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Investigations on oil shale particle reactions Lisbôa, Antonio Carlos Luz 1997

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INVESTIGATIONS ON OIL SHALE PARTICLE REACTIONS By Antonio Carlos Luz Lisboa B. Sc. (Chemical Engineering) Universidade Federal do Rio de Janeiro M . Sc. (Chemical Engineering) Universidade Federal do Rio de Janeiro A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF D O C T O R OF P H I L O S O P H Y in T H E FACULTY OF GRADUATE STUDIES CHEMICAL & BIO-RESOURCE ENGINEERING We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA October 1997 © Antonio Carlos Luz Lisboa, 1997 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Chemical & Bio-Resource Engineering The University of British Columbia 2216 Main Mall Vancouver,BC Canada V6T 1Z4 Date: Abstract Oil shale research and development has grown in the shadow of the petroleum industry. The uncertainty of petroleum prices, its growing worldwide consumption and limited availability have, motivated many oil shale rich countries to investigate means to produce and use shale oil as an alternative. On the other hand, high shale oil costs, its processing complexities and relatively stable petroleum prices have hampered the establishment of the shale oil industry. Oil is recovered from shale via endothermic reactions, heat for which is generated by combustion of the residual carbon in the spent shale. Oil shale pyrolysers and combustors have generally been designed on an empirical basis. The objective of this work was to produce working mathematical models of raw shale pyrolysis and spent shale combustion, adequate to describe the mechanism by which these reactions occur within oil shale particles, and to investigate the parameters involved. Among these, the most relevant and difficult to obtain are the kinetic ones. Verified models for single particles can then be used to describe oil shale particle reactions in any reactor configuration. A three-dimensional model was developed to describe the transient temperature profile within a cubic shaped shale particle. Also a model for shale devolatilization is presented, based on an unreacted core mechanism. Both models are especially apt for large particles, of the type used in moving bed reactors. A thorough investigation was conducted about the equipment and methods used to obtain pyrolysis kinetic parameters. A standard thermogravimetric apparatus was used to generate these data for two shales: New Brunswick shale, and shale from the Irati For-mation in Brazil. The potential of a first order—on kerogen concentration—rate equation to represent shale devolatilization was assessed. 11 A one-dimensional model was developed to describe the transient temperature profile and carbon and oxygen concentration within a particle of spent shale undergoing com-bustion. The model assumed that oxygen was able to access any part of the particle's interior. Kinetic parameters for shale combustion were also obtained by thermogravhnetry using Irati shale. The first order dependence of the combustion process ou oxygen concentration was confirmed, and kinetic parameters as a function of temperature were extracted from the results. The models were solved using the method of lines, a standard numerical method for solving sets of parabolic partial differential equations. It was implemented in conjunction with the finite difference method. Models for larger particles were verified by heating and devolatilization experiments with 1.3 cm wide particles suspended in a tube furnace. Most of the experimental work addressed two different shales; one from New Bruns-wick, Canada, and the other from the Irati Formation, in Brazil. 111 Table of Contents Abstract ii List of Tables viii List of Figures x Acknowledgment xv 1 Introduction 1 2 Objectives 6 3 Oil Shale Characterization 8 3.1 Oil Shale Physical Structure 10 4 Thermogravimetry Technique 14 4.1 Apparatus and Methods 14 4.2 Non-isothermal Thermogravimetry . . / 18 4.3 Analysis of TGA Operating Conditions 20 4.3.1 Reliability of TGA results 20 4.3.2 Gas flow rate 22 4.3.3 Gas purity 23 4.3.4 Gas nature 24 4.3.5 Particle size 24 4.3.6 Sample weight 26 i v 4.3.7 Conclusions - thermogravimetry technique 26 5 Raw Shale Particle Devolatilization Modelling 30 5.1 Previous Models 32 5.1.1 Model 1 32 5.1.2 Model 2 33 5.1.3 Model 3 35 5.1.4 Analysis of models .' 37 5.2 Characteristics of an Ideal Model 38 5.2.1 Mechanism 39 5.2.2 System of coordinates 39 5.2.3 Parameter evaluation 40 5.3 Proposed Model 40 5.3.1 Heat transfer modelling 40 5.3.2 Mass loss modelling 59 6 Kinetic Parameters for Devolatilization 71 6.1 Reaction Mechanism 74 6.2 Reaction Order 76 6.3 Isothermal TGA Tests 77 6.4 Non-isothermal TGA Tests : 87 6.4.1 Arrhenius equation method 89 6.4.2 Coats and Redfern method 95 6.4.3 Freeman and Carroll method 99 6.4.4 Integral method . . 100 6.4.5 Chen and Nuttall method 105 6.4.6 Friedman method 106 v 6.4.7 Summary of results 110 7 Spent Shale Particle Combustion Modelling 117 7.1 Previous Models 120 7.1.1 Model 1 120 7.1.2 Model 2 122 7.1.3 Analysis of models 125 7.2 Characteristics of an Ideal Model 128 7.3 Proposed Model 128 8 Kinetic Parameters for Combustion 136 8.1 Reaction Order 136 8.2 Isothermal TGA Tests 137 9 Conclusions and Recommendations 143 Nomenclature 145 Bibliography 148 A FORTRAN programs 154 A . l Temperature profile in a rectangular parallelepiped 154 A.2 Pyrolysis modelling 160 A.3 Oxidation modelling 163 B Experimental data for metallic and shale particles. 171 C Non-isothermal TGA. Expressions to obtain kinetic parameters 182 D Rate constant by the least squares method 187 v i E Experimental data for TGA F Polynomial approximations for non-isothermal T G A data G Implementation of the numerical method of lines H Pyrolysis modelling results v i i List of Tables 3.1 Oil shale analysis 9 3.2 New Brunswick oil shale analysis (wt%) 10 4.1 TGA conditions for analyses cited in Section 4.3 29 5.1 Oil shale pyrolysis parameters 41 5.2 304 stainless steel properties 49 5.3 Irati oil shale properties 55 5.4 Expected Biot numbers in gas-solid reactors 59 6.1 Sizes and amounts of particles in TGA analyses 73 6.2 Comparison of pyrolysis rate constants 83 6.3 TGA sample weight (%) at 540 °C 91 6.4 Pyrolysis kinetic parameters from Arrhenius type equation 91 6.5 Pyrolysis kinetic parameters from Coats and Redfern equation 98 6.6 Pyrolysis kinetic parameters from Freeman and Carroll equation 100 6.7 Pyrolysis kinetic parameters from Integral equation 105 6.8 Pyrolysis kinetic parameters from Chen and Nuttall equation 110 6.9 Pyrolysis kinetic parameters from Friedman equation I l l 6.10 Energy of activation for Irati shale I l l 6.11 Frequency factor for Irati shale 112 6.12 Energy of activation for New Brunswick shale 113 6.13 Frequency factor for New Brunswick shale 115 viii E. l Actual heating rates 3 (°C/min) for non-isothermal TGA 189 E.2 Sample weight (mg) for non-isothermal pyrolysis TGA 190 E.3 Sample weight W (%) versus time t (minutes) for isothermal oxidation TGA at different temperatures (°C) 200 E.4 Sample weight W (%) versus time t (minutes) for isothermal oxidation TGA at different oxygen concentrations (%) 201 G. l Numol numerical results vs. analytical ones 210 H. l Influence of Biot number on conversion times 213 IX List of Figures 1.1 Fossil fuel compositions 2 3.1 Devolatilization of oil shale 11 3.2 Sketch of an oil shale particle 12 4.1 Sketch of a TGA system 15 4.2 Irati shale TGA pyrolysis 18 4.3 Reliability of TGA results 21 4.4 Influence of gas flow rate on TGA results 23 4.5 Influence of oxygen concentration on TGA results 25 4.6 Influence of gas nature on TGA results 25 4.7 Influence of particle size on TGA results 27 4.8 TGA results for single large particles 27 4.9 Influence of sample weight on TGA results 28 5.1 Definition of the coordinate system 42 5.2 Experimental apparatus 44 5.3 Detail of reactor top for temperature measurements 45 5.4 Detail of thermocouples position and particles 45 5.5 Relation between radiation and convection 46 5.6 Significance of radiation at low heat transfer coefficients 47 5.7 Metal particle temperature profiles. Tw = 350 °C. Q = 1.13 m'^/hr 50 5.8 Metal particle temperature profiles. Tw = 350 °C. Q = 1.64 m 3/hr 50 x 5.9 Metal particle temperature profiles. Tw = 450 °C. Q = 0.59 m 3/hr 51 5.10 Sketch of fluidized bed for temperature measurements 52 5.11 Metal particle temperature profiles in a fluidized bed 52 o 5.12 Metal particle temperature profiles. Variable Tw. Q = 1.13m'/hr 54 5.13 Metal particle temperature profiles. Variable Tw. Q = 0.59 m' /hi 54 5.14 Shale particle temperature profiles. Variable Tw. Q — 0.59 m' /hr 56 5.15 Shale particle temperature profiles. Variable Tw. Q = 0.31 m'^ /hr 57 5.16 New Brunswick raw shale SEM picture (400x) 61 5.17 Irati spent shale SEM picture (50x) 61 5.18 Irati raw shale SEM picture (lOOx) 62 5.19 Irati spent shale SEM picture (lOOx) 62 5.20 Irati raw shale SEM picture (200x) 63 5.21 Irati spent shale SEM picture (200x) 63 5.22 Irati raw shale SEM picture (600x) 64 5.23 Irati spent shale SEM picture (600x) 64 5.24 Irati raw shale SEM picture (lOOOx) 65 5.25 Irati spent shale SEM picture (lOOOx) 65 5.26 Reactor top for particle weight measurements 66 5.27 Shale particle temperature and weight profiles. Q = 0.59 m'^/hr 67 5.28 Shale particle temperature and weight profiles. Q = 0.31 m'^ /hr 67 5.29 Particle pyrolysis modelling 68 5.30 Pyrolysis modelling results 70 6.1 Irati shale isothermal TGA (8 = 100 °C/min) 77 6.2 Irati shale TGA pyrolysis. Low temperatures 79 6.3 Irati shale TGA pyrolysis. High temperatures 80 xi 6.4 Influence of heating rate on reaction rate 81 6.5 Pyrolysis rate constants from isothermal TGA 82 6.6 Pyrolysis rate constants at 500 °C 84 6.7 New Brunswick shale isothermal TGA 86 6.8 Irati shale non-isothermal TGA pyrolysis - 50 °C/min 88 6.9 Irati shale non-isothermal TGA pyrolysis - 20 °C/min 88 6.10 Irati shale non-isothermal TGA pyrolysis - 10 °C/min 89 6.11 Irati shale non-isothermal TGA pyrolysis - 5 °C/min 89 6.12 New Brunswick shale non-isothermal TGA pyrolysis - 50 °C/min 90 6.13 New Brunswick shale non-isothermal TGA pyrolysis - 20 °C/min 90 6.14 New Brunswick shale non-isothermal TGA pyrolysis - 10 °C/min 92 6.15 New Brunswick shale non-isothermal TGA pyrolysis - 5 °C/min 92 6.16 Irati shale pyrolysis kinetic data plotted according to the Arrhenius type equation 93 6.17 New Brunswick shale pyrolysis kinetic data plotted according to the Ar-rhenius type equation 93 6.18 Pyrolysis kinetic data plotted according to the Coats and Redfern equation. (a): Irati Shale, (b): New Brunswick shale 96 6.19 Irati shale pyrolysis kinetic data plotted according to the Coats and Redfern equation 97 6.20 New Brunswick shale pyrolysis kinetic data plotted according to the Coats and Redfern equation 97 6.21 Irati shale pyrolysis kinetic data plotted according to the Freeman and Carroll equation 101 6.22 New Brunswick shale pyrolysis kinetic data plotted according to the Free-man and Carroll equation 101 xii 6.23 Irati shale pyrolysis kinetic data plotted according to the Integral equation - high temperature 103 6.24 New Brunswick shale pyrolysis kinetic data plotted according to the Inte-gral equation - high temperature 103 6.25 Irati shale pyrolysis kinetic data plotted according to the Integral equation - low temperature 104 6.26 New Brunswick shale pyrolysis kinetic data plotted according to the Inte-gral equation - low temperature 104 6.27 Irati shale pyrolysis kinetic data plotted according to the Chen and Nuttall equation - high temperature range 107 6.28 New Brunswick shale pyrolysis kinetic data plotted according to the Chen and Nuttall equation - high temperature range 107 6.29 Irati shale pyrolysis kinetic data plotted according to the Chen and Nuttall equation - low temperature range 108 6.30 New Brunswick shale pyrolysis kinetic data plotted according to the Chen and Nuttall equation - low temperature range • 108 6.31 Irati shale pyrolysis kinetic data plotted according to the Friedman equation. 109 6.32 New Brunswick shale pyrolysis kinetic delta plotted according to the Fried-man equation 109 6.33 High temperature pyrolysis kinetic data according to KCE theory 114 6.34 Low temperature pyrolysis kinetic data according to KCE theory 114 7.1 Effects of shale CaC0 3 content on TGA 119 7.2 Solid Conversion by Combustion Models 126 7.3 Spent shale temperature profiles during shale combustion 130 7.4 Spent shale temperature profiles during shale combustion 130 xiii 7.5 Particle combustion modelling 132 7.6 Influence of heat transfer coefficient on particle temperature in oil shale oxidation 134 7.7 Influence of oxygen diffusivity on oxidation conversion 135 7.8 Influence of reactor temperature on oxidation conversion 135 8.1 Isothermal TGA for oxidation at different temperatures 138 8.2 Experimental combustion TGA results at different temperatures 139 8.3 Experimental combustion TGA results at different temperatures 140 8.4 Experimental combustion TGA results at different oxygen concentrations (650 °C) 141 8.5 Experimental combustion TGA results (650 °C) 142 G. l Steps performed by the numerical method of lines 208 H. l Influence of particle size on core radius 215 H.2 Influence of particle size on conversion 215 H.3 Influence of particle size on particle temperatures 216 H.4 Influence of heat transfer coefficient on core radius 217 H.5 Influence of heat transfer coefficient on conversion. . . 217 H.6 Influence of heat transfer coefficient on particle temperatures 218 H.7 Influence of energy of activation on core radius 219 H.8 Influence of energy of activation on conversion 219 H.9 Influence of energy of activation on particle temperatures 220 x i v Acknowledgment I acknowledge with thanks the advice and assistance of my supervisor, Professor A.P. Watkinson, for his guidance and interest in this research. I am indebted to many others in the Chemical Engineering Department at UBC: Shelagh, Helsa and Lori from the Office Staff; John, Robert, Peter, Chris and Alex from the Shop; and Horace from the Stores. I would like to thank Dr. B. Bowen and Dr. K. Smith, from the Chemical Enginnering Department, and Dr. P. Barr, from Metals and Materials Engineering Department, who contributed with comments during my work, and critical reading of the first manuscript. I am grateful to the National Science and Engineering Research Council of Canada, which provided financial support in the form of research grants, to the University of British Columbia which provided financial support in the form of university graduate fellowships and to the Brazilian National Research Council (CNPq), which provided fi-nancial support in the form of scholarship. Many contributed in different ways to the completion of this thesis: Mrs. Goodlake, Da. Fiilvia, Dusko Posarac, Armindo and Alessandra, James and Suzanne, Philip Yue, Paul and Carmen, Elod and Cristina, Antonio Alfredo and Adriana, Keith Redford, Marcello and Sonia, Maurice and Teresa, Carlucci and Cristina, Meinardo Boizan, Ri-cardo and Cristina, Tim and Thea, Luis and Susana, Charles and Megan, Guilherme and Sandra, Isaac Hodgson, Antonio Emilio and Angela, Scott and Chris, Washington and Catarina, Calvin and Anna, Cristina Moreira, Doug and Jill, Carlos and Margarida, Rajagopal, Vivaldo and Dorte, Dilmar and Zilda, Rinaldo and Amelia. I owe a special debt to my wife, Claudia, my son Eduardo and daughters Alice and Carolina, who were a constant source of love. xv Chapter 1 Introduction Oil shale has been defined by American Society for Testing Materials as: A compact rock of sedimentary origin, with, ash content of more than 33% and containing organic matter that yields oil when destructively distilled but not appreciably when extracted with the ordinary solvents for petroleum (D288 - 47, 1947). The definition highlights some specifics of oil shale that makes it different from other carbonaceous materials like coal, peat and petroleum. Figure 1.1 shows some typical compositions. The shale band includes volatile organics between 10 and 60 %, fixed carbon around 10%, and ash content 30 - 90 %. A typical shale would contain 13 % organic carbon, 5% hydrogen and 87% mineral matter. Despite different quantitative estimates, the energy present in oil shale is generally accepted to be greater than that present in coal and petroleum on a worldwide basis. It is also true that the cost of its energy is the highest, which explains its almost nil contribution to the world energy matrix today. Nevertheless, the non-renewable character and uncertain prices of petroleum have both encouraged the study of oil shale. These studies have been carried out mainly in some countries with either a large reserve of oil shale or with low reserves of petroleum or with technology development expertise or combination of them. In the past, oil shale was largely processed in small scale retorts in China, Estonia and Scotland. Considering the last two decades, significant developments were accomplished in Brazil, USA, Canada, Australia, Israel and Japan. Most studies on oil shale reactions have been on pyrolysis. In this process oil shale 1 Chapter 1. Introduction 2 volatile matter Figure 1.1: Fossil fuel compositions. Petroleum band would lie very close to 100% volatile matter. is heated to ~ 500 °(J, which releases oil and leaves a carbonaceous residue in the solid matrix. The carbon conversion is usually not more than 50% due to lack of hydrogen. Therefore the spent shale remains suitable for combustion, which in combined systems can produce more than enough heat for the endothermic pyrolysis reactions [1]. Most recent investigations followed this route, as described later. Pyrolysis is the reaction by which the solid organic matter within shales is converted to oil, gas and char. The organic matter in shale is present as a complex combination of carbon, hydrogen, sulfur and oxygen, named kerogen, which cannot be extracted with organic solvents. A good introduction to kerogen structure and classification is given by Speight [2]. Upon heating, kerogen is initially converted to bitumen and this to oil and gas. A small amount of bound water is also released. This second step is named devolatilization, because this is the step in which matter leaves the solid matrix, making Chapter 1. Introduction 3 products available. Typical shales produce about 8% weight oil, 4% weight gas and 2% weight bound water. But the total amount of these volatiles can reach 60% for some shales, as shown in Figure 1.1. Several gas-solid contacting techniques have been used to pyrolyze oil shale. They are: moving beds, fluidized beds, spouted beds and entrained beds. Maximum reactor temperatures are kept within the range of 450 °C and 520 °C. The pressure is atmospheric. The pyrolyzed shale is also called spent or retorted shale. It contains about 50% of the original carbon and almost no hydrogen. It is rich enough in organic matter to maintain combustion even without pre-heating of the air. Spent shale can be ignited at temperatures as low as 300 °C. Bubbling or circulating fluidized beds, spouted beds and entrained beds have been used to burn shale. In Japan, oil shale development was undertaken by the government-sponsored Japan Oil Shale Engineering Company (JOSECO), established in 1981 [3]. The results obtained with three bench scale plants led to the building of a very large 12.5 t/h pilot plant facility. The process comprises an upper moving bed retort from which the oil is recovered, and a lower moving bed in which spent shale is subsequently gasified. Both layers are in the same vessel. The gasified shale is finally burned in a bubbling fluidized bed. JOSECO claims to have invested US$130 million in this project. In the USA, UNOCAL tried for almost a decade to put on stream its 10,000 bbl/day shale oil plant built in 1981 - 1983 at a cost of US$650 million [4]. Oil shale is pyrolyzed in an upward moving bed retort. Apparently the plant never reached design production levels, and was closed in March 1991. CHEVRON developed its Staged Turbulent Bed Process [5], in which a retort operating under a staged turbulent bed mode is coupled with an entrained bed combustor. The hot ash from the combustor is recycled to the retort as a source of heat. In Canada the TACIUK process, initially developed for tar sands, has been successfully Chapter 1. Introduction 4 applied to oil shale [6]. In this process, oil shale is first pyrolyzed in a rotating drum and then burned in the annulus region between this drum and a concentric external drum. In Brazil a 260 t/h oil shale, 3,200 bpd shale oil, 11 m diameter retort industrial module plant of the PETROSIX process came on stream in December 1991. This followed 10 years of successful continuous operation of a 60 t/h oil shale, 800 bpd shale oil, 5.5 m diameter retort prototype plant by PETROBRAS [7]. The retort has a pyrolysis moving bed layer followed by a cooling moving bed layer beneath it. As PETROSIX employs a moving bed retort, which requires a narrow particle size distribution, a fraction of the mined and crushed particles, around 20%, has been returned unprocessed to the mine site. Some alternatives were investigated by PETROBRAS to process these "fines" (< 6 mm). The author participated in the following programs [8], each of which entailed the operation of a pilot plant: • pyrolysis with partial combustion in a spouted bed [9]. • pyrolysis in an entrained bed [10]. • pyrolysis in a bubbling fluidized bed (plant not owned by PETROBRAS). • combustion in a circulating fluidized bed [11]. Pyrolysis followed by retorted shale combustion seems to be the favoured approach for oil shale utilization, although some work was done on gasification in the past. All the work cited above can be classified as applied research and process development and most of it was conducted on an empirical basis. The wide range of choices concerning the gas-solid contacting technique both for pyrolysis and combustion is remarkable. Besides the applied research programs cited above, several fundamental research pro-grams were carried out in the past few decades. Most of that work was done independent of the general context of designing an oil shale plant. Therefore the literature shows a Chapter 1. Introduction 5 patchwork of fundamental data about a number of oil shales. Anyone interested in design-ing a process plant for a specific oil shale would probably have to resort to data available for other shales. A conclusion that can be drawn from this introduction is that there is not a clearly stated procedure to design a plant for a given shale from fundamental data. Such a procedure could involve the following steps: 1. Obtain information for a single particle undergoing the desired reactions (pyrolysis and combustion). Required data are kinetic parameters; oil shale properties like density, thermal conductivity and specific heat; and heats of reactions. Mathematical models adequate to describe the mechanism by which these reactions occur within the particle are also needed. 2. Obtain information for each gas-solid contacting technique. Required data are, for instance, heat and mass transfer coefficients, flows and temperature profiles within the reactor and particle residence time distributions. 3. Apply semi-empirical reactor models for scale-up. This study is concerned primarily with the first step suggested. Chapter 2 Objectives To date, the design of pilot/prototype/industrial plants for oil shale processing has had to be done in spite of the lack of basic information concerning either the oil shale param-eters or the gas-solid contacting technique parameters. This, together with the intrinsic difficulty in handling solids, frequently led to trial and error procedures and substantial departures from the design point at the time of experimentation with physical models. This work aims at reducing the gap in what is known concerning reactions of oil shale particles themselves. Stated briefly the objectives of this work are: • Develop a mathematical model for a raw shale particle undergoing pyrolysis and a spent shale particle undergoing combustion. • Develop procedures, experimental and analytical, to obtain the relevant parameters to verify the mathematical model. • Execute these procedures for typical oil shales. • Use the design parameters to predict behaviour in a controlled reactor, obtain experimental data and compare the results. After two introductory chapters which discuss the characteristics of oil shale and pro-vide an assessment of the thermogravimetric technique, the work proceeds with the de-velopment of theoretical models which describe the phenomena of heat and mass transfer with chemical reaction associated with the retorting and combustion processes for a single 6 Chapter 2. Objectives 7 particle. These models indicate the relevant parameters which are subsequently obtained either by experiment or by using data in the literature. Chapter 3 Oil Shale Characterization Two oil shales from different origins were used in this work. One came from New Brunswick, Canada, and the other from the Irati Formation in Brazil. Both shales have about the same oil content: 8%. They differ primarily in their carbonate contents: the shale from New Brunswick is richer. Oil shale compositions do not vary much within a given deposit, as long as represen-tative samples are considered. In more than 30 years of mining operations in Brazil, the only significant variation observed was a welcome increase in the average oil content from 7% to 9%. Otherwise the other components remained at their known concentrations. It is interesting to observe that a representative sample from the Irati Formation in Brazil comes from shales from the. two layers present in that formation. The upper layer is poorer (5% oil) and thicker (6 m) while the lower layer is richer (12% oil) and thinner (4 m). Therefore any observed change in the average oil content could come from changes in any of these sources, i.e., thickness or richness of the two layers. The presence of carbonates imposes an upper limiting temperature in oil shale pyrol-ysis processes, to avoid the endothermic reactions of carbonate decomposition that start at 550 °C. Most shales begin to volatilize at 300 °C, some light gases evolve at even 250 °C. A typical pyrolysis process occurs at temperatures up to 500 °C. Above that, the carbonate decomposition reaction competes with the pyrolysis reactions for the available heat. Table. 3.1 shows analyses for the two shales investigated. The New Brunswick shale 8 Chapter 3. Oil Shale Characterization 9 Table 3.1: Oil shale analysis. New Brunswick Irati ultimate analysis (wt%) carbon (organic) 10.6 12.89 carbon (mineral) 2.7 0.67 hydrogen 2.05 2.11 nitrogen 0.51 0.38 oxygen 6.82 2.15 sulphur 0.92 5.08 ash 73.78 76.72 ash analysb > (wt%) S i0 2 41.9 60.87 A1 2 0 3 10.4 13.62 F e 2 0 3 4.38 9.64 CaO 9.03 2.70 MgO 3.57 3.19 oil shale Fischer assay (wt%) oil 7.76 9.01 water 1.80 1.84 gas 1.14 2.09 residue 89.30 87.06 was the same as used in a previous work by Tarn [12]. The Fischer assay analysis for the New Brunswick shale was obtained from Salib et al. [13]. The Irati shale analysis was provided by Petrobras, which also supplied a second analysis for the New Brunswick shale, shown in Table 3.2. Chapter 3. Oil Shale Characterization 10 Table 3.2: New Brunswick oil shale analysis (wt%) as function of crushed particle size ranges 1-0.84 mm 0.84 - 0.71 mm carbon (total) 13.64 14.04 hydrogen 1.30 1.30 sulphur 0.30 0.30 nitrogen 0.48 0.48 ash 79.17 79.75 high heating value (cal/g) 1380 1401 3.1 Oil Shale Physical Structure The organic matter in oil shale, can be split into two approximately equal fractions: volatilizable material and fixed carbon. The former constitutes the organic matter that leaves the shale upon heating to 550 °C in an inert atmosphere; it is basically oil, gas and a small amount of bound water. This heating process can be followed in Figure 3.1, which shows for both New Brunswick and Irati shales a material loss of ~ 10%, which compares to the values of oil plus gas plus water from the Fischer assays in Table 3.1. The Fischer assay is a non-standard analysis in which 100 g of shale < 2.4 mm (8 mesh) are placed at the bottom of a closed vessel which is heated up to 500 °C in a nitrogen atmosphere in about 45 minutes and kept at that temperature for another 45 minutes. It is used to characterize different oil shales for oil and gas yields. The assay is taken as a reference in the shale industry and research institutions to determine the efficiency of oil shale pyrolysis processes: The remaining carbonaceous material constitutes the fixed carbon fraction. Upon further heating above 550 °C in an inert atmosphere this material will react with C O 2 evolved from carbonates, if present in oil shale. This process can be again followed in Chapter 3. Oil Shale Characterization 11 100 f*° 95 90 op ' 5 3 £ 85 80 75 • Irati • New Brunswick ft \ o 8 8 8 8. o* 0 <& 300 %m°% y y g y i ^ j i 200 900 800 700 600 500 § <u 400 § 100 0 20 40 60 80 100 120 Time (minutes) Figure 3.1: Devolatilization of Oil Shale. Circular symbols (open and closed) represent Irati oil shale analyses, square symbols (open and closed) represent New Brunswick's. At 107 minutes, sweep gas was switched from nitrogen to oxygen. Figure 3.1 which shows for Irati shale a loss of ~ 4 % that corresponds to C 0 2 evolution and reaction with carbon. For New Brunswick shale, which is richer in carbonates, this loss is ~ 12%. Both losses are consistent with material balances that consider the fixed carbon and carbonate contents listed in Table 3.1. At 110 minutes of the experiment depicted in Figure 3.1, the sweep gas was changed from nitrogen to oxygen. Irati shale still had some carbon remaining and experienced a loss of mass because of its combustion. On the other hand, New Brunswick shale had no loss at all, indicating that all fixed carbon had been consumed by C O 2 . Chapter 3. Oil Shale Characterization 12 It is important to know what happens to the oil shale structure during these heating processes. First of all there is no observed change of physical size. The ~ 20% loss of organic matter is not enough to break the mineral matrix and change the oil shale volume. A direct consequence is that voidage is increased. Second, a particle cannot be easily split, broken or flaked after being processed, even after combustion. This indicates that the organic matter is not concentrated in layers or lumps, but rather is finely distributed throughout the solid matrix. Considering these facts, a simple way to represent an oil shale particle is as a porous solid consisting mostly of an inert mineral matrix with lesser amounts of organic matter scattered throughout, as sketched in Figure 3.2. The evaporation of organic matter during the heating process seems to be the major cause for the appearance of pores, which are almost not perceptible in raw shale. Figure 3.2: Sketch of an oil shale particle Chapter 3. Oil Shale Characterization 13 The shale color also changes from light gray in its raw state to black after pyrolysis to light beige after combustion. Color thus provides a qualitative indication of how far the process has been carried out. Chapter 4 Thermogravimetry Technique Most of the work done on oil shale devolatilization kinetics employs thermogravimetric analysis techniques. Standard or lab constructed pieces of equipment are used to obtain sample weight versus time data along a programmed temperature curve. These data are subsequently processed according to one of many available methods to obtain the kinetic parameters, namely the energy of activation and frequency factor. The next sections discuss these apparatuses and methods. 4.1 Apparatus and Methods Thermogravimetric analyses (TGA) have been performed using standard [32, 33, 34, 35, 36] or lab constructed [27, 28, 29, 33, 36, 37, 38] apparatuses, which should be able to heat the sample according to a prescribed temperature curve and at the same time measure the sample and/or product weights. Figure 4.1 depicts the major features of a TGA system. The sample is placed in a metallic pan or in a wire basket. The pan is supported by a wire cage. A thin wire connects the cage to a scale. During a typical analysis, the sample is placed inside a furnace, with the bottom of the pan very close to a thermocouple. The whole setup is enclosed by a glass wall within which flows a chosen gas. For oil shale devolatilization, this gas must be an inert gas, such as nitrogen, helium or argon. The analysis should be made at such conditions that the overall reaction rate should be identical to the intrinsic kinetic rate, i.e., at conditions at which transport resistances are negligible. Transport resistances arise from heat and mass transfer processes both 14 Chapter 4. Thermogravimetry Technique 15 Figure 4.1: Sketch of a TGA system. inside and outside the sample particles. To meet this condition, it is advisable to use small amounts of particles having small sizes and to promote a continuous flow of gas through or around the sample. The TGA system used in this work was a Perkin-Elmer Model TGS-2 Thermogravi-metric Analyzer. The above cited requirements were met as long as sample weights were less than 50 mg and heating rates less than 150 °C/min. The data were recorded by a data-logging system, that provided listings of sample weight and temperature with time. Most analyses were performed with ultrahigh purity nitrogen, but some were with helium. Chapter 4. Thermogravimetry Technique 16 Sample weights were kept between 10 and 30 mg. Heating rates above 100 °C/min were avoided and gas flow rates were around 100 ml/min. The temperature curve of a TGA analysis can be set to follow either a constant tem-perature program or a constant heating rate program. Most of the initial published work done on oil shale, which used non-standard TGA systems, had to rely on the constant tem-perature method, because those systems could not provide a controlled constant heating rate. The procedure to obtain the kinetic parameters from isothermal kinetic data is straight-forward. Supposing that the reaction is first order with respect to kerogen or volatile matter content, then If the conversion X is defined as the ratio between volatilized and volatilisable material, i.e., C — C X = ^!—^L (4.2) then substitution of Equation 4.2 into Equation 4.1 leads to: = * ( ! " * ) (4-3) where. X = 0 at t = 0 Integration of Equation 4.3 subject to the given initial condition produces: - l n ( l - X) = kt Thus a plot of — ln(l — X) versus t should produce a straight line whose slope is the reaction rate constant k for that temperature. If the experiment is performed for a set of Chapter 4. Thermogravimetry Technique 17 different but constant temperatures, one would obtain a set of points relating the k values to the temperature values. If the reaction rate constant follows the Arrhenius equation: k = kQ e-E'RT (4.4) then a plot of In A; versus 1/T will produce a straight line with the frequency factor kD as the intercept and the ratio —E/R as the slope. This was the procedure followed by Hubbard and Robinson [27], Diricco and Barrick [28], Allred [29] and Campbell et al. [33]. However, a flaw was detected early on in this approach, because in the heat-up period some weight loss occurs which, if ignored, introduces errors in the interpretation of the results. Thus, the data from the pioneering work of Hubbard and Robinson [27] was later reinterpreted by Braun and Rothman [30], who fitted the data to a two-step mechanism that took into account the material loss in the transient heat-up period. The difficulty of using the isothermal method is illustrated by examining Figure 4.2. The data were ob-tained at four constant heating rates, up to temperatures of 900 °C. Figure 4.2a indicates the different times the samples took to reach 900 °C and a final weight of about 86% in each case. Figure 4.2b indicates that at a given pyrolysis temperature, the remaining weight, or the lost weight, is a function of the heating rate of the experiment. In isother-mal TGA, the amount and certainly the quality of the organic matter to be devolatilized at a given temperature depends on the heating rate used to reach that temperature. For instance, at a temperature of 460 °C, the weight loss is 8% for a heating rate of 5 °C/min but only 3.5% for a heating rate of 50 °C/min. Presumably the materials left to be py-rolyzed at those temperatures are also different in quality. This could lead to inconsistent results. To correct the inherent difficulties of the isothermal method, the non-isothermal anal-ysis method, which employs constant heating rates, was devised to obtain the kinetic Chapter 4. Thermogravimetry Technique 18 100 p 98 r 96 r 94 1 light ( 92 r 90 : 88 -86 r 84 ^ 5°C/min 10°C/min 20T/min 50T/min 100 98 96 94 1 ) 92 90 88 86 84 —5°C/min + 10°C/min -20°C/min —50°C/min 0 20 40 60 80 100 120 140 160 Time (minutes) 100 200 300 400 500 600 700 800 900 Temperature (°C) Figure 4.2: Irati shale TGA pyrolysis. Curves indicate that weight loss up to a fixed temperature is a function of heating rate. parameters. This method is the subject of the next section. For the reasons explained above, it constitutes the preferred method to obtain kinetic parameters in this work. 4.2 Non-isothermal Thermogravimetry Non-isothermal TGA evolved from the fact that isothermal TGA faces a heat-up period, during which temperature is increased up to a desired value, and volatile material is lost before the testing conditions are reached. As the isothermal method requires several analyses, each one at a different temperature, it is clear that each analysis will process different information, corresponding to the material that was left after the heat-up period. To correct that, the non-isothermal technique was developed. The TGA analysis is performed at a constant heating rate 3, defined as dT 8 dt (4.5) where 3 is usually in the range of 1 - 60 C/min. During the analysis time, the sample Chapter 4. Thermogravimetry Technique 19 weight and temperature are recorded. To obtain the kinetic parameters, Equation 4.3 is modified by the introduction of Equations 4.4 and 4.5 to give: ^ = je-E,RF(l-X) (4-6) Equation 4.6 is the basic equation from which several final equations are derived, which permit the calculation of the kinetic parameters. These derivations are fully disclosed in Appendix C. These equations are: • Equation of Friedman [45] (applied to oil shale by Shih and Sohn [37] and Yang and Sohn [38]) : AY P. 1 (4.7) m ^ = - l n [ A ! o ( l - X ) ] + | i • Equation of Arrhenius type (applied by Rajeshwar [34] and Shih and Sohn [37]): (4.8) , K El ~ l Q J + R f Equation of Freeman and Carroll [46] (applied by Rajeshwar [34]): Aln(dAVdT) _ _ [ + E A( 1/T) (4.9) Aln ( l -X) RA\n(l-X) • Equation of Coats and Redfern [47] (applied by Rajeshwar [34], Thakur and Nuttall Jr. [35] and Capudi [36]): In ln(l - X) T 2 , k0R ( 2RT\ E 1 - In ( 1 — 1 + BE V E J RT • Integral method (applied by Shih and Sohn [37] and Yang and Sohn [38]): -/31n(l -X)' (4.10) RT2 + l n ( l - — J = - l n - + E • Differential method (applied by Shih and Sohn [37]): E _ fco RT2 -E/RT (i RT BE ' V dX , — = k0exp at E RT 2RT\ ~E~) (4.11) (4.12) Chapter 4. Thermogravimetry Technique 20 Equation of van Heek et al. [48, 49] (applied by Campbell et al. [33] and Wang and Noble [39]): AY r F. h RT2 1 (4.13) dX , — = k0exp dt E _ k0R  E/RT RT BE • Equation of Chen and Nuttall [50] (applied by Thakur and Nuttall Jr. [35] and Capudi [36]): - I n E + 2RT , 1 In 1 -X 4.3 Analysis of T G A Operating Conditions This section discusses the reaction of oil shale TGA results to changes in the relevant parameters of that analysis, namely gas flow rate, purity and nature; shale sample amount and particle size. Initially the reliability of the results is investigated. All analyses cited in this section were made with New Brunswick shale, and were selected from the initial exploratory runs which provided the necessary familiarity with the TGA equipment in order to conduct this study. The experimental conditions are disclosed in Table 4.1 at the end of this section. Results are presented in the figures and discussed in the text throughout this section. 4.3.1 Reliability of T G A results Reliability is taken here to mean how close a given result is from the average value of a set of results obtained under the same experimental conditions. This information helps one to define how many analyses should be made at a prescribed condition in order to get a representative result. Figure 4.3 shows results from five samples which were submitted to the same test conditions, as described in Table 4.1. Four samples followed an almost identical curve and ended at asymptotic weight (%) values between 76.5 - 78.0, with an average of Chapter 4. Thermogravimetry Technique 21 Time (minutes) Figure 4.3: Reliability of TGA results. 77.0, while sample samll followed a different, although similar, curve and yielded a final weight (%) value of 74.2. This different result is credited to experimental errors and not to sampling procedures, because all samples came from a well homogenized mixture of about 60 g of < 0.25 mm shale. The small dispersion between samples saml2, saml3, saml6 and saml7 is caused by the inherent differences in composition among shale particles, as well as the reproducibility of the TGA. A statistically representative result for the 60 g amount of shale would require more analysis than four to refine the average obtained. If one considers a mine site, adequate sampling procedures should be taken in order to get representative results; larger variance values are expected as the amount of shale to be sampled increases. The results obtained are typical of what was encountered during the progress of this study: very rarely did repeated values not match. But the situation Chapter 4. Thermogravimetry Technique 22 posed by sample samll in Figure 4.3 suggests that to obtain a reliable result, duplicate experiments should be run. Many other test results at conditions similar to those in Figure 4.3 confirm that the anomalous result of experiment samll is atypical. The above discussion leads one to conclude that oil shale TGA results are generally reliable, but a second analysis is necessary to confirm a result. 4.3.2 Gas flow rate During a TGA analysis, sweep gas continuously flows throughout the experimental setup in order to carry away any gaseous product, thereby avoiding an autogenous atmosphere around the solid sample that could affect the reaction rate. As well, the gas flow rate could influence the rate of heat transfer from the furnace wall to the solid sample. Figure 4.4 shows the results of two analyses conducted at extreme gas flow rates, but at otherwise similar experimental conditions, as listed in Table 4.1. The two curves compare well between themselves and also with the curves shown in Figure 4.3. The conclusion is that gas flow rate in the range of 60 - 300 ml/min does not affect the TGA results significantly. Most of the analyses in this work were made at gas flow rates of about 100 ml/min, which translates to gas velocities of 0.35 cm/s at ambient temperatures inside the TGA tube, whose inside diameter is 2.5 cm. This is well above the optimal value of 50 ml/min suggested by the TGA equipment manufacturer. It was observed that at flow rates of 50 ml/min there was. a buildup of condensed oil everywhere around the furnace, including on the scale wire, which led to erroneous weight determinations. Therefore, it was decided to operate at around 100 ml/min, to avoid the deleterious oil condensation on the wire and inside the scale chamber. Chapter 4. Thermogravimetry Technique 23 Time (minutes) Figure 4.4: Influence of gas flow rate on TGA results. 4.3.3 Gas purity This investigation is primarily concerned with the pyrolysis tests, which should be per-formed without oxygen in the sweep gas. Figure 4.5 shows the effect of the gradual incorporation of oxygen in the nitrogen stream. Test saml2, with ultrahigh purity nitrogen, and sam7, with technical grade ni-trogen, exhibited the same weight (%) vs. time curve, while the other two tests, sam8 and sam!5, exhibited the expected results of increased reaction rates at increasing oxygen percentage in the gas stream. Note that the reaction with pure oxygen is almost instanta-neous, and that the reaction was ignited at about 300 °(J; thereafter, the observed slower reaction is the decarbonization of spent shale-Most of the analyses conducted in this work used ultrahigh purity nitrogen, in spite Chapter 4. Thermogravimetry Technique 24 of the similar results obtained with technical grade nitrogen. 4.3.4 Gas nature The influence of the nature of the sweep gas was investigated. Figure 4.6 shows results for two experiments, which were identical except that the sweep gas was ultrahigh purity nitrogen in one case and ultrahigh purity helium in the other. As both curves are almost identical, one can conclude that there is no effect associated with the nature of these two gases. As mentioned before, all analyses in this work were performed with ultrahigh purity nitrogen. Ultrahigh purity helium stood as a second option, but was never used. 4.3.5 Particle size During a TGA analysis, the solid sample should be isothermal, i.e., inter- and intra-particle temperatures should be the same. The temperature gradient inside a particle is determined by the particle size and the thermal conductivity of the solid material, among other variables. For each material there is a minimal particle size above which isothermal conditions cannot be achieved. Therefore, TGA results are expected to be affected by the particle size of the sample. Figure 4.7 shows results for samples with < 0.25 mm (60 mesh) and < 0.074 mm (200 mesh) particle sizes. Each test was duplicated. There is apparently no difference between the weight (%) vs. time curves. Thus one can conclude that any particle size distribution with an upper size limit of 0.25 mm should yield the same results. Most of the analyses in this work were made with particle size < 0.074 mm. It was also interesting to observe the behaviour of single larger particles when analyzed by the TGA system. Figure 4.8 displays results for four different single particles, ranging in weight from 29 - 48 mg, with sizes between 2 - 4 mm. The temperature curve was Chapter 4. Thermogravimetry Technique 25 Time (minutes) Figure 4.5: Influence of oxygen concentration on TGA results. Time (minutes) Figure 4.6: Influence of gas nature on TGA results. Chapter 4. Thermogravimetry Technique 26 not included in the figure for the sake of clarity. The maximum temperature in all of these runs was 510 °C, as listed in Table 4.1. Also included in Figure 4.8 are results for beds of fine particles < 0.25 mm and < 0.074 mm for the same operating conditions. The curves for the single particles are remarkably different. This difference cannot be imputed exclusively to the particle size, but also may be influenced by the anisotropy of oil shale. Picking a representative particle in this size range seems to be quite difficult. As would be expected, the results for < 0.074 and < 0.25 mm lie as an average among the other results in Figure 4.8. 4.3.6 Sample weight The amount of solid sample in the TGA pan could affect the reaction rates, because heat transfer to the inner particles could be hampered in a large sample. Figure 4.9 indicates that for a 5.81 mg sample, which occupies about 20% of the volume of the pan, and for a 21.8 mg sample, which occupies about 80%, results are about the same. Most of the analyses in this work were made with samples of about 20 mg. This amount generated enough solid residue to be stored for future studies. 4.3.7 Conclusions — thermogravimetry technique The above investigations established the basis for the use of the available TGA system to obtain the kinetic parameters for raw shale pyrolysis arid spent shale combustion. The many exploratory experiments performed indicated both the reliability of this technique and the range of operating conditions safe from interfering effects. In summary, two analyses should be enough to obtain a representative result; the upper particle size limit should be < 0.25 mm; inert sweep gas for pyrolysis studies could be either ultrahigh purity or technical grade nitrogen; the recommended sweep gas flow rate is around 100 ml/min; and the sample weight should be around 20 mg. Chapter 4. Thermogravimetry Technique 27 100 f o o u • < .25mm , saml6 • < .25mm , saml7 ° <.074mm, sam26 * <.074mm, sam27 900 800 700 600 500 400 300 200 100 25 30 35 40 45 Time (minutes) Figure 4.7: Influence of particle size on TGA results. • • • U o 2 3 o. S <u H bp 40 60 80 100 120 140 Time (minutes) Figure 4.8: TGA results for single large particles (2-4 mm). Comparison! with small particles (< 0.25 and < 0.074 mm) Chapter 4. Thermogravimetry Technique 28 op "5 100 95 90 85 80 75 10 21.8 mg, sam26 5.81 mg, sam84 900 800 700 600 500 I <L> 400 § 300 200 100 15 20 25 30 35 40 45 Time (minutes) Figure 4.9: Influence of sample weight on TGA results. Chapter 4. Thermogravimetry Technique Table 4.1: TGA conditions for analyses cited in Section 4.3. heating maximum initial particle gas sample rate temperature weight size gas flowrate (°C/min) (°C) (mg) (mm) (ml/min) sam7 70 810 17.0 < 0.25 N 2 62 sam8 70 810 20.9 < 0.25 air 230 sam9 70 810 20.2 < 0.25 N 2 300 samlO 70 810 18.5 < 0.25 He uhp 461 samll 70 810 16.5 < 0.25 N 2 uhp 70 saml2 70 810 19.2 < 0.25 N 2 uhp -saml3 70 810 17.3 < 0.25 N 2 uhp 73 saml5 70 810 18.5 < 0.25 o 2 77 saml6 70 810 21.1 < 0.25 N 2 uhp 74 saml7 70 810 19.5 < 0.25 N 2 uhp 91 saml9 50 510 18.1 < 0.25 N 2 uhp 72 sam20 50 510 18.3 < 0.074 N 2 uhp 68 sam26 70 810 21.8 < 0.074 N 2 uhp 83 sam27 70 810 19.2 < 0.074 N 2 uhp 83 sam84 70 810 5.81 < 0.074 N 2 uhp 111 single particle samples sam69 50 510 48 2 - 4 N 2 uhp 100 sam70 50 510 30 2 - 4 N 2 uhp 98 sam71 50 510 29 2 - 4 N 2 uhp 98 sam72 50 510 38 2 - 4 N 2 uhp 97 Chapter 5 Raw Shale Particle Devolatilization Modelling Oil shale devolatilization comprises the withdrawal of oil, gas and bound water from the shale solid matrix upon heating. The devolatilization process is part of the pyrolysis process, which is defined as the conversion of kerogen into intermediates and then into oil, gas and residue. As not all of these intermediates and products are volatiles, oil shale, mostly during the first stages of heating, may undergo pyrolysis without devolatilization. The temperature at which the process of devolatilization starts is around 300 °C, and this temperature depends on the shale. Initially water, gas and light and clear hydrocar-bons are produced. Further increase in temperature up to 550 °(J produces more water, gas and heavier and darker hydrocarbons. If the shale is brought to 550 °(J and kept there for about one hour, the devolatilization of hydrocarbons ends, and the maximum amount of these volatiles is produced. At temperatures aboye 550 °C, decomposition of carbonates with release of CO2 takes place with an accompanying loss of weight. Figure 3.1, described earlier, depicts the complete devolatilization of Irati and New Brunswick oil shale. The data were obtained by thermogravimetric analysis with particles smaller than 74 fim (200 mesh) except one analysis of Irati oil shale, represented in Figure 3.1 by dark circular symbols, in which particles were < 250 /mi (60 mesh). Each analysis was repeated to indicate reproducibility. Temperatures above 550 °C should be avoided in devolatilization processes to min-imize the endothermic reactions of carbonate decomposition. The shale weight loss at these higher temperatures is related to its carbonate content. Figure 3.1 indicates that, at 30 Chapter 5. Raw Shale Particle Devolatilization Modelling 31 temperatures above 550 °C, Irati oil shale —with little carbonates— loses less weight than the New Brunswick shale —with greater content of carbonates. This is consistent with the shale analyses, shown in Table 3.1. This chapter is concerned with the devolatilization process of dry shale up to 550 °C. The mechanism of the pyrolysis process, which leads to devolatilization, is not yet completely understood. Many investigators proposed a simplified mechanism through which kerogen is initially converted to bitumen and then the bitumen is converted to oil and gas, i.e.: kerogen —> bitumen —> oil + gas (5-1) Some researchers have attempted to explain the nature of the several intermediates, and have proposed more complex mechanisms. For a complete description of the pyrolysis process, every step of each mechanism should have its own kinetic equation. To deter-mine the kinetic parameters, careful measurements of all reactants and products must be achieved. The whole pyrolysis process implies thermal decomposition of many dif-ferent reacting groups within the kerogen in such a way that reactants converted at low temperatures will not be available any more at higher temperatures. On the other hand, the simplified description of the devolatilization process implied by reaction sequence 5.1 does not require complicated measurements of the amounts of each reactant and product, including intermediates. The kinetic parameters can be obtained directly from measurements of shale weight loss, temperature and time. This is the approach adopted here. Oil shale devolatilization is a thermal decomposition process that involves no gaseous reactants. As heat flows into the particle causing an increase in temperature and products are formed, the devolatilization could either occur at a narrow reaction front or happen simultaneously throughout the particle. Models describing these concepts are presented Chapter 5. Raw Shale Particle Devolatilization Modelling 32 in the literature and are the subject of the following section. 5.1 Previous Models Three models have been presented in the literature for modelling the pyrolysis of a single oil shale particle. These models were checked against experimental data by their pro-posers. All models assumed an oil shale particle to have a concentration of volatiles Cvo at time zero. The solid is heated from T0 to increasing temperatures T up to a constant or plateau temperature, during which the concentration of volatiles Cv drops continually to a constant final value. The devolatilization reaction was considered to be first order with respect to volatile concentration, and the reaction rate constant was considered to obey the Arrhenius equation. 5.1.1 Model 1 Granoff and Nuttall [25] suggested a non-isothermal uniform conversion model. A particle initially at T0 is heated by convection and radiation in a reactor furnace by a gas at Tg, and surrounded by a wall at Tw. They assumed no intraparticle temperature gradient. Consequently the reaction proceeds uniformly within the particle. The material balance is given below, as discussed in Chapter 4: = (4.1) Taking X as the conversion, defined as the ratio between volatilized and volatilisable material, i.e., X = ^ £ _ ^ i (4.2) then substituting for Cv in terms of X in Equation 4.1 leads to: - . 1 7 = * ( ! " * ) (4-3) Chapter 5. Raw Shale Particle Devolatilization Modelling 33 where: X = 0 at t = 0. The particle energy balance is given by: dT d 7 hA. p (Ta -T) + o~evA. v ) (5.2) PpCp where: T = T0 at t = 0. Equation 5.2 neglects the heat of reaction and assumes the furnace wall is an emitter that completely surrounds the particle. The inclusion of the radiant heat transfer term was necessary to improve the match with experimental data. The values of k in Equation 4.3 vary with temperature, which is given by Equation 5.2. All values for the model were obtained from the literature, except the kinetic parameters which were obtained by fitting the model to the experimental data. The experimental data consisted of measurements of temperatures at the centre of the particle, and the conversion. Most of these experiments used cylindrical particles 1.27 cm high and 1.27 cm in diameter. The agreement between the predicted and measured temperatures at the centre of the particle was poor for the highest plateau temperatures. A reasonable match was obtained between the predicted and observed conversions. Experiments were run for plateau temperatures of 384 °C, 395 °C, 429 °C and 520 °C. 5.1.2 Model 2 Granoff and Nuttall [25] also suggested a non-isothermal shrinking core model. It was proposed based on observations of cross sections of partially pyrolyzed shale particles, which showed a light colored core surrounded by a dark layer. The core was assumed Chapter 5. Raw Shale Particle Devolatilization Modelling 34 to be unreacted. As happens with shrinking core models, the reaction was considered to occur only at the interface between the core and the outer layer. An intraparticle temperature gradient was assumed. The reaction rate is given by: Ac dt Replacing Nv = Nvo(l — X ) , one obtains ~d7 ~ Nvo ~ CV0VV ~ ~V^' By replacing Ac — Anr* and Vp = (4/3)7r?'^, the material balance equation for a spherical particle of constant radius rp and variable reaction front radius ?-c is given by: f - ^ ^ dt rl where: X = 0 at t = 0. The core radius rc is related to the conversion X by: * - ( ? ) • • Therefore, dr £ = _ ^ _ d X dt ~ 3T-2 dt " Introducing the latter result in Equation 5.3, yields dt where: £ = " * (5-4) 1'c = l'j> (lt t — 0. Chapter 5. Raw Shale Particle Devolatilization Modelling 35 (5.5) which is subject to the boundary and initial conditions: at r 0 fcpf = h(T-Tg) + o-ep(T4-T* ) at r T = T0 at t 0. Again the heat of reaction was neglected. The model was proposed for spherical par-ticles, although the experiments were carried out using cylindrical particles 1.27 cm high and 1.27 cm in diameter. The authors argue that the same conversion curves were ob-tained for these particles and for 1.27 cm spherical particles, used initially. Experimental measurements were made as described for Model 1. No comment was made about the agreement between the predicted and observed temperatures at the centre of the particle. A poor match was obtained for conversion curves at lower plateau temperatures. 5.1.3 Model 3 Pan et al. [26] proposed a model in cylindrical coordinates in which a first order reaction was supposed to occur. An intraparticle temperature gradient was assumed and the heat of reaction accounted for. The material balance is given by: dC, V dt where: Cy Cyn at t 0 and the energy balance by: (5.6) Chapter 5. Raw Shale Particle Devolatilization Modelling 36 subject to: dT dr = 0 at r = 0 i. dT = h{T - Ta) at r = rp dT dz = 0 at z = 0 I. §T = h(T - T9) at z = L/2 T — T at t = 0. The model was checked against experimental result's from cylindrical particles. The experimental results consisted of temperature measurements in the gas, at the particle surface and at the centre of the particle; and measurements of particle weight as the reaction proceeded. The latter results were then converted to weight fraction of volatiles remaining. The particles were 2.0 cm high and 2.54 cm in diameter. Kinetic parameters were obtained independently by thermogravimetric analysis. The heat transfer coefficient was also obtained for the experimental setup by replacing the oil shale particle by a steel cylinder of the same size. Values for other parameters were obtained from literature data. The temperature measurements indicated a difference of about 30 °C between the gas and the particle surface and between the particle surface and the centre of the par-ticle. This indicated the effect of the external convection and internal conduction heat transfer resistances. There was good agreement between the predicted and the observed temperatures at the centre of the particle. Inconsistent measurements of particle surface temperatures prevented comparisons with calculated values. As for the weight fraction of remaining volatiles, the model predicted well only the initial devolatilization values. The weight fractions at which the predicted and observed values departed were found to be a function of the plateau temperatures. Experiments were run at 400 °C, 415 °C, 430 °C and 445 °C. Chapter 5. Raw Shale Particle Devolatilization Modelling 37 5.1.4 Analysis of models All models disregard the mass transfer resistances inside and outside the particles. Pan et al. [26] estimated the external mass transfer rate to be 11 times greater than the reaction rate. Accordingly, the models only considered intrinsic kinetics and heat transfer resistances. Model 1 is the easiest to solve, but the assumption of uniform temperature inside the particle is expected to be valid only for very small particles. Pan et al. [26] mea-sured significant intraparticle temperature gradients for particles 2 cm in diameter. The assumptions of Model 1 are those prevailing in the application of thermogravimetric anal-ysis for kinetic studies, which use small amounts of particles, 5 - 30 mg, of small size, < 149 /mi (100 mesh). Good agreement with experimental data was obtained at the expense of small adjustments in parameter values, as well as by introduction of radiation heat transfer. The value of the energy of activation was fitted to produce a good match between predicted and observed results. Model 1 is appropriate for small particles. It is not clear up to what particle size the model is valid. Model 2 is more difficult to solve than Model 1, but could be made easier if the non-linearity at the second boundary condition of Equation 5.5 was dropped, i.e., if the contribution of radiant heat transfer were disregarded. Pan et al. [26] estimated this contribution to be less than 1% for a wall temperature of 438 °C. Granoff and Nuttall [25], the proposers of Model 2, did not disclose this temperature value for their experiments. Both Model 2 and Model 1 neglected the endothermic heat of reaction, due to pyrolysis and vaporization. The authors estimated the heat of vaporization to be less than 10% of the sensible heat supplied. Pan et al. [26] did not disregard this parameter but showed by sensitivity analysis that changing the adopted value of 375 k.I/(kg of kerogen) to one third or ten times did not affect the results significantly. The value for the energy of Chapter 5. Raw Shale Particle Devolatilization Modelling 38 activation was also obtained by fitting Model 2 to the experimental results. The energy of activation so obtained was 110 kJ/mol, compared to 148 kJ/mol obtained by fitting Model 1. Model 2, or some other variation of the shrinking core model, seems to have good potential to describe pyrolysis of oil shale particles. Observations of distinct colors between core and shell on cross sections of partially pyrolyzed shale support this model. Model 3 is the most difficult to solve. The model was proposed in cylindrical coordi-nates, the same shape as the particles submitted to pyrolysis in the experiments. Intrinsic kinetic parameters and heat transfer coefficients were obtained from independent experi-ments. It considered the heat of reaction, whose value was obtained from the literature. Other parameter values were also obtained from literature. Sensitivity analyses indicated that heat capacity and heat transfer coefficients were more influential than heat of re-action and thermal conductivity. If the sharp difference in color between the inner core and outer layer is associated with an unreached core and a reacted shell, Model 2 should be preferred over Model 3. Otherwise Model 3 could be a good choice to describe the devolatilization of oil shale particles. 5.2 Characteristics of an Ideal Model An ideal model for the pyrolysis of an oil shale particle should be based on the correct mechanism by which the reaction unfolds, be presented in a system of coordinates related to the particle shape and use the appropriate number of relevant parameters. Values for the relevant parameters should be readily determined by experiment. Chapter 5. Raw Shale Particle Devolatilization Modelling 39 5.2.1 Mechanism To describe pyrolysis or devolatilization in a single particle, a model —a set of equations— should initially take into account the mechanism by which these processes develop. Dif-ferent mechanisms lead to different sets of equations. Models 1, 2 and 3, described in Section 5.1, presumed different mechanisms and therefore their equations differ. Those proposed mechanisms were based on visual observations of the oil shale structure, such as the observation of a core in a semi-pyrolyzed particle, and arbitrary assumptions, such as the assumption of the development or not of an intraparticle temperature gradient. The validity of the models was checked by comparing the predicted results with exper-imental data. But a good match with experimental data does not necessarily mean an adequate model. For instance, Model 1 assumed uniform temperature within the particle and showed good agreement between calculated and observed results. But the authors of Model 3 identified the presence of significant intraparticle temperature gradients, which disagreed with the mechanism proposed in Model 1, at least for the 1-2 cm particle size considered. The aforementioned indicates that an ideal model should be based on quantitative information about the assumptions involved. 5.2.2 System of coordinates A model should use a system of coordinates consistent with the shape of the particle. When crushed, due to its lamellar structure, oil shale generates particles that are best ap-proximated by a rectangular parallelepiped. Therefore, although much of the laboratory work and all available modelling work so far have been done with spherical and cylindrical particles, a particle model that would be later included in a reactor model should be de-veloped in cartesian coordinates. However, because of the increased difficulty in solving Chapter 5. Raw Shale Particle Devolatilization Modelling 40 three-dimensional equations, initial development of the model could be done with a spher-ical or cylindrical system of coordinates. Accordingly, spherical or cylindrical particles could be used in the experiments. 5.2.3 Parameter evaluation Once the mechanism is known and an adequate system of coordinates is chosen, the next step is the inclusion in the model of all relevant parameters with their accurate values. The models presented in Section 5.1 regarded differently the importance of some parameters. Models 1 and 2, but not Model 3, disregarded the heat of reaction. Model 3 neglected the contribution of radiant heat transfer, that was judged important in the experimental setup used in the development of Models 1 and 2. Model 1, but not Model 2, considered the effective thermal conductivity of the particle to have an infinite or very large value. Careful consideration of all relevant parameters is important. The predictions of a model are only as good as the accuracy of its parameter values. Oil shale pyrolysis modelling poses a real challenge when it comes to parameter value estimation. Table 5.1 displays a ranking of the difficulty in obtaining values for the parameters cited in the above mentioned Models 1, 2 and 3. 5.3 Proposed Model 5.3.1 Heat transfer modelling The physical situation to be mathematically modelled is that of an oil shale particle, initially at room temperature, being exposed to an environment whose temperature is either gradually increased or is already constant at a higher temperature. This is the expected situation in oil shale devolatilization processes in fluidized, spouted, entrained or moving beds. Chapter 5. Raw Shale Particle Devolatilization Modelling 41 Table 5.1: Oil shale pyrolysis parameters. Table lists diffi-culty to obtain parameter values in oil shale particle pyrolysis modelling. Parameter easy moderate difficult particle: geometry density heat capacity thermal conductivity heat transfer coefficient kinetic parameters heat of reaction X X X X X X X The simplest situation is that in which the heat of reaction may be disregarded and the heat reaching the particle surface is transfered only by convection. This simple situation is described by the following equation, valid for constant tliermophysical properties: dT^J^fcPT &T 0*T\ dt ppcp \dx* dy2 dz*) 1 ' subject to: err dx = 0 at X = 0 I. dT = h(T - TB) at X = Lx dT dy = 0 at y = 0 I. dT dy = h(T - T9) at y dT dz = 0 at z = 0 I. dT V dz = h(T - TB) at z = LZ T - T at t = 0 in which the boundary conditions were set by placing the origin of the coordinate system at the center of the particle, as shown in Figure 5.1. Chapter 5. Raw Shale Particle Devolatilization Modelling 42 > X Figure 5.1: Definition of the coordinate system. Equation 5.7 has a semi-analytical solution, listed in many heat transfer textbooks like Arpaci's [14], given as a product of dimensionless temperature functions, i.e., 0 = 0X x 6y x 6Z (5.8) where each i = x,y,z, is the solution of the related uni-directional problems: a , for i = x,y,z (5.9) at & = 0 •t = ^ W at fc = l 0i = 1 at Ti — 0 where: Chapter 5. Raw Shale Particle Devolatilization Modelling 43 Ot = 2}_^ - — — — cos(Am&)e 71=1 ' S l V l C O S (5.10) where A„; are the roots of tan A n; = Bi,-/A. Despite the existence of an analytical solution for Equation 5.7, a numerical solution was also developed. The objective was to compare the approximate numerical results with the correct analytical ones in order to assess the accuracy of the numerical method. The numerical method adopted here was the method of lines, proposed by Schiesser [71, 72], and discussed in more details in Appendix G. The good agreement between both numerical and analytical results established the grounds to use the method of lines in other cases when analytical solutions were no longer possible. The apparatus used to obtain the experimental results for the heat transfer investi-gation is shown in Figure 5.2. The unit had controlled electrical heaters for both the pre-heater and reactor vessels. The reactor was an 80 cm high, 3.5 cm internal diameter cylindrical pipe. The top of the reactor is detailed in Figure 5.3. Three ceramic ther-mocouple insulators, each with an outside diameter of 3.2 mm, projected through the reactor lid. Each one had two cylindrical longitudinal holes to host the thermocouple wires. These had bare tips to better measure the temperatures. The thermocouple in the middle was imbedded in the particle to measure its center point temperature. The particle was attached to the insulator by a thin wire. See details in Figure 5.4. The temperatures recorded by the bare tip thermocouple immersed in the gas phase at a position before the particle is equal to the gas temperature at that place, as indicated by a steady state heat balance on the thermocouple junction. Chapter 5. Raw Shale Particle Devolatilization Modelling 44 REACTOR PRE-HEATER g a s out g a s in Figure 5.2: Experimental apparatus. The first data obtained with this apparatus, when compared with predicted results, indicated that the heat transferee! by radiation from the reactor wall played an important role. Theoretical calculations with a spherical particle demonstrated that the importance of heat transfer by radiation is a function of the convective heat transfer: the greater the heat transfer coefficient, the less the importance of radiation. This simulation, made for a 1 cm spherical particle and gas and reactor wall temperatures of 500 °(J, is shown in Figure 5.5, in which particle temperature profiles are plotted against time for two heat transfer coefficients: 10 and 500 J/sm'^K. These values are in the range of expected values for gas to particle heat transfer coefficients: 8 - 1000 J / sm 2 K. For the higher heat transfer coefficient, 500 J/s m K, the differences between the predicted results are negligible, while for the lower value, 10 J/sm'^K, it is quite significant. Another simulation was made, but now the gas and reactor wall temperatures were maintained slightly above that of the particle, i.e., the reactor wall was heated up from room temperature, and that caused the gas and particle temperatures to increase correspondingly. The result is shown in Figure Chapter 5. Raw Shale Particle Devolatilization Modelling 45 Figure 5.4: Detail of thermocouples position and particles. Chapter 5. Raw Shale Particle Devolatilization Modelling 46 500 400 5 300 n U | Z 200 100 h= 10J/sm2K -radiant heat no radiant heat 4 6 Time (minutes) S 300 S o 200 10 0.2 h= 500 J/smzK — radiant heat - no radiant heat 0.4 0.6 Time (minutes) 0.8 Figure 5.5: Relation between radiation and convection. Theoretical re-sults indicate the importance of radiation heat transfer when convection heat transfer coefficients are low. 5.6 for the lowest heat transfer coefficient, 10 J / s m 2 K . It is clear that even for the low temperature differences between particle and wall investigated, heat transfer by radiation should be regarded when the convective lieat transfer coefficient is low. The theoretical results in Figures 5.5 and 5.6 considered no intraparticle temperature variation. Low heat transfer coefficients are expected with the low gas velocities used in the present runs. The reason for using low gas velocities is that they impose low drag forces on the particles, a requirement for the subsequent tests which measured the particle weight loss during a run. Granoff and Nuttall [25] mentioned a heat transfer coefficient of 6.53 J/s m 2 K for their similar experimental conditions. The following correlation given by McAdams [15], good for large particles: Nu„ = 0.37 Re 0.6 (5.11) yields a heat transfer coefficient of 10.8 J/sm^K for the present condition. The popular Chapter 5. Raw Shale Particle Devolatilization Modelling 47 0 5 10 15 20 25 30 Time (minutes) Figure 5.6: Significance of radiation at low heat transfer coefficients. Curves indicate the effect of external temperatures. correlation of Ranz and Marshall [16, 17]: Nu p = 2.0 + 0 .6ReJ / 2 Pr 1 / 3 (5.12) usually produces higher values for low Reynolds numbers, in this case 20.4 J/sm K. The above correlations together with the ones by Bandrowsky and Kaczmarzyk [18, 19] and the one by Kato et al. [20] were analyzed by Lisboa [10] for air to shale particles heat transfer. The presence of significant radiation heat transfer in the experimental setup introduces changes to the heat transfer model, in terms of new boundary conditions at the particle surface. The particle temperature profile is still described by: Chapter 5. Raw Shale Particle Devolatilization Modelling 48 dt ~ PpCp \ dx2 + dy2 + dz2 ) ^ ' but the governing equation is now subject to the modified conditions: dT dx dT q — 0 at x = 0 9 a; kp% = h(T-Ta) + <rtp(T4-Tv]) at x = Lx If = 0 at y = 0 dT -kp^ = h(T-Tg) + aep(T4-T4) at y = Ly = 0 at z = 0 -kp% = h(T - Ta) + aep(T4 - T t 4J at z = Lz T = T0 at t = 0. Equation 5.13 does not have an analytical solution because of the introduction of non-linear boundary conditions. The numerical method of lines was able to handle the presence of radiation with only minor adjustments required. Experimental data were obtained initially for a 1.3 cm wide cubic particle made of 304 stainless steel. A 3.2 mm hole was made in the middle of one face to introduce the thermocouple insulator which measured the temperature at the particle center. There was no gap between the thermocouple insulator and the hole. A metallic particle was used first iu order to check the apparatus and methods for a material whose properties are well known and is inert under the experimental conditions. The required properties are listed in Table 5.2. It was assumed that the heat loss by conduction through the insulator was negligible. Each run started by heating the reactor wall to the desired temperature and setting the gas (air) flow rate. After steady state was reached, the reactor lid with attached thermocouples and particle was placed on top of the reactor. Temperatures were then recorded as a function of time. Chapter 5. Raw Shale Particle Devolatilization Modelling 49 Table 5.2: 304 stainless steel properties. Average values be-tween 20 °C and 500 °C. parameter values density (pp) thermal conductivity (kp) specific heat (cp) emissivity (e) 7.7 x 10+3 kg/m 3, [21, Table 3-1], [22, Table A.3-16] 16.7 J /smK, [21, Table 3-322], [22, Table A.3-16] 5.0 x 10+2 J/kgK, [21, Table 3-205], [22, Table A.3-16] 0.6, [21, Table 10-17], [23, Table 2.2] Typical experimental and predicted results are shown in Figures 5.7 and 5.8. Raw data for these curves are in Appendix B. Predicted results were calculated with program temperature, listed in Appendix A. In both runs the wall temperature was close to 350 °C. Air flow rates were respectively 1.13 and 1.64 m 3/hr at standard temperature and pressure (STP), i.e., 0 °C and 1 atm. Particle Reynolds numbers for the first run based on superficial velocity was 168. The predicted results were calculated with parameters from Table 5.2 and adjusted to the experimental data by using a heat transfer coefficient value of 7.5 J/s m 2 K. Figure 5.9 shows results for a wall temperature of 450 °C. Again, the predicted results were made to fit the experimental data by adopting a heat transfer coefficient of 2.5 J/s m K. This was a lower value than the ones used in the two previous runs, and this can be partially explained by the lower air flow rate used : 0.59 m' /hr. Another similar run was made by immersing the 304 stainless steel particle attached to a thermocouple in a sand fluidized bed at 302 °C, as shown in Figure 5.10. Experimental and predicted results are displayed in Figure 5.11. The theoretical results fitted the experimental values when the heat transfer coefficient was set equal to 500 J / sm 2 K. The output of the program used to generate the theoretical results allows one to follow the temperature profile within the x,y plane at z=0. In Figures 5.7, 5.8 and 5.9, the center Chapter 5. Raw Shale Particle Devolatilization Modelling 50 1 / , / 1 1 i : • i ' - - - " * """"""" i' / j' / i; m 1, / i; / wall it / ! / gas after particle f / gas before particle particle center 0 particle center calculated > i • particle surface calculated i . i . 0 5 10 15 20 Time (minutes) Figure 5.7: Metal particle temperature profiles. Tw — 350 °C. Q — 1.13 m /hr. 400 0 5 10 15 20 Time (minutes) Figure 5.8: Metal particle temperature profiles. Tw — 350 °C. Q = 1.64 m'*/hr. Chapter 5. Raw Shale Particle Devolatilization Modelling 51 Figure 5.9: Metal particle temperature profiles. Tw = 450 °C. Q = 0.59 m /hr. point temperatures at coordinates (0,0,0) are plotted as well as the surface temperature at coordinates (0,0.65 x 10_2,0). The surface temperature was just slightly above the center point one, which is expected for low Biot number. Another set of runs were made with the stainless steel cube, but now by increasing the reactor wall temperature from room temperature up to a desired value. In this case, to start a run, the reactor lid with the attached thermocouples and particle were placed on top of the reactor at ambient temperature conditions. Then the heaters were turned on causing the wall temperature to gradually increase. As a consequence, the gas and particle temperatures also increased. The unsteady state gas temperatures were calculated to be 1 - 4 °(J below the temperature indicated by the thermocouples immersed in the gas phase. Chapter 5. Raw Shale Particle Devolatilization Modelling 52 Figure 5.10: Sketch of fluidized bed for temperature measurements. o I i I i I i I i I i I i i i I 0 20 40 60 80 100 120 140 Time (seconds) Figure 5.11: Metal particle temperature profiles in a fluidized bed. Chapter 5. Raw Shale Particle Devolatilization Modelling 53 The results are shown in Figures 5.12 and 5.13. Air flow rates were respectively 1.13 and 0.59 m 3/hr (STP). Again the model was able to calculate the particle center temperature using a heat transfer coefficient of 7.5 J/s.m 2.K. One of the reasons to do these preliminary runs with the metallic particle was to obtain the gas to particle heat transfer coefficients in the experimental setup of Figure 5.3. The heat transfer coefficient would then be used in the shale runs. The experimental results of four out of five runs with the metallic particle could be fitted with a heat transfer coefficient of 7.5 J/s.m 2.K, despite different flow rates in the range 0.5 - 1.7 m 3/hr (STP). The calculated coefficient using the equation of McAdams, Equation 5.11, is 24.9 J/s.m 2.K, and using the equation of Ranz and Marshall, Equation 5.12, is 27.4 J/s.m 2.K, for a flow rate of 1.0 m 3/hr (STP) and air properties at 260 °C. These two empirical equations were not designed for the unique situation in the experimental setup of Figure 5.3. As mentioned before, both equations gave higher values, 10.8 and 20.4 J/s.m 2.K, respectively, for the experimental setup of Granoff and Nuttall, who obtained an experimental value of 6.53 J/s.m 2.K. The equation of Ranz and Marshall, Equation 5.12, suggests that at low enough gas flow rates, the heat transfer coefficient is independent of the flow rate, i.e., the second term on the right hand side of the equation is much lower than the first term, a constant equal to 2. This seems to be the case with the experimental setup of Figure 5.3. The heat transfer coefficient of 7.5 J/s.m .K was able to describe the experimental points for the runs with shale particles, which will be introduced next. There is no clear explanation for the run shown in Figure 5.9, which required a heat transfer coefficient of 2.5 J/s.m 2.K. The experiment done with the fluidized bed showed that the model is sensitive to these situations which require a much higher heat transfer coefficient, such as 500 J/s.m 2.K, used to fit the predicted results to the experimental points, see Figure 5.11. Following these runs, metallic particles were replaced by oil shale. A number of 1.3 cm Chapter 5. Raw Shale Particle Devolatilization Modelling 54 400 0 I . 1 1 1 . 1 . 1 . : 1 0 10 20 30 40 50 Time (minutes) Figure 5.12: Metal particle temperature profiles. Variable Tw. Q = 1.13m'/hr. 500 0 10 20 30 40 50 Time (minutes) Figure 5.13: Metal particle temperature profiles. Variable Tw. Q = 0.59 m /hr. Chapter 5. Raw Shale Particle Devolatilization Modelling 55 Table 5.3: Irati oil shale properties. parameter values density (pp) thermal conductivity (kp) specific heat (cp) emissivity (e) 2.1 x 10+3 kg/m 3 1.25 J / smK 1,126. J/kgK 0.9 wide shale cubes were shaped. Small departures from the desired 1.3 cm side length did not justify a change in the assumed length. Oil shale properties, supplied by Petrobras, are displayed in Table 5.3. All runs with shale were made with Irati shale, whose properties were known. Figures 5.14 and 5.15 compare the experimental and predicted results for two runs. Nitrogen flow rates for these runs were respectively 0.59 and 0.31 m 3/hr (STP). The theoretical results fitted the experimental points with a heat transfer coefficient of 7.5 J/s.m 2.K. The tables that follow Figures 5.14 and 5.15 show particle temperature profiles for the x - y plane at z=0 for a time of 35 minutes, when the reaction had almost reached completion. An interesting point is that the heat consumed by the endothermic reactions, which was neglected in the calculations, did not affect the predicted results. Rough estimates for the pyrolysis heat of reaction indicate that it is negligible compared to the sensible heat, as had been pointed out by Pan [26]. Heat transfer modelling during pyrolysis can be approximated by basically analysing the problem as that of heating an inert particle. Another interesting point is that for the experimental conditions tested, the intraparti-cle temperature gradient is negligible. From the conditions tested, one can infer what is Chapter 5. Raw Shale Particle Devolatilization Modelling 56 500 0 1 ' ' ' ' ' 1 • 1 0 20 40 60 80 Time (minutes) Figure 5.14: Shale particle temperature profiles. Variable Tw. Q. = 0.59 m /hr. Particle temperature profile for plane (x,y,0) at time 35 minutes. y values (mm) x coordinate values (mm) 0.000 1.625 3.250 4.875 6.500. 6.500 4.875 3.250 1.625 0.000 444.6 444.7 445.1 445.8 446.9 443.2 443.4 443.9 444.7 445.8 442.4 442.5 443.0 443.9 445.1 441.9 442.0 442.5 443.4 444.7 441.7 441.9 442.4 443.2 444.6 going to happen in a pyrolysis reactor, whatever gas-solid contacting technique is used. The parameter which governs the intraparticle gradient is the Biot number. The Biot number for a given particle material changes by changing the particle size or the heat transfer coefficient. The 1.3 cm particle size tested stands in the middle of the particle sizes used in a wide variety of reactor types. For moving beds, this size is the lower Chapter 5. Raw Shale Particle Devolatilization Modelling 57 Time (minutes) Figure 5.15: Shale particle temperature profiles. Variable Tw. Q = 0.31 m'*/hr. Particle temperature profile for plane (x,y,0) at time 35 minutes. y values (mm) x coordinate values (mm) 0.000 1.625 3.250 4.875 6.500 6.500 4.875 3.250 1.625 0.000 443.5 443.7 444.1 444.8 445.9 442.2 442.4 442.9 443.7 444.8 441.3 441.5 442.0 442.9 444.1 440.8 441.0 441.5 442.4 443.7 440.6 440.8 441.3 442.2 443.5 limit. For spouted beds, the size is not much above the average. For fluidized and en-trained beds, the size is larger than the practical size. The heat transfer coefficient used, combined with a hypothetical heat transfer coefficient by radiation, is also in the range of what is expected in a pyrolysis reactor. For moving beds, lower values are expected, Chapter 5. Raw Shale Particle Devolatilization Modelling 58 clue to lower contributions from radiation and about the same contribution from con-vection. For other types of reactors, in which particles are fed at a temperature much lower than that of the reactor, the contribution of radiation is approximately the same as in the experimental setup, while heat transferred by convection is larger. These facts lead one to deduce that the Biot number would be in about the same range as the ones tested, for all of the various reactors mentioned. If large particles are processed, such as in a moving bed reactor, combined heat transfer coefficients due to convection and radiation would have low values. On the other hand, if fine particles are used, making the entrained bed the preferred choice, high values of the combined coefficients are expected. Therefore the product of particle size and combined heat transfer coefficient, and hence the Biot number, remain approximately constant. Table 5.4 shows rough estimates of Biot number for several gas-solid contacting techniques. Heat transfer coefficients are average values for combined heat transfer coefficients. Particle radius are average values adequate for each reactor type. Biot numbers were calculated for a thermal conductivity of 1.25 J/s.m.K. Despite the fact that the numbers are just rough estimates, one can see that the Biot number should not depart much from 0.4, except in extreme cases. Heat transfer coefficients for oil shale in moving beds are given by Carley et al. [24]. Significant intraparticle temperature gradients would occur in extreme cases, for in-stance large particles in the presence of large heat transfer coefficients. For rich shales, the heat of reaction might be significant in promoting this gradient. All the above conditions could be easily simulated by the proposed model, which makes it a powerful tool for that purpose. One should also take into account that the most common large-scale reactor type for oil shale pyrolysis is the moving bed, which uses large particle sizes. This is one condition that favors intraparticle gradients. Also, it is important to mention that many endeavors have been made in the past towards developing an in situ processing technology, which would cut the cost of shale oil production by half. Chapter 5. Raw Shale Particle Devolatilization Modelling 59 Table 5.4: Expected Biot numbers in gas-solid reactors. h rp Bi (W/m 2 .K) (m) (-) entrained bed fast fluidization 1000. 0.0005 0.4 fluidized bed 500. 0.001 0.4 moving bed spouted bed 50. 0.01 0.4 The mining and crushing stages account for above 50% of oil production cost in an above ground reactor plant. Processing the shale in situ would mean dealing with large chunks of shale; in this case intraparticle temperature gradients would be expected. These are two cases in which the present heat transfer model would be useful. 5.3.2 Mass loss modelling The proposed mass loss model is based on the unreacted core model of Granoff and Nuttall [25], presented and discussed in Section 5.1.2. In this work, the implementation of that model also considered variable gas temperatures, which occur in moving bed reactors. In such situations, the contribution of radiation heat transfer is decreased, and the heat transfered to the particle surface can be approximately described by a combined heat transfer coefficient. If the heat of reaction is neglected, the partial differential equation which describe the temperature profile within the particle, Equation 5.5, can be solved independent of the differential equations which describe the mass loss, Equations 5.3 and 5.4. The basic assumption of Granoff and Nuttall, that the devolatilization process occurs Chapter 5. Raw Shale Particle Devolatilization Modelling 60 according to a unreacted core mechanism, based on convincing photographs from the interior of partially pyrolyzed particles [25], was acknowledged. A partially retorted oil shale exhibits a lighter coloured core, surrounded by a dark shell, suggesting that pyrolysis follows an unreacted core model. The fact that voidage in raw shale is small and increases during the devolatilization process, also supports the unreacted core mechanism. About 10% of the original mass is lost in the devolatilization process, as indicated in Figure 3.1. This would cause an increase in voidage within the particle. The thermal decomposition and devolatilization process occurs similarly to the shrink-ing core model of a gas solid reaction, but in the thermal decomposition process, no gaseous reactant is present. In the case of shale, as soon as heat is available to pyrolyze kerogen into bitumen and vaporize the bitumen into oil and gas, the pressure inside the particle is increased. This pressure causes cracks in the shale particle, as will be shown next. Both mass loss and cracks are responsible for a significant increase in particle voidage, even though, for the two shales studied, no volume increase was observed. However, volume increase is observed for richer shales. Figure 5.16 shows a picture, taken by a scanning electron microscope (SEM), of the surface of a New Brunswick raw shale magnified 400 times. Figure 5.17 shows the surface of Irati spent shale magnified 50 times. The latter shows the cracks that develop within the particle. These create channels which allow vaporized material to leave. Raw shale, and also the unreacted core, is essentially non-porous. Figures 5.18 to 5.25 show a sequence of pictures of the surface of both raw and spent Irati shale at magnifications of 100, 200, 600 and 1,000. The cracks exhibited by spent shale do not make the particle friable. It maintains its original shape. Mass loss experimental data were obtained for cubic shale particles using the apparatus shown in Figure 5.26. The apparatus consists of a balance placed on top of the reactor Figure 5.17: Irati spent shale SEM picture (50x). Chapter 5. Raw Shale Particle Devolatilization Modelling 62 Chapter 5. Raw Shale Particle Devolatilization Modelling 63 Figure 5.21: Irati spent shale SEM picture (200x). Figure 5.23: Irati spent shale SEM picture (600x). Chapter 5. Raw Shale Particle Devolatilization Modelling 65 Figure 5.24: Irati raw shale SEM picture (lOOOx). Figure 5.25: Irati spent shale SEM picture (lOOOx). Chapter 5. Raw Shale Particle Devolatilization Modelling 66 with the test particles hung from underneath by a thin wire. The balance was model GT410 from OHAUS, with precision of 0.001 g. The heating of the reactor followed as closely as possible the same temperature curve previously obtained in the temperature experiment. It was not possible during the same run to record both particle temperature and weight. Figures 5.27 and 5.28 show the results for two of these runs. The nitrogen flow rates were respectively 0.59 and 0.31 m'*/hr (STP). The weight loss curves followed the same pattern as those obtained in the TGA experiments shown in Figure 3.1. The experimental temperature values shown in Figure 5.27 are the same as in Figure 5.14, because the wall and gas temperature profiles of these two runs were within ± 5 C. It is assumed that the particle center temperature follows the same difference between the two runs. It is assumed that the heat loss by conduction through the hanging wire is negligible. The experimental data are tabulated in Appendix B. The experimental temperature values shown in Figure 5.28 are the same as in Figure 5.15 for the same reason. ES3 HEATERS THERMOCOUPLES E3 BALANCE Figure 5.26: Reactor top for particle weight measurements. Chapter 5. Raw Shale Particle Devolatilization Modelling 67 Chapter 5. Raw Shale Particle Devolatilization Modelling 68 Figure 5.29 shows a typical output for the pyrolysis modelling. Note that the con-version, the core radius and the particle temperatures can be followed. The results were produced for constant gas temperature and no radiation. Time (seconds) Figure 5.29: Particle pyrolysis modelling. Parameters used for the above simulation: dp — 2 cm h = 60 J/s.m 2.K k0 = 0.103 1/s (T < 423°C) k0 = 2.78 x 105 1/s (T > 423°C) PP = 2100 kg/m 3 T0 = 25 °C kp = 1.25 J/s.m.K T„ = 550 °C E = 27300. J/mole (T < 423°C) E = 105000. J/mole (T > 423°C) Cp = 1045 J/kg.K The FORTRAN code used to get the results plotted in Figure 5.29 is listed in Appendix A under the name of program pyrolysis. The parameters used to get the results are Chapter 5. Raw Shale Particle Devolatilization Modelling 69 listed in a table just after Figure 5.29. Note that the gas temperature was assumed to be constant. Kinetic parameters were based on results obtained in Chapter 6. The energy of activation and frequency factor values change at 423 °C, causing the sharp changes in the slopes of the curves core radius versus time and conversion versus time shown in Figure 5.29. The effect of particle diameter, heat transfer coefficient and energy of activation on conversion, core radius, particle surface temperature and core radius temperature is in-vestigated in Appendix G. As convection and radiation heat transfer are represented by a single heat transfer coefficient, any change in this parameter—as any change in particle diameter—will affect directly the Biot number. Figure 5.30 shows experimental and predicted results for the run shown in Figure 5.27. Program pyrolysis, listed in Appendix A, had to be modified to accept a variable gas temperature. The parameters used in the model are listed in the table after Figure 5.30. The adopted particle diameter of 1.6 cm is the one of a sphere with the same volume of a 1.3 cm cube. A combined heat transfer coefficient of 20 W/m 2 .K described well the particle center temperature. This value is above—as expected—the value of 7.5 W/m .K used for the convection heat transfer coefficient in the modelling shown in Figure 5.14, which consid-ered the correct shape of the particle and separated the heat transfer by radiation from the heat transfered by convection. The predicted particle temperature values, which are functions of radius position, depends only on the combined heat transfer coefficient. The predicted results of particle weight versus time shown in Figure 5.30 were fitted to the experimental points using the kinetic parameters shown in the table just after the figure. For reasons that are discussed in Chapter 6, it was not possible to obtain definite values for these parameters. The fitting was obtained by arbitrarily choosing values from Coats and Redfern method and subsequently changing these values to fit the experimental Chapter 5. Raw Shale Particle Devolatilization Modelling 70 Time (minutes) Figure 5.30: Pyrolysis modelling results. Particle center temperature and particle weight versus time-Parameters used for the above simulation: = 1.6 cm h = 20 J/s.m 2.K k 0 = 0.09 1/s (T < 423°C) k 0 = 1.00 x 105 1/s (T > 423°C) PP = 2100 kg/m 3 T0 = 25 °C kp = 1.25 J/s.m.K TG = 500 °C E = 27300. J/mole (T < 423°C) Cp = 1045 J/kg.K E = 105000. J/mole (T > 423°C) points. Chapter 6 Kinetic Parameters for Devolatilization Kinetic parameters for oil shale pyrolysis or devolatilization have been generally obtained by thermogravimetry through standard or laboratory constructed thermogravimetric anal-ysis (TGA) apparatuses. The major reactant in oil shale is kerogen, but minor amounts of natural bitumen can be present. Natural bitumen is the fraction of the original or-ganic matter that is soluble in organic solvents such as toluene or benzene. Usually no distinction is made between kerogen and natural bitumen in oil shale kinetic studies, but a few investigations were made after extraction of the natural bitumen. This work treats the two fractions as one single reactant. The major product is oil, followed by gas and water. In pyrolysis studies, one is also interested in the amounts of bitumen and car-bon residue produced, components that are not regarded in devolatilization studies, as explained below. Concerning pyrolysis, kinetic studies can focus on: • decomposition of organic matter [27, 28, 29, 30, 31]. In this category, kinetic pa-rameters refer to the decomposition (pyrolysis) of the organic matter (kerogen plus natural bitumen). Measurements or estimates of volatiles (oil, water and gas), bitu-men and carbon residue are required. Water and oil are collected by condensation. Bitumen is extracted from the shale matrix with an organic solvent. The amount of gas is obtained by difference between the total initial and total final weight. Carbon residue is actually considered part of the non-reacted kerogen. 71 Chapter 6. Kinetic Parameters for Devolatilization 72 • devolatilization of organic matter [32, 33, 34, 35, 36]. In this category, kinetic pa-rameters refer to the weight loss experienced by oil shale. Weight loss is measured by monitoring the weight of the shale sample. It differs from the previous cate-gory because here no non-volatile product such as bitumen and organic residue is considered. The current work falls in this category. • production of oil [33, 36, 37, 38] or gas [36]. The kinetic parameters refer to either the amount of oil or gas produced. Means should be provided to measure these quantities. • production of components of oil [39, 40] or gas [41, 42]. This category derives from the previous one. Kinetic parameters refer not to the whole amount of oil or gas produced, but to their components. • production of intermediates [43, 44]. In this category, complex mechanisms are considered, which regard a set of series and parallel reactions, each with its reactants and products that should be accounted for to obtain the many kinetic parameters involved. Two reasons make the study of the devolatilization of organic matter specially attrac-tive: the great accuracy of the experimental data and their importance for plant design. Studies of oil shale devolatilization can be performed in a standard TGA apparatus, which reduces to a minimum the effects of transport resistances. This permits the obtaining of accurate experimental data. Because the standard TGA apparatus employs small amounts of small particles, the measured weight loss kinetics represents the chemical kinetics. All of the other types of studies mentioned above involve some sort of oil collection or con-densation, which requires substantial amount of particles. Also, to obtain a uniform gas flow within a bed of these particles, they should not be too small and not of very different Chapter 6. Kinetic. Parameters for Devolatilization 73 Table 6.1: Sizes and amounts of particles in TGA analyses. size [mesh(mm)] amount (mg) Analyses with modified TGA apparatus, which enable the collection of oil produced Hubbard and Robinson [27] Diricco and Barrick [28] Allred [29] Capudi [36] Shih and Sohn [37] Yang and Sohn [38] Wang and Noble [39] 6 x 1 cm cylinder made of particles <100 (0.15) 35 (0.42) - 200 (0.07) 100 (0.15) - 115 (0.12) 28 (0.6) - 270 (0.05) 8 (2.4) - 48 (0.3) 8 (2.4) - 48 (0.3) 10 (1.7) - 16 (1.0) ~ 10,000 15,000 250 - 2,000 1,000 80,000 70,000 Analyses with standard TGA apparatus, used in studies of devolatilization of organic matter Haddadin and Mizyed [32] Campbell et al. [36] Rajeshwar [34] Thakur and Nuttall [35] Capudi [36] This work <325 (0.04) NA NA <200 (0.07) 28 (0.6) - 270 (0.05) <100 (0.15) 200 60 - 80 10 -20 25 10 - 22 10 -30 sizes. Otherwise channeling will occur and different local reaction rates will arise within the bed. The use of larger amounts and sizes of particles means increased possibilities of incorporating transport resistances into the measured (observed) kinetics, making it different from the target chemical kinetics. Table 6.1 discloses the size and amount of par-ticles employed by the studies that require oil collection and the studies of devolatilization of organic matter, which do not require oil collection. Roughly, the former require particle sizes 10 times larger and an amount of particles 1,000 times greater than the latter. Chapter 6. Kinetic Parameters for Devolatilization 74 A second attraction of devolatilization studies is that oil and gas production are of in-terest for engineering design. The studies of devolatilization of organic matter give the oil, gas and water production, but water is usually below 10%. The studies of decomposition of organic matter, production of components of oil or gas and production of intermediates would give details that are useful but not essential from the standpoint of plant design. The most useful information would come from production of oil and gas, separately, but again this would usually involve inaccuracies in the determination of the related kinetic parameters. For the reasons stated above, the kinetic parameters obtained in this study are related to the devolatilization of organic matter. These are the kinetic parameters required in the model suggested in Section 5.3. 6.1 Reaction Mechanism This section deals with the transformation that turns the original organic matter (kerogen and natural bitumen) into oil, gas, water and carbon residue. For the sake of simplicity, all the original organic matter will be represented by the word kerogen, and the word oil will also include the water produced. It is generally accepted that upon heating kerogen initially produces a soluble bitumen which, as the reaction proceeds, is converted into oil and gas: kerogen —> bitumen —> oil + gas (5-1) The above simplistic mechanism is supported by the fact that the amount of a soluble bitumen increases during the first moments of the reaction, and subsequently decreases until it vanishes completely while amounts of oil and gas are produced. However, Equation 5.1 does not address many aspects of the reaction that are still Chapter 6. Kinetic Parameters for Devolatilization 75 under debate. It is not yet clear for instance whether oil is produced in the first step, but it is more accepted that gas is produced. Also, a few researchers support the view that carbon residue is only produced in the first step, while others say it is only produced in the second step. It is not clear if kerogen is totally converted, because there has not been a reliable way to distinguish between unreacted kerogen and carbon residue. A few authors have notionally split the bitumen of Equation 5.1 into a series of intermediates, leading to a multi-step mechanism. Despite the fact that for the study of oil shale devolatilization, the simplistic mechanism represented by Equation 5.1 would be sufficient, a few extensions of that mechanism are presented next, for the sake of completeness. Hubbard and Robinson [27] suggested the following mechanism: gas gas kerogen —» bitumen —> oil (6-1) oil (?) carbon residue They believed that carbon residue is produced in the second step because the total amount of bitumen, oil and gas decreased slightly from the point of 50% conversion of kerogen. The only way that could happen is by the return of bitumen to the solid phase, i.e., to carbon residue. Allred [29] analyzed the same data and concluded that carbon residue was produced only in the first step. He considered that the total amount of bitumen, gas and oil remained constant after 50% conversion of kerogen, and proposed the mechanism: kerogen —>• gas bitumen —> carbon residue gas oil (6.2) Chapter 6. Kinetic Parameters for Devolatilization 76 Later on Braun and Rothman [30] studied the same data of Hubbard and Robinson [27] and concluded that if a heat-up period were considered the data would agree with the two step mechanism: gas gas kerogen —> bitumen —> oil (6-3) carbon residue carbon residue 6.2 Reaction Order All authors mentioned so far in this chapter obtained a first order dependency between the reaction rate and the reactant concentration, except Allred [29], who worked with his own data and those from Hubbard and Robinson [27], and obtained a better fit with the autocatalytic expression: — In I———j oc kt However, his proposal was later contested by Braun and Rothman [30], who observed that the data of Hubbard and Robinson [27] fitted well a two-step first order mechanism. The effects of the heat-up period on the data of Hubbard and Robinson was also considered by Rajeshwar and Dubow [31] who again obtained a first order dependency. Concerning the relation between the reaction rate constant and temperature, given by the Arrhenius equation, a group of researchers [27, 31, 32, 33, 37, 38] identified only one relation for the entire range of temperatures, while another group [28, 29, 30, 34, 35, 36] obtained two relations, one for the lower and another for the higher temperature ranges. Chapter 6. Kinetic Parameters for Devolatilization 77 450°C . 475°C = • 500°C • 525°C t 0 aC °° . * « 0 ° O o A • A • ° ° ° • n * A • • • • * a A a I . . . . I . . . . I . . . . I . . . . I . . . . I , , , . " 0 50 100 150 200 250 300 0 10 20 30 40 50 60 70 80 Time (minutes) T i m e (minutes) Figure 6.1: Irati shale isothermal TGA (8 = 100 °C/min). The latter is in agreement with the simplistic mechanism suggested by mechanism 5.1. It should be noted that the mechanisms and first order rate expressions suggested so far are simple representations of the complex transformations occurring within the thermal processes. Nevertheless, they provide a reasonable tool for modelling the observed effects of these processes. 100 98 96 o 350°C . 375°C ° 400°C - 425°C £ 94 92 ~ » • a n 1UU 98 96 $ I 94 "S3 92 90 R« 6.3 Isothermal T G A Tests A few analyses were made using the isothermal TGA method. They provide an insight into the oil shale devolatilization process and allow comparison with previous experimental data and with the more adequate non-isothermal TGA method, discussed in the next subsection. Figure 6.1 shows isothermal TGA results for Irati shale. The results were split into two diagrams, one covering the analyses performed at a low temperature range and the other at a high temperature range. All the analyses were made with dry shale that during a typical test was kept initially at 100 °C for 5 minutes and then brought to the test Chapter 6. Kinetic Parameters for Devolatilization 78 temperature at a heating rate of 100 °C/min; a higher heating rate would produce non-uniform temperatures within the shale sample. The higher the test temperature, the greater the fraction of volatiles that leaves the particle during the heat-up period, before the test temperature is reached. This should be taken into account when analyzing the data in Figure 6.1. The eight curves in Figure 6.1 indicate that the amount of volatilized material ap-proaches an asymptotic value that is a function of the test temperature, and increases as the temperature increases. This trend is typical of oil shale devolatilization and was earlier detected by Allred [29]. It is expected that given an infinite time the total volatiles produced would be the same, no matter at what temperature the test is done. However, as the curves indicate, the devolatilization rate decreases with time. This is very clear from the curve corresponding to the experiment done at 500 °C, which was left for a long time at this temperature: the amount of volatilized material increases with time but at progressively lower rates. The curves in Figure 6.1 also show that initially the devolatilization process occurs at a high reaction rate which accounts for more than 50% of the total conversion, depending on the temperature, and that takes less than 2 minutes. It will be seen later in this subsection that this initial high rate is about the same for all temperatures. After 2 minutes the rate starts to decrease. Capudi [36] addressed these first 2 minutes in his studies of Irati oil shale devolatilization. His data are plotted together with ours in Figures 6.2 and 6.3 for the low and high temperature ranges respectively. It should be noted that his shale samples were inserted directly into the TGA apparatus which was already at the test temperature, while ours were submitted to the 100 °C/min heating rate. Thus the heat-up periods of his samples were shorter than ours. That explains why his points lie to the left of ours in the figures. This is best illustrated in Figure 6.4, which is an enlargement of the 500 °C chart in Figure 6.3, including a new curve for the 185 °C/min Chapter 6. Kinetic Parameters for Devolatilization 79 98 £ 97 100 150 200 250 300 Time (minutes) 0 20 40 60 80 100 120 140 160 Time (minutes) 0 20 40 60 80 100 120 140 160 180 Time (minutes) 0 20 40 60 80 100 Time (minutes) Figure 6.2: Irati shale TGA pyrolysis. Low temperature isothermal tests, x: this work; +: Capudi [36] heating rate, the highest one possible with our TGA system. Points above 15 minutes were discarded to better visualize the initial weight loss. It is clear how the heating rate may affect the determination of the reaction rate constant. Also, the tests made above 475 °C indicate that Capudi [36] was dealing with a richer shale, unless the samples were not completely pre-dried. Chapter 6. Kinetic. Parameters for Devolatilization 80 100 5 10 15 20 25 30 35 40 45 Time (minutes) 100 ¥ T C £ 98 + X + 96 X 94 3 X + 92 X + X X + 90 - + - % 88 500°C *x X X 0 10 20 30 40 50 60 70 80 Time (minutes) 100 99 98 97 96 95 94 93 92 91 90 x + X X 475°C x x . * x x x X X X _J 1 J 1 5 10 15 20 25 30 35 Time (minutes) 5 10 15 20 25 30 35 Time (minutes) Figure 6.3: Irati shale TGA pyrolysis. High temperature isothermal tests, x: this work; +: Capudi [36] The reaction rate constants obtained by Capudi [36] refer to the first 2 minutes of the devolatilization process, and are displayed in Figure 6.5 as curve 1. These rate constants were recalculated according to a non-linear least squares method, disclosed in Appendix D, and the results are displayed in Figure 6.5 as curve 2. In both cases the equation used Chapter 6. Kinetic Parameters for Devolatilization 81 JS "5 100 * 98 96 94 92 90 88 + + + X X 5 0 0 ° C x 100°C/min . 185°C/min + maximum + + + X X J i I i I i I i I i I i L J i L 0 2 4 6 8 10 12 14 16 18 20 Time (minutes) Figure 6.4: Influence of heating rate on reaction rate. Points at maximum heating rate obtained by Capudi [36]. to fit the data was: X = l - e - * ' . (6.4) The present method of calculation gave smaller standard errors of estimate (s.e.e.)— which is a measure of the scatter of the points [51, Chapter 14]—than the method used by Capudi [36], but the trend of both curves is the same. Basically the reaction rate constants did not change much within the temperature range. Also shown in Figure 6.5, as curve 3, are the reaction rate constants calculated from the data in Figure 6.1 again using the non-linear least square approach. They show much lower rate constants, as expected, because their calculation also included points in the region above 2 minutes Chapter 6. Kinetic Parameters for Devolatilization 3.0 1.25 1.35 1.45 1.55 1.65 1000/T(K) Temperature (°C) curve 1 Capudi [36] curve 2 Capudi's data [36] curve 3 this work k (1/min) s.e.e k (1/min) s.e.e k (1/min) s.e.e 350 375 400 425 450 475 500 525 3.2812 0.0425 2.9798 0.0625 3.1828 0.0378 3.3799 0.0282 2.7305 0.0292 4.1656 0.0758 6.1168 0.1410 8.2757 0.1671 3.7446 0.0283 3.7060 0.0371 3.4292 0.0339 3.1680 0.0231 2.7119 0.0291 3.2062 0.0364 3.8059 0.0625 4.5803 0.0709 0.0188 0.1472 0.0285 0.0888 0.0473 0.0444 0.0940 0.0340 0.1478 0.1086 0.1910 0.1406 0.1724 0.1238 0.1895 0.1459 calculated ^experimental)2 number of points Figure 6.5: Pyrolysis rate constants from isothermal TGA. The graph and table compare the result of this work, obtained from long reaction time data, with Capudi's [36], obtained from short reaction time data, for Irati oil shale devolatilization. s.e.e. Chapter 6. Kinetic Parameters for Devolatilization 83 Table 6.2: Comparison of pyrolysis rate constants. Rate constants k min~* obtained by isothermal TGA. Numbers in parentheses denote analysis time in minutes. Hubbard Diricco Allred Campbell this Capudi Temp. and and [29] et al. work [36] (°C) Robinson [27] Bar rick [28] [33] 350 0.0188 (270) 3.2812 (2) 375 0.0061 (90) 0.0285 (180) 2.9798 (2) 400 0.0209 (150) 0.0260 (30) 0.037 0.0252 0.0473 (150) 3.1828 (2) 425 0.1107 (60) 0.0750 (12) 0.11 0.0888 0.0940 (100) 3.3799 (2) 450 0.2273 (20) 0.5000 (2) 0.35 0.1478 (40) 2.7305 (2) 475 0.5833 (15) 0.8600 (2.3) 0.77 (12) 0.1910 (35) 4.1656 (2) 500 0.8578 (10) 1.40 (7) 0.1724 (75) 6.1168 (2) 525 1.5913 (7) 2.46 0.1895 (35) 8.2757 (2) of reaction time, which exhibit lower reaction rates. In the table included in Figure 6.5, it is noteworthy that unacceptably high values of s.e.e. exist particularly for the high temperature experiments. These are a consequence of major percentages of material losses during the heat-up periods. High values of s.e.e. disqualify Equation 6.4 as a good correlation for the experimental points. In Figure 6.5, curve 3 seems to obey the Arrhenius equation, except for the two highest temperatures points. If the final points in the curves for these temperatures in Figure 6.1 were ignored in the reaction rate constant calculations, the rate cons tants at 500 °C and 525 °C would rise. Figure 6.6 shows the dependency of the reaction rate constants on the total analysis time for the 500 °C experiment. For the 500 °C experiment, consideration of the points up to 75 minutes (see Figure 6.3) led to a reaction rate constant even lower than the one obtained for 475 °C, whose calculation regarded only points until 36 minutes. The above analysis suggests that the rate constant for a defined temperature is a Chapter 6. Kinetic Parameters for Devolatilization 84 1.1 0.9 -a i 0.7 * 0.5 0.3 0.1 10 20 30 40 50 Analysis time (minutes) 60 70 Figure 6.6: Pyrolysis rate constants at 500 °C. Dependency of reaction rate constant on analysis time. Data from Figure 6.3 function of the amount of larger reaction time data considered in its calculation. A rate constant which varies by a factor of 5 with time is clearly of little use. The more weight(%) versus time points at long reaction times are considered, the lower the overall reaction rate constant will be. This important characteristic seems to have been overlooked in previous studies that used the isothermal TGA method to obtain the reaction rate constants [27, 28, 29, 33, 36]. Results are compared in Table 6.2, which displays the reaction rate constants k and their respectives TGA total analysis time spent to collect the points of weight(%) versus time. The studies differ in the following aspects: Chapter 6. Kinetic Parameters for Devolatilization 85 • Hubbard and Robinson [27], Diricco and Barrick [28] and Allred [29] used home-made apparatuses, while Campbell et al. [33], Capudi [36] and this work used standard comercial TGA apparatuses. See Table 6.1 for details. • All studies used Colorado oil shale except Capudi [36] and this work which used Irati oil shale. • Hubbard and Robinson [27], Diricco and Barrick [28] and Allred [29] studied the decomposition of organic matter, while Capudi [36] and this work regarded the devolatilization of organic matter. See introduction of this chapter for details. • All reaction rate constants refer to a first order reaction with respect to the re-actant organic matter, except those of Allred [29] which refer to an autocatalytic mechanism. • The temperatures cited in Table 6.2 refer to the nearest temperature at which the analyses were made. • All reaction rate constants were calculated by a linear method, except those by Capudi [36] and this work, which used a non-linear method. The dependency—for each temperature—of the values of the reaction rate constants on the analysis time is noteworthy. The results from Capudi [36] are remarkably higher than all the others, because the analysis time of his studies was only 2 minutes. The aforementioned results lead one to conclude that the reaction rate constants ob-tained by isothermal TGA are significantly affected by the heating rate used, and by the number of experimental points at long times considered in their calculations. It is always a good procedure to shorten the heat-up period, for instance by pre-heating the analysis environment and then placing the shale sample in it. However, it is not straightforward Chapter 6. Kinetic Parameters for Devolatilization 86 o 405°C o 422°C • 446°C x 471°C + 498°C ° ° ° ° o o 0 0 0 1 • . . I • • • 0 20 40 60 80 100 120 140 160 Time (minutes) Figure 6.7: New Brunswick shale isothermal TGA. to decide what is the most convenient analysis time to be considered for the calculation of reaction rate constants. Figure 6.6 suggests that beyond the first 20 minutes the reac-tion rate no longer depends on the analysis time. Longer reaction times benefit maximum yield. Nevertheless, for those reactors that minimize residence time instead of maximizing yield, and when particles are rapidly brought to the reactor temperatures, the method used by Capudi [36] should be followed. Isothermal TGA data for New Brunswick shale are shown in Figure 6.7, which is similar to Figure 6.1, for Irati shale. Although the values differ, the trends are similar for both shale samples. Because of the above stated drawbacks of the isothermal method, the results for the isothermal experiments with New Brunswick oil shale were not analyzed further. 1UU 98 96 A Q4 ox) • l-H ^ 92 90 88 on 0 x °„ • o a X o 0 X 0 x a + x • x ° no a + x ^ + X x x x x x x x X x + + + + + + + . . I . . . I . . . I I ° O c o o « 0 • * « 0 o o • • • D • Chapter 6. Kinetic Parameters for Devolatilization 87 6.4 Non-isothermal T G A Tests Experimental data—sample weight versus time—were obtained for Irati and New Bruns-wick oil shales by the non-isothermal TGA technique. Heating rates of 5, 10, 20 and 50 °C/min were chosen, based on similar values in applications that use moving, spouted and fluidized beds. Entrained beds usually operate at much higher heating rates that are not possible to reproduce in a TGA apparatus. The experimental data were analyzed according to the equations listed in subsection 4.2 which allow the calculation of the kinetic parameters. Figures 6.8 - 6.11 and 6.12 - 6.15 present experimental data respectively for Irati and New Brunswick oil shales for the four heating rates mentioned above. Figures 6.8(a) -6.15(a) show the raw data relating sample weight (%) and temperature with time. The data are also listed in Appendix E. Figures 6.8(b) - 6.15(b) show the relation between conversion (X) and time, where conversion is defined as: initial sample weight — sample weight initial sample weight — final sample weight The final weight was taken to be the sample weight at 540 °G. This temperature was chosen in order to avoid the weight loss contribution of decarbonization reactions that release CO2 above that temperature. Weight (%) results at 540 °C as a function of heating rates for both shales are given in Table 6.3. These results are in agreement with the results shown in Figure 3.1. Figures 6.8(b) - 6.15(b) also show plots of polynomial approximations which were fitted to the experimental points. A good continuous representation was obtained by two fourth-order polynomials linked by a straight line. Whenever necessary, the equations for these curves were used to generate the conversion value and its derivative with respect to time for any time value. Not only were the polynomials useful for that purpose, but they also provided more reliable relations than the experimental points themselves due to the Chapter 6. Kinetic Parameters for Devolatilization 88 Time (minutes) Time (minutes) Figure 6.8: Irati shale non-isothermal TGA pyrolysis - 50 °C/min. Time (minutes) Time (minutes) Figure 6.9: Irati shale non-isothermal TGA pyrolysis - 20 °C/min. observed scatter of the latter. Appendix F presents these polynomial approximations for each case. Next, the data generated by the polynomials approximations are treated by the meth-ods listed in Section 4.2 Chapter 6. Kinetic Parameters for Devolatilization 89 Time (minutes) Time (minutes) Figure 6.10: Irati shale non-isothermal TGA pyrolysis - 10 °C/min. Time (minutes) Time (minutes) Figure 6.11: Irati shale non-isothermal TGA pyrolysis - 5 °C/min. 6.4.1 Arrhenius equation method The equation of Arrhenius type: , fdX/dT\ , k0 El . . allows the calculation of the kinetic parameters with plots like the ones shown in Figures Chapter 6. Kinetic Parameters for Devolatilization 90 550 500 450 0.8 400 S 350 | | 0 6 300 g, g E O 0.4 250 H 200 150 100 0.2 6 8 10 12 14 16 Time (minutes) 9 10 11 12 Time (minutes) on 13 14 Figure 6.12: New Brunswick shale non-isothermal TGA pyrolysis - 50 °C/min. 10 15 20 Time (minutes) 8 10 12 14 16 18 20 22 24 26 28 Time (minutes) O n Figure 6.13: New Brunswick shale non-isothermal TGA pyrolysis - 20 C/min. 6.16 and 6.17. The slope of the straight line segments representing the curves give E/R while the intercept point gives — \n(k0/8). Values of dX/dT were obtained from: dX_ _ cLY_cU_ _ dX 1 dT ~ ~dTdf ~ ~dtjr The calculated values of E and kQ are displayed in Table 6.4. (6.6) One should note that at conversions approaching zero, the ordinate values in Figures Chapter 6. Kinetic. Parameters for Devolatilization 91 Table 6.3: TGA sample weight (%) at 540 °C. heating rate (°C/min) Irati New Brunswick 50 91.80 90.39 20 90.59 90.09 10 89.96 90.35 5 89.74 90.36 Table 6.4: Pyrolysis kinetic parameters from Arrhenius type equation. Determined over the linear portions of Figures 6.16 and 6.17 heating rate slope intercept E kQ (°C/min) (E/R)xl0-3 (k J/mole) (1A) Irati shale 50 15.5 -16.6 129 1.32xl07 20 12.8 -13.4 106 2.26 xlO 5 10 13.9 -15.7 116 1.84xl05 5 - 13.4 -15.4 112 3.95xl05 New Brunswick shale 50 17.1 -19.3 142 1.84x10s 20 16.9 -19.8 141 1.27xl08 10 19.2 -23.6 159 2.73xl0 9 5 15.0 -17.9 124 4.51 xlO 6 Chapter 6. Kinetic. Parameters for Devolatilization 92 0 10 20 30 40 50 60 20 25 30 35 40 45 50 Time (minutes) Time (minutes) Figure 6.14: New Brunswick shale non-isothermal TGA pyrolysis - 10 °C/min. Time (minutes) Time (minutes) Figure 6.15: New Brunswick shale non-isothermal TGA pyrolysis - 5 °C/min. 6.16 and 6.17 tend to +oo, because dX/dT approaches zero. At conversion values between zero and 0.1 the shape of the upper part of the curves in Figures 6.16 and 6.17 is strongly affected by the polynomial approximation. If the fourth order polynomials were replaced by a straight line for that conversion range, which would still provide a good representation (see Figures 6.8(b) - 6.15(b)), the u-shaped curve shown at large values of 1000/T in the figures would change to another pattern. For the four heating rates, Figures 6.8(b) Chapter 6. Kinetic Parameters for Devolatilization 93 9 1.2 1.4 1.6 1.8 2 2.2 1000/T(K) Figure 6.16: Irati shale pyrolysis kinetic data plotted according to the Arrhenius type equation. r _ _ , , , i , , , i , i i i i , , i , , u_j 1.2 1.4 1.6 1.8 2 2.2 1000/T(K) Figure 6.17: New Brunswick shale pyrolysis kinetic data plotted according to the Arrhenius type equation. Chapter 6. Kinetic Parameters for Devolatilization 94 - 6.15(b) indicate that for low conversion values, X < 0.1, dX/dT initially increases, then decreases, but eventually increases again for values of X > 0.15. This change in derivative value is responsible for the aforementioned u-shaped curves in Figures 6.16 and 6.17, for which no physical interpretation is available. For this low conversion region, considering that temperatures would also be low, the kinetic parameters obtained over the linear portions of Figures 6.16 and 6.17 provide a good approximation. Some authors [34, 37, 52] proposed to use a second straight line to connect the points close to the upper end of the straight line segments on the curves in Figures 6.16 and 6.17. This second curve would be in agreement with the two-step model suggested by Equation 5.1. At the lower ends of the curves in Figures 6.16 and 6.17, one should note that at X = 1 the ordinate value is —oo. For X ~ 0.9 the curves suffer a sharp adjustment, caused by a decrease followed by an increase in the reaction rate, also observed in Figures 6.8 -6.15. This seems to be a characteristic of Irati and New Brunswick oil shales and was not reported by previous investigators [36, 52]. This increase in rate at high conversion values should not be understood as initial liberation of CO2 from carbonates, because it happened even at 470 °(J in the 5 °C/min heating rate experiment, which is a temperature too low for decarbonization reactions. It should be rather interpreted as an increase in the release of gases and vaporized oil coming from the complex pyrolysis reactions. Anyway, the slope for this new straight line at X > 0.9 on the X - T plots is not very different from the previous one to justify the use of a different energy of activation for this region. It should be noted that the slight disturbances'on the lower ends of the straight lines segments of the curves in Figures 6.16 and 6.17 are caused by the straight lines used to connect the two fourth order polynomials that represent the experimental data. The straight line connectors caused a noticeable change in the dX/dT versus time curves, which are responsible, for the mentioned disturbances. From Table 6.4 the average value of the energy of activation (E) for Irati shale is Chapter 6. Kinetic Parameters for Devolatilization 95 115.5 kJ/mole. This is close to the 117.2 k.J/mole value obtained with the Arrhenius equation for the same shale by Lisboa and Watkinson [52], who used discrete experimental points instead of the continuous polynomial representation used in this study. They also obtained good agreement with Capudi's data [36], who worked with Irati shale and made his analyses five years before. The values in Table 6.4 for the frequency factor (kQ) for Irati shale can be compared with the average value of 1.11 x 106 1/s obtained by Lisboa and Watkinson [52]. A great discrepancy is observed in Table 6.4 among k0 values for different heating rates. Frequency factor values are strongly affected by the slope of the straight line from which they are obtained, because of the nature of the logarithmic function used to calculate them. For instance, reducing the slope of the curve for Irati shale 50 °C/min case by 10%, which is almost an unnoticeable change, would bring down E by 10%, but k0 would be reduced by 85%. This problem of association of E and k0 will be discussed at the end of this chapter. Rajeshwar [34], who also used this method to calculate E and k0 values for Colorado Green River oil shale kerogen, obtained a similar scatter among his estimated parameter values. However, the plot of the experimental data according to the Arrhenius equation parameters generated two straight line segments for each heating rate case. The same method was used by Shih and Sohn [37], who worked with Colorado Anvil Points shale. Only one straight line segment was found to represent the points in the Arrhenius plot. Again a similar scatter was found to occur among the E and k0 values. 6.4.2 Coats and Redfern method The equation of Coats and Redfern [47]: - I n - ln(l -X) = - I n k0R BE 2RT E EL RT (4.10) / Chapter 6. Kinetic Parameters for Devolatilization 96 1.2 1.4 1.6 1.8 2.0 22 1.2 1.4 1.6 1.8 2.0 2_2 1000/T(K) 1000/T(K) Figure 6.18: Pyrolysis kinetic data plotted according to the Coats and Redfern equation, (a): Irati Shale, (b): New Brunswick shale allows the calculation of the kinetic parameters with plots like the ones in Figure 6.18. The slopes of the straight line segments give E/ R. This method supposes that the first term on the right hand side of Equation 4.10 remains essentially constant. Figures 6.19 and 6.20 focus on the region where 0.9 < X < 0.1 for the four heating rates investigated. Each curve can be approximated by two straight lines represented by the dashed lines. This pair of straight lines indicates that the devolatilization process can be described by two major consecutive reactions, according to the mechanism of Equation 5.1. The calculated energies of activation are given in Table 6.5, in which E values are placed into two columns, for high temperature and low temperature ranges. At X = 0 and X = 1, the ordinate values in Figure 6.18 are respectively +00 and — 00. At X < 0.1, conversion proceeds at a very low rate. At X ~ 0.9, the rate decreases, to increase again at higher conversion values, as can be seen in Figures 6.8 - 6.15. The energy of activation values displayed in Table 6.5 could be used as an approximation to describe the reaction rates at these two end regions. This method does not allow the Chapter 6. Kinetic Parameters for Devolatilization 97 15.5 15.0 ? I 4 , X d 14.0 3 ' 13.5 13.0 12.5 1.25 1.35 1.65 1.75 1.45 1.55 1000/T(K) Figure 6.19: Irati shale pyrolysis kinetic data plotted according to the Coats and Redfern equation. 15.5 15.0 _ 14.5 d 14-0 ja i a ' 13.5 13.0 12.5 -50°C/min ^20°C/min ^10°C/min —5°C/min _i i I i i i ...i I i i i 1.25 1.35 1.65 1.75 1.45 1.55 1000/T(K) Figure 6.20: New Brunswick shale pyrolysis kinetic data plotted according to the Coats and Redfern equation. Chapter 6. Kinetic Parameters for Devolatilization 98 Table 6.5: Pyrolysis kinetic parameters from Coats and Redfern equation. heating rate temperature range E k > (°C/min) (°C) (k J/mole) (1/s) Irati shale 50 348 - 423 27.3 1.03x10 - l 423 - 488 105 2.78xl0 5 20 350 - 418 33.8 3.07x10 - l 418 - 475 100 1.02xl05 10 342 - 396 40.3 6.21x10 - l 396 - 448 101 8.93 xlO 4 5 322 - 375 30.1 4.24x10 - 2 375 - 431 89.5 7.76xl04 New Brunswick shale 50 374 - 438 55.4 35.9 438 - 485 150 8.92xl08 20 360 - 410 53.0 12.1 410 - 458 122 5.18xl06 10 352 - 397 62.0 34.1 397 - 446 150 2.76xl0 8 5 322 - 376 45.2 12.1 376 - 427 114 1.06x10s calculation of frequency factor values. The average Irati shale E value of 31.9 kJ/mole and 97.1 k.I/mole for the low and high temperature ranges are comparable to the 27.0 kJ/mole and 99.7 kJ/mole obtained by Lisboa and Watkinson [52]. The data in Figures 6.19 and 6.20 appear to describe a continuous curve. It can be seen that this graphical method of determining E values is very dependent on the range of experimental points selected to be approximated by the straight line segments. The same two region model was used by Rajeshwar [34], working with Colorado Green Chapter 6. Kinetic Parameters for Devolatilization 99 River kerogen, Thakur and Nuttall Jr. [35], working with Moroccan oil shale, and Capudi [36], who worked with Irati shale. He obtained average E values of 28.6 and 92.0 kJ/mole respectively for the low and high temperature ranges for heating rates of 2, 5, 10, 20 and 50 °C/min. 6.4.3 Freeman and Carroll method The equation of Freeman and Carroll [46]: Aln(dX/dr) £ A(i/r) Aln( l -X) + / ?Aln( l - X) 1 } allows the calculation of the kinetic parameters with plots such as the ones in Figures 6.21 and 6.22. The slope of the straight line segments give E/R. The intercept point of Equation 4.9 stands for the negative value of the order of the reaction, which was presumed to be first order in this study. That is why the constant 1 appears in Equation 4.9. Thus, the equation of Freeman and Carroll allows the calculation of the energy of activation and the order of the reaction. Figures 6.21 and 6.22 show only the straight line segments, i.e., the curves representing the regions of low and high conversions are not shown. At these end regions, as with the previous methods, the curves completely depart from a straight line pattern. Only one straight line segment was obtained for each curve. Table 6.6 presents the calculated values of energy of activation and order of reaction. The column indicating the reaction order shows how good is the initial assumption of first order reaction. Energy of activation values exhibit the usual scatter. Values for New Brunswick shale are higher than for Irati shale, which is consistent with the results from previous methods. Rajeshwar [34], who worked with Colorado shale, observed that the same straight line represented well his experimental data for both analyses at 5 and 20 °C/min, when Chapter 6. Kinetic Parameters for Devolatilization 100 Table 6.6: Pyrolysis kinetic parameters from Freeman and Carroll equation. heating rate temperature range E reaction order (°C/min) (°C) (k J/mole) Irati shale 50 400 - 480 123 0.4 20 363 - 445 104 0.7 10 352 -418 117 1.0 5 341 - 396 111 0.8 New B runswick shale 50 379 -462 149 1.3 20 354 -440 155 1.9 10 361 -428 151 0.6 5 329 - 409 129 1.3 plotted according to the Freeman and Carroll method. His range of abscissa values was very narrow, from 0 to 0.5, which corresponds to very high conversions. 6.4.4 Integral method The Integral method of calculation of kinetic parameters is based on the equation: , / 2RT\ , k0 E 1 , + l n ( l _ _ ) = _ b l ^ + _ _ (4.11) The method is praised for not requiring derivative values such as dX/d i , which are required for most of the other methods listed in Section 4.2. The kinetic parameters can be obtained from plots such as the ones shown in Figures 6.23 - 6.26. The slope of the straight line segments give EjR while the intercept points give — ln(k0/E). Figures 6.23 and 6.24 represent the high temperature region. The curves were extended to show the changes in slope that occur at high conversion values, usually above 0.9. There is -3\n(l — X) RT2 Chapter 6. Kinetic Parameters for Devolatilization 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 103A(1/T)/Aln(l-X) Figure 6.21: Irati shale pyrolysis kinetic data plotted according to Freeman and Carroll equation. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 lafyl /TVAlnCl-X) Figure 6.22: New Brunswick shale pyrolysis kinetic data plotted accon to the Freeman and Carroll equation. Chapter 6. Kinetic. Parameters for Devolatilization 102 a marked departure from the straight line pattern. Figures 6.25 and 6.26 represent the low temperature range, i.e., the early stages of the reaction. Here the data are more consistent, but show curvature which is not predicted by Equation 4.11. It should be noted that Equation 4.11 demands an initial guess for the value of E. This first guess allows the calculation of a second E value that should be compared to the first one. If the difference between the two values is greater than an acceptable increment, the new E value should be used to obtain a third E value and so on until good convergence is obtained. Table 6.7 shows the calculated kinetic parameters. The E values displayed in Table 6.7 are less than 1% different from their previous estimates. As also happened with the method of Coats and Redfern, Section 6.4.2, the graphically obtained kinetic parameters are strongly affected by the range of points considered to be represented by any straight line segment. This alone contributes to significant scatter among the kinetic parameter values, as observed in Table 6.7. Because of the nature of the logarithmic function, which is involved in the calculation of the frequency factor, the scatter of the k0 values is particularly large. Shih and Sohn [37] also used this method to calculate kinetic data for Colorado shale. When plotted according to Figures 6.23 - 6.26 their data conformed to one straight line, for each heating rate, for the whole range of temperatures, which was not possible with the data obtained in this work. Yang and Sohn [38], who worked with Chinese shale from Fushun, displayed a graph for the high temperature range, with abscissa values similar to those in Figures 6.23 - 6.26, i.e., lower than 1.48. It is presumed that at lower temperatures, the straight line behaviour was no longer followed. Chapter 6. Kinetic Parameters for Devolatilization 103 15 1.2 1.3 1.4 1.5 1.6 1000/T(K) Figure 6.23: Irati shale pyrolysis kinetic data plotted according to the Integral equation - high temperature 16 JQ I i , i , L _ j I i I I I i i , I I i I i I 1.2 1.3 1.4 1.5 1.6 1000/T(K) Figure 6.24: New Brunswick shale pyrolysis kinetic data plotted according to the Integral equation - high temperature Chapter 6. Kinetic Parameters for Devolatilization 104 -50°C/min 12.5 1.40 1.45 1.50 1.55 1000/T(K) 1.60 1.65 Figure 6.25: Irati shale pyrolysis kinetic data plotted according to the Integral equation - low temperature 16.0 _ 15.5 I 15-0 B + 14.5 14.0 M 13.5 13.0 12.5 — 50°C/min — 20°C/min ^10°C/min —5°C/min 1.70 1.50 1.55 1.60 1000/T(K) Figure 6.26: New Brunswick shale pyrolysis kinetic data plotted according to the Integral equation - low temperature Chapter 6. Kinetic Parameters for Devolatilization 105 Table 6.7: Pyrolysis kinetic parameters from Integral equation. heating rate temperature range E k0 (°C/min) (°C) (k.J/mole) Irati shale 50 353 - 429 36.1 50.0 429 - 489 110 4.18xl0 7 20 353 - 404 32.2 16.2 404 - 462 82.1 2.42 xlO 5 10 352 - 395 47.3 18.2 395 - 445 100 4.70xl0 6 5 344 - 383 47.7 12.9 383 - 431 94.2 1.19xl06 New Brunswick shale 50 367 - 414 36.8 1.01 414 - 467 101 1.16xl07 20 354 - 391 37.2 0.550 391 - 446 96.3 3.30xl06 10 352 - 392 58.5 18.7 392 - 437 133.8 1.96xl09 5 322 - 356 34.6 0.990x10 - i 356 - 412 86.7 2.67 xlO 5 6.4.5 Chen and Nuttall method The method suggested by Chen and Nuttall [50] is another iterative method of obtaining kinetic parameters. The method is based on the equation: - I n E + 2RT I n 1 , k0R E 1 (4.14) T2 "' 1 - X\ The method is similar to the Integral method, in terms of the approach needed to obtain the desired data. An initial guess for the E value is necessary to start the iter-ative procedure. Figures 6.27 and 6.28 shows the plot of the left hand side of Equation Chapter 6. Kinetic Parameters for Devolatilization 106 4.14 against 1/T for the high temperature —and consequently high conversion— values. Figures 6.29 and 6.30 do the same for the low temperature results. As with the integral method, there is a decided curvature to the plots. Figures 6.27 and 6.28 also show the parts of the curves which depart from straight line behaviour, usually for X > 0.9. Here again the kinetic parameter values are affected by the choice of points which will be ap-proximated by a particular straight line segment. The effects are greater in the k0 value calculations, due also to the logarithmic function involved in the process. Table 6.8 lists the results. Displayed values are converged to within 1%. Thakur and Nuttall Jr. [35] used this method to obtain kinetic data for Moroccan oil shale. Data for one region is listed, with no information on the temperature range. Capudi [36] reported values for the two regions —low and high temperature ranges. Average E values were respectively 29.6 and 92.7 kJ/mole, which compare with the average values of this work of 39.7 and 93.3 kJ/mole. 6.4.6 Friedman method The Friedman method [45] is characterized by the equation: _ m ^ = - m [ A : 0 ( l - X ) ] + |I (4.7) It differs from previous methods because to obtain the kinetic parameters, results from analyses at different heating rates are used simultaneously. The method requires that the first term of the right hand side of Equation 4.7 be kept constant. The only way to do that is to get dX/dt and 1/T pairs of points at the same X, which can be obtained from analyses at different heating rates. Kinetic parameters are obtained through plots such as Figures 6.31 and 6.32. Each curve required data from the analyses at the four heating rates used in this study. Arbitrarily chosen conversions between 0.2 and 0.8 were selected to generate the data Chapter 6. Kinetic Parameters for Devolatilization 107 2.5 2.0 h _2 Q i . . . . i I . , , , I . , , . I ' 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1000/T(K) Figure 6.27: Irati shale pyrolysis kinetic data plotted according to the Chen and Nuttall equation - high temperature range. Chapter 6. Kinetic Parameters for Devolatilization 108 3.3 2 g r , , . , i , , , , i , , , , i . , . , i , , , , i , ' 1.35 1.40 1.45 1.50 1.55 1.60 1000/T(K) Figure 6.29: Irati shale pyrolysis kinetic data plotted according to the Chen and Nuttall equation - low temperature range. 4.4 2 4 i . , • , i , . , , i , , , . i . , . , i , , . , i , . . , i 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1000/T(K) Figure 6.30: New Brunswick shale pyrolysis kinetic data plotted according to the Chen and Nuttall equation - low temperature range. Chapter 6. Kinetic Parameters for Devolatilization 1.25 1.30 1.35 1.40 1.45 1000/T(K) 1.50 1.55 1.60 Figure 6.31: Irati shale pyrolysis kinetic data plotted according to the Friedman equation. 4.0 0.0 r 1 • • • 1 1 1.30 1.35 1.40 1.45 1.50 1.55 1000/T(K) Figure 6.32: New Brunswick shale pyrolysis kinetic data plotted according to the Friedman equation. Chapter 6. Kinetic Parameters for Devolatilization 110 Table 6.8: Pyrolysis kinetic parameters from Chen and Nuttall equation. heating rate temperature range E k0 (°C/min) (°C (k J/mole) Irati shale 50 389 - 439 48.6 26.2 439 - 495 110 1.12x10s 20 373 - 414 36.9 1.97 414 - 466 84.3 1.55xl04 10 352 - 395 39.1 99.6 395 - 448 93.7 6.67xl04 5 338 - 379 34.4 24.2 379 - 428 84.8 9.94xl0 3 New Brunswick shale 50 379 - 414 37.1 3.03 414 - 464 92.0 9.98xl04 20 360 - 391 33.5 0.755 391 - 439 84.4 1.88xl04 10 351 - 397 54.6 26.6 397 - 442 135 1.19xl08 5 332 - 360 33.9 0.284 360 - 412 80.2 3.95xl0 3 for the curves in Figure 6.31 and 6.32. The slope of these curves give E/R and the intercept point give — ln[fc0(l — X)]. Kinetic parameter results are given in Table 6.9. Energy of activation values are higher than the ones obtained by other methods, a trend also observed in the study of Shih and Sohn [37]. 6.4.7 Summary of results Tables 6.10 and 6.11 list respectively the energy of activation and frequency factor values obtained for Irati shale. Tables 6.12 and 6.13 do the same for New Brunswick shale. Chapter 6. Kinetic Parameters for Devolatilization Table 6.9: Pyrolysis kinetic parameters from Friedman equation. Irati shale New Brunswick shale X E (kJ/mole) h (1/s) E (kJ/mole) K (1/s) 0.2 149 4.89 xlO 8 181 1.41xl0 n 0.3 160 2.62 xlO 9 188 4.56xl0 n 0.4 155 1.09xl09 191 6.93xlO u 0.5 174 2.78 x lO 1 0 161 4.98 x lO 0 9 0.6 201 2.05 x lO 1 2 190 7.92xlO n 0.7 - - 221 l.OOxlO1 4 0.8 - - 267 -Table 6.10: Energy of activation for Irati shale. Values in kJ/mole were obtained by non-isothermal TGA methods using four heating rates. Val-ues are listed for the low and high temperature reactions. method heating rate (°C/min) 50 20 10 5 Arrhenius Coats &z Redfern Freeman & Carroll Integral Chen k Nuttall. high temperature reaction 129 106 116 112 106 95.3 101 86.7 123 104 117 111 110 82.1 100 94.2 110 84.3 93.7 84.8 Coats & Redfern Integral Chen & Nuttall low temperature reaction 28.0 31.3 40.1 28.0 36.1 32.2 47.3 47.7 48.6 36.9 39.1 34.4 Chapter 6. Kinetic Parameters for Devolatilization 112 Table 6.11: Frequency factor for Irati shale. Values in 1/s were obtained by non-isothermal TGA methods using four heating rates. Values are listed for the low and high temperature reactions. method heating rate (°C/min) 50 20 10 5 Arrhenius Coats Sz Redfern Integral Chen & Nuttall high temperature reaction 1.32xl07 2.26x10s 1.84xl05 3.95xl05 3.18xl05 4.08xl0 4 4.90xl0 6 2.81xl0 5 4.18xl0 7 2.42xl0 5 4.70xl0 6 1.19xl06 1.12x10s 1.55xl04 6.67xl04 9.94xl0 3 Coats & Redfern Integral Chen & Nuttall low temperature reaction 7.20 10.7 0.603 2.64xl0- 2 50.0 16.2 18.2 12.9 26.2 1.97 99.6 24.2 The large ranges of E and and even larger ranges of k0 values are similar to reported cases in the literature concerning solid decomposition reactions such as thermal decom-position of polymers and minerals. It was observed that in most cases an empirical linear relation existed between E and k0 values of the form: In fc0 = a £ + b (6.7) where a and b are constants. The above mentioned is part of a simple theory called the kinetic compensation effect (KCE). A good recent review of this topic is given by Koga [53]. KCE arises from the effects that numerous parameters may cause in obtaining kinetic data from TGA experiments, among them: sample weight, shape, size and chemical composition, heating rate, atmosphere, and partial pressure of gaseous products. The KCE theory does not question the use of the Arrhenius equation as the starting Chapter 6. Kinetic Parameters for Devolatilization 113 Table 6.12: Energy of activation for New Brunswick shale. Values in kJ/mole were obtained by non-isothermal TGA methods using four heat-ing rates. Values are listed for the low and high temperature reactions. method heating rate (°C/min) 50 20 10 5 Arrhenius Coats & Redfern Freeman $z Carroll Integral Chen & Nuttall high temperature reaction 142 141 159 124 144 128 150 118 149 155 151 129 101 96.3 134 86.7 92.0 84.4 135 80.2 Coats & Redfern Integral Chen & Nuttall low temperature reaction 53.2 54.3 65.6 47.9 36.8 37.2 58.5 34.6 37.1 33.5 54.6 33.9 point for any method of obtaining kinetic data from TGA experiments. However, as mentioned by Garn [54, 55], the Arrhenius equation is strictly valid only for homogeneous reactions. Its use for heterogeneous reactions is an extrapolation, and so a major source of wide discrepancy in E and k0 values. However, Garn acknowledges that the Arrhenius equation might provide correct results for some heterogeneous systems. Figures 6.33 and 6.34 plot the kinetic data for the Irati and New Brunswick oil shales according to Equation 6.7. The trend towards linearity is evident. The method used in this chapter to obtain the kinetic parameters was the classical one, followed also by previous investigators, listed in Section 4.2. Considerable scattering occurs when results from different methods are compared. Within a given method, results are different for different heating rates. Also, the temperature which separates the low Chapter 6. Kinetic Parameters for Devolatilization 114 Figure 6.34: Low temperature pyrolysis kinetic data according to KCE theory. Chapter 6. Kinetic Parameters for Devolatilization 115 Table 6.13: Frequency factor for New Brunswick shale. Values in 1/s were obtained by non-isothermal TGA methods using four heating rates. Values are listed for the low and high temperature reactions. method heating rate (°C/min) 50 20 10 5 Arrhenius Coats & Redfern Integral Chen & Nuttall high temperature reaction 1.84x10s 1.27x10s 2.73xl0 9 4.51xl0 6 3.09x10s 6.81xl06 6.13x10s 1.71xl06 1.16xl07 3.30xl06 1.96xl09 2.67xl0 5 9.98xl04 1.88xl04 1.19x10s 3.95xl0 3 Coats & Redfern Integral Chen & Nuttall low temperature reaction 24.2 16.7 74.8 1.71 1.01 0.550 18.7 0.990X10- 1 3.03 0.755 26.6 0.284 temperature reaction and high temperature reaction is a function of method and heating rate. Although these results, which arise from the basic consideration that devolatilization follows a first order reaction with respect to volatilizable material concentration, could be used to describe both TGA data and large particle data, see Figure 5.30, it is not possible to predict reliable kinetic results from the procedure used. In order to minimize the error, it is suggested to obtain TGA data at the same heating rate to be used in the situation being considered. Two techniques could refine the method of obtaining the kinetic data: 1. all investigated methods used linearized forms of X versus T equations. Nonlinear fitting could improve the results as demonstrated by Chen and Aris [56], and used previously with the isothermal method, following the procedure in Appendix D. As Chapter 6. Kinetic Parameters for Devolatilization 116 temperature is not constant along a TGA run, the reaction rate constant is not also constant during a TGA run. That replaces the one-dimensional search for the best fitted fc, made with the isothermal method, for the two-dimensional search for the best fitted kQ and E. 2. the search for the best k0 and E also benefit from reparameterization techniques, as suggested by Kittrell [57] and used by Froment [58]: k0 exp RT = k0 exp M r , / exp « v r f) = k* exp M r T) where k* is a reparameterized frequency factor and T is an average temperature, or suggested by Chen and Aris [56]: £ _ B/1000 k0 exp"«r = kQ exp « T / 1 0 0 0 Reparameterization avoids situations of calculating small diferences of large num-bers, which improves accuracy. A third option would be to use the unreacted core model to interpret the TGA data, although, as shown in Chapter 4, the TGA runs followed a uniform conversion model, not affected by the mass of particles and particle sizes, for the limits investigated. Chapter 7 Spent Shale Particle Combustion Modelling Pyrolysis alone is not able to vaporize all organic matter from oil shale. Partially reacted kerogen and coke make up a carbonaceous residue or char that is left behind after all or almost all hydrogen-rich molecular fragments are released as oil during the pyrolysis reactions. Some oxygen, sulfur and nitrogen may still be present, as the spent shale analyses in Table 3.1 indicate. The mineral matrix is not expected to undergo any significant change in composition at the usual retorting temperatures which are below 550°C. On the other hand, oil shale experiences significant physical changes due to the loss of organic matter during pyrolysis. Therefore, spent shale particles are more porous than raw shale particles. This alone could affect the combustion mechanism. Thus, spent shale shows a different composition concerning organic matter, the same composition concerning mineral matter, and a larger voidage than raw shale. The combustion process is executed in an oxidizing atmosphere at temperatures above 550°C. Besides the expected reaction between carbon and oxygen: several other solid-solid and gas-gas parallel reactions occur. For instance, for shale rich in calcite and dolomite, the following reactions: C + 0 2 CO 2 (7.1) CaCO.3 # CaO + CO 2 (7.2) (7.3) 117 Chapter 7. Spent Shale Particle Combustion Modelling 118 CaCO.3 + Si0 2 ^ CaSi0 3 + CO 2 (7.4) release carbon dioxide, which can combine with carbon: C + C 0 2 2CO (7.5) to produce carbon monoxide, which then may be oxidized: CO + 1/2 0 2 ^ CO 2 (7.6) Reaction 7.5 is an important source of carbon consumption, along with Reaction 7.1. Reactions 7.2 - 7.5 are responsible for the loss of weight observed when spent shale is heated above 550°C in an inert atmosphere. This loss is indicated in Figure 3.1 for Irati and New Brunswick oil shales, when temperature was increased from 510°C to 810°C. The greater loss attained by New Brunswick spent shale is due to its greater content of carbonates. Close to the end of the experiments depicted in Figure 3.1 the sweep gas was switched from nitrogen to oxygen. New Brunswick spent shale did not show any further weight loss, indicating that all carbon in char had been consumed by Reaction 7.5, while Irati spent shale —with less carbonate content— still had some carbon to be consumed by Reaction 7.1, and showed some weight loss. According to Table 3.1, the mineral carbon content of New Brunswick oil shale is 2.7%. If it is only present as CO2 in carbonates, the amount of CO2 would be 9.9%. If this CO2 is totally reacted with the remaining carbon according to Reaction 7.5, the reacted carbon would be 2.7%. So it is expected that in going from 510°C to 810°C in an inert atmosphere, New Brunswick oil shale would face a weight loss of 2.7 + 9.9 =12.6%. This is consistent with the drop in weight displayed in Figure 3.1 from 89 to 77% of the initial weight. Another way to show the role of Reactions 7.2 - 7.5 is to compare TGA curves for raw shale and the same shale depleted of CO2. One way to withdraw C 0 2 from carbonates in raw shale is to boil it in acid solutions. Figure 7.1 shows TGA curves for New Brunswick Chapter 7. Spent Shale Particle Combustion Modelling 119 S 100 98 b 96 94 92 90 88 86 84 82 80 «-o o shale with CaCCh • demineralized shale ° 0 ° ° o O o o o D • • r , 550 500 450 400 Q 350 I & I -a 300 X ^ u 250 H -3 200 150 100 5 H 3 - & 95 h 100 90 85 80 75 o shale with CaC03 • demineralized shale on *EtD O O O O O D D OCD o o o o o o 900 800 700 600 ti 1 500 3 S3 a. 400 | 300 200 100 0 20 40 60 80 100 120 140 160 180 Time (minutes) 20 40 60 80 Time (minutes) 100 120 Figure 7.1: Effects of shale CaC0 3 content on TGA. shale containing CaCO;} and without it. The latter was produced by boiling a shale sample in a 6N HC1 solution for 15 minutes, to allow the total or partial conversion of carbonates. The graph on the left of Figure 7.1 shows that for temperatures up to 510°C, both samples exhibit approximately the same weight loss, as expected, because at this temperature the reactions are not affected by the presence of carbonates. On the other hand, the graph on the right of Figure 7.1 shows that when the temperature goes up to 810°C, the sample rich in carbonates loses more weight than the other, due to the fast carbonate decomposition and char - C O 2 reactions. Thereafter, the shale without carbonates loses weight by a slow reaction rate, possibly enabled by the initial acid treatment. Oxygen and carbon dioxide compete for the available carbon in oxidizing atmospheres, according to Reactions 7.1 and 7.5. While both overall reaction rates are functions of temperature, Reaction 7.5 is not limited by gas diffusion constraints, as happens with Reaction 7.1, which depends on the rate of oxygen diffusion into the particle, determined by the effective oxygen diffusivity inside the particle. Any model describing spent shale particle combustion should take into account all or most of Reactions 7.1 - 7.6, according to the shale composition. Also, suitable mechanisms Chapter 7. Spent Shale Particle Combustion Modelling 120 should be chosen to display how each reaction develops throughout the particle. The following section presents and discusses two different concepts for spent shale combustion modelling. 7.1 Previous Models Two models were selected from the literature to show different mechanisms by which spent shale particles are oxidized. The first model was specifically designed to describe reactions in large particles. Reaction 7.1 was found to follow a shrinking core model. Model outputs were compared to experimental results. The second model was oriented to describe combustion in particles of sizes compatible with those used in fluidized beds, namely about 1 mm. Reaction 7.1 was assumed to happen throughout the particle. No comparison with experimental data was supplied. Both models assumed that the mineral decomposition reactions and the char-CO-2 reaction occurred uniformly throughout the particle. 7.1.1 Model 1 The model proposed by Mallon and Braun [59] was designed to describe their high tem-perature experiments with spent shale cylindrical particles 15 cm in diameter and 15 -25 cm in height. Particles were brought to constant temperatures of 538°C, 620°C and 704°C by slow heating rates: 0.2 - 4°C/ min. When the particle surface reached the spec-ified test temperature, the particle centerpoint temperature was 20 - 50°C lower, and extra time was required to bring the whole particle to a uniform temperature. The au-thors considered the particles to be isothermal. In their modelling, particle temperature was supplied by the experimental data, and did not have to be calculated by an energy balance. Chapter 7. Spent Shale Particle Combustion Modelling 121 The reactions considered were 7.1, 7.2, 7.3 and 7.5, which were assumed to happen uniformly throughout the particle except Reaction 7.1 which was considered to happen at a sharp interface between an unreacted core and a reacted outer layer. The authors were driven to that conclusion based on observations and measurements made of partially retorted particles. In one experiment at 704°C and in an inert atmosphere, the particle was almost uniformly depleted of carbonates and carbon, supporting the assumption that Reactions 7.2, 7.3 and 7.5 occur homogeneously throughout the particle. In another experiment, also at 704°C but with air, two distinct regions were formed within the particle: a core and a shell, the latter with less carbon than the other. The same two-region pattern was exhibited in an experiment at 538°C with air. At this temperature no carbonate decomposition occurs, so only Reaction 7.1 was occurring. These two later experiments indicated that Reaction 7.1 followed a shrinking core model. Material balance equations were provided for each component. The rate of dolomite decomposition, from Reaction 7.3, is given by: ^ = —Kio Cdo- (7.7) The rate of calcite production, from Reactions 7.3 and 7.2, is given by: dC = / c a kcl0 Cdo — kca Cca (7-8) dt where fca is the mass stoichiometric factor for production of calcite from dolomite. The rate of char consumption in the unreacted core, from Reaction 7.5, is given by: ^ = -kchCchYCo2. (7.9) Finally, an equation was provided to describe the combustion of char, Reaction 7.1, that was assumed to follow a shrinking core model controlled by oxygen diffusion in the ash layer. The rate of oxygen consumption in the cylindrical particle is given by: dN0, „ d°oA (2nrL) -V, dt v y \ " e dr Chapter 7. Spent Shale Particle Combustion Modelling 122 The latter equation is then integrated from conditions at the particle surface, r = rp and CQ2 — Co2gi to conditions at the core surface, r = rc and Ceo = 0 to obtain dNp2 , r dt r In - = 2ir LVeCo3g. p Replacing dNo2 = (fo2 Cc)(27trcL)drc, where fo2 is the mass stoichiometric factor for the consumption of oxygen from char, one obtains drc Ve Co2g (7.10) dt fo2Ccrc\n(rc/rp) Equations 7.7 - 7.9 indicate that the reactions were assumed to be irreversible and first order with respect to each reactant. To solve the model, reaction rate constants were obtained from literature. For the oil shale char - C O 2 reaction, the rate constant was obtained from experiments with coal char - C O 2 reactions. The effective diffusiv-ity of oxygen was experimentally obtained by measuring core radius versus time in an experiment at 538°C, at which temperature only the combustion reaction occurred. 7.1.2 Model 2 The model proposed by Ballal and Amundson [60], which is similar to that of Morell and Amundson [61], was conceived to study a fluidized bed combustor development. The primary goal was to understand the char consumption as a source of heat and the carbonate decomposition as a source of calcium oxide, which works as a sulfur absorber when recycled to a fluidized bed pyrolyzer. The model is very general; it comprises heat and mass balances that give temperatures and component concentrations as functions of time and position. The reactions considered were 7.1, 7.2, 7.4, 7.5 and 7.6. They were all considered to be first order with respect to their reactants. The reversibility of the reactions involving solids were adequately considered. Chapter 7. Spent Shale Particle Combustion Modelling 123 The material balance for each of the " i " gaseous components, namely carbon dioxide, carbon monoxide, oxygen and nitrogen, in a spherical particle, is given by: jt{eCax%) = -^(r*Ni) + JZvivK (7-H) where ;/ t p is the stoichiometric coefficient of component V in reaction Rv is the rate of reaction upn. The summation of the above equations for all species 11 i" leads to: u t ' U l i=l p=l that can be introduced into the first equation to produce: U i ' U l p=l ' U l t=l p=l Replacing Ni = J, + XjN, the final mass balance equation becomes: ot 1 0 1 0 7 i=\ P =i P=I subject to: ^ = 0 at r = 0 or Ji ~\~ NX{ = kc{X{ — xiy) a t ?' = Tp x i — £%o at ^ — 0 • Similarly, for each solid component ".s", namely carbon, CaCO"3 and Si02, the material balance is given by: 9 C s o ' X s = J2 "spRp (7.13) P=I where: Chapter 7. Spent Shale Particle Combustion Modelling 124 The heat balance is given by: ± jt[eClCpt{T - Tr)] +1 §-t[Cscps(T - Tr)] = ~ ^ q r ) - £ L ^ r ' N ^ T - Tr)} where the left hand side terms account for the accumulation of heat in the gaseous and solid phases and the right hand side terms account for the flows of heat by conduction and convection. Defining an effective heat capacity as: 4 3 Cp — ^ ] £ Ci Cpi 4" ^  ] C s Cps i=1 s=l and substituting into the former equation, together with the expression for conductive heat flow qr = -kep(dT/dr) yields: By introducing Equations 7.11 and 7.13 and the following expression: 5 4 5 3 5 AHpRp = £ Cpi(T - Tr) ]T uipRp + CPS(t ~ T>-) uspRv p—\ i=\ p—1 s—l p=l the final equation is obtained: BT 1 d ( 2ledT\ 4 r c)T J ^ A T T „ t„,A. which has the boundary and initial conditions: ^ = 0 at r = 0 or ~Kl£ + t , N r c A T - T r ) = HT-Tg) + aep(T4-T*) at r = rp i=l T = T0 at t = 0. Chapter 7. Spent Shale Particle Combustion Modelling 125 The solution of the above equations demands values for several parameters, which were either assumed or obtained from the literature. The solution of the model for par-ticles 1 mm in diameter indicated that temperature is uniform within the particle. In that situation carbonate concentration would be essentially uniform. On the other hand, carbon concentration varied from zero at the particle surface to the initial carbon concen-tration at the particle centerpoint, for some specified conditions. Oxygen concentration also varied strongly from values close to bulk gas concentrations at the particle surface to close to zero at the particle centerpoint. 7.1.3 Analysis of models Model 1 succeeded in reproducing the experimental results regarding carbonate and car-bon concentrations, core radius, and CO-2 evolution during the tests. However, the exper-imental conditions were such that temperature was uniform inside the particle. Therefore the model required only material balances, Reactions 7.7 - 7.10. The tests performed and subsequent chemical analyses and observations from regions inside the particles demon-strated that the carbonate decomposition reactions, Reactions 7.2 and 7.3, as well as the char-C02 reaction, Reaction 7.5, occurred uniformly throughout the particle, while the combustion reaction, Reaction 7.1, occurred according to a shrinking core model. For the latter, it was assumed that'oxygen diffusion in the ash layer controlled the overall rate, which led to Equation 7.10. The four ordinary differential equations that make up the model can be solved by standard numerical methods. The rate constants required were obtained in the literature, at somewhat different experimental conditions. The effective diffusivity of oxygen was experimentally obtained and compared with theoretical values. Model 2 proposed to deal with a more complex situation, one in which temperature might not be constant inside the particle. The temperature profile, given by Equation 7.14, was considered to be affected not only by conduction heat transfer, but also by the Chapter 7. Spent Shale Particle Combustion Modelling 126 1 h a u o U time 2 , ' time 1 carbonate.Model 1 & 2 -carbon, Model 1 carbon, Model 2 0 1 1 0 1 Dimensionless radial position (r/rp) Figure 7.2: Solid Conversion by Combustion Models heat carried by the gas flowing inside the pores. Material balances were presented for each of the four gaseous species, namely oxygen, nitrogen, carbon monoxide and carbon dioxide, according to Equations 7.12, and also for each of the three solid species, namely calcite, silica and carbon, according to Equations 7-13. Voidage was considered to vary linearly with carbon conversion, with initial and final values of 0.25 and 0.30. Solution of the eight partial differential equations is not easily achieved. Parameter values were either obtained from the literature or arbitrarily assumed. Only theoretical results are shown, for particles 1 mm in diameter. They indicate that the particles are isothermal, carbonate and silica concentrations are uniform along the particle radius, and carbon conversion is greater in regions close to the particle surface. Model 1 and 2 agree on several points. Both models dealt with situations in which Chapter 7. Spent Shale. Particle Combustion Modelling 127 the particle was isothermal. This was imposed in Model 1, but a calculated result in Model 2, due to the small particle size considered. A consequence of that situation is that carbonate decomposition, Reactions 7.3 and 7.2, and carbonate consumption by Reaction 7.4, occur uniformly throughout the particle, because these reactions are mostly affected by temperature. This is depicted in Figure 7.2 for two different times, where time 2 is greater than time 1. Carbonate conversions increase with time, and the same conversion is expected at each radial position. Both models predict this behavior. However, the models disagree concerning carbon conversion. Carbon concentration is affected by Reactions 7.1 and 7.5. Model 1 assumed that Reaction 7.1 follows a shrinking core model, and Reaction 7.5 occurs uniformly throughout the particle. The compounded result is shown in Figure 7.2 by step-curves for the two different times. The curves indicate that the core radius moves inward with increasing time and carbon conversion inside the core increases with time. The core radius is ultimately defined by Reaction 7.1, while the carbon conversion inside the core is only affected by Reaction 7.5. For the latter, Figure 7.2 shows that it happens uniformly throughout the core, keeping the carbon concentration constant, again because of isothermal conditions. Model 2 predicts S-shaped curves for carbon conversions, at least for 1 mm particles, as indicated in Figure 7.2. The curves are also the compound effect of Reactions 7.1 and 7.5. Model 2 is more complete than Model 1, which was designed to deal with a very specific situation, and therefore it is able to handle widely different conditions. A consequence is that it requires more parameters, not only the thermal ones, but also the voidage and its variation while reaction proceeds. It is also physically more reasonable that combustion reactions vary with radius rather than occurring at a plane. Chapter 7. Spent Shale Particle Combustion Modelling 128 7.2 Characteristics of an Ideal Model An ideal model able to describe combustion reactions in an oil shale particle must com-ply with the same requirements outlined in Section 5.2 concerning mechanism, system of coordinates and parameter evaluation. The mechanism should consider each reaction independently. Correct choices should be taken about which reactions should be consid-ered, according to the oil shale mineral composition. The same list of required parameters displayed in Table 5.1 applies here. However, the list must be enlarged to include the oxygen effective diffusivity and particle voidage, which are difficult to obtain. 7.3 Proposed Model All the reported investigations on oil shale combustion processes employed small parti-cles, appropriate for bubbling or circulating fluidized beds, as well as entrained beds. It is not effective to burn large particles of the size used in moving beds, or even spouted beds. Therefore, most of the exiting research plants, or those that existed in the past, used a combination of a moving bed for pyrolysis and a fluidized type of bed for com-bustion. Within that option, spent shale particles exiting from the pyrolyzer would have to be crushed to sizes adequate for the subsequent burning equipment. In the process of crushing oil shale, the sharp edges present in large particles are smoothed, and particles approach spherical shape-Besides being more spherical and smaller, the spent shale particles are much more porous than when raw. As was seen before, many cracks develop throughout the particles during the pyrolysis process, creating channels between the particle surface and its interior. The model proposed here takes all of that into account. Due to the small size of and the presence of the cracks in each particle, the shrinking core model does not seem appropriate. It is very likely that oxygen would diffuse into the particle very easily and Chapter 7. Spent Shale Particle Combustion Modelling 129 consume carbon throughout. That does not mean that uniform conversion would hap-pen with certainty along the particle radius. The proposed model accounts for these two extreme possibilities, and of course all cases between them, characterized by carbon con-sumption throughout the particle but not uniformly. Carbon and oxygen concentrations may not have constant values within the particle during the burning process. This means that the model must have mass balances for each component. Another important parameter is the heat of reaction. In the pyrolysis process this can be disregarded due to its small contribution when compared with all the heat demanded to raise the shale temperature to reaction levels. In the burning process, the heat of reaction cannot be disregarded. To demonstrate that, two runs were made to measure the center point temperature of cubic spent shale particles of size 1.3 cm. Figures 7.3 and 7.4 show experimental results, tabulated in Appendix B, which indicate that the particle center point temperatures rise above any other temperature in the system, due to the heat of combustion that is not quickly dissipated. The heat transfer direction is changed, now occurring from the particle to the environment. It can be seen in both figures that this effect is present at temperatures below 300 °C, a fact well known for shale. This is an indication that oxygen could access the center point of this "large" particle and burn the carbon present there. This would not be possible if the physical situation were of a shrinking non-porous core. The proposed model was presented before by Lisboa and Watkinson [52] and is de-scribed by the following equations: (7.15) Chapter 7. Spent Shale Particle Combustion Modelling 130 500 wall gas after particle gas before particle particle center 20 40 Time (minutes) 60 Figure 7.3: Spent shale temperature profiles during shale combustion. 800 600 13 400 200 wall gas after particle particle center 20 40 Time (minutes) 60 Figure 7.4: Spent shale temperature profiles during shale combustion. Chapter 7. Spent Shale. Particle Combustion Modelling 131 §£=0 at r = 0 or -fcP §J = h(T -Tg) at r = rv T = T0 at t = 0 ^ e 1 5 / 2dCo2 r r 2 cV V d + # r (7.16) ^ = 0 at r = 0 - P e ^ 2 - = kc(Co2 - Co29) at r = rp Co2 — Co2o at t = 0 —S = Rr = kCcCo2 (7.17) — (^ co at t ^ 0 The model is adequate for shales of low carbonate contents, like Irati shale, because the carbonate decomposition reaction is not considered. This system of equations was solved by the numerical method of lines as presented in the Appendix G. The listing of the FORTRAN program used to solve the problem is given in Appendix A under the name oxidation. A great reduction in computational time was observed when an integration subroutine more adequate to handle stiff sets of equations was used. Figure 7.5 shows a typical output from the model, with a table listing the values adopted for the parameters. The figure shows the particle temperature, carbon conversion and oxygen concentration as functions of time at the mid-point of the particle radius. For the simulated conditions there was an insignificant gradient within the particle, either for the temperature, conversion or oxygen concentration. Therefore, the results shown refer to any point within the particle. Chapter 7. Spent Shale Particle Combustion Modelling 132 Time (minutes) Figure 7.5: Particle combustion modelling. Data refer to a point at r = rp/2. Parameters used for the above simulation: dp = 2 mm C -= 5250 mole/m3 h = 300 J/s.m 2.K PP = 2100 kg/m 3 CQ20 = 2.63 mole/m3 kc = 0.87 m/s kp = 1.25 J/s.m.K Co2g = 2.63 mole/m3 Ve = 4.4 x 10~5 m2/s Cp = 1045 J/kg.K T -± o — 25 °C AHr = 3.93 x 105 J/mole e = = 0.15 T — 650 °C k0 = 8.0 x 1010 m3/mole.s E = 245. kJ/mole The temperature increases quickly to a value slightly above the gas temperature. The high heat transfer coefficient of 300 J/s.m 2.K promotes initially a fast heating of the particle and later a rapid dissipation of the heat of reaction. The reaction proceeds at a Chapter 7. Spent Shale Particle Combustion Modelling 133 temperature just slightly above the gas temperature. The oxygen concentration, initially at the highest possible concentration, decreases sharply as the particle is heated. As the reaction proceeds, the rate of oxygen diffusion to the particle interior offsets the rate of oxygen consumption by the oxidizing reaction causing a net effect of slowly increasing the oxygen concentration. The following figures show the influence of key parameters on this basic simulation. The parameters chosen were the heat transfer coefficient (h), which affects primarily the heat transfer process, the effective diffusivity (T>e), which affects primarily the mass trans-fer process, and the gas temperature (Ty), which affects primarily the reaction process. The effect of the heat transfer coefficient is best observed for the first two minutes of the reaction, as indicated in Figure 7.6. The higher the heat transfer value, the faster the particle reaches the environment temperature, and the faster the heat from the oxidation reaction is dissipated. The effect of the oxygen effective diffusivity inside the particle is illustrated in Figure 7.7. The effective diffusivity used in the basic case shown in Figure 7.5 was 0.75 times the molecular diffusivity of oxygen in air at 400 °C, which is the maximum possible. For this high value, there is no oxygen concentration gradient within the particle. If this coefficient is reduced to 0.1 times its maximum value, a gradient is then observed, as indicated in Figure 7.7. The lower diffusivity also causes a lower conversion. Figure 7.8 shows the effect of the gas temperature on conversion. As expected, higher reaction temperatures cause increased conversions. Chapter 7. Spent Shale Particle Combustion Modelling 800 0.0 0.5 1.0 1.5 2.0 Time (minutes) Figure 7.6: Influence of heat transfer coefficient on particle temperature in oil shale oxidation. Other parameters are listed with Figure 7.5. Chapter 7. Spent Shale. Particle Combustion Modelling 135 t= 12 min t= 9 min t= 6 min - De= 4.4x10"'m'/s D = 5.9xl0"6m2/s t= 3 min — i I . I . I 0.0 0.2 0.4 0.6 0.8 1.0 Particle radius (mm) Figure 7.7: Influence of oxygen diffusivity on oxidation conversion. Other parameters are listed with Figure 7.5. 1 . 0 0.8 c 0.6 U 0.4 0.2 -^ ^ ^ ^ T = 650 c T=625 °c ^ ^ - ^ ^ T,= 600 c 1 1 1 0 5 1 0 - 1 5 Time (minutes) Figure 7.8: Influence of reactor temperature on oxidation conversion. Other parameters are listed with Figure 7.5. Chapter 8 Kinetic Parameters for Combustion For very fine particles, mass transfer resistances are negligible and intrinsic kinetics is the controlling mechanism. Oxidation of carbonaceous residue in shale appears to obey a first order reaction with respect to both carbon and oxygen concentration. This was the conclusion of Soni and Thompson [62] and Sohn and Kim [63]. This observation was used in modelling a fixed bed combustion process by Lee and Sohn [64] and a lift-pipe combustor by Dung [65]. Intrinsic kinetic parameters for combustion have been determined by a TGA appa-ratus, under conditions of gas flow and particle size which minimize ash diffusion and gas film resistances, similar to the. work done by Soni and Thompson [66]. In this study, combustion kinetic parameters were obtained by TGA. 8.1 Reaction Order The proposed model in Section 7.3 assumes that the burning of shale occurs throughout the particle. Carbon and oxygen concentrations can vary with position in the particle and with time. A simple equation to describe the rate of carbon and oxygen consumption would be: R,. = -kCo2Cc (8.1) which, by introducing a term for carbon conversion: 0 (• — (j c A'= c 136 Co Chapter 8. Kinetic Parameters for Combustion 137 becomes: dX ~dt fcC0a(l-X). (8.2) Integrating Equation 8.2 using the initial condition that X = 0 at t = 0 yields ln(l -X) = kCo2t. (8.3) The assumption in Equation 8.1 that the rate is first order with respect to both carbon and oxygen concentration is demonstrated to be valid in the next section. 8.2 Isothermal T G A Tests Isothermal thermogravimetric analyses were made with the apparatus in Figure 4.1 at temperatures of 550 °C, 600 °C, 625 °C, 650 °C and 700 °C for oxygen molar fraction of 21%. Experimental data for Irati shale are shown in Figure 8.1. A significant scatter of the data was observed, due to the very nature of the combustion process. However, non-isothermal TGA was not possible, because the particle temperatures could not be increased at a specified rate after the ignition temperature was reached. The isothermal TGA method to obtain kinetic parameters for oxidation suffers the same limitations as when it is applied to pyrolysis, as discussed in Section 6.3. Figure 8.2 presents the experimental data according to the equation: from which values for the product k Co2 can be obtained from the slopes of the straight lines passing through the points associated with each temperature. Figure 8.3 presents the data for In k Co2 vs. 1/T according to the equation: - l n ( l -X) = kCo2t (8.3) Chapter 8. Kinetic Parameters for Combustion 138 0 10 20 30 40 50 60 70 80 Time (min) Figure 8.1: Isothermal TGA for oxidation at different temperatures. from which values of k0Co2 a n d E/R can be obtained respectively from the intercept and slope of the straight line connecting the points related to each temperature. Finally the pre-exponential factor (A;0) and the energy of activation (E) were obtained as: E = 245 kJ/mole k0 = 8.0 x 1010 m3/mol.s The oxygen consumption was assumed to be first order with respect to oxygen con-centration in Equation 8.1. This was checked by doing thermogravimetric analyses at 5%, 10%, 15% and 21% oxygen in nitrogen, all at the same temperature of 650 °C. Figure 8.4 presents the experimental data according to the equation: Chapter 8. Kinetic Parameters for Combustion 139 ^550°C 600°C 625°C 650°C 700°C 60 70 80 Time (min) Figure 8.2: Experimental combustion TGA results at different tempera-tures. -\n(l-X) = kCo2t (8.3) from which values for the product k Co2 c a n he obtained from the slopes of the straight lines fitted to the points associated with each oxygen concentration. Figure 8.5 presents the data k Co2 vs. oxygen concentration. The straight line obtained indicates the linear dependency of the rate equation on oxygen concentration. Chapter 8. Kinetic Parameters for Combustion Figure 8.3: Experimental combustion TGA results at different tempera-tures. Figure 8.4: Experimental combustion TGA results at different oxygen concentrations (650 °C). Chapter 8. Kinetic Parameters for Combustion 142 Figure 8.5: Experimental combustion TGA results (650 °C). Chapter 9 Conclusions and Recommendations onclusions: • A working mathematical model which describes the transient temperature profile within a three-dimensional shale particle was developed and successfully tested against experimental data. • A working mathematical model which describes mass loss during the pyrolysis of shale was successfully tested against experimental data. The model considers that pyrolysis in the particle follows an unreacted core mechanism. • A standard thermogravimetric apparatus proved to be reliable and accurate for measurements of temperature and mass loss of shale. These data can be further used in kinetic studies, either for pyrolysis or combustion. • Devolatilization kinetic parameters were obtained for a first order—on kerogen concentration—rate equation, using non-isothermal thermogravimetric data and published relationships. It was demonstrated that these relationships can all be derived from the same source. The work indicates that kinetic data obtained this way should be used with care. • A working mathematical model which describes the temperature profile and carbon and oxygen concentration within a particle was developed for shale combustion. 143 Chapter 9. Conclusions and Recommendations 144 The model considers the particle to have enough porosity to allow oxygen access to its interior. • Combustion kinetic parameters were obtained for a first order—on carbon and oxy-gen concentration— rate equations, using isothermal thermogravimetric data. • The numerical method of lines was able to handle all computational calculations in the models presented. Recommendations for future work: • To adapt the pyrolysis mathematical method for mass loss to three-dimensional particles. • To develop better kinetic models for oil shale pyrolysis. • To obtain the effective diffusivity of oxygen inside spent shale Nomenclature Ac core surface area A-p particle surface area Bi Biot number: hdp/kp, h r p / k p or h LXtyt~/kp Cc carbon concentration Ceo initial carbon concentration C c a calcite concentration Cell char concentration Cdo dolomite concentration Cg total gas concentration C v volatile matter concentration c ^vo initial volatile matter concentration Co2 oxygen concentration Co2g bulk oxygen concentration Co2o initial oxygen concentration ce V effective heat capacity of particle Cpi heat capacity of gaseous component " i " Cp heat capacity of particle-heat capacity of solid component "s" cs concentration of solid component "s" Csol total solids concentration dp particle diameter ve effective diffusivity of oxygen E energy of activation fca mass stoichiometric factor for production of calcite from dolomite fo2 mass stoichiometric factor for consumption of oxygen from char h heat transfer coefficient Ji diffusional flux of component " i " k reaction rate constant kc mass transfer coefficient k0 frequency factor or pre-exponential factor K reparameterized frequency factor or pre-exponential factor k calcite decomposition rate constant char-CO2 reaction rate constant kdo dolomite decomposition reaction rate constant kg gas thermal conductivity kp particle thermal conductivity K effective particle thermal conductivity L length half width of particle in direction x,y,z 145 Nomenclature N total flux of gaseous components flux of gaseous component " i " Nu p particle Nusselt number: h dp/kg Nv amount of volatile matter N initial amount of volatile matter No2 amount of oxygen Pr Prandt number Q gas flow rate Qr heat flux by conduction r radial coordinate rc core radius rv particle radius R gas constant Rr reaction rate Rp rate of reaction "p" Rep particle. 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Dev., no. 24, 753 - 761 (1985). [65] Dung, N.V., A Mathematical Model of a Dilute Phase Transport Combustion of Spent Oil Shale, Fuel Processing Technology, no. 17, 235 - 262 (1988). [66] Soni,Y., Thomson, W.J., Carbon Residue Reaction Kinetics for Retorted Oil Shale, Proceedings 11 t h Oil Shale Symposium, 364 - 377 (1978). [67] Spiegel, M.R., Mathematical Handbook of Formulas and Tables, McGraw-Hill Book Co., USA (1968). [68] Turner, R.C., Schnitzer, M. , Thermogravimetry of the Organic Matter of a Podzol, Soil Science, vol. 93., 225 (1962). Bibliography 153 [69] Turner, R.C., Hoffman, I., Chen, D., Thermogravimetry of the Dehydration of Mg(OH)2, The Canadian Journal of Chem. Eng., vol. 41, 243 (1963). [70] Doyle, CD.,Series Approximation to the Equation of Thermogravimetric Data, Na-ture, vol. 207, 290 - 291 (1965). [71] Schiesser, W.E., The Numerical Method of Lines: Integration of Partial Differential Equations, Academic Press, San Diego (1990). [72] Schiesser, W.E., Computational Mathematics in Engineering and Applied Science: ODEs, DAEs, and PDEs, CRC Press, Boca Ranton (1994). [73] Forsythe, G.E., Malcom, M.A., Moler, C.B., Computer Methods for Mathematical Computations, Prentice-Hall, Englewood Cliffs (1977). [74] Press, W.H., Teukolsky, S.A., Vettering, W.T., Flannery B.P. Numerical Recipes in FORTRAN, 2 n d edition, Cambridge Univ. Press (1992). Appendix A FORTRAN programs A . l Temperature profile in a rectangular parallelepiped Below is the listing of program t e m p e r a t u r e , which was the basic program for all tem-perature profile calculations for rectangular parallelepiped. It uses the numerical method of lines to integrate partial differential equations with three space coordinates and the time coordinate. In this implementation the program solves the problem described by Equation 5.13, i.e., transient temperature profiles in a parallelepiped, subject to an environment at a different temperature than the particle's, including heat transfer by radiation. Subroutine r k f 4 5 (not listed), suggested by Forsythe et al. [73], is the ordinary dif-ferential equation integrator. The discretization of the partial differential equation is performed by subroutine d s s 0 0 4 (not listed) as suggested by Schiesser [71]. Output of this program is plotted in Figures 5.7 and 5.8. p r o g r a m t e m p e r a t u r e c T r i d i m e n s i o n a l t e m p e r a t u r e p r o f i l e i n a m e t a l i c c u b e c i n i t i a l l y a t t o a n d a t t i m e > 0 b e i n g c h e a t e d b y a n e n v i r o n m e n t a t t g . D i m e n s i o n a l n u m b e r s . c P o i n t ( 0 , 0 , 0 ) i s a t t h e c e n t e r o f t h e c u b e . c S p a c e d i f f e r e n t i a t i o n i s p e r f o r m e d b y d s s 0 0 4 c a l l e d b y c d e r v . I n t e g r a t i o n i s p e r f o r m e d b y r k f 4 5 c ( r u n g e k u t t a f e h l b e r g ) . c A d j u s t d i m e n s i o n o f y v t o > n u m b e r o f e q u a t i o n s : n x * n y * n z c i f > 1 4 0 0 a d j u s t a r r a y s i n r k f 4 5 . i m p l i c i t r e a l * 8 ( a - h , o - z ) c o m m o n / t / t i m e / y / t ( 2 1 , 2 1 , 2 1 ) / f / t t ( 2 1 , 2 1 , 2 1 ) c o m m o n / n / n x , n y , n z , i x p , i y p , i z p 154 Appendix A. FORTRAN programs 155 c o m m o n / n l / x u , y u , z u , d x , d y , d z , b i m , a l p h a , t g , t w , b i r c o m m o n / n 2 / x ( 2 1 ) , y ( 2 1 ) , z ( 2 1 ) d i m e n s i o n y v ( 1 3 3 1 ) e x t e r n a l f e n c c d a t a f o r s t a i n l e s s s t e e l ( 3 0 4 ) : c r h o = d e n s i t y ( k g / m ~ 3 ) c r k = t h e r m a l c o n d u c t i v i t y ( J / m i n m ~ 2 K / m ) c c p = s p e c i f i c h e a t , ( J / k g K ) c e m i = e m i s s i v i t y c s i g m a = s t e f a n - b o l t z m a n c o n s t a n t ( J / m i n m ~ 2 K ~ 4 ) c d a t a r h o , r k , c p , e m i , s i g m a / 7 . 7 d 3 , 1 . 0 d 3 , 5 . d 2 , 0 . 6 0 d 0 , 3 4 0 . 2 d - 8 / c o p e n ( u n i t = 7 , f i l e = ' n u m o l 4 . d ' ) o p e n ( u n i t = 8 , f i l e = ' n u m o l 4 . s ' ) o p e n ( u n i t = 9 , f i l e = ' n u m o l 4 a . s ' ) o p e n ( u n i t = 1 0 , f i l e = ' n u m o l 4 b . s ' ) c c r e a d n x ( n u m b e r o f x g r i d p o i n t s ) , c n y ( n u m b e r o f y g r i d p o i n t s ) , c n z ( n u m b e r o f z g r i d p o i n t s ) , c i x p ( c o u n t e r : p r i n t i n g i n t e r v a l f o r x ) , c i y p ( s a m e f o r y ) , i z p ( s a m e f o r z ) , c t i m e f ( m a x i m u m t i m e [ m i n ] ) , d t i m e ( t i m e i n t e r v a l f o r p r i n t i n g c r e s u l t s [ m i n ] ) , e p s ( a l l o w a b l e e r r o r ) c r e a d ( 7 , l ) n x , n y , n z , i x p , i y p , i z p , t i m e f , d t i m e , e p s 1 f o r m a t ( 6 i 5 , 3 d l 0 . 2 ) c c r e a d t o ( i n i t i a l t e m p e r a t u r e [ K ] ) , t g ( g a s t e m p e r a t u r e [ K ] ) , c t w ( r e a c t o r w a l l t e m p e r a t u r e [ K ] ) , c x u , y u , z u ( p a r t i c l e h a l f l e n g t h [ m ] i n x , y a n d z d i r e c t i o n s ) , c h ( h e a t t r a n s f e r c o e f f i c i e n t [ J / m i n m 2 K ] ) , c r e a d ( 7 , 1 0 0 ) t o , t g , t w , x u , y u , z u , h 1 0 0 f o r m a t ( 7 d l 0 . 1 ) c b i r = s i g m a * e m i / r k b i m = h / r k b i x = b i m * x u a l p h a = r k / ( r h o * c p ) c c w r i t e h e a d l i n e c Appendix A. FORTRAN programs 156 w r i t e ( 6 , 5 ) n x , i x p , x u , t 0 - 2 7 3 . , e p s , n y , i y p , y u , t g - 2 7 3 . , e m i , n z , i z p , z u , + t w - 2 7 3 . , b i x w r i t e ( 8 , 5 ) n x , i x p , x u , t o - 2 7 3 . , e p s , n y , i y p , y u , t g - 2 7 3 . , e m i , n z , i z p , z u , + t w - 2 7 3 . , b i x 5 f o r m a t ( t 2 , ' n x = ' , i 3 , t l 0 , ' i x p = ' , i 3 , t 2 0 , ' x u ( m ) = ' , d l 0 . 4 , + t 4 0 , ' t o ( C ) = ' , d l 0 . 4 , t 6 0 , ' e p s = ' , d 6 . 1 , / , + t 2 , ' n y = ' , i 3 , t l 0 , ' i y p = ' , i 3 , t 2 0 , ' y u ( m ) = ' , d l 0 . 4 , + t 4 0 , ' t g ( C ) = ' , d l 0 . 4 , t 6 0 , ' e m i = ' , f 5 . 2 , / , + t 2 , ' n z = ' , i 3 , t l 0 , ' i z p = ' , i 3 , t 2 0 , ' z u ( m ) = ' , d l O . 4 , + t 4 0 , ' t w ( C ) = ' , d l 0 . 4 , t 6 0 , ' b i x = ' , d l 0 . 4 , / ) c c s e t g r i d s p a c i n g c d x = x u / d f l o a t ( n x - 1 ) d y = y u / d f 1 o a t ( n y - 1 ) d z = z u / d f l o a t ( n z - 1 ) c c s e t s p a c e g r i d a n d i n i t i a l t e m p e r a t u r e s c d o 2 i = l , n x x ( i ) = d f l o a t ( i - l ) * d x d o 2 j = l , n y y ( j ) = d f l o a t ( j - l ) * d y d o 2 k = l , n z z ( k ) = d f l o a t ( k - l ) * d z t ( i , j , k ) = t o 2 c o n t i n u e t i m e = 0 . d O c p r i n t i n i t i a l c o n d i t i o n s c a l l p r i n t c p a r a m e t e r s t o b e p a s s e d t o r k f 4 5 n e q = n x * n y * n z d o 3 k = l , n z k l = ( k - l ) * ( n x * n y ) d o 3 j = l , n y k 2 = ( j - l ) * n x d o 3 i = l , n x y v ( k l + k 2 + i ) = t ( i , j , k ) 3 c o n t i n u e t v = t i m e t o u t = t i m e + d t i m e r e l e r r = e p s a b s e r r = e p s i f l a g = l 4 c a l l r k f 4 5 ( f c n , n e q , y v , t v , t o u t , r e l e r r , a b s e r r , i f l a g ) Appendix A. FORTRAN programs c c h e c k f o r i n t e g r a t i o n p r o b l e m s i n r k f 4 5 i f ( i f l a g . n e . 2 ) t h e n w r i t e ( 6 , 7 ) i f l a g 7 f o r m a t ( l x , ' o d e s o l u t i o n f a i l s ; i f l a g = ' , i 2 ) s t o p e n d i f c u p d a t e i n d e p e n d e n t v a r i a b l e ( t i m e ) t i m e = t v c p r i n t r e s u l t s c a l l p r i n t c c h e c k f o r e n d o f i n t e g r a t i o n i f ( t v . l t . ( t i m e f - 0 . 5 d 0 * d t i m e ) ) t h e n t o u t = t v + d t i m e g o t o 4 e n d i f s t o p e n d c s u b r o u t i n e p r i n t c p r i n t d e s i r e d r e s u l t s i m p l i c i t r e a l * 8 ( a - h , o - z ) c o m m o n / t / t i m e / y / t ( 2 1 , 2 1 , 2 1 ) / f / t t ( 2 1 , 2 1 , 2 1 ) c o m m o n / n / n x , n y , n z , i x p , i y p , i z p c o m m o n / n 2 / x ( 2 1 ) , y ( 2 1 ) , z ( 2 1 ) c p r i n t r e s u l t s k = l c k = n z / 2 + l w r i t e ( 6 , 1 0 ) t i m e , z ( k ) w r i t e ( 8 , 1 0 ) t i m e , z ( k ) 1 0 f o r m a t ( / , ' t i m e ( m i n ) = ' , d l 0 . 4 , 3 x , ' z ( m ) = ' , d l 0 . 4 , / ) w r i t e ( 6 , 2 ) ( x ( i ) , i = l , n x , i x p ) w r i t e ( 8 , 2 ) ( x ( i ) , i = l , n x , i x p ) 2 f o r m a t ( 5 x , ' x = ' , l O x , 1 0 ( l x , d l O . 4 ) , / ) d o 3 j = n y , l , - i y p w r i t e ( 6 , 2 0 ) y ( j ) , ( t ( i , j , k ) - 2 7 3 . d 0 , i = l , n x , i x p ) w r i t e ( 8 , 2 0 ) y ( j ) , ( t ( i , j , k ) - 2 7 3 . d O , i = l , n x , i x p ) 2 0 f o r m a t ( 5 x , ' y = ' , d l 0 . 4 , 1 0 ( l x , d l 0 . 4 ) , / ) 3 c o n t i n u e w r i t e ( 9 , 6 ) t i m e , t ( 1 , 1 , 1 ) - 2 7 3 . d O w r i t e ( 1 0 , 6 ) t i m e , t ( n x , 1 , 1 ) - 2 7 3 . d O 6 f o r m a t ( l x , 3 ( d l 0 . 4 , l x ) ) r e t u r n e n d c s u b r o u t i n e f c n ( t v , y v , y d o t ) Appendix A. FORTRAN programs 158 c b r i d g e b e t w e e n r k f 4 5 a n d d e r v . c i n p u t : t v : i n d e p e n t v a r i a b l e c y v : a r r a y t o b e d i f f e r e n t i a t e d c o u t p u t : y d o t : a r r a y w i t h d i f f e r e n t i a l s i m p l i c i t r e a l * 8 ( a - h , o - z ) c o m m o n / t / t i m e / y / t ( 2 1 , 2 1 , 2 1 ) / f / t t ( 2 1 , 2 1 , 2 1 ) c o m m o n / n / n x , n y , n z , i x p , i y p , i z p d i m e n s i o n y v ( * ) , y d o t ( * ) c u p d a t e i n d e p e n t ( t a u ) a n d d e p e n d e n t ( t ) v a r i a b l e s t i m e = t v d o 1 k = l , n z k l = ( k - l ) * ( n x * n y ) d o 1 j = l , n y k 2 = ( j - l ) * n x d o 1 i = l , n x t ( i , j , k ) = y v ( k l + k 2 + i ) 1 c o n t i n u e c c a l c u l a t e d i f f e r e n t i a l s t t c a l l d e r v c u p d a t e o u t p u t y d o t d o 2 k = l , n z k l = ( k - l ) * ( n x * n y ) d o 2 j = l , n y k 2 = ( j - l ) * n x d o 2 i = l , n x y d o t ( k l + k 2 + i ) = t t ( i , j , k ) 2 c o n t i n u e r e t u r n e n d c s u b r o u t i n e d e r v c c a l c u l a t e r h s f : d i f f e r e n t i a l s t t i m p l i c i t r e a l * 8 ( a - h , o - z ) c o m m o n / t / t i m e / y / t ( 2 1 , 2 1 , 2 1 ) / f / t t ( 2 1 , 2 1 , 2 1 ) c o m m o n / n / n x , n y , n z , i x p , i y p , i z p c o m m o n / n l / x u , y u , z u , d x , d y , d z , b i m , a l p h a , t g , t w , b i r c o m m o n / n 2 / x ( 2 1 ) , y ( 2 1 ) , z ( 2 1 ) d i m e n s i o n t x x ( 2 1 , 2 1 , 2 1 ) , t y y ( 2 1 , 2 1 , 2 1 ) , t z z ( 2 1 , 2 1 , 2 1 ) , + t s ( 2 1 ) , t s d ( 2 1 ) , t s d d ( 2 1 ) c c a l c u l a t e s p a c e d i f f e r e n t i a l s c c a l c u l a t e t x x d o 5 k = l , n z d o 5 j = l , n y d o 6 i = l , n x t s ( i ) = t ( i , j , k ) Appendix A. FORTRAN programs 159 6 c o n t i n u e c a l l d s s 0 0 4 ( d x , n x , t s , t s d ) s e t b o u n d a r y c o n d i t i o n s a t x = l a n d x = n x t s d ( l ) = 0 . d O t s d ( n x ) = - b i m * ( t s ( n x ) - t g ) - b i r * ( t s ( n x ) * * 4 . - t w * * 4 . ) c a l l d s s 0 0 4 ( d x , n x , t s d , t s d d ) d o 7 i = l , n x t x x ( i , j , k ) = t s d d ( i ) 7 c o n t i n u e 5 c o n t i n u e c a l c u l a t e t y y d o 8 k = l , n z d o 8 i = l , n x d o 9 j = l , n y t s ( j ) = t ( i , j , k ) 9 c o n t i n u e c a l l d s s 0 0 4 ( d y , n y , t s , t s d ) s e t b o u n d a r y c o n d i t i o n s a t y = l a n d y = n y t s d ( l ) = 0 . d O t s d ( n y ) = - b i m * ( t s ( n y ) - t g ) - b i r * ( t s ( n y ) * * 4 . - t w * * 4 . ) c a l l d s s 0 0 4 ( d y , n y , t s d , t s d d ) d o 1 0 j = l , n y t y y ( i , j , k ) = t s d d ( j ) 1 0 c o n t i n u e 8 c o n t i n u e c a l c u l a t e t z z d o 2 5 i = l , n x d o 2 5 j = l , n y d o 2 7 k = l , n z t s ( k ) = t ( i , j , k ) 2 7 c o n t i n u e c a l l d s s 0 0 4 ( d z , n z , t s , t s d ) s e t b o u n d a r y c o n d i t i o n s a t z = l a n d z = n z t s d ( l ) = 0 . d 0 t s d ( n z ) = - b i m * ( t s ( n z ) - t g ) - b i r * ( t s ( n z ) * * 4 . - t w * * 4 . ) c a l l d s s 0 0 4 ( d z , n z , t s d , t s d d ) d o 3 0 k = l , n z t z z ( i , j , k ) = t s d d ( k ) 3 0 c o n t i n u e 2 5 c o n t i n u e c a l c u l a t e t t d o 1 1 k = l , n z d o 1 1 i = l , n x d o 1 1 j = l , n y t t ( i , j , k ) = a l p h a * ( t x x ( i , j , k ) + t y y ( i , j , k ) + t z z ( i , j , k ) ) Appendix A. FORTRAN programs 160 1 1 c o n t i n u e r e t u r n e n d A.2 Pyrolysis modelling Following is the program p y r o l y s i s , which implements the unreacted core model accord-ing to Equations 5.3, 5.4 and 5.5, the former two for the mass balance and the latter for the heat balance, as discussed in Section 5.1.2. In this implementation, the boundary con-dition at 7- = rv for the heat transfer equation uses a combined heat transfer coefficient for the heat transfered by convection and radiation. By doing that the heat balance equation has a semi-analytical solution, calculated by function t e m p a in the program. Function r o o t (not listed) calculates the first 30 A roots of the equation tan A — 1 / ( l — Bi), required by t e m p a . Any subroutine adequate for root finding can be used here. The integration of Equations 5.3 and 5.4 is performed by subroutine o d e i n t (not listed), given by Press et al. [74]. This subroutine uses a Runge-Kutta-Felhberg method, subroutine r k q s (not listed), given by Press et al. [74] to advance the integration step by step. Other ordinary differential equation solvers were tried, like r k f 4 5 from Forsythe et al. [73] and s t i f f from Press et al. [74]. Function r h s f was used to calculate the right hand side functions of Equations 5.3 and 5.4. p r o g r a m p y r o l y s i s c t h i s p r o g r a m u s e s t h e r u n g e - k u t t a - f e h l b e r g m e t h o d t o s o l v e t h e c d i f f e r e n t i a l e q u a t i o n s w h i c h d e s c r i b e t h e c o r e i n t e r f a c e c p o s i t i o n ( y l ) a n d c o n v e r s i o n ( y 2 ) a s f u n c t i o n s o f t i m e ( x ) c f o r a n o i l s h a l e p a r t i c l e u n d e r g o i n g p y r o l y s i s . c d y l / d x = - k = - k o * e x p ( - e / r t ) y l ( 0 ) = r p c d y 2 / d x = - 3 * r c * * 2 / r p * * 3 * d y l / d x y 2 ( 0 ) = 0 i m p l i c i t r e a l * 8 ( a - h , o - z ) c x ( * ) = t i m e ( s ) c y ( * , l ) = c o r e r a d i u s ( m ) c y ( * , 2 ) = c o n v e r s i o n c r o ( * ) = r o o t s o f t a n ( x ) - l . d O / ( l . d O - b i ) * x Appendix A. FORTRAN programs 161 c t e m p ( * , l ) = c o r e r a d i u s t e m p e r a t u r e ( c ) c t e m p ( * , 2 ) = p a r t i c l e s u r f a c e t e m p e r a t u r e ( c ) c y a ( * ) , y o ( * ) = i n t e r n a l a r r a y s d i m e n s i o n x ( 5 1 ) , y ( 5 1 , 2 ) , y a ( 2 ) , y o ( 2 ) , r o ( 3 0 ) , t e m p ( 5 1 , 2 ) c c o m m o n t o b e p a s s e d t o f u n c c o m m o n / m / r o , c l , r p , t o , t b e x t e r n a l r h s f , r k q s c e p s = a c c e p t a b l e e r r o r i n o d e s o l v e r c h s t a r t = i n i t i a l s t e p o f i n t e g r a t i o n ( s ) c h m i n = m i n i m u m a l l o w a b l e s t e p o f i n t e g r a t i o n ( s ) c n = n u m b e r o f p o i n t s t o b e p r i n t e d c j = n u m b e r o f d i f f e r e n t i a l e q u a t i o n s d a t a e p s , h s t a r t , h m i n , n , j / l . d - 2 , 0 . 5 d 0 , 1 . O d - 6 , 5 1 , 2 / c t k = t h e r m a l c o n d u c t i v i t y ( w / m . k ) c r h o = d e n s i t y ( k g / m 3 ) c c p = h e a t c a p a c i t y ( j / k g . k ) d a t a t k , r h o , c p / l . 2 5 d 0 , 2 1 0 0 . d O , 1 0 4 5 . d O / o p e n ( u n i t = 7 , f i l e = ' s ' ) c a = i n i t i a l t i m e ( s ) t o = i n i t i a l t e m p e r a t u r e ( k ) c b = f i n a l t i m e ( s ) t b = g a s t e m p e r a t u r e ( k ) c r p = p a r t i c l e r a d i u s ( m ) h = h e a t t r a n s f e r c o e f f i c i e n t ( w / m 2 . k ) c y a ( l ) = i n i t i a l p a r t i c l e c o r e r a d i u s ( m ) c y a ( 2 ) = i n i t i a l c o n v e r s i o n a = 0 . 0 d 0 b = 2 8 2 . d 0 r p = 0 . 0 1 d 0 y a ( l ) = r p y a ( 2 ) = 0 . d 0 t o = 2 9 8 . d 0 t b = 8 2 3 . d 0 h = 6 0 . d 0 c b i = b i o t n u m b e r b i = h * r p / t k c a l l r o o t ( b i , r o ) c c l = c o n s t a n t t o c a l c u l a t e d i m e n s i o n l e s s t i m e c l = t k / ( r h o * c p * r p * * 2 ) x ( l ) = a d o 5 i = l , j y ( l , i ) = y a ( i ) 5 c o n t i n u e t e m p ( l , l ) = t o - 2 7 3 . d 0 t e m p ( l , 2 ) = t o - 2 7 3 . d 0 n m = n - l d x = ( b - a ) / n m d e p s = e p s / n m Appendix A. FORTRAN programs 162 d o 2 0 i = 2 , n i m = i - l x ( i ) = x ( i m ) + d x d o 6 k = l , j y o ( k ) = y ( i m , k ) 6 c o n t i n u e c a l l o d e i n t ( y o , j , x ( i m ) , x ( i ) , d e p s , h s t a r t , h m i n , n o k , n b a d , + r h s f , r k q s ) d o 8 k = l , j y ( i , k ) = y o ( k ) 8 c o n t i n u e t e m p ( i , l ) = ( ( t o - t b ) * t e m p a ( y ( i , l ) / r p , x ( i ) * c l , r o ) + t b ) - 2 7 3 . d O t e m p ( i , 2 ) = ( ( t o - t b ) * t e m p a ( l . d O , x ( i ) * c l , r o ) + t b ) - 2 7 3 . d O 2 0 c o n t i n u e w r i t e ( 6 , 3 0 ) e p s , n o f u n 3 0 f o r m a t ( l x , ' s o l u t i o n : ' , 5 x , ' e r r o r = ' , f 1 3 . 1 0 , 5 x , ' n o f u n = ' , i 7 / / + 4 x , ' x ' , 1 2 x , ' y l ' , 1 5 x , ' y 2 ' , / ) w r i t e ( 6 , 4 0 ) ( x ( i ) , y ( i , l ) , y ( i , 2 ) , t e m p ( i , l ) , t e m p ( i , 2 ) , i = l , n ) w r i t e ( 7 , 4 0 ) ( x ( i ) , y ( i , l ) , y ( i , 2 ) , t e m p ( i , l ) , t e m p ( i , 2 ) , i = l , n ) 4 0 f o r m a t ( l x , f 6 . 1 , 2 e l 2 . 4 , 2 f 7 . 1 ) s t o p e n d c s u b r o u t i n e r h s f ( x , y , d y d x ) c c a l c u l a t e s r i g h t h a n d s i d e f u n c t i o n s o f c d i f f e r e n t i a l e q u a t i o n s . i m p l i c i t r e a l * 8 ( a - h , o - z ) r e a l * 8 k o d i m e n s i o n y ( * ) , d y d x ( * ) , r o ( 3 0 ) c o m m o n / m / r o , c l , r p , t o , t b d a t a r / 8 . 3 1 4 d 0 / t c = t e m p a ( y ( l ) / r p , x * c l , r o ) t c = ( t o - t b ) * t c + t b i f ( t c . l t . 6 9 6 . d O ) t h e n k o = 0 . 1 0 3 d 0 * r p * * 3 . d O / y ( 1 ) * * 2 . d O / 3 . d O e = 2 7 3 0 0 . d 0 e l s e k o = 2 . 7 8 d 5 * r p * * 3 . d O / y ( 1 ) * * 2 . d O / 3 . d O e = 1 0 5 0 0 0 . d 0 e n d i f d y d x ( 1 ) = - k o * d e x p ( - e / ( r * t c ) ) d y d x ( 2 ) = - 3 . d 0 * y ( l ) * * 2 / r p * * 3 * d y d x ( l ) r e t u r n e n d c Appendix A. FORTRAN programs 163 d o u b l e p r e c i s i o n f u n c t i o n t e m p a ( r , t i , r o ) c c o b t a i n a n a l y t i c a l t e m p e r a t u r e s t = f ( r , t i ) i n a s p h e r e , c i n i t i a l l y a t t o , p l a c e d i n a n e n v i r o n m e n t a t t g . c c i n p u t : r = d i m e n s i o n l e s s r a d i a l p o s i t i o n c t i = d i m e n s i o n l e s s t i m e c r o = a r r a y w i t h 3 0 c o n s t a n t s o b t a i n e d i n r o o t c o u t p u t : t e m p a = d i m e n s i o n l e s s t e m p e r a t u r e c i m p l i c i t r e a l * 8 ( a - h , o - z ) d i m e n s i o n r o ( 3 0 ) s u m = 0 . d O i f ( r . e q . O . d O ) t h e n d o 5 i = l , 3 0 s u m = s u m + ( d s i n ( r o ( i ) ) - r o ( i ) * d c o s ( r o ( i ) ) ) * d e x p ( - ( r o ( i ) * * 2 . ) * t i ) / + ( 2 . d 0 * r o ( i ) - d s i n ( 2 . d 0 * r o ( i ) ) ) 5 c o n t i n u e t e m p a = 4 . d 0 * s u m r e t u r n e l s e d o 1 0 i = l , 3 0 s u m = s u m + ( d s i n ( r o ( i ) ) - r o ( i ) * d c o s ( r o ( i ) ) ) * d s i n ( r o ( i ) * r ) * + d e x p ( - ( r o ( i ) * * 2 . ) * t i ) / ( r o ( i ) * ( 2 . d 0 * r o ( i ) - d s i n ( 2 . d 0 * r o ( i ) ) ) ) 1 0 c o n t i n u e t e m p a = 4 . d 0 * s u m / r r e t u r n e n d i f e n d A.3 Oxidation modelling Following is the program o x i d a t i o n , which calculates particle temperatures, carbon con-centrations and oxygen concentrations in pores, as functions of time and position, for a spherical oil shale particle undergoing oxidation. The program integrates Equations 7.15, 7.16 and 7.17. These equations are integrated in subroutine o d e i n t (not listed), which in this implementation uses an ordinary differential equation solver adequate for stiff sets of equations, subroutine s t i f f (not listed), both subroutines given by Press et al. [74]. Subroutine s t i f f needs the jacobian of the differential equations, which are calculated Appendix A. FORTRAN programs 164 by subroutine j a c o b i a n . Subroutine d e r i v s calculates the right hand sides of Equations 7.15, 7.16 and 7.17. p r o g r a m o x i d a t i o n c c t e m p e r a t u r e , c a r b o n a n d o x y g e n c o n c e n t r a t i o n p r o f i l e s i n a s p h e r e , c i n i t i a l l y a t t o , p l a c e d i n a n o x y d i z i n g e n v i r o n m e n t a t t g . c v a r i a b l e o x y g e n c o n c e n t r a t i o n i n s i d e t h e p a r t i c l e , r e a c t i o n r a t e c f u n c t i o n o f c a r b o n a n d o x y g e n c o n c e n t r a t i o n , m e t h o d o f l i n e s . c i m p l i c i t r e a l * 8 ( a - h , o - z ) c r ( * ) = r a d i a l p o s i t i o n s ( m ) c s o l ( l , m ) = p a r t i c l e t e m p e r a t u r e s ( K ) c s o l ( m + 1 , 2 * m ) = c a r b o n c o n c e n t r a t i o n s ( m o l e / m 3 ) c s o l ( 2 * m + l , 3 * m ) = o x y g e n c o n c e n t r a t i o n s i n p o r e s ( m o l e / m 3 ) d i m e n s i o n r ( 5 1 ) , s o l ( 3 3 ) c o m m o n / c o / d r , h , t k , t b , r , f f , e r , c o 2 b , c p , r h o , h r , h m , + p o r o , d e , m l , m 2 , m 3 , m 4 , m 5 c o m m o n / c / a , b , d r 2 , c l , c 2 e x t e r n a l d e r i v s , s t i f f c t 0 = i n i t i a l p a r t i c l e t e m p e r a t u r e ( K ) c t b = b u l k g a s t e m p e r a t u r e ( K ) c c c O = i n i t i a l c a r b o n c o n c e n t r a t i o n ( m o l e / m 3 ) c c o 2 0 = i n i t i a l o x i g e n c o n c e n t r a t i o n ( m o l e / m 3 ) i n p o r e s c f f = f r e q u e n c y f a c t o r ( m 3 / m o l e . s ) c e = e n e r g y o f a c t i v a t i o n ( J / m o l e ) c h r = r e a c t i o n h e a t ( J / m o l e ) c t k = p a r t i c l e t h e r m a l c o n d u c t i v i t y ( J / s . m . K ) c c p = p a r t i c l e s p e c i f i c h e a t ( J / K g . K ) c r h o = p a r t i c l e d e n s i t y ( K g / m 3 ) c h = h e a t t r a n s f e r c o e f f i c i e n t ( J / s . m 2 . K ) c h m = m a s s t r a n s f e r c o e f f i c i e n t ( m / s ) c p o r o = p a r t i c l e v o i d a g e ( - ) c d e = e f f e c t i v e p a r t i c l e d i f f u s i v i t y o f o x y g e n ( m 2 / s ) c m = n u m b e r o f g r i d p o i n t s c n = n u m b e r o f t i m e p o i n t s c r m = p a r t i c l e r a d i u s ( m ) c t a u m = m a x i m u m t i m e ( s ) c e p s = a c c e p t a b e e r r o r t o b e p a s s e d t o O D E s o l v e r d a t a t O , t b , c c O , c o 2 0 , c o 2 b / 2 9 8 . d O , 9 2 3 . d O , 5 2 5 0 . d O , 2 . 6 3 d 0 , 2 . 6 3 d 0 / d a t a f f , e , r g , h r / 8 . 0 3 d 1 0 , 2 4 4 7 4 5 . d O , 8 . 3 1 4 d 0 , 3 . 9 3 d 5 / d a t a t k , c p , r h o / 1 . 2 5 d 0 , 1 0 4 5 . d O , 2 1 0 0 . d O / d a t a h , h m , p o r o , d e / 6 0 . d 0 , 0 . 8 7 d 0 , 0 . 1 5 d 0 , 4 . 4 d - 5 / d a t a m , n , r m , t a u m , e p s / l l , 5 1 , 0 . 0 0 1 d 0 , 7 2 0 . d 0 , l . d - 3 / o p e n ( u n i t = 7 , f i l e = ' s ' ) Appendix A. FORTRAN programs 165 a = t k / ( r h o * c p ) b = h r * f f / ( r h o * c p ) e r = e / r g c s e t n u m b e r o f s p a c e a n d t i m e i n t e r v a l s m m = m - 1 n m = n - l c s e t l o c a t i o n s f o r t e m p e r a t u r e ( 1 t o m l ) , c c a r b o n c o n c e n t r a t i o n ( m 2 t o m 3 ) , c a n d o x y g e n c o n c e n t r a t i o n ( m 4 t o m 5 ) d i f f e r e n t i a l e q u a t i o n s i n c a r r a y s s o l . m l = m m 2 = m l + l m 3 = 2 * m l m 4 = m 3 + l m 5 = 3 * m l c s e t i n c r e m e n t s f o r p r i n t i n g m i n c = m m / 2 n i n c = n m / 5 0 c s e t s p a c e a n d t i m e i n t e r v a l s d r = r m / m m d t a u = t a u m / n m c c o n s t a n t s t o b e p a s s e d b y c o m m o n / c / d r 2 = d r * d r c l = - 2 . d 0 * h * d r / t k c 2 = - 2 . d 0 * h m * d r / d e c a d j u s t ' a l l o w a b l e i n t e g r a t i o n e r r o r d e p s = e p s / n m c s e t i n i t i a l v a l u e a n d l i m i t s f o r i n t e g r a t i o n s t e p h s t a r t = d t a u * l . d - 4 h m i n = 0 . d O c s e t r a d i a l p o s i t i o n s d o 1 0 i = l , m r ( i ) = ( i - l ) * d r 1 0 c o n t i n u e c s e t i n i t i a l c o n d i t i o n s t a u = 0 . d O d o 1 2 i = l , m l s o l ( i ) = t 0 s o l ( m l + i ) = c c 0 s o l ( m 3 + i ) = c o 2 0 1 2 c o n t i n u e c w r i t e h e a d l i n e c w r i t e ( 6 , 2 0 ) ( r ( i ) * 1 0 0 . d O , i = l , m , m i n c ) c w r i t e ( 7 , 2 0 ) ( r ( i ) * 1 0 0 . d O , i = l , m , m i n c ) c 2 0 f o r m a t ( l x , ' t e m p e r a t u r e a n d c o n c e n t r a t i o n s a s a f u n c t i o n o f t i m e ' , Appendix A. FORTRAN programs 166 c + ' a n d p o s i t i o n - m e t h o d o f l i n e s : ' / / c + 2 x , ' t i m e ' , 1 3 x , ' r = ' , f 3 . 1 , 1 0 ( l l x , ' r = ' , f 3 . 1 ) , / ) c w r i t e i n i t i a l c o n d i t i o n s w r i t e ( 6 , 3 0 ) t a u , ( s o l ( i ) - 2 7 3 . , i = l , m l , m i n c ) , ( s o l ( i ) , i = m 2 , m 3 , m i n c ) , + ( s o l ( i ) , i = m 4 , m 5 , m i n c ) w r i t e ( 7 , 3 0 ) t a u , ( s o l ( i ) - 2 7 3 . , i = l , m l , m i n e ) , ( s o l ( i ) , i = m 2 , m 3 , m i n c ) , + ( s o l ( i ) , i = m 4 , m 5 , m i n c ) 3 0 f o r m a t ( l x , d 1 0 . 4 , 3 x , 3 d 1 6 . 9 , / , 1 4 x , 3 d l 6 . 9 , / , 1 4 x , 3 d 1 6 . 9 , / ) c s t a r t i n t e g r a t i o n l o o p d o 7 0 j = 2 , n j m = j - l c s e t t i m e t a u p = t a u + d t a u c i n t e g r a t e i n t i m e c a l l o d e i n t ( s o l , m 5 , t a u , t a u p , d e p s , h s t a r t , h m i n , n o k , n b a d , + d e r i v s , s t i f f ) c w r i t e r e s u l t s i f ( j m / n i n c * n i n c . e q . j m ) t h e n w r i t e ( 6 , 3 0 ) t a u p , ( s o l ( i ) - 2 7 3 . , i = l , m , m i n c ) , ( s o l ( i ) , i = m 2 , m 3 , m i n c ) , + ( s o l ( i ) , i = m 4 , m 5 , m i n e ) w r i t e ( 7 , 3 0 ) t a u p , ( s o l ( i ) - 2 7 3 . , i = l , m , m i n c ) , ( s o l ( i ) , i = m 2 , m 3 , m i n c ) , + ( s o l ( i ) , i = m 4 , m 5 , m i n c ) e n d i f t a u = t a u p 7 0 c o n t i n u e s t o p e n d c s u b r o u t i n e d e r i v s ( x , y , d y d x ) i m p l i c i t r e a l * 8 ( a - h , o - z ) d i m e n s i o n y ( * ) , d y d x ( * ) , r ( 5 1 ) , r r ( 5 1 ) c o m m o n / c o / d r , h , t k , t b , r , f f , e r , c o 2 b , c p , r h o , h r , h m , + p o r o , d e , m l , m 2 , m 3 , m 4 , m 5 c c a l c u l a t e r e a c t i o n r a t e s d o 5 i = l , m l r r ( i ) = f f * d e x p ( - e r / y ( i ) ) * y ( i + m 3 ) * y ( i + m l ) 5 c o n t i n u e c c a l c u l a t e t e m p e r a t u r e d e r i v a t i v e d o 1 0 i = l , m l c f o r r e q u a l z e r o i f ( i . e q . 1 ) t h e n d y d x ( 1 ) = ( 6 . d 0 * t k * ( y ( 2 ) - y ( 1 ) ) / ( d r * d r ) + h r * r r ( i ) ) / ( r h o * c p ) c f o r r n o t e q u a l z e r o e l s e c s e t b o u n d a r y c o n d i t i o n f o r r e q u a l r Appendix A. FORTRAN programs 167 i f ( i . e q . m l ) t h e n t x = - 2 . d 0 * h * d r / t k * ( y ( i ) - t b ) + y ( i - l ) e l s e t x = y ( i + l ) e n d i f d y d x ( i ) = ( t k * ( ( t x - 2 . d 0 * y ( i ) + y ( i - l ) ) / ( d r * d r ) + ( t x - y ( i - l ) ) / + ( r ( i ) * d r ) ) + h r * r r ( i ) ) / ( r h o * c p ) e n d i f 1 0 c o n t i n u e c c a l c u l a t e c a r b o n c o n c e n t r a t i o n d e r i v a t i v e d o 2 0 i = m 2 , m 3 d y d x ( i ) = - r r ( i - m l ) 2 0 c o n t i n u e c c a l c u l a t e o x y g e n c o n c e n t r a t i o n d e r i v a t i v e d o 3 0 i = m 4 , m 5 c f o r r e q u a l 0 i f ( i . e q . m 4 ) t h e n d y d x ( i ) = ( 6 . d 0 * d e * ( y ( i + 1 ) - y ( i ) ) / ( d r * d r ) - r r ( i - m 3 ) ) / p o r o c f o r r n o t e q u a l 0 e l s e c s e t b o u n d a r y c o n d i t i o n f o r r = r i f ( i . e q . m 5 ) t h e n c o 2 x = - 2 . d 0 * h m * d r / d e * ( y ( i ) - c o 2 b ) + y ( i - l ) e l s e c o 2 x = y ( i + l ) e n d i f d y d x ( i ) = ( d e * ( ( c o 2 x - 2 . d 0 * y ( i ) + y ( i - l ) ) / ( d r * d r ) + ( c o 2 x - y ( i - l ) ) / + ( r ( i - m 3 ) * d r ) ) - r r ( i - m 3 ) ) / p o r o e n d i f 3 0 c o n t i n u e r e t u r n e n d c s u b r o u t i n e j a c o b n ( x , y , d f d x , d f d y , n , n m a x ) i m p l i c i t r e a l * 8 ( a - h , o - z ) d i m e n s i o n y ( * ) , d f d x ( * ) , d f d y ( n m a x , n m a x ) , r ( 5 1 ) c o m m o n / c o / d r , h , t k , t b , r , f f , e r , c o 2 b , c p , r h o , h r , h m , + p o r o , d e , m l , m 2 , m 3 , m 4 , m 5 c o m m o n / c / a , b , d r 2 , c l , c 2 d o 1 0 i = l , m 5 d f d x ( i ) = 0 . d 0 1 0 c o n t i n u e c f i l l d f d y ( l , j ) d o 1 5 j = l , m 5 i f ( j . e q . 1 ) t h e n Appendix A. FORTRAN programs d f d y ( 1 , 1 ) = - 6 . d 0 * a / d r 2 + b * d e x p ( - e r / y ( 1 ) ) * y ( m 2 ) * y ( m 4 ) e l s e i f ( j . e q . 2 ) t h e n d f d y ( l , 2 ) = 6 . d 0 * a / d r 2 e l s e i f ( j . e q . m 2 ) t h e n d f d y ( 1 , m 2 ) = b * d e x p ( - e r / y ( 1 ) ) * y ( m 4 ) e l s e i f ( j . e q . m 4 ) t h e n d f d y ( 1 , m 4 ) = b * d e x p ( - e r / y ( 1 ) ) * y ( m 2 ) e l s e d f d y ( l , j ) = 0 . d O e n d i f 1 5 c o n t i n u e c f i l l d f d y ( i , j ) , K i < m l d o 3 0 i = 2 , m l - l d o 2 5 j = l , m 5 i f ( j . e q . i - 1 ) t h e n d f d y ( i , j ) = a * ( 1 . d 0 / d r 2 - 1 . d O / ( r ( i ) * d r ) ) e l s e i f ( j . e q . i ) t h e n d f d y ( i , j ) = a * ( - 2 . d 0 / d r 2 ) + b * d e x p ( - e r / y ( i ) ) * + e r / y ( i ) * * 2 . d 0 * y ( m l + i ) * y ( m 3 + i ) e l s e i f ( j . e q . i + l ) t h e n d f d y ( i , j ) = a * ( 1 . d 0 / d r 2 + 1 . d O / ( r ( i ) * d r ) ) e l s e i f ( j . e q . m l + i ) t h e n d f d y ( i , j ) = b * d e x p ( - e r / y ( i ) ) * y ( m 3 + i ) e l s e i f ( j . e q . m 3 + l ) t h e n d f d y ( i , j ) = b * d e x p ( - e r / y ( i ) ) * y ( m l + i ) e l s e d f d y ( i , j ) = 0 . d 0 e n d i f 2 5 c o n t i n u e 3 0 c o n t i n u e c f i l l d f d y ( m l , j ) d o 3 5 j = l , m 5 i f ( j . e q . m l - 1 ) t h e n d f d y ( m l , m l - l ) = 2 . d 0 * a / d r 2 e l s e i f ( j . e q . m l ) t h e n d f d y ( m l J m l ) = a * ( ( c l - 2 . d 0 ) / d r 2 + c l / ( r ( m l ) * d r ) ) + + b * d e x p ( - e r / y ( m l ) ) * e r / y ( m l ) * * 2 . d 0 * y ( m 3 ) * y ( m 5 ) e l s e i f ( j . e q . m 3 ) t h e n d f d y ( m l , m 3 ) = b * d e x p ( - e r / y ( m l ) ) * y ( m 5 ) e l s e i f ( j . e q . m 5 ) t h e n d f d y ( m l , m 5 ) = b * d e x p ( - e r / y ( m l ) ) * y ( m 3 ) e l s e d f d y ( m l , j ) = 0 . d 0 e n d i f 3 5 c o n t i n u e Appendix A. FORTRAN programs 169 c f i l l d f d y ( i , j ) i f r o m m 2 t o m 3 d o 5 0 i = m 2 , m 3 d o 4 5 j = l , m 5 i f ( j . e q . i - m l ) t h e n d f d y ( i , j ) = - f f * d e x p ( - e r / y ( i - m l ) ) * y ( i ) * y ( i + m l ) * e r / y ( i - m l ) * * 2 . e l s e i f ( j . e q . i ) t h e n d f d y ( i , j ) = - f f * d e x p ( - e r / y ( i - m l ) ) * y ( i + m l ) e l s e i f ( j . e q . i + m l ) t h e n d f d y ( i , j ) = - f f * d e x p ( - e r / y ( i - m l ) ) * y ( i ) e l s e d f d y ( i , j ) = 0 . d 0 e n d i f 4 5 c o n t i n u e 5 0 c o n t i n u e c f i l l d f d y ( m 4 , j ) d o 5 5 j = l , m 5 i f ( j . e q . 1 ) t h e n d f d y ( m 4 , l ) = - f f * d e x p ( - e r / y ( l ) ) * y ( m 2 ) * y ( m 4 ) * e r / y ( l ) * * 2 . d 0 / p o r o e l s e i f ( j . e q . m 2 ) t h e n d f d y ( m 4 , m 2 ) = - f f * d e x p ( - e r / y ( 1 ) ) * y ( m 3 ) / p o r o e l s e i f ( j . e q . m 4 ) t h e n d f d y ( m 4 , m 4 ) = ( - 6 . d 0 * d e / d r 2 - f f * d e x p ( - e r / y ( 1 ) ) * y ( m 2 ) ) / p o r o e l s e i f ( j . e q . m 4 + l ) t h e n d f d y ( m 4 , m 4 + l ) = 6 * d e / ( p o r o * d r 2 ) e l s e d f d y ( m 4 , j ) = 0 . d 0 e n d i f 5 5 c o n t i n u e c f i l l d f d y ( i . j ) m 4 < i < m 5 d o 6 5 i = m 4 + l , m 5 - l d o 6 0 j = l , m 5 i f ( j . e q . i - 1 ) t h e n d f d y ( i , j ) = d e / p o r o * ( 1 . d 0 / d r 2 - l . d O / ( r ( i - m 3 ) * d r ) ) e l s e i f ( j . e q . i ) t h e n d f d y ( i , j ) = ( d e * ( - 2 . d 0 / d r 2 ) - f f * d e x p ( - e r / y ( i - m 3 ) ) * y ( i - m l ) ) / p o r o e l s e i f ( j . e q . i + l ) t h e n d f d y ( i , j ) = d e / p o r o * ( 1 . d 0 / d r 2 + l . d O / ( r ( i - m 3 ) * d r ) ) e l s e i f ( j . e q . i - m 3 ) t h e n d f d y ( i , j ) = - f f * d e x p ( - e r / y ( i - m 3 ) ) * y ( i - m l ) * y ( i ) * + e r / y ( i - m 3 ) * * 2 . d 0 / p o r o e l s e i f ( j . e q . i - m l ) t h e n d f d y ( i , j ) = - f f * d e x p ( - e r / y ( i - m 3 ) ) * y ( i ) / p o r o e l s e d f d y ( i , j ) = 0 . d 0 e n d i f Appendix A. FORTRAN programs 170 6 0 c o n t i n u e 6 5 c o n t i n u e c f i l l d f d y ( m 5 , j ) d o 7 0 j = l , m 5 i f ( j . e q . m 5 - l ) t h e n d f d y ( m 5 , j ) = d e / p o r o * 2 . d 0 / d r 2 e l s e i f ( j . e q . m 5 ) t h e n d f d y ( m 5 , j ) = ( d e * ( ( c 2 - 2 . d 0 ) / d r 2 + c 2 / ( r ( m l ) + d r ) ) -+ f f * d e x p ( - e r / y ( m l ) ) * y ( m 3 ) ) / p o r o e l s e i f ( j . e q . m l ) t h e n d f d y ( m 5 , j ) = - f f * d e x p ( - e r / y ( m l ) ) * y ( m 3 ) * y ( m 5 ) / p o r o * e r / y ( m l ) * * 2 . e l s e i f ( j . e q . m 3 ) t h e n d f d y ( m 5 , j ) = - f f * d e x p ( - e r / y ( m l ) ) * y ( m 5 ) / p o r o e l s e d f d y ( m 5 , j ) = 0 . d 0 e n d i f 7 0 c o n t i n u e r e t u r n e n d Appendix B Experimental data for metallic and shale particles. For the following tables: T l = reactor wall temperature (°C) T3 = temperature (°C) of the gas after the particle T4 = temperature (°C) of the gas before the particle T5 = particle center temperature (°C) W = particle weight (g) time = time (minutes) Experimental data plotted in Figure 5.7 time T l T3 T4 T5 0.0 352. 19. 19. 19. 0.5 351. 191. 244. 58. 1.0 351. 236. 278. 78. 1.5 350. 254. 298. 109. 2.1 349. 287. 322. 136. 2.5 348. 289. 325. 151. 3.0 348. 301. 330. 162. 3.5 348. 306. 333. 182. 4.0 347. 315. 335. 195. 4.5 347. 316. 337. 211. 5.0 348. 320. 338. 220. 5.5 348. 322. 338. 235. 6.0 348. 325. 339. 244. 6.5 348. 326. 339. 257. 7.0 348. 328. 339. 263. 7.5 348. 329. 339. 273. 8.0 347. 330. 339. 278. 8.5 347. 331. 339. 287. 9.0 347. 332. 339. 291. 9.5 347. 333. 339. 298. 10.0 347. 334. 339. 302. time T l T3 T4 T5 10.5 346. 334. 339. 307. 11.0 346. 335. 339. 311. 11.5 347. 336. 339. 314. 12.0 347. 337. 339. 318. 12.5 346. 337. 339. 322. 13.0 346. 337. 338. 323. 13.5 346. 337. 339. 325. 14.0 345. 337. 339. 327. 14.5 345. 338. 339. 329. 15.0 345. 338. 339. 330. 15.5 345. 338. 339. 332. 16.0 346. 338. 339. 333. 16.5 346. 338. 339. 334. 17.0 346. 339. 339. 335. 17.5 346. 339. 339. 336. 18.0 346. 339. 339. 337. 18.5 346. 339. 339. 337. 19.0 345. 339. 339. 338. 19.5 345. 339. 339. 338. 20.0 345. 339. 339. 339. Appendix B. Experimental data for metallic and shale particles. Experimental data plotted in Figure 5.8 172 time T l T3 T4 T5 0.0 345. 18. 18. 18. 0.5 346. 192. 231. 52. 1.0 345. 226. 256. 83. 1.5 344. 251. 276. 108. 2.0 344. 270. 293. 132. 2.5 344. 281. 302. 150. 3.0 345. 292. 309. 170. 3.5 345. 298. 313. 185. 4.0 346. 304. 316. 200. 4.5 346. 308. 318. 215. 5.0 346. 312. 319. 224. 5.5 346. 315. 321. 235. 6.0 345. 318. 321. 245. 6.5 345. 320. 322. 257. 7.0 344. 321. 322. 268. 7.5 344. 323. 323. 271. 8.0 344. 324. 323. 278. 8.5 345. 326. 324. 284. 9.0 345. 327. 324. 289. 9.5 346. 329. 325. 295. 10.0 346. 330. 326. 299. time T l T3 T4 T5 10.5 346. 330. 326. 305. 11.0 346. 331. 326. 307. 11.5 345. 332. 326. 310. 12.0 345. 333. 326. 313. 12.5 344. 333. 326. 316. 13.0 344. 334. 326. 318. 13.5 344. 334. 327. 320. 14.0 345. 335. 327. 322. 14.5 345. 335. 327. 323. 15.0 346. 336. 327. 325. 15.5 346. 336. 328. 326. 16.0 346. 337. 328. 328. 16.5 346. 337. 328. 328. 17.0 345. 337. 328. 329. 17.5 344. 337. 328. 330. 18.0 344. 337. 327. 331. 18.5 344. 337. 327. 331. 19.0 344. 337. 327. 332. 20.0 345. 338. 328. 332. 20.5 346. 338. 328. 333. Appendix B. Experimental data for metallic and shale particles. Experimental data plotted in Figure 5.9 173 time T l T3 T4 T5 0.0 445. 19. 19. 20. 0.5 444. 254. 297. 59. 1.0 444. 294. 346. 98. 1.5 443. 338. 380. 134. 2.0 444. 354. 401. 177. 2.5 444. 375. 415. 196. 3.0 445. 392. 430. 248. 3.5 445. 401. 434. 267. 4.0 445. 403. 437. 291. 5.5 444. 410. 440. 319. 6.0 444. 412. 441. 338. 6.5 444. 414. 443. 352. 7.5 444. 418. 445. 366. 8.0 445. 419. 446. 379. 8.5 445. 421. 447. 385. 9.0 445. 422. 448. 395. 10.0 445. 423. 448. 403. 10.5 444. 424. 449. 409. 11.0 444. 428. 449. 413. time T l T3 T4 T5 12.0 444. 425. 449. 420. 12.5 444. 426. 450. 422. 13.0 445. 427. 451. 425. 13.5 445. 429. 451. 428. 14.0 445. 428. 451. 429. 14.5 445. 428. 451. 430. 15.0 445. 428. 451. 432. 15.5 444. 428. 451. 433. 16.0 444. 428. 451. 434. 16.5 444. 428. 451. 435. 17.5 444. 429. 452. 436. 18.0 444. 429. 452. 437. 18.5 445. 429. 452. 438. 19.5 445. 429. 452. 438. 20.0 445. 429. 452. 439. 21.0 445. 429. 452. 439. 22.0 444. 429. 452. 439. 23.5 445. 430. 453. 440. 25.5 445. 429. 452. 440. Appendix B. Experimental data for metallic, and shale particles. Experimental data plotted in Figure 5.12 174 time T l T3 T4 T5 0. 14. 18. 18. 18. 2. 38. 21. 23. 19. 3. 63. 25. 29. 20. 4. 93. 31. 38. 23. 5. 121. 42. 50. 26. 6. 136. 49. 61. 32. 7. 142. 62. 73. 37. 8. 144. 69. 81. 44. 9. 143. 79. 89. 48. 10. 148. 86. 98. 55. 11. 163. 99. 110. 61. 12. 184. 110. 126. 71. 13. 208. 130. 145. 81. 14. 224. 143. 161. 93. 15. 230. 160. 174. 103. 16. 232. 172. 187. 119. 17. 231. 182. 185. 128. 18. 230. 193. 204. 143. 19. 234. 203. 213. 150. 20. 251. 214. 227. 164. 21. 265. 227. 239. 173. 22. 272. 237. 249. 187. 23. 274. 248. 257. 197. time T l T3 T4 T5 24. 274. 256. 265. 211. 25. 273. 262. 269. 218. 26. 278. 268. 277. 230. 27. 295. 281. 290. 238. 28. 315. 294. 305. 253. 29. 322. 306. 314. 262. 30. 325. 313. 321. 274. 31. 325. 321. 327. 283. 32. 327. 325. 333. 293. 33. 341. 338. 345. 302. 34. 359. 348. 359. 315. 35. 369. 360. 367. 324. 36. 372. 367. 375. 337. 37. 372. 373. 379. 344. 38. 371. 375. 382. 352. 39. 369. 379. 383. 358. 40. 368. 380. 384. 364. 41. 367. 381. 385. 368. 42. 366. 381. 385. 372. 43. 365. 381. 385. 374. 44. 364. 380. 384. 376. 45. 362. 380. 383. 377. Appendix B. Experimental data for metallic, and shale particles. Experimental data plotted in Figure 5.13 175 time T l T3 T4 T5 0. 16. 18. 18. 18. 1. 24. 19. 22. 18. 2. 44. 21. 28. 19. 3. 70. 27. 37. 20. 4. 90. 29. 39. 23. 5. 123. 41. 53. 26. 6. 144. 51. 65. 32. 7. 150. 62. 75. 36. 8. 153. 71. 85. 44. 9. 153. 83. 94. 49. 10. 153. 90. 103. 58. 11. 160. 101. 114. 64. 12. 179. 112. 129. 74. 13. 203. 130. 148. 83. 14. 218. 139. 161. 94. 15. 232. 158. 177. 104. 16. 253. 176. 198. 122. 17. 267. 197. 215. 133. 18. 286. 220. 241. 157. 19. 296. 237. 255. 164. 20. 314. 256. 277. 187. 21. 320. 276. 291. 202. 22. 321. 286. 301. 219. 23. 320. 300. 311. 235. time T l T3 T4 T5 24. 327. 309. 323. 251. 25. 345. 326. 340. 265. 26. 363. 341. 356. 286. 27. 370. 355. 367. 300. 28. 377. 365. 380. 319. 29. 392. 382. 396. 332. 30. 413. 401. 415. 349. 31. 427. 413. 430. 369. 32.5 434. 432. 444. 389. 33. 440. 438. 453. 402. 34. 454. 454. 467. 415. 35. 462. 463. 477. 433. 36. 464. 471. 482. 442. 37. 463. 475. 485. 456. 38. 461. 477. 487. 462. 39. 459. 477. 487. 468. 40. 459. 477. 486. 472. 41. 457. 476. 486. 475. 42. 455. 473. 483. 476. 43. 454. 470. 483. 477. 44. 453. 469. 481. 476. 45. 452. 467. 480. 476. 46. 451. 465. 479. 480. Appendix B. Experimental data for metallic and shale particles. 176 Experimental data plotted in Figure 5.14. Temperature profiles also in Figure 5.27 and Figure 5.30. time T l T3 T4 T5 0. 14. 19. 19. 19. 1. 17. 19. 19. 19. 2. 33. 21. 23. 19. 3. 65. 24. 28. 22. 4. 91. 30. 36. 25. 5. 120. 38. 48. 31. 6. 137. 54. 61. 38. 7. 142. 61. 70. 48. 8. 144. 74. 80. 55. 9. 143. 83. 90. 67. 10. 143. 91. 95. 72. 11. 152. 97. 103. 82. 12. 171. 110. 114. 89. 13. 196. 123. 135. 105. 14. 213. 143. 152. 118. 15. 230. 155. 167. 134. 16. 243. 173. 183. 147. "17. 264. 189. 206. 171. 18. 271. 211. 220. 183. 19. 283. 225. 238. 203. 20. 314. 249. 260. 220. 21. 316. 266. 280. 241. 22. 323. 286. 296. 257. 23. 326. 299. 309. 277. 24. 338. 316. 325. 291. 25. 358. 333. 346. 315. 26. 367. 350. 359. 330. 27. 373. 360. 371. 347. 28. 385. 375. 386. 360. 29. 403. 389. 403. 378. 30. 416. 408. 419. 394. 31. 421. 421. 430. 414. 32. 421. 428. 435. 418. 33. 431. 434. 445. 427. 34. 451. 453. 463. 438. 35. 466. 463. 477. 451. time T l T3 T4 T5 36. 471. 473. 487. 460. 37. 472. 479. 491. 472. 38. 470. 483. 493. 476. 39. 469. 484. 495. 481. 40. 468. 483. 495. 482. 41. 466. 483. 494. 483. 42. 465. 481. 493. 482. 43. 463. 479. 491. 481. 44. 462. 477. 490. 480. 45. 461. 475. 488. 478. 46. 459. 471. 486. 476. 47. 458. 469. 484. 474. 48. 457. 466. 482. 472. 49. 455. 463. 479. 468. 50. 454. 460. 478. 468. 51. 453. 458. 476. 465. 52. 452. 454. 474. 463. 53.5 450. 451. 471. 459. 54. 450. 450. 471. 459. 55. 450. 448. 469. 457. 56.5 450. 446. 468. 455. 57.5 450. 445. 467. 453. 58. 449. 443. 466. 453. 59. 448. 442. 465. 451. 60. 448. 441. 464. 450. 61. 449. 440. 464. 449. 62. 449. 439. 464. 449. 63. 448. 437. 463. 447. 64. 447. 435. 461. 446. 65. 448. 434. 460. 446. 66. 449. 433. 460. 445. 67.5 448. 433. 460. 444. 68.5 447. 431. 460. 444. 69. 447. 431. 460. 443. 70. 447. 431. 460. 443. Appendix B. Experimental data for metallic and shale particles. Experimental data plotted in Figure 5.15. Temperature profile also in Figure 5.28. time T l T3 T4 T5 0. 14. 17. 17. 18. 1. 14. 18. 18. 18. 2. 34. 21. 23. 18. 3. 56. 23. 28. 19. 4. 84. 28. 36. 24. 5. 114. 40. 48. 28. 6. 133. 49. 61. 38. 7. 141. 62. 72. 45. 8. 144. 70. 81. 55. 9. 144. 83. 90. 62. 10. 145. 90. 98. 73. 11. 159. 104. 112. 82. 12. 176. 115. 125. 94. 13.5 208. 141. 150. 111. 14. 220. 151. 161. 125. 15. 228. 171. 175. 138. 16. 231. 185. 189. 158. 17. 231. 197. 198. 168. 18. 231. 205. 206. 182. 19. 239. 220. 219. 192. 20. 257. 232. 234. 208. 21. 277. 251. 252. 218. 22. 299. 269. 272. 236. 23. 320. 295. 295. 252. 24. 339. 313. 316. 276. 25. 359. 341. 339. 298. 26. 365. 354. 353. 323. 27. 372. 372. 367. 343. 28. 375. 380. 378. 358. 29. 389. 399. 396. 373. 30. 405. 413. 412. 391. time T l T3 T4 T5 31. 416. 428. 425. 405. 32. 421. 437. 435. 419. 33. 432. 451. 450. 428. 34. 449. 466. 467. 442. 35. 461. 480. 479. 452. 36. 468. 488. 487. 467. 37. 468. 493. 491. 474. 38. 467. 495. 493. 482. 39. 466. 495. 493. 485. 40. 464. 494. 494. 488. 41. 463. 493. 493. 487. 42.5 461. 491. 491. 486. 43. 460. 489. 490. 486. 44. 459. 488. 489. 485. 45. 458. 487. 487. 483. 46. 457. 484. 486. 481. 47. 455. 481. 482. 480. 48. 454. 479. 482. 478. 49. 453. 476. 480. 476. 50. 452. 475. 478. 474. 51. 451. 472. 476. 472. 52. 449. 470. 473. 469. 53.5 447. 467. 471. 467. 54. 446. 465. 470. 466. 55.5 446. 464. 469.' 464. 56. 447. 463. 468. 463. 57. 447. 462. 467. 462. 58.5 446. 463. 466. 461. 59.5 445. 459. 463. 460. 60. 444. 458. 464. 458. Appendix B. Experimental data for metallic and shale particles. Experimental data plotted in Figure 5.27. Weight points also plotted in Figure 5.30. time T l T3 T4 W 0. 14. 19. 19. 4.612 1. 14. 18. 18. 4.611 2. 23. 20. 20. 4.608 3. 40. 26. 23. 4.609 4. 73. 31. 28. 4.609 5. 101. 39. 37. 4.610 6. 128. 54. 48. 4.607 7. 141. 64. 60. 4.605 8. 147. 76. 70. 4.607 9. 140. 84. 81. 4.605 10. 149. 95. 91. 4.601 11. 161. 106. 105. 4.602 12. 181. 123. 119. 4.602 13. 203. 138. 137. 4.597 14. 222. 158. 153. 4.596 15. 232. 173. 171. 4.595 16. 247. 195. 191. 4.590 17. 261. 209. 208. 4.586 18. 272. 229. 225. 4.579 19. 285. 246. 245. 4.572 20. 307. 272. 267. 4.400 21. 320. 288. 285. 4.390 22. 325. 302. 298. 4.400 23.5 328. 319. 319. 4.497 24. 334. 329. 326. 4.491 25. 350. 343. 344. 4.483 26. 365. 360. 358. 4.476 27. 372. 371. 372. 4.468 28. 378. 383. 382. 4.458 29. 394. 398. 400. 4.450 30. 410. 415. 415. 4.430 time T l T3 T4 W 31. 418. 424. 426. 4.401 32. 421. 434. 434. 4.349 33. 430. 445. 447. 4.282 34. 449. 465. 463. 4.192 35. 461. 477. 476. 4.110 36. 468. 487. 485. 4.019 37. 469. 490. 459. 3.975 38. 468. 493. 493. 3.942 39. 466. 494. 494. 3.933 40. 465. 494. 436. 3.924 41. 463. 493. 494. 3.921 42. 462. 492. 492. 3.916 43. 461. 491. 491. 3.914 44. 460. 490. 490. 3.913 45. 458. 488. 487. 3.911 46. 457. 486. 463. 3.909 47. 456. 485. 484. 3.907 48. 455. 483. 482. 3.906 49. 454. 481. 480. 3.904 50. 452. 479. 478. 3.904 51. 451. 477. 476. 3.903 52. 449. 475. 474. 3.904 53. 449. 474. 473. 3.903 54. 448. 472. 471. 3.904 55. 446. 471. 469. 3.904 56. 446. 469. 468. 3.904 57. 446. 469. 467. 3.902 58. 446. 468. 466. 3.903 59. 445. 466. 464. 3.901 60. 445. 465. 463. 3.900 Appendix B. Experimental data for metallic and shale particles. Experimental data (weight vs. time) plotted in Figure 5.28. 179 time T l T3 T4 W 0. 19. 24. 24. 4.433 1. 22. 24. 24. 4.431 2. 38. 26. 67. 4.427 3. 77. 34. 36. 4.433 4. 88. 39. 39. 4.429 5. 116. 49. 49. 4.431 6. 134. 61. 61. 4.439 7.5 143. 77. 78. 4.435 8. 144. 82. 81. 4.432 9. 144. 90. 91. 4.431 10. 145. 100. 100. 4.427 11.5 160. 112. 113. 4.425 12. 174. 120. 120. 4.423 13. 189. 132. 134. 4.424 14. 215. 152. 153. 4.427 15. 232. 172. 174. 4.414 17. 258. 203. 205. 4.400 18. 268. 216. 222. 4.370 19. 278. 236. 239. 4.360 20. 295. 255. 261. 4.350 21. 318. 281. 284. 4.270 22. 320. 291. 295. 4.270 23. 322. 301. 304. 4.260 24. 328. 314. 319. 4.270 25. 342. 330. 334. 4.250 time T l T3 T4 W 26.5 355. 361. 4.260 27. 370. 363. 367. 4.255 28. 372. 369. 375. 4.242 29. 385. 381. 385. 4.237 30. 408. 402. 408. 4.219 31. 422. 420. 424. 4.201 32. 432. 431. 438. 4.185 33. 450. 451. 457. 4.100 34. 462. 465. 474. 3.980 35. 465. 474. 480. 3.880 36. 465. 479. 484. 3.790 37. 464. 483. 486. 3.750 38. 463. 484. 487. 3.730 39. 461. 484. 487. 3.700 40. 459. 484. 486. 3.709 41. 458. 483. 486. 3.709 42. 451. 482. 484. 3.703 43. 455. 480. 483. 3.705 44.5 454. 478. 480. 3.696 45. 453. 477. 479. 3.687 46. 452. 475. 478. 3.687 47. 451. 474. 476. 3.668 48. 449. 471. 474. 3.681 49. 448. 469. 472. 3.680 50. 447. 467. 469. 3.684 Appendix B. Experimental data for metallic and shale particles. Experimental data plotted in Figure 7.3 time T l T3 T4 T5 0. 19. 24. 24. 24. 1. 23. 24. 24. 24. 2. 38. 26. 28. 24. 3. 62. 33. 34. 26. 4. 87. 39. 42. 31. 5. 111. 50. 51. 36. 6. 122. 58. 61. 45. 7. 128. 67. 68. 50. 8. 133. 76. 77. 60. 9. 142. 86. 86. 65. 10. 156. 97. 97. 76. 11. 165. 110. 107. 85. 12. 168. 117. 115. 96. 13. 169. 128. 123. 104. 14. 179. 137. 135. 116. 15. 194. 151. 149. 126. 16. 201. 158. 157. 138. 17. 206. 172. 168. 150. 18. 215. 179. 178. 160. 19. 228. 193. 190. 170. 20. 240. 205. 203. 187. 21. 252. 221. 216. 198. 22. 264. 233. 229. 215. 23. 268. 245. 237. 226. 24. 273. 254. 249. 241. 25. 289. 272. 265. 254. 26. 306. 286. 281. 273. 27. 319. 304. 297. 292. 28. 324. 313. 308. 319. 29. 325. 323. 317. 333. 29.5 330. 329. 324. 345. 30. 332. 332. 329. 354. time T l T3 T4 T5 30.5 338. 341. 337. 360. 31. 347. 346. 344. 372. 31.5 357. 358. 353. 380. 32. 363. 362. 359. 392. 32.5 368. 370. 366. 401. 33. 370. 372. 370. 409. 33.5 371. 377. 373. 413. 34. 371. 379. 376. 420. 34.5 371. 383. 379. 425. 35. 371. 384. 381. 429. 35.5 370. 386. 383. 431. 36. 369. 387. 384. 435. 37. 368. 389. 386. 438. 38. 367. 386. 386. 438. 39. 365. 389. 387. 436. 40. 365. 389. 386. 432. 41. 364. 388. 386. 429. 42. 364. 388. 385. 424. 43. 363. 387. 386. 421. 44. 363. 386. 383. 421. 45. 362. 385. 382. 414. 46. 361. 383. 381. 411. 47.5 362. 382. 380. 407. 48.5 362. 381. 380. 406. 49.5 361. 380. 379. 403. 50. 361. 379. 378. 401. 51. 360. 379. 377. 399. 52. 360. 377. 376. 397. 53. 361. 377. 375. 394. 54. 361. 376. 375. 393. 55. 360. 375. 373. 390. Appendix B. Experimental data for metallic and shale particles. Experimental data plotted in Figure 7.4 181 time T l T3 T5 0. 18. 23. 23. 1. 23. 23. 23. 2. 47. 27. 24. 3. 64. 31. 26. 4. 87. 39. 30. 5. 106. 47. 36. 6. 116. 55. 40. 7. 130. 64. 48. 8. 143. 77. 57. 9. 154. 86. 65. 10. 163. 100. 74. 11. 166. 109. 86. 12. 166. 119. 95. 13. 167. 127. 105. 14. 176. 139. 113. 15. 192. 149. 125. 16. 205. 162. 136. 17. 212. 172. 149. 18. 225. 187. 160. 19. 240. 200. 176. 20. 249. 215. 189. 21.5 258. 229. 208. 22. 260. 237. 216. 23. 261. 245. 228. 24. 273. 259. 239. 25. 291. 273. 252. 26. 312. 292. 266. 27. 334. 311. 287. 28. 366. 346. 317. 29. 380. 360. 338. 29.5 394. 378. 352. 30. 404. 388. 369. 30.5 416. 405. 386. 31. 424. 411. 400. 31.5 436. 430. 418. 32. 450. 445. 445. 32.5 457. 458. 457. 33. 466. 468. 477. 33.5 477. 485. 495. 34. 488. 499. 517. time T l T3 T5 34.5 498. 514. 529. 35. 508. 525. 541. 35.5 518. 539. 549. 36. 528. 549. 561. 36.5 540. 566. 571. 37. 552. 578. 587. 37.5 557. 586. 590. 38. 567. 594. 604. 38.5 575. 606. 612. 39. 585. 615. 625. 39.5 598. 629. 636. 40. 606. 640. 647. 40.5 615. 653. 657. 41. 625. 663. 670. 41.5 634. 676. 678. 42. 644. 685. 691. 42.5 655. 712. 702. 43. 664. 706. 713. 43.5 673. 717. 721. 44. 681. 725. 732. 44.5 692. 739. 741. 45.5 707. 755. 761. 46. 716. 763. 768. 46.5 721. 768. 777. 47. 724. 771. 782. 47.5 725. 773. 786. 48. 724. 772. 788. 49. 722. 770. 788. 50. 719. 766. 786. 51. 717. 761. 782. 52. 714. 755. 778. 53. 711. 751. 773. 54. 708. 745. 767. 55. 706. 741. 762. 56. 704. 743. 757. 57. 704. 735. 753. 58.5 702. 730. 749. 59. 700. 728. 746. 60. 698. 725. 743. Appendix C Non-isothermal TGA. Expressions to obtain kinetic parameters In what follows, the symbology and equation numbering are the same as those defined for pyrolysis in Chapter 4. Nevertheless, the equations have general application. The expression for a first order reaction is: ^ = M l - X) (4.3) If it is assumed that the rate constant follows the Arrhenius equation: k = k0e-E'RT (4.4) then substituting Equation 4.4 into Equation 4.3 yields: *£ = k0e-E<"r(l-X). (Cl) Taking the logarithm of Equation C l gives - l n ^ = - h # 0 ( l - X ) ] + | i (4.7) The above equation was suggested by Friedman [45]. Plotting the left hand side term against 1/T for the same X values would produce a straight line whose slope is E/R and intercept is — ln[fc0(l — X)]. To get points dX/dt and 1 jT at same X values, experiments should be run at different heating rate values. If a constant heating rate 0, defined as / , = dT <4'5> 182 Appendix C. Non-isothermal TGA. Expressions to obtain kinetic parameters 183 is substituted into Equation C l , then one obtains: — = ^e-ElRT{l-X). (4.6) AT a y ' v ' Taking the logarithm of Equation 4.6 and rearranging the terms yields The above equation is of the Arrehnius type. Plotting the left hand side term of Equation 4.8 against 1 /T produces a straight line whose intercept is — In k0/3 and slope is +(E/R). The left hand side of Equation 4.8 can be split to obtain , dX k0 E - l n _ + N l - X > = - U . - + _ . Taking the derivative then yields: or: _ A 1 „ « + A m - ^ ) = |4 which, after rearrangement, finally gives: A\n(dX/dT) _ E A ( l / T ) A l n ( l - X) R A l n ( l - X)' The above equation was proposed by Freeman and Carroll [46]. Plotting the left hand side term of Equation 4.9 against the second term on the right hand side (without E/R), produces —E/R as the slope of the straight line. Integrating Equation 4.6 gives: Appendix C. Non-isothermal TGA. Expressions to obtain kinetic parameters 184 The right hand side integral can be integrated by parts to obtain / e-E'RTc\T = e-E'RTT - % \ % — d T . (C.3) JTO=0 R JTo=0 T To solve the integral on the right hand side, we first introduce a new variable u, such that E Thus. U=RT-T = —- and dT = — — —-du Ru R u1 where: u = oo at T = 0 u = E/RT at T = T. If u is now substituted into the last term of Equation C.3, one obtains RT E-E/RT F Q O / -7r—dT = / — du. JTO=0 1 JE/RT U The right hand side integral in the above equation is given by [67]: r o o e-u e-x / i | 2! r°° e~" e_J- / 1! Zl / — du = — 1 - - + -Jx U X \ X Xz Thus: / • o o e-u E-E/RT ( \ 2 \ JE/RT ~ d u = ~EjRT [l ~ E~jRT + (E/RT)2 -•••)• Taking the first three terms of the series and substituting them into Equation C.3 yields: [T e~E/RTdT = e~ElRTT - Ee~E/RT (i L _ + 1 ^ JTO=0 ' R E/RT V E/RT ^ (E/RT)2) RT . e / r t (. 2RT\ E _E-E/RT L 0 " ™ ) -Appendix C. Non-isothermal TGA. Expressions to obtain kinetic parameters 185 Substituting this result into C.2 and integrating then yields: - K . - X ) - ^ - r ( . - 2 f l ) . (C.4) Finally, taking the logarithms of Equation CA gives In - l n ( l - X) . k0R ( 2RT\ E 1 (4.10) The above equation was suggested by Coats and Redfern [47]. Plotting the left hand side term against 1/T gives a straight line whose slope is E/R, assuming that the first term on the right hand side is constant. Equation C.4 can be rearranged to produce: - l n ( l -X) E RT2 0 _ Z£_E-E/RT Taking the logarithm then gives: " -0\n(l -X) In RT'2 , . , 2RT\ , k0 El + l n ( l - — J = - l n - : f (4.11) E J E RT The above equation is known as the Integral method. The values of E and k0 can be obtained by an iterated least-squares fit to the experimental data. Initially a value is guessed for E and then the left hand side term is plotted against 1/T. From the slope of the straight line produced, E/R is obtained and compared to the guessed value. This algorithm is repeated until a good convergence between the two values is reached. The above equation can be rewritten as the following explicit relationship for X: X = 1- exp k0RT2 _E/RT ( 2RT\ Substituting this equation into Equation 4.3 then yields: cLY "dT = k0exp E k0RT2 RT 0E E-E/RT 1 2RT\ ~ir) (4.12) Appendix C. Non-isothermal TGA. Expressions to obtain kinetic parameters 186 The above equation constitutes the Differential Method. The values of E and k0 can be obtained by iterative least squares fitting of the equation to the experimental data. A different derivation of the Differential Method was proposed by van Heek et al. [48, 49] yielding: dX "dT exp E RT k0 RT2 E/RT BE (4.13) Another way to solve the right hand side integral of Equation C.2 is to adopt the following approximation, suggested by Turner et al. [68, 69] and discussed by Doyle [70]: rT RT2 JT0=O -E/RT E + 2RT Introducing this result into Equation C.2 and integrating then gives: In ( - K ° R T * C-E/RT \1-XJ BE + 2RT' Finally, rearranging and taking the logarithm produces: In E + 2RT 1 n , k0R E 1 (4.14) T 2 1 - XI The above equation was suggested by Chen and Nuttall [50]. The kinetic parameters are obtained by iterative fitting of the experimental data. Plotting the left hand side term, with a guessed value for E, against 1/T would produce a straight line with +E/R as the slope and — \i\{k0Rjf3) as the intercept. Appendix D Rate constant by the least squares method Problem: To obtain the best k value to fit the equation X = 1 - e~kt to m pairs of experimental points (X{,t\), (Xf,£ 2), • • (X?n,tm). The method used is the non-linear least squares method. The method searches for a value of k in order to minimize the sum of the squares of the differences between the experimental and the calculated X values. Thus the function T to be minimized is: T = [Xt - Xxf + [XI - X2f + ••• + [Xem - Xm}2 or: T = [XI - (1 - e.-kt> )]2 + [Xe2 - (1 - e-* ' a )] 2 + • • • + [X:n - (1 - e"*'"*)]2. The best k value is the one that makes the derivative of the above equation equal to zero, i.e., ~ = 2 [ X 1 e - ( l - e - f c ' 1 ) ] e - A ! t , ( - * i ) + 2 [ X 2 e - ( l - e - * < a ) ] e - ' i t 2 ( - t 2 ) + • • • + 2[Xem - (1 - e-ktm)]e-ktm(-tm) - 0 or: m £ [ ( l - e " ^ ) - . V j K ^ (/,,) = (> The problem now reduces to using a standard method to find a k value which is a root of the above equation. The standard method used in this work was an implementation of Mueller's method. 187 Appendix E Experimental data for T G A Table E.2 displays experimental data for the non-isothermal pyrolysis TGA. Listed figures are the sample weights (mg) as a function of shale origin and analyses heating rates. The numbers in the first row refer to the sample weights up to 5 min of analyses time, during which the samples were kept at 110 °C, and no change in weight was observed. Numbers are listed as collected by the data logging system. The sampling time was a function of the heating rate, as below: heating rate (°C/min) 50 20 10 5 sampling time (sec.) 2 5 10 20 With that, analyses time can be related to sample weights considering that the first row of data on Table E.2 refers to a time of 5 minutes. Analysis temperature was kept constant for the first 5 minutes at 110 °C to check previous drying. Subsequently, it was raised at nominal heating rates of 50, 20, 10 and 5 °C/min. Table E . l gives the actual heating rates 0 observed in the eight analyses. Temperature (°C) and time (min) are related by the equations: T = 110, t<5 T = (t - 5) x 0 + 110, t>5 188 Appendix E. Experimental data for TGA 189 Table E . l : Actual heating rates 3 (°C/min) for non-isothermal TGA. oil shale nominal heating rate (°C/min) 50 20 10 5 Irati New Brunswick 50 20 9.8 4.9 48 19 9.5 4.7 The data in Table E.2, converted to weight (%), are plotted in Figures 6.8(a) - 6.15(a), which also include the initial 5 minutes. Table E.3 displays experimental data, sample weight (%) versus time (minutes), for the isothermal oxidation TGA, for temperatures 550 °C, 600 °C, 625 °C, 650 °C and 700 °C. Heating rate was 100 °C/min. The data in Table E.3 are plotted in Figure 8.1 as conversion versus time. Conversion of 1 was related to a weight of 97.68%, the lowest achieved in the analyses. Table E.4 displays experimental data, sample weight (%) versus time (minutes), for the isothermal oxidation TGA, for oxygen mole fractions in sweep gas of 21%, 15%, 10% and 5%. Heating rate was 100 °C/min. The data in Table E.4 are plotted in Figure 8.4. Appendix E. Experimental data, for TGA 190 Table E.2: Sample weight (mg) for non-isothermal pyrolysis TGA. Irati shale New Brunswick shale heating rate (°C/min) 50 20 10 5 50 20 10 5 12.8500 14.6371 15.1866 17.5104 18.3547 16.4436 16.5061 18.6933 12.8500 14.6371 15.1973 17.5182 18.3547 16.4436 16.5090 18.6963 12.8695 14.6400 15.1953 17.5251 18.3547 16.4436 16.5071 18.6972 12.8695 14.6429 15.1973 17.5192 18.3439 16.4417 16.5129 18.6924 12.8695 14.6429 15.1953 17.5202 18.3439 16.4417 16.5081 18.6953 12.8695 14.6439 15.1953 17.5212 18.3439 16.4505 16.5139 18.7002 12.8461 14.6439 15.1905 17.5182 18.3517 16.4534 16.5012 18.6953 12.8461 14.6439 15.2012 17.5202 18.3517 16.4534 16.5090 18.6982 12.8461 14.6459 15.2032 17.5133 18.3517 16.4534 16.5110 18.6953 12.8461 14.6439 15.2168 17.5182 18.3517 16.4515 16.5169 18.6982 12.8569 14.6439 15.2080 17.5202 18.3498 16.4573 16.5149 18.6943 12.8569 14.6449 15.2041 17.5163 18.3498 16.4573 16.5169 18.6904 12.8569 14.6468 15.1944 17.5065 18.3498 16.4436 16.5159 18.6943 12.8803 14.6468 15.1992 17.5153 18.3498 16.4485 16.5198 18.6963 12.8803 14.6429 15.1983 17.5055 18.3527 16.4485 16.5139 18.6914 12.8803 14.6429 15.1983 17.4997 18.3527 16.4475 16.5071 18.6933 12.8803 14.6429 15.1875 17.5114 18.3527 16.4475 16.5100 18.6943 12.8666 14.6410 15.1934 17.5055 18.3527 16.4475 16.5159 18.6923 12.8666 14.6468 15.1924 17.5094 18.3537 16.4554 16.5100 18.6884 12.8666 14.6468 15.1807 17.5065 18.3537 16.4475 16.5071 18.6904 12.8666 14.6478 15.1797 17.5016 18.3537 16.4475 16.5090 18.6865 12.8773 14.6429 15.1817 17.5055 18.3537 16.4446 16.5061 18.6875 12.8773 14.6429 15.1768 17.5065 18.3537 16.4427 16.5120 18.6943 12.8773 14.6419 15.1778 17.4948 18.3537 16.4427 16.5100 18.6904 12.8773 14.6341 15.1758 17.4977 18.3537 16.4436 16.5032 18.6884 12.8637 14.6371 15.1768 17.4880 18.3547 16.4485 16.5032 18.6836 12.8637 14.6371 15.1778 17.4889 18.3547 16.4485 16.5012 18.6904 12.8637 14.6361 15.1807 17.4802 18.3547 16.4417 16.4973 18.6875 12.8637 14.6283 15.1778 17.4831 18.3547 16.4407 16.5022 18.6865 12.8578 14.6283 15.1875 17.4841 18.3537 16.4407 16.5032 18.6933 12.8578 14.6351 15.1778 17.4792 18.3537 16.4368 16.4973 18.6826 Appendix E. Experimental data for TGA 191 Irati shale New Brunswick shale heating rate (°C/min) 50 20 10 5 50 20 10 5 12.8578 14.6302 15.1778 17.4743 18.3537 16.4368 16.4993 18.6865 12.8539 14.6302 15.1739 17.4753 18.3537 16.4368 16.4944 18.6806 12.8539 14.6224 15.1719 17.4753 18.3615 16.4378 16.4983 18.6806 12.8539 14.6175 15.1797 17.4792 18.3615 16.4339 16.4954 18.6767 12.8539 14.6175 15.1651 17.4763 18.3615 16.4339 16.4925 18.6767 12.8559 14.6244 15.1573 17.4704 18.3615 16.4368 16.5042 18.6816 12.8559 14.6215 15.1690 17.4714 18.3537 16.4339 16.4964 18.6787 12.8559 14.6215 15.1680 17.4733 18.3537 16.4339 16.4973 18.6845 12.8559 14.6224 15.1622 17.4743 18.3537 16.4388 16.4964 18.6767 12.8451 14.6166 15.1573 17.4616 18.3459 16.4349 16.4973 18.6826 12.8451 14.6166 15.1602 17.4685 18.3459 16.4349 16.4983 18.6728 12.8451 14.6156 15.1514 17.4616 18.3459 16.4310 16.5012 18.6709 12.8451 14.6088 15.1456 17.4665 18.3459 16.4261 16.4925 18.6709 12.8422 14.6088 15.1465 17.4636 18.3507 16.4261 16.4983 18.6611 12.8422 14.6097 15.1553 17.4655 18.3507 16.4212 16.4895 18.6660 12.8422 14.6010 15.1465 17.4509 18.3507 16.4436 16.4944 18.6631 12.8588 14.6010 15.1368 17.4548 18.3507 16.4436 16.4964 18.6601 12.8588 14.5961 15.1446 17.4567 18.3468 16.4280 16.4915 18.6592 12.8588 14.6000 15.1368 17.4499 18.3468 16.4310 16.4856 18.6621 12.8588 14.6000 15.1514 17.4519 18.3468 16.4310 16.4925 18.6640 12.8373 14.5961 15.1475 17.4499 18.3468 16.4134 16.4915 18.6533 12.8373 14.5971 15.1417 17.4509 18.3342 16.4251 16.4934 18.6611 12.8373 14.5971 15.1368 17.4460 18.3342 16.4251 16.4905 18.6592 12.8373 14.5990 15.1417 17.4470 18.3342 16.4173 16.4925 18.6543 12.8266 14.5922 15.1368 17.4333 18.3322 16.4105 16.4925 18.6533 12.8266 14.5922 15.1348 17.4392 18.3322 16.4105 16.4798 18.6533 12.8266 14.5892 15.1446 17.4353 18.3322 16.4114 16.4876 18.6494 12.8266 14.5893 15.1378 17.4421 18.3322 16.4134 16.4798 18.6523 12.8237 14.5893 15.1397 17.4382 18.3400 16.4134 16.4807 18.6465 12.8237 14.5971 15.1348 17.4333 18.3400 16.4046 16.4827 18.6514 12.8237 14.5873 15.1339 17.4411 18.3400 16.4163 16.4788 18.6475 12.8237 14.5873 15.1378 17.4372 18.3400 16.4163 16.4759 18.6475 12.8188 14.5824 15.1319 17.4372 18.3224 16.4124 16.4778 18.6484 12.8188 14.5853 15.1260 17.4314 18.3224 16.3988 16.4856 18.6406 Appendix E. Experimental data, for TGA 192 Irati shale New Brunswick shale heating rate (°C/min) 50 20 10 50 20 10 12.8188 12.7915 12.7915 12.7915 12.7915 12.8012 12.8012 12.8012 12.8012 12.8090 12.8090 12.8090 12.8090 12.8022 12.8022 12.8022 12.7983 12.7983 12.7983 12.7983 12.7827 12.7827 12.7827 12.7827 12.7934 12.7934 12.7934 12.7934 12.7729 12.7729 12.7729 12.7729 12.7729 12.7729 14.5853 14.5727 14.5688 14.5688 14.5688 14.5648 14.5648 14.5629 14.5570 14.5570 14.5551 14.5570 14.5570 14.5570 14.5492 14.5492 14.5453 14.5434 14.5434 14.5434 14.5356 14.5356 14.5424 14.5424 14.5424 14.5356 14.5336 14.5336 14.5346 14.5258 14.5258 14.5248 14.5219 14.5160 15.1300 15.1261 15.1231 15.1173 15.1231 15.1173 15.1114 15.1192 15.1134 15.1143 15.1143 15.1085 15.1046 15.1104 15.0938 15.0958 15.0929 15.0899 15.0987 15.0870 15.0938 15.0860 15.0772 15.0821 15.0831 15.0860 15.0772 15.0802 15.0753 15.0772 15.0646 15.0694 15.0675 15.0607 17.4275 17.4216 17.4236 17.4206 17.4157 17.4167 17.4109 17.4148 17.4177 17.4099 17.4118 17.4187 17.4031 17.4060 17.3982 17.4021 17.3982 17.4011 17.4060 17.3943 17.3884 17.3845 17.3874 17.3787 17.3884 17.3933 17.3894 17.3777 17.3855 17.3894 17.3718 17.3748 17.3777 17.3718 18.3224 18.3224 18 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 3078 3078 3078 3098 3098 3098 3098 3098 3098 3098 3098 3020 3020 3020 3020 3029 18.3029 18.3029 18.2883 18.2883 18.2883 18.2883 18.2922 18.2922 18.2922 18.2922 18.2863 18.2863 18.2863 18.2863 18.2776 18.2776 16.3988 16.4027 16.3948 16.3948 16.3988 16.3949 16.3949 16.3978 16.4027 16.4027 16.3851 16.3978 16.3978 16.3958 16.3831 16.3831 16.3861 16.3792 16.3704 16.3704 16.3783 16.3714 16.3714 16.3636 16.3636 16.3636 16.3500 16.3480 16.3460 16.3441 16.3451 16.3451 16.3529 16.3519 16.4729 16.4749 16.4778 16.4768 16.4924 16.5042 16.4915 16.4700 16.4671 16.5002 16.5042 16.4758 16.4710 16.4934 16.4690 16.4798 16.4739 16.4944 16.4466 16.4866 16.4768 16.4573 16.4798 16.4680 16.4632 16.4739 16.4729 16.4271 16.4339 16.4407 16.4261 16.4261 16.4271 16.4319 18.6406 18.6445 18.6377 18.6387 18.6406 18.6338 18.6299 18.6348 18.6318 18.6260 18.6240 18.6231 18.6192 18.6133 18.6260 18.6162 18.6231 18.6123 18.6162 18.6113 18.6055 18.6143 18.6084 18.5977 18.5928 18.6074 18.5908 18.5967 18.5879 18.5987 18.5908 18.5977 18.5938 18.5947 Appendix E. Experimental data for TGA 193 Irati shale New Brunswick shale heating rate (°C/min) 50 20 10 5 50 20 10 5 12.7729 12.7749 12.7749 12.7749 12.7749 12.7475 12.7475 12.7475 12.7475 12.7632 12.7632 12.7632 12.7514 12.7514 12.7514 12.7514 12.7466 12.7466 12.7466 12.7466 12.7358 12.7358 12.7358 12.7358 12.7222 12.7222 12.7222 12.7241 12.7241 12.7241 12.7241 12.7251 12.7251 12.7251 14.5160 14.5121 14.5160 14.5160 14.5082 14.5063 14.5063 14.5004 14.5043 14.5043 14.5004 14.4956 14.4956 14.4995 14.4868 14.4868 14.4868 14.4858 14.4858 14.4848 14.4819 14.4819 14.4868 14.4838 14.4838 14.4809 14.4770 14.4770 14.4838 14.4751 14.4751 14.4741 14.4790 14.4790 15.0675 15.0636 15.0597 15.0587 15.0694 15.0587 15.0470 15.0499 15.0509 15.0568 15.0480 15.0499 15.0402 15.0402 15.0538 15.0402 15.0333 15.0363 15.0392 15.0441 15.0333 15.0392 15.0372 15.0343 15.0304 15.0216 15.0294 15.0324 15.0304 15.0294 15.0343 15.0177 15.0206 15.0265 17.3728 17.3738 17.3796 17.3826 17.3650 17.3669 17.3630 17.3777 17.3582 17.3738 17.3669 17.3552 17.3718 17.3708 17.3621 17.3562 17.3513 17.3465 17.3591 17.3552 17.3611 17.3572 17.3386 17.3523 17.3318 17.3318 17.3328 17.3260 17.3240 17.3240 17.3123 17.3123 17.3220 17.3025 18.2776 18.2639 18.2639 18.2639 18.2639 18.2551 18.2551 18.2551 18.2551 18.2531 18.2531 18.2531 18.2531 18.2483 18.2483 18.2483 18.2463 18.2463 18.2463 18.2463 18.2483 18.2483 18.2483 18.2483 18.2317 18.2317 18.2317 18.2317 18.2209 18.2209 18.2209 18.2209 18.2209 18.2209 16.3519 16.3470 16.3558 16.3558 16.3500 16.3392 16.3392 16.3431 16.3402 16.3402 16.3373 16.3285 16.3285 16.3187 16.3334 16.3334 16.3129 16.3177 16.3177 16.3197 16.3207 16.3207 16.3187 16.3197 16.3197 16.3129 16.3207 16.3207 16.3119 16.3148 16.3148 16.3168 16.3080 16.3080 16.4241 16.4212 16.4114 16.4114 16.4114 16.4114 16.4085 16.4105 16.4046 16.4036 16.4085 16.4066 16.4085 16.3958 16.4085 16.3978 16.3988 16.4124 16.3939 16.3948 16.3988 16.3870 16.3802 16.3851 16.3909 16.3831 16.3968 16.3978 16.3890 16.3978 16.3831 16.3744 16.3831 16.3831 18.5869 18.5801 18.5938 18.5967 18.5879 18.5850 18.5801 18.5772 18.5791 18.5811 18.5860 18.5879 18.5830 18.5772 18.5703 18.5694 18.5752 18.5645 18.5713 18.5772 18.5723 18.5694 18.5694 18.5694 18.5684 18.5625 18.5694 18.5674 18.5635 18.5723 18.5567 18.5606 18.5586 18.5586 Appendix E. Experimental data for TGA 194 Irati shale New Brunswick shale heating rate (°(J/min) 50 20 10 50 20 10 12.7251 12.7290 12.7290 12.7290 12.7290 12.7065 12.7065 12.7065 12.7065 12.7134 12.7134 12.7134 12.7065 12.7065 12.7065 12.7065 12.6880 12.6880 12.6880 12.6880 12.6919 12.6919 12.6919 12.6919 12.6958 12.6958 12.6958 12.6890 12.6890 12.6890 12.6890 12.6812 12.6812 12.6812 14.4770 14.47900 14.47900 14.47800 14.46720 14.4672 14.4672 14.4633 14.4633 14.4633 14.4565 14.4565 14.4575 14.4585 14.4585 14.4448 14.4458 14.4458 14.4458 14.4419 14.4419 14.4263 14.4243 14.4243 14.4097 14.4067 14.4067 14.4077 14.3941 14.3941 14.3892 14.3931 14.3862 14.3862 15.0148 15.0148 15.0080 15.0050 15.0128 15.0001 15.0011 14.9943 14.9884 14.9826 14.9875 14.9865 14.9836 14.9894 14.9748 14.9748 14.9796 14.9748 14.9679 14.9660 14.9582 14.9552 14.9592 14.9513 14.9426 14.9367 14.9377 14.9445 14.9396 14.9250 14.9211 14.9191 14.9074 14.9035 17.3094 17.2957 17.3045 17.3006 17.2996 17.2986 17.3006 17.2889 17.3025 17.2771 17.2703 17.2898 17.2723 17.2693 17.2498 17.2645 17.2645 17.2469 17.2430 17.2264 17.2332 17.2332 17.2215 17.2127 17.2020 17.2079 17.1844 17.1883 17.1630 17.1678 17.1512 17.1298 17.1317 17.1073 18.2209 18.2239 18.2239 18.2239 18.2239 18.2034 18.2034 18.2034 18.2034 18.2083 18.2083 18.2083 18.1975 18.1975 18.1975 18.1975 18.1917 18.1917 18.1917 18.1917 18.1790 18.1790 18.1790 18.1790 18.1868 18.1868 18.1868 18.1653 18.1653 18.1653 18.1653 18.1634 18.1634 18.1634 16.3070 16.3080 16.3080 16.2963 16.3148 16.3148 16.2953 16.3041 16.3041 16.3041 16.2933 16.2933 16.2904 16.3021 16.3021 16.2924 16.2826 16.2826 16.2846 16.2797 16.2797 16.2680 16.2670 16.2670 16.2582 16.2543 16.2543 16.2524 16.2543 16.2358 16.2358 16.2299 16.2240 16.2240 16.3695 16.3695 16.3763 16.3558 16.3685 16.3646 16.3539 16.3480 16.3421 16.3480 16.3392 16.3392 16.3382 16.3402 16.3392 16.3343 16.3353 16.3353 16.3119 16.3187 16.3168 16.3207 16.3177 16.3090 16.3031 16.3070 16.3099 16.2973 16.3012 16.2933 16.2924 16.2943 16.2826 16.2885 18.5469 18.5372 18.5342 18.5255 18.5235 18.5381 18.5245 18.5206 18.5059 18.5196 18.5108 18.5069 18.5040 18.5059 18.5020 18.4991 18.5011 18.4952 18.4903 18.4874 18.4679 18.4503 18.4523 18.4562 18.4454 18.4230 18.4269 18.4249 18.4171 18.4083 18.3976 18.3956 18.3917 18.4005 Appendix E. Experimental data for TGA 195 Irati shale New Brunswick shale heating rate (°C/min) 50 20 10 50 20 10 12.6812 12.6695 12.6695 12.6695 12.6695 12.6499 12.6499 12.6499 12.6411 12.6411 12.6411 12.6411 12.6372 12.6372 12.6372 12.6372 12.6294 12.6294 12.6294 12.6294 12.5963 12.5963 12.5963 12.5836 12.5836 12.5836 12.5836 12.5553 12.5553 12.5553 12.5553 12.5328 12.5328 12.5328 14.3726 14.3696 14.3696 14.3667 14.3560 14.3560 14.3579 14.3462 14.3462 14.3413 14.3218 14.3218 14.3208 14.3140 14.3140 14.2984 14.2828 14.2828 14.2701 14.2594 14.2594 14.2359 14.2125 14.2125 14.1959 14.1803 14.1803 14.1491 14.1247 14.1247 14.1091 14.0778 14.0778 14.0437 14.8918 14.8918 14.8860 14.8762 14.8655 14.8576 14.8518 14.8440 14.8401 14.8313 14.8127 14.8030 14.7845 14.7679 14.7552 14.7405 14.7278 14.7200 14.6927 14.6712 14.6478 14.6273 14.6039 14.5922 14.5541 14.5443 14.5248 14.4819 14.4819 14.4536 14.4028 14.3853 14.3648 14.3257 17.1083 17.0995 17.0829 17.0595 17.0449 17.0244 17.0205 17.0019 16.9804 16.9629 16.9414 16.9297 16.8916 16.8789 16.8409 16.8223 16.7921 16.7735 16.7267 16.7101 16.6857 16.6525 16.6154 16.6018 16.5852 16.5403 16.5198 16.4964 16.4729 16.4349 16.3978 16.3870 16.3636 16.3412 18.1634 18.1546 18.1546 18.1546 18.1546 18.1282 18.1282 18.1282 18.1204 18.1204 18.1204 18.1204 18.0892 18.0892 18.0892 18.0892 18.0911 18.0911 18.0911 18.0911 18.0433 18.0433 18.0433 18.0296 18.0296 18.0296 18.0296 17.9984 17.9984 17.9984 17.9984 17.9662 17.9662 17.9662 16.2192 16.2123 16.2123 16.1987 16.2094 16.2094 16.2006 16.1967 16.1967 16.1860 16.1938 16.1938 16.1606 16.1557 16.1557 16.1430 16.1235 16.1235 16.1216 16.1069 16.1069 16.0864 16.0591 16.0591 16.0445 16.0210 16.0210 16.0220 16.0044 16.0044 15.9722 15.9361 15.9361 15.9049 16.2748 16.2582 16.2504 16.2533 16.2465 16.2348 16.2192 16.2162 16.2231 16.2006 16.2055 16.2065 16.1860 16.1840 16.1596 16.1411 16.1362 16.1216 16.1186 16.0962 16.0767 16.0464 16.0249 16.0249 16.0093 15.9869 15.9615 15.9205 15.9078 15.9049 15.8356 15.8190 15.7946 15.7390 18.3859 18.3732 18.3576 18.3547 18.3420 18.3254 18.3137 18.3059 18.2941 18.2727 18.2610 18.2492 18.2346 18.2200 18.1965 18.1956 18.1624 18.1409 18.1243 18.0882 18.0492 18.0345 18.0043 17.9691 17.9389 17.9155 17.8803 17.8481 17.7827 17.7495 17.7300 17.6637 17.6197 17.5846 Appendix E. Experimental data for TGA 196 Irati shale New Brunswick shale heating rate (°C/min) 50 20 10 5 50 20 10 5 12.5328 14.0115 14.3101 16.3207 17.9662 15.8737 15.7136 17.5241 12.4811 14.0115 14.2867 16.3119 17.9096 15.8737 15.6980 17.4860 12.4811 13.9753 14.2555 16.3012 17.9096 15.8288 15.6326 17.4470 12.4811 13.9461 14.2369 16.2875 17.9096 15.7907 . 15.6053 17.4089 12.4547 13.9461 14.2174 16.2699 17.8628 15.7907 15.5652 17.3728 12.4547 13.9158 14.1832 16.2524 17.8628 15.7400 15.5106 17.3533 12.4547 13.8738 14.1637 16.2406 17.8628 15.6951 15.4715 17.2996 12.4547 13.8738 14.1383 16.2162 17.8628 15.6951 15.4120 17.2771 12.3962 13.8416 14.1188 16.2104 17.7866 15.6336 15.3818 17.2615 12.3962 13.7948 14.0964 16.2065 17.7866 15.5887 15.3544 17.2391 12.3962 13.7948 14.0710 16.1928 17.7866 15.5887 15.2910 17.2157 12.3962 13.7606 14.0544 16.1918 17.7866 15.5282 15.2754 17.2059 12.3318 13.7274 14.0427 16.1909 17.6861 15.4628 15.2412 17.1825 12.3318 13.7274 14.0349 16.1782 17.6861 15.4628 15.1914 17.1698 12.3318 13.6884 14.0290 16.1518 17.6861 15.4052 15.1788 17.1600 12.3318 13.6650 14.0261 16.1567 17.6861 15.3466 15.1602 17.1532 12.2781 13.6650 14.0144 16.1382 17.5768 15.3466 15.1329 17.1269 12.2781 13.6347 14.0066 16.1303 17.5768 15.2764 15.1163 17.1103 12.2781 13.5967 13.9939 16.1372 17.5768 15.2285 15.1036 17.1220 12.1961 13.5967 13.9812 16.1274 17.4382 15.2285 15.0792 17.0819 12.1961 13.5771 13.9773 16.1196 17.4382 15.1758 15.0743 17.1063 12.1961 13.5576 13.9763 16.1147 17.4382 15.1241 15.0646 17.1054 12.1961 13.5576 13.9666 16.1157 17.4382 15.1241 15.0568 17.0839 12.1170 13.5352 13.9656 16.1069 17.2869 15.0899 15.0450 17.0898 12.1170 13.5157 13.9539 16.1050 17.2869 15.0587 15.0333 17.0849 12.1170 13.4893 13.9412 16.1089 17.2869 15.0587 15.0304 17.0907 12.1170 13.4893 13.9383 16.0835 17.2869 15.0314 15.0216 17.0741 12.0341 13.4795 13.9373 16.0903 17.1064 15.0089 15.0216 17.0683 12.0341 13.4678 13.9295 16.0884 17.1064 15.0089 15.0050 17.0605 12.0341 13.4678 13.9275 16.0737 17.1064 14.9933 14.9953 17.0546 11.9511 13.4600 13.9246 16.0611 17.1064 14.9699 14.9982 17.0390 11.9511 13.4483 13.9168 16.0415 16.9668 14.9699 14.9914 17.0507 11.9511 13.4483 13.9090 16.0484 16.9668 14.9562 14.9845 17.0302 11.9511 13.4376 13.8982 16.0337 16.9668 14.9465 14.9845 17.0253 Appendix E. Experimental data, for TGA 197 Irati shale New Brunswick shale heating rate ( C/min) 50 20 10 50 20 10 11.9082 11.9082 11.9082 11.9082 11.8789 11.8789 11.8789 11.8789 11.8194 11.8194 11.8194 11.8008 11.8008 11.8008 11.8008 11.7842 11.7842 11.7842 11.7842 11.7706 11.7706 11.7706 11.7706 11.7598 11.7598 11.7598 11.7374 11.7374 11.7374 11.7374 11.7100 11.7100 11.7100 11.7100 13.4259 13.4259 13.4151 13.4200 13.4200 13.4122 13.4063 13.4063 13.3849 13.3780 13.3780 13.3790 13.3702 13.3702 13.3692 13.3673 13.3673 13.3380 13.3165 13.3165 13.2931 13.2814 13.2814 13.2619 13.2375 13.2375 13.2297 13.2170 13.2170 13.1916 13.1789 13.1789 13.1711 13.1643 13.9090 13.9109 13.8904 13.8836 13.8885 13.8787 13.8690 13.8690 13.8533 13.8397 13.8270 13.7997 13.7958 13.7801 13.7684 13.7567 13.7499 13.7333 13.7323 13.7157 13.6903 13.6903 13.6933 13.6738 13.6650 13.6669 13.6552 13.6533 13.6494 13.6318 13.6289 13.6220 13.6250 13.6103 16.0328 16.0152 16.0113 15.9869 15.9752 15.9352 15.9381 15.9156 15.9117 15.8942 15.8785 15.8688 15.8707 15.8483 15.8376 15.8278 15.8200 15.8131 15.8102 15.7995 15.7946 15.7770 15.7809 15.7683 15.7575 15.7526 15.7517 15.7458 15.7429 15.7321 15.7282 15.7175 15.7292 15.7077 16.9668 16.8458 16.8458 16.8458 16.7521 16.7521 16.7521 16.7521 16.7042 16.7042 16.7042 16.7042 16.6642 16.6642 16.6642 16.6642 16.6428 16.6428 16.6428 16.6174 16.6174 16.6174 16.6174 16.5988 16.5988 16.5988 16.5988 16.5852 16.5852 16.5852 16.5852 16.5676 16.5676 16.5676 14.9465 14.9338 14.9367 14.9367 14.9250 14.9201 14.9201 14.9074 14.8977 14.8977 14.8957 14.8879 14.8879 14.8723 14.8694 14.8694 14.8606 14.8567 14.8567 14.8489 14.8440 14.8440 14.8420 14.8371 14.8371 14.8313 14.8313 14.8176 14.8176 14.8167 14.8010 14.8010 14.8030 14.8001 14.9738 14.9709 14.9670 14.9592 14.9592 14.9552 14.9474 14.9455 14.9357 14.9338 14.9377 14.9338 14.9201 14.9269 14.9182 14.9201 14.9094 14.9045 14.9094 14.9016 14.9035 14.8957 14.8938 14.8703 14.8684 14.8762 14.8694 14.8645 14.8518 14.8586 14.8547 14.8528 14.8450 14.8352 17.0293 17.0166 17.0107 17.0088 17.0048 16.9922 16.9980 16.9853 16.9853 16.9804 16.9765 16.9648 16.9717 16.9697 16.9473 16.9531 16.9502 16.9404 16.9307 16.9307 16.9346 16.9297 16.9199 16.9102 16.9170 16.9141 16.9082 16.8916 16.8858 16.8877 16.8907 16.8848 16.8789 16.8760 Appendix E. Experimental data for TGA 198 Irati shale New Brunswick shale heating rate (°C/min) 50 20 10 5 50 20 10 5 11.6769 13.1643 13.5986 15.7087 16.5481 14.8001 14.8489 16.8555 11.6769 13.1653 13.6084 16.5481 14.7913 14.8371 11.6769 13.1506 13.6045 16.5481 14.7844 14.8440 11.6769 13.1506 13.5898 16.5481 14.7844 14.8362 11.6349 13.1506 13.5957 16.5354 14.7884 14.8313 11.6349 13.1389 13.5937 16.5354 14.7864 14.8332 11.6349 13.1389 13.5859 16.5354 14.7864 11.6212 13.1340 16.5354 14.7776 - _ 11.6212 13.1301 16.5159 14.7747 - -11.6212 13.1301 16.5159 14.7747 -11.6212 13.1262 16.5159 14.7776 - -11.6203 13.1223 16.5159 14.7708 - -11.6203 13.1223 16.5247 14.7708 - -11.6203 13.1135 16.5247 14.7649 - _ 11.6203 13.1048 16.5247 14.7640 - -11.6144 13.1048 16.5090 14.7640 -11.6144 13.1038 16.5090 14.7640 -11.6144 13.1096 16.5090 14.7679 -11.6144 13.1096 16.5090 14.7679 - _ 11.5841 ' 13.1028 16.5012 14.7552 - -11.5841 16.5012 14.7532 - -11.5841 16.5012 14.7532 - - -11.5793 16.5012 14.7493 - -11.5793 16.4934 14.7542 - -11.5793 16.4934 14.7542 - - _ 11.5793 16.4934 - - - _ 11.5734 16.4934 - - - -11.5734 16.5012 - - - -11.5734 16.5012 - - - -11.5734 16.5012 - - - -11.5754 16.5012 - - -11.5754 16.4925 - - -11.5754 16.4925 - - - _ Appendix E. Experimental data for TGA 199 Irati shale New Brunswick shale heating rate (°C/min) 50 20 10 5 50 20 10 5 11.5754 16.4925 11.5597 16.4905 11.5597 16.4905 11.5597 16.4905 11.5597 16.4905 11.5510 16.4885 11.5510 16.4885 11.5510 16.4885 11.5432 16.4885 11.5432 16.4954 11.5432 16.4954 11.5432 16.4954 11.5480 16.4954 11.5480 16.4924 11.5480 16.4924 11.5480 16.4924 11.5324 16.4866 11.5324 16.4866 11.5324 16.4866 11.5275 16.4866 11.5275 16.4749 11.5275 16.4749 11.5275 16.4749 Appendix E. Experimental data for TGA 200 Table E.3: Sample weight W (%) versus timet (minutes) for isother-mal oxidation TGA at different temperatures (°C). 550 °C 600 °C 625 °C 650 °C 700 °C t W t W t W t W t W 0.15 100 0.113 99.91 0.1 100 0.5 99.96 0.12 100 1 100 1 99.91 1 99.95 1 100 1 100 1.49 99.96 1.24 100 1.3 99.88 1.18 100 1.25 100 2 99.97 1.42 100 1.6 100 1.64 100 1.54 100 2.3 99.93 1.69 99.97 1.9 100 1.92 99.94 1.87 100 2.7 99.88 1.92 100 2.38 100 2.2 99.91 2.21 100 3 99.88 2.21 99.92 2.72 100 2.55 99.97 2.5 99.96 3.3 99.88 2.49 99.89 3 100 2.83 99.88 2.84 99.95 3.6 99.77 2.89 99.89 3.3 100 3.17 99.79 3.12 99.88 3.8 99.81 3.11 99.83 3.6 100 3.57 99.78 3.41 99.87 4.1 99.85 3.34 99.85 3.9 100 3.85 99.76 3.81 99.85 4.5 99.91 3.51 99.9 4.3 100 4.19 99.79 4.03 99.8 4.7 99.85 3.73 99.93 4.7 99.86 4.42 99.8 4.42 99.78 4.9 99.81 3.85 99.89 4.9 99.8 4.82 99.67 4.82 99.76 5.3 99.75 4.07 99.82 5.27 99.81 5.09 99.59 4.11 99.69 5.7 99.55 4.31 99.86 5.49 99.73 5.38 99.52 5.39 99.58 6.4 99.57 4.47 99.9 5.7 99.59 5.66 99.32 5.78 99.36 8 99.44 4.65 99.83 5.8 99.57 5.95 99.32 6.07 99.2 10 99.32 4.87 99.81 6 99.42 6.17 99.14 6.24 99.14 13 99.29 5.09 99.81 6.4 99.18 6.46 99 6.58 98.94 16 99.22 5.27 99.79 6.8 99.11 6.79 98.85 6.81 98.82 19 99.18 5.44 99.73 7.7 99.11 7.3 98.66 7.09 98.67 23 99.2 5.22 99.71 9.35 99.06 8.1 98.6 7.43 98.46 26 99.14 6.34 99.51 10 98.94 9 98.4 7.99 98.21 29.8 99.13 6.68 99.4 13 98.82 10.5 98.12 8.33 98.1 33.1 99.09 7.1 99.26 15 98.78 12.4 97.93 8.91 97.91 35.7 99.13 7.5 99.37 17 98.74 14 97.79 9.64 97.81 39 99.06 10 99.08 19 98.59 15.3 97.72 10 97.77 44.5 99.01 12.5 98.95 21 98.59 17 97.7 11.34 97.73 48.2 99.08 15 98.83 23 98.52 20 97.7 12.5 97.71 57.4 99.02 17.5 98.69 25.3 98.43 22.5 97.7 15 97.71 61.5 98.98 20 98.57 27 98.41 25 97.7 17.5 97.74 65.3 99.01 25 98.41 29 98.38 20 97.71 Appendix E. Experimental data, for TGA 201 550 °C 600 °C 625 °C 650 °C 700 °c t W t W t W t W t w 68.8 98.98 30 98.32 31 98.36 73 98.98 35 98.22 .33.2 98.21 40 98.24 35 98.12 45 98.18 37.4 98.15 40.4 98.1 42.5 98.05 45 97.99 47 98.05 48 97.99 49 97.95 Table E.4: Sample weight W (%) versus timet (minutes) for isother-mal oxidation TGA at different oxygen concentrations (%). 21% 15% 10% 5% t W t W t W t W 0.5 99.96 0.124 99.99 0.127 99.99 0.125 99.98 1 100 1 100 1 99.99 1 100 1.18 100 1.39 100 1.26 100 1.32 100 1.64 100 1.92 99.97 1.59 99.99 1.59 99.99 1.92 99.94 2.39 99.95 1.92 100 1.99 99.98 2.2 99.91 2.86 99.86 2.32 99.97 2.19 99.97 2.55 99.97 3.19 99.88 2.72 99.92 2.52 99.99 2.83 99.88 3.52 99.85 3.06 .99.92 2.92 99.86 3.17 99.79 3.86 99.78 3.5 99.85 3.39 99.82 3.57 99.78 4.19 99.75 3.79 99.81 3.59 99.8 3.85 99.76 4.52 99.75 4.13 99.79 3.92 99.77 4.19 99.79 4.86 99.71 4.32 99.76 4.19 99.77 4.42 99.8 5.06 99.65 4.66 99.67 4.52 99.76 .4.82 99.67 5.32 99.62 4.92 99.71 4.79 99.72 5.09 99.59 5.59 99.52 5.13 99.76 5.05 99.66 5.38 99.52 5.86 99.34 5.46 99.46 5.32 99.6 5.66 99.32 6.12 99.21 5.73 99.34. 5.59 99.46 5.95 99.32 6.39 99.09 5.93 99.31 5.79 99.39 Appendix E. Experimental data, for TGA 21% 15% 10% 5% t W t W t W t W 6.17 99.14 6.66 98.98 6.13 99.16 5.99 99.31 6.46 99 7.19 98.91 6.46 99.1 6.12 99.21 6.79 98.85 8 98.79 7.26 98.95 6.45 99.12 7.3 98.66 9.5 98.56 8.59 98.68 7.05 99.07 8.1 98.6 11.7 98.26 10 98.54 8.52 98.97 9 98.4 13.4 98.11 12 98.2 9.45 98.89 10.5 98.12 15 97.95 14 98.06 11.1 98.82 12.4 97.93 16.5 97.89 16.3 97.93 13.5 98.66 14 97.79 19 97.84 18.1 97.85 15 98.54 15.3 97.72 21 97.82 20 97.73 17 98.44 17 97.7 23.1 97.78 22.5 97.68 19 98.31 20 97.69 25.1 97.77 24 97.66 21.3 98.25 22.5 97.7 27 97.78 26.1 97.59 23 98.17 25 97.7 29.2 97.77 28 97.66 25.6 98.11 30 97.62 27 98.04 29.6 98 31.3 97.98 33.7 97.94 36 97.94 38 97.92 40 97.91 Appendix F Polynomial approximations for non-isothermal T G A data Polynomials were fitted to the experimental data, conversion (X) versus time (t), by the least squares technique. Each set of data was represented by three polynomials, two fourth order and one first order that linked the other two. Eight experimental condi-tions, determined by type of shale, Irati or New Brunswick, and heating rate, 5, 10, 20 or 50 °C/min, were available for this study, as discussed in Section 6.4. The experimen-tal data and fitted curves are displayed in Figures 6.8(b) - 6.15(b). Next is a general representation of the fitted polynomials, followed by tables displaying the constants for each particular situation. The initial conversions, occurring between times t\ and t2, and the final conversions, occurring between times t3 and t4, are obtained by fourth order polynomials. Conversions between times t2 and t3 are obtained by first order polynomials. • tx<t<t2 X = A4l x t4 + A3i x ts + A2i x t2 + Au x t + Aoi • t2<t <t3 X — A x t + B • h<t< t3 X = A4u x t4 + A3u x t3 + A2u xt2 + Aluxt + A0u The continuous curves in Figures 6.8(b) - 6.15(b) were actually obtained with a time domain sligthly larger than the one limited by the interval ti - t4 listed in the following 203 Appendix F. Polynomial approximations for non-isothermal TGA data 204 tables. The intervals t\ - t4 were so defined to avoid dX/dt < 0, that happend at time values close to the ends of the range of points used to obtain the mentioned curves. The intervals t\ - t4 were the ones used to generate the conversion and derivative values in Section 6.4. Appendix F. Polynomial approximations for non-isothermal TGA data Constants for Irati shale 205 heating rate U t2 h u A B (°C/min) (min) (min) (min) (min) (-) (-) 50 8.20000 12.5333 12.6109 13.4000 .634845' -7.32428 20 9.00000 22.0204 22.7585 26.0000 .229222 -4.55995 10 14.0000 36.8956 38.6531 48.0000 .106842 -3.53866 5 26.0000 64.6666 69.0734 92.0000 .465120X10-1 -2.65316 heating rate Mi Asi A2l Au Aoi (°C/min) (-) (-) (-) (-) (-) 50 .568533x10" -2 -.213420 2.99282 -18.5400 42.7673 20 .636382x10" -4 -.328511 x l O - 2 .615836 x l O - 1 -.481449 1.34179 10 .594574x10" -5 -.504557 x l O - 3 .155112xl0_ 1 -.197478 .897769 5 .484459x10" -6 -.735624 xlO" 4 .411462xl0-2 -.0976635 .884753 heating rate A\U A3U Mu ALU AQU (°C/min) (-) (-) (-) (-) (-) 50 -.119944 6.52223 -133.071 1207.41 -4110.20 20 .240339 x l0~ 2 -.223429 7.73868 -118.180 670.989 10 -.118368 x l O - 3 .214012X10-1 -1.44733 43.4248 -487.203 5 -.564129 x l0~ 5 .185653 x l O ' 2 -.228481 12.4781 -254.650 Appendix F. Polynomial approximations for non-isothermal TGA data Constants for New Brunswick shale 206 heating rate U h h u A B (°C/min) (min) (min) (min) (min) (-) (-) 50 7.80000 12.3912 12.6061 13.0000 .766354 -8.99053 20 10.4000 22.4790 22.9235 26.5000 .334981 -7.03368 10 23.0000 38.7575 39.5238 48.0000 .170872 -6.13123 5 26.0000 69.9764 71.4490 92.0000 .830860 xlO" 1 -5.33196 heating rate A « A3, Aii Au Am (°C/min) (-) (-) (-) (-) (-) 50 .448225 x l0~ 2 -.164277 2.24364 -13.5029 30.2094 20 .111751 x l O - 3 -.639770 xlO" 2 .134863 -1.22839 4.09858 10 .212860 xlO" 4 -.229553 x l O - 2 .923713 xlO" 1 -1.63579 10.7493 5 .454078 x 10 - 6 -.695796 x l O - 4 .388475 xlO-' 2 -.0908937 .774035 heating rate A-u A« A2U Mu AQU (°C/min) (-) (-) (-) (-) (") 50 .188831 -9.71524 186.941 -1593.92 5080.03 20 -.281553 xlO" 2 .292787 -11.4157 197.815 -1284.64 10 -.185718 x l O - 3 .339932 x l O - 1 -2.33192 71.0721 -811.243 5 -.804052 x l O - 5 .274008 x l O - 2 -.349804 19.8351 -420.744 Appendix G Implementation of the numerical method of lines The numerical method of lines (numol) was suggested by Schiesser [71, 72] as a method to integrate partial differential equations. The method requires that a space grid must be set in in the physical environment. At each point of this grid the partial differential equation is applied in discrete form, with particular expressions for the points at the boundaries. Initial condition values must be known at each grid point. Once the partial differential equation is written in discrete form, the problem becomes one of solving a set of ordinary differential equations (ODE), having as many equations as grid points in the physical environment. The sketch bellow shows how the integration advances. A time time = 0 1=0 1=L space 207 Appendix G. Implementation of the numerical method of lines 208 To implement the method, one must define a procedure to write the partial differential equation in discrete form and must select a procedure to solve the set of ODEs. The block diagram in Figure G. l shows the sequence of the method. partial differential equations + boundary conditions + initial condition discretization of partial differential equations above equations in discrete form, now a set of ordinary differential equations, one for each point of the grid; values at time t known t = t+At results for t = t + A t ODE solver Figure G . l : Steps performed by the numerical method of lines. In this work the procedure to set the spatial derivatives was: Appendix G. Implementation of the numerical method of lines 209 for first order derivative: (dy\ _ yi+1 - ijj-i \dx). 2 Ax or: (three point formula) Ox2 j . A:c 2 = 2 y . - - 2 - 1 6 y ^ + 1 6 y , - + i - 2 y , - + a > ( f i v e p o i n t f o r m u l a ) for second order derivative: 'd2y\ yi-i - 2y{ + yi+1 —, (three point formula) or: = - ^ + ^yi-1-60yi + Z2yi+1-2yi+3 \dx2)i 24A.c2 , In this work the ODE solver used was either the Runge-Kutta-Fehlberg method, as implemented by Forsythe et al. [73] or the Kaps-Rentrop method, more adequate for stiff sets of equations, as implemented by Press et al." [74] During the course of this work, the numol method was extensively tested, with the multi-dimensional cartesian coordinate system and with the spherical coordinate system. The variables which potentially affect the accuracy of the numerical results are: • the total number of grid points. • the number of grid points to establish the discrete spatial derivatives. • the accepted error in the ODE solver. Appendix G. Implementation of the numerical method of lines 210 Table G . l : Numol numerical results vs. analytical ones. Numerical results for the Equation 5.7 compared to analytical results from Equation 5.10. Biot number = 0.1, accepted error in integration = 1.0 x 10~6, number of grid points in each direction = 9. x,y,z time numerical analytical 0,0,0.5 0.2 0.9728 0.9728 0.4 0.9221 0.9221 0.6 0.8707 0.8707 0.8 0.8216 0.8216 1.0 0.7753 0.7753 1,1,0.5 0.2 0.8913 0.8913 0.4 0.8370 0.8370 0.6 0.7893 0.7893 0.8 0.7447 0.7447 1.0 0.7027 0.7027 1,0,0.5 0.2 0.9312 0.9312 0.4 0.8785 0.8785 0.6 0.8290 0.8290 0.8 0.7822 0.7822 1.0 0.7381 0.7381 The numerical method of lines was specially tested for the problem described by Equa-tion 5.7, which calculates the transient temperature profile in a rectangular parallelepiped. The latte case has an analytical solution given by Equation 5.10. The number of grid points varied from 3 to 21, both methods to calculate the discrete spatial derivatives— three or five points—were used, and accepted errors varied from 10 _ 1 to 10 - 7 . The numerical solution was also calculated for the dimensional equation; the FORTRAN code for this case is listed in Appendix A. In this case, particle size was varied from 1.3 cm to 13 cm. The method was able to produce results for all these cases. Table G. l shows a typical comparison between numerical and analytical results. Appendix G. Implementation of the numerical method of lines 211 The aforementioned simulations offered the background for the problems with no an-alytical solutions. Appendix H Pyrolysis modelling results Following are predicted results for the pyrolysis modelling. They show the effect of particle size and heat transfer coefficient on predicted results of conversion, particle core radius, and particle surface and core radius temperatures. The basic case for comparison is the one shown in Figure 5.29, with the following parameter values: dp = 2 cm h --= 60 J/s.m 2.K k0 = 0.103 1/s (T < 423°C) PP = 2100 kg/m3 T = 25 °C k0 = 2.78 x 105 1/s (T > 423°C) ks = 1.25 J/s.m.K T = 550 °C E = 27300. J/mole (T < 423°C) Cp = 1045 J/kg.K E = 105000. J/mole (T > 423°C) The FORTRAN code which generated these results is included in Appendix A under the name of program pyrolysis. For these simulations the program considered a constant gas temperature of 550 °C. The heat transfer coefficient includes both convection and radiation heat transfer. Figures H . l , H.2 and H.3 show the effect of the particle diameter on core radius, conversion and particle temperatures respectively. The larger the particle size the slower the devolatilization process will occur. Figures H.l and H.2 show that increasing ten times the particle size, from 0.6 cm, the time required for complete conversion will increase eight times. Figure H.3 shows the effect of particle size on the temperature difference between the particle surface and the core radius. As expected, as the particle diameter increases, this 212 Appendix H. Pyrolysis modelling results 213 difference increases. For dp= 6 cm, the core radius temperature reaches a maximum, a result also observed by Granoff and Nuttall [25]. The temperature gradient between the surface and the center of the particle is expected to be larger than that. Figures H.4, H.5 and H.6 show the effect of the combined heat transfer coefficient on core radius, conversion and particle temperatures. The larger the. heat transfer coefficient the faster the reaction will reach the end, as indicated by Figures H.4 and H.5, and the faster the particle temperatures will increase, as indicated by Figure H.6. Both parameters particle diameter (dp) and heat transfer coefficient (h), affect directly the Biot number. Table H. l shows how the Biot number affect the time for complete conversion for the five pairs of dv and h investigated. Table H . l : Influence of Biot number on conversion times. dp h Biot time for X ~ 1 (cm) (W/m 2 .K) (-) 00 0.6 60. 0.14 107. 2.0 30. 0.24 478. 2.0 60. 0.48 282. 2.0 150. 1.20 150. 6.0 60. 1.44 810. There is no direct relation between the Biot number and the time for complete con-version because of the two reactions involved, one at temperatures below 423 °C and the other at temperatures above 423 °C. Therefore, the Biot number is not a good parameter to analyze this modelling of the devolatilization process. Figures H.7, H.8 and H.9 show the effect of the energy of activation on core radius, conversion and particle temperatures. The lower the energy of activation value the faster the reaction will reach the end, as indicated by Figures H.7 and H.8. Appendix H. Pyrolysis modelling results 214 Figure H.9 shows particle surface temperature and core radius temperature for the three energy of activation values investigated. The curves for surface temperatures lie on the same line, as expected, because the energy of activation does not affect the heat transferred to the particle. These curves end at the time of maximum conversion for each case. On the other hand, the core radius temperature curves are functions of the energy of activation values. The objective of this appendix was to show that the pyrolysis model was able to predict the effect of key variables. In all cases investigated the model yielded reasonable predictions. The program used in this appendix is the same listed in Appendix A, used to gener-ate the data for the plot in Figure 5.29. Variations, as the introduction of variable gas temperatures—used to describe experimental data of Figures 5.30—can be incorporated to the program. Unfortunately the program is useful only for particles which can be approximated by a sphere. The application of the unreacted core model to a rectangular parallelepiped is something yet to be developed. Appendix H. Pyrolysis modelling results 3.0 ^ 2.0 ----...^ dp = 0.6 cm dp = 2 cm dp = 6 cm \ \ \ \ 1 I . I . i , i 0 200 400 600 800 1000 Time (seconds) Figure H . l : Influence of particle size on core radius. 1.0 0.8 U 0.4 0.2 -1 / 1 1 I 1 I 1 \ 1 1 / 1 / 1 / 1 1 1 I 1 \ 1 j 1 1 dp = 0.6 cm dp = 2 cm J dp = 6 cm '1 I . I , 0 200 400 600 800 1000 Time (seconds) Figure H.2: Influence of particle size on conversion. Appendix H. Pyrolysis modelling results 216 0 i • ' ' i i i i i l 0 200 400 600 800 1000 Time (seconds) Figure H.3: Influence of particle size on particle temperatures. Appendix H. Pyrolysis modelling results 217 0 100 200 300 400 500 600 Time (seconds) Figure H.4: Influence of heat transfer coefficient on core radius. 600 Time (seconds) Figure H.5: Influence of heat transfer coefficient on conversion. Appendix H. Pyrolysis modelling results 218 Figure H.6: Influence of heat transfer coefficient on particle temperatures. Appendix H. Pyrolysis modelling results 219 0 100 200 300 400 500 600 Time (seconds) Figure H.7: Influence of energy of activation on core radius. Time (seconds) Figure H.8: Influence of energy of activation on conversion. Appendix H. Pyrolysis modelling results 220 600 0 1 i i i i i i i i I 0 100 200 300 400 500 Time (seconds) Figure H.9: Influence of energy of activation on particle temperatures. 

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