above, numerous studies have amassed a body of knowledgesurrounding the overall operation and eectiveness of screw press devices with somestudies even attempting to qualitatively look at more fundamental aspects of the process;the energy reduction capacity and eect of pulp properties has been chronicled at pilotand mill scales and some qualitative work has been done to understand the changesincurred by the micro-structure of wood chips by compression. However, no completestudy probing the mechanism underlying the action of a screw press/liquid uptakeprocess has been carried out to date; by abstracting many of the concepts involved inthe industrial version of the process, it is that end which is desired to be achieved bythis work (as detailed in §1.2).2.3 Capillary Driven FlowsThe theoretical analysis of capillary driven flows which follows will serves two mainpurposes within this work:Firstly, the basic theoretical description of capillary driven flows (§§2.3.1 and 2.3.2)serves as a tool which may be used to predict liquid uptake ability of chipettes thathave undergone microscope imaging to extract a pore size distribution (see §3.3.2 forimaging details). The basic description detailed here is not posited to be a particularlyaccurate description of capillary driven imbibition of a liquid into a wood matrix,rather it oers a straightforward and simple method for the comparison of a givenchipette’s structure (by way of pore size distribution) in terms of liquid uptake ability.11CHAPTER 2. BACKGROUND AND MOTIVATIONFurthermore, as the image analysis procedure which is responsible for extracting thepore size distribution from a chipette implicitly only considers a two-dimensionalprojection of the chipette’s pore structure (see §3.1), the relatively simple (i.e., not greatlyaccurate) capillary driven flow theory used here is somewhat appropriate.The second purpose served by exploring capillary driven flow theory is that itallows for the qualitative structural interpretation of simple lab-acquired liquid uptakedata. Specifically: as the liquid uptake measurements made in the lab (see §4.1) do notmake the same two-dimensional-projection assumptions of the pore structure within awood matrix that the image analysis methods do, geometrical complexity and deviationfrom ideal cylindrical capillary tubes needs to be taken into account. By employing asimple analysis of non-ideal capillary tube geometries (see §2.3.3), the deviation fromgeometric ideality implicit in the data gathered during lab-scale uptake tests may bebetter qualitatively understood.2.3.1 Basic DescriptionsWe first seek to describe the evolution of the imbibition of fluid into a capillary tubeover time. To begin, consider a cylindrical capillary tube of radius R inclined at angleψ from horizontal with a wetting fluid (viscosity µ, density ρ, surface tension σ, andmakes a contact angle of θ with the tube surface) which is still at the very entrance(see Figure 2.3 for geometry). As the fluid contacts the tube wall, the balance of fluidsurface tension and adhesive forces between the fluid and the tube wall will curve theliquid interface which, in turn, creates a capillary pressure drop Pc across the fluidinterface. This pressure drop is related to the fluid properties and tube geometry viathe Young-Laplace equation (for a cylindrical capillary):Pc =2σ cos θR. (2.1)Viscous losses (Pv) in the flow are assumed to be described by the Hagen-Poiseuillepipe flow law:Pv =8µhR2dhdt, (2.2)and, in the general case, a static head may be present:Ps = ρgh sinψ. (2.3)Using the above expressions for driving forces and losses, it is possible to write themomentum balance for the fluid in the tube as:12CHAPTER 2. BACKGROUND AND MOTIVATIONPcPv h2RψθFigure 2.3: Generic cylindrical capillary tube geometry.2σ cos θR= ρgh sinψ +8µhR2dhdt+ddt[ρhdhdt]. (2.4)Equation 2.4 is the generalized case for capillary flow in a tube and includes therestrictive eects of gravity. Its solution may be found by separating variables andintegrating, and is implicit in time:t(h) = −hβ− αβ2ln[1− βhα], α =σR cos θ4µ, β =ρgR2 sinψ8µ(2.5)To make imbibition distance values more conveniently calculable, Equation 2.5may be made implicit in time, as per the work in [15], by employing the LambertW function. The W function is defined as the inverse to the exponential functionf(x) = xex, or: x = W (x)eW (x). Using this definition of W , it is possible to rewriteEquation 2.5 into a form which is explicit in time as:h(t) =αβ[1 +W(− exp(−1− β2tα))]. (2.6)Although appearing somewhat cumbersome, this explicit form of the imbibitionrelation is much more readily calculated with most modern numerical software suitesthan its implicit counterpart; it has been used extensively in this work and is givenhere for completeness.A useful simplification of the force balance given by Equation 2.4 comes wheninertial and gravitational eects are ignored. Such a simplification may be made forhorizontal capillaries where a developed flow exists [16]. As shown in §2.3.4, flows atscales relevant to this work (R ∼10µm) develop fully in times much less than onesecond. In making such a simplification, the force balance is reduced to a capillary13CHAPTER 2. BACKGROUND AND MOTIVATIONdriven flow with only viscous losses and has the well known solution [6, 32, 61] (initiallystationary fluid is assumed, or h(t = 0) = 0) which is explicit in time:h(t) =(σR cos θ2µt) 12. (2.7)2.3.2 Bundle of Capillary TubesIn this work wood is considered as a non-isotropic porous media. Given its hollow fibrelaminated nature, a bundle of polydisperse capillary tubes is well suited to modelingthe structure of wood insofar as fluid flow problems are concerned [5, 60]. This typeof model is also well suited to be used in conjunction with the image analysis workexperiments carried out (see Chapter 3) and is useful for qualitative interpretationof results of liquid uptake experiments (see Chapter 4). Although conceptually andmathematically somewhat simple, a bundle of polydisperse tubes may act as the basisfor the derivation of Darcy’s law and is, in practice, quite utilitarian in describingcapillary driven flows in wood [60]. Here a basic formulation of a polydisperse bundleof tubes model will be presented.We begin by recounting the result for the imbibition distance of a perfectly wettingfluid with dynamics viscosity µ into a single, horizontal, cylindrical, capillary tube ofradius r at time t as (Equation 2.7 above):h(r, t) =(σrt2µ) 12. (2.8)It follows then that the volume of liquid imbibed by that same capillary tube isgiven by:ν(r, t) = pir2h(r, t) = pi(σr5t2µ) 12. (2.9)We will restrict the model to having N unique tube radii. For the range of uniquetube size classes i ∈ [1, N ] we take there to be θi tubes of size class i and thus thetotal volume (at some time t) imbibed by all the tubes of radius class i is given by:θiν(ri, t), (2.10)and the liquid imbibed by the whole bundle at time t, V (t), is simply the summationof liquid uptakes by each tube size class, or:14CHAPTER 2. BACKGROUND AND MOTIVATIONV (t) =N∑i=1θiν(ri, t). (2.11)It is instructive to note that the expression for V given in Equation 2.11 is written fora single time; for experimental purposes it is helpful to cast the same basic equation ina time series form to obtain a vector of imbibed volumes at various, sequential times.We begin by addressing how we denote the volume of fluid imbibed by a tube sizeclass, ν(ri, t). We begin by taking a finite time interval for the flow which has beenbroken into M equal subintervals, then an imbibed volume for tube size class i at timestep j ∈ [1,M ] is denoted:νji = ν(ri, tj). (2.12)It is now possible to create a matrix containing all the imbibed volume amountsfor all the M time steps and all N size classes arranged as per:ν =ν11 ν12 · · · ν1Nν21 ν22 · · · ν2N....... . ....νM1 νM2 · · · νMN (2.13)We can also store all the individual tube size class coe cients θi in a single columnvector:θ =θ1θ2...θN (2.14)Now it is possible to compactly write the time series of imbibed volumes V as theproduct of ν and θ:V = νθ. (2.15)In practice this formalism is useful for predicting the liquid uptake of a woodsample with a known pore size distribution (e.g., a distribution obtained from imageanalysis of wood microscope sections, see Chapter 3 for details). However, a moreabstract question may also be addressed with the aid of this formalism: is a uniquebundle of tubes responsible for the given liquid uptake rate? or alternatively: isthere a wood structure which is theoretically the best at absorbing fluid from a bath?This questions stems from objective 1.2.ii as it relates to maximizing liquid uptake byaltering the structure of wood chips via compression. A proof that no single tube15CHAPTER 2. BACKGROUND AND MOTIVATIONbundle structure (i.e., wood pore structure) yields a given liquid uptake rate is given inAppendix D.2.3.3 Non-cylindrical CapillariesThe development of a momentum balance based model of capillary driven flowinto a cylindrical tube in §2.3.1 has been shown to be in good agreement withexperiments [6, 15, 61] for flows in cylindrical capillary tubes. The capillary spaces inwood represent a complication to this simple cylindrical geometry and require someattention to better model a (capillary driven) flow through it. As lumen diameter variesalong the length of a fibre with a diverging-to-converging pattern from fibre tip to tip,a model for flow into a converging or diverging capillary is desired. Such a model ispresented by Reyssat and colleagues in [45] and will be briefly summarized here.r(z)r0p0Figure 2.4: Generic non-cylindrical capillary tube geometry.We start by considering a capillary with a radial size which varies along the axialdirection according to some r(z) (as shown in Figure 2.4). We assume that the flowthrough the tube follows Darcy’s law by noting that Darcy’s law may be derived from abundle of tubes model [5] such as that developed in §2.3.2 and has been experimentallyobserved to describe the average velocity [45] in a tube similar to that which weconsider here. Stated formally, Darcy’s law is of the form:v = −kµ∂p∂z. (2.16)Note that in the above expression for average fluid velocity the permeability, k,of the tube is dependent on the shape and size of the tube’s cross section: for thecases we consider with slight and uniform radial changes k = r2/8 [5]. Since massis conserved we can write the volumetric flow rate at any time, t, as Q(t) = pir2v.Combining this with Equation 2.16 we get a modified expression of Darcy’s law:16CHAPTER 2. BACKGROUND AND MOTIVATION8µQ(t)pir(z)4=∂p∂z. (2.17)We wish to integrate Equation 2.17 from z = 0, p = p0 to the wetting fluid boundaryin the tube at z = h(t). For the liquid pressure at the meniscus we again employthe Young-Laplace relation. The capillary pressure is directly proportional to theliquid surface tension σ and the liquid interface’s curvature (which itself is inverselyproportional to the tube radius), thus we can write the liquid pressure at the meniscusas: p = p0 − cσ/r(z), where c depends on the contact angle the fluid makes with thesolid tube wall (and thus the local tube geometry). Integrating Equation 2.17 betweenthe limits as stated above, we obtain an expression for the liquid volumetric flowthrough the tube at time t to be:Q(t) =cpiσ8µr(h(t))∫ h(t)0r(z)−4 dz. (2.18)Dividing the volumetric flow by the instantaneous cross-sectional area at time t(pir(h(t))2) gives the average flow velocity as:v(t) =dhdt=cσ8µr(h(t))3∫ h(t)0r(z)−4 dz. (2.19)Equation 2.19 is a generalized version of the straight walled capillary expression.Note that when r(z) is taken to be constant the same expression for flow evolution is,importantly, recovered with h ∝ t 12 behavior. As Equation 2.19 makes restrictions as tothe form of r(z), theoretically any capillary shape with known r(z) may be analyzed,however we will restrict ourselves to somewhat physically realistic cases of linearlydiverging/converging and power law diverging/converging tubes.In the case of a capillary with linearly varying side walls, r(z) = r0 + αz,Equation 2.19 reduces to:dHdT[(1 +H)3 − 1] = 3/8. (2.20)With the dimensionless distance and time being defined as, respectively, H =αh/r0 and T = cσα2t/µr0. Solving this balance with H(0) = 0 (for an initially emptycapillary) an expression for imbibition distance evolution with time is obtained:2H4 + 8H3 + 12H2 = 3T. (2.21)The important thing to note from this result is that at long times (T 1) H (andtherefore h) grows proportionally to T14 (and therefore t14 ). This implies that even17CHAPTER 2. BACKGROUND AND MOTIVATIONfor purely capillary driven flows, time dynamics other than h ∝ t 12 are possible; thedeviation from the classical form being due to increased complexity in tube geometry(i.e., axially non-uniform tubes). Following a similar procedure for power law shapedchannels, r(z) = r0 + αzn, it may be found that long time behavior of the imbibitionfront follow a relationship of the form:H3n+1 ∝ 3n+ 18B(n)T, (2.22a)B(n) =pi6(3n− 1)(2n− 1)(n− 1)n4 sin(pi/n). (2.22b)As above, the important thing to note with this result is that dynamics other thanh ∝ t 12 occur in capillary driven flows through channels. With the greater the degree ofdivergence (the larger the value of n), the slower the imbibition dynamics. For all casesof diverging channels, the flow velocity drops with time as viscous dissipation eectsgrow relative to the capillary driving forces. As an qualitative example for convergingchannels: a discrete tube-section model was developed in [50] that shows imbibitionvelocity can vary from decelerating (in the constant or diverging tube radius case), toconstant (in a specially designed converging capillary case), to accelerating (in anotherspecially designed converging capillary’s case).2.3.4 Initially Fast ImbibitionTo better understand the relative magnitudes of the driving forces/losses present duringfluid impregnation of wood chips, an idealized case where initially fast moving fluid isintroduced into a capillary system is studied. It is thought [21, 35, 56] that followingcompression the chipette expansion, which occurs following the release of the imposedstrain by the screw, creates a pressure dierence within the wood chip (over thelength of the fibres) and in turn drives the flow of fluid into the chipette. The factorswhich determine the eectiveness of this mechanism for fluid impregnation are therate of re-expansion following strain release and the amount of re-expansion whichoccurs. The size of the channel available for flow also alters the eectiveness ofa expansion/pressure driven impregnation process but we restrict this dimension tophysically realistic fibre lumen diameters on the order of 10µm. The general approachtaken here is to impose a ‘fast’ initial imbibition velocity on the fluid flowing in thecapillary space and observe the eect of overall imbibition distance with time; in thisway the relative magnitude of initial fluid velocity (i.e., initial lumen re-expansion) onfluid uptake will be determined.Consider the same capillary geometry given in Figure 2.3 with ψ = 0. Following asimilar force balance procedure as in §2.3.1, the sum of viscous losses and capillary18CHAPTER 2. BACKGROUND AND MOTIVATIONdriving forces will equal the rate of change of momentum in the control volume (thecapillary tube) at any given time, or:2σ cos θR− 8µh¯dh¯dt¯= ρR2ddt¯(h¯dh¯dt¯)(2.23)Note that the overbar notation has been added to h and t to denote dimensionalquantities of length and time, respectively; dimensionless versions of these variableswill be denoted by plain Latin characters for this analysis. To scale the momentumbalance, a characteristic length and time scale need to be chosen. The capillary radiusR is a a convenient choice for the characteristic length, and we can define an initialphysical imbibition velocity V0 to create dimensionless length and time:h = h¯/R, t = t¯V0/R. (2.24)Making the substitution dη = d(h¯2) = h¯dh¯/2 and rearranging terms we can nowwrite the scaled momentum balance as:Red2ηdt2+dηdt=1Ca(2.25)With Re and Ca being the Reynolds and capillary numbers, respectively. They aredefined for this problem as:Re =ρRV08µ, (2.26a)Ca =2µV0σ cos θ. (2.26b)The general solution to the non-homogeneous, linear dierential momentumbalance in Equation 2.25 can be found analytically to be of the form:η = h2 = C1Re exp(−tRe)+tCa+ C2, (2.27)where C1 and C2 are constants of integration to be determined. Two initialconditions are required to determine these constants: a specified initial imbibitiondistance and rate. The initial dimensionless imbibition velocity, dh/dt|t=0, is unityas a result of the choice in scaling parameters and the inclusion of V0. We can alsospecify an arbitrary initial imbibition distance h|t=0 =  and examine the how the19CHAPTER 2. BACKGROUND AND MOTIVATIONbehavior of the solution at long times changes with various values of . Applying theseconditions we find:C1 =1Ca− 2, (2.28a)C2 = 2 − Re(1Ca− 2). (2.28b)Inserting these coe cients into the general solution we get the completed, generalsolution:η = h2 = Re(1Ca− 2)[exp(−tRe)− 1]+tCa+ 2. (2.29)Plots of h vs. t¯ over various initial imbibition rates (i.e., Re) are shown for assortedvalues of  in Figure 2.5. First let us examine the system’s behavior at short times.Generally, in the early stages of capillary imbibition, inertial eects dominate as therelatively small amount of fluid present in the tube does not allow for viscous eects tobecome palpable and imbibition distance progresses directly with time [16] (i.e., h ∝ t).In the current system, however, viscous dissipation may be observed from even smalltimes evidenced by the large variance of both imbibition distances and rates across thevarious Re and  value cases (see Figure 2.5). The introduction of momentum into thesystem through the prescribed initial imbibition velocity then has two related eects:viscous dissipation is introduced where in the initially stationary case it would not beimpactful, and the inertial stage of imbibition is altered such that h ∝ t is no longersatisfied exactly. In several mid-range Re cases, h is seen to progress roughly with t1/2(slope of lines on log-log plots correspond to exponent value) even from early timesand as such ‘eliminates’ the inertial stage. It is noted that this inertial stage bypassingoccurs when the exponential term of Equation 2.29 tends to zero; that is to say thatthis occurs for finite and non-zero V0 values. Eectively, the fast initial flow whichis imparted to the system by the rapid (though small) tube expansion acts to negate,to varying degrees, inertial lag that would otherwise inhibit fluid imbibition in smalltimes.From the plots in Figure 2.5 it is evident that this system, in long times, exhibitssimilar behavior to that given for a tube without initially moving fluid (i.e., Equation 2.7).That is to say: imbibition distance of the fluid progresses in a manner that is similarto the classical Washburn visco-capillary case, or η|=0 ∝ t as t→∞. This is in linewith the behavior expected with capillary driven flows in long times [16, 17]: at longtimes inertial eects dissipate and the capillary driving forces have only viscous lossesto balance. Fries and Dreyer in [16] calculated (via matching the intermediate timeviscous solution to the visco-capillary one) the time for a capillary system to reach thisflow regime to scale with the square of the capillary radius, R2. The fast time to reach20CHAPTER 2. BACKGROUND AND MOTIVATION10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100t¯ (seconds)10-210-1100101102103h (capillary radii)Initial Re0.01250.1250.6251.256.2512.562.5(a)  = 0.0110-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100t¯ (seconds)10-1100101102103h (capillary radii)Initial Re0.01250.1250.6251.256.2512.562.5(b)  = 0.0510-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100t¯ (seconds)10-1100101102103h (capillary radii)Initial Re0.01250.1250.6251.256.2512.562.5(c)  = 0.110-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100t¯ (seconds)100101102103h (capillary radii)Initial Re0.01250.1250.6251.256.2512.562.5(d)  = 0.510-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100t¯ (seconds)100101102103h (capillary radii)Initial Re0.01250.1250.6251.256.2512.562.5(e)  = 1.010-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100t¯ (seconds)101102103h (capillary radii)Initial Re0.01250.1250.6251.256.2512.562.5(f)  = 5.0Figure 2.5: Imbibition distance evolution for flow in an R = 10µm capillary for various initialRe/V0 and fluid penetration distances, .21CHAPTER 2. BACKGROUND AND MOTIVATIONthe visco-capillary regime in our system, then, can be seen mainly as a consequence ofthe relatively small tube radius. In the hypothetical case without an initially imposedflow and similar fluid characteristics, the system would be expected to behave as avisco-capillary system in ∼ 10−6s for R = 10µm. Our system would be expected tobehave as a visco-capillary one after a longer period of time (to allow initial fluidmomentum to dissipate) and indeed it does: imbibition distance for the various initialRe is eectively independent of Re past ∼ 10−5 − 10−3s with the short time varietybeing due to the range of initial imbibition distances, . Although broad, this rangeof times for the system to behave in a manner which is eectively independent ofinitial fluid motion is well under the threshold for industrial relevance (compressedchips/tubes are contacted with a chemical solution for an amount of time on the orderof 10’s of seconds [21, 56]).The relative insensitivity of fluid imbibition distance, at times longer than ∼ 1s, toinitial fluid motion implies that any fluid uptake caused by tube/chipette re-expansionfollowing compression can be neglected for the purposes of this problem and in furtheranalysis. Rather, it is purely the distribution of the sizes/shapes of the pores in thewood matrix which eect ultimate fluid uptake capacity.2.4 Research ApproachThe objectives listed in §1.2 are somewhat related and the approaches used in achievingthem are, similarly, complementary to one another.Concerning objective i: wood may be considered as an orthotropic cellular porousmedium [20]. The relative orthogonal independence and uniformity of the spatiallyvarying properties (e.g., density, pore size distribution) make them an excellent andconvenient choice in property to treat as representative of the structure as a whole.Specifically, the pore size distribution2 will be used as a unique descriptor of wood’sstructure/the compressive treatment which the wood sample has undergone. Poresize distributions also allow for the calculation of a predicted liquid uptake rate. Theliquid uptake rate may also be used as a descriptor of the wood structure/compressivetreatment with which it is associated, though not a unique one (as detailed inAppendix D).Concerning objective ii: the complexity of wood’s structure makes it somewhatdi cult to adequately measure explicitly in three dimensions3. Furthermore, the2The term ‘pore’ as used here simply refers to regions of void space within the wood, not to thestructural feature of the wood matrix of the same name. In a similar vein, the term ‘porosity’ used inthis work will be used to describe the proportion of a bulk solid which is void space.3Though tomographic methods of imaging a three dimensional wood fibre matrix exist [49], therelatively small spatial limitations and lengthy measurement times make its application here somewhatunattractive.22CHAPTER 2. BACKGROUND AND MOTIVATIONcomplex physical interaction of chips with one another in a bed [1, 2] which iscompressed makes it di cult to interpret any structural changes as they relate to,for example, the total strain applied to the chip, the compression rate used to doso, or the subsequent liquid interaction with the compressed bed. Both of theserealizations make it desirable to abstract the compression/liquid uptake processessomewhat in order to eliminate complexities present at pilot or mill scale and betterstudy the fundamental mechanisms of the compression/liquid uptake process. Firstly,the complex compression which occurs in a full-scale screw press will be replacedwith a one dimensional, controlled rate, controlled amount compression process at labscale on standardized (in terms of size) wood blocks replacing chips to ensure, as bestas possible, consistent compression conditions. Secondly, while recognizing that thestructure of the wood matrix as it relates to fluid flow into it is ultimately the goal ofthis work, the flow into the bed of chips will be replaced by directly measuring the flowinto single, treated blocks over time. Coupled with knowledge of how capillary flowsbehave in various capillary geometries (see §§2.3.1 and 2.3.3), structural information(i.e., porosity, degree of capillary divergence) may be back calculated from a time seriesmeasurement of liquid uptake into a wood sample. Both of these measures aim toeliminate, as much as possible, any complications to the fundamental mechanism ofaction of a screw press present at full scale: a better understanding of how the processalters single chips to make them more amenable to chemical uptake is desired.2.5 Overview of MethodsThe procedure for obtaining a pore size distribution from a wood sample requiresa mixture of lab work, digital image analysis, and model fitting. Firstly, the woodsample is sectioned, dyed, and mounted on a microscope slide so that imaging thepore structure of the thin section is straightforward with a digital microscope camera.A list of pore sizes (or more precisely, a list of contours corresponding to each of thepores in an image) in the images are then ‘harvested’ by an image analysis script. Thislist of pore sizes is then treated as a sample of a greater population and a populationmodel (pore size distribution) is fit based on the collected pore data.In the lab-scale compression tests, wood blocks are straightforwardly compressed bya hydraulic piston. Special precautions have been taken in the form of a bearing/plateapparatus to ensure strictly one dimensional compression. Both total compressionamount and compression rate is controllable with the piston. The lab-scale liquiduptake tests involve allowing water to wick into the compressed wood samples from areservoir while the mass of the reservoir is continuously monitored; the liquid uptakeof the wood sample may thus be calculated by making the assumption that no liquidlosses exist in the reservoir/tubing/wood system.233 Image AnalysisA methodology comprised of wood sample preparation, microscope imaging, and digitalimage analysis techniques has been developed to obtain a pore size distribution. Thischapter provides an explanation of this methodology and provides a discussion andanalysis of the results that followed from the analysis of chipettes from both pilot andmill-scale screw compression trials.3.1 Imaging of WoodThe acquisition of images is the first required step in the extraction of a pore sizedistribution of images of wood’s micro-structure. The uniformity and quality of theacquired image is strongly aected by both the details of the preparation of the woodensamples and the exact imaging parameters. This sensitivity to both sample preparationand image acquisition prompted a detailed sample preparation/imaging protocol to bedeveloped.30µm(a) Graphic of microtomesectioning process(b) Light transmission stereomicroscope setup(c) Dyed, mounted woodsampleFigure 3.1: Illustration of wood sample preparation process for microscope imaging.Compressively treated wood samples are first mounted on a glass slide: 30µmthick sections of wood are cut from the desired sample using a sledge microtome24CHAPTER 3. IMAGE ANALYSISafter soaking in water for several days to soften the cellulose/lignin and avoid tearingduring the cutting process. Following sectioning, the sample slices were dyed usinga 5.33 w/w% solution of Rit Navy Blue crystals1 in distilled water. After two hoursof contact time with the dye, the sections were rinsed with water, and subsequentlymounted on glass slides using Fisher Permount mounting solution. Permount is ablend of toluene and polyterpene hydrocarbon resin that is commonly used to preservebiological samples on slides. It is colorless and has a refractive index nearing that ofglass which makes it non-obstructive in terms of sample imaging. The samples wereallowed to cure and stabilize for several days before subsequent photographic work.The preceding sample preparation steps produced a uniform series of physicalsamples that were then photographed. In keeping with the goal of collecting uniformand high quality images, the camera and microscope settings were first adjusted toprovide good quality images then held constant over the course of (non-preliminary)image collection. A Nikon transmission stereo microscope with a 10Mpx top-mountdigital camera was used to collect images. A table of parameters relevant to themicro-photographic process are listed in Table A.1.3.2 Computational Methods3.2.1 Obtaining Pore Sizes from ImagesExtracting a pore size distribution from the images obtained using the proceduresoutlined in the above sections was reliant on the ability to reliably and consistentlydistinguish between solid and void regions in the images. To this end, digital imageanalysis was employed to perform this task. A single python script was created toprocess all images in a common way ensures consistent handling and interpretation ofany features in the images.The script that was created processes a specified batch of images by repeatedlycalling the function find_pores(). This function takes a raw RGB image as an inputand outputs an array of pore areas from that input image. A commented copy offind_pores() may be found in Appendix A. The overall operation find_pores()performs on an input image may be broken down into two main parts:1. Binarize the image into regions corresponding to void space/solid in the imagedsample.2. Find contours (boundaries between solid and void regions) in the binarized imagecorresponding to pores in the images sample and calculate contour areas.1Rit dyes are widely available for use as a fabric dye and is not generally known for laboratory use,however its formulation makes it ideal as a stain for all types of cellulose (e.g., cotton, wood fibres) and,in practice, useful for staining both cellulose and lignin.25CHAPTER 3. IMAGE ANALYSISThe first task is performed with the Otsu algorithm [37]. The Otsu algorithmdivides an image into light and dark regions (i.e., binarization) by minimizing theintra-region variance of the pixel intensity values. More formally: for an image withsorted, descending pixel values p = {pi}Ni=1 it is desired to find a threshold (i.e., kvalue) of pixel intensity values which divides the image into the dark and light regions0 and 1. The calculation of statistics for each of these regions based on pixel valuesis what allows for an e cient and consistent handling of the binarization step of theimage analysis pipeline. For two arbitrary regions within an image, the associatedweights ω0 and ω1 are defined as [37]:ω0 =1Nk∑i=1pi and ω1 =1NN∑i=k+1pi = 1− ω0, (3.1)the average pixel values µ0 and µ1 are defined as:µ0 =1Nk∑i=1ipiω0and µ1 =1NN∑i=k+1ipiω1, (3.2)and the intra-region variances σ20 and σ21 are defined as:σ20 =1Nk∑i=1(i− µ0)2piω0and σ21 =1NN∑i=k+1(i− µ1)2piω1. (3.3)It is then possible to calculate the within-class variance as σ2W = ω0σ20 + ω1σ21and the between-class variance as σ2B = ω0ω1(µ0 − µ1)2. The Otsu algorithm takesadvantage of the fact that the minimum total variance, σ2T = σ2W + σ2W , is achievedwhen the inter-region variance is at a maximum and the intra-region variance is at aminimum [37]. In other words, the binarization threshold to be applied to the images,k∗, is found according to:k∗ = arg maxkσ2B(k)σ2T. (3.4)Alone, the Otsu binarization procedure acting on raw images does a fair job atdistinguishing solid and void space regions from one another, although small imagingartifacts are not accounted for or dealt with and remain in the binarized productimage. This would lead to the undesired identification of false pores if not dealtwith appropriately. Examples of these artifacts are small light spots formed by duston a slide cover slip and jagged edges near a pore boundary formed by a shallowdepth of field in the imaging system. Applying a slight (i.e., kernel width of ninepixels) Gaussian blur to the raw image before Otsu binarization acted to smooth jaggededges (on the level of individual pixels) and eectively completely removed any edge26CHAPTER 3. IMAGE ANALYSISroughness present. The remaining small spots (both light and dark) in the image wereremoved by directly manipulating pixel values through morphological transforms. Theso-called morphological transforms of ‘opening’ and ‘closing’ images dilate and erodelight regions of an image in succession which acts to remove small features of the orderof the size of the kernel used while preserving larger ones (e.g., fibre lumens, cell walldelaminations, actual void space). Figure 3.2 shows the progression from a raw imagetaken in the laboratory to one with the found contours outlined. Note that contourswhich intersect the edge of an image are discarded and not counted in subsequentanalysis.With respect to the kernel size used to blur/open/close the images, it was necessaryto ensure a kernel small enough relative to the size of the features that are requiredto be preserved that important information was not lost in the processing. Thoughfeature recognition routines do exist based on pattern matching [9] that may have beenused to adaptively find the kernel size that would not disrupt important features inan image, finding ‘optimal’ kernel size for both the blur step and open/close step wastreated as a manual trial and error task. In practice a 5×5 cross-shaped kernel wasused for opening/closing and a 9×9 square kernel used for blurring; even the smallesttrue pores were observed to be of the order of 10 pixels wide.Following binarization, the contours in an image simply represent the sequenceof pixels which border the light regions corresponding to pores. As each contour is aset of points in two dimensional space bordering a region, say Ω, finding their area(and thus the area of the corresponding pore) is accomplished by calculating the 0thmoment of the region according to:a =∫ΩPxy dxdy, (3.5)where Pxy is the pixel intensity at point (x, y) with the pore region Ω. As the imagebeing operated on is binary, each bright pixel in a pore has Pxy = 1 and the integralabove reduces to a simple summation of the number of pixels contained by the contour.A conversion of pore area in terms of pixels to pore area in terms of physical dimensionswas accomplished by the use of a calibration slide. For the imaging parameters detailedin Table A.1 a 4.2px/µm conversion value was used.3.2.2 Pore Population Modeling via Gaussian Mixture ModelsThe raw product of the image analysis procedure described above is an array of poresizes which, alone, is di cult to use to compare compressive treatments against oneanother. A modeling procedure has been developed which fits a Gaussian mixturemodel to each treatment’s pore size data set obtained from image processing. Thebasic motivation to perform population modeling on the pore size data obtained comesfrom the practical constraint on laboratory imaging and sample collection: collecting27CHAPTER 3. IMAGE ANALYSIS(a) Raw image obtained from microscope cam-era.(b) Greyscale of blue channel, slightly blurred.(c) Binarized image after opening/closing to re-move very small features.(d) Greyscale of original with found contours(highlighted in orange). Scale bar is 100µm wide.Figure 3.2: Visual progression from raw image obtained from microscope camera (a) to foundcontours (d).28CHAPTER 3. IMAGE ANALYSISand imaging literally every pore (i.e., the population of pores) which undergoes a giventreatment is not logistically viable. Instead, it is desired that an estimate of the shapeof the population’s pore size distribution be obtained from a directly measured sampleof the population. A Gaussian mixture model provides exactly this utility while yielding(for a K-component model) 3K + 1 parameters which may be used to compare poresize distributions and thus treatment eectuality.It is these fitted models which are used in further analysis and discussion of poresize distributions as they describe the overall pore population and are not simply asample of the larger population which has been directly measured via image analysis.This section only provides a brief overview of the utility of, and the motivation touse, Gaussian mixture models to model wood’s pore size distribution; for a moreextensive description and discussion of the analytical and numerical techniques usedto determine model parameters, a formal mathematical description of the techniquesemployed in this work to model the pore size distribution of the whole pore populationwhich undergo a treatment is provided in Appendix C.3.2.3 Determining Adequate Pore Size Sample SizeAn ad hoc methodology for determining an adequate sample size for model fitting (seefor details on the model fitting procedure used) was followed which, in turn, governedthe number of images collected from each treatment and found that the convergenceof all models occurred at roughly 10000 pores (equivalent to roughly 15 images usingthe imaging setup detailed in Table A.1, from three chipettes). The adequate size ofpore size dataset was determined by taking increasingly large samples of data (in 500pore increments), fitting a mixture model to the sample, and examining the resultingmixture model’s mean, µ∗. The value of 10000 pores was that at which randomlyselecting 500 more pores and refitting a mixture model resulted in less than a 1%relative shift in µ∗. That is, collecting more pore areas from more images would noteect the resulting fitted model in a meaningful way. The value of 10000 pores wasthus selected as the minimum number of pores collected from each trial to fit a model.Though not directly relatable, studies of convergence of the signal from early woodvessels [3, 18] find that a sample size of the order of 30-50 were adequate to obtain aconstant signal (on a per ring basis) of vessel damage (akin to a pore size distributionfor a population) and provide credibility for the methodology used here.29CHAPTER 3. IMAGE ANALYSIS3.3 Data and Analysis of Results3.3.1 Pilot-scale ChipettesPilot-scale screw compression trials were completed at Andritz’s pilot-scale refining/R&Dfacility in Springfield, Ohio, USA and the resulting chipettes (and raw chips as a control)were collected following compression for analysis using the image analysis/modelfitting procedure outlined above. A total of six trials were carried out while varyingchip feed rate, chip presteaming pressure (saturated steam), compression screw speed,and eective total compression. All chips were steamed for an identical time beforecompression and were sourced from the same mixture of spruce/pine/fir, or SPF, woodchips. The exact conditions each of the trials were carried out at is detailed in Table 3.1.Due to time and labor constraints no repeat trials were performed, however the poresize distribution obtained from each trial did converge as per the methodology outlinedin §3.2.3. Three chipettes from each trial were randomly selected and used for imaging.Table 3.1: Summary of the screw press trial conditions for the pilot-scale trials carried out atAndritz’s pilot-scale refining/R&D facility in Springfield, Ohio, USA. ‘Pressure’ refers to the pressureof saturated steam used to heat the wood chips prior to compression. ‘Compression Ratio’ refersto a nominal ratio calculated based on inlet and outlet areas available to chip flow. For thepilot-scale screw used in the trials, the outlet area, and thus eective compression ratio, may beadjusted during operation by varying the distance which four one inch diameter pins protrudeinto the screw housing. The screw used in all trials has a compression ratio of 2.0:1 based ondesign dimensions and 0.0 inches of pin protrusion, or 2.2:1 with 0.5 inches of pin protrusion.Trial Pressure (psi) Screw Speed (RPM) Feed (t d−1) Compression RatioP1 20 36 11.8 2.0:1P2 20 36 10.9 2.2:1P3 5 36 13.1 2.2:1P4 20 36 7.0 2.2:1P5 20 24 12.6 2.2:1P6 20 12 11.9 2.2:1The pore size distributions from each of the six trials, as well as that for raw wood(O1) are given in Figure 3.4. A deviation from raw wood was observed for all of thetrials’ pore size distributions indicating that fibre-scale damage was incurred by woodat all compression conditions studied. In all cases a reduction of the relative frequencyof larger pores was observed. This is in accordance with the previous observationsof Nelsson [35] and Gorski et al. [21] which note early wood fibre wall collapse anddelamination of fibre from middle lamella as a result of compression. Though thegeneral trend of the structural deviation was similar for each of the trials, the extentof the changes to the fibre-scale morphology was observed to vary markedly from30CHAPTER 3. IMAGE ANALYSIStrial to trial. As the feed to the screw press is taken to be homogeneous and identicalfor all trials, the dierence in extent of damage incurred by the chips from trial totrial (i.e., the dierence in extent and type of deformation incurred by the (micro andmacro) structure of the chips) may then be taken to be due to dierences in stressstate incurred by the chips during compression and, in turn, dierences in appliedforces and mechanical chip properties.First consider the base operation case P1. As the chips are conveyed along thelength of the screw they are compressed, the imposed strain on the bed increases,and the radial stresses build within the chip plug. Once the buckling stress of theearly wood (i.e., larger, thinner walled) fibres is reached they begin to deform and indoing so relieve stress in the bed. This results in the reduction of the number of largerpores in the wood and relative increase in the incidence of smaller pores as shown inFigure 3.4a. This collapse translates to a 1.6% reduction in average pore size relative toraw wood. Now consider this base operation case with additional compression imposedat the outlet of the screw press, as is the case in trial P2. In this case, as chips areconveyed along the length of the screw they ultimately need to conform to a smalleroutlet size (due to addition compression provided by pins) and thus total imposedstrain is increased relative to P1. Additionally, since screw speed was not changed, thestrain rate imposed is also higher than the base case P1. Both the increased strain andstrain rate imposed on the chips lead to an increased stress in the chip bed and aneven more pronounced shift of the pore size distribution away from larger pores. Arelative increase in smaller pore (late wood) incidence is observed as a response to thelarger strain and strain rate being applied (compare Figures 3.4a and 3.4b).Figure 3.4c illustrates the eect of steam temperature on the resultant pore sizedistribution for trial P3. Trial P3 lowered (relative to P1) the steam temperature ofthe steam used to heat the chips before compression. The result is a much morepronounced (relative to an increase of compression amount, trial P2) shift of the poresize distribution towards smaller pores. An increase of roughly twice the relativeincidence of mid-range, transition wood sized pores is observable compared to rawwood. This results in the largest shift of mean pore size of all of the trials considered(6.3% relative shift in µ∗) and substantial homogenization of the distribution: a 2.2%drop in relative variance (σ2∗/µ∗) was observed from raw wood to P3 chipettes.Physically this shift is likely due to the dierence in the feed chip’s mechanicalproperties. Saturated steam at 20psi (used in P1) has a temperature of 108◦C whichis just above the lower bound of the softening temperature range (∼100-170◦C) ofwood [27], while 5psi steam is well under the lower bound with a temperature of 72◦C.The colder, stier chips used as feed for P3 would have a higher level of stress developalong the length of the screw as they are compressed as a result of their relativelylarge compression modulus. As such, they would be expected to incur more damageto larger pore structures as they buckle and collapse, as cell walls delaminate frommiddle lamella, and as cracks form in the wood matrix.31CHAPTER 3. IMAGE ANALYSIS(a) P1 (b) P2(c) P3 (d) P4(e) P5 (f) P6Figure 3.3: Selected original RGB images (in greyscale) with found contours for chipettes producedduring pilot-scale trials (P1-P6). Any pore intersecting the image boundary has not been included.Note the relative similarity between individual images and occasional omission of small poresfrom the found contours; population pore size distributions were found to converge for any giventrial when >10000 pores were used in mixture model fitting. Scale bars in the bottom rightcorners of the images are 100µm wide.32CHAPTER 3. IMAGE ANALYSIS7 6 5 4 3 2 1A log10(mm2)0.00.20.40.60.81.01.21.41.6f∗ (A|θ)P1O1(a) P17 6 5 4 3 2 1A log10(mm2)0.00.20.40.60.81.01.21.4f∗ (A|θ)P2O1(b) P27 6 5 4 3 2 1A log10(mm2)0.00.20.40.60.81.01.21.4f∗ (A|θ)P3O1(c) P37 6 5 4 3 2 1A log10(mm2)0.00.20.40.60.81.01.21.4f∗ (A|θ)P4O1(d) P47 6 5 4 3 2 1A log10(mm2)0.00.20.40.60.81.01.21.4f∗ (A|θ)P5O1(e) P57 6 5 4 3 2 1A log10(mm2)0.00.20.40.60.81.01.21.4f∗ (A|θ)P6O1(f) P6Figure 3.4: Pore size distributions of chipettes collected from pilot-scale compression trials.Probability density f∗(A|θ) vs. the logarithm of pore area A is plotted. Red ‘O1’ series correspondto fitted model for uncompressed chips, black series(histograms and fitted models) correspond tocompressed chipettes. Solid vertical lines are at µ∗ with dotted vertical lines σ2∗ away on eitherside. 33CHAPTER 3. IMAGE ANALYSISA lower chip feed rate was investigated in trial P4 by reducing the chip feed rateby 40% from the base case P1. Identical geometrical compression parameters (0.5in),steam temperature (20psi saturated steam), and screw speed (36RPM) were otherwiseused. This combination of trial parameters resulted in the smallest change to the poresize distribution observed of all of the trials and the general direction of the shiftwas similar to each of the other trials: a reduction in the relative frequency of largepores was observed while small (late wood) and mid-range (transition wood) poresexperienced a relative increase in their incidence. Contrary to the similar qualitativeshifts observed in P4’s pore size distribution, the population parameters µ∗ and σ2∗did not change in a manner similar to each of the other trials. The distribution meanµ∗ was still observed to decrease, though only by a very small amount (0.8% relativedecrease from raw wood) while the pattern of distribution homogenization observed inall other trials was broken with a 0.4% relative increase in relative variance from rawwood to P4’s pore size distribution (see Figure 3.4d). As this is the only trial that doesnot follow that general shift pattern of the pore size distribution of chipettes towards amore homogeneous one with a smaller mean pore size, it is instructive to consider themechanics of what occurs inside the chip bed during transport along the length of thescrew press.As the chips enter the screw press, the combined eects of screw speed and chipfeed rate control the starting porosity of the bed that is conveyed. Beyond the inletthe total mass of chips between each screw flight does not change and so must becompressed as the annular space around the screw decreases. For either a slowerscrew speed or an increased chip feed rate, all other things being equal, more chipswill fill between two adjacent screw flights and so the total bed compression at theoutlet will be higher (with the opposite also being true). The relatively minor, almostnegligible changes to the pore size distribution of P4 relative to raw wood is likely adirect result of this eect in practice: a 40% reduction in chip feed rate has altered thestarting bed porosity such that the total compression experienced by the bed and chipsis very mild at the outlet and results in almost no change to the pore size distribution.As the changes observed are so small, the particular combination of screw speed andchip feed rate in trial P4 seems to bring the operation very close to a threshold whereno chip-scale compression is present at the outlet of the screw at all. With respectto screw press operation, this trial has illustrated that by feeding a su ciently smallamount of chips into a screw press (or running the screw su ciently fast), no chipcompression (i.e., no fibre-scale changes going from chips to chipettes occurs) maybe possible in certain cases even though the bed of chips experiences a reduction inporosity during the compression.While trial P4’s operation achieved a reduction of total compression via reductionof chip feed rate, trials P5 and P6 investigated the eect of increased compressionvia screw speed reduction. Trials P5 and P6 reduced the screw speed by one thirdand two thirds, respectively, relative to P1 while all other operational parameters were34CHAPTER 3. IMAGE ANALYSISkept nominally constant. The general trend of a reduction in both µ∗ and σ2∗ wasobserved in both trials, although to diering degrees between the trials. First considerthe distribution obtained from a one third reduction in screw speed (as shown inFigure 3.4e). A reduction of 3.5% in average pore size was observed relative to raw woodwhile relative variance remained almost constant dropping only slightly by 0.4%. As thescrew speed was reduced to two thirds that of the base case while chip feed remainedconstant, adjacent flights on the screw of trial P5 contain 150% of the chips relativeto their faster rotating counterparts of trial P1. As such, the compression imposedalong the length of the screw to the chip bed will result in a larger strain imposed onindividual chips. This is keeping with previously observed variation of the chipettepore size distributions: relative to P1 an increased shift of large, early wood porestowards smaller, late wood pores is observed which translates to an apparent increasein mid-range, transition wood sized pores and a reduction in overall mean pore size.The eect of increased chip strain via additional chips being present betweenadjacent screw flights was investigated further in P6 where screw speed was reducedby two thirds relative to P1. The general trends of a reduction in mean pore sizeand distribution homogenization were both, again, observed with µ∗ dropping by therelatively small amount of 0.5% and relative variance dropping by the relatively largeamount of 1.8% (see Figure 3.4f). This very low screw speed (P6) resulted in a lesspronounced shift of mean pore size but a larger eect of homogenization relativeto the base operational case (P1) as well as the slow screw speed case (P5). Thisbehavior indicates that a non-linear response from the chip’s fibre-scale morphology toincreased compression. As total compression increases (P1<P5<P6), a linear responsewould result in similarly increasing shifts of mean pore size as more and more porescollapse and cell wall delamination creates new mid-range pores. This is contrary tothe observed results when only considering pore collapse and delamination, though isconsistent with a hierarchical destructuring model of wood’s structure.The hierarchical destructuring modes derives from the presence of multiple lengthscales associated with the structures in wood (i.e., early and late wood fibre diametersand wall thicknesses) which is thought to result in multiple stages of yielding duringcompression. Large, thin walled early wood fibres are first to buckle and collapse asstress increases but then, as strain is continually imposed on a sample, late wood fibresbegin to buckle and the wood matrix as a whole begins to fracture and cell walls beginto delaminate from middle lamella [19, 20]. Note that any fractures introduced to thewood matrix during compression are unique in that they introduce larger pore spaceswhile all other destructuring modes only reduce existing pore sizes/result in mid-rangepores being created. This hierarchical model eectively explains the observed changesto the pore size distributions of trials P1, P5, and P6. In this progression, at first (P1 toP5) additional compression results in an enhanced decreased of mean pore size relativeto raw wood and distribution homogenization as pores collapse and delaminate frommiddle lamella. Following this, as compression is increased further (P5 to P6), mean35CHAPTER 3. IMAGE ANALYSISpore size change is less dramatic (due to the combined eects of pore collapse andcrack/tear formation canceling the other to some extent) while homogenization eectscontinue to increase.3.3.2 Multiple Compression Stagesto refinersChip feedM0M1 M2 M3Figure 3.5: Simplified process flow diagram for Millar Western’s multi-stage screw compressionline in Whitecourt, Alberta, Canada. Each conical-screw press was operated identically at a 4:1compression ratio. Water was used for the impregnation step following each press. Chipettesample points in the process are labeled M0-M3.A second series of chipettes were collected from Millar Western’s mill in Whitecourt,Alberta, Canada where three screw presses are operated in series (see Figure 3.5 fora simplified process flow diagram). Aspen (Populus tremuloides) chips were collectedfollowing each compression stage, chips randomly selected, and the sample prepa-ration/image analysis/model fitting procedure (as per that described in §§3.1, 3.2.1,and 3.2.3 and used for the pilot-scale chipette analysis) performed. The resulting poresize distributions from the three stages of compression and raw wood are shown inFigure 3.7.The heavy bi-modality of the pore size distributions (Figure 3.7) of the multi-stagechipettes is due to the size disparity between vessels and fibre tracheids within the(hard wood) aspen chipette samples. This marked disparity results in the selectivedestructuring and buckling/collapse of vessels in the hard wood matrix. Consideringthe progression from raw wood to the chipettes samples after the second compressionstage this is made evident: a reduction in the relative incidence of the larger porestructures (i.e., vessels, represented by the right-most sub-group in Figure 3.7) isobserved while the relative incidence of smaller, fibre tracheid sized pores increaseswithout the creation of mid-range pores which would indicate vessel/middle lamelladelamination.The heavy bias during matrix destructuring towards vessel collapse in thenetwork may be explained conceptually by considering an idealized model of a singlepore within a network, such as that given in [20]. The stress which plastic collapse of asingle pore occurs at σpl is related to the wall materials yield stress σy as well as thewall thickness t and side length ` of the pore via:36CHAPTER 3. IMAGE ANALYSISσplσy∝(t`)2. (3.6)(a) M0 (b) M1(c) M2 (d) M3Figure 3.6: Selected original RGB images (in greyscale) with found contours for chipettesproduced during mill-scale, multi-stage compression trials (M0-M3). Any pore intersecting theimage boundary has not been included. As with the chipette images shown in Figure 3.3, notethe relative similarity between individual images and the failure of the image analysis procedureto find some small pores. Population pore size distributions were found to converge for any giventrial when >10000 pores (vessels or fibre tracheids) were used in mixture model fitting. Scalebars in the bottom right corners of the images are 100µm wide.For aspen, physically conservative estimates of structure dimensions of tvessel ∼ tfibreand `vessel ∼ 10`fibre [33, 55] would imply that the yield stress of a representative vesselelement is roughly two orders of magnitude less than that for a representative fibretracheid with σpl,vessel ≈ σpl,fibre/100. Qualitatively, this implies that vessels will begin37CHAPTER 3. IMAGE ANALYSIS7 6 5 4 3 2 1A log10(mm2)0.00.20.40.60.81.0f∗ (A|θ)M0M1M2M3Figure 3.7: Pore size distributions obtained from image analysis of multi-stage compressionchipettes from Millar Western’s mill in Whitecourt, Alberta, Canada. The heavily bi-modaldistributions are a result of size disparity between vessel and fibre tracheid sizes in the Aspenchips that were studied.to buckle far before fibre tracheids as the amount of applied strain increases and isconsistent with the changes observed in the pore size distributions harvested fromchips/chipettes from the first two compression stages (M1 and M2). The third stageof compression does not exhibit the same response from the vessel/fibre tracheidnetwork (Figure 3.7) if only considering average pore size and peakedness of the twosubpopulations (M3 mean pore size is 0.9% larger than that of M0), but does with respectto population homogenization (σ2∗ monotonically drops with successive compressionstages by a total of 36% for M3 relative to M0). These observed homogenization eectsas well as the tendency of a compression process to eect larger pore structuresin wood in a more pronounced way relative to smaller ones is in keeping with theobservations made from the pilot-scale chipette analysis.The results of image analysis of chipettes obtained from both pilot and mill-scalesis consistent with what has been previously, qualitatively observed [21, 22, 35]; thoughthis end is not alone useful in determining the eectiveness of a given compressionprocess. Relating the fibre-scale morphology of the chipettes produced in these trialsto their liquid uptake ability would more comprehensively serve as a measure of therelative utility of a compression process. Though this approach was not taken inthis work, directly measuring liquid uptake capacity of chipettes has been carried formultiple compression stages at pilot-scale [39] and may serve here as a validation toolfor the assumptions made when image analysis derived pore size distributions are used38CHAPTER 3. IMAGE ANALYSISto predict capillary uptake of a chipette sample.In [39], three stages of compression at both 3:1 and 5:1 compression ratios werestudied and the bulk chipette properties after each stage were measured. The overallbed uptake was observed to increase (measured as a percentage of dry chip mass) by atotal of 34% (5:1 compression stages) and 25% (3:1 compression stages). The enhancedliquid uptake ability of compressed chips has also been reported at mill scale [21, 21, 35],though the exact mechanism of this enhancement is largely speculative (see §2.2.2).Probing how eectual the role the fibre-scale morphological changes incurred duringcompression play in this observed uptake increase may be gaged by building a bundleof tubes model from the pore size distributions measured via image analysis andthen predicting the capillary driven uptake ability predicted as per the capillary flowequations outlined in §§2.3.1 and 2.3.2. To do this, specifically, the modeled pore sizedistribution f ∗(A|θ) was sampled at 10000 equally spaced points over the range ofobserved pore areas made during the image analysis ([Amin, Amax]) to determine thetube size class coe cients θi for use in a model of the form Equation 2.11. Time seriesof predicted capillary volume uptakes corresponding to the pore size distributionsobtained from chipettes from sample points M0-M3 are shown in Figure 3.8. Note thatthe volume imbibed is that associated with one ‘unit’ of the tube bundle; the uptakevolumes should only be taken as relative to one another, not absolute in nature.Each of the predicted uptake time series conforms perfectly to a V ∝ t1/2relationship as a result of the idealized nature of the tubes which make up themodel. This is not necessarily the expected behavior from a generalized porous mediawith complex pore-scale geometries [45] (e.g., wood) and may only be taken to berepresentative of the two dimensional sample taken via the image analysis procedureused. With all other structural parameters being equal however, relative uptake abilitywould be preserved regardless of out of plane complexity eects (e.g., sample tortuosity,pore inter-connectivity).Following the first stage of compression (M1), the predicted liquid uptake wasobserved to drop while staying within the same order of magnitude. Subsequentstages of compression (M2 and M3) resulted in slight successive increases in uptakeability, though the uptake capacity of raw chips (M0) was not recovered. Both thesize and overall direction of the shifts in uptake ability due to compression are indisagreement with the results observed at both mill and pilot scales [21, 22, 35, 39].The discrepancy between the measured and predicted uptake abilities of multi-stagecompression chipettes may be partially reconciled by considering the length scale offeatures that the predicted uptakes take into account in their calculation.The field of view of the photographic setup used to obtain images (and subsequentlyfind the pore size distributions used to build bundle of tubes models and predict liquiduptakes) is only of the order of one square millimeter and so does not ‘see’ changesto the chips/chipettes which occur on length scales larger than this (e.g., cracks, chipbreaking). Presumably changes on a scale greater than that taken into account by39CHAPTER 3. IMAGE ANALYSIS0 500 1000 1500 2000 2500 3000 3500Time, t (seconds)0.00.20.40.60.81.01.21.4Volume Imbibed, V (m3/tube unit)1e 5M0M1M2M3Figure 3.8: Predicted liquid uptakes for chipettes sampled from Millar Western’s multi-stagecompression process. A bundle of tubes model (see §2.3.2) was built using the pore sizedistributions obtained from image analysis of chipette samples and was used to predict capillaryuptake ability.the analysis here are those which dominate liquid uptake changes at pilot and millscales. Similarly, the disparity observed here serves to illustrate a shortcoming ofthis method in capturing chip/chipette-scale morphological changes incurred duringcompression and also the apparent unimportance of the fibre-scale morphologicalchanges measured to liquid uptake ability of a chipette. Concluding that fibre-scalemorphological changes do not play an active role in liquid uptake enhancement, though,would be premature as the small uptake changes considered here (Figure 3.8) do notexplicitly take into account the complex three dimensional structure of wood and itsresponse to compression. The influence of compression on the three dimensionalstructure and liquid uptake capacity of wood is explicitly examined in the subsequentchapter via measuring liquid uptake of compressed wood samples directly.404 Liquid Uptake ExperimentsThe fundamental elements of the screw press chip compression/liquid uptake processhave been abstracted and examined at lab scale. By removing the possibility ofchip-chip interaction within a bed by only considering a single wooden block, thefundamental eect of wood chip destructuring may be probed more readily. To thisend, a one dimensional, lab-scale compression process has been developed and usedto compress single blocks of wood to simulate screw-achieved compression. Followingcompression, the block’s liquid uptake ability has been directly measured. Liquid uptakemeasurements allow for the fitting of power curves which in turn are used to derivestructural information about the wood matrix. This chapter outlines the lab-scalecompression/liquid uptake experiments which were performed and includes an analysisand discussion of the data which was collected. A statistical approach to analyzingpower curve parameters is explained and the results are interpreted and discussed.4.1 Experimental Methods4.1.1 Wood CompressionLab-scale simulation of the wood compression process was achieved using a hydraulicpiston with a roller bearing device (see Figure 4.1). The hydraulic piston was part of anMTS laboratory material testing system setup which allowed for explicit control of totalpiston displacement and piston velocity. The piston which was used is, itself, capableof much larger, faster displacements than those that were used during testing so as tonot overload and damage the load cell at the base of the system. A total of sixteencompression conditions were investigated in a full-factorial experiment matrix with fourcompression speeds (˙ ∈ [0.1, 3.0, 4.0, 5.0]mmmin−1) and four levels of total strain(∗ ∈ [10, 20, 30, 40]%). To provide industrial relevance to these values: compressionspeeds in the pilot-scale screw press used to produce the chipettes discussed in §3.3.1ranged from roughly 2.0mmmin−1 to 6.0mmmin−1 while the total strain applied tothe bed of chips in the pilot-scale screw press ranged from roughly 45% to 50%.Strictly one dimensional loading perpendicular to the fibre axes of the sample block(i.e., a combination of tangential/radial loading relative to the wood’s growth rings) of41CHAPTER 4. LIQUID UPTAKE EXPERIMENTSthe sample was made possible by a specially designed bearing/plate assembly. Thisassembly was comprised of two hardened steel plates on either side of a series of ballbearings. The bearings were held in position relative to one another by a loose fittingauxiliary plate with holes that retained the ball bearings during operation. The topplate in the assembly was held fixed by a fixture while the bearing-plate/bearings andthe lower plate were free to move in a plane perpendicular to the piston stroke.The wooden samples used in the lab-scale compression/uptake tests were sourcedfrom a single piece of air-dried Douglas fir (Pseudotsuga menziesii). For each of thesixteen experimental compression conditions, a 50mm×50mm×12.5mm block was cutinto four equally sized sticks along the axis of fibre alignment. Each of the four stickswas then steamed at atmospheric pressure for two hours prior to compression at thegiven conditions; for each set of compression conditions four compression tests werecarried out on each of the four sticks. Following compression and prior to liquid uptakemeasurement, the samples were stored in an environmental room at 50% relativehumidity to allow moisture content to stabilize somewhat. Liquid uptake tests werenot started until the mass of the compressed blocks, or blockettes in keeping with thenaming convention which has been adopted, stabilized.4.1.2 Liquid UptakeLab-scale simulation of the chipette impregnation process was achieved by directlyconnecting blockette samples to a liquid reservoir and monitoring their mass over time.Following their compression, each of the four blockettes associated with a given set ofconditions were connected via a 0.25 inch diameter silicone tubing manifold which inturn was connected to a reservoir containing distilled water. The silicone conformedto a high degree to the outer surface of the wood, though small losses were recordedduring preliminary testing due to capillary flow first filling the space between woodand tubing before beginning to saturate the sample. To eliminate this eect, a smallamount of petroleum jelly was applied to the outside of each sample before beinginserted the end of the tubing to seal the space and prevent the occurrence of the‘parasitic’ capillary flow between tubing and wood which was observed before.Instead of directly measuring the mass of the blockettes as they imbibe liquid, themass of the reservoir was monitored over time using a python script (data retrievalscript details may be found in Appendix B) which recorded mass readings every 1.5seconds for 1.5 hours (5400 seconds). Measuring the mass of the (relatively massive)reservoir and translating the reservoir’s loss to the blockette’s gain in mass impartedstability to the signal harvested from the 0.5mg resolution balance. To mitigateevaporative losses, the samples were placed in a plastic bag during uptake and thereservoir was covered inside the balance’s draft shield.42CHAPTER 4. LIQUID UPTAKE EXPERIMENTSPistonPlatesBearingSampleBearing/Plate HolderLoad CellBaseFigure 4.1: Labeled photograph of the hydraulic piston and bearing/plate assembly used tosimulate wood chip compression at lab-scale. The bearing/plate assembly ensures no forcesperpendicular to the axis of compression are generated during a compression test; i.e., strictlyone dimensional compression is present in each test. Wood samples were located via an insetinto the base to ensure consistent loading between successive tests.Lab BalanceComputerReservoirWoodSamplesFigure 4.2: Schematic of the liquid uptake measurement laboratory setup. Each of the fourindependently compressed wood samples are connected to the liquid reservoir via 0.25 inchdiameter silicone tubing sealed with a thin coat of petroleum jelly where tubing/wood contactis made to prevent leakage. Balance readings are taken every 1.5 seconds and recorded using apython script which reads USB balance output. A copy of the relevant data retrieval code may befound in Appendix B.43CHAPTER 4. LIQUID UPTAKE EXPERIMENTS4.1.3 Power Curve FittingThe liquid uptake measurements made following the compression of a wood blockresult in a time series of masses that correspond to liquid masses the particular sampleis responsible for imbibing. Alone, these dataset only allows for qualitative, relativecomparisons to be made regarding the liquid uptake ability of a given sample and donot speak to the structure of the blockettes measured. They do, however, allow forthe fitting of a simple power curve which, when combined with the understandingof how capillary dynamics vary with channel size and shape developed in §§2.3.1and 2.3.3, allow for a semi-quantitative analysis and tracking of the fibre-scale structureof blockettes.Recall the result for the long time imbibition distance evolution in a power-lawshaped channel with exponent n (Equation 2.22):H3n+1 ∝ 3n+ 18B(n)T.Replacing the dimensionless distance and time with their respective dimensionalquantities we obtain an expression of the form:h ∝ (3n+ 1)r3n3n+10 α1−3n3n+18B(n)(cσµ) 13n+1︸ ︷︷ ︸∼ atb︷︸︸︷13n+1 . (4.1)Notice that the coe cient term preceding the time term contains only parametersrelated to fluid properties and channel geometry, while the time exponent only containsthose related to channel geometry. Simplifying the above by combining terms in thecoe cient and exponent of the above relation into power-law parameters a and b, itis possible to cast the capillary imbibition distance relation for a general power-lawshaped channel as a simple power-law fit:h = atb. (4.2)a varies monotonically with the capillary throat size r0 and in a general sense maybe thought of as representative of sample porosity. That is to say, considering all otherparameters to be equal, a larger value of a corresponds to a larger capillary diameter.Though a is also dependent on capillary shape (i.e., n), for large aspect ratio channels(small values of n) as are present in wood [5, 55] this dependence varies little.In a similar vein, b may be interpreted as representative of the capillary channelgeometry: a larger value of b corresponds to a less divergent channel or porous medium(b = 0.5 for straight walled channels as in Equation 2.7), while smaller values of bcorrespond to a more divergent capillary/porous medium network geometries (b→ 0as n→∞).44CHAPTER 4. LIQUID UPTAKE EXPERIMENTSPower-law curves were fit to the measured liquid uptake times series using a simpleleast-squares minimization routine.This simple power-law fit and parameter interpretation was used to analyze theliquid uptake time series measured in the lab according to the procedure outlined in§4.1. The parameters a and b yield insight into the fibre-scale structure of a blocketteand thus allow for the determination of a given compression process’s eectiveness intransforming blocks into blockettes with enhanced liquid uptake ability.4.2 Analysis of Measured Liquid Uptake Data4.2.1 Overall TrendsMeasured liquid uptake time series measured according to the laboratory procedureoutlined in §4.1.2 are shown in Figure 4.4. Their location in the plot is determinedby the conditions which the blockettes the uptake is associated with were subjectedto during lab-scale compression. Each sub-plot contains the time series associatedwith the particular blockette (solid, red data series) as well as that for raw wood blocks(dashed, black data series). Each uptake curve grouped by the compression rate usedto produce the blockette is given in Figure 4.3. A similar plot with all uptake curvesgrouped by the total amount of strain applied to the blockette can be seen in Figure 4.5.0 1000 2000 3000 4000 5000Time, t (s)0.00.51.01.52.02.53.03.5Mass Imbibed (g)²˙=0. 0mm/s²˙=0. 1mm/s²˙=3. 0mm/s²˙=4. 0mm/s²˙=5. 0mm/sFigure 4.3: All measured uptake curves grouped by compression rate used during compressionof the sample.45CHAPTER4.LIQUIDUPTAKEEXPERIMENTS˙ (mmmin−1)| | | |0.1 3.0 4.0 5.0∗(%)————40302010Figure 4.4: Comparison of the liquid uptake time series measures arranged according to total strain applied (∗) to the sample and compressionrate (˙) it was achieved at. Each sub-plot is a log-log plotted time series of imbibed mass of liquid vs. time with axis labels removed for clarity.Red curves (—) correspond to each compressed blockette’s uptake, black curves (—) correspond to raw wood’s uptake. The dashed black curve(- - -) corresponds to a m ∝ t1/2 line which may be used as a visual reference as to which path an ‘ideal’ capillary flow would take.46CHAPTER 4. LIQUID UPTAKE EXPERIMENTSLiquid uptake over time for the variously compressed blockettes generally exceedsthat which was achieved by raw wood, though several experiments resulted in blocketteswith less total liquid imbibed than raw wood. Selected blockettes also exhibited largedeviations from raw wood’s observed total liquid uptake as well as uptake rate. Overthe test period of 5400s, raw wood was able to imbibe 1.705g of water from thereservoir while the best performing blockette ({∗, ˙} = {20%, 3.0mmmin−1}) wasable to imbibe 3.405g in the same amount of time. The worst performing blockette({∗, ˙} = {20%, 5.0mmmin−1}) was observed to imbibe less water (1.638g) in thesame time. The use of the labels ‘best’ and ‘worst’ in the preceding assumes moretotal uptake is desired from a sample and implicitly only considers the uptake at theend of a 5400s long test, but because the capillary dynamics vary from test to test(and thus each blockettes relative position in the total uptake hierarchy) the ‘best’ and‘worst’ blockette is in fact a function of time over the course of an uptake experiment.The variability of total uptakes (both at the end of an experiment or during) as wellas uptake dynamics observed is also reflected in the range of a and b values of theassociated power-law curves that are fit to the mass uptake data.0 1000 2000 3000 4000 5000Time, t (s)0.00.51.01.52.02.53.03.5Mass Imbibed (g)² ∗ =0. 0² ∗ =0. 1² ∗ =0. 2² ∗ =0. 3² ∗ =0. 4Figure 4.5: All measured uptake curves grouped by total amount of strain each sample wassubjected to during compression.Considering power curves fit to complete mass uptake datasets, b values werefound to range from 0.44 (just under that expected for cylindrical capillary channels,0.5 [32, 45, 61]) to 0.23 (just under that expected for conical capillary channels, 0.25 [45]),while a values varied from 0.045 to 0.34. If only considering early times, however, thedynamics of the uptake process are somewhat more homogeneous: b values for all theexperiments conducted fell within the 0.5±0.05 range when power-law curves were fit47CHAPTER 4. LIQUID UPTAKE EXPERIMENTSto the first 700s of mass uptake data. This observation is keeping with the early timebehavior predicted by Equation 2.21.0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35a0.200.250.300.350.400.45b²˙=0. 0mm/s²˙=0. 1mm/s²˙=3. 0mm/s²˙=4. 0mm/s²˙=5. 0mm/sFigure 4.6: Power curve parameters a and b grouped by their associated blockette’s compressionrate. The correlation between a and b is due to their mutual involvement of channel parametern (see Equation 4.1).Considering the operation and design of a screw press or refining system, it wouldbe ideal and convenient if the response of wood’s structure (in terms of liquid uptakeability) to compression was predictable or exhibited deterministic trends. ExaminingFigure 4.4 however, the lack of such a response is evident. Progressing along a series offour tests while holding either ∗ or ˙ constant, the series where ˙ =0.1mmmin−1 is thelone series of experiments which consistently outperformed raw wood in terms of liquiduptake in an increasingly significant way. Each of the other eight series along lines ofconstant ∗ or ˙ exhibit dierent, non-monotonic trends in total uptake. The relativelystrong response of wood’s structure to the slow compression observed here is likelydue to secondary temperature eects during lab-scale compression; by consideringthe relative amounts of time taken to complete the lab-scale compression insight maybe gained into this eect. At a minimum, the series of four ˙ =0.1mmmin−1 teststake 30 times longer to complete than the other ‘fast’ compression tests and so, asall the compression was done at lab room/ambient conditions, had a minimum of 30times longer to cool down following their steaming. Though sample block temperaturewas not monitored during the tests, it is not unreasonable to expect that a steamedwood block in contact with a large mass of metal (as is the case in the compressiontests) would cool significantly over the course of the roughly 50 minute long ‘slow’compression tests. As a result of this cooling, the sample block’s compressive modulii48CHAPTER 4. LIQUID UPTAKE EXPERIMENTS0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35a0.200.250.300.350.400.45b² ∗ =0. 0² ∗ =0. 1² ∗ =0. 2² ∗ =0. 3² ∗ =0. 4Figure 4.7: Power curve parameters a and b grouped by the total strain applied to theirassociated blockettes during compression. The correlation between a and b is due to their mutualinvolvement of channel parameter n (see Equation 4.1).would then be higher than their ‘fast’ compression counterparts. These high moduliiblocks would then incur greater amounts of stress for a given imposed strain duringcompression and their structural changes would reflect this. It is interesting to note thatthis observation of a strong structural response by relatively cold wood is similar to theeect observed of increased structural damage as a result of colder chip temperatureduring pilot scale compression (pilot-scale trial P3, see §3.3.1).Other than the secondary temperature eect discussed above, the total liquid uptakevalues as well as the structural parameters a and b from fitted power curves seemsto vary randomly across their respective ranges and does not appear to correlate wellwith the compression conditions used to create the blockettes. This notion is exploredfurther in the following subsection where the statistical behavior of the structuralparameters a and b is examined.4.2.2 Statistical Perspective on Power CurvesThe apparent stochastic response from the fibre-scale structure of the blockettes (asviewed by considering power curve parameters a and b) may be investigated furtherby comparing the observed values of a and b to another sample of values which isknown to be random. This comparison may be quantitatively done by employing thetwo-sample Kolmogorov-Smirnov test [25, 41, 46, 54].49CHAPTER 4. LIQUID UPTAKE EXPERIMENTSPerforming a Kolmogorov-Smirnov test for two samples of values f(x) and g(x)containing m and n values, respectively, and with corresponding empirical cumulativedistribution functions F (x) and G(x) first requires the calculation of the samplestatistic Dmn and critical statistic Dmn,α:Dmn = maxx|F (x)−G(x)| (4.3a)Dmn,α = Γ−1(α)(m+ nmn)1/2(4.3b)In the above, Γ−1(α) is the inverse of the Kolmogorov-Smirnov distribution atconfidence level 1− α, defined as [41, 54]:Γ(α) =√2piα∞∑k=1exp[−(2k − 1)2pi28α2]. (4.4)The KS test determines the truthfulness of the null hypothesis H0 : F (x) = G(x).H0 is rejected at confidence level 1 − α if Dmn > Dmn,α. That is to say: if thetest statistic computed from the two samples is larger than the critical statistic, it isreasonable to say that the two samples were not drawn from the same distributionwith confidence level 1− α.In practice, the Kolmogorov-Smirnov test was administered by taking the followingsteps:1. Generate a random sample of 16 values from the random, uniform distributionscaled over a range of values corresponding to one which covers that of theobserved a or b values.2. Calculate Dmn and Dmn,α using the observed a (or b) values and the correspond-ing random, uniform values obtained from sampling in the previous step. A 95%confidence level was used for all the tests performed, so α was set to 0.05.3. Record whether H0 was rejected or accepted given the criteria outlined above.4. Perform test again by repeating steps 1-3.A Monte Carlo approach was taken to combat the relatively small sample sizesof a and b values: 100,000 tests were performed on both the set of a and b datasets.First considering tests on the b value data: in 95.8% of tests on the b dataset H0was accepted indicating that (with 95% confidence) it is reasonable to say that thevalues of b obtained from lab-scale liquid uptake experiments are simply randomlyand uniformly distributed. If the magnitude of b is interpreted as an indication of50CHAPTER 4. LIQUID UPTAKE EXPERIMENTSthe divergence of the capillary pores/pore network, as above, this result means thatcompression does not deterministically alter the pore network in wood in a way whichmakes it more (or less) adept at imbibing fluid to within the limits of what can bemeasured using the current laboratory setup. This result is di cult to reconcile withintuition and when taken absolutely is likely something of an overstatement of physicaltruth; in reality, though all of the blockettes studied were sourced from the same pieceof wood, the natural variation present in the pore shape/pore network geometry ofwood is great enough that any deterministic shifts caused by compression are simplyovershadowed.In contrast to the result obtained for the pore divergence parameter b, only 14.7% ofthe tests performed with the a dataset resulting in acceptance of H0. Given this value,it would not be reasonable to say that a values are uniformly, randomly distributed overtheir observed range. Indeed, a grouping of the values towards that of raw wood canbe seen in Figures 4.6 and 4.7. Physically, this result may be explained by consideringthe shift experienced by the porosity of wood as it is compressed. Three stages ofwood matrix collapse exist during a compression test [19, 20]: linear elastic regime,pore collapse, and solid compression. At small strains (∗ < 0.05) in the linear elasticregime, the wood matrix stays intact but deforms elastically. For moderate strains pastthis regime (0.05 < ∗ < 0.7), the wood matrix experiences plastic pore collapse andbegins to densify. Past this regime solid compression of the cellulose/lignin begins tooccur. For the compression tests performed in this work, the samples are all well withinthe plastic pore collapse regime. This means that although porosity will have droppedgreatly in localized areas surrounding the initially larger and now collapsed pores, thecompression applied to the samples is such that the overall sample porosity/density isnear, or grouped around, that of raw wood.515 Conclusions and Outlook5.1 On Wood (chip and fibre) Morphological ChangesDue to CompressionSingle-stage, pilot-scale compression trials were carried out and the resulting chipetteshad their pore size distributions measured using the developed digital image analysisprocedure. All of the treated chipettes that were analyzed exhibited dierences intheir pore size distribution relative to raw wood. Specifically, the collapse of largerpore structures was noted and the relative incidence of large, early wood pores inthe wood matrix was observed. These observations are consistent with the qualitativestudies performed prior [21, 22, 35] and indicate that fibre-scale destructuring is aresult common amongst wood chip compression processes. Fibre collapse as wellas wood matrix tearing/cracking and fibre/middle lamella delamination have beenobserved. The strongest response from the pore size distribution was observed inthe trial which lowered the steaming temperature (from 108◦C to 72◦C) of woodchips prior to compression. The lower steaming temperature acted to increase thecompressive modulii of the wood chips and thus altered their stress state responseduring compression. This study only considered a single alternate trial temperature andso global statements of operational eectiveness are not necessarily justified, howeverin the case studied (P3) a reduction in steaming temperature resulted in a strongdestructuring response in the wood matrix; this result indicates that varying wood chipsteaming temperature is an eective tool that may be used to strongly control level offibre-scale destructuring incurred by chips during compression.Multi-stage, mill-scale compression trials were also analyzed. Consistent with theresults obtained from pilot-scale trials, the pore size distribution of wood was observedto shift towards smaller, late wood sized pores while the overall population wasobserved to homogenize with compression. Using the obtained pore size distributions,the capillary driven liquid uptake ability of the multi-stage chipettes was predicted.Contrary to operational knowledge, the overall liquid uptake ability of chipetteswas found to decrease relative to raw wood, though successive compression stageswere predicted to result in increased liquid uptake. The disparity between these52CHAPTER 5. CONCLUSIONS AND OUTLOOKpredictions and experimentally [39] and industrially [21, 35] obtained results indicatesthat the operational eectiveness and mechanism of action of screw press machinesin enhancing liquid uptake ability of wood chips lies in their ability to alter bulk chipproperties, or at least alter properties not apparent at the scale of individual fibres (i.e.,areas of < O(1mm), or lengths of < O(100µm)).Though the fibre-scale destructuring measured in this work does not seem tostrongly or deterministically alter liquid uptake ability alone, the homogenization ofthe pore size distribution (reduction in σ2∗) which was observed in the pilot- andmill-scale trials is in keeping with the enhanced uniformity of chemical treatmentreported previously [22, 22, 35, 39].5.2 On Liquid Uptake Ability of ChipettesThe eect of compression on wood’s structure was also studied at laboratory-scale.A one dimensional compression testing scheme was devised and used to compresssingle blocks of wood at various compression rates to various total amounts. Followingcompression at lab scale, the liquid uptake ability of the blockettes was measureddirectly using an apparatus consisting, simply, of a liquid reservoir and a laboratorybalance. The liquid uptake time series that were collected allowed for the fitting ofpower-law curves to the data.With the exception of the blockettes produced by slow compression (˙ =0.1mmmin−1),no readily deducible trends were observed in the measured liquid uptakes abilities ofthe blockettes. With respect to the slow compression tests resulting in consistently wellperforming blockettes (in terms of liquid uptake), the secondary eect of temperaturedropping markedly during slow compression is thought to be responsible via alteringthe modulii of the sample blocks. This is consistent with the results from the pilot-scalechipette analysis. Overall, colder compression temperatures resulted in stronger struc-tural responses from wood both at pilot- and lab-scale; the temperature compressionis carried out at seems to represent a strong handle on the level of destructuringimparted the wood matrix via compression.The apparent random, stochastic response of the structure of wood to compression(as viewed through the lens of liquid uptake ability) prompted further statistical analysisof power curve parameters. A Monte Carlo approach was taken to confirm the uniform,random character of the capillary channel divergence parameter b while the changes toporosity incurred to the blocks during compression (a values) were shown not to berandom and uniform. Taken together, the results of the statistical testing suggest that,principally, compression processes do not have the ability to deterministically alter thestructure of wood on the fibre-scale such that it is more adept at imbibing liquid. Assuch, the success and utility of screw compression devices seems to lie within theirability to alter the morphology of chips and chip beds, not fibre-scale features of chips.53CHAPTER 5. CONCLUSIONS AND OUTLOOKThe systematic and consistent changes observed to the pore size distributions ofchipettes produced at pilot- and mill-scale indicates fibre-scale changes to the wood chipstructure do, indeed, take place. The random response of liquid uptake ability measuredin the lab indicates, though, that those fibre-scale changes are overshadowed by naturalvariation within the natural structure of wood or are otherwise not particularlysignificant in the liquid uptake process.5.3 Future WorkThe capacity of a screw press machine to alter the chemical properties of chips andso make them more amenable to subsequent refining/paper making represents anextension of this work which would likely be a productive one. The extractive removalability of a screw press machine has been studied at a high level at a mill-scale [36, 56],though fundamental studies probing the mechanism of extractive release from thewood matrix, the eect of extractive removal on overall mill process e ciency, andextractive based bio-product process integration are all largely unexplored in theliterature.An interesting empirical approach similar to that used in this work would be toconsider changes to the particle size distributions associated with various compressionconditions. A statistical approach may be adopted and population models fittedto experimental particle size distribution sample data (as was done with pore sizedistributions in this work). Alternatively, a more fundamental approach using themuch more physically based theory of comminution models could be adopted. Suchmodeling work would make bulk bed properties such as a bed’s specific surface area orporosity calculable, allowing for improved design or understanding of the impregnationstep of the overall process.As the mechanism of action and operational utility of wood chip compressionprocesses seems to lie within a given processes ability to alter chip-scale morphology,attempting to characterize wood chip compression by the changes incurred by the chipbed as a whole represents what would likely be a generally fruitful next step. As aspecific example: whereas in this work fibre/pore re-expansion following compressionwas neglected (see §2.3.4 for justification of this), investigating the re-expansion of thechip bed following compression (and any liquid flow into it driven by the re-expansion)would certainly represent a worth while endeavor. Though geometrically distinct, aporo-elastic model such as that presented in [24, 52] could be readily adapted to thecase of a bed of expanding chips in contact with a liquid bath. Eectively, only achange to the boundary conditions used in the analysis presented in [24] would berequired to make the continuum model presented applicable to a screw press processand allowing for the simulation of the chip bed liquid uptake process.54Bibliography[1] M. Alaqqad, C.P.J. Bennington, and D.M. Martinez. An estimate of the axialdispersion during flow through a compressible wood-chip bed. The CanadianJournal of Chemical Engineering, 90:1601–1610, 2012.[2] M. Alaqqad, C.P.J. Bennington, and D.M. Martinez. The permeability of wood-chipbeds: The eect of compressibility. The Canadian Journal of Chemical Engineering,90:1278–1288, 2012.[3] Estelle Arbellay, Christophe Corona, Markus Stoel, Patrick Fonti, and ArmelleDecaulne. Defining an adequate sample of earlywood vessels for retrospectiveinjury detection in diuse-porous species. PLoS one, 7:51–56, June 2012.[4] D. Atack, C. Heitner, and M.I. Stationwala. Ultra high yield pulping of eastern blackspruce. Svensk papperstidning, 81(5):164–176, 1978.[5] Jacob Bear. Dynamics of Fluids in Porous Media. Dover, 1972.[6] J.M. Bell and F.K. Cameron. The flow of liquid through capillary spaces. Journal ofPhysical Chemistry, 10:658–674, 1906.[7] Je A. Bilmes. A gentle tutorial of the EM algorithm and its application to parameterestimation for gaussian mixture and hidden markov models. Technical report,International Computer Science Institute, 1998.[8] Wout Boerjan, John Ralph, and Marie Baucher. Lignin biosynthesis. Annual Reviewof Plant Biology, 54:519–546, 2003.[9] G. Bradski. The opencv library. Dr. Dobb’s Journal of Software Tools, 2000.[10] Quang V. Cao and Qinglin Wu. Characterizing wood fibre and particle lengthwith a mixture distribution and a segmented distribution. Holforschung, 61:124–130,2007.55BIBLIOGRAPHY[11] X.F. Chang, C. Bridges, D. Vu, D. Kuan, L. Kuang, J.A. Olson, A. Luukonen, and R. P.Beatson. Saving electrical energy by alkaline peroxide pretreatment of TMP priorto low consistency refining. In PAPTAC annual meeting, 2011.[12] A.P. Dempster, N.M. Laird, and D.B. Rubin. Maximum likelihood from incom-plete data via the EM algorithm. Journal of the Royal Statistical Society. Series B(Methodological), 39:1–38, 1977.[13] D. Fengel and M. Stoll. Variation in cell cross-sectional area, cell-wall thickness andwall layer of spruce tracheids within an annual ring. Holzforchung, 27:1–7, 1973.[14] W.C. Frazier and G.J. Williams. Reduction of specific energy in mechanical pulpingby axial pre-compression of wood. Pulp and Paper Canada, 83(6):87–92, 1982.[15] N. Fries and M. Dreyer. An analytic solution of capillary rise restrained by gravity.Journal of Colloid and Interface Science, 320:259–263, 2008.[16] N. Fries and M. Dreyer. The transition from inertial to viscous flow in capillaryrise. Journal of Colloid and Interface Science, 327:125–128, 2008.[17] N. Fries and M. Dreyer. Dimensionless scaling methods for capillary rise. Journalof Colloid and Interface Science, 338:514–518, 2009.[18] Ignacio García-González and Patrick Fonti. Ensuring a representative sample ofearlywood vessels for dendrochronological studies: an example from two ring-porous species. Trees, 22:237–244, 2008.[19] Lorna J. Gibson. The hierarchical structure and mechanics of plant materials.Journal of the Royal Society Interface, 9(76):2746–2766, 2013.[20] Lorna J. Gibson and Michael F. Ashby. Cellular Solids: Structure and Properties.Pergamon Press, 1988.[21] Dmitri Gorski. ATMP Process: Improved Energy E ciency in TMP Refining UtilizingSelective Wood Disintegration and Targeted Application of Chemicals. PhD thesis,Mid-Sweden University, 2011.[22] Dmitri Gorski, Jan Hill, Per Engstrand, and Lars Johansson. Review: Reduction ofenergy consumption in tmp refining through mechanical pre-treatment of woodchips. Nordic Pulp and Paper Research Journal, 25(2):156–162, 2010.[23] Lisbeth Hellström. On the Wood Chipping Process - a Study on Basic Mechanisms inOrder to Optimize Chip Properties for Pulping. PhD thesis, Mid Sweden University,2010.56BIBLIOGRAPHY[24] Duncan R. Hewitt, Jerome A. Neufeld, and Neil J. Balmforth. Shallow, gravity-drivenflow in a poro-elastic layer. Journal of Fluid Mechanics, 778:335–360, 2015.[25] Robert V. Hogg and Johannes Ledolter. Engineering Statistics. Macmillan PublishingCompany, 1987.[26] John D. Hunter. Matplotlib: A 2d graphics environment. Computing in Science &Engineering, 9(3):90–95, 2007.[27] G.M. Irvine. The significance of the glass transition of lignin in thermomechanicalpulping. Wood Science and Technology, 19(2):139–149, 1985.[28] Lars Johansson, Jan Hill, Dmitri Gorski, and Patrik Axelsson. Improvement ofenergy e ciency in tmp refining by selective wood desintegration and targetedapplication of chemicals. Nordic Pulp and Paper Research Journal, 26:31–46, 2010.[29] Eric Jones, Travis Oliphant, Pearu Peterson, et al. SciPy: Open source scientifictools for Python, 2001–. [Online; accessed 2016-10-12].[30] Z. Koran. Energy consumption in mechanical fibre separation. In Canadian Pulpand Paper Association 66th annual meeting, Montreal, Canada, 1980.[31] K.-A. Kure, G. Dahlqvist, M.J. Sabourin, and T. Helle. Development of spruce fibreproperties by a combination of a pressurized compressive pretreatment and highintensity refining. In International Mechanical Pulping Conference, May 1999.[32] Richard Lucas. Ueber das zeitgesetz des kapillaren aufstiegs von flüssigkeiten.Kolloid-Zeitschrift, 23, 1918.[33] Michael M. Micko. Alberta aspen vs. black poplar wood quality dierences.Technical report, Canadian Forestry Service/Alberta Forest Service, 1987.[34] J. Navas, J. A. Poce-Fatou, J. J. Gallardo, and J. Martin. Evaluation method for poresize distribution by using capillary liquid suction tests. Journal of Porous Matter,17:207–215, 2010.[35] Erik Nelsson. Reduction of refining energy during mechanical pulping. LicentiateThesis, Swedish University of Agricultural Sciences, 2011.[36] Michaela Tanase Opedal. Extractives in process water from compressive pre-treatmentof chips in mechanical pulping. PhD thesis, Norwegian University of Science andTechnology, 2011.[37] Nobuyuki Otsu. A threshold selection method from gray-level histograms. IEEETransaction on Systems, Man, and Cybernetics, 1:62–66, 1979.57BIBLIOGRAPHY[38] A.J. Panshin and C. de Zeeuw. Textbook of wood technology: structure, identification,properties and uses of the commercial woods of the United States and Canada.McGraw-Hill Book Company, 1980.[39] Anita Parkinson, Patrick Tessier, and Chung L. Lee. Eects of compression ratiosof screw feeders during multistage impregnation on black spruce chips and fibres.TAPPI Journal, 79:149–156, 1996.[40] F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel,P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau,M. Brucher, M. Perrot, and E. Duchesnay. Scikit-learn: Machine learning in Python.Journal of Machine Learning Research, 12:2825–2830, 2011.[41] Wolfgang Pelz and I.J. Good. Approximating the lower tail-areas of the kolmogorov-smirnov one-sample statistic. journal of the Royal Statistical Society. Series B(Methodological), 38(2):152–156, 1976.[42] F. Peng and T. Granfeldt. Changes in the microstructure of spruce wood chipsafter screw press treatment. Journal of Pulp and Paper Science, 22(4):140–145, 1996.[43] Vincent Placet, Jo elle Passard, and Patrick Perré. Viscoelastic properties of woodacross the grain measured under water-saturated conditions up to 135c: evidenceof thermal degradation. Journal of Material Science, 43:3210–3217, 2008.[44] Richard A. Redner and Homer F. Walker. Mixture densities, maximum likelihoodand the EM algorithm. SIAM Review, 26:195–239, 1984.[45] Mathilde Reyssat, Laurent Courbin, Etienne Reyssat, and Howard A. Stone. Imbibi-tion in geometries with axial variations. Journal of Fluid Mechanics, 615:335–344,2008.[46] F. James Rohlf and Robert R. Sokal. Statistical Tables, Third Edition. W.H. Freemanand Company, third edition, 1995.[47] Lennart Salmén, Anna Tigerström, and Christer Fellers. Fatigue of wood - charac-terization of mechanical defibration. Journal of Pulp and Paper Science, 11(3):68–73,1985.[48] Gideon Schwarz. Estimating the dimension of a model. The Annals of Statistics,6:461–464, 1978.[49] Yash Sharma. Automated fibre segmentation in micro ct images of paper. Master’sthesis, The University of British Columbia, 2014.58BIBLIOGRAPHY[50] Dahua Shou and Jintu Fan. Structural optimization of porous media for fast andcontrolled capillary flows. Physical Review E, 91, 2015.[51] Dahua Shou, Lin Ye, and Jintu Fan. The fastest capillary flow under gravity. AppliedPhysics Letters, 104, 2014.[52] J.I. Siddique, D.M. Anderson, and Andrei Bondarev. Capillary rise of a liquid into adeformable porous material. Physics of Fluids, 21, 2009.[53] Bernard W. Silverman. Density Estimation for Statistics and Data Analysis, Volume26 of Chapman & Hall/CRC Monographs on Statistics & Applied Probability. CRCPress, 1986.[54] Richard Simard and Pierre L’Ecuyer. Computing the two-sided kolmogorov-smirnovdistribution. Journal of Statistical Software, 39(11), 2011.[55] Gary A. Smook. Handbook for Pulp and Paper Technologists, 2nd Edition. AngusWilde Publications Inc., 1992.[56] Michaela Tanase, Per Stenius, Lars Johansson, Jan Hill, and Christer Sandberg. Massbalance of extractives around impressafiner in mill and pilot scale. Nordic Pulpand Paper Research Journal, 25, 2010.[57] R. Thiruvengadaswamy and D. Ouellet. On the generation of structural damage inwood by cyclic compressive loading. Journal of Pulp and Paper Science, 24:247–253,1998.[58] Stéfan van der Walt, S. Chris Colbert, and Gaël Varoquaux. The numpy array: Astructure for e cient numerical computation. Computing in Science & Engineering,13(2):22–30, 2011.[59] Guido van Rossum. Python tutorial, technical report cs-r9526. Technical report,Centrum voor Wiskunde en Informatica (CWI), May 1995.[60] Georg von Arx and Marco Carrer. ROXAS - a new tool to build centuries-longtracheid-lumen chronologies in conifers. Dendrochronolgia, 32:290–293, 2014.[61] Edward W. Washburn. The dynamics of capillary flow. The Physical Review,17(3):273–283, 1921.[62] Nina Winham. Ubc research targets major industrial energy user. BC Hydro NewsRelease, 2009.[63] C.F. Je Wu. On the convergence properties of the EM algorithm. The Annals ofStatistics, 11:95–103, 1983.59BIBLIOGRAPHY[64] Amy E. Zanne, Mark Westoby, Daniel S. Falster, David D. Ackerly, Scott R. Loarie,Sarah E.J. Arnold, and David A. Coomes. Angiosperm wood structure: Globalpatterns in vessel anatomy and their relation to wood density and potentialconductivity. American journal of Botany, 97:207–215, 2010.60A Imaging/Image Analysis DetailsA.1 Image Analysis CodeA commented copy of the python function find_pores() may be found in Listing A.1.As per the docstring in the function definition, find_pores() takes an RGB image, aboolean which controls the removal of pores touching the image border, and a dyecolor code. A tuple containing the OpenCV contours and their associated hierarchiesfrom all the pores found in the image are returned for further calculation (scaling/truearea calculation). Python 2.7.11 [26, 29, 58, 59], and OpenCV 2.4.8 [9] were used.Listing A.1: find_pores() function definition1 def find_pores(img_file , interior=True , dye =2):2 ’’’3 -----4 Finds the contours coresponding to the pores in an RGB image of a thin5 section of wood that has been dyed.6 -----7 Arguments:8 -----9 img_file : RGB image of thin section.10 interior : If ’True ’, only contours that do not touch the image border will11 be returned. If ’False’ all contours will be returned.12 dye : [0, 1, 2] corresponding to [red , green , blue] layer that will be used13 as the initial mask for the image.14 -----15 Returns:16 -----17 conts , hiers [0]: Array of image contours that have been found , array of18 contour hierarchies as per OpenCV convention.19 -----20 ’’’2122 wood = cv2.imread(img_file) #read RGB image23 red , green , blue = cv2.split(wood) #split RGB channels24 color_mask = [red , green , blue]25 #woodg = cv2.cvtColor(wood , cv2.COLOR_BGR2GRAY) #grayscale of original26 pad = 42 #border size for isolating full pores27 seed = (0,0) #flood fill from the very corner2829 #make kernels for morphological transforms30 k1 = cv2.getStructuringElement(cv2.MORPH_CROSS , (3,3))31 k2 = cv2.getStructuringElement(cv2.MORPH_CROSS , (5,5))3261APPENDIX A. IMAGING/IMAGE ANALYSIS DETAILS33 #blur image to help thresholding work as it should34 wb = cv2.GaussianBlur(color_mask[dye], (9,9), 0)3536 #Otsu threshold then push a bit more to remove light spots in walls37 retval , thresh = cv2.threshold(wb, 0, 255, cv2.THRESH_BINARY+cv2.THRESH_OTSU)38 retval , thresh = cv2.threshold(wb, retval+5, 255, cv2.THRESH_BINARY)3940 #clean up small features with open/close operations41 wc = cv2.morphologyEx(thresh , cv2.MORPH_CLOSE , k2, iterations =1)42 wo = cv2.morphologyEx(wc, cv2.MORPH_OPEN , k2, iterations =2)4344 #Get rid of contours on image border if necessary45 if interior:46 #add white border , fill with black , slice off border47 w_bord = cv2.copyMakeBorder(wo, pad , pad , pad , pad , cv2.BORDER_CONSTANT ,48 value =(255 ,255 ,255))49 cv2.floodFill(w_bord , None , seed , 0)50 w_int = w_bord[pad:-pad , pad:-pad]5152 #find contours in altered image53 img , conts , hiers = cv2.findContours(w_int.copy(), cv2.RETR_TREE , cv2.CHAIN_APPROX_SIMPLE)5455 else:56 #find all contours in binarized image57 img , conts , hiers = cv2.findContours(wo.copy(), cv2.RETR_TREE , cv2.CHAIN_APPROX_SIMPLE)5859 plt.show()60 return conts , hiers [0]A.2 Camera SettingsThe settings used for the imaging of sectioned, dyed wood samples. A combinationof a Nikon transmission stereo microscope and a 10Mpx digital camera were used toacquire the images before processing to achieve a spatial resolution of 4.2px/µm.Table A.1: Parameters used when collecting images of dyed, sectioned wood samples. A 10×objective was used for imaging and the illumination was set to maximum. A daylight (‘D’) filterwas used in tandem with a neutral color balance (‘NCB’) filter as well to alter the white point ofthe halogen bulb used for sample illumination.Parameter ValueImage Resolution 3664px×2748pxExposure Time 5msBrightness Gain 1dBγ 0Contrast 0{R,G,B} Scaling {1.05, 0.98, 1.42}62B Liquid Uptake Time SeriesAcquisitionA commented copy of the function run_experiment() that was created to collect atime series of mass readings from the scale used to monitor the mass of the liquidreservoir used during liquid uptake experiments is given in Listing B.1. The function’srequired balance argument is the serial object to interface with (i.e., lab balance)which the computer sends and receives balance readings and commands to and from.This is done via a USB serial port emulation protocol made available by the balance’sfirmware. The SartScale serial object class enables the use of this protocol and itsdefinition is given in Listing B.2.To begin a new liquid uptake test, the user would typically only need to changethe log_name argument to create a new and empty datafile for that test’s data to bestored in before simply calling the function run_experiment from a terminal runningpython. This code was run during experiments using Python 3.4.4 [26, 29, 58, 59].Listing B.1: run_experiment() function definition1 def run_experiment(balance , dur=7200, dt=1.5, log_name=’TEST’, per =1):2 ’’’3 -----4 Collects weighing data from a lab balance and logs it to a file for5 later processing.6 -----7 Arguments:8 -----9 balance : SartScale object to interface with the balance with.10 dur : Total length of experiment [s].11 dt : Time interval between measurments being taken [s].12 log_name : str , Name of .csv file for data to be written to.13 per : int , number of measurments taken before printing current value to screen.14 -----15 ’’’16 #lock balance and zero , wait 5s for things to finish17 balance.lock()18 balance.zero()19 time.sleep (5)2021 times = [] #timestamps22 values = [] #measured values23 units = balance.units() #units of readings63APPENDIX B. LIQUID UPTAKE TIME SERIES ACQUISITION24 i = 0 #keep track of measurements25 end_i = int(dur/dt)26 start = time.time() #start time2728 while (time.time() - start) <= dur:29 #current time and value appended to current lists30 now = time.time()31 weight_now = balance.value()32 times.append(now)33 values.append(weight_now)34 i += 135 elapsed = now - start3637 #give periodic updates/provide sanity check38 if i%per == 0:39 print(’{} measurements taken in {:.6f}s at {}’.format(i, elapsed , time.ctime ()))4041 #wait dt seconds until next measurement gets taken42 time.sleep(dt)4344 #unlock balance and shut stuff down45 balance.unlock ()46 balance.done()4748 #squish items and values together and write to file49 data = np.column_stack ((times , values ))50 np.savetxt(’{}.csv’.format(log_name), data , fmt=[’%.6f’,’%.3f’],51 header=’Units: {}’.format(units), delimiter=’,’)52 print(’{} values written to \’{}.csv\’’.format(i, log_name ))53 print(’-----’)54 input(’Press \’Enter\’ to terminate the experiment ’)Listing B.2: SartScale object class definition1 class SartScale(serial.Serial ):2 ’’’3 -----4 Interface to a Sartorius Secura , Quintix , and Practum lab balances5 via a USB serial connection. Refer to balance manual for full list6 of serial control command specifications.7 -----8 ’’’910 def __init__(self , port=2, ** kwargs ):11 ’’’12 -----13 Initialises Sartorius Scale (SartScale) device at COM port14 ’port’ with ** kwargs being form serial.Serial objects. Usually15 shows up in COM3 (port 2).16 -----17 ’’’18 serial.Serial.__init__(self , port , ** kwargs)1920 def value(self):21 ’’’22 -----23 Sends print command to balance , returns the weight value that is24 returned. Requires balance to print a single line per print25 request of the 22 charachter variety.26 -----27 ’’’28 try:64APPENDIX B. LIQUID UPTAKE TIME SERIES ACQUISITION29 if self.inWaiting () == 0:30 self.write(’\033P\n’.encode ())3132 line = self.readline (). decode ()33 sign = line [6]34 value = line [7:17]. lstrip ()35 return np.float64(’{}{}’.format(sign , value))3637 except:38 print(’Couldn\’t read/write from port.’)39 return None4041 def units(self):42 ’’’43 -----44 Sends print command to balance , returns the weighing unit (str)45 that is returned with the value.46 -----47 ’’’48 if self.inWaiting () == 0:49 self.write(’\033P\n’.encode ())5051 line = self.readline (). decode ()52 unit = line [17:21]. strip()53 return unit5455 def tare(self):56 ’’’57 -----58 Sends ’tare’ command to balance.59 -----60 ’’’61 self.write(’\033U\n’.encode ())6263 def zero(self):64 ’’’65 -----66 Sends ’zero’ command to balance.67 -----68 ’’’69 self.write(’\033V\n’.encode ())7071 def lock(self):72 ’’’73 -----74 Sends ’lock keys’ command to balance.75 -----76 ’’’77 self.write(’\033O\n’.encode ())7879 def unlock(self):80 ’’’81 -----82 Sends ’unlock keys’ command to balance.83 -----84 ’’’85 self.write(’\033R\n’.encode ())8687 def done(self):88 ’’’89 -----90 Flush COM port buffers (in and out) and close connection to65APPENDIX B. LIQUID UPTAKE TIME SERIES ACQUISITION91 balance , confirm with print.92 -----93 ’’’94 self.flushInput ()95 self.flushOutput ()96 self.close()97 if not self.isOpen ():98 print(’Balance connection shut down successful ’)99 else:100 print(’Balance connection shut down unsuccesful ’)66C Density EstimationThe raw product of the image analysis experiments is a list of pore sizes correspondingto those found with the image analysis script. Alone, this list of numbers is not verydescriptive of the wood structure and it is thus hard to derive any meaning as tohow eectual a particular compressive treatment was from it. Furthermore, the datacollected only represents a sample of a population containing all the pores that undergoa given treatment. The problem is, then, one of density estimation: it is desired toobtain an estimate of the pore size density distribution as a function of pore size froma random sample of pore sizes. Once obtained, the pore size density estimate willallow for the comparison of wood structures to one another in a quantifiable way andthus yield insight into the changes caused by a particular treatment. An easy way tobegin to derive structure and meaning from the collected data is to employ histogramsand their generalization, kernel density estimates. A more involved method that allowsfor the detection and tracking of subpopulations within the overall population is thefitting of Gaussian mixture models (GMMs) to the data. This appendix contains adiscussion and definition of each of these three methods (each of which are relatedand build upon one another) and their relative utility is compared.C.1 HistogramsThough histograms are a common way of presenting discrete data, it is instructive toexplicitly outline what computations are required to generate one as well as outline theirrelationship to the general problem of density estimation. To construct a histogramgiven some list of values x = {xn : n ∈ [1, N ]} (i.e., data), we divide the range coveredby the values of x into some number of equally sized sub-intervals (i.e., ‘bins’ of widthdx) and then assign a bin height, fi, based on the number of occurrences of our datain the ith bin. It is clear to see that the size of each of the bins is dependent onthe sample size by way of the simple count of occurrences of data which falls intoeach bin; it is possible to normalize a histogram to obtain normalized bin sizes, f ∗i , bydividing the occurrence count by the product of the size of x and the bin width:67APPENDIX C. DENSITY ESTIMATIONf ∗i =fiNdx. (C.1)This normalization ensures that the integral of the histogram over the range of thedata is unity and transforms the raw data distribution into a discrete probability massfunction (PMF). It is thus possible to write:∫ ∞−∞f ∗ dx =∑if ∗i = 1. (C.2)f ∗ is the estimate of the probability density function given the data by way ofhistogram construction. In this construction two sources of variability in the finalhistogram shape and as such two potential data interpretation liabilities exist: thenumber of bins chosen and the way data which falls on bin boundaries is treated.Furthermore, in even the best case of a large, well-sampled dataset, histograms arenon-smooth in nature and, accordingly, features of the true population density functionmay be obscured.C.2 Kernel Density EstimatesKernel density estimates (KDE) may be thought of as a generalization of the histogramand, in all but the simplest cases, mitigate the problems associated with bin boundaryambiguity and non-smoothness. To construct a KDE (given data over the independentvariable x) we sum kernel functions, κ(xi), which have been centered on the data xiand normalize by the size of our dataset:f ∗(x) =1NN∑n=1κh(x− xn) = 1NhN∑n=1κ(x− xnh). (C.3)Where f ∗ is now an improved1 estimate of the density by way of kernel densityestimate. The utility of KDE, in general, derives from the fact that the kernel functionκ may be chosen2 from a very wide array of possible candidates, however all choicesrequire the selection of a value for the bandwidth, h. The bandwidth acts to smooth f ∗and the magnitude of h is freely adjustable su ce that h > 0. A common choice forkernel function is the standard normal density function and it is both convenient andadequate to use as an illustrative example to show the usage and relative shortcomings(for the purposes of this works’ overall goal) of KDE.1Smoothness of f∗ is only ensured when a smooth form of κ is used.2Using a uniform kernel (κ ∝ 1) yields a result which is very much similar to a histogram forcomparably selected bin width and h.68APPENDIX C. DENSITY ESTIMATIONBandwidth selection is, roughly speaking, akin to choosing the number of bins/binwidth during the construction of a histogram. As the selection of an appropriatebandwidth is key to obtaining a reasonable estimate, much work has gone intodeveloping both empirically based rules (which yield a value for h given statisticalmeasures and assumptions of the data [48, 53]) and cross-validation techniques whereina value for h is chosen based on dividing the data into parts, computing a densityestimate given some bandwidth using only a part of the data, and then determiningthe suitability of the chosen bandwidth value by calculating some metric (e.g., meanintegrated squared error, or MISE, is common and amounts to a bandwidth selectedbased on minimizing the variance of the error produced in predicting data with agiven density estimate) that describes how well the estimate fits the remaining data.These metrics are often called loss functions. Overall, the computation of a KDE givena univariate dataset results is an estimate for the population density which is smoothand tracks features better (relative to histograms) in the data that would be otherwise‘hidden’ by binning rule properties inherent to histogram construction. A KDE is aneective and easily implementable way to gain an estimate of the overall density ofthe data, but oers nothing in the way of a description of subpopulations which maybe present in the overall population.C.3 (Gaussian) Mixture ModelsAlthough eective at producing an estimate of the overall density of a populationgiven a sample from that population, kernel density estimates do not yield any insightsinto the subpopulations that may or may not be present in a population. In general,mixture models allow for just this additional layer of insight into the nature of apopulation’s ‘pieces’. As mixture models are somewhat pervasive in this work, a briefdescription of their form and properties is provided in the following. As the values ofthe model parameters which are fit (and used to drawn conclusions regarding the data)are dependent on the numerical schemes used to obtain them, a brief outline of themethod employed here (the expectation-maximization (EM) algorithm) is also given.A mixture model is a mathematical model which is composed of a weighted mixtureof some number of mixture distributions; ‘mixture distributions’ is simply the termused for probability distributions used within the mixture model context. A generalmixture model f ∗(a|θ) of a single independent variable a with associated parameterset θ and K components can be written as:f ∗(a|θ) =K∑k=1wkDk(a|θ) (C.4)69APPENDIX C. DENSITY ESTIMATIONWhere Dk is the kth mixture density function and the terms wk are the componentweights, or mixing probabilities, that satisfy wk ≥ 0. Note that the independent variableused has been changed from x to a. This change, though purely notational, is meantto provide a variable name which is more apt to our particular interests: pore areas, a.As f ∗ is itself a probability density function, it possesses all the properties of otherprobability density functions and so we can write the following:∫ ∞−∞f ∗(a|θ) da =∫ ∞−∞K∑k=1wkDk(a|θ) da =K∑k=1[wk∫ ∞−∞Dk(a|θ) da]=K∑k=1wk = 1.(C.5)The above formalism may be applied to any mixture model without loss of validity.Given that we would expect the size of pore sizes to be log-normally distributed [10, 64],the specific type of mixture model that has been applied in this work is a Gaussianmixture model. In the case of a Gaussian mixture model, the form given in Equation C.4is altered by replacing the general probability density functions Dk(a|θ) with, naturally,Gaussian density functions Nk(a|µk, σ2k) with means µk and variances σ2k :f ∗(a|θ) =K∑k=1wkNk(a|µk, σ2k). (C.6)With the Gaussian density function, Nk(a|µk, σ2k), being defined in the standardway as:Nk(a|µk, σ2k) =1√2piσkexp[−(a− µk)22σ2k]. (C.7)For the purposes of comparing several models against one another, it is useful tobe able to calculate an overall mean, µ∗, and variance, σ2∗, for the model distribution.These may be calculated by finding and manipulating the first and second moments,µ(1) and µ(2), respectively, of the mixture distribution. The first moment is the overallmean:µ(1) = µ∗ =K∑i=1wiµi, (C.8)and the second moment is the overall variance plus the squared overall mean:70APPENDIX C. DENSITY ESTIMATIONµ(2) = σ2∗+ µ∗=K∑i=1wi(σ2i + µ2i )⇒ σ2∗ =[K∑i=1wi(σ2i + µ2i )]−[K∑i=1wiµi]2. (C.9)It is instructive to note here that a Gaussian mixture model may be fit to anydata inasmuch as the fitting process is indierent to the meaning of the data beingused in the process. In this work the logarithm of physical pore area data was takenand a model composed of a mixture of Gaussian (normal) distributions was fit to thistransformed data in favor of a model composed of a mixture of log-normal distributionsto the raw data. This data transformation was done for computational convenience andas the logarithm function represents a one-to-one mapping of values for all physicallypossible positive area values this step, practically speaking, only represents a scaling ofthe numerical results, not an obfuscation. For completeness, denote the logarithm ofpore areas (a) as A, defined as:A = loga. (C.10)Thus, our problem consists of fitting a K-component Gaussian mixture model f ∗to data A of the form:f ∗(A|θ) =K∑k=1wkNk(A|µk, σ2k), (C.11)where θ is the parameter vector to be determined and contains all the required wk ,µk, and σ2k values.C.3.1 Model Parameter EstimationSelecting a Suitable K ValueOur model given in Equation C.11 requires 3K + 1 values to be specified in order to becomplete and able to predict a density (a weight, mean, and variance for each mixturecomponent as well as the mixture size, K). With a known or specified value of Kdetermining the remaining 3K parameters is a matter of numerical optimization (modelfitting ) and will be discussed in greater depth below; the selection of an appropriatevalue for K is, however, itself not a matter of model fitting but of model selection. Todraw parallels with the more commonplace fitting of a polynomial model: determining71APPENDIX C. DENSITY ESTIMATIONthe values in θ is akin to finding the coe cients of a polynomial model, whereasdetermining K is comparable to choosing a polynomial degree for a given model.To continue with the polynomial model comparison, selecting too large a K valuewould likely yield a better fit to some data, but represent an ‘over-fit’ model (too high apolynomial degree). It is required, then, to select a value for K not based on model fitbut some measure of model suitability.The measure of model suitability used in this work is the Bayesian InformationCriterion, or BIC [48]. For a model with K parameters fit to N data points with amaximized likelihood function value L∗, the BIC is defined as:BIC = K lnN − 2 lnL∗. (C.12)The best value of K is selected by fitting numerous models with various values ofK calculating the BIC corresponding to each model and choosing the best value ofK , K∗, with the lowest associated BIC value, as per:K∗ = arg minKi(Ki lnN − 2 lnL∗i ) , Ki ∈ Z+. (C.13)The property of the BIC which make it appropriate and useful in this context isits consistency [44, 48]. BIC’s consistency ensures that for an arbitrary sample size themodel predictions will converge to a single value. That is to say that instead of findingthe model best suited to the data (which has been sampled from a larger population)as an e cient criterion would aim to do, a model which describes the population as awhole will be converged upon.Maximum-Likelihood Approach to Model FittingThe concept of a likelihood function was alluded to above in the discussion of modelselection as it pertains to selecting a value for K . In general, the likelihood functionacts as an objective function of the data and parameter set that allows for an optimalparameter set to be chosen. In this work, the optimal parameter set is that whichcorresponds to a maximized likelihood function. In utilizing the maximum-likelihoodapproach, our model fitting (i.e., parameter estimation) problem becomes one ofoptimization of a multivariate likelihood function. The likelihood function, L(θ|A), isdefined as the product of probabilities given by the density model f ∗(A|θ) for somedataset A = {An : n ∈ [1, N ]}:L(θ|A) =N∏n=1f ∗(An|θ). (C.14)The ‘best’ choice of model parameter values, θ∗, is then given by the set of valueswhich maximize L to L∗:72APPENDIX C. DENSITY ESTIMATIONθ∗ = arg maxθL(θ|A) = arg maxθN∏n=1f ∗(An|θ). (C.15)The Expectation-Maximization AlgorithmImplicit in the calculation of θ∗ as outlined in Equation C.15 is an optimization (i.e.,maximization) procedure performed on L(θ|A). In theory, any multi-dimensionaloptimization procedure may be used to obtain a result, however the properties of thelikelihood function specific to Gaussian mixture models make the so-called expectation-maximization, or EM, algorithm particularly well suited for use in this case [7, 12, 44, 63].As the particulars of any numerical optimization impose (ideally very slight) dierenceson their respectively found optimums, it is instructive to outline the basic workings ofthe EM algorithm here before its fruits are used in further analysis Chapter 3. A fullderivation of the EM algorithm may be found in [7, 12]; only the main computationalpoints are given here.Firstly we note that finding the maxima of the likelihood function is equivalent tofinding the maximum of the logarithm of the likelihood function. As the likelihoodfunction often yields large, computationally unwieldy values, the log-likelihood function,Λ(θ|A), is a convenient way of keeping the numerics tractable and allows for somewhatsimplified analysis. It is defined as:Λ(θ|A) = lnL(θ|A). (C.16)We begin by defining the membership probabilities, p(An|k), as the probabilitythat data point An was selected from component k as:p(An|k) = wkNk(An|µk, σ2k)K∑m=1wmNm(An|µm, σ2m). (C.17)Now, by setting the derivatives of Λ with respect to µk, σk, and wk to zero, weobtain estimates of the model parameters corresponding to a maximized Λ:µk =∑Nn=1 p(An|k)An∑Nn=1 p(An|k), (C.18a)σ2k =∑Nn=1 p(An|k)‖An − µk‖2∑Nn=1 p(An|k), (C.18b)wk =1NN∑n=1p(An|k). (C.18c)73APPENDIX C. DENSITY ESTIMATIONThe expressions for the component means and variances are simply the samplemean and sample variances that have been weighted by membership probabilities (i.e.,the probability that datum An was generated by component k), while the expressionfor the mixing probability is the mean of the membership probabilities. It is importantto note that Equations C.18 are coupled and may not be solved explicitly. The EMalgorithm provides an iterative procedure to solve for these values.Assuming we have an initial (i = 0) guess for the values of the model parameters,{wik, µik, (σ2k)i}Kk=1, values of p(An|k)i may be calculated via Equation C.17 using thecurrent iteration of values for the model parameters. This is the ‘E’ step of the EMalgorithm. Now that we have an estimate of the membership probabilities, we cancalculate improved estimates of the model parameters as per the ‘M’ step:µi+1k =∑Nn=1 p(An|k)iAn∑Nn=1 p(An|k)i, (C.19a)(σ2k)i+1 =∑Nn=1 p(An|k)i‖An − µi+1k ‖2∑Nn=1 p(An|k)i, (C.19b)wi+1k =1NN∑n=1p(An|k)i. (C.19c)The E and M steps are repeated in succession until the model parameters convergeto a single value; these values are then deemed to be the constituents of the optimalparameter set, θ∗.74D Non-uniqueness of Tube BundlesProposition 1. There is no polydisperse capillary tube bundle which uniquely determinesa liquid uptake time series. I.e., the equation V = νθ has no unique solution, or:det(νᵀν) = 0 (for ν given by Equation 2.13).Proof. Recall the definition of ν from Chapter 2:ν = pi(σr5i tj2µ) 12. (D.1)ν may be rewritten as the product of a scalar constant representing fluid properties(say, F ) with a matrix containing the geometry and time dependent variables ri andtj , respectively, as perν = Fr2.5i t0.5j = Fνm. (D.2)where νm is the modified liquid uptake matrix with fluid properties factored out.Noting that for an M ×M matrix A, det(FA) = FM det(A), for the completionof this proof it will be su cient to show that det(νᵀmνm) = 0 as we are guaranteedF > 0 for all real fluids.Now, using the definition of a matrix product we can write νᵀmνm in terms ofindices asνᵀmνm =N∑k=1r2.5k t0.5i︸ ︷︷ ︸νᵀmr2.5k t0.5j︸ ︷︷ ︸νm(D.3)= t0.5i t0.5jN∑k=1r5k (D.4)From Equation D.4 we can now see that the system of equations represented byνᵀmνm is not linearly independent: all terms in a given row i have a common factort0.5i and thus may be transformed into those of another row i′ by scalar multiplicationby a factor t0.5i′ /t0.5i .75APPENDIX D. NON-UNIQUENESS OF TUBE BUNDLESIt then follows that the matrix νᵀmνm thus has a rank of 1, that it has det(νᵀmνm) = 0,and that it is not invertible. Thus V = νθ has no unique solution for given V and νof the form given in Equation 2.13.76
Thesis/Dissertation
eng
Chemical and Biological Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Measurement of the changes to the liquid transport properties of wood due to compression
Text