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Multiple reaction solid state kinetic parameter determination and its application to woody biomass Mochulski, David 2014

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  Multiple reaction solid state kinetic parameter determination and its application to woody biomass  by  David Mochulski   A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE  in  The Faculty of Graduate and Postdoctoral Studies  (Chemical and Biological Engineering)   THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  November 2014  © David Mochulski, 2014      Abstract The economic problem of sustainable and environmentally responsible energy production has prompted research into a number of potential alternatives to fossil fuels.  Biomass gasification has been identified as one such alternative, but incomplete characterization of the process has hindered development.  This thesis addresses the problem of predicting reaction rate behavior in the case of woody biomass and aids in identifying optimal feedstock composition.  Black cottonwood (Populus trichocarpa) and Lodgepole pine (Pinus contorta) samples were characterized in terms of their primary component composition (lignin, cellulose, and hemicellulose) and then subjected to gasification experiments.  This consisted of pyrolysis, under a nitrogen atmosphere, and then gasification, under a dry air atmosphere, while undergoing a linear temperature program in a thermogravimetric analyzer (TGA).  Inspection of the experimental data indicated the presence of three simultaneous reactions.  The data was then analyzed to recover the isoconversional activation energy trend, pre-exponential factors, and reaction mechanisms.  Results indicated that the contributions of the three reactions did not correspond directly to lignin, cellulose, and hemicellulose contents, but, in the case of nitrogen pyrolysis, could be predicted by the knowledge of these components.  Regarding air gasification, no significant correlations between reaction rate behavior and primary wood component fraction were found.  Qualitatively, the work showed that the rate at which pyrolysis occurs is increased by high cellulose and hemicellulose contents, and decreased by large lignin contents.  A detailed kinetic model describing both Poplar and Lodgepole pine pyrolysis behavior was also recovered and is reported in the body of the thesis.   ii    Preface The identification of the research program was largely in response to the research project “Gasification of BC Lodgepole pine biomass integrated with CO2 capture” proposed by BIOFUELNET CANADA.  Research design and approach was a joint effort between Professor John Grace and David Mochulski.  I was responsible for selection of samples, design/performing experiments, furthering existing theory as noted in the body, analysis/interpretation of data, and drawing conclusions.  All Poplar/Lodgepole pine samples and compositional data were supplied by Shawn Mansfield’s group in the University of British Columbia Forestry department.  Access to experimental equipment was also supplied by Shawn Mansfield’s group.   iii    Table of Contents Abstract ................................................................................................................................................................. ii Preface ................................................................................................................................................................. iii Table of Contents.................................................................................................................................................. iv List of Tables ......................................................................................................................................................... vi List of Figures ....................................................................................................................................................... vii List of Notation ...................................................................................................................................................... x Acknowledgments ................................................................................................................................................ xi Chapter 1. Introduction ...................................................................................................................................... 1 1.1 Introduction ................................................................................................................................................. 1 1.2 Gasification reactions ................................................................................................................................... 1 1.3 Previous literature ........................................................................................................................................ 3 1.4 Literature critique ........................................................................................................................................ 5 1.5 Problem definition ....................................................................................................................................... 6 1.6 Research question ........................................................................................................................................ 7 1.7 Experimental primer .................................................................................................................................... 8 Chapter 2. Theory ............................................................................................................................................ 10 2.1 Gasification reactions ................................................................................................................................. 10 2.2 Biomass composition ................................................................................................................................. 11 2.2.1 Cellulose ................................................................................................................................................. 11 2.2.2 Lignin...................................................................................................................................................... 12 2.2.3 Hemicellulose ......................................................................................................................................... 13 2.2.4 Composition summary ........................................................................................................................... 14 2.3 Solid state kinetics expression – single reaction ........................................................................................ 14 2.3.1 Lack of uniqueness in simple curve fitting ............................................................................................. 19 2.3.2 Dimension reduction in kinetic parameter estimation ........................................................................... 21 2.3.3 Estimate of activation energy ................................................................................................................ 22 2.3.4 Constraining the Arrhenius factor – method of invariant kinetic parameters ....................................... 29 2.3.5 Mechanism determination ..................................................................................................................... 34 2.4 Solid state kinetics expression – multiple reactions .................................................................................. 35 2.4.1 Reaction fraction – an additional degree of freedom ............................................................................ 36 iv    2.4.2 Multiple reaction parameter specifics ................................................................................................... 38 Chapter 3. Experiments .................................................................................................................................... 45 3.1 Equipment .................................................................................................................................................. 45 3.1.1 Equipment (simple description) ............................................................................................................. 45 3.1.2 Equipment (detailed description) ........................................................................................................... 46 3.2 Experimental design ................................................................................................................................... 50 3.2.1 Number of experiments per sample ....................................................................................................... 50 3.2.2 Heating Rates ......................................................................................................................................... 50 3.2.3 Pyrolysis and gasification ....................................................................................................................... 51 3.2.4 Experimental description ....................................................................................................................... 51 3.3 Poplar (Populus trichocarpa) samples........................................................................................................ 54 3.3.1 Nitrogen pyrolysis .................................................................................................................................. 56 3.3.2 Air gasification ....................................................................................................................................... 59 3.4 Lodgepole pine (Pinus contorta) samples .................................................................................................. 62 3.4.1 Nitrogen pyrolysis .................................................................................................................................. 63 3.4.2 Air gasification ....................................................................................................................................... 66 Chapter 4. Analysis of experimental data and discussion of results ................................................................. 68 4.1 Poplar wood reactivity ............................................................................................................................... 68 4.1.1 Nitrogen pyrolysis – simple correlations ................................................................................................ 68 4.1.2 Nitrogen pyrolysis – kinetic characterization ......................................................................................... 69 4.1.3 Air gasification ....................................................................................................................................... 78 4.2 Recovery of Lodgepole pine wood kinetic parameters .............................................................................. 81 4.2.1 Nitrogen pyrolysis .................................................................................................................................. 81 4.2.2 Air gasification ....................................................................................................................................... 85 4.3 Correlation with composition .................................................................................................................... 88 Chapter 5. Overall conclusions and recommendations ..................................................................................... 92 5.1 Overall conclusions .................................................................................................................................... 92 5.2 Recommendations for future work ............................................................................................................ 93 References ........................................................................................................................................................... 95 Appendices .......................................................................................................................................................... 97 Appendix A: Numerical methods employed ........................................................................................................... 97   v    List of Tables Table 1. Common forms of the solid state reaction mechanism (Cai, et al., 2009); (Vyazovkin, et al., 2011). ........... 16 Table 2. Common forms of generalized empirical solid state kinetics models (Cai, et al., 2009); (Vyazovkin, et al., 2011). .................................................................................................................................................................. 17 Table 3.  Example parameters for a first order reaction ............................................................................................. 31 Table 4.  Parameters for a sample which decomposes according to three parallel, independent reactions ............. 39 Table 5.  Comparison of parameter recovery performance for a system of multiple reactions using typical and enhanced methods ............................................................................................................................................. 43 Table 6.  Manufacturer quoted specifications for TA Instruments Q500 series TGA .................................................. 47 Table 7.  Summary of Poplar sample compositions ..................................................................................................... 55 Table 8. Summary of Lodgepole pine sample compositions ....................................................................................... 62 Table 9.  Summary of morphological operator magnitudes and their interpretation for Poplar wood pyrolysis ....... 70 Table 10.  Summary of Poplar pyrolysis kinetic parameters ....................................................................................... 76 Table 11.  Summary of morphological operator magnitudes and their interpretation for Lodgepole pine wood pyrolysis .............................................................................................................................................................. 83 Table 12.  Tabular representation of time required to reach 90% conversion for Lodgepole pine gasification, heating rate: 20°C/min ....................................................................................................................................... 86 Table 13.  Relative impact of major wood components on reactivity at 90% conversion .......................................... 89 Table 14.  Summary of multiple linear regression results for fitted reaction fractions vs. measured wood components ........................................................................................................................................................ 90   vi    List of Figures Figure 1. Structure of cellulose (DoITPoMS, 2014) ...................................................................................................... 12 Figure 2. Structural representation of lignin (Taiz, et al., 2014) .................................................................................. 13 Figure 3.  Simplified representative structure of hemicellulose (Wikipedia, 2014) .................................................... 14 Figure 4. Illustration of orthogonal parameter fitting ................................................................................................. 20 Figure 5. Coupled Kinetic Parameters ......................................................................................................................... 21 Figure 6. Unrolled Kinetic Parameters ......................................................................................................................... 21 Figure 7. Summary of isoconversional techniques ...................................................................................................... 23 Figure 8.  Illustration of the isoconversional activation trend in the instance of a single reaction with an activation energy of -150,000 J/mol .................................................................................................................................... 24 Figure 9.  Conversion and reactivity curves of the system shown in Table 3 .............................................................. 32 Figure 10. Graphical illustration of aligning reactivity peaks for a random activation energy .................................... 33 Figure 11. Arrhenius factors required to match peaks for arbitrary activation energies ............................................ 33 Figure 12.  Determination of Arrhenius factor using the principle of kinetic invariance ............................................ 34 Figure 13.  Compound 1 .............................................................................................................................................. 38 Figure 14. Compound 2 ............................................................................................................................................... 38 Figure 15.  Observed and component conversions of the reaction system defined in Table 4 .................................. 40 Figure 16.  Derivative curves of the composite and component conversions of the sample reaction system defined in Table 4 ............................................................................................................................................................. 41 Figure 17.  Vyazovkin method isoconversional activation energy trend for the reaction system defined in Table 4 (assuming heating rates of 15°C/min, 25°C/min, and 35°C/min) ....................................................................... 42 Figure 18.  Comparison of parameter recovery performance using derivative curves for a system of multiple reactions using typical and enhanced methods ................................................................................................. 44 Figure 19. Simplified Illustration of a thermogravimetric analyzer (TGA) ................................................................... 46 Figure 20. TA Instruments Q500 TGA .......................................................................................................................... 48 Figure 21. Annotated TGA illustration ......................................................................................................................... 48 Figure 22.  Instrument measurement error (mg) at a gas flow rate of 90 mL/min cross flow, 40 mL/min downward flow ..................................................................................................................................................................... 49 Figure 23.  Illustration of transport phenomena limitation on results at low gas flow rate (Poplar, temperature ramping at 20°C/min beginning from 50°C at time 0) ........................................................................................ 50 Figure 24.  Flowchart illustrating the experimental procedure for n samples ............................................................ 53 Figure 25.  Poplar population lignin variation and sample selection ........................................................................... 55 Figure 26.  Graphical summary of Poplar sample compositions ................................................................................. 56 Figure 27.  Poplar pyrolysis conversion curves for heating rates of 15, 20, and 25°C/min ......................................... 57 vii    Figure 28.  Poplar pyrolysis reactivity curves for heating rates of 15, 20, and 25°C/min ............................................ 58 Figure 29.  Illustration of raw data yielded by a single experiment (Lodgepole pine, 15°C/min) ............................... 59 Figure 30.  Poplar air gasification conversion curves for heating rate of 15, 20, and 25°C/min ................................. 60 Figure 31.  Poplar gasification reactivity curves for heating rate of 15, 20, and 25°C/min ......................................... 61 Figure 32.  Graphical summary of Lodgepole pine sample compositions ................................................................... 63 Figure 33. Lodgepole pine pyrolysis conversion curves for heating rates of 15, 20, and 25°C/min ............................ 64 Figure 34. Lodgepole pine pyrolysis reactivity curves for heating rates of 15, 20, and 25°C/min .............................. 65 Figure 35. Lodgepole pine gasifications conversion curves for heating rates of 15, 20, and 25°C/min ...................... 66 Figure 36. Lodgepole pine gasification reactivity curves for heating rates of 15, 20, and 25°C/min .......................... 67 Figure 37. Relative times to reach 90% conversion for Poplar pyrolysis. (“Cooler” colors indicate less time, therefore greater reactivity) – Heating rate: 15°C/min ...................................................................................... 68 Figure 38. Relative times to reach 90% conversion for Poplar pyrolysis. (“Cooler” colors indicate less time, therefore greater reactivity) – Heating rate:  20°C/min ..................................................................................... 68 Figure 39. Relative times to reach 90% conversion for Poplar pyrolysis. (“Cooler” colors indicate less time, therefore greater reactivity) – Heating rate:  25°C/min ..................................................................................... 69 Figure 40.  Time and reactivity morphological operators applied to Poplar wood pyrolysis reactivity data (15°C/min- red, 20°C/min – brown, and 25°C/min - black) ................................................................................................... 71 Figure 41.  Isoconversional activation energy trends for Poplar wood pyrolysis ........................................................ 73 Figure 42.  Minimization profile for the Vyazovkin method using a 3 experiment input for sample 500 HAL30-2/TO-10-1 ..................................................................................................................................................................... 74 Figure 43.  Graphical illustration of exhaustive tests of potential reaction mechanism combinations. Hotter colors indicate better agreement with experimental results (chosen models lie within the cube’s interior). ............. 75 Figure 44.  Conversion vs. Time curves for sample 500 HALS30-2/TO-10-1 ................................................................ 76 Figure 45.  Activation energy vs. conversion curves for sample 500 HALS30-2/TO-10-1 ............................................ 76 Figure 46.  Reactivity vs. time curves for sample 500 HALS30-2/TO-10-1 ................................................................... 77 Figure 47 Conversion vs. Time curves for sample 500 HALS30-2/TO-10-1 – incorrect activation energy trend ......... 77 Figure 48. Activation energy vs. conversion curves for sample 500 HALS30-2/TO-10-1 - incorrect ........................... 77 Figure 49. Reactivity vs. time curves for sample 500 HALS30-2/TO-10-1 – incorrect activation energy trend ........... 78 Figure 50. Relative times to reach 90% conversion for Poplar air gasification. (“Cooler” colors indicate less time and therefore are more reactive) Heating Rate: 15°C/min ....................................................................................... 79 Figure 51. Relative times to reach 90% conversion for Poplar air gasification. (“Cooler” colors indicate less time and therefore are more reactive) Heating Rate: 20°C/min ....................................................................................... 79 Figure 52. Relative times to reach 90% conversion for Poplar air gasification. (“Cooler” colors indicate less time and therefore are more reactive) Heating Rate:  25°C/min ...................................................................................... 79 viii    Figure 53. Time and reactivity morphological operators applied to Poplar wood air gasification reactivity data (15°C/min- red, 20°C/min – brown, and 25°C/min - black) ................................................................................ 80 Figure 54. Relative times to reach 90% conversion for Lodgepole pine pyrolysis.  (“Cooler” colors indicate higher reactivity) Heating Rate: 15°C/min ..................................................................................................................... 81 Figure 55. Relative times to reach 90% conversion for Lodgepole pine pyrolysis.  (“Cooler” colors indicate higher reactivity) Heating Rate: 20°C/min ..................................................................................................................... 81 Figure 56. Relative times to reach 90% conversion for Lodgepole pine pyrolysis.  (“Cooler” colors indicate higher reactivity) Heating Rate: 25°C/min ..................................................................................................................... 82 Figure 57. Time and reactivity morphological operators applied to Lodgepole wood pyrolysis reactivity data (15°C/min - red, 20°C/min - brown, and 25°C/min - black) ................................................................................ 84 Figure 58. Relative times to reach 90% conversion for Lodgepole pine air gasification. (“Cooler” colors indicate higher reactivity) Heating Rate: 15°C/min .......................................................................................................... 85 Figure 59. Relative times to reach 90% conversion for Lodgepole pine air gasification. (“Cooler” colors indicate higher reactivity) Heating Rate:20°C/min ........................................................................................................... 85 Figure 60. Relative times to reach 90% conversion for Lodgepole pine air gasification. (“Cooler” colors indicate higher reactivity) Heating Rate: 25°C/min .......................................................................................................... 86 Figure 61. Time and reactivity morphological operators applied to Lodgepole wood air gasification reactivity data (15°C/min- red, 20°C/min – brown, and 25°C/min - black) ................................................................................ 87 Figure 62. Ternary diagram showing all Lodgepole pine and Poplar pyrolysis results for experiments run at 15°C/min; colors indicate times to reach 90% conversion ................................................................................. 88 Figure 63 Ternary diagram showing all Lodgepole pine and Poplar pyrolysis results for experiments run at 20°C/min; colors indicate times to reach 90% conversion ................................................................................. 88 Figure 64 Ternary diagram showing all Lodgepole pine and Poplar pyrolysis results for experiments run at 25°C/min; colors indicate times to reach 90% conversion ................................................................................. 88 Figure 65. Time required to reach 90% conversion as a function of cellulose content ............................................... 89 Figure 66. Time required to reach 90% conversion as a function of hemicellulose content ...................................... 89 Figure 67. Time required to reach 90% conversion as a function of lignin content .................................................... 89   ix    List of Notation Symbol Definition Units (Dimensional) 𝛼𝛼 Solid state conversion None 𝐶𝐶𝐴𝐴 Concentration of A [𝑁𝑁][𝑉𝑉−1] 𝐶𝐶𝐴𝐴0 Initial concentration of A [𝑁𝑁][𝑉𝑉−1] 𝐸𝐸𝐴𝐴 Activation energy [𝐸𝐸][𝑁𝑁−1] 𝑑𝑑𝛼𝛼𝑑𝑑𝑑𝑑 Reactivity [𝑇𝑇]−1 𝑚𝑚𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆@𝑡𝑡 Mass of sample at time t [M] 𝑚𝑚𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑡𝑡𝑆𝑆𝑆𝑆𝑆𝑆 Initial sample mass [M] 𝑚𝑚𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 Final sample mass [M] 𝑚𝑚𝑆𝑆𝑓𝑓 Final sample mass of component j [M] 𝑚𝑚𝑠𝑠𝑓𝑓 Starting sample mass of component j [M] 𝑚𝑚𝑡𝑡𝑓𝑓 Mass of component j at time t [M] 𝐴𝐴 Pre-exponential factor (Arrhenius constant) Variable 𝑅𝑅 Gas constant [𝐸𝐸][𝑁𝑁−1][𝛩𝛩−1] 𝑇𝑇 Temperature [𝛩𝛩] 𝑋𝑋 Gas/liquid phase Conversion None 𝑓𝑓(𝛼𝛼) Reaction mechanism term (derivative form) None 𝑔𝑔(𝛼𝛼) Reaction Mechanism (integral form) None 𝑖𝑖 Experiment i None 𝑘𝑘 Rate constant Variable 𝑑𝑑 Time [T] 𝛽𝛽 Rate of temperature increase [𝛩𝛩][𝑇𝑇−1]   x    Acknowledgments The author would like to thank: John Grace for his continued support and encouragement. Shawn Mansfield for his gracious contribution of biomass samples and equipment time. James Campbell for being an excellent sounding board to bounce ideas off of. BIOFUELNET CANADA for the funding of this project  xi    Chapter 1. Introduction  1.1 Introduction As society’s demand for energy continues to grow, the questions of sustainability and environmental impact become more pressing.  To address these issues, numerous alternative energy production schemes have been proposed.  Many of the proposed strategies are currently areas of active research.  Amongst these, we find wood gasification; an old technology, but one with several attractive attributes.  Despite the maturity of this field, there are still large questions regarding differences in feedstock performance and the impact of process parameters.  This incomplete understanding of the process has made modeling/simulation studies inaccurate and, consequently, has made reactor design and ultimate potential determination difficult.  To address some of these knowledge gaps, this thesis attempts to provide a pathway to understand the details of wood biomass gasification kinetics.  However, before describing the project’s specific findings, we first briefly introduce the process of gasification, summarize the current literature, and formally state the project’s goals.  1.2 Gasification reactions Gasification reactions can be summarized as the use of thermochemical means to break down complex molecules into simpler building blocks.  This is accomplished by subjecting the feedstock to high temperature.  An oxidizing agent, such as oxygen, carbon dioxide, or steam, is often present to assist in bond breakage and promote certain product distributions.  If no oxidizing agent is present, gasification reduces to thermal cracking.  1    To facilitate the gasification reactions, a number of reactor designs have been attempted.  These have differed primarily in flow direction (updraft vs. downdraft) and bed dynamics – conveyed, fluidized, and stationary bed.  Reactors are typically isothermal (700-900 Celsius), and most are operated at atmospheric pressure.  The energy required to maintain isothermal conditions may be supplied externally, or through partial combustion of the feedstock within the reactor.  The reactions are multiphase (solid – gas) and may be summarized as shown by  𝐵𝐵𝑖𝑖𝐵𝐵𝑚𝑚𝐵𝐵𝐵𝐵𝐵𝐵 (𝑐𝑐𝐵𝐵𝑚𝑚𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑚𝑚𝑖𝑖𝑐𝑐𝑑𝑑𝑚𝑚𝑚𝑚𝑐𝑐 𝐵𝐵𝑓𝑓 𝐶𝐶,𝑁𝑁,𝑂𝑂,𝐻𝐻, 𝑐𝑐𝑑𝑑𝑐𝑐)𝑠𝑠𝑠𝑠𝑆𝑆𝑆𝑆𝑠𝑠 + 𝑂𝑂𝑐𝑐𝑖𝑖𝑑𝑑𝑖𝑖𝑂𝑂𝑐𝑐𝑚𝑚𝑔𝑔𝑆𝑆𝑠𝑠𝐻𝐻𝑆𝑆𝑆𝑆𝑡𝑡 𝐴𝐴𝑠𝑠𝑠𝑠𝑆𝑆𝑠𝑠�⎯⎯⎯⎯⎯⎯⎯�𝑆𝑆𝑖𝑖𝑚𝑚𝑐𝑐𝑐𝑐𝑐𝑐 𝑃𝑃𝑚𝑚𝐵𝐵𝑑𝑑𝑚𝑚𝑐𝑐𝑑𝑑 𝑆𝑆𝑐𝑐𝑐𝑐𝑐𝑐𝑖𝑖𝑐𝑐𝐵𝐵 �𝐶𝐶𝑂𝑂𝑔𝑔𝑆𝑆𝑠𝑠 + 𝐶𝐶𝑂𝑂2𝑔𝑔𝑆𝑆𝑠𝑠 + 𝐻𝐻2𝑂𝑂𝑔𝑔𝑆𝑆𝑠𝑠 +𝐻𝐻2𝑔𝑔𝑆𝑆𝑠𝑠 + 𝐶𝐶𝐻𝐻4𝑔𝑔𝑆𝑆𝑠𝑠 +𝑚𝑚𝑖𝑖𝑚𝑚𝐵𝐵𝑚𝑚 𝑐𝑐𝐵𝐵𝑚𝑚𝐵𝐵𝑑𝑑𝑖𝑖𝑑𝑑𝑚𝑚𝑐𝑐𝑚𝑚𝑑𝑑𝐵𝐵�  1   Note that Equation 1 is qualitative; neither relative amounts of product species nor an exact formula for the biomass composition is given.  This is intentional, based on the reasoning that biomass composition may vary widely and the product species distribution is dependent on both the biomass composition and reactor conditions.  The minor constituent faction is used to capture products which are present in negligible concentrations (e.g. hydrogen cyanide) or are highly variable /dependent on soil conditions (i.e. nitrogen or sulfur dioxide).  With regards to kinetics, it should be noted that the presence of alkali and alkaline earth metals are capable of catalyzing certain gasification reactions.  The extent to which these metals are present in biomass is primarily related to the specific growing environment and soil composition.  This adds further variability to the system and increases the difficulty of kinetic parameter estimation, also limiting extension of kinetic parameters to woods grown in soils of different mineral abundance.  2    1.3 Previous literature Perhaps the largest benefit of the maturity of gasification technology is that a staggering amount of literature has been written on the subject.  However, in the interest of brevity, we limit ourselves here to the discussion of key papers directly relevant to the work undertaken.  These papers are listed in a fashion which first confirms the study as a promising avenue of discovery, focuses on specific results found by other researchers, and then examines experimental procedure and analysis methods.  Together, the papers describe the state of the art of solid state kinetics, biomass gasification kinetics, and kinetic parameter determination through thermogravimetric experimentation.  Raveendran, et al.(1996) considered a number of biomass sources and attempted to model biomass pyrolysis behavior based on thermogravimetric experiments.  Biomass samples were first described in terms of their three primary components (cellulose, hemicellulose, and lignin) and then synthetic analogues were created from pure samples of the individual components.  The results showed that it is possible to model biomass pyrolysis as a mixture of pure components.  However, these results have been found to be inapplicable across strongly varying biomass types.  Even so, the significance of the work is that it supports the conclusion that the major components of biomass do not interact significantly during the pyrolysis reactions.  Burhenne, et al.(2013) again attempted to correlate pyrolysis behavior with composition.  The difference between this paper and the work of Raveendran, et al. (1996) is that the authors use correlations from ultimate analysis to obtain the lignin, cellulose, and hemicellulose fractions.  The paper is useful in its reporting of success using ultimate analysis to estimate primary component fractional composition.  However, as we will see in the next section, this approach is not without its limitations.  3    Slopiecka, et al. (2012) analyzed the thermogravimetric response of Poplar wood specifically, reporting estimates of pyrolysis activation energy using several different isoconversional methods.  Techniques employed included the Kissinger, Flynn-Wall-Ozawa, and Kissinger-Akahira-Sunose methods.  Unfortunately, no composition data were given for the tested samples.  The paper does, however, provide excellent results for qualitative comparison of isoconversional activation energy responses for Poplar wood.  Vyazovkin, et al. (2011) describes best practices for thermogravimetric data analysis and aids in defining an experimental plan which maximize the opportunity for useful results.  This document, put together by a number of senior scientists with significant experience in the area, outlines the benefits of non-isothermal experiments and concludes with suggestions for model determination through dimension reduction techniques.  Turning to mathematical techniques, the Friedman method (Friedman, 1964) was originally applied to the thermal degradation of plastics.  Since then, the model has become one of the most widely used techniques in the analysis of non-isothermal reaction data when an estimation of kinetics is required.  It has been included in a number of kinetics software packages.  The Flynn-Wall-Ozawa (OFW) method (Wall, et al., 1966) built on the ideas of the Friedman method and introduced the first widely applied integral based method for isoconversional activation energy determination.  The Kissinger-Akahira-Sunose (KAS) method (Akahira, et al., 1971), improved upon the OFW method by expanding its range of applicability.  Both the OFW and KAS methods are commonly used in practice today.  The most recent isoconversional technique added to the literature, and perhaps the most mathematically interesting, is the contribution of Vyazovkin (2000).  This paper lays out the theoretical groundwork for an advanced isoconversional method whose relative errors are significantly lower than in the older techniques used by Slopiecka, et al. (2012).  This method 4    facilitates more accurate, unbiased, estimates of activation energies at varying levels of conversion.  Detailed examinations of the Friedman, Kissinger-Akahira-Sunose, Flynn-Wall-Ozawa, and Vyazovkin methods are provided in Chapter 2.  1.4 Literature critique In general, previously published work has largely followed the thinking that, because biomass is primarily composed of three components (cellulose, hemicellulose, and lignin) and, because these three components can react independently, there should be three reactions which may be combined via superposition.  However, it may be the case that while the individual components do not interact, the decomposition reactions are more closely related to local bond strengths.  Perhaps there are three reactions, but not necessarily a single reaction per component. The foundations for this idea were originally put forward by Ozawa (1970) in the context of thermal polymer decomposition.  This would better explain observations by Beall (1971) who showed that the same major components isolated from wood behave differently.  Also, Roberts (1971) showed that the structural properties of the components influence pyrolysis characteristics.    A second deficiency of the bulk of existing gasification kinetics literature concerns the methodology by which the kinetic parameters are obtained.  Many published papers have taken the approach of simple curve fitting, to a single experiment, to obtain kinetic parameters.  Results obtained in this manner are generally not unique, and estimates of the parameters can vary widely due to a lack of constraints on parameters.  Both of these criticisms are addressed in this study.  5    1.5 Problem definition In the introduction, it was discussed that a continuing challenge in both gasifier simulation and design has been the uncertainty in how feedstock composition affects process performance; this problem is especially troubling when one considers the variety of biomass sources available.  Furthermore, this question is key in determining whether it is possible to breed specific tree varieties which could improve, or optimize, gasification performance.  Ideally, what is needed is a simple relationship which fully describes the impact of feedstock composition on the gasification process.  This would include rate of reaction, product composition distribution, internal mass transport considerations (affected by ash content), and energy balance implications.  Moreover, this relationship should require only data which are inexpensive and easy to obtain; the analysis should also be practical to complete in a reasonable period of time.  Unfortunately, several of these objectives are in conflict with each other.  To strike a balance between the various intents, we must consider what answers will provide the most utility, both currently, and in the future as a jumping-off-point.  Drawing on the subject matter presented in typical reactor design theory, the reactivity of a feedstock is solely a function of the kinetics governing the reactions it undergoes.  Admittedly, this is a blatant simplification; beyond kinetics, one must also consider heat/mass transfer properties as well as velocity profiles and mixing patterns within a reactor.  Nevertheless, the equations necessary for describing the relevant transport phenomena are well known and generally rely on knowledge of reaction rate and product distributions.  Thus, once the kinetic relationships have been determined, designing a reactor which maximizes conversion is theoretically possible from first principles.  In light of this, this project focuses on the determination of the kinetics as a predictor of feedstock reactivity.  As such, we define the problem as follows: 6    How can wood biomass be characterized to predictively estimate if/how reaction rates change as a function of wood composition; where wood composition is described in terms of the lignin, cellulose, and hemicellulose mass fractions?  Furthermore, can these data be utilized to make predictions regarding optimal feedstock composition?    1.6 Research question Having identified the specific problem, the next step is to state the research question which will attend to its specifics.    Referring back to the literature section, much of the historical difficulty in kinetic parameter estimation has come from the sympathetic effect which exists between the activation energy and the pre-exponential factor.  This has led a number of researchers to propose parameters which have varied widely, but have similar goodness-of-fit metrics.  Until this non-uniqueness in parameters is addressed, some level of doubt will remain as to the validity of the fundamental underlying relationships.  To overcome this problem, an unbiased estimator of either the activation energy or the pre-exponential factor is required.  Once this is known, other techniques, such as the kinetic compensation effect, can be used to determine the remaining parameter.  Several frameworks exist to accomplish just this.  The most advanced of these being the advanced isoconversional technique presented by Vyazovkin (2000).  We may then state our research question as: By applying the isoconversional framework laid out by Vyazovkin (2000), can we obtain consistent, unbiased estimates of the activation energy parameter for pyrolysis and gasification reactions through the analysis of non-isothermal thermogravimetric data for black cottonwood and Lodgepole pine samples?  Furthermore, can these data be coupled with the methods laid out by the kinetic compensation effect technique to better calculate unique estimates of the pre-7    exponential factor, aiding in the discovery of unique descriptions of the underlying kinetics?  Finally, from our estimations of the kinetics parameters, are we better able to model the gasification process numerically, and can we identify and quantify optimal feedstock compositions which would aid in both gasifier design and feedstock selection?   Several postulates arise naturally from these research questions; each will be addressed in the pages that follow.  They may be summarized as follows: 1. Since the Vyazovkin isoconversional method is model-free, there is less danger of obtaining parameter estimates of differing magnitudes, but of similar fit metrics.  2. Switching to an isoconversional technique will allow us to identify changes in mechanism at various stages of the reaction.  3. The improvement in kinetic parameter estimation allows for better, more consistent, numerical model development and the determination of whether the number of reactions is better correlated with the number of components or the molecular environment.  4. With consistent estimates of kinetic parameters, we will not only be able to better associate experimental observations with biomass composition, but also to reconcile results reported by other researchers with more confidence.  1.7 Experimental primer Though the experiments necessary to resolve the research question posed above will be discussed in detail in Chapter 3, it is beneficial to first describe some basic measurements.   From a review of the papers put forth by Friedman (1964), Wall, et al. (1966), Akahira, et al. (1971), and Vyazovkin (2000), it is clear that the most accurate predictions of kinetics come from non-isothermal experiments.  This suggests that experiments be performed through the 8    use of either a thermogravimetric analyzer (TGA) or a differential scanning calorimeter (DSC).  Though either instrument is suitable, a TGA was chosen for this study based on instrument availability.  The operation, accuracy, and interpretation of TGA data are beyond the scope of this section but, to fully appreciate the theory in Chapter 2, it is necessary to note that a TGA experiment records the mass of a solid reactant mixture while it undergoes conversion to products through chemical reaction as the temperature changes due to internal (enthalpy of reaction) and/or external (applied heat) effects.  9    Chapter 2. Theory Continuing with the ideas presented in Chapter 1, we can characterize the reactivity of biomass by recovering the kinetic parameters of the gasification reactions.  Accomplishment of this, however, is not a trivial matter.    2.1 Gasification reactions A review of the literature indicates that the predominant reactions taking place during gasification are relatively few, primarily consisting of reactions 2 to 10 shown below: 𝐶𝐶 + 𝑂𝑂2 ↔ 𝐶𝐶𝑂𝑂2  2 2𝐶𝐶 + 𝑂𝑂2 ↔ 2𝐶𝐶𝑂𝑂  3 2𝐶𝐶𝑂𝑂 + 𝑂𝑂2 ↔ 2𝐶𝐶𝑂𝑂2  4 𝐶𝐶𝑂𝑂 + 𝐻𝐻2𝑂𝑂 ↔ 𝐶𝐶𝑂𝑂2 + 𝐻𝐻2  5 2𝐶𝐶𝑂𝑂 + 2𝐻𝐻2 ↔ 𝐶𝐶𝐻𝐻4 + 𝐶𝐶𝑂𝑂2  6 𝐶𝐶𝐻𝐻4 + 2𝑂𝑂2 ↔ 𝐶𝐶𝑂𝑂2 + 2𝐻𝐻2𝑂𝑂  7 𝐶𝐶 + 𝐻𝐻2𝑂𝑂 ↔ 𝐶𝐶𝑂𝑂 + 𝐻𝐻2  8 𝐶𝐶 + 2𝐻𝐻2𝑂𝑂 ↔ 𝐶𝐶𝑂𝑂2 + 𝐻𝐻2  9 𝐶𝐶 + 𝐶𝐶𝑂𝑂2 ↔ 2𝐶𝐶𝑂𝑂  10 Inspecting these reactions, we notice that the species involved are comprised of carbon, carbon monoxide, carbon dioxide, methane, hydrogen, oxygen, and water.  If biomass was comprised of only carbon, these reactions would capture, to a large degree, the system’s behavior.  Unfortunately, this is not the case.  Despite this, and considering that biomass composition is much more complex, knowledge of these reactions can be useful both conceptually and practically.  We return to discuss the exact nature of their application later in this section.  10    2.2 Biomass composition Prior to further discussion of the reactions, kinetics, and modeling, it is first necessary to consider the composition of biomass.  This is dangerous territory as the term “biomass” may be used to describe a wide variety of substances which vary significantly in their compositional properties.  In light of this, we limit ourselves to woody biomass. Having narrowed the scope of investigation, we note that wood is comprised mainly of three components, cellulose, hemicellulose, and lignin.    2.2.1 Cellulose Cellulose has been estimated as being the most abundant biogenic polymer on the planet.  Fan, Gharpuray, et al. (1987) estimated resource potential at 3.24x1011 m3 globally.  This is largely due to it being the main component of woody biomass.  Of the three primary components of wood, it is also the simplest in both structure and chemistry.  Chemically, cellulose is classified as an unbranched polymer with each mer unit comprised of a glucose molecule.  The length of the glucose chains varies with biomass, but can range from 1000 for newsprint to 10,000 for cotton (Rees 1967).  This variability in chain length limits the extension of kinetic analysis between species.  Even though bond energies remain relatively consistent, the entropic effects related to chain length become increasingly significant when comparing molecules of significantly different lengths.  An illustration of three, 4-unit cellulose molecules is shown below in Figure 1.  In particular, note that both intermolecular and intramolecular hydrogen bonds strengthen the molecule appreciably. 11     Figure 1. Structure of cellulose (DoITPoMS, 2014)  2.2.2 Lignin The second most abundant component of wood is lignin, a highly complex, non-repeating polymer.  To offer a full description of the chemistry of lignin is beyond the scope of this document, but it can briefly be described as a cross-linked racemic macromolecule with an inexact degree of polymerization.  Matters are further complicated by the fact that the molecule becomes fragmented during extraction, and different types of lignin have been described depending on the means of isolation.  However, it can be stated generally that the structure of lignin is unpredictable across biomass.  One representative structure, meant to convey its complexity, is shown in Figure 2. 12     Figure 2. Structural representation of lignin (Taiz, et al., 2014)  2.2.3 Hemicellulose The final primary component of wood is hemicellulose.  Chemically, it is again a non-repeating polymer, but this time made up of simple sugars.  These include glucose, xylose, mannose, galactose, rhamnose, and arabinose.  The acidified forms of these sugars may also be present (i.e. glucuronic acid, galacturonic acid, etc.).  A simplified structure consisting of xylose, mannose, glucose, and galactose is shown in Figure 3.  13     Figure 3.  Simplified representative structure of hemicellulose (Wikipedia, 2014)  2.2.4 Composition summary Having presented the structures of the three primary components of wood, reflection upon their structures makes it obvious why traditional kinetic approaches (mechanism proposal and elimination) are unsuitable for gasification reaction characterization.  There are simply too many possible intermediate structures which may form during decomposition, and each may have the potential to interact with currently unreacted molecules or with each other.  Thus, an empirical approach is necessary.  However, to successfully generate an empirical representation of the kinetics, a detailed understanding of solid state kinetics is first required.  2.3 Solid state kinetics expression – single reaction Many of the concepts involved in solid state reaction kinetics are comparable to those of liquid and gas phase reactions.  However, despite the similarity, it is important to first define terms and expressions.  Before going on, it should be noted that the context in which the expressions are presented is that of a pure solid either decomposing to a gas or undergoing reaction with a gas.  The reason for this is that it is typical during the experimentation to measure mass loss or gain of a sample.  As for the pure component assumption, it is made at this stage for simplification purposes only and will be removed later in the chapter. 14    Similar to liquid and gas phase reactions, solid state reactions can be described in terms of conversion; this is customarily represented by the symbol 𝛼𝛼.  Conversion, in the solid kinetics literature, may be quoted in two fashions – as an ash free conversion, or as a standard conversion.  The difference between the two is best illustrated through examination of their formal definitions, shown below in Equations 11 and 12.  𝛼𝛼𝑆𝑆𝑡𝑡𝑆𝑆𝑆𝑆𝑠𝑠𝑆𝑆𝑆𝑆𝑠𝑠 =𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖−𝑆𝑆@𝑖𝑖𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖  Equation 11. Standard conversion 𝛼𝛼𝐴𝐴𝑠𝑠ℎ 𝐹𝐹𝑆𝑆𝑆𝑆𝑆𝑆 =𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖−𝑆𝑆@𝑖𝑖𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖−𝑆𝑆𝑓𝑓𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖   Equation 12. Ash free conversion  Inspection of the above two conversion definitions indicates that the difference lies in the treatment of the final mass.  The start point is zero percent for both definitions, but, under the standard conversion formula, the endpoint need not be one hundred percent if a portion of the sample is unreactive.  The ash-free conversion formula accounts for unreactive material and allows conversion to span the complete range between zero and one hundred percent.  In the event that the sample contains no unreactive material, both methods yield identical values.  This difference in the treatment of unreactive material is important when contrasting samples of differing inert content.  More importantly, use of the ash-free formula is essential when considering multicomponent solid mixtures.  The reason for this will be explained in the section which generalizes solid kinetics to multiple reactions.  For now, note that all references to conversion throughout the remainder of this thesis are to ash-free values.    Having defined a solid state conversion, we may now define the kinetics as the time rate of change of conversion.  This is written as: 𝑠𝑠𝑑𝑑𝑠𝑠𝑡𝑡= 𝐴𝐴𝑐𝑐−𝐸𝐸𝐴𝐴𝑅𝑅𝑅𝑅 𝑓𝑓(𝛼𝛼)  13 Readers familiar with kinetics will notice that we have substituted the Arrhenius expression for the rate constant, but we have also introduced another term: this is the mechanism term, 𝑓𝑓(𝛼𝛼).    15    The mechanism term is simply a placeholder for the typical mechanisms by which reactions may occur.  For example, in the fields of liquid and gas phase kinetics, where it is conventional to use X to represent conversion, this term may be replaced by such an expression as that of a first order reaction.    𝑑𝑑𝐶𝐶𝐴𝐴𝑑𝑑𝑑𝑑= 𝑘𝑘𝐶𝐶𝐴𝐴 𝐶𝐶𝐴𝐴 = 𝐶𝐶𝐴𝐴0(1− 𝑋𝑋) −𝑑𝑑𝑋𝑋𝑑𝑑𝑑𝑑= 𝑘𝑘(1− 𝑋𝑋) Equation 14. First order constant volume reaction expressed through concentration Equation 15. Concentration/ conversion relationship for constant volume Equation 16. First order constant volume reaction expressed through conversion  Likewise, for a first order solid state reaction: 𝑓𝑓(𝛼𝛼) = (1− 𝛼𝛼).  Similarly, 20 typical reaction mechanism equations have been derived under various assumptions, as summarized in Table 1.  Table 1. Common forms of the solid state reaction mechanism (Cai, et al., 2009); (Vyazovkin, et al., 2011).  Reaction Name Reaction Form (derivative form) 1. Power Law 4𝛼𝛼34�  2. Power Law 3𝛼𝛼23�  3. Power Law 2𝛼𝛼12�  4. Power Law 23𝛼𝛼−12�  5. 1 dimensional (1D) diffusion 12𝛼𝛼−1 6. Mampel (first order) 1− 𝛼𝛼 7. Avrami-Erofeev (nucleation and growth) 4(1− 𝛼𝛼)�−𝑐𝑐𝐵𝐵𝑔𝑔(1− 𝛼𝛼)�34�  8. Avrami-Erofeev (nucleation and growth) 3(1− 𝛼𝛼)�−𝑐𝑐𝐵𝐵𝑔𝑔(1− 𝛼𝛼)�23�  9. Avrami-Erofeev (nucleation and growth) 2(1− 𝛼𝛼)�−𝑐𝑐𝐵𝐵𝑔𝑔(1− 𝛼𝛼)�12�  10. 3 dimensional (3D) diffusion 32(1− 𝛼𝛼)23� �1 − (1− 𝛼𝛼)13� �−1 11. Contracting sphere 3(1− 𝛼𝛼)23�  16     Reaction Name Reaction Form (derivative form) 12. Contracting cylinder 2(1− 𝛼𝛼)12�  13. 2 dimensional (2D) diffusion �−𝑐𝑐𝐵𝐵𝑔𝑔(1− 𝛼𝛼)�−1 14. Avrami-Erofeev (nucleation and growth) 32(1 − 𝛼𝛼)(−log (1− 𝛼𝛼))13�  15. Avrami-Erofeev (nucleation and growth) 52(1 − 𝛼𝛼)(−log (1− 𝛼𝛼))35�  16. 3D diffusion (Ginstling-Brounshtein) 3(1− 𝛼𝛼)13�2 �1 − (1− 𝛼𝛼)13� � 17. 3D diffusion (Zhuralev-Lesokin-Tempelman) 3(1− 𝛼𝛼)53�2 �1 − (1− 𝛼𝛼)13� � 18. 3D diffusion (Komatsu-Uemura) 3(1 + 𝛼𝛼)23�2 �(1 + 𝛼𝛼)13� − 1� 19. Second Order Reaction (1− 𝛼𝛼)2 20. Third Order Reaction (1− 𝛼𝛼)3  Beyond the mechanisms which have been formally derived through first principles, several additional forms have been proposed which are meant to be adaptive.  The most popular of these forms are shown below in Table 2.  Table 2. Common forms of generalized empirical solid state kinetics models (Cai, et al., 2009); (Vyazovkin, et al., 2011).  Reaction Name Reaction Form  (derivative form) Number of Adjustable Parameters 1. Prout-Thomkins 𝛼𝛼(1− 𝛼𝛼) 0 2. Sestak-Berggren (1− 𝛼𝛼)𝑆𝑆𝛼𝛼𝑆𝑆(−log (1− 𝛼𝛼))𝑆𝑆 3 (n, m, p) 3. Perez-Maqueda (simplified Sestak-Berggren) 𝛼𝛼𝑆𝑆(1− 𝛼𝛼)𝑆𝑆 2 (m, n)  These generalized empirical models have seen some degree of adoption, primarily because the adjustable parameters associated with the mechanism can simply be lumped in with any other parameters to be estimated.  This approach, though attractive, greatly increases the potential 17    for over-fitting and should be avoided.  As has been said of the problem of non-uniqueness, “Give me four adjustable parameters and I can fit an elephant; give me five and I can include his tail!”.  This, of course, is in addition to the greater computational cost of increasing the potential solution space by supplementary dimensions.  Having defined 𝑓𝑓(𝛼𝛼), we now return to our consideration of the rate expression and its role in characterizing wood biomass.  As stated earlier, we wish to utilize the differences in kinetics to describe the suitability of wood types for gasification.  By necessity, the data supporting this effort must be derived experimentally and, as will be seen in the next chapter, consists of compositional data and a time series of sample mass.  Focusing now on the time series, through Equation 12, we may convert sample mass to conversion, and are free to numerically differentiate this as needed.  Thus, with regards to the rate equation, values of conversion, the time derivative of conversion, and time are known.  This leaves us with three parameters to determine: the Arrhenius factor, the activation energy, and the reaction mechanism.  This combination of unknowns is common in the field of kinetics, and is often referred to as the kinetic triplet; the relationship between these variable is illustrated in Equation 17.  Equation 17.  Kinetic triplet  At this point, the mathematically inclined may be quick to observe that the problem may be posed as a non-linear curve fitting exercise.  Indeed it can be, and a number of papers exist which have done just that.  In fact, if one is only interested in a single reaction, the curve fitting can be done directly on the differential form of the relationship (rather than solving the differential equation for 𝛼𝛼).  However, these approaches have largely been unsuccessful and have led to the reporting of parameters which vary by one or more orders of magnitude.  18    2.3.1 Lack of uniqueness in simple curve fitting The disagreements in reported kinetic parameters find their roots in both the functional form of the mechanism equation, as well as in the form of the Arrhenius expression.  We will examine both. In the case of the mechanism equation, this may be explained by considering that no matter how complex a reaction may be, or even how many reactions are involved, the overall conversion vs. time relationship will be sigmoidal in shape.  The significance of this is that it is very difficult to uniquely identify a mechanism model if each model yields similar results.    The second source of data misinterpretation comes from the form of the Arrhenius expression, whose parameters are mathematically decoupled from the time/conversion.  This implies that, because the rate constant depends on two parameters, but these parameters are decoupled from the observed data, there is the potential for the parameters to behave sympathetically towards one another.  That is, if one is increased, the other compensates by lowering itself - i.e.  Equation 18.  Conceptual illustration of the kinetic compensation effect  This leads to non-unique estimates of the parameters if no additional constraints are given.  This is also common within the field of kinetics and is known as the “kinetic compensation effect”.   If this explanation is unclear, imagine the opposite case in which no parameter compensation is possible.  To illustrate this, consider the case of a two parameter relationship in which each parameter had an orthogonal effect on the behavior.  An example of such a relationship is shown below in Figure 4. 19    𝑦𝑦 = (𝑐𝑐 − 𝐵𝐵)2 + 𝑏𝑏3   Equation 19.  An equation with independently acting  parameters Figure 4. Illustration of orthogonal parameter fitting  If we had measured data which were shifted upward relative to the case in which both “a” and “b” were zero, only a modification of the “b” parameter could cause such a result.  Likewise, because there is no possibility for interaction of the parameters, interpretation of any experimental data following the above relationship would be easy.  Unfortunately, this is not the case in kinetics.  From the above discussion, it is clear that approaching the problem of kinetic parameter estimation through simple curve fitting is undesirable.  Furthermore, we have not yet even begun to discuss the increasing computational complexity of multiple reactions!  What’s needed is a strategy that addresses these issues, while simultaneously confining the problem to a complexity within which it is practical to work.  20    2.3.2 Dimension reduction in kinetic parameter estimation As we saw previously, kinetic parameter estimation is vulnerable to misinterpretation due to lack of unique influence from each of the members of the kinetic triplet.  This leads to a rather inconvenient situation; given different initial biases on one of the three triplet members, values of the remaining members can be found which provide solutions to the problem which are valid.  Stated in another way, if no parameter is known with certainty, the properties of the system lead to a cyclical relationship; recovered parameters may vary widely, but each set explains the observed data equally well.  To overcome this problem, we need to unravel the cyclical relationship and develop a methodology which results in a unique estimate for each parameter.  This is illustrated in Figure 5 and Figure 6.   Activation Energy  Arrhenius Factor  Reaction Mechanism  Figure 5. Coupled Kinetic Parameters  Figure 6. Unrolled Kinetic Parameters  To accomplish this, it is necessary to not only have a method of uniquely estimating a single parameter, but also to have a roadmap to decouple the remaining parameters so that they depend only on the parameters which have already been uniquely determined.  Recognizing this, and after reviewing the literature of solid kinetics, we find it is most convenient to anchor the analysis around the unique and unbiased estimate of the activation energy.  From the activation energy, the Arrhenius factor can be uniquely determined using the kinetic Activation EnergyArrhenius FactorReaction Mechanism21    compensation effect.  Finally, with these two parameters known, it is a simple matter to test the various solid kinetics models for applicability.  Details on how these steps may be accomplished are provided below.  2.3.3 Estimate of activation energy To obtain a unique estimate of the activation energy, the analysis must not assume anything regarding mechanism form or the magnitude of the Arrhenius factor.  These restrictions are met by the isoconversional class of analysis algorithms.  The most common of these techniques include the Friedman method (1964), the Kissinger-Akahira-Sunose (KAS) method (1971), the Ozawa-Flynn-Wall method (1966), and the non-linear integral isoconversional method proposed by Vyazovkin (2000).  Because all subsequent calculations are built upon the accurate determination of activation energy, it is important that the scheme be as accurate as possible.  Therefore, considerable effort is devoted to justifying the choice of solely relying on the non-linear integral isoconversional method given by Vyazovkin (2000).  At a high level, the isoconversional techniques may be grouped as being differential (Equation 20) or integral (Equation 21) forms of the rate equation.  These grouping are shown below in Figure 7.   𝑑𝑑𝛼𝛼𝑑𝑑𝑑𝑑= 𝐴𝐴𝑐𝑐−𝐸𝐸𝐴𝐴𝑅𝑅𝑅𝑅 𝑓𝑓(𝛼𝛼) 𝑔𝑔(𝛼𝛼) = �dαf(α)𝑑𝑑0= � Ae−EART 𝑑𝑑𝑑𝑑𝑡𝑡0 Equation 20.  Differential form of the rate equation Equation 21. Integral form of the rate equation  22     Figure 7. Summary of isoconversional techniques   Before examining the theoretical backing of these techniques, it is useful to describe the properties common to all.  To begin, what is meant by the term isoconversional?  In essence, if a method is isoconversional, it yields a series of activation energies at varying levels of conversion.  In the context of our current discussion, regarding a single reaction, this is not particularly useful.  Since there is only a single reaction, with a single corresponding activation energy, the isoconversional trend would simply be a horizontal line indicating a constant activation energy value over the full conversion range – as shown below in Figure 8.  We will return to the significance of the isoconversional activation energy often as we expand our scope to multiple reactions.  Isoconversional MethodsDifferentialMethodsFriedman(1964)IntegralMethodsOzawa-Flynn-Wall (OFW)(1966)Kissinger-Akahira-Sunose (KAS)(1971)Vyazovkin(2000)23     Figure 8.  Illustration of the isoconversional activation trend in the instance of a single reaction with an activation energy of -150,000 J/mol  Beyond being isoconversional, all of the methods require multiple non-isothermal data sets.  Practically, this requires multiple experiments which have different thermal histories.  The need for this stems from the fact that the rate constant is decoupled from conversion within the rate equation.  This requires that a more strongly coupled variable, the temperature, be perturbed to provoke a sufficiently distinct response from the reacting system.  Furthermore, since the rate equation is time variant, the temperature perturbation must also be time variant.  This time rate of change of temperature is typically annotated as 𝛽𝛽, i.e. 𝛽𝛽 =𝑠𝑠𝑅𝑅𝑠𝑠𝑡𝑡≠ 0 22  Using this notation, let us move to an examination of the relationships on which the Friedman, Kissinger-Akahira-Sunose, Ozawa-Flynn-Wall, and Vyazovkin methods are based. 24    2.3.3.1 Friedman (1964) method The Friedman method is the oldest of the techniques mentioned and is unique; it is the only differential method in widespread use.  At its heart, the Friedman method consists of making an Arrhenius plot of the instantaneous reaction rate at each selected conversion value from experiments with different thermal histories.  This is shown mathematically in Equation 24.  Note that −𝐸𝐸𝐴𝐴/𝑅𝑅 and ln (𝐴𝐴𝑓𝑓(𝛼𝛼) are the slope and intercept, respectively, on a plot of 𝑐𝑐𝑚𝑚 �𝑠𝑠𝑑𝑑𝑠𝑠𝑡𝑡�𝑑𝑑,𝑆𝑆 vs. 1/𝑇𝑇𝑑𝑑,𝑆𝑆 .  The subscripts 𝛼𝛼 and i refer to the conversion at which we are attempting to recover the parameters and the experimental thermal history.  To recover the activation energy trend, one would make a plot at a given conversion level across multiple experiments.    Unfortunately, the Friedman method suffers from the typical difficulties of differential type techniques: it is strongly susceptible to noise.  In the event of very clean data and a large number of experiments run at multiple heating rates, the method can perform quite well.  𝑑𝑑𝛼𝛼𝑑𝑑𝑑𝑑= 𝐴𝐴𝑐𝑐−𝐸𝐸𝐴𝐴𝑅𝑅𝑅𝑅 𝑓𝑓(𝛼𝛼) 𝑐𝑐𝑚𝑚 �𝑑𝑑𝛼𝛼𝑑𝑑𝑑𝑑�𝑑𝑑,𝑆𝑆= 𝑐𝑐𝑚𝑚�𝐴𝐴𝑑𝑑𝑓𝑓(𝛼𝛼)� −𝐸𝐸𝑑𝑑𝑅𝑅𝑇𝑇𝑑𝑑,𝑆𝑆 Equation 23.  Starting relationship Equation 24. Friedman differential isoconversional method  2.3.3.2 Kissinger-Akahira-Sunose (KAS) method Both the Kissinger-Akahira-Sunose and Ozawa-Flynn-Wall methods attempt to improve upon the Friedman method by rewriting the Friedman equation in integral form.  The driver behind this was largely to improve upon the error characteristics associated with the derivative technique.  The tradeoffs, however, are unattractive.  The conversion of the Friedman method to integral form is performed as follows:  25    𝑔𝑔(𝛼𝛼) =𝑅𝑅𝐴𝐴𝐸𝐸𝐴𝐴∫𝑠𝑠𝑑𝑑𝑆𝑆(𝑑𝑑)= ∫𝐴𝐴𝛽𝛽exp �−𝐸𝐸𝐴𝐴𝑅𝑅𝑅𝑅� 𝑑𝑑𝑇𝑇𝑅𝑅00𝑑𝑑0 25  Or, replacing 𝐸𝐸/𝑅𝑅𝑇𝑇 by x,  𝛽𝛽𝑅𝑅𝐴𝐴𝐸𝐸𝐴𝐴∫𝑠𝑠𝑑𝑑𝑆𝑆(𝑑𝑑)= ∫ 𝑐𝑐−2 exp(−𝑐𝑐)𝑑𝑑𝑐𝑐∞𝑥𝑥𝑑𝑑0  26  The primary difference between the KAS and OFW methods comes from the different techniques used in evaluating the right hand side of Equation 26, which has no analytical solution.  The KAS technique addresses the RHS by using the approximation given in Equation 27.  However, it is critical to note that this approximation assumes a constant activation energy from the beginning of the reaction to the conversion of interest.  This limitation is made obvious through inspection of the limits of integration.  𝑅𝑅𝐻𝐻𝑆𝑆 = exp �−𝑥𝑥𝑥𝑥2� 𝑓𝑓𝐵𝐵𝑚𝑚 20 < 𝑐𝑐 < 50 ==> exp (−𝐸𝐸𝐴𝐴𝑅𝑅𝑅𝑅�−𝐸𝐸𝐴𝐴𝑅𝑅𝑅𝑅�2�   27   This gives: 𝑑𝑑𝛼𝛼𝑑𝑑𝑑𝑑= 𝐴𝐴𝑐𝑐−𝐸𝐸𝐴𝐴𝑅𝑅𝑅𝑅 𝑓𝑓(𝛼𝛼) 𝑑𝑑𝛼𝛼𝑑𝑑𝑇𝑇=𝐴𝐴𝛽𝛽𝑐𝑐−𝐸𝐸𝐴𝐴𝑅𝑅𝑅𝑅 𝑓𝑓(𝛼𝛼) 𝑐𝑐𝑚𝑚 �𝛽𝛽𝑇𝑇𝑑𝑑2� ≅−𝐸𝐸𝐴𝐴𝑅𝑅�1𝑇𝑇𝑑𝑑� − 𝑐𝑐𝑚𝑚 ��𝐸𝐸𝐴𝐴𝐴𝐴𝑅𝑅��𝑑𝑑𝛼𝛼𝑓𝑓(𝛼𝛼)𝑑𝑑0� Equation 28.  Starting relationship Equation 29. Intermediate form Equation 30. KAS integral isoconversional method  26    2.3.3.3 Ozawa-Flynn-Wall (OFW) method Similar to the KAS technique, the OFW method approximates the right hand side of Equation 26 and is, again, limited by the assumption of a constant activation energy from the beginning of the reaction to the conversion of interest.  The OFW approximation is:  𝑅𝑅𝐻𝐻𝑆𝑆 = exp(−1.052𝑐𝑐 − 5.33) 𝑓𝑓𝐵𝐵𝑚𝑚 20 < 𝑐𝑐 < 60 ==> exp �1.052−𝐸𝐸𝐴𝐴𝑅𝑅𝑅𝑅− 5.33�  31  This gives: 𝑑𝑑𝛼𝛼𝑑𝑑𝑑𝑑= 𝐴𝐴𝑐𝑐−𝐸𝐸𝐴𝐴𝑅𝑅𝑅𝑅 𝑓𝑓(𝛼𝛼) 𝑑𝑑𝛼𝛼𝑑𝑑𝑇𝑇=𝐴𝐴𝛽𝛽𝑐𝑐−𝐸𝐸𝐴𝐴𝑅𝑅𝑅𝑅 𝑓𝑓(𝛼𝛼) 𝑐𝑐𝑚𝑚(𝛽𝛽) ≅ 1.052−𝐸𝐸𝐴𝐴𝑅𝑅�1𝑇𝑇𝑑𝑑� − 5.33− 𝑐𝑐𝑚𝑚 ��𝑅𝑅𝐴𝐴𝐸𝐸𝐴𝐴��𝑑𝑑𝛼𝛼𝑓𝑓(𝛼𝛼)𝑑𝑑0� Equation 32.  Starting relationship Equation 33. Intermediate form Equation 34. OFW integral isoconversional method   2.3.3.4 Vyazovkin Method The Vyazovkin method is, by far, the most recent of the isoconversional methods and may be derived by starting with the rate equation, 𝑠𝑠𝑑𝑑𝑠𝑠𝑡𝑡= 𝐴𝐴𝑐𝑐−𝐸𝐸𝐴𝐴𝑅𝑅𝑅𝑅 𝑓𝑓(𝛼𝛼) 35 Separating variables and integrating between 𝛼𝛼1 and 𝛼𝛼2 ∫1𝑆𝑆(𝑑𝑑)𝑑𝑑2𝑑𝑑1𝑑𝑑𝛼𝛼 = 𝐴𝐴∫ exp �−𝐸𝐸𝐴𝐴𝑅𝑅𝑅𝑅(𝑡𝑡)� 𝑑𝑑𝑑𝑑𝑡𝑡𝛼𝛼2𝑡𝑡𝛼𝛼1 36 In this form, we still have two unknowns – 𝐴𝐴 and 𝐸𝐸𝐴𝐴.  To address this, the ratios from multiple experiments at multiple heating rates are used to eliminate the dependencies on 𝐴𝐴 and 𝑓𝑓(𝛼𝛼). 27     37 Thus, because only T is a function of time, Moreover, we may employ the fact that the minimum of 𝑆𝑆(𝑥𝑥)𝑔𝑔(𝑥𝑥)+𝑔𝑔(𝑥𝑥)𝑆𝑆(𝑥𝑥)= 2 to generalize Equation 38 for an arbitrary number of heating rate experiments: 𝑚𝑚𝑖𝑖𝑚𝑚𝑖𝑖𝑚𝑚𝑖𝑖𝑂𝑂𝑐𝑐 Φ(𝐸𝐸𝐴𝐴) = ∑ ∑∫ exp�−𝐸𝐸𝐴𝐴𝑅𝑅 𝑅𝑅𝑖𝑖(𝑖𝑖)�𝑠𝑠𝑡𝑡𝑖𝑖𝛼𝛼2𝑖𝑖𝛼𝛼1∫ exp�−𝐸𝐸𝐴𝐴𝑅𝑅 𝑅𝑅𝑗𝑗(𝑖𝑖)�𝑠𝑠𝑡𝑡𝑖𝑖𝛼𝛼2𝑖𝑖𝛼𝛼1𝑆𝑆𝑓𝑓≠𝑆𝑆𝑆𝑆𝑆𝑆=1    39 Where 𝑚𝑚 is the total number of experiments.  Inspection of the resulting equation reveals why this method is rarely applied – computational complexity.  First and foremost in regards to computational expense, the solution of Equation 39 requires minimization, most often done without derivatives.  This requires a number of iterations and, for each iteration, 𝑚𝑚 numerical integrations are needed.  Further computational cost is incurred because the equation requires the time as a function of conversion.  Considering that experiments generally obtain conversion as a function of time, this is a significant complication and requires that the measured conversion be strictly monotonic.  The slightest addition of noise resulting in non-monotonic behavior would increase the difficulty of automated integration by orders of magnitude.  This requires that a significant amount of computational time be spent upfront in preparing the data via smoothing/filtering operations.  Despite these drawbacks, modern computers are capable of completing the task handily, and solutions can generally be found within seconds. 1 =∫ exp�−𝐸𝐸𝐴𝐴𝑅𝑅 𝑇𝑇(𝑑𝑑)𝑆𝑆𝑥𝑥𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑡𝑡 1�𝑑𝑑𝑑𝑑𝑡𝑡𝛼𝛼2𝑡𝑡𝛼𝛼1∫ exp�−𝐸𝐸𝐴𝐴𝑅𝑅 𝑇𝑇(𝑑𝑑)𝑆𝑆𝑥𝑥𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑡𝑡 2�𝑑𝑑𝑑𝑑𝑡𝑡𝛼𝛼2𝑡𝑡𝛼𝛼1 38 28    From this labor, we obtain an activation energy trend which has no built-in assumptions and provides a unique, unbiased estimate of the activation energy as it changes with conversion.  Furthermore, an original observation is that the Vyazovkin method also provides the user with a built-in measure of quality.  This is simply the value of the minimized function given in Equation 39.  A failure to obtain exactly the number of permutations of experiments, at a given level of conversion, indicates the failure of the algorithm to reconcile the experimental data using a consistent activation energy.  2.3.3.5 Summary of isoconversional techniques Having briefly described each of the techniques, it is a simple matter to justify the usage of the Vyazovkin method.  The Friedman method suffers from measurement noise bias and the KAS/OFW methods make the assumption of constant activation energy up to the conversion of interest.  The Vyazovkin method on the other hand, makes no assumptions and, being an integral technique, is significantly more noise tolerant.   2.3.4 Constraining the Arrhenius factor – method of invariant kinetic parameters With an unbiased estimate of the activation energy, we recall that the activation energy and Arrhenius factor are linked through the kinetic compensation effect.  This linkage originally caused problems, since insufficient distinction in response existed to uniquely decouple the two parameters.  With one of the parameters known, we may use this linkage to uniquely determine the other.  However, before describing the process by which this is accomplished, we must first define a new term – the reactivity.  This is just defined as the rate of reaction.  Having defined the reactivity as the rate at which conversion occurs and, seeing that the conversion curve is always sigmoidal in character, we may immediately deduce that the reactivity curve will be “bump-shaped”.  If this is not immediately clear, we may borrow a similar example from the field of probability.  Consider the relationship between a cumulative 29    distribution function and its probability density function.  The reason why shape is significant will become clear below.  As explained, activation energy and Arrhenius factor are linked via the kinetic compensation effect.  This may be formalized as a log-linear relationship of the form: ln(𝐴𝐴) = 𝐵𝐵𝐸𝐸𝐴𝐴 + 𝑏𝑏  40 where 𝐵𝐵 and 𝑏𝑏 are the slope and intercept.  However before this relationship becomes useful, we must first determine how to resolve 𝐵𝐵 and 𝑏𝑏.  Luckily, this is not too difficult – we simply require two or more known values of 𝐴𝐴 and 𝐸𝐸𝐴𝐴 which satisfy the measured data; this may be approached in at least three ways.    The first approach, recommended by the ICTAC Kinetics Committee, is to make use of the method of invariant kinetic parameters.  This essentially consists of simple curve fitting experimental data to multiple mechanism (𝑓𝑓(𝛼𝛼)) models and compiling the results.  That is, one is not so much interested in using the best mechanism (indeed some matches may be terrible), but instead rely on curve fitting to deliver multiple combinations of similarly valid  𝐴𝐴 and 𝐸𝐸𝐴𝐴 values from which 𝐵𝐵 and 𝑏𝑏 can be determined.  The second method is closely related to the first, but maintains the same 𝑓𝑓(𝛼𝛼) model.  It instead produces 𝐴𝐴 and 𝐸𝐸𝐴𝐴 pairs through variation of the initial guesses used by the non-linear fitting algorithm, trusting that some of these will yield different optima.  Both of these methods work, but are suboptimal given that one is at the mercy of the success of a fitting algorithm.  A much better approach to the problem of determining 𝐵𝐵 and 𝑏𝑏 is to understand the influence of the kinetic compensation effect on reactivity and then utilize the experimental data to anchor the calculations.  This has the pleasant side-effect of clearly illustrating the problems of insufficient curve character discussed previously.  To do so, we note that the kinetic compensation effect procedure is simply finding the best pre-exponential factor, assuming a 30    given value of the activation energy, to honor the measured data.  In the first two procedures discussed, this was done in a rather naive fashion, relying on automated routines.  In the third method, we do away with automated curve fitting and instead choose an easily identifiable point in the experimental data to match to the corresponding point in the simulated data.  Here we return to the importance of the shape of the reactivity curve; because the reactivity curve has only a single maximum (for a single reaction), it is a convenient point to match.  This procedure may be better illustrated through an example:  2.3.4.1 Example 1. Though the overall procedure is identical for all reaction models, for the purposes of illustration, we adopt a first order reaction mechanism of the form 𝑓𝑓(𝛼𝛼) = 1− 𝛼𝛼.  To simulate experimental data, the parameters of Table 3 are used:  Table 3.  Example parameters for a first order reaction Parameter: Value 𝑠𝑠𝑅𝑅𝑠𝑠𝑡𝑡   15°C/min 𝐸𝐸𝐴𝐴   150,000 J/mol 𝐴𝐴   1e15 Mass fraction 100% (single component)   Full Rate Expression   Solving the rate equation described in Table 3 yields the conversion and reactivity curves shown in Figure 9. 𝑑𝑑𝛼𝛼𝑑𝑑𝑑𝑑= 1x1015exp⁡�−150,000𝑅𝑅𝑇𝑇� (1− 𝛼𝛼) 31     Figure 9.  Conversion and reactivity curves of the system shown in Table 3  Visual assessment of Figure 9 confirms that the curve exhibits a single maximum, the peak of which may be determined exactly through the regular methods; in this case, it is 686 seconds.  Once the peak value of the simulated data has been determined, we turn to determining the Arrhenius factor needed to place the simulated peak at the same point for multiple activation energies.  This can be accomplished by setting an arbitrary activation energy and iterating on the Arrhenius factor until the derivative of the reactivity is 0 (second derivative of conversion with respect to time) at the determined value of the peak, 686 seconds in this example.  This process is illustrated graphically in Figure 10.  Doing this for several activation energies, we obtain a plot similar to Figure 11.  32     Figure 10. Graphical illustration of aligning reactivity peaks for a random activation energy    Figure 11. Arrhenius factors required to match peaks for arbitrary activation energies  33    We now utilize the relationship in Equation 40 and plot the natural logarithm of the Arrhenius factor vs. the activation energy.  This yields a linear relationship from which we may derive a slope, “a” and intercept, “b”, as shown in Figure 12.  Figure 12.  Determination of Arrhenius factor using the principle of kinetic invariance  Thus, using the activation energy output by the Vyazovkin method, we apply the relation shown in Figure 12 and recover the unique Arrhenius factor necessary to honor the maximum reactivity of our sample.  Note that this procedure is a unique contribution to the field brought about by this work.  We are then left with the rather tedious problem of uncovering the reaction mechanism.  2.3.5 Mechanism determination At this point, we have uniquely estimated both the activation energy and the Arrhenius factor and are only left with the task of determining the reaction mechanism.  This may be done by testing the various models listed in Table 1 using the parameters previously calculated.  F-test 34    statistics can then be used to assess the suitability of each model and the correct model can be selected within a given statistical significance.  However, before moving on, we should point out that estimating the reaction model based on a single heating rate experiment is not using the data to its full potential.  Recall that in order to apply the Vyazovkin method, two or more experiments were required at different heating rates for a single sample.  These additional experiments should be taken into account when estimating both the Arrhenius factor and the reaction mechanism. More will be said on this in the next section.  2.4 Solid state kinetics expression – multiple reactions To this point, we have limited the discussion to a single component, single reaction system. This simplifies the explanation of many concepts, but is not broadly applicable.  To generalize the techniques discussed above, it is necessary to consider the impact of multiple reactions occurring simultaneously.    Upon first consideration, the problem of multiple reactions appears straightforward.  Simply apply the Vyazovkin technique, identify the activation energy for each reaction, pick the peaks on the reactivity curve, apply the method of invariant kinetic parameters, eliminating the Arrhenius factor for each reaction, and finally determine the reaction mechanisms through physical insight or brute force (testing each permutation of models).  However, we must now also address the problem of sample composition or, put more succinctly, the fraction of sample taking part in each reaction.  This has been termed the reaction fraction.  35    2.4.1 Reaction fraction – an additional degree of freedom Since the experimentally observed conversion will be a composite measurement when there are multiple reactions, influenced by rates of the individual reactions, it is first necessary to determine how this relates to the individual reactions.  As this does not seem to be available in the literature, the general derivation is given below. Beginning with the definition of ash free conversion given in Equation 12, we write for an n component mixture: 𝛼𝛼𝑠𝑠𝑜𝑜𝑠𝑠𝑆𝑆𝑆𝑆𝑜𝑜𝑆𝑆𝑠𝑠 =𝑆𝑆𝑆𝑆𝑖𝑖𝑆𝑆𝑆𝑆𝑖𝑖𝑆𝑆𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖−𝑆𝑆𝑆𝑆𝑖𝑖𝑆𝑆𝑆𝑆𝑖𝑖𝑆𝑆@𝑖𝑖𝑆𝑆𝑆𝑆𝑖𝑖𝑆𝑆𝑆𝑆𝑖𝑖𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖−𝑆𝑆𝑆𝑆𝑖𝑖𝑆𝑆𝑆𝑆𝑖𝑖𝑆𝑆𝑓𝑓𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖=(𝑆𝑆𝑠𝑠1+𝑆𝑆𝑠𝑠2+⋯+𝑆𝑆𝑠𝑠𝑖𝑖)−(𝑆𝑆𝑖𝑖1+𝑆𝑆𝑖𝑖2+⋯+𝑆𝑆𝑖𝑖𝑖𝑖)(𝑆𝑆𝑠𝑠1+𝑆𝑆𝑠𝑠2+⋯+𝑆𝑆𝑠𝑠𝑖𝑖)−�𝑆𝑆𝑓𝑓1+𝑆𝑆𝑓𝑓2+⋯+𝑆𝑆𝑓𝑓𝑖𝑖� 41 where the subscripts 1, 2, … , n refer to each component respectively.  Note that it is not possible to reduce Equation 41 to something useful.  Instead, to determine how the individual reactions contribute to the overall observed conversion, we must remove the ash terms 𝑚𝑚𝑆𝑆1, 𝑚𝑚𝑆𝑆2, … , 𝑚𝑚𝑆𝑆𝑆𝑆.  This is not a problem; because we will be comparing the calculated conversion to an ash free conversion, calculated from the experimental data, the ash can safely be assumed to be 0 in the simulated data.  Note that this would not have been possible if we had chosen a different form to express conversion.  However, we must also note that, by using this simplification, we are implicitly assuming that each component is composed of the same percentage of non-reactive material.  This is incorrect, but for samples with a low ash fraction, the error introduced is minimal.  Utilizing this simplification: 𝛼𝛼𝑠𝑠𝑜𝑜𝑠𝑠𝑆𝑆𝑆𝑆𝑜𝑜𝑆𝑆𝑠𝑠 =𝑆𝑆𝑆𝑆𝑖𝑖𝑆𝑆𝑆𝑆𝑖𝑖𝑆𝑆𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖−𝑆𝑆𝑆𝑆𝑖𝑖𝑆𝑆𝑆𝑆𝑖𝑖𝑆𝑆@𝑖𝑖𝑆𝑆𝑆𝑆𝑖𝑖𝑆𝑆𝑆𝑆𝑖𝑖𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖=(𝑆𝑆𝑠𝑠1+𝑆𝑆𝑠𝑠2+⋯+𝑆𝑆𝑠𝑠𝑖𝑖)−(𝑆𝑆𝑖𝑖1+𝑆𝑆𝑖𝑖2+⋯+𝑆𝑆𝑖𝑖𝑖𝑖)(𝑆𝑆𝑠𝑠1+𝑆𝑆𝑠𝑠2+⋯+𝑆𝑆𝑠𝑠𝑖𝑖)=𝑆𝑆𝑠𝑠1−𝑆𝑆𝑖𝑖1𝑆𝑆𝑠𝑠1+𝑆𝑆𝑠𝑠2+⋯+𝑆𝑆𝑠𝑠𝑖𝑖+𝑆𝑆𝑠𝑠2−𝑆𝑆𝑖𝑖2𝑆𝑆𝑠𝑠1+𝑆𝑆𝑠𝑠2+⋯+𝑆𝑆𝑠𝑠𝑖𝑖+⋯+𝑆𝑆𝑠𝑠𝑖𝑖−𝑆𝑆𝑖𝑖𝑖𝑖𝑆𝑆𝑠𝑠1+𝑆𝑆𝑠𝑠2+⋯+𝑆𝑆𝑠𝑠𝑖𝑖  42  We then multiply by 𝑚𝑚𝑠𝑠1/𝑚𝑚𝑠𝑠1, 𝑚𝑚𝑠𝑠2/𝑚𝑚𝑠𝑠2, … , 𝑚𝑚𝑠𝑠𝑆𝑆/𝑚𝑚𝑠𝑠𝑆𝑆 respectively.  This gives:  36    𝑆𝑆𝑠𝑠1(𝑆𝑆𝑠𝑠1−𝑆𝑆𝑖𝑖1)(𝑆𝑆𝑠𝑠1+𝑆𝑆𝑠𝑠2+⋯+𝑆𝑆𝑠𝑠𝑖𝑖)𝑆𝑆𝑠𝑠1+𝑆𝑆𝑠𝑠2(𝑆𝑆𝑠𝑠2−𝑆𝑆𝑖𝑖2)(𝑆𝑆𝑠𝑠1+𝑆𝑆𝑠𝑠2+⋯+𝑆𝑆𝑠𝑠𝑖𝑖)𝑆𝑆𝑠𝑠2+⋯+𝑆𝑆𝑠𝑠𝑖𝑖(𝑆𝑆𝑠𝑠𝑖𝑖−𝑆𝑆𝑖𝑖𝑖𝑖)(𝑆𝑆𝑠𝑠1+𝑆𝑆𝑠𝑠2+⋯+𝑆𝑆𝑠𝑠𝑖𝑖)𝑆𝑆𝑠𝑠𝑖𝑖=𝑆𝑆𝑠𝑠1(𝑆𝑆𝑠𝑠1+𝑆𝑆𝑠𝑠2+⋯+𝑆𝑆𝑠𝑠𝑖𝑖)∗�1 −𝑆𝑆𝑖𝑖1𝑆𝑆𝑠𝑠1�+𝑆𝑆𝑠𝑠2(𝑆𝑆𝑠𝑠1+𝑆𝑆𝑠𝑠2+…+𝑆𝑆𝑠𝑠𝑖𝑖)∗ �1−𝑆𝑆𝑖𝑖2𝑆𝑆𝑠𝑠2�+⋯+𝑆𝑆𝑠𝑠𝑖𝑖(𝑆𝑆𝑠𝑠1+𝑆𝑆𝑠𝑠2+…+𝑆𝑆𝑠𝑠𝑖𝑖)∗ �1−𝑆𝑆𝑖𝑖𝑖𝑖𝑆𝑆𝑠𝑠𝑖𝑖�  43  By inspection, we see that the 1 −𝑆𝑆𝑖𝑖𝑖𝑖𝑆𝑆𝑠𝑠𝑖𝑖 term is simply the conversion of a single component.  Thus, 𝛼𝛼𝑠𝑠𝑜𝑜𝑠𝑠𝑆𝑆𝑆𝑆𝑜𝑜𝑆𝑆𝑠𝑠 =𝑆𝑆1𝑆𝑆1+𝑆𝑆2+⋯+𝑆𝑆𝑖𝑖𝛼𝛼1 +𝑆𝑆2𝑆𝑆1+𝑆𝑆2+⋯+𝑆𝑆𝑖𝑖𝛼𝛼2 +⋯+𝑆𝑆𝑖𝑖𝑆𝑆1+𝑆𝑆2+⋯+𝑆𝑆𝑖𝑖𝛼𝛼𝑆𝑆          44  Equations 41 through 44 show that the correct way to interpret the observed conversion for multiple, simultaneous reactions is to combine conversions on a mass fraction weighted basis.  Furthermore, by inspection, we see that the reactivity curves may also be combined through simple superposition, weighted by mass fraction.  In many cases, this is all that is necessary to define the reaction fraction.  In the case of gasification, however, we are interested in the thermal breakdown of complex, high molecular weight, organic biopolymers which consist of relatively few elements (C, H, O).  Re-examining Figure 1, Figure 2, and Figure 3, it is not difficult to imagine that at least some of the intermediates will be of similar structure.  This implies that a number of the decomposition reactions, though their reactants may been derived from different parent molecules, will be duplicated across primary components.  This has the rather inconvenient consequence of it being inappropriate to define the reaction fraction based solely upon primary component composition.  Instead, we must shift our interpretation from thinking of the reactions as consisting of the decomposition of lignin, cellulose, and hemicellulose, to thinking of the reactions necessary to break particular types of bonds.  To illustrate this concept, consider Figure 13 and Figure 14.  If we were investigating the kinetics of breaking the C=C double bond in a mixture of compounds 1 and 2, we may initially attempt to model this using two separate reactions weighted by their relative fractions.  However, as the chain length, N, becomes very 37    large, the influence of Cl/Br atoms on the double bond becomes diminishingly small.  The consequence of this is that, even though the mixture is multicomponent, the investigator would only be able to identify a single composition and a single reaction.  The interpretation of this would then be that the decomposition reaction consists 100% of breaking the C=C bond type.    Figure 13.  Compound 1 Figure 14. Compound 2  2.4.2 Multiple reaction parameter specifics Now that the usage of the reaction fraction parameter has been defined, we return to the specifics of parameter determination in the event of multiple reactions.  To do so, let us consider the sample system of reactions specified by Table 4, assuming that each of the reactions follows first order kinetics (Equation 6, Table 1).           38    Table 4.  Parameters for a sample which decomposes according to three parallel, independent reactions  Parameter: Value Initial Temperature 25°C 𝑠𝑠𝑅𝑅𝑠𝑠𝑡𝑡   15°C/min Reaction 1.   𝐸𝐸𝐴𝐴   -150,000 J/mol 𝐴𝐴   1e18 Mass rxn fraction 50%   Reaction 2.  𝐸𝐸𝐴𝐴   -200,000 J/mol 𝐴𝐴   1e26 Mass rxn fraction 10%   Reaction 3.  𝐸𝐸𝐴𝐴   -250,000 J/mol 𝐴𝐴   1e30 Mass rxn fraction 40%  Solving the rate equation for each of the reactions and combining using the relationship derived in Equation 44, we obtain the conversion curves shown in Figure 15. 39     Figure 15.  Observed and component conversions of the reaction system defined in Table 4  Converting the curves in Figure 15 to derivatives, we obtain Figure 16.  By inspection, we note that: 1. The simulated experimental derivative curve contains one peak per reaction.   2. As predicted by Equation 44, the simulated experimental derivative curve can be obtained by superposition of the individual contributing reaction derivative curves.  3. The observed peaks do not line up exactly with the peaks of the contributing reactions, but are very close. Statement 3 is a consequence of the superposition effect and indicates that even though Equation 40 can still be applied in a fashion analogous to that seen previously, the result will be slightly shifted.  Statement 1, though technically true, may not always be obvious.  The ability to identify unique peaks will depend upon the exact values of the kinetic triplet.  In cases where individual peaks 40    are indistinguishable, the parameters determined will be a fraction weighted average for the composite reaction.  This is generally not a problem since the recovered rate law will still honor the experimental data.  However, care should be taken when extrapolating results beyond the temperature range covered in the experiments.   Figure 16.  Derivative curves of the composite and component conversions of the sample reaction system defined in Table 4  Turning to the activation energy, if we assume that we have run multiple experiments at different heating rates, we may apply the Vyazovkin method to calculate the isoconversional activation energy trend.  As an example, by simulating experiments run at heating rates of 15, 25, and 35°C/min and applying the Vyazovkin technique, Figure 17 is obtained.  The trend recovered is significantly more complex than that in Figure 8.  By inspection, it is obvious that determination of the exact activation energies of the individual reactions is 41    difficult.  Based solely on inspection of the plot, a reasonable estimate of the component reaction activation energies may be -1.8e5, -1.45e5, and -2.50e5.  Since these are simulated data, we know that these, with the exception of one, are incorrect.  Note that this is not always the case and, depending on the properties of the system being investigated, it may be possible to pick the true activation energies directly off the isoconversional plot.  Figure 17.  Vyazovkin method isoconversional activation energy trend for the reaction system defined in Table 4 (assuming heating rates of 15°C/min, 25°C/min, and 35°C/min)  Thus, in the case of multiple reactions, applications of the method of invariant kinetic parameters and the Vyazovkin advanced isoconversional technique alone are not enough to fully recover the Arrhenius factors and activation energies from experimental data.  However, the method of invariant kinetic parameters does yield an excellent estimate of the Arrhenius factor, and the complex character of the Vyazovkin isoconversional trend adds further uniqueness to the signature of a system of simultaneous reactions (while also providing estimates of potential activation energies).  It then follows that if we apply typical nonlinear parameter estimation techniques to both the experimentally measured conversion and the 42    Vyazovkin isoconversional trend, using the relationship in Equation 40 to provide initial guesses of the Arrhenius factors based on the inexact activation energies derived from the isoconversional trend, we obtain a high accuracy estimate of the true model parameters.  Using the initial guesses for parameters obtained through the methods discussed above and solving, we obtain excellent parameter recovery performance using the technique discussed.  This is illustrated in Table 5 and Figure 18.   Table 5.  Comparison of parameter recovery performance for a system of multiple reactions using typical and enhanced methods Parameter: Value   Initial Temperature 25°C 𝑠𝑠𝑅𝑅𝑠𝑠𝑡𝑡   15°C/min  Initial Guess Simple Match Match using Vyazovkin trend Correct Value Reaction 1.      𝐸𝐸𝐴𝐴(J/mol)   -145,000  -164,719 -152,433 -150,000 𝐴𝐴   1e18 2.02e19 2.02e18 1e18 Mass rxn fraction 30% 65% 50% 50%      Reaction 2.     𝐸𝐸𝐴𝐴 (J/mol)  -200,000  -179,401 -196,000 -200,000 𝐴𝐴   1e26 1.11e23 2.5e25 1e26 Mass rxn fraction 30% 16% 10% 10%      Reaction 3.     𝐸𝐸𝐴𝐴 (J/mol)  -250,000  -235,970 -250,508 -250,000 𝐴𝐴   1e30 3.65e29 1.15e30 1e30 Mass rxn fraction 40% 19% 40% 40%  43     Figure 18.  Comparison of parameter recovery performance using derivative curves for a system of multiple reactions using typical and enhanced methods  Finally, it should be mentioned that although the enhanced parameter recovery technique has performed quite well, we have not taken advantage of all available experimental data.  The process above uses the conversion data from a single experiment, but to calculate the isoconversional trend, we would have had to conduct multiple experiments at multiple heating rates.  By incorporating these data into the parameter recovery process, the exact reaction parameters can be recovered in many fewer experiments when using the enhanced technique.  This technique will be employed in analyzing the experimental results presented in the next chapter.   44    Chapter 3. Experiments  3.1 Equipment As we are interested in gasification, a thermochemical process, we must first decide upon an experimental system suitable for determining kinetic parameters.  From our survey of the literature, we have noted that thermogravimetric experimentation is an acceptable method of accomplishing this.  A simplified illustration of a thermogravimetric analyzer is shown in Figure 19.  3.1.1 Equipment (simple description) In the non-isothermal thermogravimetric technique, a small solid sample, while being flooded with either an inert or reactive gas, is heated under a linear temperature program while its weight is measured.  If the sample is reactive, the experimenter notes a characteristic time series which captures the rate at which mass is lost.  This mass loss curve can then be transformed into a dimensionless conversion curve which is suitable for deriving kinetic information; the relationship describing this is given in Equation 12.    To capture a true estimate of only the kinetic parameters, it is essential that the measurements be done on a system which is as close to being transport limitation free as possible.  To accomplish this, the solid sample is ground very finely and only a very small amount is used in the instrument.  Despite these precautionary measures, it is essential that work be done to verify transport limitations are indeed absent from the experimental results.   45     Figure 19. Simplified Illustration of a thermogravimetric analyzer (TGA)  3.1.2 Equipment (detailed description) Though the instrument description given in 3.1.1 covers the general idea of TGA operation, there are several specifics that are critical to experimental success and that vary with instrument details.  In this study, all experiments were run on a TA Instruments, Q500 series, TGA.  3.1.2.1 TA Instruments Q500 Series Thermogravimetric Analyzer specifications The TA Q500 is a versatile instrument, allowing for the variation of several experimental parameters.  As mentioned in section 3.1.1, TGA analysis consists of heating a sample under a linear temperature program, flooding the reaction chamber with gas, and measuring the sample mass.  This leads to several parameters which are subject to variation by instrument capability: heating rate, flood gas type, sample size (volume) limit, and sample mass limit.  Manufacturer quoted information regarding these parameters are summarized in Table 6.   46    Table 6.  Manufacturer quoted specifications for TA Instruments Q500 series TGA  Low Limit High Limit Temperature Range (°C) Ambient 1000 Isothermal Temp Accuracy (°C) ±1 Heating Rate (°C/min) 0.01 100 Flood gases 𝐶𝐶𝑂𝑂2, 𝑁𝑁2, Air, He, Ar, CO Flood gas rates (mL/min) Dependent on source Dependent on source Sample Weight (mg) 0 1000 Mass measurement (%) ±0.01 Mass baseline drift (µg) <50 Maximum Sample size (µL) 100  Beyond the parameters stated in Table 6, the layout of the reaction chamber differs slightly from that shown in Figure 19.  Most importantly, it should be noted that the Q500 TGA utilizes a true cross-flow design with the reaction gas entry and exhaust ports located directly adjacent to the sample.  This serves to minimize mass transport effects which may arise due to localized variations in atmosphere composition caused by poor mixing.  Furthermore, although it is primarily a feature of convenience, the Q500 TGA also allows for automated sample acquisition by providing a sample queue.  An experimenter may prepare up to 16 samples, set the desired experimental conditions, and the machine will independently run each experiment in turn.  Photographs of the instrument are shown in Figures 20 and 21.  47      Figure 20. TA Instruments Q500 TGA Figure 21. Annotated TGA illustration  3.1.2.2 Wood pyrolysis/gasification reactions – TA Q500 Instrument Accuracy  Despite manufacturer claims of measurement accuracy, which are valid in the case of very low gas flow rate, variable accuracies were found at high gas flow rates.  The significance of this is that in the course of determining the conditions necessary to avoid transport effects, it was found that moderately high gas rates flow were necessary.  Under these conditions, the instrument error was evaluated by running a mock experiment with an empty pan (no sample but tared) and evaluating its weight behavior.  As expected, it was found that the measured weight centered on zero.  However, it was also found that the weight measurement oscillated slightly, ranging from -0.0028 to 0.0024 mg.  This behavior is shown for an experiment having a heating rate of 25°C/min in Figure 22.  A likely explanation is slight swaying of the pan caused by the moderate (cross-flow) gas velocities needed to eliminate transport effects. 48     Figure 22.  Instrument measurement error (mg) at a gas flow rate of 90 mL/min cross flow, 40 mL/min downward flow  3.1.2.2.1 Transport effect elimination Since we are interested in a true estimate of reaction kinetics only, it is necessary to determine the proper experimental conditions to ensure that this is indeed what we are measuring.  By running samples at several gas flow rates, it was found that the potential for mass transport limitation does exist at low gas flow rates.  To eliminate the effect completely, it is necessary to use a gas flow rate which exceeds that found to adversely affect measurement accuracy during the gasification stage.  Thus, a balanced approach was taken and a gas flow rate which eliminated ~95% of the transport limitations, but did not cause undesirable measurement errors in the gasification portion of the experiment, was chosen.  This was found to be 40 mL/min in downward flow and 60 mL/min cross-flow.  Figure 23 illustrates the observed transport limitation.  Note that the downward gas flow rate was held constant at 40 mL/min. 49     Figure 23.  Illustration of transport phenomena limitation on results at low gas flow rate (Poplar, temperature ramping at 20°C/min beginning from 50°C at time 0)  3.2 Experimental design 3.2.1 Number of experiments per sample To apply the isoconversional techniques introduced in section 2.3.3, a minimum of two experiments with unique thermal histories must be run.  Given the potential for noise corruption, it is beneficial to exceed this minimum.  A three experiment program allows for not only a better weighted analysis, but also aids in flagging experiments which are potentially faulty.  For example, under a three experiment regiment where experiments are denoted “a”, “b”, and “c”, isoconversional trends can be calculated using the combinations “a” and ”b”, “a” and ”c”, “b” and ”c”, and “a”, ”b”, and ”c”.  These may then be compared to verify consistency.  3.2.2 Heating Rates The maximum resolution obtainable from non-isothermal experiments can be obtained when each of the experiments is maximally separated by heating rate.  This can be explained by considering that signals will be most easily identifiable as unique when obtained at very 50    different heating rates.  However, the benefits of a large separation in heating rates is balanced by the limitation that high heating rates are in direct competition with heat transfer limitations; as the rate of temperature increase is raised, the sample has less time to reach thermal equilibrium.  It is also important to observe that the equipment itself also has limitations.  The TA Q500 has an upper temperature limit of 1000°C and a maximum heating rate of 99°C/min.  The higher the heating rate, the sooner the temperature limit is reached.  Considering the implications of the discussion above, experimental heating rates were chosen as 15, 20, and 25°C/min.  These provide for a 25-33% increase in heating rate from experiment to experiment, while also providing a slow enough rate of temperature increase that sufficient data could be collected for analysis.  3.2.3 Pyrolysis and gasification Though the pyrolysis and gasification kinetics were measured separately, it is critical to recognize that there is the potential for gasification properties to be path dependent.  To maintain relevance of results, only char samples derived from wood which had gone through pyrolysis under an identical temperature program were used in the gasification experiment.  3.2.4 Experimental description Combining the arguments of the previous sections, a generalized experimental procedure was created to measure each sample’s reactivity.  This is described graphically in Figure 24.  Note that due to the hygroscopic nature of woody biomass, a 60 minute drying time was included in the procedure. 51    It should also be observed that, together with the pyrolysis information, we also obtained the gasification data from the same experimental run.  Referring again to Figure 24, we see that once pyrolysis has been completed, indicated by the sample mass no longer changing, the sample is allowed to cool to 100°C under nitrogen.  The reaction gas is then switched to air and the heating profile repeated.  This leads to a combined measurement of both pyrolysis and gasification behavior which must be separated before analysis can occur.  To separate the data, the experimental data are plotted and separated using the procedure summarized in Figure 29.  52     Figure 24.  Flowchart illustrating the experimental procedure for n samples    53    3.3 Poplar (Populus trichocarpa) samples Fourteen Poplar samples were selected from a trial population maintained by Shawn Mansfield’s group in the University of British Columbia (Faculty of Forestry).  These were from trees grown in a common garden and were the offspring of wild trees growing over a large geographic area, spanning from Washington State to Northern British Columbia.  Samples were prepared by milling and sieving (using a screen size of 60 mesh) and were stored at room temperature.    Compositional analysis had been done on the population beforehand, permitting for the selection of samples with the greatest compositional variation.  To cover the compositional range, samples were chosen by first binning and plotting against lignin content.  The results of this exercise, along with the sampled points, are shown below in Figure 25.  A full breakdown of the sample compositions is shown in Table 7.  A fraction plot, showing the information graphically, is provided in Figure 26. 54     Figure 25.  Poplar population lignin variation and sample selection  Table 7.  Summary of Poplar sample compositions    55     Figure 26.  Graphical summary of Poplar sample compositions   3.3.1 Nitrogen pyrolysis 3.3.1.1 Conversion and reactivity curves Following the experimental procedure laid out in Figure 24, the ash free conversion curves shown in Figure 27 were obtained for the three heating rates. 56     Figure 27.  Poplar pyrolysis conversion curves for heating rates of 15, 20, and 25°C/min  Taking the data in Figure 27 and differentiating with respect to time, reactivity data were obtained, shown below in Figure 28.  Note that peaks can be distinguished at approximately 9, 11, and 14 minutes for the 25°C/min curve.  This indicates that the system is made up of at least three reactions. 57     Figure 28.  Poplar pyrolysis reactivity curves for heating rates of 15, 20, and 25°C/min  58    3.3.2 Air gasification 3.3.2.1 Conversion and reactivity curves As mentioned in section 3.2.4, an air gasification experiment was conducted immediately following pyrolysis, requiring the data to be separated after the fact.  Figure 29 illustrates this, presenting experimental data measured from a Lodgepole pine experiment run at 15°C/min.  Note the drift in mass occurring at ~215 minutes.  This is a common problem among samples, occurring in ~30% of experiments, but it did not seem to be correlated with heating rate, gas flow, or sample.  Thus, the drift is removed prior to analysis.    Figure 29.  Illustration of raw data yielded by a single experiment (Lodgepole pine, 15°C/min)  Separated Poplar air gasification conversion curves are shown in Figure 30.  Note the sharp changes in slope exhibited in several of the curves.  This behavior was found to occur in many of the experiments, but was not correlated with particular times, temperatures, or compositions.   59     Figure 30.  Poplar air gasification conversion curves for heating rate of 15, 20, and 25°C/min  60     Figure 31.  Poplar gasification reactivity curves for heating rate of 15, 20, and 25°C/min 61    3.4 Lodgepole pine (Pinus contorta) samples Unfortunately, no large Lodgepole pine populations with predetermined compositions were available for this study.  Thus the sampling approach taken for the Poplar could not be used in Lodgepole pine selection.  Instead, ten samples, provided by Shawn Mansfield’s group in the University of British Columbia (Faculty of Forestry), were chosen at random and analyzed.  These were from wild trees growing throughout British Columbia.  Samples were prepared by milling and sieving (using a screen size of 60 mesh) and were stored at room temperature.  Detailed compositional results for each of the samples can be found in Table 8 and graphically in Figure 32.  Table 8. Summary of Lodgepole pine sample compositions   62     Figure 32.  Graphical summary of Lodgepole pine sample compositions  3.4.1 Nitrogen pyrolysis 3.4.1.1 Conversion and reactivity curves Derived using an identical procedure as for the Poplar samples, conversion and reactivity curves for Lodgepole pine are shown in Figure 33 and Figure 34 respectively. The tightness of the curves, relative to the Poplar samples, is a consequence of random sampling from what is likely a normal distribution of compositions.  Note, however, that the Lodgepole pine curve shapes are identical to those of the Poplar curves. 63     Figure 33. Lodgepole pine pyrolysis conversion curves for heating rates of 15, 20, and 25°C/min    64     Figure 34. Lodgepole pine pyrolysis reactivity curves for heating rates of 15, 20, and 25°C/min 65    3.4.2 Air gasification 3.4.2.1 Conversion and reactivity curves   Figure 35. Lodgepole pine gasifications conversion curves for heating rates of 15, 20, and 25°C/min   66     Figure 36. Lodgepole pine gasification reactivity curves for heating rates of 15, 20, and 25°C/min   67    Chapter 4. Analysis of experimental data and discussion of results  4.1 Poplar wood reactivity 4.1.1 Nitrogen pyrolysis – simple correlations To begin to understand the impact of composition on kinetic behavior, we plot the time taken to reach 90% ash free conversion as a function of composition using a ternary diagram.  This is shown for the case of Poplar pyrolysis in Figure 37, Figure 38, and Figure 39.  Note that composition is indicated by each point’s location while time required to reach 90% conversion is represented by the patch color.  Inspecting the figures, we see that the relative magnitude of time, for each sample, to reach the benchmark conversion remains consistent.  This is an important observation since it indicates that each of the samples follow a constant reaction path.  A second observation is that, generally, samples with higher hemicellulose and cellulose fractions reach 90% conversion sooner.      Figure 37. Relative times to reach 90% conversion for Poplar pyrolysis. (“Cooler” colors indicate less time, therefore greater reactivity) – Heating rate: 15°C/min  Figure 38. Relative times to reach 90% conversion for Poplar pyrolysis. (“Cooler” colors indicate less time, therefore greater reactivity) – Heating rate:  20°C/min  68     Figure 39. Relative times to reach 90% conversion for Poplar pyrolysis. (“Cooler” colors indicate less time, therefore greater reactivity) – Heating rate:  25°C/min  Having considered the simple graphical relationships above, we must note that it is not necessarily the case that reactivity correlations remain constant over the entire course of reaction.  In fact, it is quite likely that this will not be the case.  In order to more fully describe the relationship between reactivity and composition, it is necessary to rigorously investigate the fundamental kinetic relationships.  4.1.2 Nitrogen pyrolysis – kinetic characterization Kinetic parameter recovery for multi-reaction systems is not a trivial matter.  Appropriate groundwork must be done before the process can begin if the results are to have any validity.  As explained in Chapter 2, we begin by first calculating the isoconversional activation energy trend using the method specified by Vyazovkin (2000).  However, before doing so, we must first be convinced that the suite of reactions occurring is consistent over the individual heating rate experiments.  A novel approach to this is to apply a morphological operator to both the time and reactivity scales to ensure that the effect of temperature is acceleratory and that no changes in mechanism occur.  The specifics of this test involve calculating the dilation/contraction operators for the time and reactivity scales which minimize the sum of 69    squared error between experiments which occur at constant composition, but varied thermal history.  The results of this test are shown in Figure 40.  Note that it was only necessary to apply a single dilation factor to each the time and reactivity axis to achieve the plots given in Figure 40.  This is a critical point as it not only indicates that the set of reactions are consistent across the temperature dimension, but across the composition dimension as well.  The magnitudes of the dilatational operators are provided in Table 9.  Table 9.  Summary of morphological operator magnitudes and their interpretation for Poplar wood pyrolysis Axis Dilation Operator Magnitude Interpretation  15°C/min 20°C/min 25°C/min  Time 1 130% 159% Reaction occurs 23.1%  and 37.1% faster at 20°C/min and 25°C/min, respectively, than at 15°C/min Reactivity 1 77% 62% Maximum rate is 23.0%  and 38.0% higher at 20°C/min and 25°C/min, respectively, than at 15°C/min .70     Figure 40.  Time and reactivity morphological operators applied to Poplar wood pyrolysis reactivity data (15°C/min- red, 20°C/min – brown, and 25°C/min - black)  Examination of Figure 40 reveals that reactions are consistent across heating rates, except for the 25°C/min heating rate experiments of the 8th and 12th samples.  The reason for this inconsistency is a slightly different starting temperature which shifts the time axis.  This is an easy phenomena to correct for, but in the interests of preserving the data in raw form, has not been done in the plot above.  71    4.1.2.1 Isoconversional activation energies Having verified that the experimental data are consistent, we apply the technique given by Vyazovkin (2000) to the experimental data and obtain the activation energy trends shown in Figure 41.  Scrutiny of the figure reveals a common feature, the data stretching either upwards or downwards at conversions less than ~2% or larger than ~88%.  Given the discussion and results presented in Section 4.1.2, these data may be misleading.    To diagnose whether or not the high and low conversion data are valid, we apply the technique outlined at the end of section 2.3.3.4, calculating the result of Equation 39 for each value of conversion.  This gives an error curve which indicates that the trends indeed follow those profiles, but the algorithm fails to determine a consistent activation energy which reconciles experimental observations at the extremes.  An example of this is shown in Figure 42; note that there were three experimental runs; thus, the theoretical minimum, indicating no error, is 6.    A potential explanation for these observations is that low conversion data exhibiting high or low activation energies are relatable to incomplete drying: low when water is trapped in large pore spaces, high when water is trapped in very small pore spaces.  As for the deviations at high conversions, this is interpreted as partial mineral volatilization since the temperature during this portion of the experiments was close to 900°C.72     Figure 41.  Isoconversional activation energy trends for Poplar wood pyrolysis 73     Figure 42.  Minimization profile for the Vyazovkin method using a 3 experiment input for sample 500 HAL30-2/TO-10-1  Recognizing the limitations in accuracy of the low and high conversion values, we follow the methodology laid out in Chapter 2 to interpret potential activation energies and constrain the Arrhenius factor based on the kinetic compensation effect.  With this done, the reaction mechanism, for each of the three reactions, was determined through an exhaustive trial and error procedure in which every permutation (repeats allowed) of mechanisms was tested for suitability.  A graphical representation of the results of this exercise is shown in Figure 43 in which “hotter” colors indicate better agreement with experimental data.  Observe that, because we have interpreted there to be three derivative peaks, the full solution space for three simultaneous reactions is a cube.  74        Model 1 Power Law 2 Power Law  3 Power Law  4 Power Law  5 1D diffusion  6 Mampel (first order)  7 Avrami-Erofeev (nucleation and growth) 8 Avrami-Erofeev (nucleation and growth) 9 Avrami-Erofeev (nucleation and growth) 10 3D diffusion  11 Contracting sphere  12 Contracting cylinder 13 2D diffusion  14 Avrami-Erofeev (nucleation and growth) 15 Avrami-Erofeev (nucleation and growth) 16 3D diffusion (Ginstling-Brounshtein) 17 3D diffusion (Zhuralev-Lesokin-Tempelman) 18 3D diffusion (Komatsu-Uemura) 19 Second Order Reaction  Figure 43.  Graphical illustration of exhaustive tests of potential reaction mechanism combinations. Hotter colors indicate better agreement with experimental results (chosen models lie within the cube’s interior).   Probing the results shown in Figure 43, a single set of mechanisms was able to simultaneously honor both the time-conversion trend, as well as the conversion-activation energy trend across the full suite of compositions tested.  This mechanism combination was that of 3D diffusion, first order reaction, and contracting cylinder; models 17, 6, and 12 in Table 2.  Combining the mechanism, isoconversional activation energy, and derivative peak information, a final description of Poplar pyrolysis kinetics was reached, this is presented in Table 10.   75    Table 10.  Summary of Poplar pyrolysis kinetic parameters Parameter Arrhenius Factor, 90% confidence interval Activation Energy (J/mol), 90% confidence interval Reaction fraction (approximate) Reaction 1 – 3D diffusion 2.74e12±39.6% -172310±0.580% 0 - 10% Reaction 2 – First Order Reaction 1.77e8±11.8 % -113500±0.881% 15 – 30% Reaction 3 – Contracting Cylinder 4.06e8±10.2% -135000±0.741% 60 – 80%  An example of the results of the model are shown in Figure 44, Figure 45, and Figure 46.  Note the small mismatch in conversion near 100% in Figure 44.  This is due to the residual effect of correcting the experimental data to remove the portion interpreted as mineral volatilization.  It is possible to match the trend exactly, as shown in Figure 47, Figure 48, and Figure 49, but this requires an activation energy inconsistent with the results of the measured isoconversional trend and is thus rejected.   Figure 44.  Conversion vs. Time curves for sample 500 HALS30-2/TO-10-1 Figure 45.  Activation energy vs. conversion curves for sample 500 HALS30-2/TO-10-1  76     Figure 46.  Reactivity vs. time curves for sample 500 HALS30-2/TO-10-1     Figure 47 Conversion vs. Time curves for sample 500 HALS30-2/TO-10-1 – incorrect activation energy trend Figure 48. Activation energy vs. conversion curves for sample 500 HALS30-2/TO-10-1 - incorrect  77     Figure 49. Reactivity vs. time curves for sample 500 HALS30-2/TO-10-1 – incorrect activation energy trend   4.1.3 Air gasification Following the same analysis procedure as for pyrolysis, we begin by plotting times to reach 90% conversion on a ternary diagram, shown in Figure 50, Figure 51, and Figure 52.  Though relative relationships are preserved across the temperature dimension, examination of the results indicates little correlation with composition.  Note that in the case of gasification, 0% conversion is defined as being 100% char.  78      Figure 50. Relative times to reach 90% conversion for Poplar air gasification. (“Cooler” colors indicate less time and therefore are more reactive) Heating Rate: 15°C/min Figure 51. Relative times to reach 90% conversion for Poplar air gasification. (“Cooler” colors indicate less time and therefore are more reactive) Heating Rate: 20°C/min   Figure 52. Relative times to reach 90% conversion for Poplar air gasification. (“Cooler” colors indicate less time and therefore are more reactive) Heating Rate:  25°C/min  Application of the morphological operator technique to test for reaction consistency leads to Figure 53.  Note that dilatational factors were calculated using sample 69 LNZK28-4/TO-15-24 as a basis.  Echoing the results of the ternary diagrams, we interpret the morphological results as indicating that reactions are not consistent across compositions.  This is likely due to the catalytic effect of trace inorganic species which vary from sample to sample.  For this reason, it is not useful to attempt to characterize the kinetics of these reactions in terms of reaction mechanisms since no broad generalizations can be made.  79     Figure 53. Time and reactivity morphological operators applied to Poplar wood air gasification reactivity data (15°C/min- red, 20°C/min – brown, and 25°C/min - black) 80    4.2 Recovery of Lodgepole pine wood kinetic parameters The structure of analysis for the Lodgepole pine samples mirrors that of the Poplar analysis.  Thus only results are discussed.  4.2.1 Nitrogen pyrolysis Examining the ternary diagrams given in Figure 54, Figure 55, and Figure 56, we see the consequences of random sampling.  The lack of variance in composition between samples makes it difficult to draw conclusions but, as in the case of Poplar pyrolysis, we again recognize that samples tend to be largely consistent across both temperature and composition.    Figure 54. Relative times to reach 90% conversion for Lodgepole pine pyrolysis.  (“Cooler” colors indicate higher reactivity) Heating Rate: 15°C/min Figure 55. Relative times to reach 90% conversion for Lodgepole pine pyrolysis.  (“Cooler” colors indicate higher reactivity) Heating Rate: 20°C/min  81     Figure 56. Relative times to reach 90% conversion for Lodgepole pine pyrolysis.  (“Cooler” colors indicate higher reactivity) Heating Rate: 25°C/min  To examine the data with higher resolution, the morphological procedure is applied using the parameters given in Table 11.  Note that the magnitude of each parameter is identical to that of the corresponding parameter used to adjust the Poplar pyrolysis data.  This, along with the visual similarity of curve shape, is clear evidence that the pyrolysis reactions are consistent between the Poplar and Lodgepole pine samples.  Results are shown in Figure 57.  The inconsistency seen for sample LP_45 can be explained as a slight offset in effective experiment start time; results are left uncorrected in this case to maintain the purity of the experimental data.  Note also the higher level of oscillations in the data relative to those of the Poplar samples.  This was caused by unknown factors and, when investigated, did not correlate to any experimental variables.  Finally, calculating kinetic parameters in an analogous fashion to that applied to the Poplar samples, it was found that all parameters lay within the confidence intervals found for the Poplar samples.      82    Table 11.  Summary of morphological operator magnitudes and their interpretation for Lodgepole pine wood pyrolysis Axis Dilation Operator Magnitude Interpretation  15°C/min 20°C/min 25°C/min  Time 1 130% 159% Reaction occurs at 23.1%  and 37.1% faster at 20°C/min and 25°C/min, respectively, than at 15°C/min Reactivity 1 77% 62% Maximum rate is 23.0%  and 38% higher at 20°C/min and 25°C/min, respectively, than at 15°C/min  83     Figure 57. Time and reactivity morphological operators applied to Lodgepole wood pyrolysis reactivity data (15°C/min - red, 20°C/min - brown, and 25°C/min - black)   84    4.2.2 Air gasification Under air gasification, sample composition variability is insufficient to draw any qualitative conclusions from the ternary diagrams shown in Figure 58, Figure 59, and Figure 60.  If we instead look at these data in tabular form, Table 12, we see that the reaction times vary widely - despite the similarities in composition.  This is further evidence that gasification kinetics are strongly influenced by factors other than those measured during the course of this study.  Application of the previously described morphological technique, given in Figure 61, agrees.    Figure 58. Relative times to reach 90% conversion for Lodgepole pine air gasification. (“Cooler” colors indicate higher reactivity) Heating Rate: 15°C/min Figure 59. Relative times to reach 90% conversion for Lodgepole pine air gasification. (“Cooler” colors indicate higher reactivity) Heating Rate:20°C/min    85     Figure 60. Relative times to reach 90% conversion for Lodgepole pine air gasification. (“Cooler” colors indicate higher reactivity) Heating Rate: 25°C/min  Table 12.  Tabular representation of time required to reach 90% conversion for Lodgepole pine gasification, heating rate: 20°C/min Sample Name Cellulose (mass %) Hemicellulose (mass %) Lignin (mass %) t@90% (min) 20°C/min LP_B40 44.9 24.0 28.6 6.167 LP_B45 46.2 25.2 28.6 9.042 LP_B51 45.5 24.8 29.7 8.775 LP_B52 45.6 25.6 29.7 8.088 LP_B54 48.1 23.9 28.4 7.217 LP_C32 45.0 25.5 28.8 7.833 LP_C40 42.5 27.3 28.8 7.508 LP_C41 45.6 26.4 29.4 9.117 LP_C42 44.6 25.4 30.3 7.342 LP_C44 45.9 26.1 28.2 6.300  86     Figure 61. Time and reactivity morphological operators applied to Lodgepole wood air gasification reactivity data (15°C/min- red, 20°C/min – brown, and 25°C/min - black) 87    4.3 Correlation with composition Combining results for Lodgepole pine and Poplar samples, we obtain Figure 62, Figure 63, and Figure 64; note that the color indicates relative times to reach 90% conversion.  Though they fail to account for cross correlative behavior, x-y plots are also helpful in visualizing the data and are provided in Figure 65, Figure 66, and Figure 67.  From these, we build the qualitative descriptors given in Table 13.   Figure 62. Ternary diagram showing all Lodgepole pine and Poplar pyrolysis results for experiments run at 15°C/min; colors indicate times to reach 90% conversion Figure 63 Ternary diagram showing all Lodgepole pine and Poplar pyrolysis results for experiments run at 20°C/min; colors indicate times to reach 90% conversion   Figure 64 Ternary diagram showing all Lodgepole pine and Poplar pyrolysis results for experiments run at 25°C/min; colors indicate times to reach 90% conversion  88      Figure 65. Time required to reach 90% conversion as a function of cellulose content Figure 66. Time required to reach 90% conversion as a function of hemicellulose content   Figure 67. Time required to reach 90% conversion as a function of lignin content  Table 13.  Relative impact of major wood components on reactivity at 90% conversion Major component Impact on reactivity behavior Increasing cellulose Increasing reactivity Increasing hemicellulose Slightly decreasing Increasing lignin Decreasing reactivity  Turning back to the kinetic model derived from the Poplar pyrolysis experiments, we have already determined the invariant parameters necessary to characterize pyrolysis.  The final step is to state the fashion in which the fraction of component involved in each reaction can be determined from knowledge of the cellulose, hemicellulose, and lignin fractions.  To do this, we 911131517192123252735 40 45 50 55 60 65 70Time to reach 90% Conversion (min)Cellulose Content (mass %)15°C/min Data20°C/min Data25°C/min Data911131517192123252715 17 19 21 23 25 27 29 31Time to reach 90% Conversion (min)Hemicellulose Content (mass %)15°C/min Data20°C/min Data25°C/min Data911131517192123252715 20 25 30 35Time to reach 90% Conversion (min)Lignin Content (mass %)15°C/min Data20°C/min Data25°C/min Data89    take the reaction fractions necessary to fit the experimental data and correlate them against the cellulose, lignin, and hemicellulose composition using multiple linear regression.  Doing so, and calculating 90% confidence intervals, we find the results given in Table 14:  Table 14.  Summary of multiple linear regression results for fitted reaction fractions vs. measured wood components  Reaction Mass fraction Cellulose Coefficient Mass fraction Hemicellulose Coefficient Mass fraction Lignin Coefficient Comments Rxn 1 Not Significant Not Significant Not Significant No correlation with any component Rxn 2 -0.12±0.08 0.72±0.28 0.67±0.26 88.4% variance explained Rxn 3 1.05±0.11 Not Significant 0.62±0.26 67.3% variance explained  𝑅𝑅𝑐𝑐𝑚𝑚 𝑓𝑓𝑚𝑚𝐵𝐵𝑐𝑐 1 =  𝑁𝑁𝐵𝐵 𝑐𝑐𝐵𝐵𝑚𝑚𝑚𝑚𝑐𝑐𝑐𝑐𝐵𝐵𝑑𝑑𝑖𝑖𝐵𝐵𝑚𝑚  45 𝑅𝑅𝑐𝑐𝑚𝑚 𝑓𝑓𝑚𝑚𝐵𝐵𝑐𝑐 2 =  −0.12 ∗ 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑚𝑚𝑐𝑐𝐵𝐵𝐵𝐵𝑐𝑐 + 0.72 ∗ ℎ𝑐𝑐𝑚𝑚𝑖𝑖𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑚𝑚𝑐𝑐𝐵𝐵𝐵𝐵𝑐𝑐 + 0.67 ∗ 𝑐𝑐𝑖𝑖𝑔𝑔𝑚𝑚𝑖𝑖𝑚𝑚  46 𝑅𝑅𝑐𝑐𝑚𝑚 𝑓𝑓𝑚𝑚𝐵𝐵𝑐𝑐 3 =  1.05 ∗ 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑚𝑚𝑐𝑐𝐵𝐵𝐵𝐵𝑐𝑐 + 0.62 ∗ 𝑐𝑐𝑖𝑖𝑔𝑔𝑚𝑚𝑖𝑖𝑚𝑚  47  By keeping in mind that the sum of mass fraction coefficients, for each of the three components of wood, must sum to unity across the three reactions, examination of Table 14 leads to several important conclusions: 1. Reaction 1, interpreted as a diffusion process, is not well correlated with any component.  2. The hemicellulose coefficient confidence interval very nearly includes unity for reaction two and is absent in reaction three.  It is quite likely that hemicellulose decomposes primarily through first order kinetics with residual variance captured by a diffusion process.  90    3. The cellulose coefficient in Reaction two is negative, but very near zero, while the cellulose coefficient in reaction three is nearly one.  It is likely that cellulose decomposes via kinetics which follow contracting cylinder kinetics, with additional variance explained by the diffusion process.  4. Lignin decomposes by a combination of first order and contracting cylinder kinetics.   5. It is interesting to note that the calculated reaction fraction for the diffusion reaction varied between 5-7%.  Recalling the results presented in Figure 23, the gas flow rate chosen eliminated ~95% of transport effects.  If experimental data integrity could be maintained at higher gas flows, perhaps the diffusion mechanism could be eliminated.   Combining the recovered compositional relationships with the kinetic parameters of Table 10 and the mechanisms found through Figure 43, a final description of pyrolysis kinetics for Poplar and Lodgepole pine is reached, given in Equation 48. 𝑃𝑃𝑦𝑦𝑚𝑚𝐵𝐵𝑐𝑐𝑦𝑦𝐵𝐵𝑖𝑖𝐵𝐵 𝑠𝑠𝑑𝑑𝑠𝑠𝑡𝑡= �2.74𝑐𝑐1012 ∗ 𝑐𝑐−172310𝑅𝑅𝑅𝑅 ∗32(1− 𝛼𝛼)23� �1− (1− 𝛼𝛼)13� �−1∗ 0.06�+ �1.77𝑐𝑐108 ∗𝑐𝑐−113500𝑅𝑅𝑅𝑅 ∗ 1− 𝛼𝛼 ∗ (−0.12 ∗ 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑚𝑚𝑐𝑐𝐵𝐵𝐵𝐵𝑐𝑐+ 0.72 ∗ ℎ𝑐𝑐𝑚𝑚𝑖𝑖𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑚𝑚𝑐𝑐𝐵𝐵𝐵𝐵𝑐𝑐 + 0.67 ∗ 𝑐𝑐𝑖𝑖𝑔𝑔𝑚𝑚𝑖𝑖𝑚𝑚)�+ �4.06𝑐𝑐108 ∗𝑐𝑐−135000𝑅𝑅𝑅𝑅 ∗ 2(1− 𝛼𝛼)12� ∗ (1.05 ∗ 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑚𝑚𝑐𝑐𝐵𝐵𝐵𝐵𝑐𝑐 + 0.62 ∗ 𝑐𝑐𝑖𝑖𝑔𝑔𝑚𝑚𝑖𝑖𝑚𝑚)�  48   91    Chapter 5. Overall conclusions and recommendations  5.1 Overall conclusions The primary purpose of this project was to determine if an optimal feedstock could be identified for the purpose of gasification.  Secondary objects included both determining if the Vyazovkin method could be used, in conjunction with the kinetic compensation effect, to uniquely determine kinetic parameters and also the full characterization of gasification process kinetics.  Determining optimal feedstock composition is a complex question and depends on many parameters, ranging from desired product distribution to cost of feedstock.  Despite these complications, this thesis brings the community one step closer to answering this question.  From solely a rate of pyrolysis perspective, it is desirable to limit lignin content in order to provide the highest reactivity and thus the largest conversion to char.  On the other hand, the kinetics of char conversion, under air gasification, were found to be largely uncorrelated to wood composition.  This is interpreted as air gasification kinetics being dominated by the catalytic activity of active minerals.  However, it should be observed that, in all the cases tested, air gasification occurs significantly faster than pyrolysis (2-3 times faster).  The technique described by Vyazovkin (2000) was found to aid tremendously in the determination of a unique kinetic triplet.  Without the additional information it provided, the conversion data could have been fitted by a multitude of parameter combinations using a variety of mechanisms.  Beyond this, it aided in providing good initial estimates of the final activation energies, making parameter recovery significantly easier.  This is clearly evident when comparing Figure 45 with Figure 48.  Furthermore, application of the Vyazovkin technique fit in well with the application of the kinetic compensation effect.  The combination of these two techniques were synergistic and, though they did not allow for a completely 92    deterministic approach, they narrowed the solution space to a dimensionality which could be explored in a practical timeframe.  The final goal of the study was to uniquely determine the kinetic parameters relevant in converting raw wood to syngas.  This was largely accomplished with the kinetic parameters given in Table 10 and the reaction fractions correlation with composition given in Table 14.  However, it was found that air gasification kinetic parameters could not be clearly identified from the data.  This is likely due to the influence of mineral species like potassium with catalytic properties.  Finally, it is worthwhile mentioning that this thesis has made four unique contributions to the field: 1. Interpretation of reaction fraction and the understanding of multiple, simultaneous, solid state reactions.  2. The approach to determining the Arrhenius factor using the kinetic compensation effect based on derivative peak information.  3. The inclusion of the isoconversional activation energy trend as a method of adding uniqueness to kinetic parameter/mechanism determination.  4. The technique of morphologic operators to verify reaction consistency across multiple dimensions (composition and temperature in this case).  5.2 Recommendations for future work Though the results presented in this thesis advance the understanding of gasification, and more broadly, solid state kinetics, there is always the need for work which could further explore the conclusions.  Recommendations for especially useful studies follow: 93    1. With the observation that pyrolysis kinetic behavior can be related to lignin, cellulose, and hemicellulose content for Poplar and Lodgepole pine wood, it would be useful to test whether or not these same correlations hold true for other biomass types, non-woody as well as other wood species.  2. The results of this thesis showed that air gasification cannot be explained purely through examination of cellulose, hemicellulose, and lignin proportions.  It was speculated that the catalytic influence of inorganic mineral matter is the cause of this.  Future experiments could be run with an excess of pre-prepared ash, of common origin, to investigate air gasification kinetics further.  3. One of the key suppositions made during this thesis was that pyrolysis could be viewed as a collection of reactions which acted to overcome the average bond energies of the molecules.  Given the activation energies put forward, it would be insightful to attempt to calculate bond enthalpies based on the structures of lignin, cellulose, and hemicellulose to determine whether a correlation exists.  4. The experimental results on which this thesis rests were based on the measure of true kinetics, free from the influences of bulk phase transport limitation.  However, because pyrolysis reactions are complex, and many intermediates are possible, further work should be done to investigate if kinetic parameters vary at sample sizes large enough for mass transport to play a significant role.  If these experiments indicate a change in activation energy, this could be due to interaction of said intermediates.  5. Though this work considered nitrogen pyrolysis and air gasification only, many other gas combinations are industrially important.  Further study should be undertaken by applying the same workflows to characterize reactions involving the other industrially important gasification atmospheres such as 𝐶𝐶𝑂𝑂, 𝐶𝐶𝑂𝑂2, and 𝐻𝐻2𝑂𝑂.   94    References Abramowitz M and Stegun I A Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables [Book]. - [s.l.] : Dover publications, 1964. Akahira T and Sunose T Method of determining activation deterioration constant of electrical insulating materials [Journal] // Res. Rep. Chiba Inst. Technol. (Sci. Technol.). - 1971. - Vol. 16. - pp. 22-31. Beall F C Differential calometric analysis of wood and wood components [Journal] // Wood Science and Technology. - 1971. - 3 : Vol. 5. - pp. 159-175. Burhenne L [et al.] The effect of the biomass components lignin, cellulose and hemicellulose on TGA and fixed bed pyrolysis [Journal] // Journal of Analytical and Applied Pyrolysis. - 2013. - Vol. 101. - pp. 177-184. Cai J and Liu R Kinetic analysis of solid-state reactions: a general empirical kinetic model [Journal] // Industrial Engineering Chemistry. - 2009. - Vol. 48. - pp. 3249-3253. DoITPoMS [Online] // University of Cambridge. - 02 05, 2014. - http://www.doitpoms.ac.uk/tlplib/wood/structure_wood_pt1.php. Friedman Kinetics of thermal degradation of char-forming plastics from thermogravimetry. Application to a phenolic plastic [Journal] // Journal of Polymer Science Part C: Polymer Symposia. - 1964. - 1 : Vol. 6. - pp. 183-195. Hemicellulose [Online] // Wikipedia. - 02 05, 2014. - http://en.wikipedia.org/wiki/Hemicellulose. Lagarias J C [et al.] Convergence properies of the Nelder-Mead simplex method in low dimensions [Journal]. - [s.l.] : Journal of optimization, 1998. - 1 : Vol. 9. - pp. 112-147. Ozawa T Kinetic analysis of derivative curves in thermal analysis [Journal] // Journal of thermal analysis. - 1970. - 3 : Vol. 2. - pp. 301-324. 95    Raveendran K, Ganesh A and Khilar K C Pyrolysis characteristics of biomass and biomass components [Journal] // Fuel. - 1996. - 8 : Vol. 75. - pp. 987-998. Roberts A Thirteenth Symposium (International) on Combustion [Conference] // The Combustion Institute. - 1971. - p. 893. Slopiecka K, Bartocci P and Fantozzi F Thermogravimetric analysis and kinetic study of poplar wood pyrolysis [Journal] // Applied Energy. - 2012. - Vol. 97. - pp. 491-497. Taiz L and Zeiger E A Companion to Plant Physiology [Online] // Plant Physiology Online. - 03 12, 2014. - http://5e.plantphys.net/article.php?id=24. Vyazovkin S [et al.] ICTAC Kinetics Committee recommendations for performing kinetic computations on thermal analysis data [Journal] // Thermochimica Acta. - 2011. - 1-2 : Vol. 520. - pp. 1-19. Vyazovkin S Modification of the integral isoconversional method to account for variation in the activation energy [Journal] // Journal of Computational Chemistry. - 2000. - 2 : Vol. 22. - pp. 178-183. Wall L A and Flynn J H General Treatment of the Thermogravimetry of Polymers [Journal] // Journal of Research of the National Bureau of Standards - A. Physics and Chemistry. - 1966. - 6 : Vol. 70A. - p. 487.    96    Appendices  Appendix A: Numerical methods employed Conversion Simulated conversion data was obtained by solving the general rate equation: 𝑠𝑠𝑑𝑑𝑠𝑠𝑡𝑡= 𝐴𝐴𝑐𝑐−𝐸𝐸𝐴𝐴𝑅𝑅𝑅𝑅 𝑓𝑓(𝛼𝛼)  1A This was done by first separating variables: ∫1𝑆𝑆(𝑑𝑑)𝑑𝑑𝛼𝛼 = 𝐴𝐴∫𝑐𝑐−𝐸𝐸𝐴𝐴𝑅𝑅𝑅𝑅(𝑖𝑖) 𝑑𝑑𝑑𝑑  2A The left hand side of 2A is easily determinable by integrating any of the potential 𝑓𝑓(𝛼𝛼) functions given in Table 1 or Table 2.  Thus, the left hand side of 2A can be denoted as 𝑔𝑔(𝛼𝛼) (shown in 3A) 𝑔𝑔(𝛼𝛼) = ∫1𝑆𝑆(𝑑𝑑)𝑑𝑑𝛼𝛼  3A  The right hand side of 2A was obtained by first recognizing that the temperature is a linear function of time, as given in 4A. 𝑇𝑇 = 𝐵𝐵𝑑𝑑 + 𝑇𝑇0  4A Re-writing, we obtain 5A: ∫1𝑆𝑆(𝑑𝑑)𝑑𝑑𝛼𝛼 = 𝐴𝐴∫𝑐𝑐−𝐸𝐸𝐴𝐴𝑅𝑅(𝑖𝑖𝑖𝑖+𝑅𝑅0) 𝑑𝑑𝑑𝑑  5A This may then be solved to yield the form given in 6A. 𝐴𝐴∫𝑐𝑐−𝐸𝐸𝐴𝐴𝑅𝑅(𝑖𝑖𝑖𝑖+𝑅𝑅0)𝑑𝑑𝑑𝑑 = 𝐴𝐴�𝑆𝑆−𝐸𝐸𝐴𝐴𝑅𝑅(𝑖𝑖𝑖𝑖+𝑅𝑅0)∗𝑅𝑅∗(𝑆𝑆𝑡𝑡+𝑅𝑅0)+𝐸𝐸𝑖𝑖�𝑆𝑆−𝐸𝐸𝐴𝐴𝑅𝑅(𝑖𝑖𝑖𝑖+𝑅𝑅0)�𝐸𝐸𝐴𝐴𝑆𝑆𝑅𝑅�  6A 97    Where 𝐸𝐸𝑆𝑆 is the exponential integral, defined as: 𝐸𝐸𝑆𝑆(𝑂𝑂) = −∫𝑆𝑆−𝑖𝑖𝑡𝑡∞−𝑧𝑧𝑑𝑑𝑑𝑑   7A The LHS and RHS sides, once converted, may then be reconciled through bisection - iterating on conversion.  Reactivity The reactivity, or the time derivative of conversion, was calculated using the first order backwards difference formula given by 8A. 𝑓𝑓′ =𝑆𝑆(𝑥𝑥𝑖𝑖)−𝑆𝑆(𝑥𝑥𝑖𝑖−1)ℎ  8A where h is the step size.  Note that up to a 4th order centered difference scheme was also tried for the experimental data; this did not improve the results in a meaningful way.  Solution of Equation 39 Numerical integration were performed using 64 point Gaussian quadrature using the typical definition given in 9A and a weight of 1.  Relevant weights and abscissae were taken from Abramowitz, et al. (1964). ∫ 𝑊𝑊(𝑐𝑐)𝑓𝑓(𝑐𝑐)𝑑𝑑𝑐𝑐 ≅ ∑ 𝑤𝑤𝑆𝑆𝑓𝑓(𝑐𝑐𝑆𝑆)𝑁𝑁𝑆𝑆=1𝑜𝑜𝑆𝑆  9A   The derivative-free minimization of Equation 39 was accomplished using the Nelder-Mead algorithm as described by Lagarias, et al. (1998). 98  

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